Handbook of Gravitational Wave Astronomy [1 ed.] 9789811643057, 9789811643064, 9789811643071

Table of contents :
Preface
Contents
About the Editors
Section Editors
Contributors
Part I Introduction
1 Introduction to Gravitational Wave Astronomy
Contents
Introduction
Historical Development
Content of This Chapter
Propagation of Gravitational Waves
Linearized Einstein Equations in Vacuum
Gauge Transformations
Plane Wave Solutions
The TT Gauge
The Effect of a Gravitational Wave on a System of Test Particles
Generation of Gravitational Waves
Linearized Einstein Equations with Matter: The Quadrupole Formula
Example: Equal Mass Circular Binary
Energy of Gravitational Waves
Beyond the Quadrupole Formula
The Gravitational Wave Field of an Orbiting Binary
Inspiral Rate and Time to Merger
Eccentricity Reduction by Gravitational Waves
Detection of Gravitational Waves
GW Detection Facilities
Data Interpretation and Determination of Source Parameters
GW Observations: Contributions to Physics and Astrophysics
Fundamental Physics
Cosmology
Astrophysics
Conclusion
Cross-References
References
Part II Gravitational Wave Detectors
2 Terrestrial Laser Interferometers
Contents
Introduction: A Historical Perspective
Resonant Mass Detectors
The Beginnings of Laser Interferometry
Principles
Coupling of Gravitational Waves to a Michelson Interferometer
The Michelson Interferometer Response
Gravitational Waves as Phase Modulation
Extensions to the Michelson Interferometer
Noise Sources and Noise Reduction Strategies
Quantum Noise
Thermal Noise
Seismic and Gravity Gradient Noise
Noise from Technical Constraints
Some Enabling Technologies
Lasers
Vacuum Systems
Seismic Isolation
Feedback Control
Things Not Yet Mentioned
Precision Optics
Simulation and Diagnostic Methods
Robustness
Calibration
Laser Interferometers World-Wide
LIGO
Virgo
KAGRA
GEO600
Outlook
LIGO A+ and Virgo AdV+ Upgrades
LIGO-India
Third-Generation Detectors
Final Words
Cross-References
References
3 Space-Based Gravitational Wave Observatories
Contents
Introduction
LISA
Interferometry
Gravitational Reference Sensor
Data Analysis of Space-Based Observatories
Time-Delay Interferometry
Introduction
Spacecraft Jitter Measurement and Subtraction
Suppressing Laser Frequency Noise
Suppressing Clock Noise and TTL Using TDI
Impact on Unsuppressed Noises
Noise Quasi-uncorrelated TDI Generators
Instrument Response to GW and Sensitivity
Data Analysis Strategies
Matched Filtering and Bayesian Analysis
Global Fit Strategies
Robust Analysis and Other Analysis
Instrumental Artifacts and Noise Characterization
Ground Segment Design
GW Sources for Space-Based Observatories
Compact Binaries in the Milky Way
Stellar-Origin Black Hole Binaries
Massive Black Hole Binaries
Extreme Mass Ratio Inspirals
Cosmological Sources
Science with Space-Based Observatories
Astrophysics
Compact Binaries in the Milky Way
Stellar-Origin Black Hole Binaries
Massive Black Hole Binaries
Extreme Mass Ratio Inspirals
Cosmology
Probes of the Early Universe
Cosmography with Standard Sirens
Fundamental Physics
Elucidating Dark Matter
Testing the Foundations of the Gravitational Interaction
Testing the Nature of Black Holes
Prospects for Space-Based Observatories
Cross-References
References
4 Pulsar Timing Array Experiments
Contents
Introduction
Radio Emission from Pulsars
Pulsar Life Cycle and Spin Properties
Pulsar Timing and Pulsar Timing Arrays
Template Profiles
Template Matching
Timing Model Determination
Interstellar Propagation Delays
Timing Noise
Other Noise Sources
Pulsar Timing Software
Gravitational Waves and Other Correlated Signals
Correlated Signals in Pulsar Timing Data
Effect of Gravitational Waves
GW Sources in the PTA Band
Present PTA Constraints
Recent and Ongoing Improvements in PTA Sensitivity
Pulsar Surveys
IISM Studies
Sensitivity Predictions
Summary
Cross-References
References
5 Quantum Sensors with Matter Waves for GW Observation
Contents
Introduction
Atom Interferometry and GW Detection
Principle of GW Detection Using Matter Waves Interferometry
Space-Based and Terrestrial Instruments
Classes of Quantum Sensor-Based Gravitational Wave Detector
Two-Photon Transition-Based Interferometers
Optical Clocks and Single-Photon-Based Interferometers
Noise Sources
Interferometer Arrays for Rejecting Gravity-Gradient Noise
Long-Baseline Atom Interferometers
MIGA: Matter-Wave Laser Interferometric Gravitation Antenna
Functioning and Status of the MIGA Antenna
MIGA Sensitivity and Prospects
MAGIS: Mid-band Atomic Gravitational Wave Interferometric Sensor
ZAIGA: Zhaoshan Long-Baseline Atom Interferometer Gravitation Antenna
AION: Atom Interferometry Observatory and Network
VLBAI: Very Long-Baseline Atom Interferometry
Role of Atom Interferometry in GW Astronomy
GW Sources in the Atom Interferometry Detection Bandwidth and Multiband Astronomy
Outlook
Roadmap to Increase Sensitivity
ELGAR: European Laboratory for Gravitation and Atom-Interferometric Research
Cross-References
References
6 CMB Experiments and Gravitational Waves
Contents
Introduction
CMB B-Modes and Foregrounds
Primordial B-Modes
Reionization
Lensing B-Modes
Foregrounds
Key Instrument Technologies
Detectors
Optics, Cryogenics, and Multiplexing
Overview of Experiments
Large-Aperture Telescopes
ACT
SPT
CLASS: A Small-Aperture Telescope
Spider: A Balloon-Borne Telescope
Satellite Telescopes
WMAP
Planck
Future Experiments
Simons Observatory
CMB-S4
Future Outlook and Summary
References
7 Third-Generation Gravitational Wave Observatories
Contents
Introduction
3G Science Targets
Extreme Gravity and Fundamental Physics
Extreme Matter
Observing Stellar-Mass Black Holes Throughout the Universe
Sources at the Frontier of Observations
Cosmology and Early History of the Universe
From the Second to the Third Generation
Cosmic Explorer (CE)
A Brief History of CE
CE Detector Instrumentation
CE Status and Timing
Einstein Telescope (ET)
A Brief History of ET
ET Detector Instrumentation
ET Status and Timing
Down Under
The Path to 3G Observatories
References
8 Research and Development for Third-Generation Gravitational Wave Detectors
Contents
Introduction
The Three Generations of Gravitational Wave Detectors
Quantum Noise
Input/Output Relations
Squeezing
Squeezing for Silicon-Based Interferometers
Radiation Pressure Noise
Frequency-Dependent Squeezing
Interferometer Topologies
Speedmeters
Interferometers with Non-linear Elements
High-Power Lasers
Consequences of High Circulating Optical Power
Thermal Noise
Fluctuation Dissipation Theorem
Application to More Complex Systems
Dissipation from Heat Flow
Coating Thermal Noise
Multilayer Dielectric Coating
Brownian Thermal Noise
Thermo-optic Noise
Coating Materials
Low mechanical loss
Young's modulus
Thermo-optic parameters
Contrast of refractive index
Optical absorption
Ta2 O5/SiO2 Coating
Other Coating Materials
Multi-material Coating
Nano-layer Construction
Crystalline Coating
Large Laser Beams
Khalili Cavity
Substrate Thermal Noise
Substrate Brownian Noise
Substrate Thermoelastic Noise
Substrate Thermo-refractive Noise
Candidate Materials
Fused Silica
Sapphire
Silicon
Suspension Thermal Noise
Thermal Noise Formula
Losses in a Suspension
Dilution Effect by Gravity
Fused Silica Fibers
Cryogenic Suspension Materials
Experimental Methods for Thermal Noise Study
Quality Factor Measurement
Direct Measurement of Thermal Noise
Cryogenic Technologies
Extraction of Heat from Mirrors
Conductive Cooling
Radiative Cooling
Black Coatings
Heat Injection into Mirrors
Laser Power Absorption
Room Temperature Radiation
Cooling Engines
Preventing Vibrations
Molecular Layer Formation
Cryogenic Compatible Sensors and Actuators
Seismic Noise
Seismic Isolation
Newtonian Gravitational Noise
Seismic Sources
Atmospheric Sources
Seismic Management Through Architecture
Seismic Meta Materials
Technical Noise
Prototypes
Controls
Vacuum Technology
Conclusion
References
9 Squeezing and QM Techniques in GW Interferometers
Contents
Introduction
Quantum Fluctuations of the Electromagnetic Field and States of Light
Expectation Values of Quantum Fluctuations of the Electromagnetic Field
States of Light
Quadrature Noise Estimation
Generation of Nonclassical States of Light
Quantum Noise in the Interferometric Gravitational Wave Detectors
Quantum Noise in a Michelson Interferometer
Quantum Noise in Power Recycled Interferometers
Quantum Noise in Dual Recycled Interferometers
Quantum Noise in Presence of Squeezed Light
Quantum Noise in Real Interferometers
State of the Art
Advanced Methods for Quantum Noise Reduction
Variational Readout
Speed Meters
EPR Squeezing
Other Methods
Cross-Reference
References
10 Environmental Noise in Gravitational-Wave Interferometers
Contents
Introduction
Gravitational-Wave Interferometer at a Glance
Environment Monitoring
Environmental Sensors
Physical Environment Monitors
Geophysical Monitors
Infrastructure Monitors
Sensors Integration
Methods for Investigating Environmental Noise
Noise Hunting
Data Mining Techniques
Experimental Techniques
Coupling Functions
Validation of Gravitational-Wave Events
Seismic Noise
Earth Crust Deformations
Earthquakes
Sea and Wind
Anthropogenic Seismic and Acoustic Sources
Sound and Vibrations
Sound and Vibration Sources
Vibration Noise Reduction
Vacuum Pumping System
Cooling and Climatization System
Clean Room Areas
Optical Benches
Vibration Isolation
Tuned Damper
Beam Jitter Noise
Scattered Light
Scattering Processes
Scattered Light Noise Model
Morphology of Scattered Light Noise
Scattered Light Hunting Methods
Inspection and Tapping
Noise Injections
Switch-Off Tests
Scattered Light Coupling Measurement
Scattered Light Mitigation
Electromagnetic Noise
EM Noise Coupling to Electronics
Noise from the Utility Mains
Uninterruptible Power Supplies
Switching Devices
Power Supplies
Digital Devices
External Sources of RF Noise
Magnetic and Electric Fields Coupling to Test Masses
Global Magnetic Noise
Magnetic Field Influences
Note on Barkhausen Noise
Electric Field Influences
Charging and Discharging Processes
Cosmic Rays
Gravity Gradient Noise
Newtonian Noise from Ground Density Fluctuations
Newtonian Noise from Air Density Fluctuations
Environmental Noise Considerations in Site Selection and Site Facilities
Site Selection Considerations for Minimizing Environmental Noise
Site Facilities Considerations for Minimizing Self-Inflicted Environmental Noise
Buildings
Site Roads and Parking Areas
Mains Electrical Considerations
HVAC and Other Equipment
Conclusions
References
11 Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum
Contents
Introduction
Experimental Frontiers
DECIGO
B-DECIGO and Technology Developments
Lunar-Based Experiments
Summary
Cross-References
References
Part III Gravitational Wave Sources
12 Binary Neutron Stars
Contents
Introduction: General Description of Neutron-Star—Neutron-Star Mergers
Gravitational Waves from the Pre-merger Phase
Tidal Effects and Their Relation with the Neutron-Star Equation of State
Other Finite-Size Effects
Detectability and Detection of Pre-merger Gravitational Waves
Gravitational Waves from the Merger and Post-merger Phases
Spectral Properties and Their Relation with the Neutron-Star Equation of State
Detectability of Post-merger Gravitational Waves
Investigating Phase Transitions with Post-merger Waveforms
Cross-References
References
13 Isolated Neutron Stars
Contents
Introduction
Continuous Gravitational Wave Emission from Rotating Sources
Multipole Radiation
Gravitational Wave Strain
Composition and Material Properties of Compact Stars
Phases of Dense Matter
Rigidity and Shear Modulus
Viscous Damping
Heat Transport and Cooling
Gravitational Waves Due to ``Mountains''
Crustal Mountains
Magnetic Deformations
Exotic Matter and Core Deformations
Gravitational Wave Seismology
Orthogonal Oscillation Modes and Instabilities
f- and r-Modes in Newborn and Young Sources
r-Modes in Recycled Sources
Multi-messenger Observations
Gravitational Wave-Driven Spin Evolution
Impact of Oscillation Modes on the Thermal Evolution
Continuous GW Searches and Current Bounds
Present Searches
Electromagnetic Constraints
Gravitational Wave Constraints
Conclusions
Cross-References
References
14 r-Process Nucleosynthesis from Compact Binary Mergers
Contents
Introduction
Matter Ejection from Compact Binary Mergers
Ejecta from Binary Neutron Star Mergers
Ejecta from Neutron Star-Black Hole Mergers
Ejecta Expansion and Thermodynamics
r-Process Nucleosynthesis in Compact Mergers
Compact Binary Mergers as r-Process Site
The Working of the r-Process in Compact Binary Mergers
The NSE Phase
The r-Process Nucleosynthesis Phase
The Neutron Freeze-Out and the Decay Phases
The r-Process Peaks and the s-Process Nucleosynthesis
Nucleosynthesis in High-Entropy and Fast-Expanding Ejecta
Nuclear Physics Input and Detailed Network Calculations
Detailed Network Calculations and Nucleosynthesis Yields from Compact Binary Mergers
Nucleosynthesis Yields from Compact Binary Mergers
Observables of Compact Binary Merger Nucleosynthesis
Electromagnetic Signatures of r-Process Nucleosynthesis in Compact Binary Mergers
What Is a Kilonova?
r-Process Nucleosynthesis and Kilonovae
Modeling Kilonovae
GW170817 and Its Kilonova
Compact Binary Mergers and the Chemical Evolution
Summary and Outlook
Cross-References
References
15 Black Hole-Neutron Star Mergers
Contents
Introduction
Black Hole-Neutron Star Binary Population
Event Rates
Binary Parameters
Dynamics of BHNS Mergers
Tidal Disruption
Disk Formation
Post-merger Evolution
Gravitational Wave Signals
Observing BHNS Mergers Through Gravitational Waves
Waveform Models and Their Accuracy
Detectability and Detection Biases
Current BHNS Merger Candidates
R-Process and Kilonovae
Nucleosynthesis in BHNS Mergers
Radioactively Powered Transients: Kilonovae
Kilonova Models
UV/Optical/IR Follow-Up of BHNS Merger Candidates
Short Gamma-Ray Bursts
Other EM Counterparts to BHNS Mergers
Radio Emission from Mildly Relativistic Outflows
Extended X-Ray Emission
Pre-merger Electromagnetic Signals
Conclusions
Cross-References
References
16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes
Contents
Introduction
BH Mergers in Globular Clusters
Dynamical Processes in Globular Clusters
Merger Probability in Globular Clusters
Merger Rate Density
GW Frequency and Eccentricity Distribution for Globular Clusters
BH Mergers in Galactic Nuclei
Mergers Driven by Binary-Single Encounters
Single-Single Gravitational Wave Captures
Mergers Triggered by the Kozai-Lidov Effect
Gas-Assisted Mergers in Active Galactic Nuclei
Further Observational Diagnostics of the Dynamical Channel
Mass, Spin, and Redshift Distributions
Universal Gravitational Wave Statistics
Observed Merger Fraction and Branching Ratios
Conclusions
Cross-References
References
17 Formation Channels of Single and Binary Stellar-Mass Black Holes
Contents
Introduction: Observational Facts About Gravitational Waves
The Formation of Compact Remnants from Single Stellar Evolution and Supernova Explosions
Stellar Winds and Stellar Evolution
Core-Collapse Supernova (SN) or Direct Collapse
Pair Instability and the Mass Gap
The Mass of Compact Remnants
Compact Object Spins
Natal Kicks
Binaries of Stellar Black Holes
Mass Transfer
Common Envelope (CE)
Alternative Evolution to CE
BBH Spins in the Isolated Binary Evolution Model
Summary of the Isolated Binary Formation Channel
The Dynamics of Binary Black Holes (BBHs)
Dynamically Active Environments
Two-Body Encounters, Dynamical Friction, and Core Collapse
Binary: Single Encounters
Hardening
Exchanges
Stellar Mergers and BHs in the Pair-Instability Mass Gap
Direct Three-Body Binary Formation
Dynamical Ejections
Formation of Intermediate-Mass Black Holes by Runaway Collisions
Hierarchical BBH Formation and IMBHs
Alternative Models for Massive BHs and IMBH Formation in Galactic Nuclei
Kozai–Lidov Resonance
Summary of Dynamics and Open Issues
BBHs in the Cosmological Context
Data-Driven Semi-analytic Models
Cosmological Simulations
Summary and Outlook
Cross-References
References
18 The Gravitational Capture of Compact Objects by Massive Black Holes
Contents
Introduction: Why Is This Important?
Fundamental Science
Extreme-Mass Ratio Inspirals
A Long Story Short
Stellar Tidal Disruptions
Relaxation Theory
The Loss-Cone
Formation of EMRIs via Relaxation
Formation of EMRIs via Tidal Separation of Binaries
Geodesic Motion and Relativistic Precession
Geodesic Motion Around a Schwarzschild Black Hole
Relativistic Precession
The Kerr Case
Evolution in Phase-Space
Accumulated Phase Shift
Event Rate of Relaxation EMRIs
Intermediate-Mass Ratio Inspirals
Intermediate-Mass Black Holes
Wandering of IMBHs
Numerical Simulations of IMRIs
Event Rate of IMRIs
Multi-bandwidth IMRIs
Modelling IMRIs
Extremely Large Mass Ratio Inspirals
A Relativistic Fokker-Planck Algorithm
Newtonian Motion Around a Newtonian Potential
Relations Between the Relativistic and Newtonian Parameters for a Schwarzschild SMBH
Relations Between the Relativistic and Newtonian Parameters for a Kerr SMBH
A Possible Scheme
Cross-References
References
19 Massive Black-Hole Mergers
Contents
Introduction
Dark Matter Halos
Basic Quantities
Formation Time, Fast Collapse vs. Slow Accretion
Median and Average Halo Mass Growth
Radial Mass Profile and Pseudo-evolution
Halo Spin
Halo Mass Function
Halo Merger Trees
Halo Merger Rates
Baryons and Black Holes
Black-Hole Mass Function at High Redshift
Delays Between Galaxy and Black-Hole Mergers
Predictions for LISA and PTAs
Future Prospects
Cross-References
References
20 LISA and the Galactic Population of Compact Binaries
Contents
Introduction
LISA Summary
Population Modeling
Field Evolution
Dynamical Evolution
Source Classes/Observed Systems
Detached Binary White Dwarfs
Interacting Binary White Dwarfs
Neutron Star-White Dwarf Binaries
Binary Neutron Stars
Black Hole-Neutron Star Binaries
Binary Black Holes
Detection
Resolved Systems
Confusion-Limited Signal
Conclusion
Cross-References
References
21 Gravitational Waves from Core-Collapse Supernovae
Contents
Introduction
Basic Overall Picture
The Road to Core Collapse
Core Collapse and Road to Explosion
Generation of Gravitational Waves
Non-rotating and Slowly Rotating Case
Early Quasi-periodic Signal
PNS Convection
Neutrino-Driven Convection
SASI
Protoneutron Star Pulsations
Explosion Phase Signal
Rapidly Rotating Case
Bounce and Ring-Down Signal
Non-axisymmetric Instabilities
Collapse to Black Hole
Anisotropic Neutrino Emission
Quark Deconfinement Phase Transition
Multi-messenger Aspects
GW Searches
Neutrino Searches
Combined Searches
Conclusion and Prospects
Cross-References
References
22 Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range
Contents
Introduction
Astrophysical Sources of Gravitational Waves Detectable by Ground-Based Detectors
Binary Systems of Compact Objects
Core Collapse of Massive Stars
Isolated Neutron Stars
Expected Electromagnetic Counterparts of Gravitational Wave Signals
Gamma-Ray Bursts
Jet Launch by the Merger Remnant
The Dominant Form of Energy in the Jet
Jet Interaction with the Progenitor Material
Prompt Emission
Afterglow Dynamics and Emission
The Kilonova
Mass Ejection from Compact Binary Mergers: A Variety of Mechanisms, Compositions, and Morphologies
R-Process Nucleosynthesis and Ejecta Heating by Nuclear Decay
Kilonova Emission Features
Interaction with the Interstellar Medium: The Kilonova Radio Remnant
The Electromagnetic Follow-Up of Transient Gravitational Wave Sources
Low-Latency Search for Gravitational Wave Signals
The Gravitational Wave Alert System: Distribution and Alert Contents
The Gravitational Wave Sky Localization
Source Classification and Properties
Identification and Localization of the Counterpart
Electromagnetic Counterpart Search Strategies
Wide-Field Optical/IR Observatories for Candidate Detection
Wide-Field X-Ray and Gamma-Ray Observatories
Counterpart Classification and Follow-Up
Optical/NIR Observatories for Photometric and Spectral Classification
Radio Observations
High-Energy Follow-Up Observations
GW170817 and Its Electromagnetic Counterparts
References
23 Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravitational Waves
Contents
Introduction
GW Observations in a Multi-messenger Context
The Physical Parameters Derived from GW Observations
Complementarity of Electromagnetic, Gravitational Wavesand Particle Observations
High-Energy Neutrino Emission from Gravitational Wave Sources
UHECR Emission from Gravitational Wave Sources
The GeV-TeV Astronomy with the Gravitational Waves
The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts
GeV-TeV Instruments
Space Gamma-Ray Observatories
Ground-Based Gamma-Ray Telescopes: Cherenkov Telescopes and Particle Detectors
High-Energy Gamma Rays from GRBs and Their Implications on Gravitational Wave Observations
GRB190114C and GRB160821B at the TeV Energies
Observation Strategies with Neutrino and Gamma-Ray Instruments
Neutrino Follow-Up of Gravitational Wave Candidates
GeV-TeV Follow-Up of GW Candidates
Summary on Open Questions
References
24 Mission Design for the TAIJI Misson and Structure Formation in Early Universe
Contents
Introduction
A Survey of Gravitational Wave Sources
Compact Binary Star Systems
Binary Black Hole Mergers
Intermediate-Mass Black Hole Binaries at High Redshift Universe
Supermassive Black Hole Mergers
Extreme Mass Ratio Inspirals
Intermediate-Mass Ratio Inspirals
Gravitational Waves from Early Universe
Burst Signals
Mission Design
Simulations of Cosmic Growth and Merger of Black Holes and Event Rate Estimates
Event Rate Estimate for the Detection of IMRIs in Dense Star Clusters
Concluding Remarks
References
25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin
Contents
Introduction: Gravitational Waves in Friedmann-Lemaître-Robertson-Walker Backgrounds
Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties
Characterization of a Stochastic Gravitational Wave Background
Evolution of a Stochastic Gravitational Wave Background in the Expanding Universe
Signal from a Generic Stochastic Source of Gravitational Waves
Cosmological Gravitational Wave Backgrounds: Relevant Examples
First-Order Phase Transitions
Examples of Cosmological First-Order Phase Transitions
Elements of the Dynamics of the Expanding Bubbles and the Surrounding Fluid
Parametrization of the GW Signal
Cosmic Defect Networks
Irreducible Gravitational Wave Emission Due to Scaling
Gravitational Wave Background from the Decay of Cosmic String Loops
References
26 Primordial Gravitational Waves
Contents
Introduction
Cosmological Inflation as Source of Primordial Gravitational Waves
Motivations for Cosmological Inflation
The Basic Mechanism of Gravitational Wave Production by Inflation
Realizations of Inflation with a Single Scalar Field
Beyond the Standard Mechanism of Inflationary Gravitational Wave Production
Propagation of Inflationary Gravitational Waves Through Cosmic History
Conclusions
References
27 Primordial Black Holes
Contents
Introduction
Primordial Black Holes as Dark Matter
Conclusions
References
28 Testing the Nature of Dark Compact Objects with Gravitational Waves
Contents
Introduction
Models of Exotic Compact Objects
ECO Phenomenology
Multipole Moments
Testing the Nature of Compact Objects with Multipole Moments
Tidal Heating
Tidal Deformability
Ringdown
Ergoregion Instability of ECOs
Gravitational-Wave Echoes
Conclusions and Open Issues
References
29 Quantum Gravity and Gravitational-Wave Astronomy
Contents
Introduction: Why Quantum Gravity?
Stochastic GW Background
Basics
Results in Quantum Gravity
String-Gas Cosmology
New Ekpyrotic Scenario
Brandenberger–Ho Non-commutative Inflation
Multi-fractional Spacetimes
Modified Dispersion Relation and Propagation Speed
Basics
Results in Quantum Gravity
Luminosity Distance
Basics
Results in Quantum Gravity
Strain Noise
Results in Quantum Gravity
Conclusions
References
30 LIGO, VIRGO, and KAGRA as the International Gravitational Wave Network
Contents
Introduction
Gravitational Wave Detectors
Basic Concepts
LIGO
VIRGO
KAGRA
International Gravitational Wave Network (IGWN)
Observing Runs
Observational Science Highlights
Observing Run 1 (O1)
Observing Run 2 (O2)
Observing Run 3: The First 6 Months (O3a)
Outlook
References
Part IV Gravitational Wave Modeling
31 Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspirals
Contents
Introduction
Goal and Relation to Other Chapters
Notations
The Essence: Quadrupole Radiation from a Mass in Circular Orbit
Adiabatic Approximation
Stationary Phase Approximation
Post-Newtonian Gravitational Waveforms for Spinning, Nonprecessing Binary Black Holes
PN Binding Energy, Energy Flux, and BH Horizon Flux
Balance Equation for Slowly Evolving Black Holes
Accuracy of the Post-Newtonian Approximants
Time- and Frequency-Domain Inspiral Templates
Taylor Time-Domain Approximants
TaylorT1
TaylorT4
TaylorT2
TaylorT3
TaylorT5
Taylor Frequency-Domain Approximants
TaylorF1
TaylorF2
Beyond Spinning, Nonprecesssing Binary Black Hole Cases
Full Inspiral-Merger-Ringdown Waveform Models
Effective-One-Body (EOB) Approach
SEOBNR Family
TEOBResumS
Phenomenological (IMRPhenom) Models
IMRPhenomD Family
IMRPhenomX Family
IMRPhenomTP
GIMR for Modified Theory of Gravity
Conclusion
Cross-References
References
32 Effective Field Theory Methods to Model Compact Binaries
Contents
Introduction
Notation
The Setup
Classical Path Integral
Method of Regions
Gravitational Self-Interactions, Wick Contraction, and Power Counting
The Conservative Sector
Divergences
Infrared Divergences
The Radiative Sector
Conclusions
Cross-References
References
33 Repeated Bursts
Contents
Introduction
Eccentric Dynamics
Detection Strategies
Matched Filtering
Power Stacking
Modeling of Black Hole Binaries
Numerical Models
Numerical Relativity
Effective Kerr Spacetime
Inspiral-Merger-Ringdown waveforms
Analytic Models
Adiabatic Waveforms
Effective Flyby Approach
Timing Models
Tests of General Relativity
Binaries with Neutron Stars
Tidal Interactions and Resonances
Post-merger and Remnants
Cross-References
References
34 Numerical Relativity for Gravitational Wave Source Modeling
Contents
Introduction
The Role of Numerical Relativity in Gravitational Wave Astronomy
A Brief History and the Current Status of Numerical Relativity
The Core Difficulties of Numerical Relativity and Current Solutions
The Partial Differential Equation Formalism
The Singularity Inside the Black Hole
The Problem of Multiple Physical Scales
The Problem of Boundary Condition
The Problem of Coordinate Choice
The Problem of Parallel Computation
The Problem of Gravitational Waveform Extraction
Application of Numerical Relativity to Gravitational Wave Source Modeling
Binary Black Hole (BBH) System
Neutron Star-Black Hole Binary and Binary Neutron Star
Summary and Outlook
References
35 Black-Hole Superradiance: Searching for Ultralight Bosons with Gravitational Waves
Contents
Introduction
Black-Hole Superradiance in a Nutshell
Superradiant Instabilities in the Presence of Ultralight Fields
Evolution of the Superradiant Instability
Evolution of Superradiant Instabilities for Astrophysical BHs
Bounds on Ultralight Bosons from BH Spin Measurements
GW Signatures of the Superradiant Instability
Direct GW Emission from Boson Clouds
GW Emission from Level Transitions
GW Bursts from Bosenova Explosions
Signatures in Binary Systems
Open Questions
References
36 Black Hole Perturbation Theory and Gravitational Self-Force
Contents
Introduction
Perturbation Theory in General Relativity
Isolated, Stationary Black Hole Spacetimes
Metric
Null Tetrads
Symmetries
Black Hole Perturbation Theory
The Teukolsky Formalism and Radiation Gauge
Geroch-Held-Penrose Formalism
Teukolsky Equations
Teukolsky-Starobinsky Identities
Reconstruction of a Metric Perturbation in Radiation Gauge
Gravitational Waves
GHP Formalism in Kerr Spacetime
Mode-Decomposed Equations in Kerr Spacetime
Sasaki-Nakamura Transformation
Metric Perturbations of Schwarzschild Spacetime
Alternative Tensor Bases
Regge-Wheeler Formalism and Regge-Wheeler Gauge
Regge-Wheeler Formalism in the Frequency Domain
Transformation Between Regge-Wheeler and Zerilli Solutions
Transformation Between Regge-Wheeler and Teukolsky Formalism
Gravitational Waves
Metric Reconstruction in Regge-Wheeler Gauge
Lorenz Gauge
Lorenz Gauge Formalism in the Frequency Domain
Lorenz Gauge Metric Reconstruction from Regge-Wheeler Master Functions
Gravitational Waves
Small Objects in General Relativity
Matched Asymptotic Expansions
Tools of Local Analysis
Local Solution: Self-Field and an Effective External Metric
Equations of Motion
Skeleton Sources: Punctures and Particles
Orbital Dynamics in Kerr Spacetime
Geodesic Motion
Constants of Motion, Separable Geodesic Equation, and Conventions
Quasi-Keplerian Parametrization
Fundamental Mino Frequencies and Action Angles
Fundamental Boyer-Lindquist Frequencies and Action Angles
Resonant Orbits
Accelerated Motion
Evolution of Orbital Parameters
Method of Osculating Geodesics
Perturbed Mino Frequencies and Action Angles
Perturbed Boyer-Lindquist Frequencies and Action Angles
Multiscale Expansions, Adiabatic Approximation, and Post-Adiabatic Approximations
Transient Resonances
Solving the Einstein Equations with a Skeleton Source
Multiscale Expansion
Structure of the Expansion
Multiscale Expansion of Source Terms and Driving Forces
Snapshot Solutions and Evolving Waveforms
Adiabatic Approximation
First Post-Adiabatic Approximation
Mode Decompositions of the Singular Field
Example: Leading Order Puncture for Circular Orbits in Schwarzschild Spacetime
Conclusion
Cross-References
References
37 Distortion of Gravitational-Wave Signals by Astrophysical Environments
Contents
Introduction
Standard Siren
Mass-Redshift Degeneracy
Effects of Astrophysical Environments
Strong Gravitational Lensing
The Effect of Motion
Deep Gravitational Potential
Peculiar Acceleration
The Effect of Gas
Summary
References
38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories
Contents
Theory of Gravity and Its Extensions
Generation of Gravitational Waves
f(R)-Models
The Chameleon Mechanism in f(R)-Gravity
The Gravitational Wave Stress-Energy (Pseudo) Tensor
Application to Binary Systems and Observational Constraints
Time Variation of the Orbital Period in General Relativity
Periastron Advance in General Relativity
Other Post-Keplerian Parameters and Observational Constraints on General Relativity
First-Time Derivative of the Orbital Period in f(R)-Gravity
Observational Constraints on f(R)-Gravity from Binary Systems
Periastron Advance in f(R)-Gravity
Constraining Alternative Theories of Gravity Using GW150914 and GW151226
Constraints from the Shapiro Delay
Conclusions
Cross-References
References
39 Testing General Relativity with Gravitational Waves
Contents
Introduction
Parameterized Tests
Formalism
Parameterized Waveforms
Parameter Estimation
Current Status
Future Prospects
Applications to Specific Theories
Inspiral-Merger-Ringdown Consistency Tests
Formalism
Current Status
Future Prospects
Applications to Specific Theories
Gravitational-Wave Propagation
Modified Dispersion Relation
Graviton Mass and GW Propagation Speed
Generic GW Propagation Tests
Amplitude Birefringence in Parity Violation
Open Questions
Cross-References
References
40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves
Contents
Introduction
Brief Introduction to Dynamical Systems
Continuous and Discrete Dynamical Systems
Hamiltonian Systems and Integrability
Poincaré Surfaces of Section
Stability of Orbits in Maps and Continuous Systems
Fixed Points in Discrete Dynamical Systems
Stability of Periodic Trajectories and Fixed Points in Continuous Systems
Stable and Unstable Manifolds
Stability of Generic Trajectories
KAM, Poincaré-Birkhoff Theorem, and Chaos
KAM Theory and Birkhoff Chains
Chaotic Layers
Tools to Study Resonances
Inspirals Through Resonances
A Generic Inspiral
Non-resonant Motion
Near-Resonant Motion
Error Budget
Additional Perturbations
Orbital Motion in Kerr Spacetimes and Perturbations
Deviating Spacetimes
Bumpy Black Holes
External Matter Deformation
Spinning Particle
Impact of Non-integrability on Extreme-Mass-Ratio Systems
Resonance Growth
Prolonged Resonances
Discussion
Cross-References
References
Part V Data Analysis Techniques
41 Principles of Gravitational-Wave Data Analysis
Contents
Introduction
Statistical Theory of Signal Detection and Parameter Estimation
Random Variables and Random Processes
Hypothesis Testing
Bayesian Approach
Neyman-Pearson Approach
Likelihood Ratio Test
The Matched Filter in Gaussian Noise: Known Signal
Cameron-Martin Formula
Stationary Noise
Matched Filtering
Estimation of Parameters
Fisher Information
Bayesian Estimation
Maximum Likelihood Estimation
Lower Bounds on the Variance of Estimators
Likelihood Ratio Statistic
Application of the Likelihood Ratio Statistic to Detection of Gravitational-Wave Signals
Signal-to-Noise Ratio and Fisher Information Matrix
False Alarm and Detection Probabilities
False Alarm Probability
Detection Probability
Number of Templates
Suboptimal Filtering
Deterministic Gravitational-Wave Signals
Signal from a Rotating Neutron Star
Signal from an Inspiralling Binary System
Signal from a Supernova Explosion
Colored Noise
Cross-References
References
42 Inferring the Properties of a Population of Compact Binaries in Presence of Selection Effects
Contents
Introduction
Source Production, Rates, Populations
Selection Effects
Hierarchical Likelihood
Data Partitioning
Calculating the Likelihood of Data Without a Trigger
The Good: p(bardjS0B0 |Λ)
The Bad: p(bardjS1B0 | Λ)
The Ugly: p(bardjS0B1 | Λ)
Putting the Pieces Together
Calculating the Likelihood of Data with a Trigger
We Got a Source: p(diS1B0 | Λ)
We Got a Background: p(diB1S0 | Λ)
Putting the Pieces Together
Finishing Off
Special Cases
Constant Background Rate
Constant Background and Source Rate
Constant Source Rate and No Background
Hierarchical Posterior
The Full Hyper-posterior
Analyzing Sources with Different Data Quality
The Hyper-posterior of the Shape Parameters
Handling Missing Parameters
A Simple One-Dimensional Example
A Gravitational-Wave Example
What Happens Next?
Conclusions
Glossary and Main Symbols
Cross-References
References
43 Machine Learning for the Characterization of Gravitational Wave Data
Contents
Introduction
Characterization of Gravitational Wave Data
Discrete Gaussian Processes
Autoregressive Parametric Model and Whitening Technique
Detecting and Characterizing Transient Noise Signals
Transient Signal Detection and Wavelet Detection Filter
Challenges for Machine Learning
Machine Learning for Transient Signal Classification
Application of Deep Learning for Glitch Classification
Data Preparation
Testing Classification Performance to Simulated Data
Classification of Glitches in Real Data
Classification of Unmodeled Sources
Searches of the Unknown
Summary
Cross-References
References
44 Advances in Machine and Deep Learning for Modeling and Real-Time Detection of Multi-messenger Sources
Contents
Introduction
Machine Learning and Numerical Relativity for Gravitational Wave Source Modeling
Machine Learning for Gravitational Wave Data Analysis
Deep Learning for Gravitational Wave Data Analysis
Deep Learning for the Classification and Clustering of Noise Anomalies in Gravitational Wave Data
Deep Learning for the Construction of Galaxy Catalogs in Large-Scale Astronomy Surveys to Enable Gravitational Wave Standard-Siren Measurements of the Hubble Constant
Challenges and Open Problems
Convergence of Deep Learning with High Performance Computing: An Emergent Framework for Real-Time Multi-messenger Astrophysics Discovery at Scale
Cross-References
References
45 Measuring Cosmological Parameters with Gravitational Waves
Contents
Introduction and Overview
Standard Cosmological Model
The Hubble Tension
Distances from Standard Sirens
Determining Redshift and Further Physics
Outline
Basics on Standard Sirens
Characteristic Scales
From the Waveform to the Hubble Constant
Statistical Methods to Extract Cosmological Parameters with Standard Sirens
Factors Limiting the Accuracy of Measurements
Bayesian Statistical Method: An Outline
Methods with Redshift Information from Independent Observations
Redshift from Identified Hosting Galaxy
Redshift from Galaxy Catalogues
Methods for Determining the Redshift from the GW Alone
Redshift from Tidal Deformation and Postmerger Signal
Astrophysical Distributions
Current Results and Projections for the LIGO, Virgo, and KAGRA Network
Standard Sirens with EM Counterparts
Present Results
Projections
Improving the H0 Measure with Information on Inclination Angle
Standard Sirens Without Counterpart
Present Results
Projections
Cosmology with Gravitational Waves and Third-Generation Detectors
Projections with LISA and Space-Based Interferometry
Projections with Planned Ground-Based GW Detectors ET and CE
Tests of GR with Standard Sirens
Cross-References
References
Index

Citation preview

Cosimo Bambi Stavros Katsanevas Konstantinos D. Kokkotas Editors

Handbook of Gravitational Wave Astronomy

Handbook of Gravitational Wave Astronomy

Cosimo Bambi • Stavros Katsanevas • Konstantinos D. Kokkotas Editors

Handbook of Gravitational Wave Astronomy With 381 Figures and 24 Tables

Editors Cosimo Bambi Department of Physics Fudan University Shanghai, China

Stavros Katsanevas Université de Paris and European Gravitational Observatory Paris, France

Konstantinos D. Kokkotas Institute for Astronomy and Astrophysics Eberhard Karls University of Tübingen Tübingen, Germany

ISBN 978-981-16-4305-7 ISBN 978-981-16-4306-4 (eBook) ISBN 978-981-16-4307-1 (print and electronic bundle) https://doi.org/10.1007/978-981-16-4306-4 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Gravitational waves are ripples of spacetime propagating at the speed of light. They were among the first predictions of the theory of general relativity of Albert Einstein, but for a few decades, the community discussed whether these gravitational waves were a real physical phenomenon or only an artifact of the coordinate system. The puzzle was solved only at the end of the 1950s, and immediately after Joseph Weber started developing the first gravitational wave detector. In 1974, Russell Alan Hulse and Joseph Hooton Taylor, Jr. discovered the first binary pulsar, which led to the first indirect evidence of the existence of gravitational waves. Accurate radio observations of the Hulse–Taylor binary system in the 1970s showed indeed that the decay of its orbital period matched well with the predictions of general relativity for the emission of gravitational waves. Such a discovery, which led Hulse and Taylor to receive the 1993 Nobel Prize in Physics, boosted the efforts to develop facilities for the direct detection of gravitational waves. In February 2016, the LIGO-Virgo collaboration announced the first direct detection of gravitational waves: the signal was produced by the coalescence of two black holes with masses of 29 and 36 solar masses and the event was observed on 14 September 2015. Rainer Weiss, Kip Thorne, and Barry Barish received the 2017 Nobel Prize in Physics for their contribution in such a breakthrough. On August 17, 2017, the detection of another type of event – a neutron star merger – triggered a follow-up campaign with about 100 observatories detecting signals using electromagnetic messengers ranging from high energy photons to visible light and radio waves. In particular, the quasi-simultaneous detection of gravitational waves from a binary neutron star and of a gamma-ray burst solved the longstanding problem of the origin of the short gamma-ray bursts and, at the same time, can be seen as the beginning of the era of multi-messenger astronomy. For these discoveries, LIGO and Virgo were named historical Milestones of technology by the Institute of Electrical and Electronic Engineers (IEEE) on February 3, 2021. The possibility of detecting gravitational waves from astrophysical sources has opened a completely new window for the study of the Universe. This is a new and extremely promising line of research for the next decades. With the latest upgrades, the LIGO and Virgo facilities can now detect an event every week, and this detection rate will increase with the next upgrades to a few per day and by orders of magnitude for the next generation of gravitational wave detectors. These detections are in the process of revolutionizing fundamental science, impacting astrophysics, cosmology, v

vi

Preface

particle and nuclear, and physics. From the detection of gravitational waves from compact binaries, we can even test fundamental physics, in particular Einstein’s theory of general relativity in the strong and dynamical regime. The Handbook of Gravitational Wave Astronomy provides an updated comprehensive description of gravitational wave astronomy. It has four parts: Gravitational Wave Detectors, Gravitational Wave Sources, Gravitational Wave Modeling, and Data Analysis Techniques. In Gravitational Wave Detectors, we review all observational facilities for the detection of gravitational waves, from ground- and space-based laser interferometers to pulsar timing arrays and indirect detection from the cosmic microwave background. In the second section, Gravitational Wave Sources, we discuss a number of astrophysical and cosmological gravitational wave sources, including black holes, neutron stars, possible more exotic objects, and sources in the early Universe. The third section reviews the methods to calculate gravitational waveforms in general relativity and in other theories of gravity. The fourth section, Data Analysis Techniques, covers techniques employed in gravitational wave astronomy data analysis. We hope that the Handbook of Gravitational Wave Astronomy can become a valuable reference work for graduate students and researchers in the gravitational wave communities for the next two decades. With the living edition, we plan to keep the handbook always updated. We are grateful to all authors and section editors for their contributions in this project as well as for their future efforts to update the chapters.

Contents

Volume 1 Part I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

3

Introduction to Gravitational Wave Astronomy . . . . . . . . . . . . . . . . . Nigel T. Bishop

Part II

Gravitational Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2

Terrestrial Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katherine L Dooley, Hartmut Grote, and Jo van den Brand

37

3

Space-Based Gravitational Wave Observatories . . . . . . . . . . . . . . . . . Jonathan Gair, Martin Hewitson, Antoine Petiteau, and Guido Mueller

85

4

Pulsar Timing Array Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. P. W. Verbiest, S. Osłowski, and S. Burke-Spolaor

157

5

Quantum Sensors with Matter Waves for GW Observation . . . . . . . Andrea Bertoldi, Philippe Bouyer, and Benjamin Canuel

199

6

CMB Experiments and Gravitational Waves . . . . . . . . . . . . . . . . . . . Livia Conti and Benjamin R. B. Saliwanchik

243

7

Third-Generation Gravitational-Wave Observatories . . . . . . . . . . . . Harald Lück, Joshua Smith, and Michele Punturo

283

8

Research and Development for Third-Generation Gravitational Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert L. Ward, Bram J. J. Slagmolen, and Yoichi Aso

301

9

Squeezing and QM Techniques in GW Interferometers . . . . . . . . . . Fiodor Sorrentino and Jean-Pierre Zendri

361

10

Environmental Noise in Gravitational-Wave Interferometers . . . . . Irene Fiori, Anamaria Effler, Philippe Nguyen, Federico Paoletti, Robert M. S. Schofield, and Maria C. Tringali

407

vii

viii

11

Contents

Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kiwamu Izumi and Karan Jani

Part III

479

Gravitational Wave Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493

12

Binary Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luca Baiotti

495

13

Isolated Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brynmor Haskell and Kai Schwenzer

527

14

r-Process Nucleosynthesis from Compact Binary Mergers . . . . . . . A. Perego, F.-K. Thielemann, and G. Cescutti

555

15

Black Hole-Neutron Star Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francois Foucart

611

16

Dynamical Formation of Merging Stellar-Mass Binary Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bence Kocsis

661

Formation Channels of Single and Binary Stellar-Mass Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michela Mapelli

705

The Gravitational Capture of Compact Objects by Massive Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pau Amaro Seoane

771

17

18

19

Massive Black-Hole Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enrico Barausse and Andrea Lapi

851

20

LISA and the Galactic Population of Compact Binaries . . . . . . . . . . Matthew Benacquista

885

21

Gravitational Waves from Core-Collapse Supernovae . . . . . . . . . . . Ernazar Abdikamalov, Giulia Pagliaroli, and David Radice

909

Volume 2 22

23

Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marica Branchesi, Antonio Stamerra, Om Sharan Salafia, Silvia Piranomonte, and Barbara Patricelli Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . Antonio Stamerra, Barbara Patricelli, Imre Bartos, and Marica Branchesi

947

993

Contents

ix

24

Mission Design for the TAIJI Misson and Structure Formation in Early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 Shanquan Gui, Shuanglin Huang, and Yun-Kau Lau

25

Stochastic Gravitational Wave Backgrounds of Cosmological Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041 Chiara Caprini and Daniel G. Figueroa

26

Primordial Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 Gianmassimo Tasinato

27

Primordial Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 Juan García-Bellido

28

Testing the Nature of Dark Compact Objects with Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 Elisa Maggio, Paolo Pani, and Guilherme Raposo

29

Quantum Gravity and Gravitational-Wave Astronomy . . . . . . . . . . 1177 Gianluca Calcagni

30

LIGO, VIRGO, and KAGRA as the International Gravitational Wave Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 Patrick Brady, Giovanni Losurdo, and Hisaaki Shinkai

Part IV

Gravitational Wave Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227

31

Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 Soichiro Isoyama, Riccardo Sturani, and Hiroyuki Nakano

32

Effective Field Theory Methods to Model Compact Binaries . . . . . . 1279 Riccardo Sturani

33

Repeated Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 Nicholas Loutrel

34

Numerical Relativity for Gravitational Wave Source Modeling . . . . 1347 Tianyu Zhao, Zhoujian Cao, Chun-Yu Lin, and Hwei-Jang Yo

35

Black-Hole Superradiance: Searching for Ultralight Bosons with Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377 Richard Brito and Paolo Pani

36

Black Hole Perturbation Theory and Gravitational Self-Force . . . . 1411 Adam Pound and Barry Wardell

37

Distortion of Gravitational-Wave Signals by Astrophysical Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1531 Xian Chen

x

Contents

38

Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553 Mariafelicia De Laurentis and Ivan De Martino

39

Testing General Relativity with Gravitational Waves . . . . . . . . . . . . 1591 Zack Carson and Kent Yagi

40

Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625 Georgios Lukes-Gerakopoulos and Vojtˇech Witzany

Part V

Data Analysis Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1669

41

Principles of Gravitational-Wave Data Analysis . . . . . . . . . . . . . . . . . 1671 Andrzej Królak

42

Inferring the Properties of a Population of Compact Binaries in Presence of Selection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709 Salvatore Vitale, Davide Gerosa, Will M. Farr, and Stephen R. Taylor

43

Machine Learning for the Characterization of Gravitational Wave Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1769 Elena Cuoco, Alberto Iess, Filip Morawski, and Massimiliano Razzano

44

Advances in Machine and Deep Learning for Modeling and Real-Time Detection of Multi-messenger Sources . . . . . . . . . . . . . . . 1793 E. A. Huerta and Zhizhen Zhao

45

Measuring Cosmological Parameters with Gravitational Waves . . . 1821 Simone Mastrogiovanni and Danièle A. Steer

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1873

About the Editors

Cosimo Bambi received his laurea from Florence University (Florence, Italy) in 2003 and his doctoral degree from Ferrara University (Ferrara, Italy) in 2007. He worked as a postdoctoral research scholar at Wayne State University (Detroit, Michigan) in 2007–2008, at IPMU at the University of Tokyo (Kashiwa, Japan) in 2008–2011, and in the group of Gia Dvali at LMU Munich (Munich, Germany) in 2011–2012. He joined Fudan University (Shanghai, China) at the end of 2012 as associate professor under the Thousand Young Talents Program of the State Council of the People’s Republic of China. He was promoted to full professor at the end of 2013 and named Xie Xide Junior Chair Professor of Physics in 2016. In 2015, he was awarded a Humboldt Fellowship to conduct research at Eberhard-Karls Universitaet Tuebingen (Tuebingen, Germany), which he regularly visited from 2015 to 2018. In 2016, he was awarded an Invitation Fellowship from the Japan Society for the Promotion of Science (JSPS) to conduct research at Kyoto University (Kyoto, Japan). Professor Bambi has received numerous awards, including the Magnolia Silver Award from the Municipality of Shanghai for outstanding contributions to Shanghai’s development. Professor Bambi has worked on a number of topics in the fields of high-energy astrophysics, particle cosmology, and gravity. He has published around 200 papers in refereed journals and authored/edited several academic books with Springer: Introduction to Particle Cosmology: The Standard Model of Cosmology and its Open Problems (Springer-Verlag Heidelberg Berlin, 2016), Astrophysics of Black Holes: From Fundamental Aspects to Latest Developments (Springer-Verlag Heidelberg Berlin, 2016), Black Holes: A Laboratory xi

xii

About the Editors

for Testing Strong Gravity (Springer Singapore, 2017), Introduction to General Relativity (Springer Singapore, 2018), and Tutorial Guide to X-ray and Gamma-ray Astronomy: Data Reduction and Analysis (Springer Singapore, 2020). The book Introduction to General Relativity was published in Chinese by Fudan University Press in 2020 and in Spanish by Editorial Reverte’ in 2021. Professor Bambi has also written a popular science book, Niente e’ impossibili: Viaggiare nel tempo, attraversare i buchi neri e altre sfide scientifiche (in Italian), published by Il Saggiatore in 2020.

Stavros Katsanevas, professor exceptional class Université de Paris and director of the European Gravitational Observatory (2018-) hosting the Gravitational Wave (GW) antenna Virgo in Pisa/Italy. In the pasrt years, he has been director of the Laboratory of Astroparticle Physics and Cosmology (APC) and co-director of the Laboratory of Excellence (LabEx) UnivEarths (APC and IPGP: Institute of the Physics of the Globe) combining scientific expertise on Earth and environmental sciences, with astroparticle physics and cosmology (2014–2017); coordinator of the cluster of Sciences and Engineering of the community of universities and research institutions that eventually merged in the University of Paris (UdP). First chairman of the European Consortium of Astroparticle Physics (APPEC) gathering the European funding agencies funding the field and proposer and first coordinator of its precursor the EU-funded Network ASPERA (2006– 2012). Deputy director (2002–2012) responsible for Neutrino and Astroparticle Physics and Cosmology of the National Institute of Nuclear and Particle Physics (IN2P3) of CNRS and Director of the CNRS interdisciplinary program “Particules et Univers”. President of the administration council of the European Gravitation Observatory (EGO, 2002–2012), chairman of the resource board of the AUGER High Energy Cosmic Ray Observatory. Member of the resource boards of the FERMI satellite, the ANTARES Deep Ocean Observatory, the LSM Laboratoire Souterrain de Modane, the Double-CHOOZ Reactor Neutrino Experiment, CTA and KM3NET. Also, member of

About the Editors

xiii

the Committee of Large Research Infrastructures of CNRS, the Evaluation Committee for Space Research (CERES) of CNES, and founding Principal Investigator (2007-) of the Institute of Physics and Mathematics of the Universe (KAVLI-IPMU) at Tokyo. Initiator and chief executive of the Public-PrivatePartnerships (GIS) between CNRS and the companies PHOTONIS on photodetection (2005–2009) and SAGEM on extreme optics (2007–2011). His recent research concerns the possibilit of a gravitational wave antenna deplyed at the Moon (project LSGA), the synergies between GW research and Geosciences (project APOGEIA), muon radiography issues of geoscientific and archaeological structures, distributed fiber and/or robotic sensor networks. He has also been the coordinator of the GW International Committee (GWIC) Third Generation (3G) working group on governance. Education and academic career: Diploma of Physics U. of Athens (1975), DEA of Theoretical Physics ENS/Paris7/Paris 11 (1976), Doctorate of 3e cycle Ecole Polytechnique (1979) and PhD from the U. of Athens (1985). Fellow of the French (1976–1979), and Greek Ministry (1979– 1982), of CERN (1983–1986) as well as CERN associate scientist (1991–1992) and corresponding Fellow (1995–1996). Lecturer and associate professor at the U. of Athens (1982–1996), professor for the U. of Lyon (1996–2004) and U. Paris Diderot/U. de Paris (2004-). He has co-authored the paper “GW Physics and Astronomy in the 2020s and 2030s”, published by Nature. In the previous century he has worked on particle and astroparticle physics (standard model, supersymmetry and neutrino) at CERN (LEP, ISR, BEBC), Fermilab(E537), Gran Sasso (OPERA) and Greece (Neutrino telescope NESTOR). He is coauthor of over 400 publications, and his scientific achievements include, notable publications on Higgs search and supersymmetry at LEP, QCD at Fermilab, neutrino oscillation at BEBC and OPERA at CERN, the highly used and cited supersymmetry generator at LEP: SUSYGEN, the first design of the CERNGran Sasso neutrino beam, one of the first instalments (2000) of an intelligent sensor network with 1000 nodes (OPERA). He has taught Physics Courses in U. Paris Diderot, Lyon I and Athens, and Ecole Normale

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About the Editors

Supérieure de Paris et de Lyon. He has supervised 6 theses. He is also interested on the links between art and science. He is a member of the orientation board of the Foundation Carasso (Paris) and coordinator of a project gathering artists and scientists (Univers 2.0). He has been curator of the exhibition “The Rhythm of space” in Museo delle Grafica , Pisa (2018), and he has written many essays and provided insight for artist exhibitions (Saraceno, Csorgo, Kavalieratos,. . . ). He has obtained the physics prize of the Academy of Athens for his works on superymmetry (2000) and the title of Chevalier de l’Ordre National du Mérite (2011-).

Konstantinos D. Kokkotas is Professor (Chair) of Theoretical Astrophysics, Institute of Astronomy and Astrophysics, University of Tübingen. BSc in mathematics (1981, U. of Thessaloniki), MSc in applied mathematics and astronomy (1985, U. of Wales, Cardiff), PhD in physics (1988, U. of Thessaloniki). From 1990 to 2007, he served in the Physics Department at the University of Thessaloniki as Lecturer, Assistant, and Associate and Full Professor of Relativity. In 2007, he moved to the present position at the University of Tübingen. Since 2010, he has been an adjunct professor at Georgia Institute of Technology, and in 2017, he was elected as honorary professor at the University of Thessaloniki. Professor Kokkotas has supervised over 50 BSc and MSc theses, as well as 14 PhD thesis, and mentored 30 postdoctoral researchers. He has taught general relativity, numerical methods, relativistic astrophysics, classical field theory, and first-year mathematics and physics in both the Universities of Thessaloniki and Tübingen. Professor Kokkotas has served as core member of various international research initiatives (VESF, EU Marie Curie networks, COST actions), PI and spokesperson for the section “Gravitational Waves: Theory and Sources" of the European program “Integrated Large Infrastructure for Astroparticle Physics (ILIAS),” and elected member of national and international scientific societies such as the following: member of the Governing Council of International Society

About the Editors

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for General Relativity and Cosmology, chairman of the Hellenic Society for General Relativity and Cosmology, member of scientific advisory boards of institutes and departments in various European countries, and managing editor for IJMPD. His research interests are related to theoretical studies of gravitational waves sources with emphasis on the dynamics of neutron stars and black holes. He has published over 200 research articles in refereed journals and more than 20 invited review articles.

Section Editors

Prof. Dr. Mairi Sakellariadou Theoretical Particle Physics and Cosmology Group, Physics Department, King’s College London, University of London, Strand, London, UK

Laura Cadonati Associate Dean for Research, College of Sciences, Professor, School of Physics, Georgia Institute of Technology, Atlanta, Georgia

Prof. Dr. Konstantinos Kokkotas Theoretical Astrophysics (TAT), Eberhard Karls University of Tuebingen, Tuebingen, Germany

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Section Editors

Prof. Dr. Gerhard Schäfer Institute of Theoretical Physics (retired) Friedrich Schiller University Jena, Jena, Germany

Prof. Valeria Ferrari Professor of Theoretical Physics (retired) Physics Department, Sapienza University of Rome, Rome, Italy

Prof. Leonardo Gualtieri Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Roma, Italy

Prof. Stavros Katsanevas Université de Paris and director of the European Gravitational, Observatory hosting the Gravitational Wave (GW) antenna Virgo in Pisa, Paris, France

Section Editors

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Prof. Pablo Laguna Department of Physics, The University of Texas at Austin, Austin, TX, USA

Prof. Deirdre Shoemaker Center for Gravitational Physics, Department of Physics, The University of Texas, Austin, TX, USA

Prof. Patrick Sutton School of Physics and Astronomy, Cardiff University, Cardiff, Wales, UK

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Section Editors

Prof. Masaki Ando Department of Physics, University of Tokyo, Tokyo, Japan

Prof. Edward Porter Laboratoire Astroparticule et Cosmologie, Université de Paris, Campus des Grands Moulins Bâtiment Condorcet 10, rue Alice Domon et Léonie Duquet, Paris, France

Prof. David Reitze LIGO Laboratory, California Institute of Technology, Pasadena, CA, USA

Contributors

Ernazar Abdikamalov Department of Physics, School of Sciences and Humanities, Nazarbayev University, Nur-Sultan, Kazakhstan Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan, Kazakhstan Pau Amaro Seoane Institute of Multidisciplinary Mathematics, Universitat Politècnica de València, València, Spain Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing, China Kavli Institute for Astronomy and Astrophysics, Beijing, China Yoichi Aso Gravitational Wave Science Project, National Astronomical Observatory of Japan, Mitaka, Japan Luca Baiotti International College and Graduate School of Science, Osaka University, Toyonaka, Japan Enrico Barausse SISSA – Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy IFPU – Institute for Fundamental Physics of the Universe, Trieste, Italy INFN-Sezione di Trieste, Trieste, Italy Imre Bartos University of Florida, Gainesville, USA Matthew Benacquista University of Texas Rio Grande Valley, Brownsville, TX, USA Andrea Bertoldi Laboratoire Photonique, Numérique et Nanosciences (LP2N), Université Bordeaux – IOGS – CNRS:UMR 5298, Talence, France Nigel T. Bishop Department of Mathematics, Rhodes University, Grahamstown, South Africa Philippe Bouyer Laboratoire Photonique, Numérique et Nanosciences (LP2N), Université Bordeaux – IOGS – CNRS:UMR 5298, Talence, France

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Contributors

Patrick Brady Department of Physics, Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA Marica Branchesi INFN, Laboratori Nazionali del Gran Sasso, Gran Sasso Science Institute, L’Aquila, Assergi, Italy Richard Brito Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy S. Burke-Spolaor Department of Physics and Astronomy, West Virginia University, Morgantown, WV, USA Center for Gravitational Waves and Cosmology, West Virginia University, Morgantown, WV, USA Canadian Institute for Advanced Research, CIFAR Azrieli Global Scholar, Toronto, ON, Canada Gianluca Calcagni Instituto de Estructura de la Materia – CSIC, Madrid, Spain Benjamin Canuel Laboratoire Photonique, Numérique et Nanosciences (LP2N), Université Bordeaux – IOGS – CNRS:UMR 5298, Talence, France Zhoujian Cao Department of Astronomy, Beijing Normal University, Beijing, China Chiara Caprini Université de Paris, CNRS, Astroparticule et Cosmologie, Paris, France Zack Carson Department of Physics, University of Virginia, Charlottesville, VA, USA G. Cescutti INAF, Trieste Astronomical Observatory, Trieste, Italy Xian Chen Astronomy Department, Peking University, Beijing, P. R. China Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing, P. R. China Livia Conti Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, Padova, Italy Elena Cuoco European Gravitational Observatory (EGO), Pisa, Italy Scuola Normale Superiore and INFN, Pisa, Italy Mariafelicia De Laurentis Dipartimento di Fisica, Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Napoli, Italy Ivan De Martino Universidad de Salamanca, Departamento de Fisica Fundamental, P. de la Merced, Salamanca, Spain

Contributors

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Katherine L. Dooley Cardiff University, Cardiff, UK Anamaria Effler LIGO Livingston Observatory, Livingston, LA, USA Will M. Farr Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA Daniel G. Figueroa Instituto de Física Corpuscular (IFIC), CSIC-Universitat de Valencia, Valencia, Spain Irene Fiori European Gravitational Observatory, Cascina, Pisa, Italy Francois Foucart Department of Physics & Astronomy, University of New Hampshire, Durham, NH, USA Jonathan Gair Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam-Golm, Germany Juan García-Bellido Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, Spain Davide Gerosa Dipartimento di Fisica “G. Occhialini”, Universitá degli Studi di Milano-Bicocca, Piazza della Scienza, Milano, Italy INFN, Sezione di Milano-Bicocca, Piazza della Scienza, Milano, Italy School of Physics and Astronomy, Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham, UK Xuefei Gong Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences Beijing, China Hartmut Grote Cardiff University, Cardiff, UK Shanquan Gui School of Physical Science and Technology, Lanzhou University, Lanzhou, China Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China Brynmor Haskell Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland Martin Hewitson Institut für Gravitationsphysik der Leibniz Universität Hannover, Hannover, Germany Shuanglin Huang Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Science, Beijing, China University of Chinese Academy of Sciences, Beijing, China

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Contributors

E. A. Huerta Data Science and Learning Division, Argonne National Laboratory, Lemont, IL, USA Department of Computer Science, University of Chicago, Chicago, IL, USA Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, USA Alberto Iess Università di Roma Tor Vergata and INFN Sezione Roma, Roma, Italy Soichiro Isoyama School of Mathematical Sciences, University of Southampton, Southampton, UK Kiwamu Izumi Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa, Japan Karan Jani Department Physics and Astronomy, Vanderbilt University, Nashville, TN, USA Bence Kocsis Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK Andrzej Królak Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland Andrea Lapi SISSA – Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy IFPU – Institute for Fundamental Physics of the Universe, Trieste, Italy INFN-Sezione di Trieste, Trieste, Italy INAF-Osservatorio Astronomico di Trieste, Trieste, Italy Yun-Kau Lau Morningside Center of Mathematics and Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Science, Beijing, China Chun-Yu Lin National Center for High-Performance Computing, Hsinchu, Taiwan Giovanni Losurdo Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy Nicholas Loutrel Department of Physics, Princeton University, Princeton, NJ, USA Harald Lück Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck Institut für Gravitationspyhsik, Max-Planck Gesellschaft, Germany Georgios Lukes-Gerakopoulos Astronomical Institute of the Czech Academy of Sciences, Prague, Czech Republic Elisa Maggio Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy

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Michela Mapelli Dipartimento di Fisica e Astronomia Galileo Galilei, Padova, Italy INFN, Sezione di Padova, Padova, Italy ˆ ˆ Simone Mastrogiovanni Artemis, Université Cote d’Azur, Observatoire de la Cote d’Azur, CNRS, F-06304 Nice, France Université de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France Filip Morawski Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland Guido Mueller Department of Physics, University of Florida, Gainesville, FL, USA Hiroyuki Nakano Faculty of Law, Ryukoku University, Kyoto, Japan Philippe Nguyen University of Oregon, Eugene, OR, USA S. Osłowski Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC, Australia Giulia Pagliaroli Gran Sasso Science Institute, L’Aquila, Italy Paolo Pani Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy Federico Paoletti INFN, Sezione di Pisa, Pisa, Italy Barbara Patricelli European Gravitational Observatory, Cascina, Italy Albino Perego Department of Physics, University of Trento, Trento, Italy INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy Antoine Petiteau AstroParticule et Cosmologie (APC), Université de Paris/CNRS, Paris, France Silvia Piranomonte INAF – Osservatorio Astronomico di Roma, Monte Porzio Catone, Italy Adam Pound School of Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton, UK Michele Punturo Istituto Nazionale di Fisica Nucleare Perugia, Perugia, Italy David Radice Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA, USA Department of Physics, The Pennsylvania State University, University Park, PA, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA, USA

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Contributors

Guilherme Raposo Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy Massimiliano Razzano University of Pisa, Pisa, Italy INFN Sezione di Pisa, Pisa, Italy Om Sharan Salafia INAF – Osservatorio Astronomico di Brera, Merate, Italy Benjamin R. B. Saliwanchik Yale University, New Haven, CT, USA Robert M. S. Schofield University of Oregon, Eugene, OR, USA Kai Schwenzer Science Faculty, Department of Astronomy and Space Sciences, Istanbul University Istanbul, Turkey Hisaaki Shinkai Osaka Institute of Technology, Hirakata City, Osaka, Japan Bram J. J. Slagmolen Centre for Gravitational Astrophysics, Research Schools of Physics, and of Astronomy and Astrophysics, The Australian National University, Acton, Australia Joshua Smith Nicholas and Lee Begovich Center for Gravitational-Wave Physics and Astronomy, California State University, Fullerton, CA, USA Fiodor Sorrentino INFN, Genova, Italy Antonio Stamerra INAF, Osservatorio Astronomico di Roma, Monte Porzio Catone (Roma), Italy SNS, Scuola Normale Superiore di Pisa, Pisa, Italy SSDC, Space Science Data Center, Roma, Italy Danièle A. Steer Université de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France Riccardo Sturani International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal, Brazil Gianmassimo Tasinato Physics Department, Swansea University, Swansea, UK Stephen R. Taylor Department of Physics and Astronomy, Vanderbilt University, Nashville, TN, USA F.-K. Thielemann Department of Physics, University of Basel, Basel, Switzerland GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany Maria C. Tringali European Gravitational Observatory, Pisa, Italy Jo van den Brand Nikhef and Maastricht University, Amsterdam, The Netherlands J. P. W. Verbiest Fakultät für Physik, Universität Bielefeld, Bielefeld, Germany Max-Planck-Institut für Radioastronomie, Bonn, Germany

Contributors

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Salvatore Vitale Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA Robert L. Ward Centre for Gravitational Astrophysics, Research Schools of Physics, and of Astronomy and Astrophysics, The Australian National University, Acton, Australia Barry Wardell School of Mathematics and Statistics, University College Dublin, Dublin, Ireland Vojtˇech Witzany School of Mathematics and Statistics, University College Dublin, Dublin, Ireland Shengnian Xu Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, China Kent Yagi Department of Physics, University of Virginia, Charlottesville, VA, USA Hwei-Jang Yo Department of Physics, National Cheng-Kung University, Tainan, Taiwan Jean-Pierre Zendri INFN, Padova, Italy Tianyu Zhao Department of Astronomy, Beijing Normal University, Beijing, China Zhizhen Zhao Department of Electrical and Computer Engineering, Coordinated Science Laboratory, Department of Mathematics, Department of Statistics, National Center for Supercomputing Applications, Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Part I Introduction

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Introduction to Gravitational Wave Astronomy Nigel T. Bishop

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Content of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearized Einstein Equations in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The TT Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Effect of a Gravitational Wave on a System of Test Particles . . . . . . . . . . . . . . . . . . . . . Generation of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearized Einstein Equations with Matter: The Quadrupole Formula . . . . . . . . . . . . . . . . . Energy of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond the Quadrupole Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gravitational Wave Field of an Orbiting Binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inspiral Rate and Time to Merger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eccentricity Reduction by Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Detection Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Interpretation and Determination of Source Parameters . . . . . . . . . . . . . . . . . . . . . . . . GW Observations: Contributions to Physics and Astrophysics . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 6 7 7 9 10 11 12 14 14 17 20 22 22 25 26 26 28 30 31 32 32

N. T. Bishop () Department of Mathematics, Rhodes University, Grahamstown, South Africa e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_1

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Abstract

This chapter provides an overview of gravitational wave (GW) astronomy, providing background material that underpins the other, more specialized chapters in this handbook. It starts with a brief historical review of the development of GW astronomy, from Einstein’s prediction of GWs in 1916 to the first direct detection in 2015. It presents the theory of linearized perturbations about Minkowski spacetime of Einstein’s equations and shows how gauge transformations reduce the problem to the standard wave equation with two degrees of freedom or polarizations, h+ , h× . We derive the quadrupole formula, which relates the motion of matter in a source region to the far GW field. It is shown that GWs carry energy, as well as linear and angular momentum, away from a source. The GW field of an orbiting circular binary is found; and properties of the evolution of the binary including rate of inspiral and time to coalescence are calculated. A brief review is given of existing and proposed GW detectors and of how to estimate source parameters in LIGO or Virgo data of a GW event. The contributions that GW observations have already made to physics, astrophysics, and cosmology are discussed. Keywords

Gravitational waves; Quadrupole formula; Orbital inspiral; LIGO; LISA

Introduction Historical Development Soon after proposing the general theory of relativity (GR), Einstein showed, by linearizing the field equations, that the theory implies the existence of gravitational waves (GWs) [1, 2] and obtained what has become known as the quadrupole formula. However, the concept of a GW actually predates GR. Newtonian gravitation theory can be expressed as an elliptic equation with the gravitational field changing instantaneously if the source changes. Since special relativity had established the speed of light as a universal speed limit, this suggests that the gravitational theory should be expressed as a hyperbolic system, so implying the existence of GWs. During the early years, i.e., from about 1920 to 1960, there was uncertainty about the nature of GWs. It was not clear whether GWs carry energy since GW energy at a given event in spacetime cannot be defined; it was not clear whether the quadrupole formula could be applied to gravitationally bound systems; and it was argued that the full nonlinear theory of GR does not permit GW solutions. For a detailed discussion of these issues, see [3]. The mathematical theory of GWs became well-established in the 1960s. The above issues were resolved by expressing the Einstein equations of the full nonlinear theory of GR in a suitable coordinate system [4, 5] and also by the “shortwave

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approximation” averaging procedure [6]. Once it was clear that the emission of GWs causes a loss of energy in the emitting system, the dynamics of an orbiting binary and the resulting inspiral could be calculated using linearized theory and the quadrupole approximation [7, 8]. In this period, there were also important developments in the experimental area. The theory of stellar evolution had long indicated that the end-state of a massive star would be a neutron star or a black hole. However, the first observation of a neutron star was of a pulsar in the Crab nebula in 1969, and the first evidence of a massive (≈15M ) compact object that could only be a black hole was Cygnus X-1 in 1971. The confirmation of the existence of these compact objects was important: their inspiral and merger would be both powerful and in the frequency range ≈10 to ≈ 1000 Hz which would be suitable for a terrestrial detector. The first attempt to detect gravitational waves was reported in 1969 using a bar detector [9]. This paper actually reported the detection of a number of GW events, although subsequently it became clear that these events were not astrophysical but rather the result of experimental error. In 1979, observations of the binary pulsar system PSR 1913+16 showed that the orbit was inspiraling at a rate consistent with GW emission in GR, thus providing the first experimental evidence for the existence of GWs [10]. The most powerful GW sources, and therefore those most likely to be detected, are the merger of two compact objects, but this case does not satisfy the conditions of the quadrupole approximation and there is a need to go beyond linearized theory. This can be achieved by means of a series expansion, adding terms of quadratic, cubic, and higher orders to the linearized expressions. This approach was considered in the early years of GR [11, 12] and in the 1980s formalized as the post-Newtonian method. For the actual merger, a numerical simulation of the full Einstein equations is needed, normally with the spacetime foliated into a sequence of spacelike hypersurfaces [13]. In 1977, the GWs from a head-on collision of two Schwarzschild black holes were computed [14], but it was only in 2005 that codes were able to make a stable evolution of the physically realistic problem of the inspiral and merger of two black holes [15]. Since then, many other groups have successfully evolved black hole spacetimes, as well as neutron star mergers and supernovae, often using a combination of post-Newtonian and numerical methods. The possibility of using laser interferometry to detect GWs was suggested in the 1960s, and simple prototypes were constructed at that time. In 1980, the US National Science Foundation provided funding for the construction of certain prototypes, as well as for a study of the technical issues and the costs of building an interferometer with arms several km in length. This eventually led to the construction of LIGO (Laser Interferometer Gravitational-Wave Observatory) facilities at Hanford and Livingstone, USA, with 4 km arms; scientific studies commenced in 2002. Also at this time, the much smaller detector GEO600 (600 m arms) started operation, and in 2007 Virgo in Italy, which has 3 km arms and is a project of the European Gravitational Observatory consortium, made its first science run. These instruments underwent a number of upgrades to improve the sensitivity, and the first direct detection of GWs was made on September 14, 2015, by LIGO Hanford and Livingstone [16].

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Content of This Chapter This chapter provides an overview of GW astronomy and describes aspects of the basic theory of GWs that form the foundation of the field. It provides the background material that underpins the specialized chapters in this handbook. The discussion is mainly within the context of the Einstein equations linearized about Minkowski spacetime. This theory is sufficiently straightforward that it can be presented within a single chapter yet is also widely applicable. The magnitude of GWs is dimensionless, and those propagating past the Earth that have been detected can be characterized as being O(10−22 ); thus linearized theory certainly applies. The later phase of compact object inspiral is driven by GW emission; this process lasts millions or billions of years and, apart from the final seconds or minutes, is accurately described by linearized theory. Even though linearized theory does not provide an accurate waveform for the actual merger of two compact objects, it remains useful as it provides a simple guide in analytic form. The theory of linearizing Einstein’s equations about Minkowski spacetime is developed in section “Propagation of Gravitational Waves”. A key issue is the use of gauge transformations (which, in this context, are coordinate transformations that are almost the identity transformation) to simplify the equations to a form that is manifestly the wave equation. The simplest solution to the wave equation is a plane wave, i.e., in Cartesian coordinates (t, x, y, z) a function f (t, z) that satisfies the wave equation; this solution well describes GWs in the solar system produced by a source many Mpc away. We construct the plane wave solution and make further gauge transformations to the transverse traceless, or TT, gauge. In this way, the ten components of a symmetric tensor are reduced to two independent components or polarizations, usually denoted as h+ , h× . Finally, this section investigates the effect of a plane GW on a system of test particles. In the TT gauge, the coordinates of the particles do not change, but that does not mean that there is no movement: a coordinate independent quantity such as the proper distance between two of the particles does vary with time. It is also shown that the Riemann tensor has nonzero components, so the spacetime is not flat. The generation of GWs by the motion of matter is discussed in section “Generation of Gravitational Waves”, and the quadrupole formula is derived. As an example, the formula is applied to the astrophysically important problem of a binary comprising two masses in circular orbit around each other. The two polarization modes, h+ , h× , are evaluated with respect to spherical polar coordinates. We next discuss the energy, as well as the linear and angular momenta, carried away by GWs. These effects are beyond the scope of linearized theory, being quadratic in the perturbations, and the calculations are only outlined with much detail omitted. This section also summarizes methods that are used for situations where the quadrupole formula is inadequate: post-Newtonian approximations, numerical relativity, the theory of quasinormal modes of a black hole, and extreme mass ratio inspirals (EMRIs). Now that we know what GWs carry away from a system, conservation laws are used in section “The Gravitational Wave Field of an Orbiting Binary” to find the

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evolution of an orbiting circular binary. The orbital diameter and wave period slowly decrease, and we find an expression for the time to coalescence. We also find a relation between the chirp mass, which is a function of the individual masses of the binary, and an expression involving the wave frequency and its rate of change; this means that the chirp mass is entirely determined by observational GW data. It is also shown that GW emission causes an eccentric orbit to circularize, so the focus on circular orbits is for reasons of physics rather than for mathematical convenience. Section “Detection of Gravitational Waves” provides an outline of various existing and planned GW detection facilities, including terrestrial laser interferometers, satellite systems, and pulsar timing. We indicate the frequency range in which a detector is sensitive and some actual or expected astrophysical sources. Then, for the LIGO and Virgo network, we outline the process of determining basic source parameters of an observed GW event representing a compact object inspiral and merger. Some contributions that GW observations have already made to physics, cosmology, and astrophysics are outlined. This chapter ends with a conclusion, section “Conclusion”.

Propagation of Gravitational Waves Linearized Einstein Equations in Vacuum The essential idea is to consider spacetimes that comprise small perturbations about Minkowski spacetime. More precisely, the metric tensor gαβ is written gαβ = ηαβ + hαβ ,

(1)

where ηαβ = is the metric of Minkowski spacetime which in Cartesian (t, x, y, z) coordinates is diag(−1, 1, 1, 1) and  hαβ is a small perturbation. In the linearized 2 are neglected. The contravariant metric approximation, terms of order O hαβ is written as g αβ = ηαβ − hαβ , and the identities δγα = g αβ gβγ = ηαβ ηβγ imply that hαβ = ηαγ ηβδ hγ δ .

(2)

Thus, the indices of quantities of order O(hαβ ) are raised and lowered using the background metric ηαβ rather than the full metric gαβ . The first step toward constructing the Einstein equations is to determine the metric connection μ

Γ αβ =

1 μν 1 μ η (∂β hνα + ∂α hνβ − ∂ν hαβ ) = (∂β hμα + ∂α h β − ∂ μ hαβ ) . 2 2

(3)

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Then the Ricci tensor is Rμν = ∂α Γ αμν − ∂ν Γ αμα =

1 (∂α ∂ν hμα + ∂α ∂μ hν α − ∂α ∂ α hμν − ∂μ ∂ν h) , 2

(4)

where h = hαα = ηαβ hαβ . The Ricci scalar is thus R = Rμμ = ∂α ∂β hαβ − ∂α ∂ α h ,

(5)

and therefore Einstein’s equations Gμν = Rμν − Rgμν /2 = 8π Tμν are  1 α ∂ ∂ν hμα + ∂ α ∂μ hνα − ∂α ∂ α hμν − ∂μ ∂ν h − ημν (∂ α ∂ β hαβ − ∂ α ∂α h) 2 = 8π Tμν .

(6)

Equation (6) may be somewhat simplified by changing the metric perturbation hαβ to its trace reversed form, that is 1 1 h¯ αβ = hαβ − ηαβ h so that h¯ = −h and hαβ = h¯ αβ − ηαβ h¯ . 2 2

(7)

Einstein’s equations are then Gμν = −

 1 ∂α ∂ α h¯ μν + ημν ∂ α ∂ β h¯ αβ − ∂ν ∂ α h¯ μα − ∂μ ∂ α h¯ να = 8π Tμν . 2

(8)

The first term ∂α ∂ α h¯ μν is the wave operator applied to h¯ μν , and the problem would reduce to a standard wave equation if the other terms could be made to disappear. This can be achieved on applying the Lorentz gauge condition ∂ α h¯ αβ = 0 ,

(9)

1 − h¯ μν = 8π Tμν , 2

(10)

in Eq. (8) to obtain

where the operator  = ∂α ∂ α , and clearly in vacuum h¯ μν = 0. Imposition of the Lorentz gauge condition is a constraint on the coordinates being used and is not a restriction on the geometry of the spacetime. This matter is discussed further in the next section.

1 Introduction to Gravitational Wave Astronomy

9

Gauge Transformations A gauge transformation is a coordinate transformation of the form x (NEW)α = x (OLD)α + ξ α (x (OLD)β )

(11)

where ∂β ξ α is of the same order of smallness as hαβ ; terms of order O(hαβ )2 , O(hαβ ∂β ξ α ), O(∂β ξ α )2 are neglected. Applying the coordinate transformation (11) to the metric (1) gives (OLD) h(NEW) αβ = hαβ − ∂β ξα − ∂α ξβ ,

(12)

and then the change to trace-reversed form leads to μ ¯ (OLD) h¯ (NEW) αβ = hαβ − ∂β ξα − ∂α ξβ + ηαβ ∂μ ξ .

(13)

Requiring h¯ (NEW) αβ to satisfy the Lorentz gauge condition then implies ξβ = ∂ α h¯ (OLD) αβ .

(14)

The right hand side of Eq. (14) is regarded as given, and then Eq. (14) comprises separate wave equations for each of the four unknowns ξ0 , · · · , ξ3 . The existence of solutions to these four wave equations follows from the theory of partial differential equations. Note however that the solutions are not unique, and we are free to write ξ α = ξ (0)α + ζ α ,

(15)

where ξ (0)α is a solution to Eq. (14) and where ζ α satisfies ζ α = 0. The possibility to make gauge transformations means that a metric that appears to be wavelike may not represent a GW. For example, suppose that the spacetime is Minkowski (and so does not contain gravitational waves) with h¯ (OLD) αβ = 0 and that ξα = (0, 0, 0, cos(t − x)). Then ⎛

h¯ αβ

0 0 ⎜ 0 0 =⎜ ⎝ 0 0

sin(t − x) − sin(t − x)

⎞ 0 sin(t − x) 0 − sin(t − x) ⎟ ⎟. ⎠ 0 0 0 0

(16)

A simple test to determine whether or not a vacuum spacetime is Minkowski in unusual coordinates is to evaluate the Riemann tensor: R αβγ δ = 0 if and only if the spacetime is Minkowskian.

10

N. T. Bishop

Plane Wave Solutions Let us now make the ansatz   h¯ αβ =  Aαβ exp(ikα x α ) ,

(17)

where the kα are real constants and the Aαβ are complex constants; note that a general wave may be expressed as a Fourier sum of terms with the above form. Substituting into the vacuum field equations h¯ αβ = 0 leads to ηαβ kα kβ = 0, so that kα is a null vector, and usually k0 is written as ω. Since Aαβ is symmetric, it has ten independent components. Now, the Lorentz gauge condition ∂ β h¯ αβ = 0 implies four conditions Aαβ kβ = 0 ,

(18)

reducing the number of independent components of Aαβ from 10 to 6. Then a gauge transformation can be made where ξ α = (B α exp(ikμ x μ ))

(19)

with B α constant; since ξ α satisfies the identity ξ α = 0, the condition ∂β h¯ αβ = 0 is not affected. The four constants B α are used to reduce the independent components of Aαβ from 6 to 2. In order to apply the above and construct in a transparent way an explicit form of Aαβ with only two independent components, we align the coordinates with the direction of propagation of the wave. Let the wave be propagating in the z−direction with frequency ω/(2π ) so that kα = (ω, 0, 0, −ω). Then the conditions (18) imply that Aα0 = Aα3 , so we can write ⎛

Aαβ

A00 ⎜ A01 =⎜ ⎝ A02 A00

A01 A11 A12 A01

A02 A12 A22 A02

⎞ A00 A01 ⎟ ⎟, A02 ⎠ A00

(20)

which has six independent components A00 , A01 , A02 , A11 , A12 , A22 . Differentiating the gauge transformation (19) gives ∂ α ζ β = (iB α k β exp(ikμ x μ )), so that from Eq. (13),   h¯ αβ =  (Aαβ − iB α k β − iB β k α + ηαβ iB μ kμ ) exp(ikμ x μ ) .

(21)

Now, Aαβ , B α , (k α − k α ), (x α − x α ) are small quantities, and so the  product of any pair is negligible. Thus we can write h¯ αβ =  (Aαβ exp(ikμ x μ ) , and then Aαβ = Aαβ − iB α k β − iB β k α + ηαβ iB μ kμ .

(22)

1 Introduction to Gravitational Wave Astronomy

11

A00 = A00 + iω(B 0 + B 3 ) ,

A01 = A01 + iωB 1 ,

A02 = A02 + iωB 2

A11 = A11 + iω(B 3 − B 0 ) ,

A22 = A22 + iω(B 3 − B 0 ) ,

A12 = A12 . (23)

The constants B α may be chosen so as to simplify Aαβ in any desired way, and common practice is to use them to set A00 = A01 = A03 = 0 , A22 = −A11 ,

(24)

so that Aαβ is ⎛

Aαβ

0 ⎜0 =⎜ ⎝0 0

0 0 11 A12 A 12 A −A11 0 0

⎞ 0 0⎟ ⎟. 0⎠ 0

(25)

The TT Gauge The resulting metric h¯ αβ is transverse: the wave travels in the z-direction, and its effects are in the x- and y-directions. More generally, the transverse property is expressed as h¯ αβ k α = 0 .

(26)

¯ h¯ α α = h = 0,

(27)

Further the metric is traceless, i.e.,

so that the metric and its trace-reverse form are the same, h¯ αβ = hαβ . A metric satisfying Eqs. (26) and (27) is said to be in the transverse traceless or TT gauge. From now on, we will drop the  to denote a quantity in this gauge, and in circumstances where there is a need to differentiate the gauge of quantities, we will use the superfix T T to denote a quantity in the TT gauge. The GW derived above has two independent components h11 , h12 . However, there is no essential difference between the components, as can be seen by making a coordinate transformation which, geometrically, corresponds to a rotation of the (x, y) axes about the z-axis through an angle θ = π/4. More precisely, (t, x, y, z) → ((t  , x  , y  , z ) where x  = x cos θ + y sin θ , and here cos θ = sin θ = to

y  = x sin θ − y cos θ ,

(28)

√ 2/2. Application to the metric g αβ = ηαβ − hαβ leads

12

N. T. Bishop

h11 = −h22 = h12 ,

h12 = h21 = −h11 .

(29)

The two independent components of a GW, h11 , h12 , are usually denoted by h+ , h× , respectively, and are called the two polarization states of the GW. There is an analogy to electromagnetic waves which are also transverse to the direction of propagation and which also have polarization states, although in this case the angle between the two states is π/2 rather than π/4. As shown above, the distinction between h+ , h× is observer dependent, but in some circumstances, the polarization can be described in an observer-independent way. We write h+ = (A+ exp (iω(t − z))) , h× = (A× exp (iω(t − z))) .

(30)

Now, A+ , A× are complex constants (i.e., independent of the coordinates x α ), and consider the ratio A× /A+ = RA which must also be constant. There are two interesting special cases: (1) suppose that the ratio RA has zero imaginary part, then h× / h+ is constant, and so we may change coordinates by rotating the (x, y) axes about the z-axis through some angle θ to achieve h× = 0; in this case the GW is said to be linearly polarized. (2) Suppose that RA = ±i, then from the identity cos2 s + sin2 s = 1, it follows that the magnitude of the GW h2+ + h2× is constant, and the wave is said to be circularly polarized. The physical properties leading to these names are discussed in the next subsection.

The Effect of a Gravitational Wave on a System of Test Particles We now consider the physical effects of a GW in the TT gauge as described in α = 0. The the previous subsection. Since hα0 = 0, it follows from Eq. (3) that Γ00 geodesic equation is μ ν d 2xα α dx dx , = −Γμν 2 dτ dτ dτ

(31)

where τ is the proper time; if a particle is initially at rest, dx μ /dτ = (1, 0, 0, 0) so that d 2 x α /dτ 2 = 0 and the particle remains at rest. But this is just a consequence of the coordinates being used, and does not mean that the GW has no physical effect. In order to analyze this further, we use the equation of geodesic deviation. Consider two nearby particles, A and B, each with four-velocity U α , and let cα be a vector connecting A and B and orthogonal to U α ; then d 2 cα α + Γβγ U β U γ = −R αβγ δ U β U δ cγ . dτ 2

(32)

Now suppose that we use proper coordinates with A located at the origin; then U α = (1, 0, 0, 0), cα = (0, ci ) = (0, xBi ) (with i = 1, 2, 3 denoting a spatial index), the metric is

1 Introduction to Gravitational Wave Astronomy

  ds 2 = ηαβ + O(|x i |2 ) dx α dx β ,

13

(33)

α = 0, τ = t, and Eq. (32) simplifies to Γβγ

d 2 xBi j = −R i0j 0 xB . dt 2

(34)

It is important to note that R αβγ δ is gauge-invariant; that is if we make a gauge transformation given by Eq. (11), then in the new gauge R (NEW)αβγ δ = R (OLD)αβγ δ . This result follows since R αβγ δ = O(hαβ ), and the coordinate transformation matrix is ∂x (NEW)α = δβα + O(hαβ ) , ∂x (OLD)β

(35)

so to O(hαβ ) only the δβα is used in the coordinate transformation. Thus, we may use the TT gauge to evaluate the Riemann tensor in Eq. (34), and we find 1 1 R 1010 = −R 2020 = − ∂t2 h+ , R 1020 = R 2010 = − ∂t2 h× , 2 2

(36)

with the remaining components of the form R i0j 0 being zero. Then Eq. (34) simplifies to d 2 xBi = dt 2

 1 2 1 2 1 2 1 2 1 2 1 2 ∂ h+ xB + ∂t h× xB , ∂t h× xB − ∂t h+ xB , 0 . 2 t 2 2 2

(37)

i + i (t) with i of order O(h) and replace x i by x i We now write xBi (t) = xB0 B B0 on the right hand side of Eq. (37) (as terms of order O(h2 ) are ignorable) so that Eq. (37) can be integrated to give

= i

 1 1 1 1 1 2 1 2 h+ xB0 + h× xB0 , h× xB0 − h+ xB0 , 0 . 2 2 2 2

(38)

In order to facilitate the interpretation of Eq. (38), let us suppose that the (x 1 , x 2 ) 2 = 0, use Eq. (30), and suppose that the origin axes have been rotated so that xB0 of the spacetime coordinates is such that A+ is real. Then in the case of a linearly polarized wave

i =

1 1 x A+ cos ωt (1, RA , 0) , 2 B0

(39)

so that B moves in a straight line; and in the case of a circularly polarized wave

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N. T. Bishop

Fig. 1 The effect on a circle of particles transverse to a GW. Upper row, circular polarization; lower row, linear polarization. In the horizontal axis, T is the period of the GW. The particles are in their unperturbed state at T /4 and 3T /4 in the lower row. In the circular polarization case, it may appear that the whole ellipse is rotating, but actually particles are moved a little from the unperturbed state to give this impression, in the same way that a wave moving across the ocean is caused by water moving up and down, not horizontally

i =

1 1 x A+ (cos ωt, sin ωt, 0) , 2 B0

(40)

so that B moves in a circle. The effect of a GW, for both linear and circular polarization states, is illustrated in Fig. 1. The preceding analysis brings out two important general points. Firstly, since R αβγ δ = 0, the spacetime is not Minkowskian. Secondly, suppose that the particles A and B are massive and are connected by a damping system; then the relative motion and acceleration will mean that work is done on the damping system and therefore its temperature must increase. These considerations show that GWs are real physical effects that contain energy that must have originated in some source and which can, in principle, be extracted from the GWs.

Generation of Gravitational Waves Linearized Einstein Equations with Matter: The Quadrupole Formula From standard wave equation theory, the solution to Eq. (10) representing an outgoing wave is h¯ μν (t, x i ) = 4



T μν (t − |x i − x i |, x j ) 3  d x , |x i − x i |

(41)

where (t, x i ) is the event at which the metric perturbation h¯ μν is evaluated, x i represents a spatial point inside the source, and |x i − x i | is the Euclidean distance between x i and x i . Suppose now that the source is matter localized near the origin in a region r = |x i | < r0 , and we are evaluating the metric perturbations where

1 Introduction to Gravitational Wave Astronomy

15

r = O(rE ) with r0 rE ; then Eq. (41) simplifies to 4 h¯ μν (t, x i ) = r



T μν (t − r, x j )d 3 x  .

(42)

The discussion in section “Plane Wave Solutions” showed that the ten components of h¯ μν have only six independent components, so it is sufficient to determine the spatial components h¯ ij . In the linearized approximation, the conservation condition ∇α T 0α = 0 simplifies to ∂t T 00 + ∂i T 0i = 0, and applying this relation twice gives ∂i ∂j T ij = ∂t2 T 00 .

(43)

Using the above together with the rules of basic calculus gives ∂t2 (T 00 x i x j ) =(∂k ∂ T k )x i x j =∂k ∂ (T k x i x j ) − 2∂k (T ki x j + T kj x i ) + 2T ij , from which it follows that  T ij d 3 x =

1 2 ∂ 2 t

(44)

 T 00 x i x j d 3 x ,

(45)

  since the remaining terms ∂k ∂ (T k x i x j ) − 2(T ki x j + T kj x i ) can be transformed by the divergence theorem into a surface integral over the boundary of the volume of integration where T αβ = 0. Aside from the gravitational field being weak with hμν 1, we now assume that throughout the source region, the relative speed of the matter flow is much less than that of light. Then T 00 can be replaced by the matter density ρ so that Eq. (41) becomes 2 d 2 I ij (t − r) h¯ ij (t, x i ) = , r dt 2

(46)

where the three-dimensional tensor  I ij (t) =

ρ(t, x k )x i x j d 3 x

(47)

is the second moment, also called the quadrupole moment, of the mass distribution. In the case that the matter distribution is treated as a system of N particles, I ij (t) =

N  a=1

j

Ma xai xa .

(48)

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N. T. Bishop

Example: Equal Mass Circular Binary Consider two particles, A and B, each of mass M in circular orbit around each other with orbital diameter r0 , and suppose that the angular velocity of each particle is ω. Choosing coordinates so that the orbit is in the x − y plane, the position vector of particle A is xAi =

r

0

2

cos ωt,

 r0 sin ωt, 0 , 2

(49)

and for particle B, xBi = −xAi . Then

I ij

⎛ ⎞ cos2 ωt cos ωt sin ωt 0 Mr02 ⎝ cos ωt sin ωt = 0⎠ . sin2 ωt 2 0 0 0

(50)

Then using Eq. (46), it follows that ⎛ ⎞ cos 2ω(t − r) sin 2ω(t − r) 0 2 ω2 2Mr 0 ⎝ sin 2ω(t − r) − cos 2ω(t − r) 0 ⎠ . h¯ ij = − r 0 0 0

(51)

For an observer on the z-axis, Eq. (51) is already in the TT gauge, but for any other observer, a transformation to the TT gauge is needed. We now construct the transformation for an observer on the x-axis, by amending the theory developed in section “Plane Wave Solutions”. The wave propagation vector becomes kμ = (2ω, −2ω, 0, 0); then using Eq. (18) and noting the zero entries in Eq. (51), we have ⎛

Aαβ

A11 ⎜ A11 =⎜ ⎝ A12 0

A11 A11 A12 0

A12 A12 A22 0

⎞ 0 0⎟ ⎟. 0⎠ 0

(52)

Equation (22) is now applied, giving A11 =A11 + i2ω(B 0 + B 1 ) ,

A12 = A12 + i2ωB 2 ,

A22 =A22 + i2ω(B 1 − B 0 ) ,

A23 = 0 ,

A13 = i2ωB 3

A33 = i2ω(B 1 − B 0 ) . (53)

The TT condition A22 +A33 = 0 is satisfied upon setting i2ω(B 1 −B 0 ) = −A22 /2, and then (B 0 + B 1 ), B 2 , B 3 are chosen so that A11 = A12 = A13 = 0. Thus,

1 Introduction to Gravitational Wave Astronomy

⎛ ⎞ 0 0 0 2 ω2 Mr 0 ⎝ 0 cos 2ω(t − r) ⎠. h¯ ij = 0 r 0 0 − cos 2ω(t − r)

17

(54)

Finally, in order to be able to make a consistent comparison of the waveform in different directions, we need to rotate the (x, z) coordinates about the y-axis through an angle π/2 so that the GW is travelling in the z-direction. The result is ⎛ ⎞ − cos 2ω(t − r) 0 0 2 ω2 Mr 0 ⎝ h¯ ij = 0 cos 2ω(t − r) 0 ⎠ . r 0 0 0

(55)

In the general case, the observer’s position is expressed using spherical polars (r, θ, φ); without loss of generality, we set φ = 0 since a nonzero φ is equivalent to changing t to t − φ/ω. The wave propagation vector becomes kμ = (ω, −ω sin θ, 0, −ω cos θ ), and the transformation to the TT gauge proceeds in a similar way to that described above, but the details are omitted. In order for the TT form to be explicit, we must also make a coordinate transformation that represents a rotation through an angle θ of the x, z coordinates about the y−axis. The result for a general circular binary with masses M1 , M2 is h+ = −

2ω2 μr02 (1 + cos2 θ ) cos(2ω(t − r) − 2φ) r

h× = −

4ω2 μr02 cos θ sin(2ω(t − r) − 2φ) , r

(56)

where μ = M1 M2 /(M1 + M2 ) is M/2 for the equal mass case, and μ is called the reduced mass of the binary. The magnitudes of both GW components h+ , h× are maximum at θ = 0, π (i.e., for an observer seeing the binary “face-on”) and minimum for an observer at θ = π/2 (i.e., for an observer in the plane of the binary). Note further that at θ = 0, π the GW is circularly polarized, whereas at θ = π/2, it is linearly polarized.

Energy of Gravitational Waves The energy associated with gravitational waves is a second-order effect, and so its justification goes beyond the linearized approximation considered so far. Here, we outline the calculation, but with many details omitted [6]. Equation (1) is amended to [B] [2] + h[1] gαβ = gαβ αβ + hαβ ,

(57)

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N. T. Bishop

[2] where: h[1] αβ is hαβ in Eq. (1) and is the first-order perturbation; hαβ is the second-

[B] order perturbation; and gαβ is the background metric and is a little different to the Minkowski metric ηαβ because it includes the response of the background geometry to the first-order perturbations. Evaluation of the Ricci tensor for the metric of Eq. (57) gives a similar expansion [B] [B] [1] [1] [2] [1] [1] [2] Rαβ = Rαβ (g ) + Rαβ (h ) + Rαβ (h ) + Rαβ (h ) ,

(58)

     2 [1] [1] where Rαβ (h ) is of order O h[1] and the last two terms are of order O h[1] . [1] [1] Then solving Eq. (58) to first-order in h[1] gives Rαβ (h ) = 0, from which the linearized theory discussed in the preceding sections of this chapter is obtained. The next step is to split the second-order part of Rαβ into a part that varies on scales much larger than the wavelength of the GWs and a remainder that contains the local fluctuations. Defining < x > to be the average of the quantity x over a suitable region (in a sense made precise in [6]), then [B] [B] [2] [1] Rαβ (g )+ < Rαβ (h ) > = 0 ,

(59)

[1] [2] [2] [1] [2] [1] Rαβ (h ) + Rαβ (h )− < Rαβ (h ) > = 0 .

(60)

We can then define [GW ] Tαβ

1 =− 8π

 1 [B] [2] [1] [2] [1] < Rαβ (h ) > − gαβ < R (h ) > , 2

(61)

since, from Eq. (59), the Einstein tensor for the background metric can be written as

 1 [B] [B] 1 [B] [B] [2] [1] [2] [1] g g = R − R = − < R (h ) > − < R (h ) > . G[B] αβ αβ αβ 2 αβ 2 αβ (62) [GW ] Evaluation of Tαβ leads to a simple form when h[1] satisfies the Lorentz and αβ TT gauge conditions [GW ] Tαβ =

1 < (∂α hμν )(∂β hμν ) > . 32π

(63)

In the case of GWs moving radially outward from a source which is far away, we can write [GW ] [GW ] [GW ] = −T01 = T11 = T00

=

1 < (∂t hμν )(∂t hμν ) > 32π

1 < (∂t h+ )2 + (∂t h× )2 > . 16π

(64)

1 Introduction to Gravitational Wave Astronomy

19

[GW ] The stress-energy tensor Tαβ can be written in the form ρkα kβ with kα being the

[GW ] null vector (−1, 1, 0, 0) and ρ being the right hand side of Eq. (64); thus Tαβ can be interpreted as being sourced by dust comprising massless particles moving radially outward at the speed of light. The total power crossing the two-surface S at r =constant, t = constant is then

L[GW ] =

 S

T [GW ]01 r 2 dΩ =

r2 16π

 S

(∂t h+ )2 + (∂t h× )2 dΩ ,

(65)

with the averaging removed since h+ , h× vary slowly in the angular directions. For example, in the case of a circular binary with h+ , h× given by Eq. (56), L[GW ] =

32 2 4 6 μ r0 ω . 5

(66)

Equation (65) describes a power output, and conservation of energy means that it should be balanced by an energy loss somewhere else. Indeed, there is a precise result to this effect [4, 5], which we briefly outline. The mathematical framework used is coordinates based on outgoing null cones. For example, the metric of Minkowski spacetime is ds 2 = −du2 − 2du dr + r 2 (dθ 2 + sin2 θ dφ 2 )

(67)

in which the coordinate transformation t → u = t − r has been applied to the usual Minkowski metric in spherical polar coordinates (ds 2 = −dt 2 + dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 )). For a general spacetime, the metric is of the form Eq. (67) plus additional terms (which may not be small). However, the key assumption made is that the geometry is asymptotically flat, which means that the additional terms → 0 at least as fast as 1/r as r → ∞. In the general case, the Einstein equations are very complicated, but if we consider only the leading-order term (in an asymptotic expansion in 1/r), then they simplify enormously and it can be shown that dm = −L[GW ] . du

(68)

In general, the physical meaning of m is ambiguous, but in the case that the spacetime is static, the physical meaning is clear: it is the Schwarzschild mass of the system. We can therefore consider a system that is static until time u1 then emits GWs until time u2 from when it is again static:  m(u2 ) − m(u1 ) = −

u2

L[GW ] du .

(69)

u1

If the dimensionless quantity dm/du 1, then the system is regarded as quasistatic, and m in Eq. (68) is treated as the Schwarzschild mass. What does it mean to

20

N. T. Bishop

say that the total mass of a system, for example, an orbiting binary, is decreasing? In general relativity, all forms of energy contribute to the total mass. If the system is approximately Newtonian and satisfies the conditions for the quadrupole formula to be valid, then the total mass is the sum of the rest masses of the orbiting objects (which does not change), together with the Newtonian kinetic plus potential energies. This matter is pursued further in section “Inspiral Rate and Time to Merger”. Aside from energy, GWs may also carry angular momentum and linear momentum. The formula for the linear momentum P i is 

dP i r2 =− dt 16π

S

  ni (∂t h+ )2 + (∂t h× )2 dΩ ,

(70)

where ni is the unit vector normal to S. When the motion of the source is in the x − y plane, then the angular momentum Ji is in the z-direction and is r2 Jz = 16π

 S

∂t h+ ∂φ h+ + ∂t h× ∂φ h× dΩ .

(71)

These formulas are derived, for example, in [17]. Further, even when the source is not weak, sufficiently far from the source h+ , h× are small and so in the limit as r → ∞ Eqs. (65), (70), and (71) still apply.

Beyond the Quadrupole Formula The derivation of the quadrupole formula assumed that velocities are small, but that does not apply for compact objects near merger. In this case, an accurate GW calculation requires going beyond the quadrupole formula. The various options available are summarized here. The post-Newtonian (PN) method gives a formula for the GWs as a series with the first term in the series being the quadrupole result; thus, the method is an extension of the quadrupole formula. The series can be regarded as being in terms of powers of v; however, the formalism used in PN work normally uses SI (with G, c = 1) rather than geometric units, and a term in the series expansion of the form 1/c2n is said to be of PN order n. The quadrupole term is PN order 1, and other terms are PN order 32 or higher. PN methods accurately describe the inspiral of two compact objects; further, results are obtained simply by evaluating analytic formulas, and so the computation of the evolution of the system is very quick. However, as merger approaches, the gravitational field becomes highly nonlinear and velocities may approach that of light, and the PN approximation ceases to be accurate. The PN method is discussed further elsewhere in this book; see also the review article [18]. In numerical relativity, the full nonlinear Einstein equations are solved. The spacetime is foliated into a sequence of 3D hypersurfaces, and data on an initial

1 Introduction to Gravitational Wave Astronomy

21

hypersurface is evolved to the next hypersurface; the process is then repeated until the desired region of the spacetime has been covered. While numerical relativity is accurate even when the fields are highly nonlinear, it can require substantial computational resources. Thus, the GWs emitted by a compact binary system are normally calculated using the PN method during the inspiral, followed by numerical relativity for the final stages of the inspiral and the merger. As with PN methods, numerical relativity is a huge field, but it is not discussed further here. It is covered elsewhere in this book and also in the textbooks [17, 19]. Quasinormal modes are perturbations on a Kerr background that satisfy regularity conditions at the black hole horizon and infinity. After merger, and assuming a black hole is formed, a system will emit GWs at a frequency and decay rate that depend only on the final mass and angular momentum of the black hole. Details can be found in [20, 21]. This process is called the “ring-down” and is quite short compared to the inspiral and merger phases. Physically, it represents a distorted black hole at merger relaxing to the static Kerr solution. The ring-down phase is normally included in a numerical simulation with the properties compared to the analytic solution as a check. The search for GWs in detector data requires a large number (O(104 )) of waveform templates to cover the parameter space of a binary merger. It is not practical to generate these templates using numerical relativity since each run takes several weeks. Thus, methods have been developed that use the limited numerical relativity results available to set certain parameters in formulas that are either algebraic or the solution of an ordinary differential equation, thus enabling the rapid generation of the required waveform templates. The most commonly used methods are the effective one-body (EOB) approach (see, e.g., [22]), which can be regarded as being mainly an analytical modification of post-Newtonian theory, and phenomenological waveform models (e.g., [23]), in which the waveform during the merger phase is constructed as a best fit to the numerical relativity data. An extreme mass ratio inspiral (EMRI) is a compact binary inspiral and merger when the mass of one companion is much lower than that of the other [24]. Astrophysically, such events occur when a supermassive black hole captures an object with a mass of order a few M . Such events are not in the LIGO/Virgo frequency band but are expected to be observed by LISA (see section “Detection of Gravitational Waves”). The result of GW calculations is often reported in terms of ψ4 (rather than h+ , h× ), and the result decomposed into functions denoted by s Y ,m . What are these quantities? The Newman-Penrose quantity ψ4 is a complex quantity defined in terms of the Weyl tensor and a null tetrad and is related to h+ , h× by ψ4 = ∂t2 (h+ − ih× ) .

(72)

The s Y ,m are spin-weighted spherical harmonics that can be regarded as generalizations of the familiar spherical harmonics Y ,m to be able to represent vector and tensor quantities over the unit sphere. In order to describe quadrupolar GWs, only a limited set of the s Y ,m will be needed, specifically

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N. T. Bishop

r0

m1 v1

r1

C

r2

v2

ω

m2

M=m1+m2 , μ=m1m2/M

Fig. 2 Variables used in the orbiting circular binary calculations

2Y

2,2

−2 Y

2,2

√ √ 5 5 2 2,−2 = √ exp(2iφ)(1 − cos θ ) , 2 Y = √ exp(−2iφ)(1 + cos θ )2 8 π 8 π √ √ 5 5 2 2,−2 = √ exp(2iφ)(1 + cos θ ) , −2 Y = √ exp(−2iφ)(1 − cos θ )2 . 8 π 8 π (73)

Then, for example, Eq. (56), which gives the GW emission of a circular binary, can be reexpressed as √ 32μr02 ω4 5π × ψ4 = 5r      cos(2ωt) −2 Y 2,−2 + −2 Y 2,2 + sin(2ωt) i −2 Y 2,−2 − i −2 Y 2,2 .

(74)

We refer to the literature for a fuller discussion of the definition and properties of the Newman-Penrose scalars and of spin-weighted spherical harmonics, e.g., [17, 19, 25].

The Gravitational Wave Field of an Orbiting Binary Inspiral Rate and Time to Merger We now consider the long-term behavior of a binary system in circular orbit, with the GW field given by Eq. (56) and consequently the energy loss LGW given by Eq. (66). The variables used are illustrated in Fig. 2: there are two masses m1 , m2 with velocities v1 , v2 , respectively, in circular orbit of diameter r0 and angular velocity ω, and let M = m1 + m2 , μ = m1 m2 /M. Let r1 , r2 be the distances from m1 , m2 , respectively, to the center of mass of the system C so that r1 + r2 = r0 and r1 m1 = r2 m2 ; solving these two equations leads to r1 = r0 m2 /M, r2 = r0 m1 /M. In Newtonian theory, gravitational attraction is balanced by the orbital centripetal acceleration, which leads to

1 Introduction to Gravitational Wave Astronomy

ω2 =

23

M r03

(75)

,

and then the Newtonian kinetic plus potential energy of the system is E=−

μM 1 μM μM μM + ω2 r02 μ = − + =− . r0 2 r0 2r0 2r0

(76)

Now, conservation of energy gives ∂t E + L[GW ] = 0 with L[GW ] given by Eq. (66), so that ∂t r0 = −

64M 2 μ 5r03

,

(77)

which easily integrates to give a time to coalescence tc (when r0 (t) = 0) tc =

5r04 . 256M 2 μ

(78)

Equations (77) and (78) lead to important results in GW astrophysics. Consider the example of a binary system comprising two 10M black holes in circular orbit such that the time to coalescence tc is 109 years: what is the initial diameter of the orbit r0 ? The value of r0 for which tc is 109 years is astrophysically important, since if the initial separation was larger (by a factor of 2), then tc would be greater than the age of the universe and so GW emission alone would not cause the binary to coalesce. Inverting Eq. (78) gives 

3 109 years × 2000M r0 = 4 5

1 4

= 7.5 × 106 km ,

(79)

where we have used the conversion factors 1 year = 0.95 × 1013 km, and 1M = 1.48 km. By astronomical standards, the distance 7.5 × 106 km is very small – it is about 5 solar diameters. Thus, the formation of coalescing black hole binaries must involve astrophysical processes other than GW emission. The derivation of Eqs. (77) and (78) assumed that the system is quasi-static, but that is not the case at, and just before, coalescence. From the quadrupole formula, the magnitude of a GW scales is v 2 (where v is the orbital velocity), and the error is O(v 4 ). The problem being considered determines an acceptable magnitude for the error. Suppose that we take vmax = 0.1 (so that the error in the GW is O(10−4 )), and we find that for the equal mass binary example, the orbital diameter when v = vmax is r0 = 50M. Working out the time to coalescence gives 12s; of course, this figure is not accurate, but if the time to coalescence is 109 years, any error occurring during the last few seconds of the inspiral is negligible. Differentiating Eq. (75) gives

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N. T. Bishop

3 ∂t r0 ∂t ω =− , ω 2 r0

(80)

and we can then replace ∂t r0 , r0 in Eq. (77) by ∂t ω, ω to obtain the chirp mass formula M ≡

(m1 m2 )3/5 = μ3/5 M 2/5 = M 1/5

5 −8/3 −11/3 π f (∂t f ) 96

3/5 ,

(81)

where f is the wave frequency, related to the orbital angular velocity by ω = πf . Equation (81) will be discussed further in section “Data Interpretation and Determination of Source Parameters”: the key point is that both f and ∂t f are determinable entirely from LIGO/Virgo data, and thus so is the chirp mass M . Observed changes to the orbital frequency of the Hulse-Taylor binary pulsar [10], and other systems, have provided confirmation of this formula since the 1970s. The decrease in the Newtonian orbital energy of the binary (Eq. (76)) can be regarded as being caused by a radiation reaction force (also called the “self-force”). A straightforward Newtonian calculation shows that the radiation reaction is of magnitude F =

32 2 3 5 μ r0 ω , 5

(82)

and is applied to each object in the binary in the opposite direction to that of its orbital motion. Since the radiation reaction comprises two equal and opposite forces a distance r0 apart, it imposes a torque of magnitude r0 F which is equal to the rate of change of angular momentum of the system. Since angular momentum is conserved, the GWs must be carrying angular momentum away at a rate r0 F =

32 2 4 5 32μ2 M 5/2 μ r0 ω = . 7/2 5 5r0

(83)

Now the angular momentum J of the system is  J = ω(m1 r12 + m2 r22 ) = r02 ωμ = μ Mr0 ,

(84)

so that √ ∂t J = μ

M∂t r0 32μ2 M 5/2 =− √ 7/2 2 r0 5r

(85)

0

which is consistent with Eq. (83). The rate at which GWs transport angular momentum may be expressed in a form similar to Eq. (65), giving the value found in Eq. (85) for a circular binary. This also applies to the transport of linear momentum; although it is zero for the binary being considered here, it can be substantial for

1 Introduction to Gravitational Wave Astronomy

25

two black holes with misaligned spins. These results are not discussed here but are given, for example, in [19].

Eccentricity Reduction by Gravitational Waves In general, binary orbits are elliptical rather than circular, but an effect of GWs is to circularize an elliptic orbit. Thus close to merger (when the GW emission is strongest and most likely to be detected), binary orbits have a very low eccentricity and can be treated as circular; the only exception would be the unlikely scenario that the binary is located in a system containing other massive objects that increase the eccentricity of the binary. The eccentricity reduction formula was first derived in [8], and here we just outline the calculation. An elliptical orbit is described by its semimajor axis a and its eccentricity e. In Newtonian theory, these are constants of the motion, and here they are treated as slowly varying functions of time. The orbital equation is r=

2a(1 − e2 ) , 1 + e cos φ

(86)

where r is the distance between the orbiting objects. A circular orbit is recovered on setting e = 0, a = r0 /2 so that r = 2a = r0 . The Newtonian energy and Newtonian angular momentum are E=−

 μM , J = μ 2Ma(1 − e2 ) , 4a

(87)

and the rates of energy and angular momentum emission by GWs are 

 73e2 37e4 μ2 M 3 1 + + , 24 96 5a 5 (1 − e2 )7/2 √  

 dJ 7e2 2 2μ2 M 5/2 1 + = − 7/2 , dt 8 5a (1 − e2 )2

 L[GW ] = −

(88) (89)

where  denotes an average over one orbit. The above formulas reduce to those for a binary in circular orbit on setting e = 0, a = r0 /2. Substituting Eqs. (87) into Eqs. (89) leads to equations for ∂t a, ∂t e, which are combined to give 

from which it follows that

  da 12a  = 1 + O(e2 ) , de 19e

(90)

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N. T. Bishop

e ≈ e0

a a0

19/12 (91)

.

During the time that GW emission starts to drive an inspiral until the GWs become detectable, the size of the orbit (i.e., a) decreases by a factor of about 104 , so that e ≈ 10−6 e0 < 10−6 .

Detection of Gravitational Waves GW detectors are sensitive only in a specific frequency range, and so key factors for the detection of a GW are its magnitude and frequency, and it is useful to have simple order of magnitude estimates of these quantities. From Eqs. (56) and (75), it follows that 

2μM |h+ , h× | = O , (92) r0 r which is maximized just before merger when r0 ∼ 2M so that |h+ , h× | = O

μ r

≈ 0.5 × 10−19

The GW frequency is estimated from Eq. (75) as ω =

μ/1M . r/1Mpc

(93)

√ 2/(4M) so that

√

f ∼

2 M ≈ 2 × 104 Hz . 4π M M

(94)

The GW from a compact binary inspiral slowly increases in magnitude and frequency until reaching the maximum values indicated above. The various options for measuring GWs are described in detail elsewhere in this book, and here only an outline is presented. The frequency ranges in which detection systems are sensitive, and the corresponding astrophysical processes are summarized in Fig. 3.

GW Detection Facilities Ground-based laser interferometer detectors comprise two perpendicular arms each of length several km, and laser light travels along each arm and back again (see Fig. 4); then the difference in travel time between the two arms is measured as an interference fringe. These detectors are sensitive in the frequency range ≈20 to ≈2000 Hz and have observed compact binary mergers (with the compact objects being neutron stars or stellar-mass black holes) since September 2015 [16]. Other potential GW sources that may be detected are discussed in section

1 Introduction to Gravitational Wave Astronomy

27

Fig. 3 LIGO, LISA, and pulsar timing frequency ranges and target astrophysical events

Test mass

L

Test mass Beam splitter

Test mass

Test mass L

Laser source Photodetector

Fig. 4 Schematic representation of a terrestrial laser interferometer detector. The arm length L is 4 km for LIGO and 3 km for Virgo

28

N. T. Bishop

“GW Observations: Contributions to Physics and Astrophysics”. Detector sensitivity has been continually improved since the first science run in 2001. This is important, since a tenfold increase in sensitivity means a tenfold increase in the distance at which a source may be detected (see Eq. 46), which increases the volume searched and therefore the expected detection rate by a factor of 1000. Currently the network comprises operational detectors GEO600 in Germany (this is the least sensitive detector), KAGRA in Japan, LIGO Livingstone in the USA, LIGO Hanford in the USA, and VIRGO in Italy, with LIGO-India scheduled to be operational in 2024. In order for there to be a high probability that an observation of a GW event is real rather than an instrument glitch, it is important that it be observed in at least two detectors. Further, the accuracy of sky localization increases with the number of (well-separated) detectors in which the event is observed; accurate sky localization improves the chance that an EM counterpart can be identified, leading to a fuller picture of the astrophysics of the event. The Laser Interferometer Space Antenna (LISA) is a mission of the European Space Agency and is scheduled for launch in 2034. It comprises three satellites at the vertices of an equilateral triangle with sides 2.5 × 106 km in an Earth-like orbit around the Sun. It will be sensitive to GWs in the frequency band 10−5 to 1 Hz. The GW events that are expected to be observed include the merger of supermassive black holes and EMRIs. LISA should also detect GWs from close binaries comprising stellar-mass black holes, neutron stars, or white dwarfs and should also be able to predict the end point of an inspiral, i.e., the time of a compact binary merger event detectable by a ground-based facility. Millisecond pulsars can be regarded as highly accurate clocks. Pulsar timing projects measure accurately the time of arrival of each pulse from a particular pulsar and compare it to the expected time of arrival assuming a uniform time between the pulses. This difference is recorded as the timing residual. (In practice, the process is rather more complicated, because it is only when averaged over a number of pulses that uniform pulse emission occurs, and further there may be glitches in pulse emission; thus the data analysis is statistical rather than direct.) These measurements are repeated for a number of pulsars distributed over the sky. If a GW passes the Earth, then the timing residuals will oscillate according to the form of the GW. Pulsar timing is sensitive to GWs at very low frequencies, about 10−7 to 10−9 Hz corresponding to periods of a few months up to a decade, generated, for example, by supermassive black hole binaries well before merger. Data collection has been ongoing since 2004 but to date has only been able to set upper limits.

Data Interpretation and Determination of Source Parameters A major aspect of laser interferometer data analysis is to identify GW signals that are hidden by noisy data. The methods used vary according to the type of signal being searched for; these matters are not pursued here but are discussed in detail elsewhere in this book. The presentation here is limited to an outline of the determination of the astrophysical parameters from GW data of a binary black hole merger.

1 Introduction to Gravitational Wave Astronomy

29

Suppose that a GW signal is identified in detectors A and B and that it is observed in detector B a time tAB after it is observed in detector A. Then if ΔAB is the light travel time from A to B, the direction to the GW source must be at an angle

α = arccos

tAB ΔAB

 (95)

to the line AB. In practice, there is an error involved in the measurement of tAB , which means that the source is located within an annulus, rather than on a circle, on the sky. Further, as well as the timing of the GW signal, we also know its magnitude in each detector, and when combined with a model of the source, this may lead to a further restriction on the direction to the source; i.e., the annulus may be reduced to a partial annulus. In the case that the GW event is seen in three or more detectors, the sky localization reduces to a point if there are no measurement errors and in practice to a small region of the sky. For the next steps, we assume that the quadrupolar formulas remain valid. The chirp mass M is determined using Eq. (81). Then, using Eq. (75) and ω = πf in Eq. (56) gives 2M 5/3 (πf )2/3 (h+ , h× ) = − r   × (1 + cos2 θ ) cos(2ω(t − r) − 2φ), 2 cos θ sin(2ω(t − r) − 2φ) . (96) Thus, a measurement of the polarization modes h+ , h× would lead to the ratio h× / h+ which can be solved for the angle θ (which is the polar coordinate of the Earth in the reference frame of the source), and then Eq. (96) may be solved to find the distance r between source and Earth. If the event is well-localized on the sky, then the plane N transverse to the propagation direction is known. Within N, let eθ , eφ be unit vectors in the θ, φ directions of the source frame, with respect to which h+ , h× of Eq. (96) are defined. Let e1x , e1y be unit vectors representing the directions of the arms of a detector D1 projected into N , and let the angle between eθ and e1x be ψ. The angle between e1x and e1y is known, so that between eθ and e1y is known in terms of ψ. These quantities are illustrated in Fig. 5. Now, given h+ , h× and ψ, we can use Eq. (28) to rotate the coordinate axes and find the change in length of each arm of D1 due to the GW, and the difference between these changes is the detector response. Thus the measured detector response leads to one equation involving the three unknowns h+ , h× and ψ. Measurements in detectors D2 , D3 (and if possible, more detectors so as to reduce error bars) then provide sufficient information to close the system and solve for the unknowns. The remaining astrophysical parameters are determined using features of the waveform beyond the quadrupole formula. A template bank of merger waveforms over the parameter space to be explored is needed. For black hole mergers, the frequency of the GWs scales as the inverse of the total mass M, and the magnitude

30

N. T. Bishop

Fig. 5 Detector arms projected into plane N

scales as M; then the parameter space comprises the mass ratio q = m1 /m2 and the spins S 1 , S 2 . The mismatch between the calculated waveform (hc ) and the observed waveform (ho ) is defined as M (hc ) = 1 − hc , ho 

(97)

where the inner product is evaluated in the Fourier domain and includes the detector response function of each detector in which a signal is observed. It is normalized so that it satisfies the conditions 0 ≤ h1 , h2  ≤ 1 with h1 , h2  = 1 iff h1 = constant ×h2 . Then M (hc ) is minimized over the template bank. Since ho is subject to measurement error, the values found for the various astrophysical parameters are also subject to error.

GW Observations: Contributions to Physics and Astrophysics Fundamental Physics The inspiral and merger of a binary neutron star system was observed by LIGO/Virgo as GW170817 [26, 27]. It was followed, 1.7 s after merger in the GW signal, by a gamma-ray burst GRB 170817A. The distance to the system is 40 Mpc, so we can deduce that the speed of GWs (cGW ) is very close to being exactly that of light (cEM ) [28] 1 − 3 × 10−15 ≤

cGW ≤ 1 + 7 × 10−16 . cEM

(98)

In general relativity, the speed of GW propagation is exactly the same as that of light so this result is a strong test of the theory. Further, all GW waveforms observed to date have been consistent with the predictions of general relativity. These waveforms

1 Introduction to Gravitational Wave Astronomy

31

have been produced by compact object mergers and include the strong field regime. (Previously, experiments have only been able to test the theory in the regime of small deviations from Newtonian theory.) Again, this amounts to a strong test of general relativity.

Cosmology Data for the binary neutron star event GW170817 was taken by three detectors enabling the angle θ to be constrained; then Eq. (96) is used to provide an independent distance estimate. This is an example of a compact binary merger being a standard siren [29]. Since an optical counterpart was identified, the redshift of the system is known to good precision, and thus the expansion rate, i.e., the Hubble constant, can be measured. The result is 70.0+12.0 −8.0 km/s/Mpc. The uncertainty in the value should be reduced in the future as more events are discovered. The value obtained is consistent with estimates from both cosmic microwave background data and supernovae (SN1a), and does not yet shed light on the tension between these values. Astrophysics The first GW detection, GW150914[16], was the merger of two black holes of mass approximately 36M and 29M to give a 62M black hole. Until then, the most massive black hole that had been observed was ≈25M . As more and more such events are detected, the mass distribution of black holes in the local universe, as well as the rate at which mergers occur, will be well-constrained, and this information will enable improved modeling of black hole, and black hole merger, formation channels. Similar remarks apply to neutron star mergers, although to date only two such events have been reported. Event GW190814 [30] was particularly interesting because it involved the merger of a 23M black hole with a 2.6M compact object, which could be either the most massive neutron star or the smallest black hole to have been observed. Besides compact object mergers, current detectors are expected to observe GWs from supernovae, as well as the continuous signal from any asymmetry in a millisecond pulsar (which is a rapidly rotating neutron star). Such signals have not yet been detected, but their observation should lead to significant improvement to the modeling of these systems.

Conclusion GWs in general relativity were first predicted in 1916, but it was nearly 100 years until the first direct detection of a GW event in 2015. Detections of new events are becoming more and more frequent, and the results to date have already had an impact on astrophysics, cosmology, and physics. This impact will grow as detector sensitivities are improved and new facilities come online, so that more events, and of different types, are detected. The future of GW astronomy is, of course, unknown,

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but we can expect that it will bring some fascinating results and probably some surprises.

Cross-References  LIGO, VIRGO, and KAGRA as the International Gravitational Wave Network  Machine Learning for the Characterization of Gravitational Wave Data  Measuring Cosmological Parameters with Gravitational Waves  Numerical Relativity for Gravitational Wave Source Modeling  Post-Newtonian Templates for Gravitational Waves from Compact Binary

Inspirals  Principles of Gravitational-Wave Data Analysis  Pulsar Timing Array Experiments  Space-Based Gravitational Wave Observatories  Terrestrial Laser Interferometers

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16. Abbott BP et al (2016) Observation of gravitational waves from a binary black hole merger. Phys Rev Lett 116:061102 17. Alcubierre M (2008) Introduction to 3+1 numerical relativity. Oxford University Press, Oxford 18. Blanchet L (2006) Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev Relativ 9:4 19. Baumgarte TW, Shapiro SL (2010) Numerical relativity: solving Einstein’s equations on the computer. Cambridge University Press, Cambridge 20. Chandrasekhar S (1998) The mathematical theory of black holes. Oxford University Press, New York 21. Kokkotas K, Schmidt B (1999) Quasi-normal modes of stars and black holes. Liv Rev Relativ 2:2 22. Damour T, Nagar A The effective one-body description of the two-body problem, pp 211–252. Springer Netherlands, Dordrecht (2011) 23. Santamaría L et al (2010) Matching post-Newtonian and numerical relativity waveforms: Systematic errors and a new phenomenological model for nonprecessing black hole binaries. Phys Rev D 82:064016 24. Poisson E, Pound A, Vega I (2011) The motion of point particles in curved spacetime. Liv Rev Relativ 14:1–190 25. Bishop NT, Rezzolla L (2016) Extraction of gravitational waves in numerical relativity. Liv Rev Relativ 19:2 26. Abbott BP et al (2017) GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys Rev Lett 119:161101 27. Abbott BP et al (2017) Multi-messenger observations of a binary neutron star merger. Astrophys J Lett 848:L12 28. Abbott BP et al (2017) Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys J Lett 848:L13 29. Schutz BF (1986) Determining the hubble constant from gravitational wave observations. Nature 323:310–311 30. Abbott BP et al (2020) GW190814: gravitational waves from the coalescence of a 23 solar mass black hole with a 2.6 solar mass compact object. Astrophys J Lett 896:L44

Part II Gravitational Wave Detectors

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Terrestrial Laser Interferometers Katherine L Dooley, Hartmut Grote, and Jo van den Brand

Contents Introduction: A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonant Mass Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Beginnings of Laser Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling of Gravitational Waves to a Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . Extensions to the Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Sources and Noise Reduction Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Enabling Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Things Not Yet Mentioned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser Interferometers World-Wide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virgo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KAGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GEO 600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGO A+ and Virgo AdV+ Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGO-India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Third-Generation Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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K. L. Dooley () · H. Grote Cardiff University, Cardiff, UK e-mail: [email protected]; [email protected] J. van den Brand Nikhef and Maastricht University, Amsterdam, The Netherlands e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_2

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Abstract

Terrestrial laser interferometers for gravitational-wave detection made the landmark first detection of gravitational waves in 2015. We provide an overview of the history of how these laser interferometers prevailed as the most promising technology in the search for gravitational waves. We describe their working principles and their limitations and provide examples of some of the most important technologies that enabled their construction. We introduce each of the four large-scale laser interferometer gravitational-wave detectors in operation around the world today and provide a brief outlook for the future of ground-based detectors. Keywords

Gravitational waves · Laser interferometer · LIGO · Virgo · KAGRA · GEO600 · Fabry-Perot cavity

Introduction: A Historical Perspective Albert Michelson was reportedly a “hard-core” physicist, dedicating pretty much all of his time to research. He was interested early on in improving methods to measure the speed of light, and to this end, he developed the instrument carrying his name today, the Michelson interferometer. His invention is best known in the history of physics for the null result testing the ether hypothesis via the attempt to measure differences in the speed of light traveling in different directions. While it is disputed to what extent this famous null result triggered the development of special relativity, it certainly lent credence to Einstein’s theory of 1905. By 1915, Einstein had developed the general theory of relativity, which predicted the existence of gravitational waves, though it took decades to convince most physicists of the existence and the possibility to measure these waves [1]. The interesting twist here is that much-enhanced successors of Michelson’s interferometer first detected gravitational waves in 2015. These km-scale terrestrial laser interferometers of today are more than ten orders of magnitude more sensitive than the model that Michelson and Morley used for their ether experiment. In this chapter, we will examine how this astonishing improvement was achieved.

Resonant Mass Detectors Notwithstanding the title of this chapter, we would like to emphasize here the pioneering work of Joseph Weber, which started the field of experimental gravitationalwave physics. In the late 1950s, Weber contributed to the forming consensus that gravitational waves could indeed be measured, and he set out on a program to attempt the feat with so-called resonant mass detectors. These detectors are massive objects of cylindrical or spherical shape whose mechanical eigenmodes may be excited by passing gravitational waves. Weber claimed to have detected gravitational

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Fig. 1 A gravimeter on the Moon, which Weber had convinced NASA to deliver on their last Apollo mission. The gravimeter was to measure vertical accelerations of the Moon’s surface that could be caused by gravitational waves coupling to the Moon’s quadrupolar eigenmode. The instrument can be seen in the foreground, with wires running to service stations further back. (Photo courtesy: NASA)

waves with his detectors in the late 1960s, which spurred several research groups around the globe to attempt replication. In Fig. 1, we highlight a less well-known episode of Weber’s work, where he attempted to use the Moon as a resonant mass detector [2]. By the mid-1970s, no other group had been able to confirm Weber’s claims, despite having developed significantly more sensitive detectors. Most scientists today think that Weber was mistaken in the way he analyzed his data. Not only was there no confirmation by other groups, but the claimed signal sizes would have meant that most of the mass of the Milky Way would have been converted to energy in the form of gravitational waves. Furthermore, once the sensitivity of laser interferometers had far surpassed that of resonant mass detectors, they also could not confirm the existence of events of the magnitude Weber had claimed he saw. Once set on this exciting adventure, many research groups did not want to let go of the fascinating prospect of detecting gravitational waves. Subsequently, the experimental community split into two branches. One continued to perfect resonant mass detectors to unprecedented sensitivity levels by cooling ton-scale masses to millikelvin temperatures [3]; by 2016, however, all of the operating resonant mass projects had stopped taking data. The other branch that started to develop a new technology would ultimately be successful: laser interferometry.

The Beginnings of Laser Interferometry The idea of using a Michelson interferometer to measure gravitational waves appears to have surfaced among various scientists independently of one another.

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According to Joseph Weber’s lab notes, this idea came to him soon after the Chapel Hill Conference in 1957, and he spoke of it in a telephone conversation with his colleague Robert Forward in September 1964 [4]. The idea had been published in Russian in a 1962 work by Mikhail Gertsenshtein and Vladislav Pustovoit; see [5] for an English translation. Because one could make interferometers long, it was recognized that they had the potential to be sensitive to strains – a length change proportional to distance – of 10−21 or less, a level where detections were deemed possible. But assessing the feasibility of building such an instrument required extensive analyses of noise sources and of the technology that was available. Experimental physicist Rainer Weiss carried out much of this early work after he began to think about interferometers as gravitational-wave detectors in 1969. He calculated how sensitive such an instrument could be and how the influence of various sources of noise could be minimized [6]. Weiss cites the work of Felix Pirani, a British theoretical physicist, and the running of an undergraduate seminar as two of his inspirations. The theoretical physicist Kip Thorne was interested in gravitational waves early on in his research and was an enthusiastic supporter of Weber. Initially, Thorne was not convinced about developing interferometers for the purpose of gravitationalwave detection. In 1970, in a standard textbook on gravity co-authored with Misner and Wheeler [7], he writes: Such detectors have such low sensitivity that they are of little experimental interest.

At the time, lasers, in particular, were still very unstable, and existing interferometers were far from being sensitive enough to detect gravitational waves. In spite of his skepticism, Thorne maintained contact with Weiss, who considered the use of interferometers as feasible, in principle. Robert Forward from Hughes Aircraft Research Laboratory in Malibu, California, and a former member of Weber’s team, was the first scientist to begin building an interferometer as a prototype in 1971. With a simple folded arm of effective length of 4.25 meters, this instrument achieved about the same sensitivity to gravitational waves as Weber’s cylinder. [8] The difference was that the interferometer was sensitive across a broad frequency band, providing a distinct advantage over cylinders which were sensitive in only a narrow band around 1660 Hz. The further development of Forward’s interferometer was discontinued, however, as he turned his attention to other areas of study. Beginning in 1972, at the Massachusetts Institute of Technology (MIT) in Cambridge, MA, Weiss attempted to obtain research funding from the National Science Foundation (NSF). It was finally granted in 1975, but initially, Weiss had difficulties getting PhD students to work on his project because it involved lengthy development work. At that time, the resonant mass antennas had been established, and the future of interferometers was still uncertain. In 1975, Weiss said in an interview [4]: We [at MIT] are in a physics department. And . . . engineering is not considered respectable physics. To build something and show that it works as predicted, but without making a measurement of anything new does not really count as any achievement.

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Despite this obstacle, Weiss started with a prototype arm length of 1.5 meters and was able to secure funding in 1981 for a study to build a much larger detector with arm lengths in the kilometer range. In 1974 in Munich, Germany, a group led by Heinz Billing turned from resonant mass detectors to interferometers and began to build a laboratory-sized prototype with an arm length of 3 meters. Using the concept of delay lines, beams passed through each arm of the interferometer up to 138 times. This was the world’s leading interferometer for many years and served as the prototype for the development and successful demonstration of important new interferometer techniques. This included the idea of hanging the mirrors as pendulums by Karl Maischberger to avoid mechanical resonances; the invention of the mode cleaner by Albrecht Rüdiger and others to suppress laser beam movements; the development of a comprehensive theory of the effect of scattered light by Walter Winkler; and the concept of power recycling, which was proposed at about the same time by both Roland Schilling in Munich and by Ronald Drever in Glasgow. In 1983, the construction of a much larger and improved prototype with an arm length of 30 meters began on the Garching science campus near Munich. This prototype was the first of its kind in the world to reach shot noise, an important limitation to the sensitivity of optical interferometers that had previously only been theoretical. This achievement was to be of decisive importance for the funding of the American LIGO project. By the end of the 1980s, the peak strain sensitivity of the Garching detector was about 10−19 . This was an improvement of a factor of one-thousand over Weber’s cylinders of 20 years earlier, in addition to having a much wider bandwidth. The 30-meter prototype was in use until 2002, and in its final years, it was the first interferometer to demonstrate the combination of power and signal recycling, an optical configuration known as dual recycling (see Section “Extensions to the Michelson Interferometer”). It also served as a test facility for the GEO 600 detector in Germany, prompting the development of numerous techniques such as the ability to keep the suspended mirrors at the correct angle during a measurement. In Glasgow, Scotland, beginning in 1975, Drever turned his attention to interferometry, initially studying it in order to achieve a more precise readout from resonant mass antennas. In 1976, Drever began constructing a prototype interferometer with an arm length of 10 meters and applied the concept of Fabry-Perot resonators in the arms. In 1979, following an invitation from Thorne, Drever also led the construction of a 40-meter arm-length prototype at the California Institute of Technology (Caltech) in Pasadena, CA. After Drever moved to Caltech permanently in 1983, Jim Hough took over the management of the 10-meter prototype in Glasgow, where Brian Meers would develop the concept of signal recycling [9]. In addition to these first significant prototypes, another noteworthy facility is the Australian International Gravitational Observatory (AIGO), located north of Perth. Originally, the construction of an interferometer with arms several kilometers long was planned, but, despite concerted effort, the necessary funds could not be obtained. AIGO is currently a prototype with an arm length of 80 meters and is used for testing high-intensity laser power in interferometers [10].

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In the mid-1980s, after gaining experience with laser interferometer prototypes and developing new techniques, groups in the United States, the United Kingdom, and Germany, and later in France and Italy, began applying for research funding for kilometer-sized systems. This was a tall order given that at least $100 million would be needed to build instruments that were perceived as having only a small chance of ever measuring gravitational waves. Interferometer technology was not yet a mature science, and uncertainty remained as to whether or not such large facilities would function sufficiently well. In the remainder of the chapter, we will introduce some of the principles of laser interferometry and the development of today’s terrestrial gravitational-wave detectors. In Section “Principles,” we look at the basics of how laser interferometers can be used for gravitational-wave detection, including optical design considerations, relevant noise sources, and enabling technologies. In Section “Laser Interferometers World-Wide,” we introduce the large-scale terrestrial laser interferometers, with a focus on their individual histories and some particularities. We conclude with an outlook in Section “Outlook” but also refer the reader to the chapters in this handbook on research for future detectors and third-generation detector technologies. Many more details on advanced interferometric gravitational-wave detectors can be found in the two-volume book of the same name [11]. A more compact scholarly overview of gravitational-wave detectors is provided by Saulson’s book [12] and by a few overview papers [13, 14]; an introduction to gravitational-wave detection for a public audience can be found in [15].

Principles We’ll start this section by taking a step back from the assumption that a Michelson interferometer is an appropriate tool to sense gravitational waves and build up the reasoning for this central design choice. At their core, terrestrial gravitational-wave detectors are instruments that must be capable of measuring a strain in spacetime of the order 10−21 at frequencies of hundreds to thousands of Hertz. The basic design element of today’s detectors makes use of the unique property of light – that its speed is absolute – to probe the distance between two inertial masses that act as markers of spacetime coordinates. A passing gravitational-wave modulates (ΔL) the separation (L) of these so-called test masses, when placed several kilometers apart, typically by less than 10−19 m around 100 Hz. The test masses are mirrors, which provide a means to reflect the laser light dozens of times back and forth. Use of these mirrors as either a delay line or an optical cavity effectively increases their separation, making any induced spacetime strain from gravitational waves (h = ΔL/L) result in all the larger a change in the light travel time (There is a limit to just how big the effective mirror separation should be. After all, if the light experiences both a stretching and shrinking of spacetime, the net length change sensed will be reduced. The optimum situation is therefore when the light samples the mirror separation for exactly one half period of the gravitational wave. Critically, this does mean that not all frequencies of gravitational waves can be simultaneously optimally detected.).

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To ensure the mirrors are protected from external forces other than gravity, a careful consideration of all potential disturbances and a scheme to reduce them is necessary. For terrestrial gravitational-wave detectors, the motion of the Earth’s surface itself is the most egregious of such disturbing forces. Part of the art of the field of designing and building gravitational-wave detectors lies in devising ways to mitigate everything that physically displaces the mirrors as well as everything that prevents the achievement of the most fundamental precision of displacement measurements. The result has been the construction of detectors that have pushed the limits of modern technologies, from state-of-the-art seismic isolation systems to low loss optical coatings to nonclassical light sources and ultrahigh-vacuum systems. The Michelson interferometer is often mistakenly assumed as a theoretically required necessity of the design of a gravitational-wave detector based on interferometry principles. It is not. In principle, gravitational waves can be measured with a single cavity and a perfect clock. However, a perfect clock does not exist, and thus two sets of two mirrors are used and their respective separations measured simultaneously such that one set can act as a reference for detecting common irregularities of the timing measurement. Here, the arrangement of these cavities as two perpendicular arms of a Michelson interferometer is critical, with the reason rooted in a particular feature of gravitational waves: that they induce strains in spacetime in a quadrupole configuration (see  Chap. 1, “Introduction to Gravitational Wave Astronomy” of this Handbook). By simultaneously measuring the stretching of one arm and the shrinking of the other, it is assured that sensed length changes resulting from common clock irregularities can be decoupled from the differential effect of gravitational waves. In addition, these two simultaneous length measurements make the response of the detector to gravitational waves up to a factor of two larger compared to a single cavity (The precise response depends on the orientation of the detector with respect to the propagation direction of the gravitational wave.). In this section we derive how gravitational waves couple to a Michelson interferometer (Section “Coupling of Gravitational Waves to a Michelson Interferometer”); we present and motivate the various extensions to a Michelson interferometer design, namely, the use of optical cavities (Section “Extensions to the Michelson Interferometer”); we discuss the primary noise sources, both fundamental and technical, that limit the sensitivity of the instruments (Section “Noise Sources and Noise Reduction Strategies”); and we describe the principles of a few select enabling technologies (Section “Some Enabling Technologies”). For additional resources summarizing the technologies, techniques, and theoretical models used in designing gravitational-wave detectors, we direct the reader to [12–14].

Coupling of Gravitational Waves to a Michelson Interferometer Two fundamental ways of looking at how a gravitational wave affects the interferometer are useful to distinguish because of the significant effect they have on how one thinks about the functioning of the detector. In a viewpoint which is always valid, gravitational waves change the metric describing the spacetime between two

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freely falling test masses. The coordinates of the test masses and their coordinate separation do not change, although the changing metric does make the proper distance change. If, however, one views the test masses in a proper reference frame, the effect of the gravitational waves is to exert a force on the test masses. Their coordinates do change. This viewpoint is only valid for test mass separations that are small compared to the gravitational-wave wavelength. Another duality to ponder for a moment is that of light as both a particle and a wave. A question asked by many a thoughtful student that arises from thinking about the effect of gravitational waves on spacetime is: “If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves?” As before, different reference frames will answer this question differently, and ultimately, the answer is that the wavelength is irrelevant as explained in [16–18]. It becomes clear that a measurable effect exists if we walk through a frameindependent argument of thinking about light as a photon. Consider two wave packets leaving the beam splitter of a basic Michelson interferometer (see Fig. 2) at the same time, each heading down a different arm. If a gravitational wave is present (It should be noted that h is treated as a constant in Eqs. 1 and 2, which assumes that the wavelength of the gravitational wave is much larger than the interferometer arm length. In this case, the temporal variation of h(t) is negligible during the time it takes the photon to make its round trip.), then the amount of time the wave packet takes to make one round trip down a stretched arm and back is τrt+ =

  h 2L 1+ . c 2

(1)

Likewise, the round-trip travel time for a compressed arm is   h 2L τrt− = 1− . c 2

Fig. 2 A basic Michelson interferometer with arm lengths Lx and Ly . Either output port can be used to obtain the gravitational-wave signal. A design convention is to use the anti-symmetric port

(2)

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There is a nonzero 2Lh/c difference in arrival times at the beam splitter, a quantity one could measure with an accurate clock. The detector at the beam splitter is not a clock, however, but a photodetector which physically measures the power of the recombined light, and therefore is a proxy for the relative phase of the two returning beams. It is thus informative to express the difference in arrival times as a difference in phase. To do so, we must move away from the photon model and think about the wave model of light where its phase is given by φ = ωτ , with τ the proper time and ω the angular frequency of the light. Then, the difference in phase between the two light beams after each has completed its round trip is Δφrt = φrt+ − φrt− =

2L ωh = 2kLh c

(3)

where k = 2π/λ is the wave number. We can already gain an appreciation for the magnitude of sensitivity required by the gravitational-wave detectors. Let’s consider the very first gravitationalwave detection, GW150914, of a merger of two ∼30 M black holes ∼400 Mpc away, which produced a strain on Earth of 10−21 at 100 Hz about 10 ms before their merger [19]. We can use the static strain approximation of Eqs. 1 and 2 because the wavelength of the gravitational wave is about three orders of magnitude greater than that of the kilometer scale detectors. For simple 4-km-long Michelson interferometers, the difference in arrival times of wave packet returning from one arm compared to the other is a mere 2.6 × 10−26 seconds.

The Michelson Interferometer Response Basic interferometry can be studied using monochromatic, scalar, plane waves as we will do here. A more realistic model must include at least the shape of the beam and additional frequency components, some of which will be discussed later in the chapter. Using the convention of describing an electromagnetic wave by its electric field, and assuming perfectly reflecting end mirrors and a perfect 50/50 beam splitter, one can derive the field at the symmetric and anti-symmetric ports of the interferometer (see Fig. 2): E0 2ikLx [e − e2ikLy ] 2 E0 2ikLx i[e = + e2ikLy ] 2

ES = EAS

(4) (5)

and easily verify that energy is conserved. Because of the quadrupole nature of gravitational waves, their effect on the interferometer is to change the differential arm length, ΔL = Ly −Lx . It is therefore more interesting to express the fields as a function of ΔL and L¯ = (Lx + Ly )/2, the common arm length:

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Fig. 3 The response of a basic Michelson inteferometer to a differential arm length change (ΔL), such as that caused by gravitational waves. Modern gravitational-wave detectors are operated at or close to the dark fringe

¯

ES = E0 ie2ik L sin(kΔL) ¯

EAS = E0 ie2ik L cos(kΔL).

(6) (7)

This response, as measured by a photodetector and normalized by the input laser power P0 is shown in Fig. 3. We choose to use the anti-symmetric port to extract a signal, and we operate the detectors either at or slightly off from the anti-symmetric port dark fringe. The choice of operating point is an important and subtle aspect of interferometer design, though beyond the scope of this chapter, so we refer the reader to [11, 14]. Nonetheless, the basic principle is clear: a modulation of power at the anti-symmetric port can be directly linked to a modulation of the differential arm length.

Gravitational Waves as Phase Modulation A different, more comprehensive perspective of describing the effect of gravitational waves on a Michelson interferometer is as a phase modulation of the light in the arms. When a gravitational wave impinges the interferometer, some energy is shifted from one frequency of the laser light, the carrier, to other frequencies, the sidebands. We must expand our model of the electromagnetic field to no longer be monochromatic, and we must consider non-static gravitational-wave strains. If we extend the model of Eq. 3 to represent non-static gravitational waves, we see that gravitational waves contribute to the total phase picked up by the laser field traversing each arm of the Michelson by the amount φGW (t) = ±kLh0 cos(ΩGW t), where ΩGW is the angular frequency of the gravitational wave and h0 its amplitude (Without loss of generality, we treat here the gravitational wave as monochromatic.). The electric field thus experiences a phase modulation of eiφGW , which can be expanded using Bessel functions because h0  1:

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eiφGW = 1 ± i

kLh0 kLh0 exp (−iΩGW t) ± i exp (+iΩGW t) ± . . . 2 2

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(8)

The first, zero-th order term represents the so-called carrier field which oscillates at the laser frequency, and the other two terms are the lower and upper sideband pair of order one. Gravitational waves thus create new electric fields in the Michelson arms with amplitudes proportional to both the carrier field and kLh0 /2 and with frequencies shifted away from the carrier frequency by ±kΩGW (Expansion to the second order is necessary to see that power is indeed transferred from the carrier to the sideband fields.). For stretched arms, the phase of the sideband fields is rotated +90 deg with respect to the carrier field, and for compressed arms, they are rotated −90 deg. When combined at the beam splitter, the two sets of gravitational-wave sidebands create a field at the anti-symmetric port that oscillates at a frequency of ΩGW with respect to the carrier and has an amplitude proportional to h0 . This signal is too small to be useful on its own. For the toy example presented above of the effect of GW150914 on a simple Michelson, the amplitude of the gravitational-wave sidebands is only 10−11 that of the carrier. If a 200 W laser were used, these sidebands would produce less than one photon per second at the antisymmetric port. For gravitational waves like GW150914 that generate strains of 10−21 for only 10 ms, less than one out of every 100 passing gravitational waves would actually produce a photon! Solutions for how to measure and amplify these signals include the introduction of local oscillator fields and Fabry-Perot cavities, respectively. The concept of increasing the signal with the use of cavities is addressed in the next section (Section “Extensions to the Michelson Interferometer”). Here, we introduce the technical trick of using a local oscillator to measure small signals. By adding a large field, ELO , to the signal, EGW , at the anti-symmetric port, the power measured by a photodetector becomes 2 2 + 2ELO EGW + EGW . PAS = ELO

(9)

The first term is large and static, and the third term (the pure gravitational-wave sidebands) oscillates at 2ΩGW , but is proportional to h20 and thus negligibly small, as seen in our example above. The middle term is where the benefit of the local oscillator field is relevant: it’s a strong signal due to the local oscillator, yet is proportional to h0√ and oscillates at ΩGW . Re-calculating our toy example for a local oscillator field of 10 mW, we now obtain an order of 1010 photons per second for the gravitational-wave signal (The size of the local oscillator field is determined by technical considerations and, once large enough to dominate other noise sources, does not affect the maximal signal-to-noise ratio that can be achieved.). The decision of what field to use as a local oscillator is intricately connected to several technical considerations. Current detectors use a small offset to the dark fringe operating point. In the past, radio-frequency sidebands have been used, and in the near future, local oscillator fields split off from the carrier light in the interferometer may be used.

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Fig. 4 A Fabry-Perot cavity. On resonance with the incoming light, the light field inside the cavity is resonantly enhanced by multiple reflections. The reflectivity coefficients of the mirrors set the enhancement level. (Image courtesy: Rob Ward)

Extensions to the Michelson Interferometer The typical optical configuration used in today’s generation of gravitational-wave detectors is an extension of the basic Michelson interferometer. The most prevalent additional feature is that of optical cavities, which are included at nearly every port, either for fundamental or technical reasons. Optical cavities can serve several functions that range from creating an effectively longer interferometer arm to increasing the laser power to filtering out unwanted spatial modes, laser frequencies, and even polarizations of light. Figure 4 shows an optical cavity consisting of two mirrors, known as a FabryPerot cavity, with a light field entering and leaving the cavity on the left-hand side. Assuming monochromatic light of a precise wavelength, the light field is resonantly enhanced between the two mirrors if the round-trip length of the cavity is a multiple integer of the wavelength. This resonant enhancement within a cavity is typically used to increase the precision of an interferometric phase measurement. When the cavity is instead used for spatial and temporal mode-filtering, the reflectivities of the two mirrors are made equal, which results in the filtered light being transmitted to the right-hand side. For an excellent original source on the basics of Fabry-Perot cavities, see [20]. In considering extensions to the basic Michelson, we assume the interferometer is operated at or close to the dark fringe, such that nearly all carrier power leaves toward the symmetric port and all signal toward the anti-symmetric port. There are two particular extensions to the Michelson interferometer that serve to enhance the gravitational-wave signal in two independent ways: the power and signal recycling mirrors. A third extension, Fabry-Perot cavities in the arms, combines the effects of power and signal recycling and, in combination with power recycling, also accommodates constraints arising from imperfections of the mirrors and the beam splitter. The power recycling mirror is located at the interferometer’s symmetric port and forms an optical cavity with each of the Michelson arms. By reflecting back into the interferometer all of the light leaving in the direction of the symmetric port, the laser power in the interferometer can be resonantly enhanced. This is important because

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as we saw in Eq. 8, the amplitude of the sidebands generated by the gravitational waves is proportional to the amplitude of the carrier field in the Michelson arms. One may question, however, where all of the laser power ultimately goes, since after all, the power buildup in the interferometer does not approach infinite levels even though the laser is always on. It is the reality that the mirrors are not perfect – they scatter and absorb light – that means there is a finite maximum circulating power. To reach that maximum power, the transmissivity of the power recycling mirror is designed to match the arm losses and create a nearly critically coupled (impedance-matched) power-recycling cavity. Today’s large interferometers achieve resonant power enhancement factors of about 5000, resulting in hundreds of kW of light power in the arms for input powers of order 100 W. The counterpart to the power recycling mirror is the signal recycling mirror, which is located at the interferometer’s anti-symmetric port and which forms cavities with each of the Michelson arms. The signal recycling mirror sends the gravitational-wave sidebands back into the interferometer such that they can constructively interfere with the new sidebands being created. The reflectivity of the signal recycling mirror determines how long (on average) the sidebands stay in the interferometer and therefore determines whether the sideband amplitude gets maximally resonantly enhanced or ultimately averaged away or something in between. The signal recycling cavity can therefore be designed to enhance the gravitational-wave signals at the frequencies of greatest scientific interest, at the cost of less signal at other frequencies. This is done by carefully choosing the reflectivity of the signal recycling mirror. Power- and signal recycling can be combined, a configuration called dual recycling [9]. A similar but different extension to the basic Michelson interferometer is the creation of Fabry-Perot cavities in the arms. By combining power recycling with arm cavities, the power in the interferometer can be concentrated in the arms and away from the beam splitter. This is important because the beam splitter substrate is transversed by only one of the two beams prior to their interference, making the beam splitter a dominant source of asymmetric optical loss in the interferometer. Any asymmetric loss disrupts the ideal condition that all carrier light leaves toward the symmetric port and none toward the anti-symmetric port (The most important reason to maintain perfect decoupling of the two ports is that any carrier light that leaks to the anti-symmetric port adds to the shot noise of the measurement and thus decreases the signal-to-noise ratio. A secondary reason is that any carrier power heading to the anti-symmetric port means all the less that goes to the symmetric port for recycling.). Power recycling alone can thus typically not achieve the high power levels in the interferometer that are possible when arm cavities are included. In this case, the reflectivity of the power recycling mirror can be adjusted to match the total losses of each arm, ensuring most of the laser power gets dumped into the losses of the arms, which are set to be as equal as possible through careful selection of the available optics. Traditionally, one often thinks of arm cavities as an effective way to lengthen the arms. This view is equivalent to the effect of signal recycling, in that the storage time of the gravitational-wave sidebands is increased. If such arm cavities

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Fig. 5 A diagram adopted from Seiji Kawamura [11] which organizes the concepts of the power and signal recycling cavities in relation to the finesse of the arm cavities. The configurations that have been used by various gravitational-wave detectors are as follows: (A) GEO 600; (C) Initial LIGO and Initial Virgo; (D) Advanced LIGO, Advanced Virgo and KAGRA. In (B) signal recycling is added to the use of arm cavities in a way that further enhances the signal sidebands at low frequencies. Option (C) describes the case where the frequency response of the arm cavities is precisely as desired such that no further shaping with a signal recycling mirror is required. In (D) the signal recycling is operated in RSE mode, decreasing the arm cavity finesse for the signal sidebands, but not for the carrier light. Option (E) describes the extreme where all power buildup is achieved by the arm cavities. Note that in the ideal case, the power in the arms is the same in all options, though in practice this is difficult to achieve in (A). (Figure credit: Advanced Interferometric Gravitational-wave Detectors. Reitze, Saulson and Grote, editors. c Copyright 2019 by World Scientific Publishing Co. Pte. Ltd)

are used on their own, they combine the effects of power and signal recycling, but the two parameters cannot be adjusted independently. When using power recycling and arm cavities, the distribution of power in the interferometer is coupled to the frequency response of the interferometer by choice of the reflectivity of the arm cavity mirrors. However, when arm cavities and signal recycling are combined, they allow the distribution of power in the interferometer and the frequency response to be independently tuned, where the finesse of the arm cavities determines how power is distributed between the power recycling cavity and the arms. This results in a particular bandwidth of the arm cavities, which can then be modified by the signal recycling mirror. Figure 5 organizes the parameters characterizing these three types of optical cavities into a single diagram. Here we see how increasing the arm cavity finesse keeps power away from the beam splitter. But due to an increase of losses in the arms

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with higher finesse, the power recycling gain (the power enhancement factor due to the power recycling cavity) must simultaneously be lowered in order to maintain impedance matching of the power recycling mirror to the arms. If arm cavities are present, the modification of bandwidth by the signal recycling mirror can happen in two different ways, either as resonant enhancement of signal sidebands, as in the case of signal recycling without arm cavities, or as resonant de-enhancement, also called resonant sideband extraction (RSE). In RSE mode the signal recycling mirror (or RSE mirror, as it may be called in this case) serves to effectively lower the arm cavity finesse, but only for the signal sidebands, not for the carrier field.

Noise Sources and Noise Reduction Strategies As outlined above, the interferometers exhibit a particular frequency-dependent response to an incident gravitational wave, which, in short, we refer to as a signal. As in any measurement, the sought-after signal has to compete with noise sources inherent to the particular measurement undertaken, and we therefore want to maximize the signal-to-noise ratio of the measurement. While optical cavities can be used to maximize the signal, we turn our attention here to the noise sources and the techniques we use to minimize them. The sources of noise that contaminate the detector’s output may be loosely grouped into two categories: displacement noise and sensing noise. Displacement noises are those that create real motion of mirror surfaces. At very low frequencies, seismic noise and the related gravity gradient noise are the dominant displacement noises. In the low- to mid-frequency range, thermal motion of the dielectric coatings of the mirrors and of their suspensions dominate, as does quantum radiation pressure noise. Sensing noises are those that arise during the process of measuring the positions of the mirrors. The primary sensing noise is shot noise, which, in a semiclassical picture, arises from the Poisson statistics of photon arrival time at the photodetector. Figure 6 depicts a typical noise plot showing the inverse of the signal-tonoise ratio, which may be calibrated to displacement or strain. Here we highlight the contributions of some of the most fundamental noise sources for a modern gravitational-wave detector in comparison to the total measured displacement noise. The noises are treated as independent of one another and thus are added in quadrature when modeling their total contribution. One will note that the fundamental noises do not fully account for the total measured noise. This is because there are also technical noise sources, such as the angular control noise highlighted in Fig. 6. The subsections that follow describe each of the primary types of noise sources, both fundamental and technical.

Quantum Noise Quantum noise is the primary limiting noise source for most of the sensitive band of gravitational-wave detectors. At frequencies above about 50 Hz, the dominant

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form of quantum noise is shot noise, the manifestation of Poisson statistics for the counting of detected photons. Shot noise is thus a sensing noise, and it limits all position measurements that use classical (i.e., not quantum-manipulated) light. At frequencies below about 50 Hz, quantum noise manifests as radiation pressure noise, where momentum transfer from all photons recoiling from the freely suspended test masses gives rise to position noise. √ Both shot noise and radiation pressure noise scale with n for the number of photons n. In order to increase the signal-to-shot noise ratio, increasing laser power in the interferometer is a viable concept because the gravitational-wave signal is proportional to n, and thus increases more than the shot noise, which is proportional √ to n. An increase in laser power, however, leads to an increase in radiation pressure noise at low frequencies, which acts as pure displacement noise. To keep the radiation pressure effect at bay, one can increase the inertial mass of the test masses up to limits imposed by the materials and fabrication techniques. Given a fixed test mass, there is then an optimum power to use for any given gravitationalwave target frequency, since shot- and radiation pressure noise are inversely linked by the laser power in the interferometer.

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At a deeper level, both shot- and radiation pressure noise can be understood as the result of the vacuum fluctuations of the electromagnetic field entering the interferometer from the output port. They are therefore subsumed under the common label of quantum noise. In this description, the randomly fluctuating vacuum field beats with the laser carrier field at the output upon detection to manifest as shot noise and at the test mass to manifest as radiation pressure noise. In pointing this out in a classic paper [22], Carlton Caves also realized that in reducing the vacuum fluctuations entering the interferometer, one can reduce one of these noises at the expense of the other. This can be realized through the use of squeezed vacuum states of light, which have been implemented in GEO 600 since 2010 [23], and as a permanent addition in LIGO and Virgo since 2019 [24, 25]. The two quadratures of the electromagnetic vacuum field, which can be labeled as amplitude and phase, are not independent, and their product has to obey Heisenberg’s uncertainty principle. Since the amplitude and phase uncertainties relate to radiation pressure and shot noise, respectively, they are relevant at different frequency regions. By reflecting the squeezed beam off a suitable optical resonator, one can achieve a low noise level in the phase quadrature for high frequencies and in the amplitude quadrature at low frequencies. As a result, these so-called filter cavities enable a reduction of quantum noise across the entire frequency band of a gravitational-wave detector [26]. For a deeper consideration of quantum noise, the reader is referred to the dedicated chapter within this handbook.

Thermal Noise Thermal noise in terrestrial laser interferometers is particularly important in the middle of the sensitivity band for terrestrial gravitational-wave detectors and has been subject to decades of research. It is relevant not only to gravitational-wave detectors but also to other precision metrology devices where cavity-stabilized lasers are exploited. However, the requirements of gravitational-wave detectors have pushed the understanding of the theory and the development of thermal noise mitigation techniques. In general, thermally driven fluctuations of mechanical systems can be derived from the application of the fluctuation-dissipation (F-D) theorem [27]. The F-D theorem provides a link between the fluctuating motion of a system in equilibrium and the dissipative (real) part of the admittance (the inverse of the impedance). An early approach to the estimation of thermal noise in mechanical systems of gravitational-wave detectors was to consider them in terms of their vibrational modes, treating each mode as a simple damped harmonic oscillator [28]. For a terrestrial laser interferometer, it is the thermal noise affecting the surface positions of the test masses that matters most, critical for the measurement of gravitational waves. In this case, surface position fluctuations arise from the internal vibrational modes of the mirrors, the vibrational modes of the fibers suspending the mirrors (also called violin modes), and the pendulum motion of the suspended mirrors. The uncorrelated sum of noises associated with each of these degrees of freedom constitutes the total thermal noise.

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It was later understood, however, that the approach of independently treating and adding the noises of all degrees of freedom assumes spatially uniform dissipation mechanisms. In particular, for the optically coated mirrors of a gravitational-wave detector, this condition proved to be too much of an idealization, such that other methods had to be found in order to estimate thermal noise accurately. Today, the most prevalent method in use for gravitational-wave detectors was developed by Yuri Levin [29], in which an oscillatory pressure resembling the spatial profile of the laser beam is imagined to be applied to the front surface of a mirror. The average power dissipated in the mirror due to this pressure then gives rise to mirror surface position fluctuations, again by applying the F-D theorem. There are different manifestations of thermal noise to be considered. In general, the term Brownian thermal noise describes noise arising from internal friction, which can come, for instance, from material defects. Internal friction within materials leads to damping of the harmonic oscillator modes, resulting in an increase of motion at the off-resonant frequencies of each mode. As a consequence, ultralowloss materials have to be used to minimize this motion, and the resonant frequencies (which still exhibit the highest motion) need to be shifted out of the observation band wherever possible. Another class of thermal noise stems from thermoelastic dissipation, caused by temperature fluctuations or temperature gradients. When a suspension fiber bends, a temperature gradient at the bended flexure develops, which in turn leads to a dissipative heat flow that generates mechanical noise via the F-D theorem. Statistical fluctuations of temperature, as caused by heat dissipation, for example, also cause mechanical displacement fluctuation via the linear thermal expansion coefficient of materials. Both of these processes are subsumed as thermoelastic noise. Temperature fluctuations can also lead to refractive index fluctuations. This effect is called thermorefractive noise. Thermoelastic and thermorefractive noises are particularly important for the optical coatings on the mirrors that provide the high reflectivity required for the test masses. They can be categorized together as thermooptic noise, and they partially cancel each other out when treated coherently [30]. Despite this partial cancellation, coating thermal noise is still the most relevant and limiting thermal noise source in current room-temperature detectors such as Advanced LIGO and Advanced Virgo. Research is ongoing to identify improved coating materials and designs and also to identify coating options for other test mass materials in addition to cryogenic operation.

Seismic and Gravity Gradient Noise Seismic ground vibrations limit the performance of all terrestrial interferometric gravitational-wave detectors. They are a dominant noise source at frequencies below about 20 Hz, and when they are particularly large, they can cause a loss of resonance of the Fabry-Perot cavities and significantly reduce the detectors’ observing time. Seismic noise predominately originates from ocean and ground water dynamics, earthquakes, wind, and human-induced activities, as well as slow gravity drifts and the atmosphere. These ground and atmospheric disturbances couple to the test

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masses in two ways: by mechanically moving the mirror suspensions and by direct gravitational attraction. Figure 7 shows the typical amplitude of ground displacements as a function of frequency at each of the advanced detector sites. The influence of these ground disturbances on the test masses is suppressed through the use of innovative vibration attenuation systems which decrease seismic noise by up to more than twelve orders of magnitude at frequencies above 10 Hz. These passive and active seismic isolation systems are described in Section “Seismic Isolation.” The direct coupling to the test masses of density fluctuations in the atmosphere and in the ground is known as gravity gradient noise or Newtonian noise. Although gravity gradient noise cannot be directly attenuated, accurate models can be used to subtract its effect from the gravitational-wave detector output. Early analytical estimates of gravity gradient noise can be found in Weiss’s and Saulson’s work [6, 32], and the most recent modeling is based on information from seismic surveys with large sensor arrays placed on the surface of the ground. Simulations on how to optimize the sensor array configuration to achieve a more accurate estimate of gravity gradient noise are an ongoing area of research. Mitigation techniques for seismic noise such as the vibration isolation systems and gravity gradient noise subtraction need not be so heavily relied upon if the detector sites themselves could be carefully selected (which is important for

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potential future detectors as discussed in Section “Outlook”) and if care is taken to minimize machine-induced vibrations, infrasound, or acoustics from turbo pumps, HVAC systems, water lines, etc. If one has the opportunity to visit a site, you may well be impressed by just how still it is inside the instrument halls that house the main interferometer components. For a more in-depth review of environmental noise couplings to terrestrial laser interferometers, we refer the reader to the chapter on the subject within this volume.

Noise from Technical Constraints Although the more fundamental noises (quantum noise, thermal noise, and seismic noise) place ultimate constraints on detector sensitivity, a plethora of technical noises often limit what can actually be achieved in some frequency bands, and a significant portion of interferometer commissioning is dedicated to reducing them. What we call technical noises, in loose distinction to fundamental noises, is everything that arises from the realities and imperfections of the technologies used. Imperfect sensors, the need for control systems, thermal effects, and crosscouplings of different kinds all create pathways for either displacement or sensing noises. Describing each specific noise source (as appears in the complete noise budget shown in Fig. 11) is not within the scope of this chapter, but we present here examples of some of the means by which technical noises come about: • Feedback loops (see Section “Feedback Control”) can impress the intrinsic noise of auxiliary sensors on the gravitational-wave readout channel. Even the best current efforts to suppress test mass motion locally with seismic preisolation systems leave the test masses with low-frequency motion around 1 Hz or below, which is too large for interferometer operation. It therefore needs to be suppressed further by alignment feedback, using signals derived from the global operation of the interferometer. These feedback loops currently need a minimum bandwidth of a few Hz, which results in imprinting additional noise that stems from the sensing (shot) noise of these auxiliary sensors. Constraints on the stability of these feedback loops constrain the ability to filter this noise contribution from the gravitational-wave readout. • Cross-coupling of degrees of freedom can be significant for several subsystems because it provides a multitude of pathways for noise couplings. At the highest level, some of the technical noise sources that affect the strain sensitivity stem from sensing noise of auxiliary length degrees of freedom (length of the power- and signal recycling cavities, for instance) that cross-couple to the differential arm length. At a lower level, within some subsystems, the hardware itself can have mechanical cross-couplings. This shows up, for instance, in the multistage test mass suspension systems, where cross-couplings between mechanical degrees of freedom cause some degrees of freedom, which may be less well isolated from ground motion, to add noise to others that are more relevant for the longitudinal test mass motion. Sensors themselves may also be imperfect in distinguishing degrees of freedom. Seismic acceleration sensors, for

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instance, are plagued by cross-coupling from tilt motion of the ground, which is currently a limitation to the performance of seismic pre-isolation systems. Crosscoupling terms can be linear, bilinear, or nonlinear. A form of bilinear coupling (involving two linear coupling parameters that multiply) is the residual motion of laser beam spot position on a test mass, which, when combined with test mass angular motion, creates a change in the longitudinal position of the test mass as sensed by the laser beam. • Scattered light can be generated by unevenness or dust contamination on the test mass surfaces. If this vagabonding light finds its way back into the main beam of the interferometer, it inserts a small additional phase shift, contaminating the gravitational-wave measurement in the form of sensing noise. Scattered light can also disturb the position of the test masses through the radiation pressure it exerts, which acts as a displacement noise. Substantial effort goes into the design and implementation of baffles to absorb scattered light before it does any harm and into further improving the optics to reduce the generation of scattered light. • High power in the interferometer is technically very challenging to achieve because laser light absorbed by the optics creates thermal distortions, which in turn produce higher-order spatial modes of the laser beam. These higher-order modes can deteriorate the interferometer performance by creating new crosscoupling paths or contaminating sensing signals. Thermal compensation systems have been developed to mitigate this problem and push the boundary of highpower application.

Some Enabling Technologies While noise sources set fundamental or technical limits on obtainable sensitivity for a given detector design, enabling technologies are those that are required to construct an instrument of that sensitivity in the first place. Here we briefly introduce some of these enabling technologies.

Lasers The development of the laser in the 1960s enabled decisive progress in interferometry. Because of their unique design, lasers produce very-high-intensity light, emitted as a bundled beam in one direction. In addition, laser light is monochromatic to a very high degree, which means it has a very well-defined wavelength, making it perfect for use in interferometers. Simply put, the more monochromatic the light, the easier it is to accurately measure its phase, which is the purpose of an interferometer. In the 1980s and early 1990s, the most ubiquitous type of laser used for the research on gravitational-wave detectors was the argon ion gas laser. While output powers of up to 10 W were available, these lasers were not operationally reliable over longer periods of time, as was required for gravitational-wave observatories. Substantial progress was made with the development of the nonplanar ring oscillator (NPRO) [33], a core element of laser-diode-pumped solid-state lasers. The NPRO design provides high reliability as well as very low intrinsic frequency and

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amplitude noise. For the widely used wavelength of 1064 nm, neodymium-doped yttrium aluminum garnet (Nd:YAG) crystals are used as the lasing medium. Driven by the needs for higher power to reduce shot noise, as described in Section “Quantum Noise,” the NPRO-based lasers have been extended with single-pass amplifiers and/or injection lock cavity-based amplifiers to yield output powers of up to 200 W or more [11]. While the intrinsic noise properties of the NPRO are excellent with respect to other lasers, the requirements for their use in gravitational-wave detectors demand active pre-stabilization techniques. Active feedback control is mandatory for stabilizing the frequency and amplitude of the laser light, and an optical feedback-controlled resonator (pre-mode-cleaner) is used to improve the spacial purity of the laser mode. Further stabilization of frequency and amplitude noise is achieved by nested feedback control, using sensors within (or intrinsic to) the main interferometer [34]. In particular, the km-long arms of a gravitational-wave detector with their seismically isolated mirrors provide the best possible frequency reference within the gravitational-wave frequency band of interest. The stabilized lasers thus achieve frequency noises down to micro-Hertz, 20 orders of magnitude below the lasing frequency. Research is ongoing for the use of fiber-based laser amplifiers and configurations of ever higher output power, as required for future gravitational-wave detectors.

Vacuum Systems Gravitational-wave detectors feature comprehensive ultrahigh vacuum (UHV) systems which serve several purposes. First, fluctuations in the effective refractive index of residual gas can mask or imitate the expected gravitational-wave signals. A high enough vacuum can reduce this phase noise from residual gas density fluctuations along the beam path to an acceptable level. Moreover, the vacuum environment isolates the test masses and other optical elements from acoustic noise and reduces test mass motion excitation due to residual gas fluctuations. It reduces gas damping in the mirror suspensions, leading to lower-suspension thermal noise, and contributes to the thermal isolation of the test masses and their support structures. Finally, the UHV environment contributes to preserving the cleanliness of optical elements. The vacuum systems of gravitational-wave detectors are composed of two basic elements: chambers and beam tubes. The vacuum chambers house hardware including electrical, mechanical, and optical systems. The chambers are equipped with pumping stations and instrumentation for vacuum measurement and contain vacuum isolation valves and access doors as frequent access is required. The beam tubes connect the chambers over kilometer-long distances. Excellent vacuum must be maintained, and the beam tubes are never vented. Figure 8 shows the beam tubes of the LIGO Hanford detector emanating out from vacuum chambers that house the optics. So-called cryolinks are used to connect the beam tubes to the corner stations. The cryolinks are big cryo-pumps for pumping water vapor and allow the interferometers

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Fig. 8 The laser and vacuum equipment area at the corner station of the LIGO Hanford Observatory. (Photo courtesy: Caltech/MIT/LIGO Lab)

to obtain pressures as low as several 10−10 mbar. Magnetically levitated turbomolecular pumps are employed for initial evacuation only, while ion pumps assisted by non-evaporable getters are used in normal operation. No rotating or vibrating machinery is permitted in the vicinity of the test masses during interferometer operation. While the construction of the vacuum chambers is reasonably conventional, this is not the case for the beam tubes, which are highly customized. The impressive engineering achievement of constructing four 4-km-long stainless steel tubes for LIGO – among the largest UHV systems in the world – required factory-like plants to be constructed at each of the Hanford and Livingston sites. The LIGO vacuum tubes were produced by coil spiral-welding steel from rolls to tubes 1.2 m in diameter and 16 m long. Each 16 m section was cleaned and leak-checked and an FTIR analysis carried out to confirm that the tubes were free from hydrocarbons. The sections were then butt-welded together in the field using a traveling clean room, yielding over 50 linear km of weld. The raw stock material, stainless steel 304 L of 3.2 mm thickness, was air baked for 36 hours at a temperature of 455◦ C, resulting in a final hydrogen out-gassing rate of L/2 is independent of the spacecraft distance. In this regime, longer arms increase the displacement δl = hL, but as the shot noise limit scales with P −1/2 ∝ L, the strain sensitivity does not change. When λGW becomes comparable to, or shorter than, the arm length, the effect of the gravitational wave averages away following a sinc function with zeros at λGW = L/2n (n ∈ N) for optimally aligned observatories. Averaging over all angles will wash out these zeros and give rise to the “wiggles” in LISA’s sensitivity curve at higher frequencies. As shown in Fig. 3, the current LISA design uses a telescope diameter of D = 0.3 m to send a PT = 2 W laser beam of wavelength λ = 1064 nm to an identical telescope on the receiving spacecraft. If we assume an efficiency of η ≈ 0.5

Fig. 3 Due to diffraction, the laser beam will have grown in radius ωrec by several orders of magnitude before it reaches the far spacecraft. The final size depends on the initial Gaussian beam radius ω0 = 0.446 D where D is the diameter of the telescope, λ is the wavelength, and L is the distance. Shown here are the current LISA design parameters (λ = 1064 nm)

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which takes unavoidable optical losses, quantum efficiencies, contrast defects, and also power lost to additional frequency components modulated onto the field in the final beat signal into account, the effective received power in the relevant frequency component is then Prec

 η   D 4  2.5 Gm 2  1064 nm 2  P  T ≈ 570 pW 0.5 0.3 m L λ 2W

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√ leading to a shot noise limit of δ l˜SN (f ) ≈ 3 pm/ Hz. Shot noise is neither the only limitation of the interferometric measurement system nor would the system even remotely be able to detect gravitational waves without subtracting several other contributions to the beat signals. Laser frequency noise was already added in Equation 2. The historical Michelson interferometer used equal arm lengths (L1 = L2 ) and a single thermal light source (δω1 = δω2 ) to become insensitive to frequency variations. Ground-based observatories use nearequal arm lengths and a single laser whose frequency is stabilized to the average or common arm of the interferometer. As discussed before, in LISA, the arm lengths are constantly changing and will reach macroscopic length differences of up to 35,000 km which would place unrealistic demands on the levels of laser frequency noise needed. Still, LISA will use state-of-the-art laser frequency stabilization √ systems and is expected to reach a laser frequency noise floor of below 30 Hz/ Hz above 2 mHz, increasing with f −2 below 2 mHz. This is about eight orders of magnitude too high for a direct strain measurement. Instead, LISA will eliminate laser frequency noise in post-processing using time-delay interferometry (TDI) to form a quasi-equal arm Michelson interferometer signal based on the knowledge of the light travel time between the spacecraft [99, 103, 180, 215, 217]. Laser ranging will be used to measure the distances between the spacecraft with sub-meter accuracy which then leads to an apparent length noise caused by laser frequency noise of δ l˜
7, except when delay times are long and mergers occur at later times. The number of events observed in the heavy-seed models is largely independent of the exact configuration of the space-based detector, since these events are so loud, but some light-seed

Fig. 13 Contours of constant signal-to-noise ratio for observations of MBH mergers with mass ratio 1 : 5 with the space-based detector LISA, as a function of the redshifted total mass of the binary (horizontal axis) and distance/redshift (vertical axis). Space-based gravitational wave detectors will be able to observed MBH mergers of the right mass to very high redshift. Stars indicate points for which LISA mission requirements were set in order to ensure detection of these sources

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models predict many light mergers at high redshift, so a more sensitive detector can detect significantly more of these, with important astrophysical implications [154]. Space-based gravitational wave observations will constrain the intrinsic parameters of observed MBH binaries to very high precision, driven by the fact that the signals can be observed with very high signal-to-noise ratio and for many thousands of cycles of phase evolution. The (redshifted) masses of the binary components can be determined to a precision better than 1% for about half of the observed events, and the spin magnitude of the primary (secondary) black hole measured to 1% (10%) for about ten percent of the observed events [154]. A significant fraction of systems will have the primary spin magnitude and the misalignment of the spin of the binary components with respect to the orbital plane of the binary constrained to the percent level. The fraction of observed systems for which this is possible could be as high as ∼25%, but this is dependent on the details of the spin distribution of astrophysical MBHs, which is extremely uncertain. A few events per year should also have well-measured spins for the remnant black hole created during the merger [154]. The typical expected precision of sky localization is tens of square degrees, and that of luminosity distance is tens of percent. However, as many as a few tens of events per year at z < 5 could be localized to better than 10 square degrees and 10% in distance [154]. These well-localized events at lower redshift are good targets for follow-up electromagnetic observations, to identify any counterparts. It is also expected that there will be at least one event, and perhaps a few tens, that will be at high redshift, z > 7, and have distance measured to ∼30% [154]. These events are important as the distance precision is enough that we will be sure they are at high redshift and hence provide important constraints on models of MBH formation and evolution.

Extreme Mass Ratio Inspirals The MBHs in the centers of galaxies that were described in the previous section are typically surrounded by clusters of stars. Stars in these clusters follow the usual evolutionary path, leading to the eventual formation of a compact remnant, which will be a black hole, neutron star, or white dwarf, depending on the mass and the metallicity of the original star. These galacto-centric stellar clusters are dense, and the stars within them undergo frequent encounters which can leave these compact objects on orbits that pass very close to the central MBH. Such objects can get captured onto orbits bound to the central MBH and then gradually inspiral into the MBH via emission of gravitational waves [21]. Typically the ratio of the mass of the stellar-origin compact object that is falling into the MBH to the mass of the MBH is ∼10−5 , so these events are called extreme mass ratio inspirals or EMRIs. Over the past two decades, observations of the stellar cluster around the black hole in the center of the Milky Way have revealed a number of unexpected features [128, 130], indicating that the physics of stellar clusters around MBHs is poorly understood. EMRI observations will explore a much larger sample of these stellar environments in the universe. In addition, EMRIs offer an exciting new way to probe

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fundamental physics. Due to the extreme mass ratio, each EMRI emits detectable gravitational waves for hundreds of thousands of waveform cycles, while the small object is in the strong gravitational field close to the central MBH. The emitted gravitational waves encode a map of the spacetime structure that can be used to test general relativity [125]. This will be discussed further in section “Testing the Nature of Black Holes”. The capture scenario for the formation of an EMRI described above is the “standard” formation channel [141, 207] and leads to EMRIs that have moderate eccentricity (∼0.2) at plunge and are on orbits that are inclined with respect to the orbital plane of the central MBH. However, a number of alternative scenarios have also been suggested. Binary stars in the vicinity of the MBH can come close enough to the MBH to undergo a three-body interaction that splits the binary and leaves one component bound to the MBH [178]. Massive stars that similarly come close to the MBH can have their outer envelope stripped, leaving the white dwarf core bound to the central objects [98]. In both these scenarios, the compact object is captured with random inclination, but sufficiently far from the central MBH that the orbit will have circularized before the object enters the band of space-based detectors as an EMRI. A final alternative scenario is the formation of compact objects in an accretion disc around a MBH. In this scenario, parts of the disc collapse to form massive stars which then evolve as normal and leave compact remnants in orbits around the MBH that eventually inspiral as EMRIs [161]. As in the previous two scenarios, EMRIs formed in this way are predicted to be on circular orbits, but now also in the equatorial plane of the MBH. The relative importance of these various scenarios in the universe is currently unknown, but gravitational wave observations could elucidate the different channels through measurements of the orbital properties of the objects. We refer the reader to [21] and references therein for more details. The complexity of the physics of stellar clusters means that the rate at which EMRIs occur in the universe is very uncertain. Of particular importance is the fraction of compact object captures that lead to gradual inspiral into the MBH, which would be observable as EMRIs, versus those that plunge directly into the MBH, the poorly known scaling of the EMRI rate with MBH mass, and the uncertain number of MBHs in the range relevant to space-based gravitational wave detectors. The impact of these various uncertainties was extensively explored in [33], where it was shown that the number of EMRIs observed by LISA could be anywhere between 1 and several thousand per year. The most pessimistic and the most optimistic models were deliberately chosen to be extreme, but the more reasonable models spanned a range from a few tens to almost a thousand events per year. These rates assume that a signal-to-noise ratio of ∼20 is required for the detection of an EMRI, which is somewhat larger than the ∼8 that is typically assumed for MBH binaries and other sources. This is driven by the expected complexity of the very long EMRI waveforms, which means that the number of independent waveform templates across the parameter space is very large [120]. Preliminary results from the Mock LISA Data Challenges suggest that this threshold might be pessimistic [32, 34], but those uncertainties are negligible compared to the much greater astrophysical uncertainties. If the rate of EMRI events is at the high end

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of predictions, then in additional to these individually resolvable EMRI events, there could be a stochastic foreground generated by the population of unresolved EMRIs, similar to the expected foreground from ultra-compact binaries in the Milky Way [38, 63]. Due to the eccentricity and inclination of the orbits, EMRI waveforms show a very rich structure that is a superposition of the orbital frequency and precession frequencies of the periapse and orbital plane. This is illustrated in Fig. 14. This complexity, combined with the long duration and hence large number of cycles

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observed for a typical EMRI, facilitates extremely accurate measurements of the parameters of the source. Using a Fisher matrix approach, it was shown that a single EMRI observation will typically provide estimates of the masses of both components, the spin of the MBH and the eccentricity of the orbit to fractional accuracies of ∼10−6 –10−5 [33]. The location of the EMRI on the sky can be determined to better than 10 square degrees in most cases and less than a square degree in a good fraction of cases. The luminosity distance of the EMRI will typically be measured to a precision of ∼1%–10%. If the MBH has near-extremal spin, i.e., the rotation rate is close to the maximum allowed value of 1, the spin measurement improves by another order of magnitude [73], allowing the confident identification of near-extremal systems if they exist. These precise measurements have been verified through Bayesian posterior estimation within the context of the Mock LISA Data Challenges [32, 34] and have important implications for science with EMRIs, which will be discussed in section “Science with Space-Based Observatories”. We conclude this section by mentioning a couple of related sources which could also be observed with space-based gravitational wave detectors. In the standard EMRI formation picture, the inspiralling object begins on a highly eccentric orbit, with periapse quite close to the MBH. Until the source has inspiralled sufficiently to be radiating continually in the millihertz GW band, the periapse remains approximately fixed, while the apoapse decays. The emission in this period is characterized by periodic bursts of gravitational waves emitted each time the compact object passes the MBH. These GW bursts from systems in this early stage of inspiral in the center of the Milky Way [50], or nearby galaxies [51], could potentially be seen by a space-based detector, but this is dependent on the properties of the MBHs in the local universe and on the astrophysical EMRI event rate. Another related source has been termed an extremely large mass ratio inspiral or “XMRI” [18]. These are the inspirals of brown dwarfs, which have mass a few hundredths of a solar mass. Brown dwarfs are more abundant than compact objects in galactic centers and inspiral more slowly, so there could be many of these in the process of inspiral at any given time. These are only detectable in the local universe, and the exact number that will be observed is highly uncertain, but a space-based detector like LISA could observe O(10) signals from such systems.

Cosmological Sources Processes occurring at high energies in the early universe can generate stochastic backgrounds of gravitational waves. A detection of this radiation would be of great importance in understanding early universe cosmology, since it can freely propagate from earlier times than the cosmic microwave background, which is the earliest that can be probed with EM observations. The frequency of GWs generated by cosmological processes is determined by the horizon scale, and hence temperature, of the universe at the time of production and by the amount of expansion of the universe between the time of production and today. A number of physical processes

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have been proposed that could generate GWs in the 0.1–100 mHz band. These include cosmological first-order phase transitions at energy scales of 0.1 to 100 TeV. Such phase transitions happen when the plasma in the universe changes phase via bubble nucleation. These bubbles expand, perturb the plasma, and collide, creating in this way gravitational waves at close to the horizon scale [104, 143, 157, 229]. A detection of these gravitational waves with a space-based detector could provide constraints on new physics, such as self-coupling of the Higgs field, supersymmetry, or conformal dynamics [77, 78]. Another scenario is the existence of extra spacetime dimensions. At the TeV scale, the Hubble length is about 1 mm, so GWs on this scale could probe the dynamics of warped extra dimensions, as predicted in some string theory scenarios [144, 193]. The Planck scale is not far above the TeV scales in some braneworld scenarios, in which case space-based GW detectors could probe inflationary reheating [106, 113, 152]. GWs at millihertz frequencies could also be produced through the amplification of quantum vacuum fluctuations in some unconventional inflationary models, such as the pre-big bang and bouncing brane scenarios [45, 71, 72]. A final mechanism for producing stochastic gravitational waves in the millihertz backgrounds is through the interactions of cosmic string networks. Cosmic strings are topological defects created by phase transitions, initially on microscopic scales, which are then stretched to astronomical scales by cosmological expansion [60,91]. These strings can interact, forming cusps and loops that decay through emission of GWs. The emitted gravitational waves will form a background that is distinct from backgrounds generated via any other source, with nearly constant energy per logarithmic interval in frequency over many decades in frequency [60]. Space-based detectors are the most sensitive probes for these objects [29]. If strings are not too light, GW bursts from individual cosmic string cusps could also be detected, providing firm evidence of the cosmic string origin of the cusp. All of the scenarios outlined above are somewhat speculative, so there is no guarantee that a space-based GW detector will see a stochastic cosmological background. Nonetheless, if it did, the implications for the physics of the early universe would be profound. Detection of a stochastic background in a gravitational wave detector is somewhat different to detection of individual sources. It will rely on cross-correlation of two data channels with independent noise. For space-based interferometers like LISA, the TDI channels are noise-independent and so can be used for this purpose. The idea is that in the cross-correlation, the cosmological noise component combines constructively, while the instrumental noise does not. Detectability also depends on the shape of the stochastic background, since the stochastic signal is broadband and can be integrated over the range of sensitivity of the detector. The detectability of backgrounds of various shapes and for various specific scenarios was explored in detail in [29, 45, 75–78, 116]. Broadly speaking, a space-based detector like LISA would be able to detect any background which contains more than ∼10−5 of the closure energy density of the universe [20]. Broadband backgrounds with logarithmic energy density at 1 mHz in excess of a few ×10−14 should be detectable. For more detailed results under specific assumptions, we refer the reader to [29, 45, 75–78, 116].

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Science with Space-Based Observatories In this section, we will highlight some of the science applications of observations with space-based gravitational wave detectors, which range across astrophysics, cosmology, and fundamental physics.

Astrophysics Compact Binaries in the Milky Way The formation of ultra-compact binaries depends on various astrophysical processes, such as stellar formation and binary stellar evolution, including the poorly understood common envelope phase [148]. Characterizing the ultra-compact binary population will thus shed light on open astrophysical questions about the Milky Way stellar population. Currently only a few tens of ultra-compact binaries are known, and only a couple of these have periods shorter than 10 min [170]. A space-based gravitational wave detector such as LISA will discover several thousand additional systems, expanding the known population by two orders of magnitude and making a complete survey of such systems at the shortest periods. These observations will provide key insights into the total number of such systems and hence their merger rates. The observed distribution of individually resolved systems and the distribution of the unresolved population inferred from the modulation of the stochastic foreground will resolve the structure of the Milky Way, including the thin and thick disc, the halo, and globular clusters [13,156]. Gravitational waves provide a unique probe for this as they do not suffer from dust obscuration and can thus “peer through” the galactic center. Finally, joint observations of ultra-compact binaries with gravitational waves and electromagnetic observations at high signal-to-noise ratio will provide key insights into the complex physics of interacting binaries, including tidal interactions and mass transfer [206]. Stellar-Origin Black Hole Binaries By identifying the time of merger and approximate sky location of SOBHB systems well in advance of the event, space-based observatories can trigger follow-up observations with electromagnetic telescopes and ground-based GW detectors [201]. GW detectors are not pointable, so the pre-localization on the sky is not important, but pre-determination of time of coalescence would allow the ground-based detectors to avoid scheduling maintenance at the time of the event. Triggers to EM facilities would allow deep searches for associated EM emission both pre- and post-merger. Detection of any EM emission would reveal properties of the material in the vicinity of the SOBHB and hence shed light on the astrophysical environment of such systems. Detection of residual eccentricity in the binary would provide crucial clues as to the origin of such systems. SOBHBs could form in the field as the end point of isolated binary evolution, but could also form in the dense environments of

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globular clusters through dynamical capture, or in the vicinity of a MBH through Kozai-Lidov hardening of binaries created via mass segregation [24]. The residual eccentricity would be larger in the latter two cases, and could be detectable in an observation by a space-based detector, but would be too small by the time the source reached the band of ground-based detectors to be measurable. It was shown in [185,186] that just a handful of SOBHB observations with a space-based detector could identify binaries formed in the vicinity of a MBH, so this should certainly be possible. Several tens of systems would be needed to distinguish the isolated binary and dynamical capture scenarios, so this will only be possible if the number of SOBHB systems observed is at the high end of current ranges. The formation channels of SOBHBs are currently hotly debated, and the information obtained from space-based observatories could be crucial to resolving it [65].

Massive Black Hole Binaries As described in section “Massive Black Hole Binaries”, massive black holes are observed to exist very early in cosmic history, and it is generally assumed that these black holes start as seed black holes and then grow through mergers and accretion. However, there are a number of plausible models for the formation of those seeds that are consistent with current data. A space-based GW detector operating in the millihertz band will measure the masses and spins of MBHs in merging binaries out to very high redshift, directly probing the epoch of formation and early evolution of black holes. This epoch cannot be easily probed with EM observations, so GWs may provide unique insight into the nature of black hole seeds and their early growth through accretion. It was shown in [202] that LISA will be able to distinguish between a wide variety of seed black hole models and identify mixed populations, determining the mixture fraction up to a precision of ∼ ± 0.2 [19, 202]. In addition to the properties of the individual black holes, the number and redshift at which the MBH binaries are observed to merge encodes important astrophysical information. MBH mergers follow mergers between their host galaxies, so the merger distribution tracks mergers between galaxies and the early growth of structure [138, 155]. LISA observations of MBH mergers out to high redshift will provide indirect constraints on the rate of galaxy mergers in the early universe and the relative fraction of “major” or “minor” mergers, i.e., the fraction of mergers in which the MBHs have similar masses or not. Measurements of the spin of these black holes will provide clues to the nature of accretion in galaxy halos at early and later times [56]. The distribution of events in redshift will encode clues to the delay time between the galaxy merger and the MBH merger and hence the efficiency with which the MBH binary is brought to the center of the merged galaxy. Finally, if these binaries are observed to have significant residual eccentricity, it could suggest the presence of a third MBH in the vicinity of the binary, which can excite eccentricity through the Kozai-Lidov resonance [64]. The fraction of LISA mergers observed to occur in triple systems is another important clue to build up a picture of the early evolution of cosmic structure. The astrophysical impact of GW observations of MBH binaries will be significantly enhanced if multi-messenger observations are made [174]. Observing an

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EM counterpart to the GW event will provide complementary information about the material in the vicinity of the black hole. The environment of a black hole plays a key role in its evolution, driving spin, and mass growth through accretion and can play a role in driving the coalescence of MBH binaries [182]. While it is not certain that EM counterparts will be detected for any MBH binary, the discovery potential of multi-messenger observations is huge, as already demonstrated by joint observations of the binary neutron star merger GW170817 with ground-based detectors [3].

Extreme Mass Ratio Inspirals EMRI observations will probe MBHs of similar mass to MBH mergers, but the primary black holes in EMRI events will be “quiescent” MBHs at lower redshifts, rather than MBHs undergoing highly dynamical interactions during mergers. Comparing the properties of the quiescent MBH population to the dynamical MBH population will provide further clues to the evolution of the MBH population. The observation of EMRI events would provide constraints on the MBH population in the interval where EM observations are poor or missing [124]. EMRI observations could also constrain the occupation fraction of MBHs in low-mass galaxies without relying on accretion signatures [224]. EMRI measurements of black hole spins will constrain the spin distribution of low-mass black holes up to moderate redshift [33], providing a more complete census than can be obtained through, for example, accretion disc measurements, which are restricted to actively accreting MBHs, which are a minority. EMRI observations will also provide precise measurements of the masses and orbital properties of compact objects in galactic nuclei [33]. These observations will reveal the mass spectrum of stellar-origin black holes in galactic nuclei, which can be compared to the corresponding mass spectrum observed in SOBHBs, which is now being constrained by ground-based GW detectors [11]. Differences or similarities between the observed populations will shed further light on stellar evolution in different astrophysical environments and on the origin of the SOBHB population. The number of EMRI events observed as a function of black hole mass will encode information about mass segregation in galactic nuclei [22], while the observed eccentricity and inclination distributions provide direct constraints on the EMRI formation channel [21]. Taken together, EMRI observations will build up a comprehensive picture of the complex physical processes that govern the dynamics of stars in galactic nuclei [17]. EMRI-like systems in which the smaller object is an intermediate mass black hole (IMBH) of ∼102 –104 M could also be observed by space-based detectors [16]. These are often called intermediate mass ratio inspirals or IMRIs. Binaries of two IMBHs could also potentially be observed. The existence of IMBHs is not yet conclusively established observationally [134], but space-based GW detectors would provide mass measurements that are precise enough to robustly identify black holes that lie in the IMBH range. GW observations of such systems from space and from the ground thus offer a unique way to understand the astrophysics of these objects if they exist [123].

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Cosmology Probes of the Early Universe The detection of a stochastic background of gravitational waves and the measurement of its amplitude and slope would have profound implications for our understanding of the early universe. As described in section “Cosmological Sources”, there are a number of non-standard scenarios that could produce such a background, including first-order phase transitions [77, 78, 104, 143, 157, 229], warped extra dimensions [144, 193], inflationary reheating in braneworld scenarios [106, 113, 152], non-standard inflation including pre-big bang and bouncing brane scenarios [45, 71, 72], or cosmic string networks [60, 91]. The spectra generated under these various scenarios are distinct and can be constrained by space-based GW observations [29, 45, 75–78, 116], so if a background were to be detected, it would provide insight into this new physics. In addition, because the GW background is generated before Big Bang nucleosynthesis (BBN), it can probe earlier epochs than any that have been constrained so far, even indirectly. In cosmological models that differ from the standard model prior to BBN, the GW background spectrum can change dramatically, providing a smoking gun for new physics during that epoch [29]. Cosmography with Standard Sirens An important application of gravitational wave observations of SOBHBs, MBH binaries, and EMRIs is for cosmography, i.e., to probe the expansion of the universe over cosmological history. To probe the expansion of the universe, we need to measure the rate of expansion of the universe, characterized by redshift, as a function of distance, characterized by luminosity distance. The luminosity distanceredshift relation depends on the cosmological model and the matter and energy content of the universe and hence can be used to constrain these properties. Various EM sources, including type IA supernovae [195], have been used for this purpose. These sources are referred to as standard sirens, since the basis of the approach is to assume that the intrinsic luminosity of the source is known and hence the observed luminosity provides a measure of distance. Redshift can be measured directly from the shift in frequency of spectral lines. The notion of using GW sources as standard sirens for the same purpose was first suggested in [199]. As described earlier, the strain of a GW source scales with the ratio of its (redshifted) mass to its luminosity distance. The redshifted mass also impacts the GW phasing and so can typically be measured very accurately from the GW data, so the observed amplitude gives a direct measurement of the luminosity distance. This is appealing since, in contrast to EM probes of cosmology, these measurements do not need to be calibrated to the local distance ladder. However, GW observations do not provide direct measurements of redshift. If the GW event has an EM counterpart, the redshift can be obtained from the EM observation. This was exploited for GW170817, the first binary neutron star merger observed by ground-based interferometers [8], for which a kilonova counterpart

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was observed [3], enabling the first gravitational wave constraint on the Hubble constant [7]. For space-based detectors, the only source for which counterparts are thought to be possible are MBH mergers [174]. MBH mergers at low redshift, z ∼ 1–2, can be localized to a few square degrees, permitting searches for EM counterparts [154]. These sources will have luminosity distance measurements of ∼1%, so any event with an associated counterpart will provide a percent-level constraint on cosmological parameters. An EMRI in which the smaller object is a white dwarf and the larger object is a low-mass, rapidly spinning black hole could generate an observable counterpart when the white dwarf is tidally disrupted toward the end of the inspiral [169, 204]. However, the event rate of such systems is likely to be very low [21, 33]. In the absence of a counterpart, cosmological constraints can be obtained statistically by comparing the locations of observed GW events with catalogues of galaxy redshifts. This has also been done using observations with ground-based detectors [9, 114, 208]. It has been shown that this statistical approach could yield constraints on the Hubble constant at the level of 1%, if 20 EMRIs are observed at redshift z < 0.5 with a LISA-like space-based GW detector [168]. That analysis assumed somewhat optimistic EMRI localization volumes, but this will be partially compensated by the larger number of EMRI events predicted in current models [33]. Statistical cosmological constraints using observations of SOBHBs will achieve comparable precision on H0 , if the number of observed events is at the higher end of predictions [101, 159], while observations of MBH mergers will achieve slightly worse precision on the Hubble parameter, but will permit estimates of the matter content of the universe and the equation of state of dark energy [190]. There is a third approach to cosmology with gravitational wave sources, which is to use the GW measurement of the redshifted mass to estimate the redshift of the source. This can be done if assumptions are made about the distribution of masses of the observed signals. This was initially proposed in the context of observations of binary neutron star mergers with ground-based detectors, where it is justified by the narrow observed mass distribution of neutron stars in compact binaries [213, 214]. The mass distributions of EMRIs and MBHs are not expected to be sufficiently compact to permit interesting constraints in this way. However, the same procedure can be applied when the distribution has a sharp feature, such as the presence of the mass gap in SOBHBs. Exploiting this feature with observations of SOBHBs in future ground-based detectors could yield interesting cosmological constraints that are independent of all EM information [111], and so it is possible that something could also be done with SOBHB observations by space-based detectors. However, the lower expected number of events and evidence that the mass distribution is more complicated than a truncated power law [10, 11] suggest that, for space-based detectors, this approach will not be competitive with the counterpart or statistical approaches. To finish this section, we note that the cosmological constraints described here, although competitive with current EM constraints, will probably be surpassed by EM data obtained between now and the launch of LISA. These measurements are nonetheless interesting as they provide a completely independent verification of

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the EM results and are subject to a completely different set of systematic errors. In addition, space-based GW observations are one of the few approaches that can obtain constraints over a wide range of redshifts, probing redshift values that cannot be measured by electromagnetic probes. This could be crucial for resolving current tensions between low redshift [194, 195] and high redshift [14] cosmological measurements.

Fundamental Physics Gravitational wave sources observed by both ground-based and space-based detectors can provide powerful tests of fundamental physics, i.e., whether the evolution of the binary and the observed gravitational wave emission are consistent with the predictions of general relativity. These tests are possible because GW observations provide very precise measurements of the waveform phase and hence can identify very small changes to the phasing arising from new physics. Observations with space-based detectors are particularly powerful, because the sources typically have very high signal-to-noise ratio and are also long-lived. There are three distinct types of tests of gravitational physics that have been proposed, which we now briefly summarize.

Elucidating Dark Matter Only ∼15% of the universe is composed of “normal” baryonic matter [212]. For decades, astronomers and particle physicists have been struggling to understand the nature of the other 85% of “dark matter” (DM). Observations with GW detectors will be able to shed light onto the nature of dark matter in a number of ways. Measurements of the distribution of masses and spins of MBHs can reveal the existence of DM due to the effect of DM interactions on these distributions (see [66] for a review and further references). If EMRIs or MBH mergers are taking place in an environment containing significant amounts of DM, this will impact the observed phasing of the emitted GWs in a measurable way [42]. The emitted waveforms can also be used to identify if the central MBH is in fact a self-gravitating DM structure [82, 164]. Clouds of ultra-light DM particles around spinning black holes can also generate GWs, either continuously or as bursts, that could be directly detected by GW detectors [67, 68]. Finally, DM interacting directly with the space-based interferometer could lead to measurable signatures [135, 183]. Testing the Foundations of the Gravitational Interaction Departures in the physics of gravity away from the predictions of general relativity can lead to differences in how gravitational waves are generated and how they propagate through the universe. These differences change the phasing of the GWs, which can be detected in observations with gravitational wave detectors. A large number of alternatives to general relativity have been proposed, each varying one or more of the physical assumptions that underlie GR. Examples of alternatives that lead to measurable deviations in gravitational waveforms include massive

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gravity theories [179, 228], the existence of large or compact extra spacetime dimensions [79, 105, 232], variation of Newton’s coupling constant over cosmic time [211, 237], violations of parity or chirality [15, 234], violations of Lorentz invariance [136, 238], violations of the equivalence principle [53, 231, 233], or the existence of additional scalar or vector fields that generate alternative polarization states for gravitational waves [218]. We refer the reader to [41, 52, 125] for reviews and further references. There are two basic approaches to constraining these physical effects with spacebased observations of GWs. The first is to construct model waveforms in these alternative physical scenarios and use them within the framework of Bayesian inference calculations to place constraints on the parameters that occur in these specific theories. The alternative approach is “model-free” in the sense that no reference is made to a specific alternative theory. Instead, generic modifications are made to the waveform models, which are then constrained with observations. Two different formalisms have been proposed in the literature. The first is to directly measure, or constrain modifications to, the post-Newtonian phase coefficients, i.e., the numerical factors multiplying different powers of the GW frequency in an expansion of the GW phase [28]. Space-based detectors should be able to identify departures from the GR values in observed MBH merger waveforms at the level of one tenth of a percent in the low-order coefficients. Alternatively, generic additional terms can be included in the amplitude or phase of the gravitational waveform, allowing deviations in not only the size of the terms but also their frequency dependence. This is termed the parameterized post-Einsteinian (ppE) formalism [236]. MBH merger observations with space-based detectors will be able to place constraints across a wide range of the ppE parameter space, with particularly strong constraints at higher orders in frequency which cannot be well constrained by EM observations of Newtonian binaries in the Newtonian regime [92, 188]. Similar tests are also possible with SOBHBs, by exploiting the possibility of observing the same system by both space-based and ground-based detectors. The two observations provide snapshots of the waveform at two different epochs. Small differences in the rate of inspiral evolution of the binary can lead to measurable changes in the time separation between the two observations. Multi-band SOBHB observations with LISA and ground-based detectors can provide constraints on various alternatives to general relativity that are much better than are currently available [188]. This includes a six orders of magnitude improvement in constraints on the emission of dipole radiation [43]; several orders of magnitude improvement in constraints on Brans-Dicke theory, Einstein-dilaton-Gauss-Bonnet gravity, and dynamical Chern-Simons gravity [132]; and an order of magnitude improvement in merger-ringdown consistency checks (see next section) [83]. To conclude this section, we will mention the memory effect. A prediction of general relativity is that after a merger, the spacetime will retain a permanent shift in its zero point and hence have a “memory” of the fact that a merger took place. There is both linear and nonlinear memory, and while the final spacetime offset is at zero frequency and hence unobservable, it is in principle possible to directly observe the buildup of the memory through the GW observation. Space-based detectors

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will measure the nonlinear memory buildup in MBH coalescences with sufficient significance to also test this aspect of gravitational theory [112, 147].

Testing the Nature of Black Holes Black holes in general relativity are completely characterized by two parameters – a mass and a spin. All higher “moments” in an expansion of the gravitational field are determined by these two parameters, and the resulting spacetime structure is described by the Kerr metric. This is often referred to as the “no-hair property.” If general relativity does not describe the structure of black holes, or one of the auxiliary assumptions that lead to the uniqueness of the Kerr metric, such as the energy conditions or the formation of a horizon, is violated, then the spacetime structure could have “hair” and deviate from the Kerr metric [82, 119, 125]. GW observations can be used to construct a map of the spacetime in the vicinity of a black hole and hence test the no-hair property. For space-based detectors, observations of EMRIs and of the ringdown signal following the merger of two MBHs provide the cleanest tests. For an EMRI observation, the information comes from tracking the waveform phase over many hundreds of thousands of waveform cycles. Small changes in the multipole structure lead to changes in the rate of inspiral of the binary that accumulate over the observation. This has been studied extensively, starting from a direct extraction of successive multipole moments described in [196]. More recent work has focused on the ability of space-based detectors to quantify the size of “bumps” away from the Kerr metric. Relevant early works include [40, 74, 87, 122, 131], but we refer the interested reader to [82, 125, 230] for a comprehensive summary of the literature. Figure 15 shows results, first reported in [33], on the precision with which EMRI observations can detect departures in the quadrupole moment, ΔQ, of the central MBH from the value predicted by its mass and spin. These results are shown for 12 different models of the astrophysical population of EMRIs, labelled M1–M12, and for 2 different models of the EMRI waveform, labelled “AK” and “NK.” The two EMRI waveforms both use the analytic kludge model described in [39], but differ in where the inspiral is terminated as the small object plunges into the MBH. Full details can be found in [33]. Departures at the level of 10−4 can be detected if they are present. Other information about the nature of the object that can be extracted from EMRI observations includes the presence of a horizon [151], the nature of the tidal coupling interaction [162], and the influence of perturbing matter [42] or nearby stars [235]. These effects should be distinguishable from those arising from differences in the nature of the central object. After a MBH merger, the remnant black hole that forms settles down from a highly perturbed state to a quiescent Kerr spacetime through a process called ringdown. The ringdown radiation is a superposition of damped sinusoids, the frequency and decay time of which are uniquely determined by the mass and spin of the remnant black hole. Observation of two or more ringdown modes allows a consistency check between the frequencies and damping times that directly tests the no-hair property of the remnant [54, 55]. Recently a framework for model-independent ringdown constraints, similar to the ppE formalism, has

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Fig. 15 Distributions of accuracy with which EMRI observations can constrain deviations in the quadrupole moment of the central MBH from the value predicted by the Kerr metric. Results are shown for various models of the EMRI population, as described in [33]. (Figure also reproduced from [33])

been developed [81, 171, 175]. Ringdown constraints probe a different regime to inspiral constraints and are therefore somewhat complementary. Certain types of modification to the spacetime structure are better probed by one approach or the other [57, 59]. A final approach to testing the nature of the black holes in GW observations is to look for consistency in the properties of the black holes inferred from the inspiral and those inferred from the ringdown [58,129]. Any differences between the observed properties of the merger and ringdown and those predicted from the inspiral using general relativity would reveal new physics during the highly dynamical merger phase. This approach has been applied to observations with ground-based detectors [6], so far revealing no evidence for deviations from the predictions of general relativity. A closely related idea is to look for additional signals, or “echoes,” in the data after an observed event. If they are seen, these echoes could indicate the existence of new physics near the horizon of black holes [80]. Claims have been made for evidence of these echoes in LIGO observations [12], but these are more likely to have been due to instrumental noise [227]. Future space-based detectors will shed further light onto this ongoing debate.

Prospects for Space-Based Observatories LISA will certainly not be the last space-based gravitational wave observatory. Missions beyond LISA have already been proposed in white papers and peerreviewed scientific journals since at least the first years of this century [48, 49, 95]. Several new concepts were submitted to the Decadal 2020 review in the USA [88] and ESA’s Voyage 2050 long-term plan [36, 37, 89, 200, 203]. They often target

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the frequency range between LISA and LIGO and are optimized between 1 mHz and 1 Hz [37, 200]. This frequency range is very interesting for mergers between intermediate mass black holes beyond LISA, for typical LIGO mergers at higher redshifts, and for transient signals passing from the LISA band into this band and then finally merging in the ground-based band. This frequency range might also be the most promising range for detecting the gravitational wave background radiation formed just after the Big Bang. Most of these proposals are loosely based on LISA technologies and will likely be limited by the same noise sources than LISA; one notable exception is atom interferometer-based observatories [107] which are outside of the scope of this chapter but are discussed in  Chap. 5 “Quantum Sensors with Matter Waves for GW Observation”. Here we want to look at ways to start the design of such missions, how basic mission parameters are used to derive target sensitivities, and glance over technological improvements that will be required to enable these missions. Figure 16 shows how acceleration noise δ a˜ and interferometer or sensing noise δ l˜ define the sensitivity expressed as a linear spectral density of the strain: "  # 2  2 1 δ a ˜ π Lf/c ˜ )∼ + δ l˜ h(f L sin (π Lf/c) 4π 2 f 2

(54) sky locations

Fig. 16 The generic sensitivity curve shows the two standard constituents, acceleration noise δ a˜ ˜ and how they limit the strain sensitivity. Improvements in acceleration and interferometer noise δ l, noise push the red curve down and improve the sensitivity at low frequencies; improvements in interferometer noise push the blue curve down and improve the sensitivity at high frequencies. Increasing the arm lengths pushes both curves to the left, reducing it to the right. Increasing the laser power or the diameter of the telescopes and decreasing the wavelength will reduce shot noise and δ l˜ as long as the sensing system allows shot noise-limited detection

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For simplicity and following most mission proposals, the acceleration noise and the interferometer noise are assumed to be frequency independent. As discussed before, frequency-independent or white acceleration noise causes a displacement of the test masses which scales with 1/f 2 and limits the performance at low frequencies. At higher frequencies, the periods of the gravitational waves become comparable to the light travel time in the arms, and the stretching and squeezing of spacetime compensate each other. This leads to a sinc function in the response of the instrument or an inverse sinc function in the strain sensitivity for otherwise optimally aligned gravitational waves when the propagation direction is normal to the constellation. The zeros in the response at fGW = nc/L wash out when the sensitivity is averaged over all possible sky locations which turns the sharp peaks of the sinc into these wiggles. The sweet spot is in the frequency range where acceleration noise and sensing noise become comparable and before the sinc function starts to matter: between approximately 3 and 30 mHz in this generic LISAlike sensitivity curve. Pushing this range to lower (higher) frequencies requires to lengthen (shorten) the arms, and, as long as all other basic parameters – laser power, telescope diameter, and laser wavelength – stay the same and sensing noise is dominated by shot noise, both curves move to the left (right) without changing their relative position. For shorter arms, the received laser power increases, and the shot noise limit will decrease. As shown in Equations 3 and 5, a shorter wavelength improves the displacement sensitivity for the same phase sensitivity. The light is also better collimated, and the amount of received light increases. However, as every photon is also more energetic, the number of received photons only increases linearly with the inverse wavelength and not quadratically. Therefore, the sensitivity scales with λ−3/2 if the laser power stays the same. Increases√in the laser power without changing the wavelength improve the sensitivity with P , while the diameter D of the telescope enters quadratically. Future mission proposals which plan to take advantage of the lower shot noise have to assume that the technology progresses enough that the interferometer measurement system continues to be limited by it and not by technical noise. While this seems to be overly optimistic, many technical noise sources are driven by temperature changes which will be significantly smaller at higher frequencies. The shorter arms will also reduce the dynamics within the constellation which reduces the Doppler shifts and potentially reduces the beat frequencies and the timing requirements within the phasemeter. It is expected that significant improvements in acceleration noise beyond what is shown in Equation 10 are more likely at higher frequencies than at lower frequencies. The reason is again temperature which rises faster than f −2 toward lower frequencies and residual spacecraft motion which couples gravitationally to the test mass and is also expected to be smaller at higher frequencies. Also voltage noise in actuators and capacitive sensors improves at higher frequencies, while frequency-independent noise sources typically scale with somewhat controllable environmental parameters such as pressure and absolute temperature. Many forces scale with the surface and not the volume of the test mass and can be reduced by increasing the mass of the test mass itself. Interferometric readouts can be

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significantly more sensitive than capacitive readouts. Employing those for all degrees of freedom would improve our ability to measure, calibrate, and subtract spacecraft motion. The ambitious goals spelled out in these proposals require a broad range of technical improvements across several decades in frequency space. They will not be easy to reach, but that has always been the case, and compared to 30 years ago, the community has a much better understanding of the challenges ahead of them. In the context of future mission proposals, DECIGO plays a somewhat special role [150]. It targets the 0.1 to 10 Hz frequency range with a total of 12 spacecraft to form 4 equilateral triangular clusters. All clusters are placed in a heliocentric orbit, the center of two of them are collocated but rotated by 60◦ deg with respect to each other. The others are placed 120◦ deg offset in heliocentric orbits as shown in Fig. 17. The Japanese project plans to form 106 m long optical cavities between free-falling 1-m-diameter mirrors. The higher measurement frequency allows the application of forces to the mirrors at lower frequencies that can be used for stationkeeping to keep the cavities on resonance with their interrogating laser beams. For this reason, DECIGO is often seen as a mission which brings ground-based technologies to space which might be needed for orders of magnitude improvements in displacement sensitivity. A last mission which we want to mention here is known as the Folkner mission [118] which was proposed to NASA during the SGO studies [133] 6 years prior to the LPF launch. In this mission, the three spacecraft are placed in heliocentric orbits separated by 120◦ deg, similar to the locations of the DECIGO clusters. The arm length of 250 Gm was expected to compensate for increased acceleration noise should LPF fail. Since LPF was successful and a similar GRS could be used for the Folkner mission, the entire sensitivity curve would (ideally) be shifted to the left and probe frequencies 100 times lower than current LISA. A similar proposal, μAres,

Fig. 17 DECIGO is one of the most ambitious proposed future missions. It uses 12 spacecraft which form 4 equilateral triangular clusters. Two clusters are collocated in their heliocentric orbits, while the two others are distributed around the sun as shown on the left. The right graph shows a conceptual design of a single constellation in which each arm is defined by two mirrors forming an optical cavity [150]

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was submitted to the Voyage 2050 call [203]. These missions would bridge the gap between pulsar timing and LISA for supermassive black hole mergers. Galactic ultra-compact binaries are also expected to create a gravitational wave background in this frequency range which such missions could study, but this background would somewhat limit the distance to which massive black hole binaries could be resolved.

Cross-References  LISA and the Galactic Population of Compact Binaries  Massive Black-Hole Mergers  Quantum Sensors with Matter Waves for GW Observation  Stochastic Gravitational Wave Backgrounds of Cosmological Origin  Terrestrial Laser Interferometers  Testing the Nature of Dark Compact Objects with Gravitational Waves  The Gravitational Capture of Compact Objects by Massive Black Holes

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4

Pulsar Timing Array Experiments J. P. W. Verbiest, S. Osłowski, and S. Burke-Spolaor

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radio Emission from Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsar Life Cycle and Spin Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsar Timing and Pulsar Timing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Template Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Template Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timing Model Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interstellar Propagation Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timing Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulsar Timing Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Waves and Other Correlated Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Sources in the PTA Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Present PTA Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recent and Ongoing Improvements in PTA Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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J. P. W. Verbiest () Fakultät für Physik, Universität Bielefeld, Bielefeld, Germany Max-Planck-Institut für Radioastronomie, Bonn, Germany e-mail: [email protected] S. Osłowski Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC, Australia S. Burke-Spolaor Department of Physics and Astronomy, West Virginia University, Morgantown, WV, USA Center for Gravitational Waves and Cosmology, West Virginia University, Morgantown, WV, USA Canadian Institute for Advanced Research, CIFAR Azrieli Global Scholar, Toronto, ON, Canada © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_4

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Pulsar Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IISM Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Pulsar timing is a technique that uses the highly stable spin periods of neutron stars to investigate a wide range of topics in physics and astrophysics. Pulsar timing arrays (PTAs) use sets of extremely well-timed pulsars as a galaxyscale detector with arms extending between Earth and each pulsar in the array. These challenging experiments look for correlated deviations in the pulsars’ timing that are caused by low-frequency gravitational waves (GWs) traversing our galaxy. PTAs are particularly sensitive to GWs at nanohertz frequencies, which makes them complementary to other space- and ground-based detectors. In this chapter, we will describe the methodology behind pulsar timing; provide an overview of the potential uses of PTAs; and summarize where current PTA-based detection efforts stand. Most predictions expect PTAs to successfully detect a cosmological background of GWs emitted by supermassive black hole binaries and also potentially detect continuous-wave emission from binary supermassive black holes, within the next several years.

Keywords

Pulsars · Pulsar timing · Timing array · Black holes

Introduction Neutron stars are the collapsed cores of massive stars that have undergone a supernova explosion after the end of nuclear burning and are supported from further collapse by neutron degeneracy pressure [12, 52, 114]. Since neutron stars are far more compact than their progenitor stars, they tend to exhibit very short rotational periods and extremely strong magnetic fields, as shown in Fig. 1. Generally the magnetic axis is not aligned with the spin axis, so magnetic dipole radiation that is created in the neutron star’s atmosphere is swept around in space, somewhat like the beam of a lighthouse (this is the so-called lighthouse model). Depending on the orientation of the beam and its width, Earth may fall within that radiation beam once per rotation – which then causes the neutron star to be detected as a source of pulsed radiation, otherwise referred to as a “pulsar.”

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1e-10

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Fig. 1 P − P˙ diagram for all pulsars known. Shown are the spin period and spin-period derivative for all pulsars included in the ATNF pulsar catalogue version 1.61 [105]. Solitary pulsars are shown as full dots, and pulsars in binary systems are shown as open circles. The gray dotted lines slanting downwards from the left represent surface magnetic field strengths of 1013 G, 1011 G, and 109 G from top to bottom; and the black dot-dashed lines slanting upwards to the right represent characteristic ages of 104 yr, 108 yr, and 1010 yr respectively, also from top to bottom

Radio Emission from Pulsars Following the lighthouse model described above, it is natural to expect that pulsars appear to the observer as so-called “pulse trains”: pulses of emission separated by a fixed period that equals the spin. These appear with a shape defined by the plasma properties in the pulsar’s magnetosphere, which can differ greatly from one pulsar to the next (see Fig. 2). The emission mechanism of pulsars is understood in broad terms [see, e.g., 109]. A few intriguing observational facts have been identified over the half century since the first pulsars were discovered. Most importantly, it has been shown that for most pulsars, the exact shape of individual pulses changes randomly from one period to the next. In contrast, however, the average shape of the pulsed emission is typically stable on timescales from minutes up to decades [56]. This implies that for any given pulsar, the pulsed emission can be averaged after accounting for the pulsar’s rotational period. The average pulse shape that can be obtained in this way is unique for the pulsar and can be thought of as a fingerprint. At

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Fig. 2 Pulse profile shapes vary across pulsars and observing frequencies. Shown are pulse profiles for two traditional PTA pulsars: PSR J0437−4715 (left column) and PSR J1022+1001 (right column); and for each pulsar this profile is shown at three different observing frequencies: at ∼ 726 MHz (top row), ∼ 1369 MHz (middle row), and ∼ 3100 MHz (bottom row). Total intensity profiles are shown in full black lines, linear polarization in dashed red lines, and circular polarization in blue dotted lines. The tendency of getting sharper profile shapes at higher frequencies causes the timing precision to increase at those frequencies, although generally the noise level also increases due to the steep spectrum of pulsars [20, 65]. Consequently, most pulsar timing to date has been carried out at intermediate frequencies around 1.4 GHz. Finally, while this figure only shows profiles for MSPs, the evolution with frequency has been shown to be far more extreme in the case of slow pulsars [79]. (These plots were made based on the public data published by Dai et al. [39] and contain hundreds of hours of observing time at the higher frequencies, causing the radiometer noise to be barely visible)

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radio wavelengths this average and reproducible pulse shape is typically called the pulse profile of the pulsar, whereas the term light curve is more common at higher frequencies (gamma and X-rays). The shape of the profile is defined by the emitting geometry in the pulsar magnetosphere. Since it is expected that the emission height is different for photons with different frequencies [36], it stands to reason that the shape of the pulse profile also typically differs with observing frequency (again, see Fig. 2). Not all pulsars have a stable pulse profile, and not all pulsars emit radiation all the time. Indeed, a veritable zoo of pulsar emission phenomena has been discovered, studied, and described throughout the years. There are so-called “nulling” pulsars [15], which often turn off, only to reappear at some point after. Some pulsars null for minutes on end, others for hours – some even turn off for months or years (at which point they are also called “intermittent” pulsars), suggesting an almost continuous distribution all the way up to so-called RRATs (rotating radio transients), which only sporadically emit one or several pulses of radiation [29, 108]. Another category of pulsar emission is displayed by the “moding” pulsars [14]. These do not have one characteristic fingerprint, but two or three – and they arbitrarily change between them: while one day their pulse profile may look one way, the next day it may look different, only to go back to its original state on day three. Moding can also have a wide range of possible timescales, from single pulses all the way up to months or years between mode changes. Finally, there are drifting pulsars [46]. These “drifters” also have a well-defined pulse period that is readily and repetitively measurable on timescales of minutes to hours, but on timescales of seconds that pulse period seems to be overestimated, as the pulse appears to come a bit too late after each rotation, only to “reset” after a few dozen rotations, leading to the more stable long-term periodicity. Luckily for pulsar astronomers, the nulling, moding, and drifting pulsars have turned out to be the exception rather than the rule, and the majority of pulsars manifest themselves as predictably repetitive pulses of emission that have arbitrary pulse shapes on timescales of seconds but stable and well-defined pulse shapes on timescales from minutes upwards.

Pulsar Life Cycle and Spin Properties Whereas the first few pulsars that were discovered appeared to be a fairly homogeneous group of objects, extensive surveys have continuously expanded the parameter space in which pulsars have been discovered; and consequently a wide variety of pulsar types is now known. After the formative supernova explosion, pulsars start their new life as so-called “young” pulsars with spin periods of a few tens of milliseconds and a magnetic field strength of order 1012 G at their surface [24, 132]. The emission of magnetic dipole radiation does make them lose angular momentum, and consequently their rotation slows down gradually, typically by about 10−13 seconds per second. The youngest and most well-known example of this class of pulsars is the Crab pulsar which was

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formed in a supernova explosion in 1054 CE [97, 136], which has been recorded by several civilizations across the world. After the first few thousand years, the spin period of young pulsars has slowed down sufficiently to have an appreciable impact on the spindown itself, which slows down their evolution. At this point they turn into the first discovered type of pulsar: the so-called “slow” pulsars or “canonical” pulsars. These are pulsars with rotation periods between about a tenth of a second and roughly ten seconds. Their magnetic fields are thought to have strengths of roughly 1010 G to 1013 G, and they are expected to have formed thousands to hundreds of millions of years ago (see Fig. 1). Their spin period gets longer by about 10−14 s every second, as they lose angular momentum slowly but steadily. This loss in angular momentum causes them to eventually rotate too slowly to produce detectable amounts of radio waves, and so after about a billion years they “turn off” and become undetectable [125]. Most of the slow pulsars are solitary objects that were either born as solitary stars or broke free from their companion stars during their supernova explosion. A small subset, however, do have companion stars that are almost without exception mainsequence stars. When these main-sequence stars evolve and turn into red giants, it is not uncommon that their outer atmospheric layers stray into the gravitational well of the pulsar and cause it to accrete matter and, along with the matter, angular momentum. These pulsars are then spun up while their magnetic field decays. The result is a “millisecond” pulsar (MSP), with spin periods between 1 ms and about 30 ms. Due to the accretion process, MSPs have relatively weak magnetic fields (109 G or less), and consequently their spindown rates are also far lower than for slow pulsars (typically of the order of 10−20 s/s). As a result the rotational and emission properties of MSPs are not thought to appreciably evolve over their lifetimes. For an overview of formation and evolution of MSPs, see [3, 22]. Given the extremely small spin periods, stable pulse profiles for MSPs can already be obtained in a matter of minutes or even seconds. Furthermore, to date, only very few MSPs [101] have been demonstrated to show any of the anomalous emission properties (nulling, moding, drifting) mentioned in the previous section that some of the slow pulsars display. Finally, due to the much larger angular momentum, MSPs have turned out to be far more stable clocks than slow pulsars. These are the reasons why MSPs have become known as “nature’s gift to physics”: the perfect Einstein clock that can be used to test a wide range of relativistic predictions.

Pulsar Timing and Pulsar Timing Arrays Pulsar timing is a method that exploits the highly regular spin period of pulsars and their predictable pulse shapes, to study a wide range of questions in physics and astrophysics. In essence, when doing pulsar timing, one monitors the times at which subsequent pulses from a pulsar arrive at an observatory. These observed pulse arrival times or ToAs are then compared to a mathematical model that attempts to quantify all the factors that impact the travel time of the electromagnetic waves on

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their way from the pulsar to the Earth. In practice, a number of complications need to be dealt with, as outlined below. A more advanced approach is to use multiple pulsars – an “array” of pulsars – to look for signals that correlate between different pulsar pairs. Such experiments are called “pulsar timing arrays” (PTAs).

Template Profiles In its simplest form, pulsar timing could be based on the times when the peaks in a train of pulses are detected. In order to increase the measurement precision, one could also take the intensity-weighted average arrival time of any given pulse. A far more powerful method, however, is to use the information encoded in the shape of the pulse, to measure the arrival time relative to a standardized pulse shape. This can be thought of as the ultimate, noise-free pulse profile. Obtaining such a standardized pulse shape or “template profile” is necessarily an iterative process. Fundamentally, as many pulses should be averaged together as possible. However, in order to align said pulses, an accurate pulsar timing model should be used to predict the phase of subsequent pulses to a precision far better than the time resolution afforded by a typical observation. To give an example, we aim to predict arrival times to a precision of nanoseconds, while typical observations have phase resolution on the order of microseconds. The use of all of the information contained in a complex pulse profile, through the use of a template profile, allows one to achieve this necessary magnitude of improvement. Since the creation of the template profile itself is the very start of the path towards a functional pulsar timing model, typically the entire process gets iterated so that the template profile and the timing model can both improve until their solutions converge. A danger in the creation of template profiles is so-called self-standarding. This is a phenomenon that occurs when the data that are being used in the timing are also used to construct the template profile. Specifically, if the number of observations that are combined to construct the template profile is too low (a typical rule of thumb is that “too low” is less than a few thousand), then the noise within the observation can be “recognized” in the template profile, leading to inaccurate offset measurements [as illustrated clearly in Appendix A1 of 64]. Consequently, many timing experiments make use of analytic models to describe the template profile. These may not always be able to perfectly model all the pulse-shape features, but they avoid timing corruptions caused by self-standarding. Alternatively, a smoothing filter may be applied to the template, in order to reduce the correlating noise [41]. It was mentioned earlier that pulse profile shapes typically change with the observing frequency. This should ideally be taken into account when constructing the template profile, i.e., the template profile should effectively have a dependence on observing frequency, too. In most published works this has not been the case because the bandwidth of pulsar observations used to be sufficiently narrow that frequency evolution of the pulse profiles was effectively undetectable, but over the last decade (fractional) bandwidths of observing systems have increased

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so significantly that so-called “frequency-resolved” template profiles are rapidly becoming the norm rather than the exception. Also in this case, one can either create an analytic description of the profile in two dimensions [95, 115] or use a template purely based on accumulated data [45].

Template Matching Once a template profile has been created, it can be used to calculate the ToAs of the various observed pulse trains. Since most pulsars are so faint that individual pulses cannot be detected and because single pulses are usually not all alike, standard pulsar timing experiments do not time individual pulses, but average subsequent pulses modulo the pulse period. (At this point the question of which pulse is being timed exactly, is in principle an arbitrary choice, but the most commonly used pulsar timing software uses a pulse in the center of the observation.) This averaging procedure is commonly referred to as folding – it reduces the time resolution of the observations while phase resolution is maintained. Typical values for the time resolution of pulsar timing data after folding are anywhere from minutes to one hour, depending on the brightness of the pulsar and the goal of the experiment. Phase resolution is defined by the number of bins across the profile, with typical values ranging from 128 to 2048, depending on the sensitivity of the telescope and the bluntness or sharpness of the pulse profile. The folded pulse profiles could be cross-correlated with the template profile in order to achieve the phase of the observation – which can then be added to the observation’s time stamp in order to achieve a ToA. In practice this measurement is commonly undertaken in the Fourier domain, as explained in detail in the appendix of Taylor [143]. Since an offset in the cross-correlation is equivalent to a phase gradient in the cross-power spectrum, typically ToAs are determined by leastsquares fitting the phase gradient in the cross-power spectrum of the template profile and the folded observation. The phase offset resulting from this is then added to the observation’s time stamp. The measurement uncertainty of these phase offsets – and hence of the ToAs – is an important value as well, since many pulsars appear to have highly variable flux densities (a process caused by the interstellar medium, called scintillation), which means that not every ToA carries as much information and hence should not be weighted equally. Specifically, the pulse profile that is to be timed will contain a certain amount of white noise called radiometer noise, which depends on the observational properties of the pulsar and the observing system as follows [96]: −1 σToA



Tpeak ∝ S/N = β np tint Δf Tsys

√ W (P − W ) , P

(1)

−1 where σToA is the ToA uncertainty; S/N is the signal-to-noise ratio of the observation; β is a factor which describes instrumental losses, e.g., due to digitization; np ,

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tint , and Δf are respectively the number of polarizations combined, the integration time, and the bandwidth of the observation; Tpeak is the brightness temperature of the pulsar at the peak of its profile; and Tsys is the brightness temperature (i.e., noise) of the observing system, which typically includes corrections for cable losses, instrumental gain, spillover, and sky noise, among other things. W is the equivalent width of the pulse profile, defined as the integrated pulse intensity divided by the peak intensity, and P is the pulse period. The radiometer noise is the most fundamental factor limiting pulsar timing precision, in the sense that it is present in all observations and is determined to a large degree by the fixed properties of the pulsar and the technical capabilities of the telescope. Traditionally it was quantified as the formal uncertainty of the phase-gradient fit described above, although it has been shown that in the low-S/N regime this can cause irregularities [see 7, App. B], leading people to either remove ToAs below a certain S/N level (e.g., requiring S/N > 8) or apply more advanced, Monte Carlo-based uncertainty estimations, as was proposed as “good pulsar timing practice” by Verbiest et al. [153].

Timing Model Determination Once the ToAs have been measured, they need to be compared to predicted arrival times provided by a pulsar timing model. A timing model is a mathematical formula that predicts the arrival time of a pulse based on a set of timing-model parameters. Generally, the timing model is defined in two steps [see also 47, 143]. In the first step, the measured pulse arrival time tobs is *cast to* the time of emission at the pulsar tPSR . This is achieved by accounting for all known propagation and geometric delays: tPSR = tobs − Δ − ΔIISM − ΔBin . Specifically, first the pulse ToA is transferred to the Solar System barycenter (i.e., corrected by a delay Δ ), which is the inertial reference frame most commonly used in pulsar timing. This transformation includes correction factors for relativistic effects caused by the mass distribution in the Solar System and the Earth’s orbital and rotational velocity, for atmospheric propagation delays (note that these have mostly been neglected to date but will become important with the next generation of radio telescopes), for light-propagation times (the so-called Roemer delay), parallax effects, frequency-dependent propagation delays induced by the Solar Wind, and corrections for the observatory clock. After the ToAs have been transferred to the Solar System barycenter, interstellar propagation delays (ΔIISM ) are corrected for. As described in more detail in the next section, these delays have long been treated as dependent on the observing frequency, but constant in time. With increased measurement precision provided by wider bandwidths and lower observing frequencies, time-variable models of interstellar dispersive delays are now becoming more common.

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For pulsars that inhabit binary systems, there is one further transformation, namely, from the barycenter of the binary system to the pulsar (delay ΔBin ). This includes the Roemer delay based on a Keplerian description of the binary orbit but can also contain relativistic effects such as the Shapiro delay (time dilation caused by the companion star’s gravitational field), the Einstein delay (time dilation caused by the pulsar’s gravitational field and gravitational redshift), or a host of other more complex effects, dependent on the binary’s properties. [See 47, for a complete listing.] Once the time of emission tPSR is determined, it can be converted to a rotational phase based on a spindown model that is usually simply described as a Taylor expansion: 1 φ (tPSR ) = ν (tPSR − t0 ) + ν˙ (tPSR − t0 )2 + . . . , 2

(2)

where ν is the spin frequency of the pulsar, ν˙ its first derivative, and t0 an arbitrary reference epoch. Standard electromagnetic theory predicts a second frequency 2 derivative ν¨ = 3νν˙ , but in practice this is immeasurably small in the case of MSPs and is typically obscured by other effects (so-called timing noise, see further) in most slow pulsars. Consequently, by default pulsar timing models contain a spin frequency and frequency derivative but not usually any higher-order spin frequency derivatives. Initial timing models are derived from pulsar-search observations. These are raw time series that are not folded, but instead are Fourier transformed in order to obtain an instantaneous pulse period. By monitoring the pulse frequency evolve over several such observations, an initial estimate of the spindown and of the binary orbit can be determined. This then constitutes an initial timing model that can be used to predict the pulse frequency for future observations, allowing the data to be folded in real time, which makes observations much less demanding in terms of data-storage and processing power requirements. When a basic timing model has been constructed that is able to predict the arrival time of future observed pulses to well within a pulse period, it is said that “phase connection” has been achieved. From this point forward, the phases calculated in Equation 2 can be used to improve the timing model. Effectively, these phases should all be zero if the timing model was perfect and no corrupting noise sources were present. Consequently, any deviation from zero highlights effects which do affect the observations, but are not taken into account (correctly) by the timing model. These differences between the observation and the model are called the timing residuals, and they lie at the core of pulsar timing analyses. Indeed, the art of pulsar timing is to optimize and extend the timing model to decrease these timing residuals. In the optimal case, the timing residuals will only consist of white noise which is accurately quantified by the uncertainties of the ToAs. In this case the timing is fully dominated by the timing residuals described earlier. Each parameter that is contained in the timing model affects the timing residuals in a well-defined way, which is called the timing residuals of this parameter.

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Fig. 3 Examples of pulsar timing residuals. The top-left figure shows a typical PTA data set taken from [106]. The residuals are not purely white in this case, most likely due to a combination of uncorrected variations in interstellar propagation delays and interstellar dispersion. In the top-right figure, the impact of a 1% change in the spindown is demonstrated, leading to a clear quadratic trend in the residuals as the spin period gets increasingly incorrect as time progresses. The bottomleft figure shows a positional offset of 0.1 arcsec in both right ascension and declination, leading to an annual sine wave with constant amplitude. The bottom-right plot shows what happens in contrast if the position is correct (at the reference epoch near the start of this data set) but the proper motion is 10% incorrect. This causes a linear increase in positional error and hence induces an annual sine wave with linearly growing amplitude

Consequently, simply by visual inspection of the timing residuals, particular errors can sometimes easily be picked out, as shown in Fig. 3. In most practical cases the uneven spacing between observations, the variability in the ToA uncertainty, and the combination of multiple timing signatures in incomplete or outdated timing models make the analysis of timing residuals rather more complex than these simplified examples suggest.

Interstellar Propagation Delays A particular source of difficulty when analyzing pulsar timing data is the impact of the ionized part of the interstellar medium (also referred to as the IISM). The

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refractive index of the interstellar medium is determined by the plasma frequency, which is dependent on the local electron density [96]:  fp =

√ e2 ne ≈ 8.5 kHz ne , π me

where e is the electron charge, ne is the electron density in units of cm−3 , and me is the electron mass. Given that plasma frequency, refractive index for a photon  the  2 fp with frequency f can be determined as: μ = 1 − f . Since the group velocity of electromagnetic waves is dependent on the refractive index (vg = μc0 ), this leads to a frequency-dependent group velocity which is observable as a frequencydependent propagation delay: Δt =

f1−2 − f2−2 DM, K

where DM is the “dispersion measure” defined below and the constant K = 1 −4 −2 −3 D = 2.41 × 10 MHz pc/cm /s is the inverse of the “dispersion constant.” Theoretically, the dispersion constant could be determined to higher precision 2 (specifically, D ≡ 2πeme c ), but in pulsar timing it has traditionally been fixed as given above [80]. The dispersion measure is straightforwardly defined as the integrated electron content along the line of sight between the telescope and the pulsar:  DM =

D

ne dl, 0

where ne is the electron density in cm−3 , D is the pulsar distance in pc, and DM is typically expressed in units of pc/cm−3 . However, given that in most pulsar timing software D has been defined fixed at the abovementioned value and since the actual observable is D×DM, in practice most DM measurements would require a slight correction before being interpreted as physical electron density measures, as described by Kulkarni [80]. Due to the typically high spatial velocities of radio pulsars [up to 1000 km/s and beyond 33], the line of sight along which the pulses travel to Earth sweeps through interstellar space; and due to the numerous turbulent structures that are present throughout that space [5], this motion causes the DM to change as a function of time. While such variability has long been known to exist, it was mostly detectable as a slowly varying, red-noise process. Since the turn of the century, however, with improved instrumental sensitivity, ever more precise measurements of DM variations in time have been detected, and accurate modelling of DM(t) is becoming a complex and essential part of pulsar timing experiments [70, 74].

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One particular component that contributes to temporal variations in DM is the Solar Wind. Due to the annual motion of lines of sight – particularly for pulsars near the ecliptic plane – the additional dispersion caused by the Solar Wind has a clear annual signature, which is typically modelled straightforwardly by assuming the Solar Wind to be homogeneous and spherically symmetric [47]. Generally, it is assumed that such a straightforward model would suffice for the purposes of highprecision pulsar timing, except perhaps closest to the Sun, so in addition to the spherical models, PTAs have tended to remove ToAs for observations that took place within 5–10 degrees of the Sun [153]. It has been attempted to extrapolate optical observations of the Sun to derive a more detailed, inhomogeneous model of the Solar Wind for pulsar timing purposes [157], but while this model was shown to provide an accurate spectrum of inhomogeneities, it does not provide accurate corrections for pulsar timing experiments [146]. In addition to dispersion, the IISM introduces a host of other propagation effects, as recently reviewed by Stinebring [138]. In most cases these effects are not limiting timing precision yet, although time-variable scattering (also referred to as multi-path propagation – a phenomenon that widens the pulse shape through increased travel path lengths) has been shown to be relevant in the timing of at least one MSP [92].

Timing Noise Probably the hardest effect to mitigate in pulsar timing is the so-called timing noise. This term is generically applied to any timing residuals that are not white noise and cannot be corrected for by any of the deterministic timing-model parameters or by frequency-dependent DM models. Presumably this typically long-term noise is caused by inherent rotational instabilities in the neutron star itself [78], although the physical mechanism is as yet not known. Timing noise has been studied extensively in slow pulsars [59, 99], where it is highly common. In MSPs, timing noise has been shown to be far less common or to exist only at much lower levels [91, 152]. Nevertheless, as the length of pulsar timing data sets grows and the timing precision increases, the prevalence of timing noise – and the importance of mitigation techniques – continues to increase even in MSP timing projects [10].

Other Noise Sources After the timing model is optimized and the IISM variations are modelled and corrected for, ideally the timing residuals should be spectrally white and normally distributed. In practice a wide range of effects can negatively affect the timing, as recently reviewed by Verbiest and Shaifullah [151], although in practice the primary impact aside from the IISM and timing noise is pulse phase jitter, also known as SWIMS. The work by Lam et al. [81] on 37 MSPs shows that jitter is relevant primarily at low frequencies (particularly at observing frequencies below

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1 GHz) but less so at higher frequencies, where high-precision pulsar timing is most commonly done. Since the importance of pulse jitter is strongly dependent on the sensitivity of the telescope (and on the available bandwidth), jitter will become an even more important source of noise in the next generation of radio telescopes [94]. One approach to avoid jitter-dominated timing would be to divide up large interferometric telescopes into less-sensitive sub-arrays which allows more effective scheduling with lower jitter noise [135]. Another approach is to mitigate jitter noise in post-processing as demonstrated in Osłowski et al. [112, 113].

Pulsar Timing Software All of the analysis described above tends to be carried out with two separate types of data analysis tools. Firstly one needs software that can process the raw observation files: the so-called archives that store radio wave intensity as a function of polarization, pulse phase, frequency, and time. Secondly one needs model-fitting software that analyzes the ToAs and the related timing models. The primary software package that is used globally to analyze pulsar archives in the context of pulsar timing is PSRCHIVE [63, 150] (http://psrchive.sourceforge. net). The only exception to the use of PSRCHIVE for the analysis of pulsar archives and the creation of ToAs is the possible creation of broadband ToAs with analytic, frequency-dependent templates. For this purpose, the purpose-built PULSEPORTRAITURE software [115] (https://github.com/pennucci/PulsePortraiture) is used increasingly commonly. For the analysis of ToAs and timing models, however, a larger variety of software packages has been developed. The most common ToAanalysis package is TEMPO2 [62] (https://bitbucket.org/psrsoft/tempo2/), which is fundamentally a C/C++ translation of the much older, Fortran 77-based TEMPO package [111] (https://github.com/nanograv/tempo), which is still being used, often in parallel with TEMPO2. More recently, the PINT package was developed [98] (https://github.com/nanograv/PINT), which is an independent timing package written in Python and which is primarily used in North America. Extending pulsar timing software to constrain or detect correlated signals, such as those from GWs, is a non-trivial effort. Whereas frequentist methods have occasionally been implemented as part of packages like TEMPO2, it has become far more common to build independent software packages that are specifically aimed at Bayesian analyses of the timing model – including correlated signals. These packages tend to be written in Python and use Python wrappers around the source code of the standard pulsar timing packages mentioned above – most commonly TEMPO2. The most recently used package for such advanced ToA analysis (including GW analyses) is ENTERPRISE [48] (https://github.com/nanograv/ enterprise); two commonly used earlier packages are TEMPONEST [89] (https:// github.com/LindleyLentati/TempoNest) and PICCARD [149] (https://github.com/ vhaasteren/piccard).

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Gravitational Waves and Other Correlated Signals In the previous sections, we focussed on pulsar timing effects and phenomena that affect each pulsar independently in fully unrelated ways. This is typical of the traditional single-pulsar timing experiments that have been the mainstream of pulsar timing research ever since the first pulsar discovery. However, throughout the 1980s the realization developed that some signals might have timing signatures that correlate between pairs of pulsars [50, 57, 123]. These signals would require a new, more complex analysis as they require a joint analysis of a larger number, say an array, of pulsars. The concept of the PTA was born, but would not come to full fruition until the start of the new millennium, after the number of known pulsars (and the number of PTA-worthy pulsars) had dramatically increased following a couple of highly successful surveys, primarily in the inner galaxy [103, 104]. In the following paragraphs, an overview will be given of typical correlated signals in pulsar timing data, with particular focus on the signature of gravitational waves.

Correlated Signals in Pulsar Timing Data As described above, pulsar timing is effectively a model-fitting exercise where deterministic parameters get optimized as increasing amounts of data constrain the timing model. In addition to the deterministic components of the timing model, there are non-deterministic effects like DM variations or jitter noise that affect the timing differently for each pulsar. Finally, there are signals – both deterministic and nondeterministic – that affect all pulsars in similar ways. Three types of such correlated signals have been described to date [50, 123]: • a monopolar signal that would most likely arise from imperfections in the reference clock [60, e.g.], • a dipolar signal that is most likely related to the Solar-System ephemerides [53, e.g.], and • a quadrupolar signal that is predicted to arise from gravitational waves [57]. These correlated signals require more complex approaches: Solar-System ephemeris models could be updated with the pulsar timing data [32, 53, 147], but clock signals and gravitational-wave signatures are in essence random, and hence the actual correlations between timing of different pulsars need to be used in order to determine those. For monopolar (or clock) signals, this has been successfully achieved a number of times [60, 122], but quadrupolar (or gravitational wave) signals have so far not been unambiguously detected, partly also because all these signals interact, making a clear detection of the highest-order correlated signal (i.e., the gravitational waves) dependent on accurate determination of all other correlated signals [145].

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Effect of Gravitational Waves The effect of GWs on pulsar timing was first described by Detweiler [44] and more recently clearly summarized by Sesana and Vecchio [130]. Fundamentally, a GW passing over the Earth-pulsar system will introduce a time-variable redshift into the pulsed signal:     ˆ − ν0 /ν0 , z t, Ωˆ ≡ ν(t, Ω) where Ω is the direction of propagation of the GW, ν0 is a reference frequency, ˆ is the observed pulse frequency which is defined by the geometry of the and ν(t, Ω) system (pulsar and GW position with respect to the observer) and the GW properties (polarization and amplitude). The integral of these redshifts quantifies the impact on the timing residual:  r(t) =

t

ˆ  z(t  , Ω)dt

0

for an observation taken a time t after the first observation in our data set. Furthermore, the observed redshift is only dependent on the perturbation of the space-time metric at the position of the pulsar at the time when the pulse was emitted and the space-time perturbation at the location of Earth when the pulse was received. A more straightforward way of putting this is that the GW impact on the pulsar timing residuals has two equally large components: a “pulsar term” that quantifies the impact of the GW on the emission of the pulse and an “Earth term” which quantifies the impact of the GW on the detection of the pulse on Earth. For nonevolving, sinusoidal GWs, the Earth and pulsar terms will be identical except for a phase offset between the two. Furthermore, the Earth term will affect all pulsars equally (albeit modulated in a quadrupolar way, as described by the so-called Hellings and Downs curve [57], see Fig. 4), whereas the pulsar term will have a different phase offset for each pulsar, since the phase of the pulsar term depends on the distance to the pulsar. For such mono-chromatic signals, this phase offset could in principle be measured along with the pulsar distance, allowing extremely precise localization of both the GW source and the pulsars in the array [87]. Such an experiment would, however, require a level of timing precision that is not realistically achievable with present-day telescopes (but may be achievable with the next-generation Square Kilometre Array or SKA). At present, therefore, the pulsar term is typically considered a noise term, while the Earth term is the actual quadrupolar signal we hope to detect. The amplitude of the correlated signal will give insights into the GW’s origins (see the next section), whereas the shape of the correlation curve could constrain fundamental physics of gravitational waves, like their polarization properties [86] and propagation speed [84]. Furthermore, deviations from the theoretically expected Hellings and Downs curve can be expected for anisotropic backgrounds of GWs, where a few bright sources stand out above the background

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0.7 0.6 0.5 Expected correlation

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0

0.5

1

1.5 2 Angular separation [rad]

2.5

3

Fig. 4 Correlated impact of a stochastic background of GWs on pulsar timing residuals. Shown is the so-called Hellings and Downs curve [after 57] as solid black line, which quantifies the correlation induced into pulsar timing residuals of sets a pulsars as a function of the angular separation between those pulsars on the sky. For a single GW, the shape would be similar but would be dependent on the orientation of the pulsar pair with respect to the GW source; for a stochastic background, only the angle between the pulsars matters. Square points show simulated measurements for an optimistic realization of a PTA experiment

and cause a correlation function that is not only dependent on the angle between pulsars but also on their location on the sky [144].

GW Sources in the PTA Band The sensitivity of PTAs to GWs is limited in terms of the GW frequency by the length of the observing time span and the cadence of the observations. Specifically, PTAs are most sensitive near a frequency of 1/T where T is the length of the data set, i.e., on the order of a decade or more, which corresponds to a frequency on the order of nanohertz. Since the GW impact is a change in the pulse frequency [44], but the timing residuals are phases, i.e., effectively integrals over pulse frequency, the sensitivity of PTAs decreases with increasing frequency down to the Nyquist frequency 2/C where C is the cadence of the observations, typically of the order of weeks or a month, which corresponds to a frequency cutoff of the order of microhertz. This frequency range makes PTAs particularly complementary with other GW detectors like LIGO [10 Hz–10 kHz, 107] and LISA [0.1 mHz–1 Hz, 4] and competitive with proposed GW detection methods based on space-based VLBI [26, 28]. The difference in GW frequencies furthermore implies that different

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sources of GWs can be expected to be detectable with PTAs. Specifically, four types of sources are anticipated, as described below. Example timing residuals induced by these four types of GW are shown in Fig. 5.

Fig. 5 Example timing residuals for four GW types on three different pulsars. These four panels show what the timing residuals that are caused by GWs could look like. The top-left panel shows the influence of a GWB (with characteristic strain amplitude 10−15 and spectral index −2/3), the top-right panel that of a CW (from a 109 M equal-mass binary supermassive-black hole at redshift z = 0.01), the bottom-left panel shows a BWM (with strain amplitude 5 × 10−15 ), and the bottom-right panel shows a GW burst (without memory and with arbitrary waveform). The three different colors show the impact on different pulsars (i.e., different sky locations), with red showing the timing residuals of PSR J0437−4715, blue those of PSR J1012+5307, and black those of PSR J1713+0747. The simulated measurement uncertainty is 1 ns, and no intrinsic spin noise or DM variations were included in the simulations. The top figure in each panel shows the pre-fit residuals (i.e., the raw GW signature), whereas the bottom plot shows the post-fit residuals. The difference between these is caused by fitting of the standard timing-model parameters. As can be seen, most of the power of the GWB is absorbed in the timing-model fit, since its long-term signature resembles the timing signatures of the pulse period and spindown, which absorb most of the signal. Such absorption of GW residuals in common timing-model parameters is less likely to happen for CWs or bursts, unless they happen to have a periodicity that is close to a year (or an integer fraction or multiple thereof) or to the orbital period of the pulsar being timed

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Gravitational wave background (GWB): A GWB is a superposition of GWs from a large number of GW sources that add incoherently. The most likely GWB in the PTA frequency range is widely expected to arise from supermassive black hole binaries, and predictions for its spectrum and amplitude are based on simulations such as the Millennium Run [129] or the Illustris simulation [75]. Specifically, the power spectrum of this GWB is expected to have a power-law shape with slope −2/3 and is likely to flatten or even tip over at GW frequencies lower than ∼0.1 yr−1 [34]. Alternative backgrounds have been proposed [see, e.g., 27, 127, 128], with differences in both spectral shapes and amplitudes, implying that an eventual detection would be able to differentiate between the origins of the GWB or would be able to place constraints on the galaxy-merger history of the Universe. Continuous waves (CWs): Continuous GWs in the PTA band are expected from single supermassive black hole binaries that are close enough to Earth to stand out beyond the GWB. With improved sensitivity, a detection of CWs is generally expected to follow within a few years from a GWB detection [110, 124]. GW bursts: Single GW events like close encounters between supermassive black holes or some cosmic string interactions [40] could result in a single burst of GWs, which might be detectable provided the burst itself lasts sufficiently long for it to affect multiple subsequent pulsar observations (i.e., at the very least days long) and provided the burst is sufficiently bright to stand out of the noise. Bursts with memory (BWMs): Bursts that are too faint to be detected directly could still be detected as a “memory event” or a BWM. In this case it is the permanent deformation of the space-time metric [49] that has a lasting impact on the pulse frequency, causing the impact on the timing to accumulate over time.

Present PTA Constraints Around the world, collaborations have emerged to carry out the large-scale observational campaigns required for GW detection with PTAs (Table 1). The Australian Parkes PTA [PPTA, see 106] was the first one, commencing observations in 2005 and revising some of the original theoretical work [67, 68]. Centered around the 64-m Parkes radio telescope, they have been monitoring between 20 and 30 pulsars using both dedicated observations [76] and archival data [152]. The PPTA last placed a limit on the GWB in 2015 [131], finding that the normalized amplitude at a frequency of 1 yr−1 must be lower than 10−15 with 95% confidence – which started to get into the area of theoretically predicted amplitudes at the time; this limit is still the most constraining bound on a GWB in the PTA band to date. Most recently,

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Table 1 Summary of present limits on GWs from PTA data GW Type

EPTA 3.0 × 10−15 GWBa Lentati et al. [90] – BWMc – 1.5 × 10−14 CWd Babak et al. [13]e

NANOGrav PPTA 1.45 × 10−15 1.0 × 10−15 Arzoumanian et al. [9] Shannon et al. [131] 1.5/yr 0.75/yr Arzoumanian et al. [8] Wang et al. [154] 3.0 × 10−14 1.7 × 10−14 Arzoumanian et al. [6] Zhu et al. [159]

IPTA 1.7 × 10−15 Verbiest et al. [153]b – – – –

a Limits

on the GWB are typically given as upper limits on the dimensionless strain amplitude at a frequency of 1/yr b Verbiest et al. [153] note that the limit derived from the IPTA data set was only indicative and not rigorous as a full analysis was deferred to a future paper c BWMs can be quantified in many ways. In order to provide some comparative measure, this table presents the upper limit on the burst rate for bursts with normalized characteristic strain amplitude 10−13 d Limits for CWs are given as upper bounds on h , at a GW frequency of 10 nHz 0 e Babak et al. [13] don’t give a specific value; the number given is based on their Figure 3

the PPTA has focussed its efforts on alternative sources of GWs, such as ultralight scalar-field dark matter [118] as well as CWs and BWMs [100], alongside a more intensive commitment to the global International PTA (see below) and instrumental development [61]. The European PTA [EPTA, see 43] also commenced in 2005, soon after the PPTA, and also relies on a combination of specific PTA data and archival monitoring data, adding up to a total of 42 MSPs [43]. To date, the EPTA has primarily used data from the Jodrell Bank 76.2-m Lovell Telescope, the 100-m Effelsberg Radio Telescope, the 300-by-35-m Decimetric Radio Telescope at Nançay Observatory, and the Westerbork Synthesis Radio Telescope which is an interferometric array consisting of 14 antennae of 25-m diameter. Their most recent limit also dates back to 2015 [90], but is slightly less constraining, at A1/yr < 3.0 × 10−15 . The same data set has been used to place constraints on CWs [13] and possible anisotropies in the GWB [144]. Finally, dedicated, high-cadence, data were used to place constraints in the microhertz regime [116]. In recent years, the EPTA has primarily focussed on instrumental development with new data-recording systems [83, e.g.], the commissioning of a new 64-m radio telescope in Italy [SRT, see 119], and the interferometric combination of all major EPTA telescopes in the “Large European Array for Pulsars” (LEAP) project [19]. The EPTA has also had a strong involvement in the commissioning of the LOw-Frequency ARray [LOFAR, see 148], which has shown to be a useful telescope for monitoring not only MSPs [77] but particularly DMs [45, e.g.].

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The third major PTA is the North American Nanohertz Observatory for Gravitational Waves [NANOGrav 41]. NANOGrav has been particularly involved in pulsar searches, and as such their source list has been continuously expanding, counting 48 MSPs at their current data release [https://data.nanograv.org and 82]. The latest and most constraining limit on the amplitude of the GWB is A1/yr < 1.45 × 10−15 [9], slightly above the PPTA limit. Within the last few years, NANOGrav has also placed significant bounds on BWMs [2] and on CWs [1] and placed a specific limit on the proposed binary black hole system in the radio galaxy 3C66B [11, 142]. The three original PTAs mentioned above joined forces to further increase sensitivity and in an attempt to decrease the time to the first GW detection in the nHz regime. As described by Manchester and IPTA [102], the first joint PTA meeting took place in 2008, but the formal establishment of the International PTA (IPTA) did not happen until 2011. Since then, the IPTA has released two combined data sets [117, 153] and has indicated that an improvement in GWB sensitivity by a factor of about two should be expected, but no full GW analysis has been carried out on IPTA data so far, given the complexities involved with the highly inhomogeneous nature of the data set. These inhomogeneities and complexities [discussed and listed in 153] are a specific challenge for any combined PTA experiment and often require additional research regarding detailed aspects of the analysis or development of new methods that are able to deal with such inhomogeneous data. A lot of progress has been made in recent years in this regard, particularly with the advent of Bayesian analysis software packages like TEMPONEST [89,91] and ENTERPRISE [48], among others. So far, the first IPTA data combination has been used by Hobbs et al. [60] to construct a pulsar-based time standard (thereby solving for any monopolar correlations in the data), while Caballero et al. [31] used it to constrain errors in the Solar-System ephemeris models used. A comprehensive GW analysis is planned for the second data release. Meanwhile, analysis tools are being tested on mock data challenges [17, 54, 151]. As a new generation of telescopes is being constructed on the pathway to the SKA, a number of new PTAs have recently emerged. Specifically, the Indian PTA [InPTA, 71] has been formed in 2018 and uses high-precision timing data from the upgraded Giant Metre-wave Radio Telescope (uGMRT) and low-frequency data from the Ooty Radio Telescope (ORT). The Chinese PTA [CPTA 85] uses the Five-hundred-meter Aperture Spherical Telescope (FAST), the Xinjiang Qitai 110m Radio Telescope (QTT), and a network of 100-m-class radio telescopes across China and is predicted to go well below current sensitivity limits after even a few years of observing. Finally, the South African MeerKAT telescope [69] is being used by the international MeerTIME consortium [16] to produce (among other things) a highly sensitive PTA data set, part of which will be taken at relatively high radio frequencies, between 1.7 and 3.5 GHz, thereby limiting the impact of interstellar effects.

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Recent and Ongoing Improvements in PTA Sensitivity Much of the present interest in PTA experiments derives from the work by Jenet et al. [67] who predicted that a GWB should be detectable after a mere 5 years of timing on 20 pulsars, with a timing residual RMS of 100 ns – a level of precision that had only recently been demonstrated to be achievable in practice [141]. Subsequently, it became clear that the ideal PTA (20 pulsars, 5 years, and 100 ns RMS) was unlikely to ever become a reality since the pulsar population is by nature highly inhomogeneous, implying a few pulsars would be likely to be timed at better precision than 100 ns, but most probably would not. Consequently, scaling relations were derived, first by Jenet et al. [68] and later by Siemens et al. [133], to allow fine-tuning of PTA experiments with a view to optimizing sensitivity and shortening detection timescales. Siemens et al. [133] showed that the S/N of a GWB in a PTA data set scales with typical properties of the data set as follows: S/N ∝ NC 3/26 A3/13 T 1/2 σ −3/13 ,

(3)

where N is the number of pulsars, C is the cadence of the observations, A is the amplitude of the GWB, T is the length of the timing data set, and σ is the RMS of the timing residuals. While this analysis makes some basic simplifications in terms of data homogeneity, it does show clearly the very strong dependence of PTA sensitivity on the number of pulsars in the array. Consequently, a large number of pulsar surveys have been undertaken in recent years to increase the number of PTAuseable MSPs, as described below. The dependence on all other parameters is far less significant, except for the data length, which still adds considerably. In the lowS/N regime, Siemens et al. [133] show that a PTA’s sensitivity scales most strongly with the data length, S/N ∝ T 13/3 , in close agreement with the earlier findings of Jenet et al. [67]. For long data sets, the sensitivity is however affected by the level at which other long-period signals can be mitigated. Most significantly this refers to DM variations which can be mitigated provided the observing setup has been well chosen. Pulsar timing noise is equally important, but in the absence of predictive models or independent estimates of this noise source, the only possible approach is to limit its impact by post facto modelling and subtraction. Finally, in the intermediate-S/N regime (i.e., as we get closer to detections rather than mere limits), the sensitivity to a GWB is only weakly related to the timing precision of the MSPs, but for single sources of GWs the timing precision is still the dominant factor; consequently some efforts are being made to further improve the timing precision of MSPs by instrumental improvements and building of new telescopes. At the end of this section, a brief overview is given about various studies that quantify how all of these improvements are likely to affect the time to the first GW detection with PTAs – which mostly agree a detection within years to at most a decade is highly likely.

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Pulsar Surveys In order to increase the sample of MSPs that can be used in PTA experiments, a number of pulsar surveys have been undertaken in recent years and are being planned for the near future. Specifically, two long-lasting surveys have been running on the Arecibo radio telescope for most of the past decade: the P-Alfa survey [37] and the Arecibo drift-scan survey [42]. In addition, the Effelsberg and Parkes radio telescopes (in Germany and Australia respectively) are continuing their all-sky partner surveys HTRU North and South [18, 73]. Parkes has simultaneously been equipped with cutting-edge processing technology as part of the SUPERB [72] survey, which aims to do real-time searches for pulsars and fast radio bursts. Data from the Green Bank Telescope (GBT) continues to be analyzed in the Green Bank Northern Celestial Cap (GBNCC) survey [140]. At low frequencies, the LOFAR telescope [148] is finishing processing of the LOFAR Tied-Array All-Sky Survey [LOTAAS, 126], and at the GMRT the GMRT High-Resolution Southern Sky Survey (GHRSS) is ongoing [23]. New telescopes are also getting up to speed on pulsar surveys, with the first successful discoveries published by the FAST telescope [120, 158], as part of the Commensal Radio Astronomy FAST Survey (CRAFTS), and survey observations recently commenced for the MeerKAT “TRAPUM” survey [137].

IISM Studies In order to increase sensitivity of pulsar timing data sets to long-term signals like those expected from a GWB, it is of utmost importance to understand, mitigate, and model any long-term signals that may be affecting the timing. Most such effects are deterministic effects contained in the timing model, but two more complex sources of red noise exist: timing noise and IISM noise. As discussed earlier, the origin of timing noise has not been unequivocally determined, and consequently most models are rather ad hoc power-law descriptions of uncorrelated timing signatures. IISM noise (or DM variations) is different, since it is the only known effect that causes a frequency dependence in timing residuals. The frequency dependence of IISM noise implies that in principle it can be measured and modelled independently from all the other timing-model parameters and correlated signals, because it can fully rely on the frequency resolution of the data. Specifically, three scenarios could be envisaged for measuring and correcting DM variations in pulsar timing data [66]: • Multiple different observing bands: When more than one observing band is used, the frequency difference between the bands can be used as a lever arm that enables high-precision DM estimates. This idea has been implemented both with co-axial receivers and, more recently, with ultra-broadband receivers such as the UWL in Parkes [61] and similar observing systems at the Effelsberg and Green

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Bank radio telescopes, often not using actual instantaneous observations, but by determining an average DM over a range of dates [74]. In this scenario, care must be taken in the allocation of the observing time, since the DM sensitivity of the data will scale with the square of the observing wavelength, but the timing precision may not, depending on the spectral index of the pulsar. Specifically, since on average pulsars have a spectrum that is flatter than ν −2 [65] and since the sky background noise is steeper than typical pulsar spectra [155], it is likely that the low-frequency band will still have superior sensitivity to DM, but have worse timing precision overall. For observations beyond 1.4 GHz, however, the situation often reverses, in that higher-frequency bands may have both less sensitivity to dispersion and lower timing precision [see, e.g., 83]. This implies a change in integration time depending on the observing frequency may be in order. A more extensive analysis of post-correction ToA precision in this scenario is given by Lee et al. [88]. • Low-frequency monitoring: As a possible way to mitigate the complexities of balancing DM and timing precision, one could attempt to monitor pulsars at low frequencies (generally at or below 400 MHz) and derive independent DM time series from those low-frequency data, to correct the higher-frequency data. This has the additional advantages that the DM modelling is now fully independent from the higher-frequency timing; and at low frequencies DM corrections could often be measured within a single observation, which avoids correlations with effects like timing noise or other timing-model parameters. This approach has its own drawbacks because other IISM effects like scattering also become more pronounced at low frequencies and so the DM measurements may be biased or corrupted. Finally, in some cases the differences in the Fresnel scale (the Fresnel scale is a basic measure for the size of the Fresnel zone, which in turn is the region of space a signal can travel through between emitter and receiver.) at the top and bottom of the observing band make it possible that the actual space probed by electrons at different frequencies is slightly different – and hence the DM as well. This results in a frequency-dependent DM, which has been theoretically predicted [38] and observed [45], but the overall impact of this phenomenon on PTA sensitivity has so far not been accurately quantified. • High-frequency timing: Finally, with sufficiently sensitive telescopes, the main observing frequency could be moved to higher frequencies, where the IISM effects are weaker. Pulsars also tend to be fainter at those frequencies, but in the case of highly sensitive telescopes, this may be a blessing since it implies jitter noise will be less significant, as more pulses will need to be averaged per observation. However, with present telescopes, this approach may require too much observing time [83] – with more sensitive, future, telescopes it may become a realistic option. When it comes to effects of the IISM, dispersion is only the peak of the iceberg. Things get a lot more complex when we consider multi-path propagation or scattering. This phenomenon arises when some of the photons get deviated from the straight line between the pulsar and Earth due to refraction and later get refracted

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back into the line of sight. In its simplest form, for a thin scattering screen with density inhomogeneities that follow a Kolmogorov spectrum, this should cause a delay in photon arrival time which scales with the observing frequency to the fourth power: τscat ∝ ν −α , where the scattering index α is theoretically expected to be 4.4. In practice, Bhat et al. [21] have measured an average spectral index of the scattering strength of α = 3.9 ± 0.2, slightly inconsistent with theory. More detailed studies at lower frequencies [51] have shown that in many cases the frequency scaling of the scattering was more consistent with a highly anisotropic scattering screen than with the typical Kolmogorov screen. Regardless of the frequency scaling, the primary observable effect of scattering in pulsar timing is that the pulse profile gets smeared out and gets a characteristic exponential tail. This worsens timing precision since it can wash out features and it may corrupt the DM measurement (although absolute DM measurements may not be necessary for pulsar timing experiments anyway), but in principle this would not affect GW sensitivity as long as the effect is constant in time. The strength of scattering does change in time, though, as can most readily be seen by inspecting a “dynamic spectrum”: a plot of pulsed intensity as a function of frequency and time. Such time-variable spectra often show changes in pulse intensity – a phenomenon known as scintillation. Diffractive scintillation (as shown in Fig. 6) occurs when photons travelling from the pulsar to Earth meet refractive structures and get slight phase shifts due to location-dependent refractive indices. These phase shifts cause constructive and destructive interference which are seen as bright and dark patches in the dynamic spectrum. Since scattering and diffractive scintillation are effectively two different observables caused by the same turbulent and diffractive structure, it should not come as a surprise that they are related – in fact, they are inversely proportional [93]. An extensive analysis of PTA data carried out by Levin et al. [93] quantified the variations in diffractive scintillation and consequently estimated how variable scattering is in typical PTA observations. This analysis showed that at present levels of sensitivity, scattering variations are only rarely a real concern [with one exception studied in detail by 92], but in the next era of highly sensitive telescopes (notably MeerKAT, FAST, and SKA), this will probably change. A further complication could arise from a higher-order effect where scattering and diffractive scintillation combine in what are known as “scintillation arcs” [139]. This primarily occurs in highly anisotropic media, when the majority of the radiation comes from the pulsar (essentially a point source), but significant amounts of energy come from other, typically straight and narrow, structures on the sky. It causes ripples across the dynamic spectrum, which are more easily noticed in the 2D Fourier transform of the dynamic spectrum – this is also called the “secondary spectrum” (see right-hand side of Fig. 6). The initial discovery of secondary spectra is still relatively recent, and since these tend to be faint features which require high

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Fig. 6 Dynamic and secondary spectrum of PSR J0826+2637 (PSR B0823+26). Left: the dynamic spectrum of a half-hour observation on PSR J08216+2637 with the LOFAR core. Shown is the pulsed intensity on an arbitrary color scale (units are uncalibrated) for a segment of 5-MHz bandwidth, centered on 147.5 MHz. The bright patches are called scintles and are caused by diffractive scintillation. Less clear is the higher-frequency corrugations that run diagonally across this dynamic spectrum and which are caused by a combination of diffractive and refractive scintillation. Right: this figure shows the secondary spectrum of the left-hand plot, i.e., the 2D Fourier transform, whereby the power levels are shown on a logarithmic scale. A highly asymmetric arc is visible, extending out to fractions of a millisecond at negative fringe frequencies (also called “Doppler rates”) but only out to about 0.1 ms at positive fringe frequencies. (Figure courtesy of Ziwei Wu).

sensitivity and high resolution (i.e., large data rates), their study has only developed slowly. However, early studies by Hemberger and Stinebring [58] already showed how these phenomena can impact timing significantly. Turning this around, Reardon et al. [121] showed how scintillation arcs can actually be useful for timing, as they can provide independent constraints on timing-model parameters. The study of how to use scintillation arcs, or what effects they really do have on PTA experiments, has only just begun, so at present their impact is not fully clear yet. A final concerning occurrence that the IISM may create is frequency-dependent DMs (also often – and confusingly – named “chromatic DMs”). The principle behind frequency-dependent DMs is as follows: in wide-band observations, photons with a wide range in wavelengths are observed. These photons did not all travel through the same space – in fact, since the Fresnel scale is frequency-dependent, there is a bias that causes lower-frequency photons to be able to travel through a wider region of space than higher-frequency photons. If the variations in electron density are sufficiently extreme – or if measurement precision is sufficiently high – this would imply that the high-frequency photons may sample a different electron distribution than the low-frequency photons. Consequently, the DMs measured at the top and bottom of the observing bands may be different because they refer to different parts of space.

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While the concept of frequency-dependent DMs had been known for much longer, the theoretical description was first laid out by Cordes et al. [38]. Detection of such a phenomenon is naturally complex given the many other frequencydependent effects described earlier, but by looking at the time difference of DMs measured at opposite parts of a low-frequency observation, Donner et al. [45] succeeded in making the first clear detection of such chromatic effects. These initial results showed rather a more complex picture than the theory had predicted, clarifying that further research into frequency-dependent DMs is required before a conclusive understanding of their possible impact on PTAs can be drawn. Given the continuous coverage over extremely wide frequency ranges of new telescopes like the uGMRT, ngVLA, MeerKAT, and the SKA, a much clearer understanding is bound to arise within the next decade.

Sensitivity Predictions As mentioned earlier, as PTAs edge closer to a GW detection, the number of pulsars in the array is of key importance for the PTA’s sensitivity. Since predicting pulsar discoveries is infamously hard, it is equally hard to make accurate predictions of PTA sensitivity given future pulsar discoveries, yet several papers have demonstrated the validity of the adage “More pulsars is more sensitivity.” Most recently, Kelley et al. [75] showed that – assuming ongoing regular discoveries of MSPs that can be timed at high precision – all PTAs could hope to detect GWs within a few years; and all were virtually guaranteed a detection within a decade. Based on the most up-to-date predictions for a GWB in the PTA band and realistic numbers from existing PTA experiments, Rosado et al. [124] showed that the most likely scenario would be that a GWB would be detected in about one to two decades time. A detection of CWs was also not unrealistic, but would probably take somewhat longer. Rosado et al. [124] did not investigate the impact of increasing the number of MSPs in the array, but did evaluate the impact of more sensitive systems – the SKA in particular – and found that with the full SKA, a GWB should be detectable within a few years and CWs within about a decade. The idea that a highly sensitive telescope could detect GWs within a few years, even with existing pulsars, was also demonstrated by Lee [85], who drew essentially the same conclusion for the CPTA with its unprecedentedly sensitive set of telescopes. Also Lazarus et al. [83] investigated the impact of improved sensitivity on PTA detection timescales, this time in the context of improved receiver and recording systems. Specifically, they estimated that the new recording system at Effelsberg would improve the telescope’s sensitivity to a GWB by a factor of up to three compared to the status quo, in only four years’ time. They furthermore continued the work by Lee et al. [88] to demonstrate how wide-band and low-frequency observing systems might aid the detection of GWs by efficiently correcting DM variations and thereby keeping the timing RMS low. Verbiest et al. [153] approached the sensitivity improvements in a very complementary way, showing that global collaboration and sharing of data would lead to approximately a factor of two improvement in GW sensitivity.

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It is also possible that the coming years continue to bring non-detections; this could occur if we encounter an unexpected instrumental noise floor, such as intrinsic pulsar noise or a high level of ephemeris uncertainty. However, this scenario is unlikely to be an issue. Pulsar noise can be overcome by targeted noise modelling [as in 55], or by simply adding more pulsars to a PTA, which will beat down noise that is uncorrelated between different pulsars, thus still raising our sensitivity to the correlated GW signals. Regarding uncertainties in the SolarSystem ephemerides, as previously noted on page 171, ephemeris uncertainties are correlated. The signal, however, is dipolar; thus, we can attempt to measure and remove it, even if there is some leakage between dipolar and quadrupolar signals [145]. In addition, it has been demonstrated that PTAs are already breaching the accuracy of published ephemerides, and techniques have been developed to overcome such uncertainties [147]. Regardless, current upper limits on the GWB are already impacting our understanding of the evolution of galaxies and their supermassive black hole residents [9, 34, 131, 134], in addition to placing novel constraints on cosmic strings [9, 156] and exotic forms of dark matter [e.g., 35]. For the interested reader, the wealth of accessible science with GW limits and detections is the subject of another large review [30]. For the purposes of this review, it suffices to say that if our limits continue to improve around an order of magnitude beyond their current point, it would be astrophysically surprising. This is because even the most pessimistic simulations of supermassive binary black hole evolution (where no galaxy merger results directly in a binary supermassive black hole due to inefficient inspirals) still result in signals detectable at the hyr > 10−16 level [25]. See also the  Chap. 19, “Massive Black-Hole Mergers” of this edition by E. Barausse and A. Lapi. In summary, there are a variety of ways in which PTAs can further gain sensitivity, and all of these ways are being explored. Essentially all predictions conclude a detection is likely within the next few years or at most within the next decade – regardless of which particular improvement is being studied. This is not surprising given that our most reliable predictions on the strength of the GWB are right up against our most constraining limit [9, 131], which strongly implies a detection is bound to be imminent.

Summary The extreme properties of the spinning neutron stars called pulsars enable a unique experiment to detect GWs in a spectral range that is highly complementary with other mature GW projects like LIGO and LISA. PTAs are expected to make the first detection of nHz GWs within the next few years, and in doing so will allow new and unprecedented constraints on galaxy formation and evolution scenarios. The way forward is long and hard, however, as data sets are complex and highly heterogeneous and a variety of noise sources need to be dealt with. Instrumental upgrades and extension and improvement of observing schedules and source lists

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are underway to further enhance sensitivity – a process that will culminate in the ultimate PTA to be ran on the telescope of the future: the SKA. With the added collecting area and the larger number of pulsars that the SKA will be able to time at high precision, GW astronomy in the nHz range can be expected to properly take off.

Cross-References  Massive Black-Hole Mergers

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5

Quantum Sensors with Matter Waves for GW Observation Andrea Bertoldi, Philippe Bouyer, and Benjamin Canuel

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atom Interferometry and GW Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principle of GW Detection Using Matter Waves Interferometry . . . . . . . . . . . . . . . . . . . . . Space-Based and Terrestrial Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classes of Quantum Sensor-Based Gravitational Wave Detector . . . . . . . . . . . . . . . . . . . . Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interferometer Arrays for Rejecting Gravity-Gradient Noise . . . . . . . . . . . . . . . . . . . . . . . Long-Baseline Atom Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MIGA: Matter-Wave Laser Interferometric Gravitation Antenna . . . . . . . . . . . . . . . . . . . . MAGIS: Mid-band Atomic Gravitational Wave Interferometric Sensor . . . . . . . . . . . . . . . ZAIGA: Zhaoshan Long-Baseline Atom Interferometer Gravitation Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AION: Atom Interferometry Observatory and Network . . . . . . . . . . . . . . . . . . . . . . . . . . . VLBAI: Very Long-Baseline Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Role of Atom Interferometry in GW Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Sources in the Atom Interferometry Detection Bandwidth and Multiband Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roadmap to Increase Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ELGAR: European Laboratory for Gravitation and Atom-Interferometric Research . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Bertoldi () · P. Bouyer · B. Canuel Laboratoire Photonique, Numérique et Nanosciences (LP2N), Université Bordeaux – IOGS – CNRS:UMR 5298, Talence, France e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_5

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Abstract

Quantum sensors exploiting matter waves interferometry promise the realization of a new generation of gravitational wave detectors. The intrinsic stability of specific atomic energy levels makes atom interferometers and clocks ideal candidates to extend the frequency window for the observation of gravitational waves in the mid-frequency band, ranging from 10 mHz to 10 Hz. We present the geometry and functioning of this new class of ground and space detectors and detail their main noise sources. We describe the different projects undertaken worldwide to realize large-scale demonstrators and push further the current limitations. We finally give the roadmap for achieving the instrumental sensitivity required to seize the scientific opportunities offered by this new research domain. Keywords

Atom interferometry · Cold atoms · Gravity gradiometry · Gravity-gradient noise · Mid-band gravitational wave detection · Atom-laser antenna · Multiband GW astronomy · Cold atoms · Quantum sensors

Introduction The observation of gravitational waves (GWs) [1] has opened a new era of GW astronomy that can bring new insight for the study of general relativity in its most extreme regimes, dark matter, or the exploration of the early universe, where light propagation was impossible. It is becoming possible to study a large range of GW sources and frequencies, from well-understood phenomena [2] to more speculative ones [3, 4]. To widen the reach of GW astronomy, it is necessary to explore a frequency range beyond that accessible by state-of-the-art detectors [5,6], which currently span from 10 Hz to 10 kHz [7]. While planned detectors will either push GW sensitivity in the current band with the third-generation ground-based laser interferometer (Einstein Telescope – ET) [8, 9], or investigate GWs sources at very low frequency with the space-based Laser Interferometer Space Antenna (LISA) [10], the critical infrasound (0.1–10 Hz) band [2, 11–13] is left open to new concepts. Sensors based on quantum technologies, such as atom interferometers (AIs) [14, 15] or atomic clocks [16, 17], can provide a plausible answer to this challenge. The continuous development over more than three decades of techniques to manipulate and coherently control ultracold atomic samples has pushed these sensors to unprecedented levels of accuracy and stability. Atomic clocks reach today a stability at about the 10−18 level [18–25], and their precision continues to improve so that one day it will be feasible, for instance, to utilize them for a direct detection of the gravitational field [26]. Atom optics and matter waves manipulation also pushed the development of new generation of force sensors exhibiting unprecedented sensitivity and accuracy [27, 28]. They can nowadays address many applications,

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such as probing inertial forces [29–31], studying fundamental physics, and testing gravitational theories [32–37]. The availability of these sensors motivated the emergence of concepts for ultrahigh precision measurements of time and space fluctuations, with direct application to tests of general relativity [14, 38, 39] and in particular with a potential to open an inaccessible window in GW detection. While the first proposals [40] were purely speculative, concepts have recently evolved to extended proposals for space [16,41] or ground [42, 43] with the current developments of lower scale version of what could be a future generation infrasound GW detector.

Atom Interferometry and GW Detection Principle of GW Detection Using Matter Waves Interferometry Since the first AIs were realized almost three decades ago, these elegant experimental demonstrations of quantum physics have evolved to instruments at the leading edge of precision measurements. They allow for measuring inertial or gravitational forces affecting the propagation of matter waves with a sensitivity and precision comparable to or even better than existing classical sensors. Their present performance and technological maturity provide breakthrough capacities in a variety of fields [33] from applied to fundamental sciences [27] such as navigation and gravimetry. They are nowadays developed both by academic teams and industry, with specific focus in miniaturization [44] and transportability [45,46] without compromising their performances. Examples are gyroscopes, gravimeters, gradiometers, with applications in navigation, geophysics, metrological determination of fundamental constants, and tests of GR. The reported precision and 9 sensitivity of these inertial sensors √ are parts in 10 of Earth gravity and rotation −9 rates at the verge of 10 rad/s/ Hz. These values compare favorably with current technologies, even outside the very quiet environment of a laboratory. In general, an AI uses a succession of light pulses tuned to a particular atomic frequency resonance that act as coherent beam splitters separated by a time T [47]. The first pulse splits the incoming matter wave into two wavepackets that follow different paths. In direct analogy with light, the accumulation of phase along these two paths leads to interference at the last beam splitter. Each output channels will then exhibit complementary probability as a sinusoidal function of the accumulated phase difference, Δφ. Most AIs follow the Mach-Zehnder design: two beam splitters with a mirror inserted inside to fold the paths (see Fig. 1). The sensitivity of such an interferometer is defined by the area enclosed by the two atomic trajectories. When atoms are subject to acceleration or rotation along their trajectory, their speed along this trajectory is modified, which modifies their de Broglie wavelength and ultimately leads to a variation of Δφ. The output of the AI blends then together the effects of rotation and acceleration, as well as unwanted contributions from wave front distortions and mirror vibrations. For instance, if the platform containing the

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Fig. 1 Space-time diagram schematic of an AI using light pulses. The atomic trajectories are represented in black: the solid lines refer to the propagation in state |1, the dashed ones in state |2. The propagation paths are represented as straight lines, whereas uniform gravity makes them parabolic. The two states have a momentum separation given by the two-photon momentum exchange imparted by the interferometric pulses, represented by the sinusoidal red lines. (Taken from [15] under CC BY 4.0)

laser beams accelerates, or if the atoms are subject to an acceleration a, the phase shift becomes: Δφacc = keff aT 2 ,

(1)

where keff is the effective wave number of the coherent manipulation laser. When the laser beams are vertically directed, the interferometer measures the acceleration due to gravity g. Remarkably, ppb-level sensitivity, stability, and accuracy have been achieved with such gravimeter [28]. Combining two such interferometers separated in space and using a common laser beam is well-suited to measure gravity gradient [30] as a result of the differential phase shift. In this way, major technical background noises are commonmode rejected, which leads to nearly identical phase shifts if each interferometer is subjected to the same acceleration. This configuration can measure Earth’s gravity gradient, as well as the modification of gravity from nearby mass distributions [48]. In the laboratory, gravity gradiometers have achieved resolutions below 10−9 s−2 and allowed the precise determination of the gravitational constant G [49–51]. The dependence of Δφ to inertia can more generally be seen as a direct consequence of the propagation of matter waves in curved space-time [38, 52], thus leading to consider atom interferometry as a potential candidate for precision tests of general relativity and consequently for GW detection [40]. A GW affecting the AI would typically induce a phase shift Δφ ∼ h(t)T vkeff /2π , where h(t) is the GW strain amplitude and v the velocity of the atoms entering the interferometer [53]. For this effect to be large considering a single atom, the instrument would have to be of unreasonable size (the largest AIs are of meter scales, whereas hundreds of meters would be needed) and use high velocity (besides the fact that sensitivity improves with cold-slow atoms). These limiting factors vanish when the AI is not solely used to read out the GW dephasing but to differentially measure the effect of the GW on

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Fig. 2 Atom interferometry-based GW detector using a gradiometric configuration of baseline L. The AIs are placed at Xi,j . A common interrogation laser is retro-reflected by a mirror at position MX . (Taken from [15] under CC BY 4.0)

the propagation of light (see Fig. 2), either by measuring how the light propagation time can be affected [16] or how the laser phase can be modulated [14]. This connects directly to the current GW detectors based on laser interferometry [5,6,10], but with the use of matter waves acting as quasi-perfect, free-falling, proof masses. The resulting spacecraft requirements for space-based missions are significantly reduced, and access is granted, on Earth, to frequencies usually hindered by seismic and more general geophysical noise [42, 54, 55].

Space-Based and Terrestrial Instruments There are a strong interest and expected synergies to ultimately use concomitantly both terrestrial and space-based GW detectors. Today, GW observation is performed thanks to three running ground-based detectors, and new ones are currently under development or nearly ready. Location on the ground provides an ideal environment to develop and further enhance novel concepts to either improve current instruments (such as the use of quantum states of light in LIGO [56] or VIRGO) or develop new ones [8, 57]. There are two major challenges in order to improve the detectors sensitivity and expand it to lower frequency. First, vibrations (or seismic noise) need to be mitigated. In LIGO/VIRGO, this is achieved by levitating the mirror proof masses thanks to suspension systems which set the current lower band limit to about 10 Hz. Atom interferometry solves this problem by using free-falling atoms (that are naturally isolated from vibrations) and by tuning the AI so that its maximum sensitivity is at the desired low frequency [14, 60]. Second, even without vibrations (for instance, by setting the instrument deep underground), the fluctuations of mass around the instrument lead to fluctuations of the gravitational forces that typically forbid any signal to be detected below a few Hz [54]. This gravity-gradient noise (GGN) or Newtonian noise (NN) may finally be mitigated by mapping it using a network of ultraprecise accelerations sensors that can correlate the noise to the measurement itself [55]. Space, on the other hand, provides the perfect environment to detect and monitor GW at very low frequencies. Spacecraft can be set drag-free to be in perfect free fall, and GW signals can be extracted by monitoring the propagation time of the light between two satellites over very large distance (kilometers to millions of kilometers), as shown in Fig. 3. This gives access to very low frequency (mHz

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Fig. 3 Space- and ground-based detectors. (left) In the DOCS proposal [58], spacecrafts 1 and 2 are set to one of the Earth-Moon Lagrangian points and to deep space (L = 1.5 AU), respectively. Each of the spacecrafts has an optical clock on board. A radio signal with a nominal frequency ν0 is transmitted from spacecraft 1 (or 2) to spacecraft 2 (or 1), namely, using two one-way link (Taken from [58] under CC BY 3.0). (right) Diagram of the setup for a terrestrial experiment, taken with permission from [59], copyright by the American Physical Society. The straight lines represent the path of the atoms in the two IL ∼ 10 m interferometers I1 , and I2 separated vertically by L ∼ 1 km. The wavy lines represent the paths of the lasers

to Hz) GW signatures that can be precursors of larger signals later monitored by ground-based detectors [12]. Space provides also an ideal environment for atom interferometry, since the lack of gravity opens for enhanced sensitivities, with larger interrogation times T and interferometer arms separations keff T that approach the initial ideas of [40].

Classes of Quantum Sensor-Based Gravitational Wave Detector Before performing a phase measurement with an AI, many steps must be performed [33]: prepare an ensemble of cold atoms with the proper velocity distribution, i.e., temperature and state, perform the appropriate sequence of matter wave interferometry, and finally read out the interferometer signal and extract the information about the phase. A light-pulse interferometer sequence uses, to produce the interference, a series of laser pulses applied on atoms that follow free-fall trajectories (see Fig. 1). This process relies on the exchange of momentum h¯ keff between the atoms and the lasers, while at the same time avoiding to drive spontaneous emission from the laser excitation. Several schemes can achieve this. They either rely on two-photon

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processes – where the atoms are always in their ground, non-excited states – such as Bragg diffraction [61] or two-color Raman processes [47] or on single-photon excitation of a nearly forbidden, long-lived transition [41].

Two-Photon Transition-Based Interferometers Bragg and Raman two-photon diffraction processes both rely on the same principle: two lasers are far detuned from an optical transition, and their frequency difference is set equal to the atom’s recoil kinetic energy plus any internal energy shift. During a pulse of light, the atom undergoes a Rabi oscillation between the two states |1, p and |2, p + keff ). A beam splitter results when the laser pulse time is equal to a quarter of a Rabi period ( π2 pulse), and a mirror requires half a Rabi period (π pulse). In the specific case of Raman diffraction, the internal and external degrees of freedom of the atom are entangled, resulting in an energy level change and a momentum kick (see Fig. 4). The typical AI configuration of Fig. 1 follows a beam splitter-mirror-beam splitter ( π2 − π − π2 ) sequence [47]: the initial beam splitter ( π2 ) pulse creates a superposition of states which differ in velocity by keff /m. The resulting spatial separation after a time of flight T sets the interferometer’s sensitivity to gravity along the direction of keff . The mirror (π ) pulse reverses the relative velocity, and the final ( π2 ) pulse, applied at time 2T , interferes these overlapping components. The interferometric fringes are then detected using light-induced fluorescence detection. If the detection is limited only by the quantum projection noise of the atoms (atom shot noise), a sensitivity √ the phase differences Δφ can be measured with √ below 10−3 rad/ Hz, which corresponds to the shot noise limit 1/ N for 106 atoms in the interferometer. Higher atom number, or the use of squeezed atomic states instead of uncorrelated thermal atom ensembles, can increase further this sensitivity. If two light-pulse interferometer sequences are performed simultaneously at different positions (see Fig. 2) in a gradiometer configuration, the gravitational wave of strain amplitude h and frequency ω will typically produce a differential acceleration signal ∼hLω2 on the interferometers, √ as shown in detail in section “Noise Sources”. A√ phase sensitivity of 10−5 rad/ Hz can target√ an acceleration sensitivity of 10−15 g/ Hz leading to a strain sensitivity of 10−18 / Hz for L = 1 km.

Fig. 4 (left) Stimulated Raman transition between atomic states |1 and |2 using lasers of wavelength λ1 and λ2 . (right) Rabi oscillations between states |1 and |2. A π2 pulse is a beam splitter since the atom ends up in a superposition of states |1 and |2, while a π pulse is a mirror since the atom changes from state |1 to state |2

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Each interferometer sequence must be repeated in order to record the GW signal. The measurement repetition rate will put limits of the GW detection frequency range; usually it is limited by the time to produce the sample of cold atoms and can thus increase when using colder samples. On Earth, typical repetition rates of 10 Hz or higher must be achieved for allowing to target frequencies up to 5 Hz without reaching sampling problems. In space, where target frequencies can be as low as ∼10−3 − 1 Hz, the sampling rate can be reduced below 1 Hz. To fulfill this requirement, interleaved measurement sequences will be adopted [62, 63]. As for light-based interferometer detection, laser frequency noise is one limiting factor in this gradiometer configuration. Usually, a configuration of two orthogonal interferometer arms can exploit the quadrupolar nature of gravitational radiation to separate gravitational wave-induced phase shifts from those arising from laser noise. For the gradiometer configuration of Fig. 2), laser frequency noise is suppressed since the same laser beams interrogate both ensembles of atoms along a common line-of-sight. Nevertheless, the time delay between the two AIs and the need of two counter-propagating laser beams for each AIs [30] leaves a residual sensitivity to laser frequency noise, or to optical elements vibrations, and the impact of this effect increases with the baseline length L. One solution consists in eliminating the retroreflected laser beams altogether by driving optical atomic transitions with a single laser [41], as recently demonstrated in [66] (Fig. 5).

Optical Clocks and Single-Photon-Based Interferometers In contrast to Raman or Bragg two-photon diffraction, driving a narrow-linewidth clock transitions only requires a single resonant laser beam. In these clock transitions, for instance, the strontium 1 S0 → 3 P0 one, the spontaneous emission loss from excited state decay can be neglected thanks to the 150 s lifetime [64], and if needed minimized by evolving both paths of the interferometer in the ground state by means of extra pulses [41]. As for optical atomic clocks [21], interferometry pulses can be performed by driving a Rabi oscillation with a low-linewidth laser frequency stabilized to an optical cavity, allowing to build an AI following similar sequences as for two-photon transitions [17, 65, 66]. In a single-photon Mach-Zehnder interferometer, the shot noise-limited accel√ eration sensitivity is given by δa = (nk NT 2 1)−1 , where k = 2π/λ is now the wave number of the clock laser of wavelength λ. To increase the sensitivity, one can adopt large momentum transfer (LMT) techniques resulting in increasing n [17]. Due to the small dipole moment of the clock transition, high-intensity laser beams (1 kW/cm2 ) are generally needed to obtain a sufficiently high Rabi frequency. Moreover, a high-spectral purity laser source is necessary to drive a high-quality factor Q optical transition [21, 66]. With this in mind, various concepts of space GW detectors have been proposed [13, 67], first relying on the extension of previous space gradiometry concepts [68], and finally raising the idea that there is little conceptual difference between the atom interferometry-based differential phase readout and time delay heterodyne measurements [69, 70]. This led to concepts for space mission relying on optical clocks [16, 25, 58] as pictured in Fig. 3.

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Fig. 5 Experimental apparatus of an ultracold strontium AI (a) using interferometry pulses on the clock transition at 698 nm shown in red on the energy diagram (b). Laser radiation at 698 nm is frequency stabilized in two steps to ultra-stable optical cavities (c). MOT, magneto-optic trap; PDH, Pound-Drever-Hall; MOPA, master oscillator power amplifier; PM, polarizationmaintaining; ECDL, external-cavity diode laser; DM, dichroic mirror; PMT, photomultiplier tube. (Taken from [66] under CC BY 3.0)

Single-photon AIs are also central to ground-based projects currently under development [71, 72], where the key idea is to exploit the laser frequency noise immunity to develop single-arm GW detectors that could extend their sensitivity and open new fields such as dark matter or dark energy surveys. While clock transitions can open prospect of a major leap in sensitivity and might allow for using very long resonant cavities, there are still many open questions about the noise-induced hindering of GW signatures that are shared with the current designs relying on Raman transitions [15].

Noise Sources In this section, we will consider the main noise sources related to the geometry and functioning principle of an AI GW detector. Reaching a given GW target sensitivity

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will require to consider other backgrounds impacting the sensitivity, which are discussed in detail in [73]. These main noise sources can be understood by analyzing the standard geometry introduced earlier, i.e., the gradiometric configuration based on the interrogation of distant atoms clouds using two-photon transitions as shown in Fig. 2. In this setup, the AIs are created by pulsing a common interrogation laser retro-reflected to obtain the two counter-propagating electromagnetic fields that drive the twophoton transitions. At the time of each pulse, the difference of the laser phase Δϕlas between the two interrogation fields is imprinted on the atomic phase. At the output the interferometer, the measured variation of atom phase Δφ results from the fluctuation of the interrogation laser phase during the duration of the interferometer [38]; these fluctuations may arise from a GW signal or from different noise sources affecting the laser phase. The specific number and sequence of pulses used to manipulate the atomic wavefunction define the sensitivity of the measurement. A given AI geometry is thus associated with a sensitivity function g(t) [74] which provides the interferometer output Δφ as a function of Δϕlas :  Δφ(Xi , t) = n

∞ −∞

Δϕlas (Xi , τ )g  (τ − t)dτ + (Xi , t) ,

(2)

where 2n is the number of photons coherently exchanged during the interrogation process and (Xi , t) is the detection noise, i.e., the atom shot noise [75] that we now further explain. At the output of the interferometer, the transition probability P between the two atom states coupled by the interrogation process is given by a two-wave interference formula P = 1/2 [1 − cos(Δφ)], and the transition probability is usually measured by fluorescence of the atomic ensemble to recover the atomic phase. During this measurement process, the wavefunction of each atom is projected on one of the two states with a probability, respectively, of P and 1 − P . The noise in the evaluation of P, and thus Δφ, follows a Poissonian statistics and is inversely proportional to the square root of the number of atoms used for the measurement. As an example, an √ atom flux of 1012 atoms/s will result in an atom shot noise of S = 1μ rad/ H z. As seen in section “Classes of Quantum Sensor-Based Gravitational Wave Detector”, the use of an entangled source of atom can improve this limitation potentially up to the 1/N Heisenberg limit. In the following numerical applications, we will consider the same source with 20 √dB squeezing [76, 77] that enables to reach a detection limit of S = 0.1μ rad/ H z. We now derive the GW strain sensitivity of an atom gradiometer: in addition to detection noise, we also consider the main noise sources affecting state-of-the-art GW detectors [5, 6] such as interrogation laser frequency noise δν(τ ), vibration of the retro-reflecting mirror δxMX (τ ), and NN that introduces fluctuations of the mean atomic trajectory along the laser beam direction δxat (Xi , τ ). These effects affect the local variation of Δϕlas and Eq. (2) can be written as [15]:

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Δφ(Xi , t) =

209

h(τ )  (MX − Xi ) ν 2 −∞    MX − Xi  δxMX (τ ) − δxat (Xi , τ ) g  (τ − t)dτ + δxMX (τ ) − c ∞

2nkl

 δν(τ )

+

+ (Xi , t) ,

(3)

where h(τ ) is the GW strain variation and kl = 2πc ν is the wave number of the interrogation laser. By considering the gradiometric signal ψ(Xi , Xj , t) given by the difference of the outputs of the two AIs placed at positions Xi and Xj , we obtain:  ψ(Xi , Xj , t) = Δφ(Xi , t) − Δφ(Xj , t) =

∞ −∞



2nkl

 δν(τ ) ν

+

 h(τ ) δxMX  − L 2 c

+ δxat (Xj , τ ) − δxat (Xi , τ ) g  (τ − t)dτ + (Xi , t) − (Xj , t) , (4) where L = Xj − Xi is the gradiometer baseline. Considering that the detection noise of the two AIs is uncorrelated, the PSD of this signal is then given by:  S (ω) S (ω) ω2  δν h + + S (ω) L2 δx M X 4 ν2 c2  + SN N1 (ω) |ωG(ω)|2 + 2S (ω) ,

Sψ (ω) = (2nkl )2

(5)

where S. denotes the power spectral density (PSD) operator, G is the Fourier transform of the sensitivity function g, and SN N1 is the PSD of the relative displacement of the atom test masses due to Newtonian noise: NN1 (t) = δxat (Xj , t) − δxat (Xi , t). Using this gradiometric configuration, the signal-to-noise ratio (SNR) for detecting GWs at a given frequency is given by dividing the GW term of Eq. (5) by the sum of all the other terms. The strain sensitivity at a given frequency is then obtained by considering the GW strain corresponding to an SNR of 1 and is given by: Sh =

2 4Sδν (ω) 4SN N1 (ω) 4ω SδxMX (ω) 8S (ω) + + + . ν2 L2 c2 (2nkl )2 L2 |ωG(ω)|2

(6)

The detector should be ideally designed such that the dominant noise is the detection limit coming from atom shot noise, last term of Eq. (6). This term is depending on the transfer function of the AI, which must be chosen mainly with respect to sensitivity to spurious effects and compatibility with LMT schemes. As an example, we report in Fig. 6 the strain sensitivity curves for standard 3-pulses “π/2-π -π/2” and 4-pulses “π/2-π -π -π/2” interferometers of respective

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Strain (Hz-1/2)

10-18

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Frequency (Hz) Fig. 6 GW strain sensitivity of a 16.3 km atom gradiometer for 3-pulse and 4-pulse geometries. Parameters of calculation in the text

  and |ωG4p (ω)|2 = sensitivity transfer functions |ωG3p (ω)|2 = 16 sin4 ωT 2   2 4 ωT 64 sin (ωT ) sin 2 [74, 78]. The curves are calculated for a total interferometer time of 2T = 600 ms, a√number of photon transfer n = 1000, a shot noise level of S = 0.1μ rad/ H z, and a gradiometer baseline of L = 16.3 km. These parameters are considered for the matter wave-based GW detector proposed in [55], see section “Interferometer Arrays for Rejecting Gravity-Gradient Noise”. We observe that the strain sensitivity for the two interferometer √ reaches respectively√ configurations about 2.1 × 10−22 / Hz and 2.7 × 10−22 / Hz at a corner frequency (lowest frequency at which the maximum sensitivity of the detector is reached) of 1.7 and 2.2 Hz. For a given AI geometry, the corner frequency, and thus the detection bandwidth, is inversely proportional the total interferometer time. We now discuss the other noise terms listed in Eq. (6). For the seismic noise contribution, we observe that the position noise of the retro-reflecting mirror δxMX (see Eq. (3)) is a common noise for the two AIs and is thus rejected in the gradiometer signal. Still, seismic noise can impact the measurement through residual sensitivity to mirror velocity and induce a strain limitation of 4ω2 /c2 SδxMX (ω) (third term of Eq. (6)). √ As an example, considering a seismic displacement noise at 1 Hz of 10−9 m/ Hz, typical √ of sites with good seismic conditions, the strain limitation will be 2 × 10−17 / Hz. An atom gradiometer with parameters of Fig. 6 would therefore require a suspension system to reach the atom shot noise limit. Remarkably, such system would have less stringent requirements with respect to the suspensions needed in an optical GW detector [79, 80]: considering, for example, the case of the 4-pulse AI of Fig. 6, an isolation factor of only 2.5 × 104 is needed at a frequency of 1 Hz. This difference stems from the important common mode rejection factor introduced by the gradiometric configuration. For what concerns frequency noise of the interrogation laser, we observe that the relative fluctuations it causes are indistinguishable from the GW effect. Examining

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again the example of Fig. 6, the required laser stability in order to be shot noise limited is about five orders of magnitude beyond the state of the art of pre-stabilized lasers [81]. This issue can be solved adopting, as in optical interferometry, a two orthogonal arm configuration, see later section “ELGAR: European Laboratory for Gravitation and Atom-Interferometric Research”. Local gravity perturbations of different geophysical origins, commonly referred to as Newtonian noise, create spurious gradiometric signals. Indeed, the atom gradiometer configuration is similar with an optical GW detector with the atoms as test masses instead of mirrors. Thus, any differential gravity perturbations between the test masses will impact the signal in the same way as GWs do. Since the first generation of GW detectors, NN has been identified [82] as an important source of noise and extensively studied [54]. It will constitute a potential limiting factor for third-generation detectors such as the Einstein Telescope [8] in their low-frequency detection window, at a few Hz. As seen in Fig. 6, atomic gradiometers target a detection bandwidth centered at even lower frequencies, and NN represent a critical issue to reach the ultimate detector performances linked to atom shot noise. In the next section, we give NN projections on atom gradiometers and present dedicated methods and detector geometries developed to reduce its impact.

Interferometer Arrays for Rejecting Gravity-Gradient Noise Any anthropological or geophysical process implying a mass transfer or a fluctuation of density of the medium around the detector can be a source of NN. Among these different sources, the main stationary components early identified as possible limitations for GW detector in the infrasound domain come from medium-density fluctuations due to the local atmospheric and seismic activity [82], respectively, named infrasound and seismic Newtonian noise (INN and SNN). Figure 7 [55] shows projections of both contributions on the strain measurement of a single gradiometer with a baseline of L = 16.3 km (dashed black and blue curves). The density variations are calculated from [82, 83] using as input an air pressure fluctuation spectrum of Δp2 (ω) = 0.3 × 10−5 /(f/1Hz)2 Pa2 /Hz and a seismic noise of 1 × 10−17 m2 s−4 /Hz at 1 Hz. We observe that for frequencies < 1 Hz both noise curves stand well above the shot noise limit of a single gradiometer as discussed for Fig. 6. To reduce the impact of NN on atom interferometry-based GW detectors, it has been proposed to use not a single gradiometer but an array of them [55], with a geometry optimized to statistically average the NN. Indeed, GWs and NN signals will have a different spatial signature over the gradiometers of the array: while GWs have a strict plane wave structure, NN has a coherence length of a few kilometers typically in the infrasound domain for the sources considered. We now detail further the method and performances obtained using the detector geometry of Ref. [55], shown in Fig. 8. The detector is formed by two symmetric arms along orthogonal directions, interrogated by a common laser. Each arm is formed by N = 80 gradiometers of baseline L = 16.3 km and the separation between the

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strain sensitivity (Hz-1/2 )

10 -20 INN

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10 -24 10 -1

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frequency (Hz) Fig. 7 Dotted-dashed black (dashed blue): INN (SNN) of a single gradiometer of baseline L. Green: atom shot noise strain limitation for a 2D gradiometer array. Black (blue) line: residual strain INN (SNN) using the averaged signal of the array. Red line: overall strain sensitivity of the array. (Taken from [55])

gradiometers is set to δ = 200 m. Averaging the signals from different gradiometers in each arm averages the contribution of NN but maintains the GW contribution. We therefore consider the difference between the average signals of the gradiometers of each arm, given by: HN (t) =

N 1 ψ(Xi , XN +i , t) − ψ(Yi , YN +i , t). N

(7)

i=1

As for an optical GW detector, this two arm configuration is immune to frequency noise of the interrogation laser for a (+) polarized GW impinging on the detector. Using the method presented in the previous section and neglecting seismic noise (see details in Ref. [15] for discussion on this term), the strain sensitivity using the average signal of Eq. (7) can be written as: Sh (ω) =

SN N (ω) 4S (ω) + , 2 L N(2nkl )2 L2 |ωG(ω)|2

(8)

where N N (t) is the differential displacement on the test masses of the array induced by NN given by:

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L d

d

d

d Fig. 8 Proposed detector geometry, based on a 2D array of atom gradiometers of baseline L regularly separated by a distance δ. (Taken from [55])

N  1  δxat (XN +i , t) − δxat (Xi , t) − δxat (YN +i , t) + δxat (Yi , t) . N i=1 (9) √ Figure 7 shows the residual NN strain limitation of the array SN N (ω)/L from INN and SNN (resp. black and blue lines). We observe that both contributions are much reduced with respect to the strain limitations they induce on a single gradiometer (dashed black and blue lines). At 1 Hz, the gain on the INN and SNN is respectively of √ 30 and 10: the proposed method averages NN with a reduction factor better than 1/ N in some frequency range. Indeed, the spacing between gradiometers, set by the parameter δ, is chosen to have anti-correlation of the NN for the two AIs placed at this distance. A variation of delta negligible with respect to the NN correlation length will not spoil the rejection efficiency. The method presented here opens the way toward NN reduction for ground-based GW detector and is now considered in the design of the ELGAR instrument, see section “ELGAR: European Laboratory for Gravitation and Atom-Interferometric Research”.

NN (t) =

Long-Baseline Atom Interferometers Long-baseline sensors based on the coherent manipulation of matter waves are increasingly considered as disruptive tools to investigate fundamental questions related to the nature of the universe. They propose to drastically advance the scientific knowledge in three specific contexts: (i) the search for dark matter [37,84– 86]; (ii) the potential interplay between quantum mechanics and general relativity [52]; and (iii) the detection of GWs in the mid-frequency band uncovered by the LISA and LIGO-Virgo interferometers. The size of the experimental apparatus has direct implications on the ultimate sensitivity curve of atom gravity gradiometers, i.e., the configuration commonly adapted to measure tiny variations of the gravitational field: increasing the distance between the two atomic sensors improves linearly the instrument sensitivity to

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strain, given that the baselines typically considered are much shorter than the GW wavelength. Allowing for a wider wavefunction separation has two potential outcomes: (i) exploited to increase the interrogation time, it shifts the sensitivity curve to lower frequencies, and (ii) used to transfer a larger momentum separation with LMT techniques shifts the sensitivity curve vertically, thus improving it. All these configurations benefit from a longer baseline and justify the recent trend to build longer and longer instruments. The first important leap in the size of AIs is represented by the 10 m tall atomic fountains realized about 10 years ago, like those in Stanford [87] and Wuhan [88]. Other instruments of similar size are being developed nowadays both with the coherent manipulation along the vertical direction (e.g., Hanover [89] and Florence) or the horizontal one (e.g., MIGA prototype). We are now well into the second dramatic increase of the instrument size, with several experiments being built or proposed to realize on Earth AIs with baselines from a few hundred meters to a few tens of kilometers. At the same time, several actions are being carried out to study the potential scientific outcome of a future long-baseline instrument operated in space; different aspects – like specific technical solutions, instrument configurations, orbit selection, measurement protocols, and target signals – are being investigated for measurement baselines ranging from a few km to several millions of km. Within the actual initiatives to realize long-baseline atom gravity gradiometers on Earth, we can mention the following ones: • The Matter-wave laser Interferometric Gravitation Antenna (MIGA) experiment [42] (see section “MIGA: Matter-Wave Laser Interferometric Gravitation Antenna”), a horizontal interferometer that is being realized in the underground environment of the “Laboratoire Souterrain à Bas Bruit” (LSBB) [90]. Cold rubidium atomic ensembles launched in free fall and coherently manipulated with two horizontal, vertically displaced, and 150 m long laser beams will measure the differential gravity acceleration between the two extremes of the setup. The first-generation instrument will reach a strain sensitivity of 2×10−13 Hz−1/2 and will be a demonstrator for instruments of a later generation, characterized by an improved sensitivity thanks to more advanced atomic manipulation techniques, a longer baseline, and better protocols to reduce GGN. • The Mid-band Atomic Gravitational Wave Interferometric Sensor (MAGIS) experiment [43, 91] developed by a US consortium (see section “MAGIS: Mid-band Atomic Gravitational Wave Interferometric Sensor”); it plans a threephase experimental activity on Earth to prepare a future space mission for an instrument capable of detecting GWs in the frequency band [30 mHz–10 Hz]. The first two phases are already being developed, and they consists, respectively, in a 10 m atomic fountain in Stanford (CA-US) and in a 100 m vertical detector being realized in an existing vertical shaft at Fermilab (IL-US). The third detector has been proposed with a vertical baseline of 1 km and could be located at the Sanford Underground Research Facility (SURF in SD-US). The common solutions pursued for the three preparatory phases are the vertical interrogation

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configuration, which naturally opens toward long interrogation intervals, and the adoption of ultracold strontium atoms as gravitational probe. • The Zhaoshan long-baseline Atom Interferometer Gravitation Antenna (ZAIGA) experiment in Wuhan (China) [92] (see section “ZAIGA: Zhaoshan Long-Baseline Atom Interferometer Gravitation Antenna”), which consists in an underground facility for experimental research on gravitation and which will also host an horizontal, three 1 km arms on a triangle GW detector using rubidium atoms. • The Atom Interferometry Observatory and Network (AION) experimental program (see section “AION: Atom Interferometry Observatory and Network”), consisting of four successive phases to realize successively a 10 m, 100 m, 1 km, and a satellite mission targeting dark matter search and GW detection in the midfrequency band. AION proposes also to be operated in a network configuration with other GW detectors to optimize the scientific output by implementing multiband GW astronomy techniques [7] or exploiting the uncorrelation of far located instruments to look for stochastic background [93]. • The Very Long-Baseline Atom Interferometer (VLBAI) experiment (see section “VLBAI: Very Long-Baseline Atom Interferometry”), 10 m long atomic fountain, where ultracold ytterbium and rubidium atoms will be used to test several pillars of quantum mechanics and general relativity, looking for the intrinsic nature of the decoherence mechanisms, violation of the equivalence principle, and developing at the same time the enhanced atom interferometry tools to achieve the sensitivity required for detecting GWs. Other projects are in their very preliminary study phase, as is the case of the Italian project MAGIA-Advanced, which is studying the feasibility of a vertical instrument with a baseline of a few 100s meters to be installed in a former mine shaft in Sardinia [15], or are at the stage of study proposals such as the European Laboratory for Gravitation and Atom-interferometric Research (ELGAR) [15, 73] and the Atomic Experiment for Dark Matter and Gravity Exploration in Space (AEDGE) [13]. ELGAR, described in depth in section “ELGAR: European Laboratory for Gravitation and Atom-Interferometric Research”, proposes an underground array of gravity gradiometers with a total baseline of 32 km, adopting advanced atom interferometry techniques to mitigate Newtonian noise and achieve the required sensitivity to detect GWs in the [0.1–10 Hz] frequency band with a terrestrial instrument. AEDGE studies different space configurations exploiting matter wave sensors to push the boundaries of fundamental science, most notably concerning the nature of dark matter and the search of GWs in the frequency band intermediate to the maximum sensitivity of LISA and LIGO-Virgo.

MIGA: Matter-Wave Laser Interferometric Gravitation Antenna The MIGA [42] is a French ANR funded “Equipement d’Excellence” project to build a large-scale infrastructure based on quantum technologies: a hybrid atom and laser interferometer using an array of atom sensors to simultaneously measure the

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Fig. 9 The three AIs of the antenna will be located at (a), (b), and (c). The optical setups for cavity injection will be hosted in room (a). The vacuum vessel is pre-equipped to add other AIs in the future and reach a total of five measurement positions

gravitational waves (GWs) and the inertial effects acting in an optical cavity. The instrument baseline is designed to reach high sensitivity in the infrasound domain with a peak sensitivity at 2 Hz. This infrastructure will be installed at LSBB [90], in dedicated galleries located 300 m deep from the surface inside a karstic mountain. This site demonstrates a very low background noise and is situated far from major anthropogenic disturbances determined by cities, motorways, airports, or heavy industrial activities. A scheme of the antenna at LSBB can be seen in Fig. 9. Two horizontal cavities are used for the atom interrogation. The optics of the resonators are placed at the extremities of the detector inside vacuum towers (cylinders of 1 m height and diameter shown on panels (a) and (c)). A vacuum vessel with a diameter of 0.5 and 150 m long hosts the cavity fields that correlate three AIs regularly placed along the antenna (see details (a), (b), and (c) in Fig. 9). The optical benches for the cavity injection are hosted in room (a). In addition to this infrastructure, a shorter version of the instrument, an 6 m cavity gradiometer, is in construction at the LP2N laboratory in Talence. This equipment will be used to test advanced atom manipulation techniques that will be implemented later on the antenna. We now describe the working principle of the antenna by focusing on the measurement process of each AI, detail the status of its construction, and give prospects on the scientific results to pursue once the antenna will be operative.

Functioning and Status of the MIGA Antenna MIGA will require the simultaneous interrogation of the matter wave interferometers placed along its baseline by time-modulation of the laser injected in the cavity. The geometry of each AI, shown in Fig. 10, consists in a 3-pulse interferometer “π/2-π -π/2”. After a cooling and trapping sequence, a 87 Rb cloud is launched

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Fig. 10 (left) View of an AI of MIGA. A 87 Rb atom source uses a combination of a 2D and 3D magneto-optic traps. After being trapped and cooled, the atoms are launched on a parabolic trajectory and enter in the interferometric region where they are manipulated using a set of two cavity beams. (right) Geometry of the 3-pulse AI, taken from [42] under CC BY 4.0

on a vertical parabolic trajectory. When the atoms reach the lower cavity beam, they experience a first “π/2” Bragg pulse that creates a balanced superposition of the external states |±hk. When the cloud reaches the upper cavity beam at the ¯ apex of its trajectory, the atoms experience a “π ” pulse that reverses the horizontal atomic velocity. When the falling atoms reach again the lower cavity beam, a second “π/2” pulse closes the interferometer. The state occupancy of the two states at the interferometer output is then measured by fluorescence detection to extract the transition probability and so the interferometric phase Δφ. This signal is determined by the phase difference along the two paths followed by the matter waves inside the interferometer, which is in turn related to the variation of the phase Δϕlas of the counter-propagating cavity field. The response of the AI can then be obtained from Eq. (2) using the sensitivity function g(t) of the three pulse interferometer [74]. The GW strain variation induced on the phase of the cavity resonating field can be determined with a gradiometric measurement on two atomic sources separated by a distance L along the antenna. Considering the noise sources detailed in section “Noise Sources”, we obtain the following strain sensitivity [42]: 4Sδν (ω) 1  Sh (ω) = +

2 ν ω2 1 + ω2



4SNN (ω) 8ω2 Sx (ω) 8S (ω) + + 2 2 2 2 L ωp L (2nkl ) L2 |ωG(ω)|2

,

p

(10) where G(ω) is the Fourier transform of g(t) and ωp is the cavity frequency pole. When compared to the free-space gradiometric configuration treated in section “Noise Sources”, we observe that the MIGA cavity setup has a similar strain sensitivity for frequencies smaller than the pole of the cavity, whereas it shows a higher sensitivity to seismic noise Sx (ω), due to the cavity geometry that amplifies the impact of mirror displacement noise on the measurement.

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Fig. 11 (a) MIGA gallery within LSBB. (b) Standard 6 m long section under vacuum test. (c) Vacuum tower in production at SAES Parma (Italy)

We now describe the status of the realization of the antenna. Starting from 2017, heavy infrastructure works were carried out at LSBB for the installation of MIGA, and two new perpendicular 150 m long galleries were bored. These operations lasted till the beginning of 2019. One of the two galleries is being used to install the equipment, and the other one will host tests of mass transfer reconstruction and will later allow the instrument upgrade toward a 2D antenna. The MIGA galleries can be seen on Fig. 11a; they have a depth ranging from 300 to 500 m and their access is at about 800 m from the laboratory main entry. The vacuum vessel of the antenna is a set of 50 cm diameter SS304 pipes produced by SAES in Parma (Italy) during 2020; see Fig. 11b, c. It is mainly composed of sections of 6 m long connected using helicoflex gaskets and is designed to reach a residual pressure better than 10−9 mbar after a baking process up to 200 ◦ C. A 87 Rb cold atom source for the antenna can be seen in Fig. 12b. A combination of 2D and 3D magneto-optic traps prepares about 109 87 Rb atoms at a temperature of a few μK. An optical system based on Raman transitions selects from the cloud a narrow velocity class and a pure magnetic state before entering in the interferometric region. The detection of the atom state is then carried out by fluorescence of the cloud. This system works in combination with a robust and remotely operable fibered laser system [94] shown in Fig. 12a and a modular hardware control system [95].

MIGA Sensitivity and Prospects √ MIGA aims at an initial stain sensitivity of 2 · 10−13 / Hz at 2 Hz, which is several orders of magnitude short of targeting the GW signals expected in the band. In this sense, MIGA will be a demonstrator for GW detection, and a significant upgrade of both its baseline and the adopted laser and atom optics techniques will be required to fill the sensitivity gap. The initial instrument will be used to study advanced measurement strategies and atom manipulation techniques that could be directly implemented on the experiment so as to improve its sensitivity. These prospects are

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Fig. 12 (a) Fiber laser system developed by the Muquans company. (Image from [94]), (b) Cold 87 Rb atom source. (Image from [42] under CC BY 4.0)

Fig. 13 Atom shot noise limited sensitivity for the initial (dashed green line) and improved (plain green line) MIGA detector. Projections of strain infrasound GGN using the Bowman atmospheric pressure model [96] (dashed black lines) and using data measured at LSBB for calm (blue) and windy (red) periods. (Taken from [97])

10 -10

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illustrated in Fig. 13, which shows the MIGA strain sensitivity for its initial and upgraded version – the latter with an envisaged use of√LMT of 2 × 100 photon transitions and an improved detection noise of 0.1 mrad/ Hz. In the short term, the antenna will provide extremely high sensitivity measurements of the local gravity over large baselines which can be used to study how networks of atom gravimeters can resolve the space-time fluctuations of the gravity field. These studies are important for future GW interferometers, since GGN will be a limiting factor for their operation. As an example, we see in Fig. 13 that infrasound GGN could be detectable in the decihertz range with the upgraded version of the antenna.

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Fig. 14 Proposed site for the MAGIS-100 experiment, which will exploit the existing NuMI (Neutrino MAin Injector) vertical tunnel at Fermilab, indicated in yellow. (From [43], with permission of T. Kovachy and J. M. Hogan)

MAGIS: Mid-band Atomic Gravitational Wave Interferometric Sensor The MAGIS project [43, 91] will realize in different phases very long-baseline AIs, with scientific targets ranging from search for dark matter and new forces to test of quantum mechanics and general relativity at distances not yet investigated. One of the most important objectives is the detection of GWs in the 0.1–10 Hz frequency band, in the sensitivity gap between the Advanced LIGO and LISA experiments. The program will have different phases, each based on the development of a gravity gradiometer with increasing length: the first prototype is being already realized in Stanford, and it consists in a 10 m atomic strontium fountain, where key sensitivity enhancement techniques have been already demonstrated, as is the case of LMT clock atom interferometry [17]. The second step will consist in the realization of a vertical, 100 m long atomic fountain, called MAGIS-100, which will exploit an existing vertical shaft at the Fermilab (see Fig. 14). The planning and construction of the instrument have already started, backed by a consortium formed by Fermilab, Stanford University, Northern Illinois University, Northwestern University, Johns Hopkins University, and the University of Liverpool. The long-baseline instrument will have several challenging scientific targets, like creating quantum superposition states with unprecedented spatial and momentum separation, search for ultralight dark matter candidates, and test the equivalence principle of general relativity. The primary objective will be however to complement present and future instruments based on optical interferometry in the measurement of GWs, providing high strain sensitivity in the mid-frequency band (see Fig. 15). MAGIS-100 will be the precursor of a terrestrial instrument with an even longer baseline, possible 1 km, for which the SURF laboratory in South Dakota could be a potential candidate site. The final target is a space-based mission [91], where the gradiometric measurement uses two atomic sensors on dedicated spacecrafts placed at a distance of 4.4×104 km; the very long baseline, together with the elimination of the disturbance represented by the GGN thanks to the quiet environment

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Fig. 15 GW strain sensitivity curves for different generations and configurations of the MAGIS experiment, both ground- and space-based. For comparison purposes, the expected GGN limiting terrestrial detectors and the sensitivity curves of TOBA [98], Advanced LIGO, and LISA are also shown. (From [43], with permission of T. Kovachy and J. M. Hogan)

obtained with the orbiting satellites, will permit to deploy the full potential of atom interferometry. A strain sensitivity 1500), the spectrum shows a damping tail due to photon diffusion: the wavelength of the cosmological perturbation becomes smaller than the photon mean free path so that the sound wave cannot propagate (Silk damping), since it cannot be supported by the radiation pressure. At large angular scales (l < 100), a plateau is observed (the Sachs-Wolfe plateau) due to the fact that perturbations at these super-Hubble scales have not evolved and so directly reflect the initial perturbation. At these large scales, the measurements are affected by cosmic variance, due to the fact that we apply statistical analysis to only one sky (i.e., the only one we see).

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Fig. 1 Experimental CMB power spectra from the Planck mission [7] for temperature (top), the temperature-polarization cross-spectrum (middle), the E-mode of polarization (bottom left), and the lensing potential (bottom right). Error bars represent the 1σ uncertainties from the Planck 2018 c dataset; the blue line is the fitted power spectrum. (Reproduced with permission ESO)

CMB B-Modes and Foregrounds At early time in the universe’s history, following the thermal distribution the typical energy of a photon was much larger than the energy required to ionize the hydrogen atom. Thus, neutral atoms could not form, and the universe was filled with the socalled photon-baryon fluid with free electrons strongly interacting with radiation mainly through Compton scattering and with hydrogen ions through Coulomb

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interactions. The low-energy limit (i.e., for hν me c2 so that the recoil of the electron can be neglected and for non-relativistic electrons) of Compton scattering is the (elastic) Thomson scattering: the features of this process, namely, the fact that it is not isotropic and that it can polarize radiation, have a large impact on the CMB. The Thomson scattering differential cross section dσ/d is defined as the radiated intensity per unit solid angle normalized to the incoming intensity per unit area. For linearly polarized initial radiation: dσ |pol = d



e2 me c 2

2 sin2

(6)

where is the angle between the direction of oscillation of the electron and the outgoing direction: the preferred direction for the outgoing radiation is perpendicular to the input linear polarization and the polarization of scattered light peaks at angles parallel to that of incident light. The proportionality of the Thomson cross section with the inverse of the square of the particle mass allows us to focus on the interaction of the radiation with only the free electrons and not the ions. If unpolarized radiation impinges on a free electron, then, by treating it as an incoherent superposition of two orthogonally polarized waves, one finds the differential cross section to be: dσ |unpol = d



e2 me c 2

2

 1  1 + cos2 θ 2

(7)

where θ = π/2− is the angle between the directions of propagation of the incident and scattered radiation. Because of the angular dependence cos2 θ ∝ cos 2θ , the radiation has a quadrupolar pattern, i.e., it has π periodicity. It is important to note that the scattered radiation acquires some degree of polarization : =

1 − cos2 θ 1 + cos2 θ

(8)

So, in the case of incident unpolarized radiation, the polarization state of the scattered radiation depends on the direction of observation. Light scattered parallel to the input direction is seen unpolarized, since all directions in the orthogonal plane are equivalent. However, looking perpendicularly to the input direction, the symmetry is broken, and the emerging light is 100% linearly polarized along the direction perpendicular to the scattering plane. If unpolarized radiation incident on an electron comes uniformly from all directions with equal intensity (as seen in the rest frame of the electron), the outgoing (i.e., scattered) radiation would be unpolarized as seen by any observer. This is also true in the case where the radiation has a dipole anisotropy. However, if the intensity distribution of the unpolarized incident radiation has a non-vanishing quadrupolar symmetry, then the scattered radiation would have some net degree of

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linear polarization. In general, the polarization pattern will reflect the quadrupole anisotropy of the radiation: a didactic discussion of the CMB polarization pattern is offered in Hu et al. [8]. At early times in the history of the universe, before recombination, the tight coupling between the photons and the electrons and baryons makes the CMB radiation isotropic in the rest frame of the electron: the development of quadrupole fluctuations of the radiation field seen by the electrons is prevented so no net polarization can develop. However, as time passes and recombination proceeds, i.e., the capture of electrons by protons becomes efficient, and the redshifted photons are less efficient in ionizing the newly formed neutral hydrogen atoms. As a result, the density of free scattering electrons (i.e., those which have not recombined yet) decreases, the Thomson mean free path becomes larger than the Hubble radius, and the photons begin to free stream. In such condition (occurring at redshift z about 1100), the remaining free scattering electrons see an anisotropic radiation field with local quadrupole components: these become sources of polarization. A significant degree of polarization requires on the one hand an adequate number of free scattering electrons, but on the other hand, it also requires a low optical depth (otherwise the polarization would be averaged out). So any polarization observable would come from a short length of time before the last scattering: after photons have started free streaming but before the end of recombination. If recombination is instantaneous, i.e., the surface of last scattering has vanishing width, then the polarization is zero. In general, if quadrupolar temperature anisotropies are present, they produce a degree of polarization. Polarization fluctuations can originate in the presence of a background of tensor perturbations in the metric at the time of recombination: the gravitational waves (GW). Measurements of CMB radiation polarization can hence offer information about the presence of a significant cosmological GW background, predicted by inflationary models, as well as on the ionization history of the universe. The ratio of total polarization to total anisotropy depends on the duration of recombination and on later reionization [9]: Standard Cosmological Model expectations are around 10% polarization.

Primordial B-Modes To describe the polarization of the CMB, we follow the discussion in Kamionkowski et al. [10]. Let us consider first an electromagnetic wave of angular frequency ω travelling along the zˆ axis. Its components are: Ex = ax cos(ωt − θx )

,

Ey = ay cos(ωt − θy )

(9)

The polarization status of the wave is defined by the relation between θx and θy . In particular, the polarization of the EM radiation can be described via the Stokes parameters (I, Q, U, V ) (The polarization of a wave with a constant or slowly varying polarization can be described with the aid of the polarization ellipse.

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However, for light which is partially polarized, and hence that can be considered as the sum of a completely polarized component, and a randomly polarized component, the polarization ellipse would be varying too fast in time or be undefined. For such cases, one can introduce time-averaged quantities derived from the polarization ellipse, which are the Stokes parameters.): I =< ax2 > + < ay2 > Q =< ax2 > − < ay2 >

(10)

U =< 2ax ay cos(θx − θy ) > V =< 2ax ay sin(θx − θy ) >

where represents the time average. Here I is the intensity of the wave; Q and U are related to the shape and orientation of the polarization ellipse: Q is the linear polarization along the x and y axes (Q/I = +1and − 1 for light linearly polarized along the x and y axes, respectively), U describes the linear polarization along the axes rotated by 45 degrees counterclockwise relative to the x and y axes, and V describes the handedness of the ellipse and thus the circular polarization. If the intensity of the unpolarized incoming radiation is expanded in spherical harmonics as: I  (θ, φ) =

l ∞  

(11)

alm Ylm (θ, φ)

l=0 m=−l

then after Thomson scattering the outgoing radiation would have Stokes components [11]: Q ∝ σT Re(a22 )

,

U ∝ σT I m(a22 )

,

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(12)

This shows that Thomson scattering creates polarized radiation fields from an unpolarized quadrupolar radiation field. Note also that Thomson scattering can never create circular polarization, so V = 0 in cosmology. While the observables I and V are independent of the coordinate system, both Q and U depend on the coordinate system. Thus, the representation of polarization with the Stokes parameters is not useful if one wants to quantify the physics in a coordinateindependent way. This is why the polarization tensor is introduced to describe the linear polarization states. In a curved sky, that is, on a sphere where the metric is:

gab

  1 0 = 0 sin2 θ

(13)

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one can define a traceless (g ab Pab = 0), symmetric (Pab = Pba ) tensor (STF) to represent the linear polarization on a plane tangent to photon propagation along direction nˆ = (θ, φ) as: P(n) ˆ =

  1 Q(n) ˆ −U (n) ˆ sin θ ˆ sin θ −Q(n) ˆ sin2 θ 2 −U (n)

(14)

Similar to the expansion of the temperature in spherical harmonics (as in Eq. (1)), the polarization tensor itself can be decomposed in terms of tensor spherical harmonics as the sum of a “gradient” (labelled as E, in analogy to the electric field which can be written as the gradient of a scalar function) and “curl” (labelled as B, in analogy to the magnetic field) components [10]: l  ∞    Pab E E B B alm = Y(lm)ab (n) ˆ + alm Y(lm)ab (n) ˆ Tcmb

(15)

l=2 m=−l

where the coefficients are: E a(lm) = B a(lm)

=

1 TCMB 1 TCMB

 

E ab ∗ ˆ Y(lm) (n) ˆ d nP ˆ ab (n)

(16)

B ab ∗ ˆ Y(lm) (n) ˆ d nP ˆ ab (n)

(17)

E B and where the basis functions Y(lm)ab and Y(lm)ab are the covariant derivatives on the sphere of the spherical harmonics Ylm and are a complete basis set for expanding E-type and B-type STF tensors, respectively. This E/B decomposition plays a key role in studying CMB polarization. T , a E , and a B can completely describe In particular, the multipole coefficients alm lm lm the CMB map, in terms of both temperature and polarization. Note that because T , Q, and U are real: x∗ x alm = (−1)m al−m

for x ∈ {T , E, B}

(18)

So the two-point statistics of the CMB are fully characterized by six sets of power spectra: 



x∗ x alm al  m  = Clxx δll  δmm

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(19)

Because the gradient and curl components of the polarization tensor have opposite parity (respectively, (−1)l and (−1)l+1 )), the parity preserving nature of the physics that creates CMB fluctuations implies that ClT B = ClEB = 0. Therefore, the statistics of the CMB temperature and polarization fluctuations, under the hypothesis of zero-mean Gaussian random fields, are fully determined by the four

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Multipole, Fig. 2 Expected BB spectra from primordial GWs (gray lines) for three values of the tensor-toscalar ratio, r = 0.1, 0.01, and 0.001, and from lensing (blue line): the reionization bump at small multipole l (i.e., l = 2–8) is visible. The colored bands show the dust and the synchrotron power spectra measured in different sky regions and at different frequencies, as indicated in the plot. c (Figure from [23], reproduced with permission ESO)

sets of spectra ClT T , ClT E , ClEE , andClBB . Experimental measurements of the T T , T E, and EE spectra are shown in Fig. 1, where DlXY are defined in analogy with Eq.(5). Among all, ClBB is the one with the lowest expected values, as is clear by comparing experimental spectra of Fig. 1 with the theoretical expectations shown in Fig. 2. As said above, understanding the polarization pattern of the CMB gives information about the quadrupolar temperature fluctuations at the last scattering surface. The orthogonality properties of the spherical harmonics ensure that no multipole moment other than l = 2 can generate polarization via Thomson scattering. Thermal fluctuations produced by primordial fluctuations can have three different geometrical patterns [8]: • Scalar (or compression-like) perturbations are perturbations in the energy density of the cosmological fluid: they generate fluctuations that dominate at large scales and generate photon bulk flows, or dipole anisotropies, from hot to cold temperature regions. They are the only fluctuations which are important for structure formation. At linear order in perturbation theory, they cannot produce any B-mode perturbations. The quadrupole produced by scalar perturbations has an m = 0 pattern. • Vector perturbations are vortical motions of matter. Since they do not produce density perturbations, vector modes are not involved in structure formation. They can produce polarization in both E and B patterns. Since gravity does not

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enhance vorticity, these perturbations are damped by the expanding universe; thus, in general, they are not dominant at recombination, and thus, their contribution to CMB polarization is considered negligible. The quadrupole produced by vector perturbations has an m = ±1 pattern. • Tensor perturbations are expected from the primordial gravity waves predicted in the inflationary scenario: the primordial GWs are the tensor perturbations in the background metric during inflation. Since they do not produce density perturbations, tensor modes are not involved in structure formation. Transversetraceless distortion provides temperature quadrupoles with m = ±2. They can generate polarization patterns of both E and B types. The strength of the GW signal is usually parameterized as the power ratio of tensor modes to scalar modes in the CMB, which is referred to as the tensor-toscalar ratio, r: r≡

t s

(20)

This ratio provides a convenient means of discriminating between different inflationary theories, which may produce different ratios of tensor and scalar modes. The most natural class of inflationary theories predict r > 0.001. A measurement of r would also provide a measurement of the potential of the inflaton field, Vφ , the field which drives inflation: 1/4

Vφ

≈ 1016 GeV

 r 1/4 . 0.01

(21)

Furthermore, if the inflaton is a slowly changing “slow-rolling” field, then r is also a measurement of the change in the amplitude of the inflaton field during inflation: φ = MP



r , 0.01

(22)

where MP is the Planck mass. In the end, a curl component (B-mode) in the CMB arises at linear order in perturbation theory only from primordial GWs. Tensor modes are a robust prediction of inflation. Hence, measuring B-mode polarization in the CMB would represent a so-called smoking gun in favor of inflation theory, would allow us to measure the energy scale of inflation, and would offer valuable insight into relevant inflationary parameters, constraining inflationary field models. At present, there is no direct measurement of the primordial BB spectrum, though a wide range of experimental efforts are being pursued, as described later in this chapter. The experimental enterprises need to take into account that the CMB polarization can be altered after recombination by two processes before it can actually be detected: these are reionization and weak gravitational lensing of the CMB.

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Reionization Reionization is the second major change in the ionization state of hydrogen in the universe. In the history of the universe, reionization is the period which followed the “dark ages” after the CMB (a period from 1100 < z < 30 in which no optical light is produced in the universe) and the “cosmic dawn” (when the first stars and quasars form, at z ∼ 30). At this point, the Lyman-α UV photons emitted by the first stars and quasars almost completely ionized the H gas which formed from recombination. The epoch of reionization occurred between z ∼ 15 and z ∼ 6. During this period, the cosmic gas changes from being almost completely neutral to almost completely ionized. Detailed knowledge of the reionization process and development is still beyond our reach. The late-time reionization starting at about z ∼ 6–7 causes about 10% of the CMB photons to be rescattered by the free electrons produced by reionization, thus damping the primordial fluctuations at small angular scales (i.e., at scales smaller than the horizon size at the epoch of reionization, l  20) by a factor e−2τ , τ being the electron scattering optical depth which is proportional to the integrated electron density along the line of sight and the Thomson scattering cross section. For τ = 0.054 (as measured by Planck [12]), this means a reduction of about 10%. This effect on the CMB power spectrum is very similar to a change in the primordial amplitude of scalar perturbations making it difficult to disentangle the two on the basis of the temperature spectrum alone. Furthermore, the free electrons produced by reionization can also induce largescale polarization of the CMB in addition to the polarization produced at the surface of last scattering at recombination, due to their Thomson scattering with CMB photons. The scattered power depends on the square of the optical depth τ and peaks at scales larger than the horizon at reionization (l  20). The “reionization bump” is visible in the range l = 2–8 in the expected BB spectrum shown in Fig. 2. If the hydrogen were homogeneously ionized, then only E-mode polarization could be created by quadrupole temperature anisotropies. However, if inhomogeneities in the ionization fraction were present, they would generate B-mode polarization. Indeed, numerical simulations of the reionization epoch suggest that the ionization process was significantly inhomogeneous. Following Furlanetto et al. [13], inhomogeneities are schematized as fully ionized bubbles in the neutral intergalactic medium, which then grow and merge to form the completely ionized universe. This is motivated by the fact that the photoionization cross section is very large at energies >13.6 eV and so the UV photons are absorbed close to their sources, creating “bubbles” of ionized hydrogen. In this patchy reionization scenario, different lines of sight would probe different scattering histories, and hence the value of τ would be direction-dependent. The reionization bump was first observed by WMAP, using measurements of the T E and EE power spectra at l < 20. Using these spectra, WMAP also obtained the first measurement of the optical depth τ at reionization. More recent data comes from the Planck collaboration, who measured τ = 0.054 ± 0.007 [12].

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Lensing B-Modes Gravitational lensing is a major source of distortion of the polarization power spectrum. For a comprehensive overview, see, for instance, Lewis et al. [15] and Hanson et al. [16]. It was recognized since the work of Zaldarriaga et al. [14] in 1998 that gravitational lensing can produce a mix between E and B type of polarization, thus changing the pattern of polarization. This lensing is experienced by the CMB photons in their travel through the inhomogeneous universe. The CMB photons can be deflected by the spacetime perturbations caused by galaxies and galaxy clusters when passing nearby: lensing distorts the pattern of polarization by shifting the positions of the photons in the plane of the sky relative to the last scattering surface. The sum of the many small deflections that the CMB photons experience between the surface of last scattering and us is called the weak gravitational lensing effect. A “weak” gravitational lens is a lens whose Newtonian gravitational potential  is small: /c2 1. This is the case of cosmological interest, e.g., for a galaxy cluster /c2 < 10−4 1; on the other hand, this is not valid in general, e.g., not for the lensing by a black hole. Under this approximation, according to general relativity, for a point lens (i.e., a point mass in the lens plane, for which the potential is  = − GM r ), the deflection angle experienced by a photon is: |α| = 4

4GM  = 2 2 c c b

(23)

with b the impact parameter. Note that this expression is linear in M so deflection angles of ensembles of lenses can be linearly superimposed; this expression is twice the value that can be computed with standard Newtonian gravity. Thus, one can expect that a single lens produces deflection angles θ  4 × 10−4 rad = 1 arcmin. In Lewis et al. [15], a heuristic argument is made for the number of lenses one can expect a CMB photon has experienced before reaching us: the characteristic size of potential wells is given by the scale of the peak of the matter power spectrum, i.e., ∼300 Mpc, while the distance of the last scattering surface is ∼14 GPc (all co-moving distances). Therefore, one would expect about 14/0.3 ∼ 50 individual masses (i.e., lenses) along a line of sight; since they are uncorrelated, their effect would sum up incoherently, and thus one would expect a total deflection angle θ  √ 50 × 4 × 10−4 rad = 10 arcmin. This corresponds to angular scales l  1100 which is where we expect lensing to dominate the observed power. It is estimated that most of the lensing effect is caused by dark matter structures around redshift z ∼ 2 (see, e.g., Lewis et al. [15]). The power leaking from E-modes into B-modes produces a term in the BB power spectrum independent of the primordial GWs. Indeed, for realistic (and small) values of the tensor-to-scalar ratio r, this can be the dominant term, and specific “delensing” strategies [17] need to be implemented in the search for the GW imprints from the early universe. Delensing aims at subtracting the lensing portion of the B-mode polarization and thus leaving as residual any potential inflationary

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gravitational wave background. The gravitational lensing B-mode spectrum was first detected in the CMB using the first-season data from the polarization-sensitive receiver on the South Pole Telescope (SPTpol), at a significance level of ∼8σ [18]. Figure 2 shows the BB spectrum expected from primordial B-modes for different levels of the tensor-to-scalar ratio r, as well as from weak gravitational lensing; the effect of the reionization bump is also visible. In addition to contributing to the BB power spectrum, and hence contaminating the signal from primordial B-modes, weak gravitational lensing is also effective in flattening the peaks and troughs of the C T T , C T E , and C EE power spectra. On the positive side, beyond these effects on the power spectra, gravitational lensing imprints non-Gaussian correlations in the observed CMB map which can be used to extract information about the projected large-scale structure potential and hence derive constrains on the late time evolution of the universe, the properties of dark energy, and the portion of the energy budget in massive neutrinos [20]. This procedure is known as lensing reconstruction.

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Experiments aiming at mapping the CMB collect all microwave radiation arriving at the detector, either of cosmological origin or from subsequent galactic and extragalactic sources. Although they carry invaluable information, these signals are considered a contaminant to the CMB and need to be removed before the clean map can be studied. These astrophysical contamination emissions are called foregrounds, as opposed to backgrounds, since the CMB is the most distant photon source observable in the universe. Foregrounds can be in the form of point sources or diffuse emissions. Of course, many different sources and mechanisms can produce foreground contributions; some have a limited degree of polarization and hence are relevant mainly the T T maps. Figure 3 shows the foreground contributions

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Fig. 3 Foreground contributions for temperature (left) and polarization (right) as a function of frequency, in the microwave range. Gray vertical bars show the Planck observing bands. (Figure c from [7], reproduced with permission ESO)

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for temperature (left) and polarization (right) as a function of frequency, in the microwave range. There are two main terms that affect the polarization of the CMB: diffuse synchrotron radiation and thermal dust emission, the cross-over frequency being ∼70 GHz. Synchrotron radiation is the bremsstrahlung radiation emitted by the population of relativistic cosmic ray electrons and positrons (CR), originating mainly from supernova explosions and subsequent shock acceleration, accelerated by the weak magnetic field of the galaxy (on the order of tens of μG [21]). The brightness temperature of the emitted radiation depends on the component of the magnetic field perpendicular to the line of sight and on the energy distribution of the accelerated charged particles; as a function of frequency, it is usually modelled as a power law: Tsynchr ∝ ν βsynchr

(24)

where the index βsynchr ≈ −3 is related to the slope of the power law of the CR energies. However, since the propagation of cosmic rays depends on the structure of the magnetic field, the observed synchrotron radiation in a particular direction on the sky is the line of sight integral of the bremsstrahlung produced by different CR populations in different environments: therefore, variations in the spectral index across the sky are expected and have indeed been observed. Theoretically, for the spectral index of βsynchr ≈ −3 observed at microwave frequencies, one could expect synchrotron emission to have polarization as high as 0.75%, with the direction of polarized emission orthogonal both to the line of sight and to the magnetic field lines. However, a number of effects are at play that reduce the degree of polarization: in particular, the Faraday rotation effect which scales as the inverse of the square of the frequency and hence is important at ν  10 GHz. The Faraday rotation effect is related to the component of the magnetic field along the line of sight and is due to the different refractive indices experienced by the two circular polarization components when propagating in the magnetized interstellar medium of non-relativistic thermal electrons [22]. At high frequencies, the dominant source of polarized foreground emission is dust, along the galactic plane as well as at intermediate latitudes. Dust grains in the interstellar medium can come in different sizes (from nm to μm), shapes, and compositions (including polycyclic aromatic hydrocarbon molecules and amorphous silicate and carbonate compounds); for CMB studies, the relevant emission mechanism is thermal gray body emission, i.e., re-emission by dust grains which are heated by stellar radiation to about 20 K. In particular, the emission increases with frequency as a power law Tdust ∝ ν βdust with βdust ≈ 2 and varying across the sky. Aspheric dust grains can produce polarized emissions as they align with their long axis perpendicular to the magnetic field. The smallest dust particles do not align with the magnetic field due to their low inertia and hence do not contribute to the polarization. The dust emission efficiency is greatest along the long axis, leading to partial linear polarization perpendicular to the magnetic field (similar to synchrotron emissions). The level of polarization is usually small (3–4%) but reaches 10% in particularly dense dust clouds.

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Figure 2 from [23] shows the B-mode power spectrum vs multipole for the dust components measured at 95 and 150 GHz, the synchrotron radiation measured at 95 GHz, and the CMB B-modes from primordial gravitational waves (gray lines) expected for three values of the tensor-to-scalar ratio r = 0.1, 0.01, and 0.001 and from lensing (blue line).

Key Instrument Technologies The impressive improvements in measurements of the CMB, and in the resulting cosmological constraints, in the first two decades of the twenty-first century are largely due to improvements in instrumentation that have led to orders of magnitude improvements in the sensitivity of surveys. The ongoing search for the signal of primordial gravitational waves in the CMB is fueled by these instrumental improvements as well. Here we will describe some of the key developments in instrumentation that have enabled this work across the field of CMB science.

Detectors Ultimately, the key to detecting primordial gravitational waves in the CMB is sensitivity. The CMB itself is blackbody radiation with a characteristic temperature of 2.725 K, and the anisotropies in the CMB are even smaller, one part in 105 compared to the DC level of the CMB. The CMB is only 10% polarized, resulting in an order of magnitude decrease in the power in the polarized EE power spectrum compared to the T T power spectrum. The lensing BB signal is another order of magnitude lower, and the primordial signal is lower magnitude still. In order to measure these signals, and with sufficient precision to disentangle them from foregrounds many orders of magnitude higher in amplitude, extremely sensitive detectors are necessary. For many years, the best technology available was high-electron-mobility transistors (HEMTs), which were developed in the late 1970s. HEMT can function at higher frequencies than most transistors, up to tens of gigahertz. This makes them useful for microwave and radiofrequency applications such as radar, satellite communications receivers, cell phones, and microwave astronomy. HEMTs were used on the WMAP and Planck CMB satellite missions. In the 1990s and early 2000s, a new detector technology was developed, which allowed experiments to access higher frequencies with more sensitivity: transitionedge sensor (TES) bolometers. In a TES bolometer, photons are coupled from free space into an antenna, typically with either a feedhorn or an anti-reflection coated lenslet. The antenna types vary, from unpolarized total power-sensitive antennas, to simple crossed dipole antennas which serve as microwave orthomode transducers, to recently developed broadband log-periodic sinuous antennas. An example of a TES bolometer with a sinuous antenna from the SPT-3G experiment is shown in Fig. 4. The microwave frequency power is transmitted on microstrip line waveguides and deposited on a thermally isolated structure called the TES “island.” The island

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Fig. 4 A scanning electron microscope image of a three-band multichroic microwave frequency pixel from the SPT-3G experiment. The dual-polarization log-periodic sinuous antenna has a twooctave bandwidth. Gigahertz power is split into three bands with a lumped element triplexer and coupled into three transition-edge sensor (TES) bolometers. In the TES, the photon energy is converted into heat, which is read out as a varying current passing through a resistive element in the TES. (Figure courtesy of the SPT Collaboration)

is usually physically suspended above the base structure supporting it by long legs formed by photolithographic etching, whose geometry and thermal properties are tuned to give a desired thermal conductance. On the island, a normal-phase metal resistor element is used to convert the microwave power into heat, which is deposited on the island. This heat is measured by means of the transition-edge sensor itself: a metal alloy whose transition temperature is tuned such that the heat from the flux of CMB photons, isolated from the thermal bath by the island support legs, will heat the TES to exactly its superconducting transition temperature. The island heat capacity is tuned with a larger amount of normal-phase metal, such that the fluctuations in CMB flux will produce temperature changes within the TES transition. Because of

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the extremely sharp phase transition between the superconducting and normal states, fluctuations in the island temperature on the order of millikelvin can be read out as changes in the resistance of the TES on the order of 1 . A close view of the TES bolometer from SPT-3G is shown in Fig. 5. The TES is current biased and held in an analog or digitally controlled negative feedback loop so that it can be maintained in the superconducting transition. That is, when the CMB photon flux increases, the readout current is decreased to compensate, so that the island TES remains in the transition. The varying readout current is amplified with a superconducting quantum interference device (SQUID), a quantum-limited amplifier. SQUIDs are actually magnetometers, capable of detecting a single quanta of magnetic flux and producing highly amplified current proportional to the amount of flux detected. The readout current is passed through an inductor adjacent to the amplifier SQUID, and the resulting magnetic field is used to produce the amplified current which is read out by the warm electronics. TES bolometers are “quantum noise limited”: in essence, you cannot build a more sensitive microwave antenna; all you can do to increase instrument sensitivity is build more of them. The need for more sensitive instruments has therefore led to successive generations of telescopes fielding focal planes with orders of magnitude more detectors, as described in the following section. However, these detectors have finite size related to their wavelength, and there are effectively limits to how

Fig. 5 (left) Two bolometers from the SPT-3G experiment. The bolometer island is thermally isolated by suspending it on SiN legs above a Si substrate. The island layered by photolithographic processes and released from the substrate with a chemical etching process. (right) Closeup view of the bolometer island. The load resistor converts gigahertz frequency electromagnetic waves into heat, which raises the temperature of the island, and changes the resistance of the transition-edge sensor, which is held in a superconducting phase transition with an electrothermal feedback loop. The Pd layer increases the heat capacity of the TES to slow down its thermal response. The gigahertz frequency microstrip waveguides and readout traces are niobium, which is superconducting at the operating temperature of the bolometer (500 mK). (Figure courtesy of the SPT Collaboration)

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large one can make a focal plane, due to aberrations in optics incurred on large focal planes and the prohibitive cost of larger optical elements. Another method of increasing the effective number of detectors, and thus the sensitivity of an instrument, is to increase the bandwidth of the antenna and the number of frequency channels read out by it. This is the motivation for the development of broadband sinuous antennas: they can be used for “multichroic” pixels, an antenna connected to two or three different TESs tuned to different frequency bands, providing effectively two or three pixels in one [71]. A new technology being developed that may have the potential to further revolutionize the field of microwave astronomy is the microwave kinetic inductance detector (MKID). In an MKID, microwave power from the antenna is coupled into a superconductor, where the incident power breaks Cooper pairs, creating excess quasiparticles. The kinetic inductance of a superconductor is inversely proportional to the density of Cooper pairs, so variations in photon flux produce variations in inductance. The inductor is then coupled to a capacitor to create an LC resonant circuit whose resonant frequency varies with the photon flux. The advantage here is that the readout is now natively in the microwave frequency, and the resonant bands are very narrow, enabling potentially hundreds or thousands of detectors (and thus CMB frequency bands) to be read out on a single broadband microwave channel with frequency-division multiplexing (also called frequency-domain multiplexing) [45]. MKIDs may therefore be able to provide another two to three orders of magnitude increase in the number of detectors in a focal plane and two to three orders of magnitude increase in instrument sensitivity.

Optics, Cryogenics, and Multiplexing Microwave astronomy can use reflective and refractive optics, though the materials used differ from those used at other wavelengths. Reflective elements are similar to those at other wavelengths: polished conductive mirrors. The required surface smoothness is significantly lower than at optical wavelengths, however. Surfaces must be smooth on scales significantly smaller than the wavelength of the photons being observed (a common criteria is an RMS of λ/50), which means mirror surfaces can have ∼1000 times higher surface fluctuations when observing at microwave frequencies (λ ≈ 1 mm) than at optical frequencies (λ in the hundreds of nanometers). While large primary reflective optics are usually limited to being warm, secondary or refractive optics are often cooled to cryogenic temperatures to reduce the thermal radiation they produce. Similarly, focal planes are all cryogenically cooled. In some cases, liquid cryogens are used, typically liquid nitrogen to cool to 77 K and liquid helium to cool to 4 K. However, helium in particular is a non-renewable resource and is increasingly expensive as a result of the finite supply, leading most experiments to move to closed-cycle helium refrigeration systems. Pulse tubes employ a compression and expansion Carnot cycle with helium as the refrigerant and typically have stages that cool to 50 K or 4 K. To reach lower temperatures, there

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are a variety of technologies. Helium sorption fridges use He3 and He4 under large pressure and pump on a liquid reservoir of helium using adsorption and absorption of helium onto a carbon trap to produce cooling power. Helium sorption fridges can reach temperature of ∼250 mK. Adiabatic demagnetization refrigerators (ADRs) produce cooling power using the magnetocaloric effect: a paramagnetic material is held in a strong magnetic field, so that its magnetic domains align with the field. Then, as the magnetic field strength is decreased, and the domains realign to random orientations, they absorb energy from the system in order to alter their alignment. ADRs can reach temperatures on the order of 10 mK, but have very low pumping power. They are typically used for testing detectors or readout systems, but not for larger applications such as telescope focal planes. The current best method for cooling focal planes to temperatures below 100 mK is a dilution refrigerator. Dilution refrigerators use a combination of He4 and He3, which undergoes a spontaneous phase separation at temperatures below 870 mK into a He3 rich and He3 diluted phase. He3 flows from the rich phase to the diluted phase, but passing through the phase boundary is an endothermic process, which extracts heat from the surrounding system. Dilution fridges can reach temperatures below 10 mK, but with thermal loading, the achieved base temperature is usually between 10 and 100 mK. The focal plane and optics are then typically contained inside a cryostat which is pumped out initially by turbopump to ∼10−3 torr, and in operation, the pressure is further reduced by cryopumping to ultra-high vacuum pressures (∼10−9 torr). Focal planes are supported by thin fiberglass or carbon fiber supports to reduce conductive heat load and surrounded with many layers of aluminized mylar or multilayer insulation (MLI) to reduce radiative load. Refractive optical elements are typically constructed out of materials like silicon, polyethylene, polytetrafluoroethylene (Teflon), and sapphire, which are transparent at microwave frequencies. Anti-reflection (AR) coats can be created from thin layers of similar plastics. Additionally, several current experiments use metamaterial AR surfaces created by drilling holes or cutting grooves in the surface of a refractive optical element on scales smaller than the wavelength of light observed. This combination of a low index of refraction material (vacuum) and a higher one, mixed on sub-wavelength scales, creates an “effective index of refraction” intermediate between the two materials, determined by the filling fraction of the solid material. A difficulty incurred by the cryogenic containment of the focal plane, and the large number of detectors in modern telescopes, is that the focal plane mush be read out by some system that does not negate the thermal isolation of the focal plane and swamp the cryogenic pumping power of the cooling mechanism. Essentially, 1000 or 10,000 TESs cannot by read out by individual wires, because the conductive heat load would be too great to manage. The solution that has been unanimously adopted in CMB experiments is signal multiplexing: combining the signals from many detectors on a single wire. Two methods are used in different experiments: time-division multiplexing (TDM) and frequency domain multiplexing (fMux). In a TDM system, detectors are grouped together into a two-dimensional array, with each column of detectors sharing a dedicated readout SQUID amplifier chain. The rows are addressed cyclically at a rapid pace to read out the entire array. When one row is addressed, a single detector in each column is read out to the

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corresponding column amplifier. In this way, many detectors can be read out on a single cold wire. The number of detectors on a single readout wire is referred to as the multiplexing factor of the readout system. The time delay between subsequent reads must be less than the thermal time constants of the detector, so that the changing thermal state of the detector can be monitored. This in combination with the switching speeds possible with the SQUID amplifiers limits the number of detectors that can be read out on a single wire. In an fMux system, detectors are grouped together as well, but each one is modulated with a unique LC resonant circuit so that each detector transmits on a different frequency. In this manner, all detectors in a group can transmit simultaneously, and the signals can be separated in software in the warm readout electronics. The limitations to how many detectors can be read out in this system are imposed by how narrow the frequency channels can be formed, how accurately they can placed, and the total bandwidth available. Both methods have currently achieved 64× multiplexing, and further improvements are expected for future experiments.

Overview of Experiments A number of experiments have sought to measure the polarization of the CMB, including CBI [63], DASI [4], BOOMERanG [60], QUAD [37], WMAP [6], CAPMAP [34], BICEP [38], QUIET [68], BICEP2 [33], Keck [75], Planck [66], POLARBEAR [55], Spider [70], EBEX [62], ABS [58], ACTPol [81], Advanced ACTPol [51], SPTpol [52], SPT3G [32], GroundBIRD [61], LiteBIRD [50], QUIJOTE [69], BICEP3 [57], PIPER [59], CLASS [49], and QUBIC [30]. Here we will discuss a subset of these experiments to highlight some of the varied experimental architectures used for these measurements and the strategies employed to overcome the observational difficulties involved. Generally, the different morphologies of CMB telescopes can be grouped into four categories: large and small-aperture ground-based telescopes, balloon-borne telescopes, and satellite telescopes. Examples of each category are described in the following sections and are depicted in Fig. 6.

Large-Aperture Telescopes Microwave frequency large-aperture telescopes (LATs) resemble many other traditional optical and radiofrequency telescopes: they consist of a large-diameter primary mirror to collect and focus photons and typically one or more secondary optical elements, which may be reflective or refractive, to further focus the incoming light and correct for various aberrations. They have the advantage of having large collecting area, which increases sensitivity, and high angular resolution, which can be essential for studying structures with small angular sizes on the sky. In the case of the CMB, the objects of interest were initially galaxy clusters, which can be observed in the microwave band due to the Sunyaev-Zel’dovich effect, by which a spectral distortion of the CMB is produced by photons inverse-Compton

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Fig. 6 (top left) ACT, a large-aperture telescope (section “Large-Aperture Telescopes”), showing the large primary mirror, co-moving side shields, and fixed ground shield used to reduce ground pickup and related systematics. Photo credit Mark Devlin. (top right) The CLASS telescope, a small-aperture telescope (section “CLASS: A Small-Aperture Telescope”). Each mount contains two separate telescopes with wide field of view refractive optics, optimized for measuring the large spatial scales of CMB B-modes. Photo credit Matthew Petroff. (bottom left) The Spider balloon-borne telescope (section “Spider: A Balloon-Borne Telescope”). The cryostat contains six independent optics tubes, in order to minimize the size, weight, and complexity of the optical elements. The gondola is carried by a NASA long-duration balloon. Photo credit Johanna Nagy. (bottom right) Artist’s impression of the WMAP satellite (section “Satellite Telescopes”), showing the two primary mirrors of its unique back-to-back Gregorian design and distinctive golden sun shield below the instrument to shield it from solar radiation and allow for passive cooling to cryogenic temperatures. (Figures courtesy of the respective collaborations)

scattering off the hot electrons in the ionized intracluster medium [78]. However, high angular resolution is also proving essential for measuring the spatial structure of polarized foregrounds from dust, one of the key foregrounds to the much sought after primordial B-mode signal.

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Several CMB telescopes fall into this category, including CAPMAP, ACT, SPT, and POLARBEAR. Here we will describe ACT and SPT, which demonstrate several different instrumentation solutions to some of the challenges presented by the search for B-modes.

ACT The Atacama Cosmology Telescope (ACT) is located on the Cerro Toco mountain, in the Atacama Desert of northern Chile, at an altitude of 5190 m. The high altitude and arid environment are crucial for reducing the precipitable water vapor (PWV) at the site: the integrated depth of water in a column of air, if all that water were to precipitate out as liquid. Atmospheric water is the chief source of atmospheric emission and opacity across much of the millimeter wavelength band and is therefore one of the most important selection criteria in choosing an observing site. The ACT instrument consists of a 6-m-diameter primary mirror, and a 2-mdiameter secondary mirror, in an off-axis Gregorian configuration. This telescope configuration is common for millimeter wave instruments with reflective optics, as it is designed to give a sky view which is unobstructed by the secondary element and receiver, which are highly emissive in-band due to thermal radiation. The compact optics also allow for fast scanning speeds, which are useful for filtering out atmospheric fluctuations in the Fourier domain, therefore improving instantaneous signal-to-noise ratio and eventual survey mapping depth. Another important source of emissions is ground pickup, because the ground is also thermally emissive in the observing band. To combat this source of noise, ACT is designed with two “ground screens”: one large stationary screen surrounding the telescope and one co-moving inner screen that connects the sides of the primary and secondary mirror to capture rays that would “spill over” either mirror and terminate on the ground. The screens are constructed out of reflective material and designed to redirect spillover to the sky, which is effectively a much colder load than the ground. ACT observes in constant elevation scans to prevent scan synchronous signals from changing air mass. Scanning east and west along the arc from the south celestial pole to the zenith also provides “cross-linked” observations of the same sky location in different orientations, which can be used to reduce scanning-induced systematic affects in the resulting maps. The scan speed is set such that the beam moves across the sky faster than the 1/f knee of the low-frequency sky noise, but faster than the thermal equilibrium time constants of the detectors. This filters out sky emissions, but ensures the detectors are well equilibrated at each pixel on the sky. For ACT, this means that the entire 40 metric-ton structure of the telescope must rotate at 1.5 deg /s while holding the elevation constant to within 2 of the desired pointing. ACT uses an array of three focal planes, which together with their secondary reimaging optics are cryogenically cooled to reduce thermal loading on the detectors. The array of three focal planes allows the cryogenic containment for each focal plane to be separate, resulting in smaller-diameter vacuum windows and optical elements, which are easier to manufacture and cool. It also allows each focal plane

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to observe at a separate frequency, reducing the necessary bandwidth for the antireflection (AR) coatings on the optics, which are difficult to produce in wide bands. The reimaging optics consist of two high-purity silicon lenses, a series of thermal filters to reduce loading on the optics and focal plane, and a cold Lyot stop to define the illumination of the secondary and primary mirror. High-purity silicon was chosen for its high index of refraction (n = 3.416) for focusing ability and high thermal conductivity at cryogenic temperatures for thermal uniformity. The lenses are AR coated with several hundred μm of Cirlex (n = 1.8). The vacuum window must have low loss and emissivity in band and be strong enough to withstand the pressure differential between the ambient external pressure and the interior vacuum of the cryostat. The vacuum windows were constructed of ultra-high molecular weight polyethylene, with Teflon AR coats, and were measured to be 93–96% transmissive across the observing band. The lenses were cooled to 4 and 1 K, and the focal plane was cooled to 300 mK [79]. Three cameras have been installed on ACT to date, the initial Microwave Bolometer Array Camera (MBAC), the ACTPol camera which was the first to incorporate polarization sensitivity and improved the sensitivity of the instrument [81], and the Advanced ACTPol camera with further improved polarization sensitivity [51]. The first ACT camera had focal planes which observed at three different frequency bands, centered at 148, 218, and 277 GHz, each with ∼20 GHz bandwidth. Each focal plane consisted of 1024 pixels in a 32 × 32 element array. Detectors were fabricated by the Detector Development Lab at NASA Goddard Spaceflight Center. The detectors were TES√bolometers with Tc = 510 mK and a total array sensitivity (NET) of 30–40 μK/ Hz [79]. The detectors were read out with 32× time-domain multiplexing(also called time-division multiplexing). This camera was not polarization sensitive and so could not measure gravitational lensing or place limits on the scalar-to-tensor ratio r. It observed from 2008 to 2013. The second ACT camera, called ACTPol, was polarization sensitive. It consisted of three polarization-sensitive arrays: PA1, PA2, and PA3. PA1 and PA2 consisted of 512 feeds, each containing 2 bolometers coupled to a single linear polarization and observing at 148 GHz. PA3 contained dichroic pixels, each observing simultaneously at 97 and 148 GHz, and consisted of 255 antennas each coupled to 4 bolometers, one for each unique combination of frequency and linear polarization. The detectors were read out with 32× time-domain multiplexing(also called timedivision multiplexing). For ACTPol, the system optics and cryogenics were also upgraded, notably by the addition of a dilution refrigeration system to cool the focal planes to 100 mK, further reducing thermal noise. Given the new cooling system, the critical temperature of√these detectors was reduced to 150 mK. The median array sensitivity was ∼20 μK/ Hz [81]. ACTPol observed from 2013 to 2015. The third- and current-generation camera on ACT, Advanced ACTPol, consists of 4 multichroic arrays: the high-frequency array (150 and 230 GHz), 2 mediumfrequency arrays (90 and 150 GHz), and 1 low-frequency array (28 and 41 GHz), with a total of 5612 detectors, read out with 64× time-domain multiplexing(also called time-division multiplexing). The target Tc was 160 mK, and the array

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√ sensitivity was ∼12 μK/ Hz [39]. It is projected to have a final map sensitivity of 7 μK-arcmin at 150 GHz. Advanced ACTPol was deployed in stages between 2016 and 2018 and continues to observe in 2020 [51].

SPT The South Pole Telescope (SPT) is located at the Amundsen-Scott South Pole Station, located at the geographic South Pole, at an altitude of 2835 m above sea level and a pressure elevation in excess of 3350 m due to atmospheric thinning from the low temperature at the site and from atmospheric bulging at the equator. SPT is a 10 m off-axis Gregorian telescope, with a 1 m cryogenically cooled secondary mirror and a 1deg2 field of view. A mirror shield surrounds the primary mirror, co-moving side screens connect the primary mirror to the receiver cabin, and a conical “baffle” surrounds the vacuum window to reduce spillover. The secondary mirror, refractive optics, and focal plane are contained in two joined cryostats which share a vacuum space. The vacuum window is constructed of Zotefoam, a closed cell HDPE foam. The secondary mirror, cold stop, and single AR-coated silicon lens are cooled to 10 K with a PT-415 pulse tube cooler. The focal plane is further cooled to 250 mK with a helium sorption refrigerator. SPT observes in constant elevation scans, which, due to the location at the geographic South Pole, correspond to scans at constant right ascension. The 300 metric-ton upper structure of the telescope rotates at 0.25 deg /s for scans, but can rotate safely at up to 2 deg /s between scans [36]. Three cameras have observed on the SPT: the initial SPT-SZ camera, the polarization-sensitive SPTpol camera, and the multichroic and polarizationsensitive SPT-3G camera. SPT-SZ consisted of a single focal plane with detectors at three frequency bands, centered at 95, 150, and 220 GHz, to provide spectral discrimination against galactic foregrounds and radio and infrared extragalactic sources. Detectors were tiled into a hexagonal focal plane in 6 triangular wafers, each containing 161 detectors for a total of 966 detectors: 1 wafer of 95 GHZ, 1 of 220 GHz, and 4 of 150 GHz detectors. The detectors were “spiderweb” temperature-sensitive antennas, with AlTi √ TES bolometers. The detectors had an individual detector sensitivity of 600 μK/ Hz √ √for the 95 GHz detectors, 400 μK/ Hz for the 150 GHz detectors, and 1200 μK/ Hz for the 220 GHz detectors. The detectors were read out with 8× frequency domain multiplexing. SPT-SZ observed between 2007 and 2011, surveying a 2500 deg2 field to a depth of ≤18 μK-arcmin [36]. The polarization-sensitive SPTpol camera consisted of 7 wafers of 150 GHz pixels, with 84 pixels per wafer, and 180 individually assembled 95 GHz pixels, for a total of 1536 bolometers [52]. The detectors had Tc = 470 mK and an √ array sensitivity of 15 μK/ Hz [28]. The detectors were read out with 12× digital frequency multiplexing (DfMux) [71]. SPTpol surveyed a 500 deg2 field to a depth of 5 μK-arcmin between 2011 and 2015. SPTpol has detected B-mode power at very high significance; Sayre et al. report an 18σ detection and a constrain of σ (r) = 0.22 on the tensor-to-scalar ratio from SPTpol data alone [72].

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The SPT-3G deployed multichroic broadband log-periodic sinuous planar antennas, each of which coupled to three frequency bands centered at 95, 150, and 220 GHz. The optics chain of the South Pole Telescope was also reworked to accommodate a focal plane with ∼4× the area as previous focal planes. The modifications included a new warm aluminum elliptical secondary mirror and three cold alumina lenses. Together these improvements resulted in over an order of magnitude increase in the number of detectors, to 16,140 bolometers, and a focal √ plane sensitivity of 3.4 μK/ Hz. The detectors are read out with a 64x DfMux system. Due to the increased number of detectors, and the decreased thermal loading from the optics rework, the mapping speed of SPT-3G is 17× greater than that of SPTpol, in both temperature and polarization. SPT-3G is currently observing, as of 2020, and expects to reach a survey depth of ∼3.5 μK-arcmin at 150 GHz in E and B, over a 2,500 deg2 field. SPT-3G expects to reach a 150σ detection of gravitational lensing of the CMB and reach a constrain of σ (r) = 0.01 on the tensorto-scalar ratio. This estimate includes the effects of foregrounds, atmospheric and instrumental noise, E-B separation, and delensing [32]. SPT was optimized for fine angular resolution, in order to study galaxy clusters. This means it and other large-aperture telescopes (LATs) do not have as much mapping speed on the larger angular scales of gravitational waves. Their raw constraining power on metrics such as the tensor-to-scalar ratio tends to be lower than for small-aperture wide field of view instruments, in the absence of foregrounds. However, fine angular resolution is useful for characterizing foregrounds such as polarized dust and therefore is valuable for studies of gravitational waves. It is unknown currently whether raw mapping speed or foreground removal will be more influential in detecting primordial gravitational waves, and for this reason, future observatories typically are envisioned as combinations of large- and small-aperture instruments, as discussed in section ‘Future Experiments”.

CLASS: A Small-Aperture Telescope Small-aperture telescopes (SATs) have many potential advantages over largeaperture instruments. They typically use refractive optics systems, which are more simple than the reflective+refractive optics systems of LATs, making their instrument systematics potentially easier to understand. They are smaller and lack large-diameter precision optical elements and are thus less expensive to construct. And they naturally have large fields of view, which is useful for studying the degreescale structure of the CMB. Many current and former CMB experiments fall into this category, including CBI, DASI, QUAD, BICEP, QUIET, BICEP2, Keck, ABS, GroundBIRD, QUIJOTE, BICEP3, QUBIC, and CLASS. Here we will describe one example of a current SAT, the Cosmology Large Angular Scale Surveyor (CLASS). With a large field of view and survey area (65% of the southern sky), CLASS is optimized to observe the l ≤ 10 recombination peak of the BB power spectrum, which should be less obscured by polarized foregrounds. The primordial B-mode signal is predicted to dominate over foregrounds at l < 20 if r > 0.01. CLASS

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consists of four separate telescopes, one observing at 40 GHz, two at 90 GHz, and one at 150 and 220 GHz, located in the Atacama Desert of Chile. The two 90 GHz telescopes are designed for observation near the minimum in the polarized galactic emission, while the others will aid in foreground removal. The 40 GHz telescope will measure polarized synchrotron emission, and the 150/220 GHz telescope will measure polarized dust foregrounds [49]. A key technology in the CLASS instrument is the variable-delay polarization modulator. Polarization-sensitive CMB instruments must modulate incoming light to separate the CMB signal from detector, instrument, and atmospheric drifts. Most other CMB instruments employ either fast azimuthal scanning (as in SPT) or a rotating half-wave plate (as in ACT). However, both these methods have disadvantages. Azimuth scanning modulates both the polarized and temperate anisotropies synchronously, so another method must be used to separate them. Half-wave plates add the spectral transmission of the wave plate to the polarized signal, which must then be adequately understood to be removed. A VPM in theory circumvents both these limitations. It consists of a polarizing wire array in front of a coplanar movable mirror. The incoming light is separated into its two linear polarization components: the component polarized parallel to the wire array is reflected, while the component perpendicular passes through and is reflected by the mirror. The separation between the two elements results in a phase delay between the two polarizations. By moving the mirror, the relative phase can be modulated, and a direct measurement of the polarization Stokes parameters can be obtained [46]. The CLASS optics consist of an ultra-high molecular weight polyethylene (UHMWPE) vacuum window, AR coated with polytetrafluoroethylene (PTFE), several metal mesh filters to reduce thermal loading, and two cold lenses (cooled to 4 and 1 K) constructed of high-density polyethylene (HDPE). The cold lenses use simulated dielectric layer AR coats, in which holes drilled in the lens with spacing smaller than the wavelength produce an effective dielectric constant lower than that of bulk HDPE. The optics are cooled by pulse tubes, with the focal plane further cooled to a base temperature around 70 mK by a dilution refrigerator. The lower base temperature allows for a lower operational temperature of the TESs, around Tc = 150 mK, which reduces thermal noise. Dilution refrigerators can also operate continuously, as opposed to helium sorption fridges, which provides improved observing efficiency. The CLASS 40 GHz camera has a 14◦ × 19◦ field of view (FOV), with 1.5◦ pixel beam full width at half maximum (FWHM), substantially greater than the ∼1◦ FOV and ∼1 arcmin FWHM of most CMB LATs. The focal plane consists of 36 dualpolarization pixels, for a total of 72 bolometers. It is read out with√an 11× TDM SQUID amplified readout system. It has an array NET of 181 μK/ Hz, which is estimated to produce a survey depth of 36 μK-arcmin after a 3-year nominal survey, 4× lower than the Planck sensitivity at 40 GHz [27]. The 90 GHz camera contains 259 dual-polarization pixels, with a 380 square degree FOV and 40’ √ FWHM. It is read out with an 11× TDM system. It has an array NET of 14 μK/ Hz [43]. The final camera contains 255 dichroic (150/220 GHZ ) dual-polarization pixels

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coupled to√1020 TESs. It is read out with √ a 44× TDM system. It has an array NET of 17 μK/ Hz at 150 GHz and 51 μK/ Hz at 220 GHz [44]. The different CLASS telescopes came online between 2016 and 2018 and are still observing as of publication. CLASS is aiming for a final constrain on the tensor-toscalar ratio of σ (r) = 0.005 [49].

Spider: A Balloon-Borne Telescope Balloon-borne telescopes are an intermediate class between ground-based instruments and satellites. They avoid the majority of atmospheric noise by observing at ∼36 km of altitude, and their cost is substantially less than that of a satellite mission. The trade-off is the limited observing time at float altitude. NASA’s longduration ballooning (LDB) flights are usually limited to approximately 2 weeks of observing, limited by balloon mechanics and flight path. LDB missions are typically flown from McMurdo base in Antarctica, so that the balloon will be guaranteed to fly over uninhabited terrain, and so that the Antarctic circumpolar winds will return the balloon to a location near its origin for ease of retrieval. Conveniently, 2 weeks of observation in the lower noise environment of balloon float altitude is roughly comparable to a year of observations from the ground, so the trade-off in observing time can be worthwhile. Other advantages of this mode of observation include unobstructed sight lines to the full sky, reduced loading from the ground, and a natural testbed for technologies that are being prepared for the rigorous requirements imposed on satellite telescopes. Spider is one such balloon-borne CMB telescope. Spider consists of six refracting telescopes contained in a single liquid helium-cooled cryostat. Pulse tubes, while they are closed cycle and provide continuous cooling, require large amounts of electrical power to operate and so are not viable on a balloon. The telescopes and associated electronics are housed in a carbon fiber gondola, with a total payload mass of 3500 kg. Power is provided by a 2 kW solar panel array. Command, telemetry, and location are communicated to the ground by several radio link antennas. It is not possible to transmit all of the detector data over the radio link due to bandwidth limitations. A small subset of data is transmitted for monitoring, but the on-board hard drives must be recovered after the flight to obtain the full science data. Telescope pointing is slightly more complicated in a balloon than in a groundbased telescope and is accomplished with reaction wheels to regulate the angular momentum of the payload and allow for scanning in azimuth and elevation at up to 4 deg/s. A combination of star cameras, GPS receivers, sun sensors, and gyroscopes provides pointing, with an instantaneous accuracy of 2σ detection of GWs if r > 10−3 or a >6σ

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detection if r > 3 × 10−3 , by the mid-2030s. Proposed next-generation space-based telescopes such as LiteBIRD [50], Pico [48], or CORE [42] would potentially extend these constraints even lower, to r > 10−4 , in the range 2035–2050. In principle, the GWs produced during inflation are also directly measurable by interferometric GW detectors as a stochastic gravitational wave background (SGWB), since these GWs are expected to extend in frequency with a slightly red spectrum from the current horizon size to the horizon size at the time of inflation, 10−19 Hz < f < 1011 Hz. CMB instruments probe the lowest-frequency region, at f < 10−16 Hz, and within that region, their constraining power is unparalleled. The expected constraint of r > 10−4 from planned CMB observations corresponds to a constraint on the energy density of the SGWB of GW > 2 × 10−19 h−2 , which is significantly below the sensitivity of any currently envisioned interferometric GW detector [74]. So for the foreseeable future, CMB anisotropies will provide the only method of exploring primordial gravitational waves in this frequency range. A detection of primordial GWs through CMB anisotropies would be incredibly influential for modern physics. It would be the first evidence of the quantization of gravity and would provide information on inflationary field physics near the energy scale associated with grand unified theories. It would effectively probe energy scales significantly beyond the reach of the LHC or indeed any currently conceivable collider experiment. Such a measurement would have far-reaching implications for many aspects of fundamental physics, including the unification of forces and key aspects of string theory. In the case of non-detection, a 95% confidence upper limit of r < 0.001 would still significantly improve our understanding of the inflaton field and the inflationary epoch. It would exclude large classes of models, requiring significant re-evaluation of the physics behind the early universe. In either case, the results will be illuminating. The decade of the 2020s will see revolutionary observations that probe the earliest times and the highest energy scales in the universe and change our understanding of fundamental physics and the nature of the cosmos.

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Third-Generation Gravitational-Wave Observatories Harald Lück, Joshua Smith, and Michele Punturo

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3G Science Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme Gravity and Fundamental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Stellar-Mass Black Holes Throughout the Universe . . . . . . . . . . . . . . . . . . . . . . Sources at the Frontier of Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmology and Early History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From the Second to the Third Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmic Explorer (CE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brief History of CE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CE Detector Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CE Status and Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Einstein Telescope (ET) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brief History of ET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ET Detector Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ET Status and Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Down Under . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Path to 3G Observatories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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H. Lück () Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck Institut für Gravitationspyhsik, Max-Planck Gesellschaft, Germany e-mail: [email protected] J. Smith Nicholas and Lee Begovich Center for Gravitational-Wave Physics and Astronomy, California State University, Fullerton, CA, USA e-mail: [email protected] M. Punturo Istituto Nazionale di Fisica Nucleare Perugia, Perugia, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_7

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Abstract

The second generation of gravitational-wave (GW) detectors has opened a new window on the cosmos, yet leaves much of the gravitational universe unexplored. A greatly more sensitive third generation of observatories is currently being planned: The Einstein Telescope in Europe and Cosmic Explorer in the USA. The Einstein Telescope is envisaged as an underground observatory in the shape of an equilateral triangle with a side length of 10 km located in Europe. Cosmic Explorer, on the other hand, aims for two above-ground L-shaped detectors with up to 40 -km-long arms in the USA. These detectors will be capable of observing the post-merger phase of neutron star mergers, detecting a large fraction of all of the stellar mass binary black hole (BH) mergers in the universe – including some with incredible precision, and of mapping the stellar evolution of the universe. These projects are currently in the design stage and are expected to start operations around the mid-2030s. The concepts are slightly different but have many technical similarities. Keywords

Einstein Telescope · Cosmic Explorer · Voyager · NEMO · Third generation · 3G

Introduction Gravitational wave (GW) physics has experienced a rapid upswing following the historic first detection of GWs from a pair of merging black holes (BH), GW150914, in 2015 by the two advanced LIGO detectors [1]. Further, the 2017 discovery by advanced LIGO and advanced Virgo of the binary neutron star merger GW170817 opened an era of multi-messenger GW astronomy and received significant appreciation by the wider scientific community. The scientific findings from the advanced LIGO and advanced Virgo observation runs completed to date (O1, O2 and O3) have inspired scientists by providing further insights into previously unexplored areas. In the first two observation runs in the period from 2015 to 2017, a total of 11 GW events were observed [2]. Analysis of the first half of the third observation run (O3a, 1 April to 1 October 2019) produced another 39 events from coalescing pairs of BH or a BH with a neutron star (NS) [3]. The second half of this observation run (O3b, 1 November 2019 to 27 March 2020) is still being analysed with about two dozen more candidates. Despite the incredible sensitivity of these instruments and the impressive scientific results already obtained, the detection range and number of observations is still too small for deep explorations of the universe, and the signal-to-noise ratio obtained comes mainly from the inspiral phase of coalescence of binary systems and is not yet sufficient to satisfactorily study the merger process and ring-down of the emerging new object. To exploit more of the potential of GW astronomy, the sensitivity of the detectors must be further increased. For the future, it is planned to increase the sensitivity by one order of magnitude and over a wide frequency range

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beyond the design sensitivity of current detectors. In parallel, the frequency range will be extended down to a few Hz in order to observe interesting sources emitting GWs in this low-frequency band. GW astronomy is still in its infancy and just about to take the first steps. For the next decade, the scientific program of the network of the LIGO, Virgo and KAGRA detectors foresees two more data taking periods (O4 in 2022–2023 and O5 in 2025–2026), alternating with periods of detector upgrades, until ∼2026–2027. During this period, the detector sensitivity will increase by up to a factor ∼3–5 for Virgo and ∼3 for LIGO, i.e. about a factor of two beyond their design sensitivities. The Japanese detector KAGRA joined the network in April 2020, and LIGO India, a third version of the advanced LIGO detectors currently being constructed in India, will become operational in the second half of the 2020s. Plans for the period after 2027 are currently being discussed. However, it is likely that upgraded versions of the three advanced LIGO detectors, advanced Virgo and KAGRA (five detectors in total) will be operating until the end of the next decade. It became obvious quite early that the infrastructure of the LIGO and Virgo detectors built in the 1990s would offer only limited possibilities beyond these upgrades. The infrastructures themselves, with the necessary machinery, are neither designed nor suitable for the desired increase in sensitivity (see also  Chap. 8 “Research and Development for Third-Generation Gravitational Wave Detectors”). The planned retrofits of existing detectors within the next decade will push the limits of the infrastructures, both in terms of durability and performance, and therefore no significant further improvements in the existing infrastructures will be possible. In the long run, any further increase in sensitivity can only be achieved by increasing the size of the detectors and building dedicated ultralow-noise infrastructures. Furthermore, the scientific community is quite keen to conduct regular and frequent data runs to obtain new data for continued new scientific discoveries. A radical reconstruction of the existing infrastructures would not only lead to considerable costs but also to unwanted downtime of the observatories and delays in the observation runs. A third generation (3G) of instruments with significantly longer baselines and facility lifetimes of 50 years is currently being planned. Two leading 3G concepts have emerged: the Einstein Telescope in Europe and Cosmic Explorer in the USA. The Einstein Telescope is envisaged as an underground observatory in the shape of an equilateral triangle with a side length of 10 km located in Europe. Cosmic Explorer aims for two above-ground L-shaped detectors with up to 40 -km-long arms in the USA. At the time of writing (2021), we are in the fortunate position of having highly sensitive detectors capable of detecting GWs at a rate of about one merger event per week. So far, these are all from binary star systems with BHs, neutron stars or other unknown objects. The signals have provided us with remarkable scientific results, but the capabilities of GW astronomy are far from exhausted; quite the contrary – the age of GW astronomy is just beginning. By peering more deeply into the gravitational universe, we will observe a variety of sources and unlock a treasure of scientific results. Below we summarise the main scientific objectives that will be pursued with the next generation of GW detectors.

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3G Science Targets Following a study [4] made by the Gravitational Wave International Committee (GWIC), the targets of the next generation of GW detectors can be grouped into the following topics: • • • • •

Explore new physics in gravity and in the fundamental properties of compact objects. Determine the properties of the hottest and densest matter in the universe. Reveal the merging BH population throughout the universe and search for massive black hole seeds. Understand the physical processes and mechanisms that underlie the most powerful astrophysical phenomena. Investigate the particle physics of the primeval universe and probe its dark sectors.

The immense scientific potential of a network of third-generation GW detectors is already evident in the enormous distance at which it can still observe compact merging binary systems. With the design sensitivities of the Einstein Telescope and CE2 as shown in Fig. 3, binary systems of merging BH can be observed up to a redshift of about 100 (see Figs. 1 and 2). This goes far beyond the distances at which electromagnetic telescopes can observe individual sources. The next generation of GW detectors offers a unique window on the earliest moments of structure

Fig. 1 Astrophysical reach for equal-mass, non-spinning binary systems. Curves for 3G GW detectors ET, shown in green, and CE (CE), shown in pink, compared with the design sensitivity of advanced LIGO, shown in blue. The detection horizon is the farthest distance to which an optimally oriented source can be seen. The shaded curves represent the sensitivity to sources with differently distributed sky locations and random inclination. Graphics kindly provided by E. Hall (MIT)

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100 Redshift 10 1 0.1 GW150914 GW170817

Fig. 2 Astrophysical detection horizon for compact binary systems. The dotted lines show the detection horizons for the different detectors. The detection horizon is the farthest distance to which an optimally oriented source can be seen. The yellow and white dots are for a simulated population of binary neutron star mergers and binary BH mergers, respectively. Note that beyond z ∼ 10, i.e. in the Dark Ages of the universe, no sources are shown. Graphics kindly provided by E. Hall and Salvatore Vitale (MIT)

formation in the universe. The following overview of science targets is based on the community’s contribution to the GWIC 3G science case. More detail and thorough references can be found in that document [4].

Extreme Gravity and Fundamental Physics Detectable GWs originate from regions where large, compact masses are strongly accelerated and the curvature of space-time is accordingly strong and changes rapidly. The GWs carry information about the dynamics of the mass distribution almost unaltered to our detectors on Earth, where the signals can be recorded, analysed and interpreted. They thus open up the possibility of testing general

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relativity (GR) under the most extreme conditions of these regions, which cannot be produced in any terrestrial laboratory and is not found anywhere in the vicinity of Earth. A network of third-generation GW detectors, which can observe a large number of sources with sometimes considerable accuracy, enables us to test GR in strong fields as found near BH. Violations of GR would show up in differences between the signals calculated from GR and the ones observed. Also the presence of unknown (dark) particles or effects of quantum gravity can have observable effects on the orbital dynamics of BH binaries and spin properties of BH, and even during the long propagation path of GWs from the sources to us, birefringence can alter the observed signals. Probably the sources of most of the binary coalescences observed by advanced LIGO and advanced Virgo are BH, but all signals observed so far lack the signal-tonoise ratio for a detailed analysis of resonances in the ring-down phase of the newly formed object to distinguish between BH and other putative objects with similar properties, so-called BH mimickers (see also  Chap. 28 “Testing the Nature of Dark Compact Objects with Gravitational Waves”). However, with the excellent sensitivity of a 3G network, signals may reveal features such as higher order modes or GW echoes that are incompatible with the predictions of signals from BH. Galactic dynamics and observations of gravitational lenses call for a large amount of dark matter of hitherto unknown nature. The 3G network may detect such dark matter and help distinguishing different forms through the effects on the inspiral phase of compact binary coalescences.

Extreme Matter In neutron stars, matter is present in its densest form possible. According to current understanding, any further compression leads to an unstoppable collapse into a BH. In the initial inspiral phase, the dynamics of a neutron star coalescence can be approximated as that of two point masses, but as they approach, the deformations of the neutron stars caused by their mutual tidal forces have an increasing influence on the orbital motion of the stars around each other and thus on the GWs emitted. By analysing the GW signals, it is thus possible to draw conclusions about the deformations and dissipation processes in neutron stars and so to study the behaviour of matter under these extreme conditions (see also  Chaps. 12 “Binary Neutron Stars” and  13 “Isolated Neutron Stars”). Furthermore, due to the enormous centrifugal forces, a hypermassive neutron star could be kept in an unstable state of ultrahigh-density matter for a short moment immediately after the merger, providing insights into new, otherwise inaccessible physics. Although many of the coalescences of double neutron stars will be at the edge of the range that can be observed by third-generation GW detectors (as indicated in Fig. 2) and the data will therefore have a moderate signal-to-noise ratio, with a total expected merger rate of more than 100,000 per year, there will be several events that will allow to study the merger process with excellent signal-to-noise ratios. These

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observations will allow us to study the behaviour of matter at extreme densities not even found in atomic nuclei.

Observing Stellar-Mass Black Holes Throughout the Universe The gravitational waves first detected in 2015 originated from a source at a distance of 1.3 billion light years. This may seem very far away, but with a redshift of only 0.1, it is actually in our neighbourhood on cosmic scales. 3G GW detectors will be able to observe similar sources up to a redshift of about 100 (see Fig. 1). The formation mechanism of many of the observed binary star systems is still unclear, although there are different models for their evolution. Standard models of stellar evolution do not predict the existence of BH in the observed mass range. Systems like the observed ones may have been formed by multiple capture of isolated BH each being the final result of the evolution of individual massive stars, or from direct formation in multiple star systems, or from density fluctuations in the very early universe. With the sensitivity of 3G observatories, it will be possible to observe compact binary coalescences (CBC) from the childhood of the universe. The first stars, known as population III stars, were likely very massive due to lack of metal. Observing the BH remnants formed from Pop. III stars would provide a glimpse of the life and death of the first stars and inform models for how our universe evolved from that point. Even earlier than this is a dark period, known as the Dark Age, in which the emission of the cosmic microwave background had faded, but before the first galaxies and stars were formed about 100 million years after the Big Bang and thus before the formation of compact double star systems. The 3G GW network can help to shed some light onto these Dark Ages. No CBC signals are expected from these times as indicated in Fig. 2 (see also  Chaps. 12 “Binary Neutron Stars” and  15 “Black Hole-Neutron Star Mergers”). Observing BH mergers in the Dark Ages would mean that these BH, so-called primordial BH (see also  Chap. 27 “Primordial Black Holes”), formed in the early moments of the universe. Such a finding could fan the recently rekindled debate about the role of primordial BH as the long sought-after dark matter.

Sources at the Frontier of Observations So far some of the sources of GWs, although known to exist, cannot yet be observed with the sensitivity of the current detectors. Fast rotating neutron stars, observable as pulsars with radio telescopes, show glitches in their rotation period, which are associated with stellar quakes. Such changes in mass distribution should emit GWs measurable with 3G GW detectors. If rotating neutron stars show deviations from the spherical shape, they can emit sinusoidal GWs over a long period of time (see also  Chap. 13 “Isolated Neutron Stars”). The eccentricity can either be left over from the formation process of the neutron star or be caused by the accretion of

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mass on the neutron star. Mass from an accretion disk is guided along the magnetic field lines to the poles of the star, just like the aurora borealis (just that the magnetic fields are up to 100 billion times stronger) on Earth, and can heat the magnetic poles, causing eccentricity and adding angular momentum, such that a high constant GW emission can be maintained despite the energy loss in GWs. 3G observatories can help to find out whether this balance between energy gain and radiation could be responsible for the observed maximum rotational frequency of X-ray pulsars. The interior of a supernova explosion is shielded from observation with electromagnetic telescopes by a large amount of ejected matter. The dynamics of this large, fast-moving mass during the collapse, the subsequent explosion, the resulting shock front and the turbulent convection processes generate GWs, which can be observed by 3G observatories at distances up to a few MPc and which provide valuable information about the ongoing processes (see also  Chap. 21 “Gravitational Waves from Core-Collapse Supernovae”).

Cosmology and Early History of the Universe GW sources of compact binary coalescences (CBCs) can be used as “standard sirens” – the GW equivalent of standard astronomical candles – to determine distances. If the event is observed with multiple detectors to break the inclinationstrength degeneracy (can be colocated like ET), the source strength of the GW of a CBC can be calculated from the waveform of the observed signals. By comparing this to the signal strength observed by our detectors on Earth, we can determine the distance of the source. If the redshift can also be derived for this event, for example, from EM counterparts or identifying the associated galaxy with statistical methods, the expansion rate of the universe over time, i.e. the Hubble parameter, can be derived from the multitude of signals from different cosmic distances observable with 3G instruments. The observed waveforms of a CBC are usually degenerate in total rest-frame mass and redshift, i.e. from just observing the inspiral waveform, we cannot distinguish whether it is a heavier system or whether it is further away and hence more redshifted. If features in the merging process which break the massredshift degeneracy can be observed, like the tidal correction to the GW phase in the late-inspiral signal of binary neutron star systems or BH neutron star systems, redshifts and luminosity distances can be obtained without an EM counterpart. These observations of GW with the 3G network will allow to distinguish between different cosmological model. The integration of GWs into the multi-messenger astronomy, i.e. observing the universe with a multitude of electromagnetic telescopes and neutrino and cosmic ray observatories, already yielded a wealth of scientific findings in the observation of GW170817. The interaction of the multi-messenger instruments operating in the 2030s together with the 3G-GW network can produce revolutionary astronomical discoveries that none of the instruments alone would be able to achieve. And, furthermore, GW from the earliest moments of the universe and its phase transitions are expected to provide a stochastic background of GWs, much like

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the microwave electromagnetic background (see also  Chap. 25 “Stochastic Gravitational Wave Backgrounds of Cosmological Origin”). The detection and measurement of such a background would provide insights into particle physics at energies that will never be attainable in any terrestrial experiment.

From the Second to the Third Generation When GW detectors gain another order of magnitude in sensitivity, we intend to call it a new generation. The first generation was initial LIGO, Virgo, GEO600 and TAMA300. After upgrading to advanced LIGO (aLIGO), advanced Virgo (adVirgo) and newly constructing the underground KAGRA detector, we call it the second generation. The current detectors have not yet fully reached their design sensitivity (see Fig. 3), as some of the last upgrade steps are still pending, such as signal recycling for advanced Virgo, or design parameters have not yet been fully reached, such as the used laser power for advanced LIGO. On the other hand, however, techniques that go beyond the design features of the second generation are already being applied, e.g. squeezed light, as it has demonstrated excellent performance in the smaller GW detector GEO 600. The short-term development road map with observation objectives and sensitivity progress of the network of currently active GW detectors is described in [5]. Such innovations and technological advances will make it possible to further increase the detectors’ sensitivity by a factor of about three before the constraints of the infrastructures prevent any further progress. This intermediate step between the “advanced” detectors and the third generation is usually called “A+”, i.e. the “Plus” version of the advanced detectors. According to [5], the upgrades planned are these: “The A+ upgrade to the aLIGO instruments will include higher power, frequency-dependent squeezing and, crucially, new test masses with improved coating thermal noise. Facilities modifications to incorporate the filter cavity required for frequency-dependent squeezing will begin after O3. The full A+ configuration, adding improved test masses and balanced homodyne readout, is expected to be in place for O5. The AdV+ upgrade will occur in two phases. Phase 1 installation will begin after O3 and will involve adding signal recycling, frequency-dependent squeezing, higher input laser power (to 50 from 20 W currently) and cancellation of Newtonian noise. Phase 2 will be implemented between O4 and O5 and will include input laser power increase to 200 W, 100 kg test masses and better optical coatings.” Until now, interferometers have been operated at room temperature, but their sensitivity is increasingly limited by thermal noise, an effect which as the name suggests has to do with temperature. One way of reducing this noise is therefore to lower the temperature. At lower temperatures, however, fused silica as it is currently used in LIGO and Virgo is no longer suitable due to high mechanical loss and the resulting thermal noise such that a different mirror material must be chosen. KAGRA, already pioneering the path of cryogenic interferometers, uses sapphire mirrors. Another promising option is silicon mirrors. This, however, requires a transition to longer laser wavelengths as it is not transparent at the current laser

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Fig. 3 The strain sensitivity of various second- and third-generation detectors, plotted as an amplitude spectrum of detector noise as a function of the frequency. Above – the second generation and just beyond: Advanced Virgo (dark grey) and advanced LIGO (silver pink) in O3, the design of adVirgo (black) and aLIGO (pink) and the A+ upgrades (blue). Below – the “2.5th” and third generation: LIGO Voyager (orange), Cosmic Explorer (purple), the Einstein Telescope (green) and the Australian detector NEMO (olive), with advanced LIGO (light grey) and the A+ upgrade (blue) repeated from above for comparison. Shown are the sensitivity curves for circularly polarised and overhead sources (except for CE, which is shown for 15 degrees from zenith). Graphics kindly provided by E. Hall (MIT)

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wavelength of 1064 nm. Promising candidates are 1550 nm, a wavelength that is widely used in telecommunications, or 2 μm, where optical absorption is even lower. The challenge of combining high laser power with the operation of cryogenic optics in combination with the emergent problems of point absorbers, which then tend to cause unacceptable mirror deformations due to local heating, led to the concept of “Voyager” [6]. According to this proposal, an interferometer, operating at a temperature of 123 K, a laser wavelength of about 2 μm and heavy silicon mirrors, is to be installed in the LIGO infrastructure. This technological concept is also being discussed for the second stage of Cosmic Explorer (see next section). Another concept that goes beyond the second generation is being developed by Australian researchers. A proposal for a new GW detector, NEMO (Neutron Star Extreme Matter Observatory) [7], abandons high sensitivity in the low frequency range for cost reasons and concentrates on the frequency range around a few kilohertz. This frequency range is particularly suited for studying the nature of neutron stars. The key design aspects of NEMO are the usage of high light power in the interferometer, recycling techniques optimised for high frequencies and the heavy use of squeezed states. A suitable location on the Australian continent has yet to be found. The third-generation GW observatories will start operation, at earliest, in the first half of the 2030s. In the meantime, the existing detectors will be upgraded, while new ones will be built. The Voyager project is one of the options under evaluation, exploring the possibility to anticipate some of the third-generation cryogenic technologies in the LIGO detectors. Based on experience with the current detectors, we anticipate that the biggest challenge in reaching the target low frequency sensitivity for 3G detectors (set by suspension thermal noise, seismic noise and Newtonian noise (see also  Chap. 2 “Terrestrial Laser Interferometers”)) will be to reduce the multitude of technical noises (such as light scattering and control noises) that can afflict the interferometers. The sensitivities of the instruments from the current state to the third generation are shown in Fig. 3.

Cosmic Explorer (CE) A Brief History of CE The first intellectual steps toward the project now known as Cosmic Explorer grew out of efforts within the LIGO Scientific Collaboration (LSC) to conceptualise future GW detectors. Devised in the form of a challenge, three teams of scientists, code-named Red, Green and Blue, developed concepts with differing technological centerpieces. At the conclusion of this process, a conglomerate team with the code-name Lavender presented a fourth concept, a detector much like advanced LIGO, only significantly longer. The details of this were further worked out by a team involving members from MIT and Syracuse University. The last presentation of the GWADW2013 on Elba on 24 May 2013 summarised the vision of this

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team for a “Long Uncomplicated Next-Generation Gravitational-Wave Observatory (LUNGO)” [8]. By 2015, this project became known by the name “Cosmic Explorer”, which comes from David McClelland of the Australian National University and represents the project’s goal to explore the Universe with GWs on cosmological scales. 3G science and research in the USA has mostly been carried out within the LSC. In 2017, many of the technical noise sources of Cosmic Explorer were worked out, including evaluations of how those noise sources scale with arm length. Detector options with room-temperature fused silica test masses (current technology) or cryogenic silicon test masses were considered, and noise curves were presented that accounted for optimistic and pessimistic progress by the community on technological research and development [9]. The project’s first dedicated federal funding was awarded the following year through a National Science Foundation award [# 1836814] as a collaborative grant to five US institutions to study the science case for an international network of next-generation GW detectors and the potential avenues for construction of a 3G detector in the USA. As part of these activities, with members of the LIGO lab, this Cosmic Explorer team wrote and submitted “Cosmic Explorer: The U.S. Contribution to GravitationalWave Astronomy beyond LIGO”, an Astro2020 Decadal Survey APC ground-based technology development white paper, which clarified the timeline and plans for CE [10]. This year (2021) has seen a maturation of the CE concept and the community supporting it. The Cosmic Explorer Consortium was formed in Fall 2020 to provide an open and efficient way for members of the broader physics and astronomy communities to contribute to the conceptualisation, design and future use of CE. The CE consortium has a few hundred members now and is holding regular monthly research and development remote meetings. The first major step toward building a CE community was the First Cosmic Explorer Meeting, a five-day remote conference, held in October 2020 with broad community participation to discuss the technical design and the science case for CE.

CE Detector Instrumentation Cosmic Explorer is currently envisioned as two surface-based facilities with arm lengths of 40 km and 20 km located in the United States (USA) with a projected lifetime of 50 years. The CE Horizon Study team, has released a study in 2021 describing the plans ( [11]) Cosmic Explorer would be commissioned in stages, in facilities that could accept detector upgrades similar to the upgrade from Initial LIGO to Advanced LIGO. Within this facility, Cosmic Explorer would be built in stages, with upgrades similar to what was done for initial LIGO, enhanced LIGO, advanced LIGO and advanced LIGO+. The sensitivity of Cosmic Explorer, compared with other current and future detectors, is shown in Fig. 3.

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Cosmic Explorer, is being developed for operations in the 2030s using a relatively low-risk first interferometer based on proven advanced LIGO+ technology, but scaled up to the 40 -km and 20 km baseline facilites, to achieve roughly an order of magnitude sensitivity improvement over advanced LIGO. The optical layout would be that of current interferometers, an L-shaped dual-recycled Fabry-Perot Michelson Interferometer (DRFPMI), using 1064 -nm-wavelength laser light, roomtemperature fused silica optics and quadruple pendulum suspensions from active and passive seismic isolation systems. Current studies anticipate that CE will use suspensions similar to LIGO but scaled up and modified to reduce thermal, seismic and control noises, 1.5 MW of circulating power, 6 dB of frequency-dependent squeezing and improved inertial sensors for its active seismic isolation. The sensitivity of this detector could then be improved with increased Newtonian noise subtraction, even more sensitive inertial sensors and 10 dB of frequencydependent squeezing. An alternative path to reaching Cosmic Explorer’s target sensitivity is to use the Voyager technology [6] being developed for a possible upgrade to the existing LIGO facilities using cryogenic silicon optics, 2-micron lasers and lower-loss optical coatings. Should CE use Voyager technology, it would capitalise on the experience with its implementation in LIGO and again primarily scale up the optics and suspensions to match the 40 -km arms and larger facilities. Ideas for stages beyond the Cosmic Explorer sensitivity shown in Fig. 3 have not yet been thoroughly developed; they will rest upon research and development breakthroughs in the coming decades. For this reason, the CE facility will be built with enough room and flexibility to allow moving of vacuum chambers and other equipment to accommodate different configurations (take, for instance, the current work happening for advanced LIGO+ where vacuum tanks and tubes are being added to realise 300 -m quantum noise filter cavities – a technology which was not yet mature when advanced LIGO was proposed).

CE Status and Timing The current focus for a Cosmic Explorer is on drafting and soliciting feedback on a “Horizon Study” white paper [11] has been released to the scientific community that details the scientific goals and capabilities and the design, community, and management considerations for CE. Current ongoing research efforts include a “trade study” aimed at evaluating how well the science goals of CE can be met by various detector configurations and global networks; the development of a reference design and an associated cost-schedule estimate for the construction of CE; the identification of land large enough to host a CE detector while minimising earth-moving costs; civil engineering costing for three reference sites that will be parameterised to estimate project costs at other candidate locations; investigation of novel and low-cost vacuum systems for the long arms; and a more detailed evaluation of the noise sources that will limit CE’s performance. In parallel, possible funding avenues for CE are being investigated.

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Einstein Telescope (ET) A Brief History of ET The idea of a third-generation (3G) GW observatory was elaborated for the first time in 2004 by a working group studying the future of GW detectors in the European project ILIAS (2004–2008), funded by the European Commission in the Sixth Framework Programme (FP6). An exploratory workshop was funded by the European Science Foundation (ESF) in 2005, where the basis of the Einstein Telescope proposal was set. The name “Einstein Telescope” and its short form “ET”, which nicely hints at the exclusively extraterrestrial origin of our signals, was selected in March 2007, after a long debate considering a wide variety of (sometimes most peculiar) names and acronyms. The first crucial step for the Einstein Telescope project was the approval and then funding of the ET conceptual design study proposal by the European Commission within the Seventh Framework Programme (FP7). The Conceptual Design Report (CDR) for the ET project was delivered in 2011 [12]; the CDR elaborates the concept of Research Infrastructures or Observatory, focusing the attention on the infrastructure to be designed and realised for ET with the capability to allow future upgrades of the hosted detectors for decades. The following years, mainly dedicated to the upgrades, operation and data analysis of the second-generation GW detectors, advanced Virgo and advanced LIGO, allowed the ET scientific community to grow, the “3G idea” to spread and some of the basic technologies to be developed through a series of small grants in Europe. The second crucial step for the ET project was taken in September 2020. The proposal by an alliance of five national governments (Belgium, Poland, Spain and the Netherlands, led by Italy) and a consortium of 41 institutions, including some from Germany, Hungary, Norway, Switzerland and the UK, to include ET in the 2021 update of the ESFRI Roadmap [13], which describes the key European research infrastructures for the decade 2021–2030, was accepted and ET will be included in the update of the ESFRI Roadmap.

ET Detector Instrumentation The Einstein Telescope will be a new GW observatory with a unique design as detailed in the “Einstein Telescope: Science Case, Design Study and Feasibility Report” [14]. ET will improve the sensitivity by an order of magnitude with respect to the design sensitivity of advanced Virgo and advanced LIGO and extend the observation band towards lower frequencies, i.e. down to about 3 Hz compared to ∼10 Hz for the advanced Virgo and advanced LIGO design. Its design combines the proven concepts from current detectors, i.e. a modified Michelson interferometer, with Fabry-Perot cavities in the arms and power recycling and signal recycling techniques with advanced upgrades in an ultralow-noise infrastructure designed to accommodate several technology upgrades over a period of 50 years (Fig. 4).

7 Third-Generation Gravitational Wave Observatories Fig. 4 The Einstein Telescope will consist of three nested detectors (shown in blue, green and red) in a triangular arrangement. Each detector consists of two interferometers, one optimised for low-frequency (solid) and one for high-frequency sensitivity (dashed)

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Specific ET design concepts are: Triangle ET will consist of three nested detectors in a configuration of an equilateral triangle, pairwise sharing a 10 -km-long tunnel, resulting in an angle between the detector arms of 60◦ . With a minimum of tunnelling, this configuration will enable ET to resolve the GW polarisation, provide a null stream and allow for continuous operation during maintenance. 10 km The ET detectors will have 10 -km-long arms, to increase the signals produced by the GWs compared to current detectors, thereby reducing the impact of virtually all of the sensitivity-limiting noise sources. Xylophone Each of the three ET detectors will be composed of a pair of complementary interferometers, one with a peak sensitivity at low frequencies and the other with a sensitivity optimised for higher frequencies. The reason is to separate the challenges related to the use of high power stored in the arms (needed to reduce the photon shot noise) such as thermal and radiation pressure effects, from those related to achieving the targeted low-frequency sensitivity (limited by Brownian noise, quantum back-action noise and radiation pressuredriven control noise). The high-power detector (3 MW) ET-HF, operating at room temperature, covers the high frequency range (>35 Hz) and largely uses today’s advanced detector technology, pushing it to its physical and technical limits. The low-power detector ET-LF, which operates at a temperature of 10–20 K and a low power of circulating light (18 kW), is optimised for low-frequency ( 10−4 . Because AlGaAs has larger coefficients of thermal expansion and thermorefraction, thermo-optic noise could be large enough to hinder the benefit of lower Brownian noise. However, as explained in section “Thermo-optic Noise”, this problem can be circumvented by carefully adjusting the thickness of the coating layers to cancel out the thermoelastic and thermo-refractive noises. Such a coating design was proposed and confirmed to work as expected [129]. The crystalline coatings have already been successfully employed for lowthermal-noise optical reference cavities [152], macroscopic quantum measurement experiments [60, 158], and other applications. The largest remaining challenge is to scale up the coating size to the level required for future gravitational wave detectors (40 cm diameter or more). There is also a research on developing a lattice matched crystal of GaP/AlGaP directly grown on a silicon substrate [159].

Large Laser Beams Random displacement of mirror surface arising from thermal noise is averaged over the beam spot area when sampled by a laser beam. This is why most mirror-related

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thermal noises are inversely proportional to the beam radius wm (see (21), (22), (24), (26)). Therefore, using a larger beam is a straightforward way to reduce these thermal noises. For a given length of arm cavities, increasing the beam size makes the cavity closer to unstable region. Then the radiation pressure-induced angular instability is enhanced [116]. Therefore, there is a limit in increasing the beam size as long as we use the fundamental (TEM00) Gaussian mode of laser beam. Alternative beam shapes are proposed to be used in place of the TEM00 mode. One possibility is to use Laguerre-Gaussian (LG) higher-order modes, especially LG33 mode. These higher-order modes spread more widely in space compared with the TEM00 mode. Therefore, thermal noise can be averaged out better. We can use usual spherical mirrors to resonate LG modes. However, a simulation study found that they are extremely susceptible to mirror imperfections [160]. Combinations of higher-order modes can create fancier beam shapes. Mesa beam is one of such beams with a flat mesa-like intensity profile [161]. The mesa beam is found to be less susceptible to mirror imperfections [160]. However, we need to make non-spherical mirrors for resonating mesa beams. Fabrication of such mirrors is a significant challenge at the moment.

Khalili Cavity In order to reduce the coating thermal noise, fewer layers of coating are desirable. However, such a coating does not have required high reflectivity. A method to increasing the reflectivity while maintaining a fewer numbers of coating layers on the mirror was proposed [162]. This scheme utilizes the fact that a Fabry-Perot cavity kept at anti-resonance shows a higher reflectivity than the reflectivity of the input mirror alone. As shown in Fig. 4, an end test mass mirror, which needs to have a highest reflectivity in an interferometer, is replaced by a mirror with a small number of coating layers accompanied by another mirror at the end to form an optical cavity. By keeping the cavity at anti-resonance, the reflectivity of the combined end test mass can be made very high. This cavity is called Khalili cavity after the name of the proposer. The Khalili cavity scheme has a number of technical difficulties, such as the control noise of the cavity length and the noise from alignment fluctuation. Fig. 4 Schematic of a Khalili cavity

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Therefore, an improved scheme called Khalili etalon was proposed [163]. In this scheme, an additional reflective coating is applied on the back side of a mirror, where usually an anti-reflective coating is applied, to form an etalon. The temperature of the substrate is controlled so that the etalon is kept at anti-resonance for the laser beam. This scheme mostly eliminates the problem of the cavity control noises. However, thermal lensing is shown to be still a serious problem for room temperature mirrors [163].

Substrate Thermal Noise Thermal noises of mirror substrate are not the dominant noise compared with the coating thermal noise in the current generation of GW detectors. However, as we push our detectors toward higher sensitivity, the substrate noises need to be lowered along the way. As in the case of coating, there are several types of thermal noises of substrate.

Substrate Brownian Noise This is the noise caused by the mechanical loss of energy in the substrate. For the case of half-infinite mirror, the power spectrum density of the noise is given by [122], 4kB T 1 − σ 2 Sx (ω) = √ φsub (ω, T ), π ω Y wm

(24)

where Y is the Young’s modulus and σ is the Poisson’s ratio of the substrate material. wm is the beam radius on the mirror and φsub is the loss angle of the substrate. Correction to this formula for finite-sized mirrors is given by Liu and Thorne [164]. Within the usual range of the aspect ratio of mirrors used by GW detectors, the correction is small (a few tens of %). However, the thermal noise becomes significantly higher than (24) for thin mirrors.

Substrate Thermoelastic Noise The substrate thermoelastic noise takes distinctive forms at low- and high-frequency regions. The cut-off frequency fc which separates these regions is given by: fc =

κ , 2 πρCwm

(25)

where κ is the thermal conductivity, ρ is the density, and C is the heat capacity per volume. Below this frequency, the adiabatic approximation is valid, and the noise power spectrum is given by [165]:

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α 2 kB T 2 κ 16 Sadi (ω) = √ (1 + σ )2 , 3 ω2 (ρC)2 wm π

325

(26)

where α is the coefficient of thermal expansion and σ is the Poisson’s ratio. At frequencies lower than fc , the semi-isothermal approximation gives us the following form [166, 167]: Siso (ω) =  J (ω) =

2 π3



∞ ∞ 0

−∞

4kB T 2 α 2 (1 + σ )2 wm J (ω) √ πκ

(27)

u3 e−u /2  dvdu.    2 u2 + v 2 u2 + v 2 + ω2

(28)

2

Substrate Thermo-refractive Noise Thermo-refractive noise is caused by the temperature fluctuation inside the substrate coupled to dn/dt. The magnitude of this noise is given by [165, 168]: SSTR (ω) 

4kB T 2 β 2 κl , 4 ω2 π(ρC)2 wm

(29)

where β is dn/dt and l is the thickness of the substrate. The thermo-refractive noise only affects the light transmitting through the substrate. However, the most sensitive part of our measurement happens inside the arm cavities, where substrate is not involved at all. Therefore, this noise is often not considered for sensitivity estimates of a GW detector.

Candidate Materials The most popular substrate material among the currently operating large-scale gravitational wave detectors is fused silica. LIGO and Virgo use fused silica for all the mirrors. KAGRA uses sapphire for its cryogenically cooled test mass mirrors [169]. Another material considered for use in future GW detectors is monocrystalline silicon [51]. From the viewpoint of thermal noise, material properties such as mechanical loss, Young’s modulus, coefficient of thermal expansion, thermal conductivity, and so on are important. However, for the choice of substrate, other factors also chime in. Optical loss is one of the most important properties for substrate. Availability of high-quality and large sized bulks is also important as future GW detectors are likely to use much larger mirrors than the current generation ones. Transmission properties such as the homogeneity of refractive index and birefringence must be carefully checked, especially for crystalline materials. For cryogenic operation, thermal conductivity also needs to be high. We will review the three materials from these viewpoints.

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Fused Silica Fused silica has been known to have excellent optical properties, such as subppm/cm level of absorption [170] and very low loss angle (down to 5×10−9 ) [131]. This is why it is used very widely in interferometric gravitational wave detectors. However, fused silica is known to have a broad increase of mechanical loss at low temperatures peaked at around 40K [171]. Together with the relatively low thermal conductivity and large thermal expansion at low temperatures, fused silica is not considered to be a good material for cryogenic operation. On the other hand, fused silica shows lower thermoelastic noise at the room temperature compared with sapphire and silicon. This material shows good transparency for wavelength up to 2000 nm. Therefore, even for future cryogenic interferometers, fused silica will remain as the first choice for non-cryogenic mirrors. Sapphire Sapphire is a crystalline form of aluminum-oxide (Al2 O5 ). It has an excellently low mechanical loss angle (φ < 10−8 ) at cryogenic temperatures [172] and high thermal conductivity peaked around 30K [173, 174]. Thermal expansion also gets small at low temperatures [175]. These properties make sapphire a good material for cryogenic mirrors. Because of the high thermal conductivity κ, the cut-off frequency of thermoelastic noise, given in (25), becomes high for sapphire. Therefore, the semi-isothermal approximation given in (27) can be used to explain the thermoelastic noise of sapphire. Especially at low temperatures, the noise can be written as, α 2 kB T 2 1 2 Siso (ω) = √ (1 + σ )2 √ √ . π κω ρC

(30)

The low thermal expansion α and high thermal conductivity κ contribute to low thermoelastic noise. Sapphire is transparent at 1064 nm. Therefore, we can use the proven technologies of 1064 nm lasers and optics with sapphire mirrors. This is an advantage over silicon, which is opaque at 1064 nm. However, even if it is transparent in a usual sense, the optical absorption of sapphire is relatively high compared with high-quality fused silica. The lowest measured value for KAGRA-sized (22 cm diameter, 15 cm thick) sapphire bulk is 23 ppm/cm [176]. This is a much larger value than the sub-ppm absorption of fused silica. As discussed in section “Cryogenic Technologies,” larger absorption generates more heat inside the substrate. If the heat generation exceeds the cooling capacity, we cannot keep the cryogenic operation. Crystalline materials tend to have birefringence, i.e., refractive index depends on the direction of polarization. Sapphire is a uniaxial crystal. As long as light propagates along the c-axis, there should be no birefringence. However, in reality, there is finite birefringence from the misalignment of the crystal axis with the geometric (cylindrical) axis of a mirror. Moreover, it was found in KAGRA mirrors that there are position-dependent fluctuations of birefringence in a sapphire crystal.

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Even though the birefringence is small (n ∼ 10−7 [177, 178]), because of the rather thick substrate (15 cm for KAGRA), the polarization rotation is nonnegligible (A round trip travel in a substrate of 15 cm thick gives about 10 degree phase difference between polarizations). The single bounce conversion from S-polarization into P-polarization in a KAGRA input test mass mirror is up to 10%. Unlike a misalignment of the optical axis, which induces uniform birefringence, position-dependent fluctuations of birefringence cause scattering of the incident beam into higher-order modes. While the carrier light benefits from the mode healing by the high-finesse arm cavities, thus relatively immune from the polarization rotation, RF sidebands, which are used for the sensing of auxiliary degrees of freedom of an interferometer, are directly affected by the birefringence in the input test mass substrates. Currently, the origin of this birefringence fluctuation is not well understood. It is postulated that the residual internal stress from the crystal growth may be the cause of the birefringence. Active research is ongoing to reduce the birefringence fluctuation in sapphire crystals. Production of large sapphire boule has industrial interest, for example, for making large wafers to manufacture LEDs. Therefore, progress in the availability of large size sapphire crystal is ongoing. However, the quality of the crystal necessary for our application is usually not pursued by industry. Therefore, production of high-quality sapphire crystals with large size still requires further research and development.

Silicon Mono-crystalline silicon is known to have a very low mechanical loss both at room and cryogenic temperatures [179]. Therefore, this is one of the most actively studied substrate materials for the next generation cryogenic GW detectors. A notable property of silicon is that the coefficient of thermal expansion (CTE) crosses zero at two temperatures, 123 and 18 K [180]. This property makes the thermoelastic noise to vanish at those temperatures. The fact that we can eliminate the thermoelastic noise at a relatively high temperature of 123 K opens a possibility of using radiative cooling instead of heat conduction (see section “Cryogenic Technologies”). For sapphire, the thermoelastic noise is too high at 123 K because the CTE is still high. Therefore, we need to cool down the mirror below 30 K or so to prevent the thermoelastic becoming dominant. A caveat of silicon is that it is not transparent at 1064 nm. Therefore, we need to use longer laser wavelengths, such as 1550 and 2000 nm. This means we cannot use familiar optical components and sensors optimized at 1064 nm. In terms of thermal noise, longer wavelength requires thicker coating layers. Therefore, the coating thermal noise is increased. Since silicon is the most important material for the semi-conductor industry, there is a strong incentive to produce larger silicon crystals. However, large silicon crystals need not to satisfy the stringent quality requirements of GW detectors. Especially, optical absorption strongly depends on the crystal growth method. Floating zone (FZ) method is known to produce the highest-quality silicon crystals.

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The optical absorption of FZ silicon can be a few ppm/cm [181]. However, the FZ method cannot produce crystals larger than 20 cm diameter. Czochralski (CZ) method is capable of producing much larger crystals. However, the optical absorption of CZ crystals measured at 1550 nm is more than 1000 ppm/cm [181] probably due to impurities. Magnetic Czochralski method may be able to produce a large and low absorption crystals [51]. Birefringence of crystalline silicon is reported to be of the order n ∼ 10−7 [182]. This is a similar value as KAGRA’s sapphire mirrors. Special variation of the birefringence is not measured in [182]. Therefore, further investigation is in need.

Suspension Thermal Noise Thermal Noise Formula A pendulum suspension of a mirror is well approximated by a harmonic √ oscillator at low frequencies. Using the resonant frequency of the pendulum ω0 = k/m in (12), the thermal noise of a pendulum can be written as,

Sp (ω) =

ω02 φp 4kB T ,  2  mω ω − ω2 2 + ω4 φ 2 0 0 p

(31)

where m is the mass of the pendulum bob (mirror) and φpend is the mechanical loss angle of the pendulum. A real suspension has higher-order resonant modes of the suspension wires, called violin modes. Each of the violin modes has a similar thermal noise shape as (31) with ω0 replaced by the resonant frequency of the mode and m by the reduced mass of the mode. The total thermal noise of the suspension is given by summing up all the contributions from the higher-order modes. For complex suspension systems, such as a monolithic fused silica suspension having tapered wires and silicate bonded ears, Levin’s direct approach explained in section “Application to More Complex Systems” is used to accurately account for different losses from different parts of a suspension.

Losses in a Suspension Mechanical loss of a suspension can arise from several different mechanisms. A suspension wire bends during pendulum motion. The internal stress and strain in the wire material induce energy dissipation. A thin suspension wire has a large surface area compared to its volume. Because surface of a wire can have much worse loss angle than the bulk part because of the roughness and contamination, for suspensions using low loss materials, such as fused silica, the surface contribution to the loss can be much larger. The contribution of surface loss φsurf is related to the intrinsic loss angle φs of the lossy surface layer by,

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φsurf 

8hφs , d

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(32)

where h is the thickness of the lossy layer and d is the diameter of the suspension fiber [183]. Development of manufacturing process of fibers with better surface quality is a key challenge in making a low thermal noise suspension. Improvement of surface smoothness also improves the thermal conductivity of fibers used for cryogenic suspension [173]. When a suspension wire is bent, the wire material is compressed or expanded depending on the location within the cross section. This stress distribution creates a temperature gradient, then the resultant heat flow dissipates energy. Such a thermoelastic damping mechanism is analyzed by Cagnoli and Willems [184]. The loss associated with the thermoelastic damping is given by: φTE (ω) =

YT ρC

    ωτ β 2 α − σo , Y 1 + (ωτ )2

(33)

where α is the coefficient of thermal expansion, σo is the static stress in the suspension fiber, ρ is the density, C is the heat capacity per volume, Y is Young’s modulus, and τ is the relaxation time of the heat flow. For a circular cross section [185], τ=

1 ρCd 2 , 4.32π κ

(34)

where κ is the thermal conductivity. β is the thermoelastic coefficient given by, β=

1 Y



dY dT

−1 .

(35)

For most materials, α is positive and β is negative. However, fused silica has a positive β for broad range of temperatures [186]. Therefore, the term in the squared parentheses of (33) can cancel out by choosing an appropriate static stress σo . In the case of silicon, α changes the sign at 123 and 18 K [180]. Therefore, it is also possible to cancel out the thermoelastic damping near those temperatures [187]. Suspension fibers need to be fixed at both ends to something, such as a mirror or a penultimate mass, through clamping, bonding, and so on. This fixing mechanism has its own loss φfix . Great care needs to be taken to design a fixing mechanism whose loss is negligible compared with the other losses.

Dilution Effect by Gravity When a pendulum moves toward its maximum amplitude, energy is stored in two forms: gravitational potential energy from the elevation of the pendulum bob and elastic strain energy from the bending of pendulum wires. Gravity is lossless. Therefore, there is no associated energy dissipation in the gravitational potential.

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If Ug and Ue are the energy stored as gravitational potential and elastic strain, the effective loss of the pendulum is: φpend =

Ue φe , U g + Ue

(36)

where φe is the loss angle of the elastic part of the pendulum, which includes all the losses described above. In order to increase this dilution effect, ribbon-like geometries are considered [187–189]. For a fixed cross-sectional area required to hold the mirror load, a thin ribbon with rectangular cross section is more flexible in one direction than the standard circular cross-section fibers. This reduces elastic energy stored in the bending of the ribbon, resulting in higher dilution effect.

Fused Silica Fibers Fused silica is the best suspension material we know so far for room temperature suspensions, because the material has an extremely low intrinsic mechanical loss. Moreover, as explained in section “Losses in a Suspension,” the unusual temperature dependence of the elastic modulus, that is getting higher at higher temperature, makes it possible to cancel out the thermoelastic loss. In order to null the thermoelastic noise, the static stress of the bending point of the fiber needs to be 184 MPa. However, this value is much smaller than the tensile strength of fused silica fibers (order of GPa). Because we want the violin mode frequencies to be as high as possible and the vertical bounce mode frequencies as low as possible, thinner fibers are preferred. In practice, a tapered design is employed to have a diameter optimized for thermoelastic cancellation around the bending points of the fibers while the diameter is thinner in the middle [190]. A monolithic silica suspension is constructed using hydro-catalysis bonding [191] and laser welding [192] to fix the fibers at both ends. Mechanical losses associated with the bonding and welding parts are investigated [193, 194]. Due to its large mechanical loss at low temperatures and poor thermal conductivity, fused silica is not a good material for cryogenic suspensions. Cryogenic Suspension Materials Sapphire and mono-crystalline silicon have very low mechanical losses and high thermal conductivity at low temperatures. Therefore, these are the primary candidate materials for cryogenic suspensions. KAGRA uses sapphire for its cryogenic mirrors and suspensions [169]. LIGO Voyager and Einstein Telescopes are considering to use silicon [51, 195]. When operating at temperatures where radiative cooling is not effective, suspension fibers need to serve as the main heat extraction path. In this case, the diameter of the fibers needs to be large enough to conduct required heat. Shorter fibers are also preferred for better heat conduction. However, such a fiber dimension reduces the dilution factor by gravity. Violin mode frequencies also tend to become lower, and vertical bounce modes get higher deteriorating vertical vibration isolation.

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Therefore, a suspension design needs to be optimized to strike a balance between the benefit of reducing the temperature and the adversary effects of thicker and shorter fibers. As discussed in section “Losses in a Suspension,” cancellation of thermoelastic noise happens at two temperatures for silicon. However, when used as heat extraction path, diameter of suspension fibers may not be optimized for thermoelastic cancellation. Sapphire does not have any cancellation temperature of thermoelastic loss. Even though, ever decreasing thermal expansion at lower temperatures and very high thermal conductivity of those materials generally make the thermoelastic loss negligibly small. When a mirror is cooled, there is inevitable temperature gradient, especially in the suspension fibers. Such a system is in a stationary state but not in thermal equilibrium. Since the fluctuation dissipation theorem is derived for systems in equilibrium, applicability of the FDS is in question for a system with temperature gradient. An extension of the FDT to a system with inhomogeneous temperature distribution is proposed by Komori [196].

Experimental Methods for Thermal Noise Study Quality Factor Measurement In order to estimate the magnitude of thermal noises, material properties of coating, substrate, and suspension must be known. Out of many relevant properties, mechanical loss angle is the most elusive one. A standard method to measure the loss angle of a material is the mechanical ring down experiment. Consider a mechanical resonator made of single material. When a resonant mode is excited, the displacement of the resonator shows decaying oscillations of the following form: x(t) ∝ e−t/τ sin ωt,

(37)

where τ is the decay time of the damped oscillation. By monitoring the oscillation with an appropriate sensor, τ can be measured. The loss angle of the material and τ is related to φ = 2/(τ ω0 ), where ω0 is the resonant frequency of the oscillator. Mechanical loss of mirror substrate materials has been measured with this method [131, 172, 197, 198]. Special care has to be taken when performing ring down measurements to minimize external losses such as friction with the supporting structure of the oscillator. For example, nodal support scheme is used in [197], where a cylindrical substrate is supported at its center where an oscillation node appears for certain resonant modes. Because the oscillation amplitude is zero at the node, the external loss induced by the support is minimized. Thus, the measured decay time is dominated by the internal loss of the substrate. For measuring the loss angle of coating materials, the standard method is to apply a coating on a thin oscillating plate or disk made of low mechanical loss substrate materials, such as fused silica for room temperature and sapphire or silicon

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for cryogenic temperatures [128, 138, 139, 199]. Then a ring down measurement is performed to measure τ . In this case, the resonator is composed of two materials. The measured loss angle φtotal and the loss angles of individual components have the following relation: Utotal φtotal = Us φs + Uc φc ,

(38)

where Us and Uc are the maximum elastic energy stored in the substrate and the coating during the oscillation and Utotal = Us + Uc , φs , and φc are the loss angles of the substrate and the coating materials. By performing a separate ring down measurement of the substrate before the coating is applied, we can measure φs . Combined with the measured φtotal , φc can be deduced from (38). Us and Uc are usually computed numerically with the finite-element method.

Direct Measurement of Thermal Noise From ring down measurements, we can infer the magnitude of the Brownian thermal noises. However, estimate of thermo-optic noise requires detailed knowledge of other properties like coefficient of thermal expansion and thermo-refractive coefficient. Sometimes, the values of these coefficients are not reliably available, especially at low temperatures. Therefore, direct confirmation of thermal noise level is desirable. A Fabry-Perot cavity is the standard device to measure tiny displacements of mirror surface. However, if we only use a single cavity, the measurement result is usually contaminated by the laser frequency noise. Therefore, the so-called twin-cavity configuration is used. Differential measurement of two nearly identical cavities cancels out the laser frequency noise, and the residual fluctuation of the measurement contains the thermal noise information. Since thermal noises are incoherent between the two cavities, they do not cancel out [200, 201]. Another approach is to use a longer cavity for frequency stabilization and use this stabilized laser to measure the length change of a shorter cavity which contains mirrors of interest [152]. While the twin cavity method can cancel out the laser frequency noise, the method still suffers from incoherent noises between the two cavities. Especially, the effect of seismic vibration can be different for the two cavities due to small mechanical asymmetries even if they are located close together. A novel scheme to circumvent this problem was proposed and demonstrated [202,203]. In this scheme, two higher-order modes (TEM20 and TEM02) are resonated in a single folded cavity at the same time. Because those two modes are orthogonal to each other, they sample different areas of the mirror surface. Therefore, even though they are resonating in the same cavity, the thermal noise sensed by those modes is incoherent. Meanwhile, seismic vibration moves the entire mirrors at frequencies lower than the first internal resonance of the mirror. Therefore, the seismic noise can be coherently cancelled out. The folded cavity configuration enables us to focus the beam on the flat folding mirror in the cavity. In this case, thermal noise contribution of the flat mirror becomes dominant over the other two mirrors (input/output mirrors). With

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this configuration, exchange of the sample mirror is relatively easy, making this method suitable for surveying many different coating materials.

Cryogenic Technologies Cooling down the mechanical components of an interferometer obviously reduces thermal noise. This has been the main motivation to pursue cryogenic in gravitational wave detectors. In fact, most of the resonant bar detectors were operated at cryogenic temperatures [204]. Cryogenic operation has another benefit for interferometric detectors. Very low thermal expansion and high thermal conductivity of sapphire and silicon diminish the thermal lensing to an almost negligible level [205, 206]. In this section, we review some of the key technical challenges associated with cryogenic mirrors and suspensions.

Extraction of Heat from Mirrors In order to cool down mirrors, heat must be somehow extracted from them. Since there is no convection in vacuum, there are two options left for heat transfer: conduction and radiation.

Conductive Cooling By connecting heat links to a mirror, we can extract heat. Suspension fibers are the most obvious choice for this cooling path. Sapphire and silicon are good materials for such fibers because they have excellent thermal conductivities at low temperatures. From the viewpoint of heat extraction, thick and short fibers are necessary. For example, KAGRA uses sapphire fibers of 1.6 mm diameter and 35 cm long [169]. However, such fibers have lower dilution from gravity. Therefore, the improvement of suspension thermal noise by going to a low temperature is hampered by the poor dilution factor. One could consider to attach soft heat link wires other than the suspension fibers to the mirror for extra heat extraction, thus allowing us to use thinner suspension fibers. However, soft materials with good heat conduction, such as pure aluminum or copper, have high mechanical losses. Therefore, the overall loss of the suspension can be severely compromised. So far, no feasible design to attach extra-heat links directly to mirrors is found. A mirror suspension is constructed by bonding together several components, such as fibers, ears, and a mirror. The thermal resistance of the bonding layers must be small enough to allow for efficient cooling of the mirror. In KAGRA, hydrocatalysis bonding (HCB) and gallium bonding are used [169]. Indium bonding is also studied [207]. The thickness of those bonding layers must be kept minimum not to compromise the thermal conductivity of the heat extraction path.

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Introduction of mechanical vibrations through the heat links to the mirror is another disadvantage of conductive cooling, mitigation of which will be discussed in section “Preventing Vibrations.”

Radiative Cooling According to the Stefan-Boltzmann law, the radiation power is proportional to T 4 . Therefore, cooling by radiation is a viable option only for operation at relatively high temperatures such as 123 K. A proposed scheme for LIGO Voyager uses a black coating on the barrel side of a mirror to increase the emissivity [208]. Cryogenic shields surrounding the mirror are also black coated except for the outer most surfaces. Cooling power of 5 W is expected for a mirror at 123 K with a 83 K inner shield. The radiative cooling allows us to use thinner suspension fibers. It also does not introduce vibrations from the cryogenic system directly into the suspension. Black Coatings Black coatings are important for cryogenic applications mainly for increasing emissivity to efficiently transport heat with radiation. Another reason for black coating is to reduce scattered light noise from vibrating cryogenic components near interferometer mirrors. Many black coatings have porous or hairy surface structure. Therefore, they serve as good reservoir of residual gas molecules, imposing a long lasting outgas problem. Search for vacuum and cryogenic compatible black coatings is ongoing. KAGRA uses diamond-like carbon (DLC) for the mechanical components of the cryogenic mirror suspensions [169, 209]. For cryogenic baffles, they use a nickel phosphide-based plating called Solblack [210]. DLC has good vacuum compatibility and surface durability. Solblack has better absorption coefficient but lesser vacuum compatibility and durability.

Heat Injection into Mirrors After the initial cooling from the room temperature, a mirror reaches an equilibrium temperature determined by the balance between the cooling power and the continuous heat injection into it. Main sources of continuous heat injection are the followings.

Laser Power Absorption Extremely high laser power is circulating in arm cavities of an interferometric GW detector (0.4 MW for KAGRA, 3 MW for Voyager, and 18 kW for ET LF). Small optical absorption in the coatings can generate considerable amount of heat. The optical absorption within the input test mass substrate can also be a large contribution, even though the laser power transmitting through the substrate is hundreds of times smaller than the one inside the arm cavities. This is because the

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mirror substrate is thick and the absorption coefficient is larger for sapphire and silicon compared with fused silica. In the case of KAGRA, 0.7 W of heat is injected into each mirror from the absorption of laser [169].

Room Temperature Radiation In order to protect a cryogenic mirror from room temperature thermal radiation, it is usually covered by multiple layers of radiation shields, which are walls made of low emissivity materials also cooled down. However, laser beams need to get in and out of this radiation shield. Therefore, some holes have to be open in the shield. Because the openings to the room temperature world are directly visible from the mirror, room temperature thermal radiation also directly bombard the mirror. The amount of heat injected this way depends on the solid angle of the opening seen from the mirror. Therefore, the further away the hole is, the smaller the heat injection. For this purpose, a short section of the beam duct near a cryogenic chamber is cooled down (Fig. 5). Such a section is called cryogenic duct. While a long cryogenic duct can reduce the solid angle, if the inner surface of the duct is smooth, thermal radiation can reflect inside the duct multiple times and reach the mirror in the end [211]. This funneling effect can be reduced by putting baffles inside the cryogenic duct [212, 213].

Cooling Engines We need to create a cold part in a vacuum chamber to extract heat from a mirror. There are mainly two options for this: cryofluid and cryocoolers. Cryofluids such as liquid helium and nitrogen are widely used in cryogenic experiments. For small experiments running for a short time, cryofuid is a convenient cooling method. However, for continuous operation of a GW detector with large heat extraction requirement, it will be necessary to build a facility to generate cryofluid on site. A large-scale plumbing may be necessary to send cryofluid to

Fig. 5 A conceptual drawing of a cryogenic system for a mirror. A ray of room temperature radiation is blocked by an inner baffle of the cryogenic duct. As indicated by the dashed arrow, it will otherwise make multiple bounces and reach the mirror

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cryostats separated by a few km or more and recover used one. Safety management of cryofuid is also a challenge, especially for underground sites. KAGRA uses another method, cryocoolers, as its cooling engines. Compact cryocoolers are suitable for a sparsely distributed cryogenic system. Disadvantage of cryocoolers compared with cryofluid is larger vibration from the compression and expansion of the working gas. For KAGRA, specially designed pulse-tube cryocoolers are used to reduce the vibration from the cooler as much as possible [214].

Preventing Vibrations For conductive cooling, a suspension system must be connected to a cold head at some point near the mirror to minimize the distance of heat transport. However, cold heads are usually not mechanically quiet, being connected to a vibrating head of a cryocooler or a cryofluid pipe mounted on the ground. Therefore, vibrations injected from the cold head need to be attenuated. Even for a radiative cooling system, inner shields surrounding a mirror must be well isolated from vibrations of ground and cold heads, to reduce scattering noise [208]. In order to conduct heat without transmitting mechanical vibration, soft heat link wires made of pure metals, such as pure aluminum and copper, are used. Purity is the key for making highly conductive heat links. While copper is a standard material for cryogenic heat conduction, KAGRA developed high-purity (99.9999%) aluminum braid for heat links [215]. The Young’s modulus of pure aluminum is about a half of pure copper. Therefore, we can make softer heat links. The bulk thermal conductivity of copper is better than aluminum. However, for thin heat link wires, the size effect limits the heat conductance; thus the overall heat extraction performance of an aluminum heat link is comparable to the one made of copper. Heat links are anchored to masses which are suspended as pendulum for vibration isolation. However, there are competing factors. For better vibration isolation, multiple stages of anchor masses are needed. However, this will elongate the heat path, increasing the thermal resistance. Also larger anchor masses are better because they have large inertia to resist introduced vibration. However, larger masses are slower to cool down (Fig. 6). One possible way to circumvent the conflicts is to use active control similar to the suspension point/platform interferometer technique [216,217]. Heat links can be anchored to a suspended mass. Another mass, called reference mass, is suspended by a multiple stage vibration isolation system, which does not have to be cryogenic, next to the anchor mass. The relative motion between the anchor mass and the reference mass is monitored by sensors, such as shadow sensors or interferometric ones. Then feedback control is applied to the anchor mass to follow the motion of reference mass. This way, the quiet behavior of the well-isolated reference mass is copied to the anchor mass, without the need for multiple anchor stages. A similar

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Fig. 6 Active vibration isolation of heat links

idea is proposed and tested for the vibration isolation of inner shield for radiative cooling [208].

Molecular Layer Formation When a mirror is cooled down, residual gas molecules randomly hitting the mirror are adsorbed on its surface. This process will eventually form a layer of adsorbed molecules called cryogenic molecular layer (CML). Interferometric gravitational wave detectors have a unique reason to seriously suffer from the CML formation. While most residual gas particles around the mirror are quickly adsorbed by cryogenic walls of the cryostat, there is a constant molecular flow from the room temperature part of vacuum through the openings in the radiation shields to let through the interferometer laser beam. This is in stark contrast to, for example, cryogenic cavities for laser frequency stabilization, where the mirror surfaces of the optical cavity are completely enclosed by cryogenic walls [218]. Formation rate of CML has been studied [219, 220]. For KAGRA, the formation rate is found to be very high (∼27 nm/day). This is attributed to the bad vacuum level of current KAGRA due to much less number of vacuum pumps currently in operation [220]. A CML can scatter or absorb light to deteriorate the interferometer sensitivity. A recent study shows that optical absorption can be particularly serious for future detectors using longer wavelengths, such as 1550 and 2000 nm [221], because the absorption coefficient of amorphous ice, which is expected to be the main

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component of CML, is much larger for those wavelengths than 1064 nm. Even a few nm of CML can induce too much optical absorption to prevent cryogenic operation. Another study pointed out that CML can also increase the coating thermal noise by introducing extra mechanical loss [222]. Since the mean free path of residual gas molecules in ultra-high vacuum environment of GW detectors is more than 100 km, all the molecules fly in straight lines, like laser beams. Therefore, similarly to the room temperature radiation, cryogenic ducts and baffles can reduce the injection of room temperature molecules. The length of cryogenic ducts needs to be selected to make the CML formation rate low enough. Alternatively, we may be able to locally heat up the mirror surface with, for example, a CO2 laser to evaporate CML once in a while.

Cryogenic Compatible Sensors and Actuators Actuators are attached to mirrors for controlling the mirror position and orientation to keep the interferometer locked. Local sensors are deployed throughout the suspension system to damp its resonant modes [223]. For cryogenic suspensions, these sensors and actuators need to be low temperature compatible. This means that (1) they have to work at cryogenic temperatures and (2) working ranges of them need to be large enough to tolerate large thermal drift (mainly shrinkage) during initial cooling. KAGRA uses LED-based reflective shadow sensors for its cryogenic payloads [169]. They tested many LEDs and photodiodes to find ones which work at cryogenic temperatures. Samarium cobalt magnets are known to retain their magnetism at low temperatures [224], making them a suitable choice for cryogenic coil-magnet actuators. While the the coil-magnet actuators work fine at cryogenic temperatures, the noise level of the cryogenic shadow sensors used currently in KAGRA is not satisfactory for low noise damping control. For future cryogenic interferometers, development of low noise and wide-range sensors with cryogenic compatibility, such as fiber-coupled interferometric sensors, is necessary.

Seismic Noise Ground motion varies around the world; however there is a universal reference to a nominal maximum and nominal minimum level of ground motion [225]. These limits are the maximum and minimum outlays of the power spectral density estimates of about 75 seismometer stations worldwide. These are the spectral densities of the vertical ground motion and are referred to as the new low noise model (NLNM) and the new high noise model (NHNM). Particular locations can have increased or reduced ground motion. The general features in defining the noise models are relatively well understood [226]. At frequencies below 1 mHz, the Newtonian attraction of the atmosphere

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above the seismic sensor dominates the recordings [227]. The tidal mechanics, such as the Earth’s rotation with respect to the moon, can be observed in this frequency range. From about 1 to 30 mHz the spectra have multiple overlapping contributions. Up to 7 mHz contribution of the fundamental dominates, while features up to 30 mHz come from the Rayleigh waves, circling around the Earth. In the frequency range from 30 mHz to 1 Hz is the microseism, originating from the interaction between ocean waves and the seafloor and coast, and can increase during ocean storms. Seismic motion at frequencies above 1 Hz is predominantly caused by anthropogenic noise. This can be from factories, truck and trains, or local buildings. Even at the level of the new low noise model, the motion will need to be much √ reduced to reach a residual displacement level of the test masses of ∼10−19 m/ H z at 3 Hz. This is achieved by a sequential number of pendulums, such as that in Advanced LIGO [228, 229], Virgo [230], and KAGRA [231]. These are complex mechanical systems utilizing a range of active and passive techniques to isolate the test masses from seismic motion.

Seismic Isolation The vibration isolation system for the test masses is a hybrid of mechanical oscillators such as inverted pendulums, vertical springs, and linear pendulum. Their response to an external force is that they have a mechanical resonant frequency, with the region below where there is a fixed flat response related to the effective spring constant and the region above which shows a reduced response. Placing these mechanical oscillators in series will reduce the response by 102n well above the highest resonant frequency, where n is the number of stages. The challenge is to engineer these systems to make them physically possible and keep them operational, so when they are installed they are set and forget. The design of the vibration isolation and suspension systems in the current detectors provides sufficient test mass isolation from 10 Hz and higher. A variety of local sensors and actuators are used to maintain the isolation system at their operating point. Depending on the approach, such as a more passive implementation as used at the Virgo observatory, they can utilize small forces at the top of the system, to locate and control large suspended mass. An alternative approach is to use a more active sensing and control implementation, such as used at the LIGO observatories. Both implementations use local sensors ranging from displacement, velocity, and acceleration, commercial or custom built, to maintain and reduce the effective motion of the test mass. Future detectors aim to reduce the lower frequency bound from 10 Hz down to 3 Hz [10] or 5 Hz [11,12]. This will require more R&D to improve the performance of these already complex systems. In addition to use novel techniques to enhance the control of these systems [232], the alternative is to improve the sensitivity of the local sensors, by using new sensor technology used within these vibration isolation and suspension systems. There is active research toward improving inertia sensors

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[233–235], 6D sensors [236], and tiltmeters [237, 238], which all can be integrated into the complex√isolation system. Displacement√sensors with sensitivities as low at 2 × 10−14 m/ Hz at 70 Hz to 7 × 10−11 m/ Hz at 10 mHz [239] are under development. The challenge is to integrate these units into inertia sensors with suspended masses to characterize their response. The actual suspended mass motion and unintended misalignment in the optical readout techniques can greatly affect their performance. Other technologies, such as suspension point interferometers (or suspension platform interferometers), can aid in reducing the effective relative motion between the various cavity mirrors, such as the test masses [240,241]. Other implementations achieving reduction of the test mass motion are the arm length stabilization system in LIGO [242, 243] or similar implementations using alternative techniques [244, 245]. There is a challenge to improve the sensitivity of the readout technique, at the same time√as lowering the signal bandwidth lower bound, with targets of less than 10−12 m/ Hz down to or lower than 100 mHz, below which one could be limited by Johnson noise from the readout electronics or mechanical thermal noise limited. Development for lower noise displacement sensors, extending their sensitivity down to lower frequencies, is of great interest for future seismic isolation system for future gravitational wave detectors.

Newtonian Gravitational Noise The Newtonian attraction from a moving mass at close proximity around the test mass has the potential to disturb the nominal position of that test mass. The interferometer readout cannot distinguish motion due to the Newtonian force from terrestrial masses to that of the effective force from the cosmic gravitational wave. In addition one cannot shield the test mass from this force. To mitigate potential localized Newtonian gravitational forces during an observation run, on-site activities are greatly reduced to lower the number of moving bodies (truck, cars, humans, etc.). Machinery to keep the observatories’ operation is tuned to minimize their effect on the detector. However the natural background motion of the ground and atmospheric activities above and around the interferometer can still generate Newtonian forces onto the test masses. It is these seismic- and atmospheric-induced Newtonian noise contributions at low frequencies which can set the limit of third-generation gravitational wave detectors. Newtonian noise from seismic activities has been studied [246–249] and covers much of the challenges. So far Newtonian noise from seismic or atmospheric origin has not directly been detected. There are experiments underdevelopment to directly measure Newtonian noise [250, 251]. To mitigate the effect of the Newtonian noise onto the detectors, R&D is needed to develop such technologies. They consist of sensor arrays to infer the density

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changes which cause the Newtonian force and then to estimate its influence on the test mass or detector readout. Locating the observatory underground can greatly mitigate the effect of Newtonian noise from seismic surface wave. The Japanese KAGRA detector is located underground in the Kamioka mine in Japan to mitigate seismic noise, gravity gradient noise, and other environmental fluctuations such as temperature and humidity [252–254]. In addition it is the only observatory operating at cryogenic temperatures to reduce thermal noise [255]. Also the European proposed Einstein Telescope will be located underground, to mitigate Newtonian noise from surface waves and improve the sensitivity down to 3 Hz [10]. The US-led Cosmic Explorer next generation detector will use innovative sensor technology and potential infrastructure design to mitigate Newtonian noise; however the target low frequency will be down to 5 Hz [11, 12].

Seismic Sources The Newtonian noise from seismic sources is from the local density change due to the seismic waves. Predominantly the Newtonian noise from surface waves will be dominant in interferometric gravitational wave detectors. Much R&D is underway to investigate and develop technologies on mitigating this effect. A seismometer arrays with tilt sensors are under active development and show much promise in reducing the influence by many factors [256, 257]. In particular the determination of the optimal number and seismometers distribution around the test mass is important. A key component of such arrays are tilt meters, which record the ground tilt [238, 258, 259]. The tilt of the ground provides an ability to sense the phase of the surface wave, while the seismometers can register the amplitude. Correlations between a seismometer array, tilt meter, and the LIGO detector show strong correlation with the ground tilt [251]. Atmospheric Sources Atmospheric-induced Newtonian noise would predominantly be dominant at frequencies below 10 Hz. There are different mechanism which can produce atmospheric Newtonian noise and can come from infrasound, turbulence, and advected temperature fluctuations, as discussed in Fiorucci et al. [260]. Even sensing these mechanisms with sufficient sensitivity to infer their Newtonian noise contributions is difficult. Infrasound microphones are used to measure within the gravitational wave observatories; however when used outside, they are challenging as general wind noise can dominate and spoil the measurements. Other means to measure wind, temperature, and pressure variation of the atmosphere above the test mass are difficult. This is not only to record these parameters of a volume, sensing these parameters in 3D, but also the current sensor sensitivity and technology are currently limiting their use and implementation.

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Seismic Management Through Architecture Locating the detector underground will be the most effective method to mitigate or reduce the effect of seismic surface waves and their contribution to Newtonian noise and atmospheric-induced coupling. However there is a large cost overhead for going underground, but it is potentially worth it for the increased astrophysical reach. Even so, careful design of the infrastructure is still as relevant as with surfacelocated future observatories. When designing and constructing the infrastructure of the gravitational wave observatory, consideration should be taken to minimize self-inflicted noise. Care needs to be taken to mitigate the coupling of vibrations from infrastructure machinery into the ground floor and walls of the observatory. Vibrations from improper installed isolated HVAC systems, air compressors, etc. can introduce unwanted excitation within observatory [261]. These vibrations can couple into the vibration isolation systems via ground floor tilt, wall motion-induced floor motion or tilt, or even direct acoustic coupling. Although not directly Newtonian noise coupling, even wind can contribute to ground tilt, via buildings and structures [262], and can potentially be reduced by wind fences which is under investigation at the LIGO Hanford Observatory. Not only the HVAC system needs to be installed properly, e.g., isolated from the ground and walls, but also the air flow needs to be non-turbulent within the ducts as well as at the release and extraction vents. This will mitigate any acoustic thumping or shock waves into the observatory hall.

Seismic Meta Materials Reducing the actual natural ground motion at the detector site would even further reduce the effect of seismic-induced noise (vibration or Newtonian). This has been an active geophysics research area for the last decade, potentially to direct damaging seismic surface waves from earthquakes [263, 264]. Constructing a metamaterial using boreholes, they showed a reduction of ground motion by up to 50%. One can imagine to construct seismic metamaterials [265, 266] around the detector site to reduce or deflect the inbound seismic waves. A forest of trees has been shown to influence the propagation of Rayleigh waves [267]. The coupling of the vertical resonances of the trees with the vertical ground motion greatly attenuates the Rayleigh waves. This could potentially be implemented as vertical pillars with vertical resonances below 10 Hz to reduce Rayleigh waves entering the detector sites [268]. Care needs to be taken so that tilt of these pillars is not adding to the ground motion. Another approach to construct a seismic metamaterial by burying an array of large mass-spring resonators into the ground, which convert surface waves into shear bulk waves, away from the surface [269]. Using this implementation they showed a reduction of surface ground motion of up to 50%.

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The use of seismic metamaterials to reduce the ground motion at the detector can potentially be envisaged. However, most results so far are effective in a narrow frequency band of a few Hertz, using monochromatic sources. However, the attenuation frequency can be tailored toward the main building resonances (walls or floors). Further R&D should see to investigate how the construct seismic metamaterials tailored toward the use in gravitational wave detectors.

Technical Noise The three preceeding sections have focused on what have been historically referred to in the gravitational wave community as “fundamental” noises, although as we have seen they are amenable to technological developments. Here we will also briefly describe some key “technical” issues that have plagued the current and previous generations of gravitational wave detectors. As gravitational wave detectors are complex devices, these can be mundane issues or devilishly intricate. A key aspect is that many of them are not necessarily the result of one particular subsystem but rather come from the interactions of subsystems; this means they can only be resolved during the integrated testing process (also called commissioning), once installation of all components is complete. The commissioning process can take years – the Advanced LIGO interferometers finished installation in 2014 and as of 2020 have not reached their design sensitivity.

Prototypes Instrumented prototypes are the best way to get some advance notice of difficulties. They are of course limited, often particularly in terms of sensitivity, but nonetheless they provide an opportunity to observe the interaction of subsystems which are often developed separately. The LIGO lab operates two prototypes at Caltech and MIT. The Caltech prototype [270] is a fully integrated prototype and serves as a mini-LIGO. It is currently being upgraded to a cryogenic silicon interferometer, where optical, thermal, and controls issues will be studied for Voyager [51]. The MIT prototype (LASTI) is an advanced systems testbed, where, for example, the Advanced LIGO quadruple pendulum suspension systems were tested. The Glasgow 10 m prototype [91] has been testing speedmeter techniques [92], which include control issues and design and integration of low thermal noise suspension systems. A prototype facility at the Albert Einstein Institute [271] is testing advanced quantum methods, including radiation pressure [272], and has also been used as a testbed for suspension platform interferometry [273], which could prove crucial for future detectors. Further prototypes exist in Australia [274] for testing high optical power techniques and in Japan, where the old TAMA300 detector has been repurposed as a prototype [68]. There is an Einstein Telescope pathfinder at the University of Maastricht, which will test key techniques for the Einstein Telescope.

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Controls Gravitational wave detectors are designed as highly precise measurement devices to detect the motion of freely falling test masses. The suspensions which keep the test masses from falling to the laboratory floor also have resonances; furthermore the interferometer is composed of multiple mirror endowed test masses, along with extra mirrors (such as the recycling mirrors). High-precision interferometric sensing has non-linear readout. All together, the system has multiple degrees of freedom with non-linear sensors, where each sensor can be influenced by any (or all) of the degrees of freedom. As the suspensions are driven by seismic noise, there must be a control system to keep sensors within their linear range and to prevent secondary degrees of freedom from polluting the gravitational wave signal. This in addition to control systems requires to damp any optical instabilities [18, 116]. Despite significant effort, at low frequencies, all gravitational wave detectors have been limited by excess noise from their control systems [102]. This is despite significant efforts to model the more complex aspects such as alignment sensing and control [275]. More research is required, and controls problems must be considered at the design stage, for third-generation detectors to reach their target sensitivity.

Vacuum Technology For the first- (and second-) generation detectors, the high-quality vacuum infrastructure was a significant portion of the total project cost. As the third-generation detectors will be significantly longer, research and development is required to find more cost-effective ways of achieving similar levels of vacuum performance. Collaborative work is likely to yield results that will benefit all the third-generation projects. A first workshop in a planned series on this topic was hosted by the US.National Science Foundation in early 2019, with proceedings available [276].

Conclusion Gravitational wave astronomy is an exciting new field, and it rests on decades of experimental effort to build the gravitational wave detectors that opened the field. The scientific potential of gravitational wave astronomy is vast, and the thirdgeneration detectors will open the cosmos. It remains the case, however, that our ability to explore that cosmos is limited by the sensitivity of our detectors. In the case of the second-generation detectors Advanced LIGO and Advanced Virgo, mature technical designs were ready, and the projects funded, even before the first-generation detectors had reached their design sensitivity. In the case of both the first- and second-generation detectors, more than a decade passed between the first technical designs and the functioning detectors; from this perspective, there is urgency to conduct the necessary research to make the third generation

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of gravitational wave detectors possible with the least delay between the operations of second- and third-generation detectors. This research is underway in laboratories around the world and will no doubt benefit from the existing spirit of collaboration and robust organizations that built the successful second-generation detectors.

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268. Colombi A, Roux P, Guenneau S, Gueguen P, Craster RV (2016) Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances. Sci Rep 6(1):19238 269. Palermo A, Krödel S, Marzani A, Daraio C (2016) Engineered metabarrier as shield from seismic surface waves. Sci Rep 6(1):39356 270. Ward RL, Adhikari R, Abbott B, Abbott R, Barron D, Bork R, Fricke T, Frolov V, Heefner J, Ivanov A, Miyakawa O, McKenzie K, Slagmolen B, Smith M, Taylor R, Vass S, Waldman S, Weinstein A (2008) dc readout experiment at the caltech 40m prototype interferometer. Class Quant Grav 25(11):114030 271. Westphal T, Bergmann G, Bertolini A, Born M, Chen Y, Cumming AV, Cunningham L, Dahl K, Gräf C, Hammond G, Heinzel G, Hild S, Huttner S, Jones R, Kawazoe F, Köhlenbeck S, Kühn G, Lück H, Mossavi K, Pöld JH, Somiya K, van Veggel AM, Wanner A, Willke B, Strain KA, Goßler S, Danzmann K (2012) Design of the 10 m aei prototype facility for interferometry studies. Appl Phys B 106(3):551–557 272. Junker J, Oppermann P, Willke B (2017) Shot-noise-limited laser power stabilization for the AEI 10m prototype interferometer. Opt Lett 42(4):755–758 273. Dahl K, Heinzel G, Willke B, Strain KA, Goßler S, Danzmann K (2012) Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout. Class Quant Grav 29(9):095024 274. Zhao C, Blair DG, Barrigo P, Degallaix J, Dumas J-C, Fan Y, Gras S, Ju L, Lee B, Schediwy S, Yan Z, McClelland DE, Scott SM, Gray MB, Searle AC, Gossler S, Slagmolen BJJ, Dickson J, McKenzie K, Mow-Lowry C, Moylan A, Rabeling D, Cumpston J, Wette K, Munch J, Veitch PJ, Mudge D, Brooks A, Hosken D (2006) Gingin high optical power test facility. J Phys Conf Ser 32:368–373 275. Barsotti L, Evans M, Fritschel P (2010) Alignment sensing and control in advanced LIGO. Class Quant Grav 27(8):084026 276. Dylla HF, Weiss R, Zucker ME (2019) Proceedings: Nsf workshop on large ultrahigh-vacuum systems for frontier scientific research. Technical Report LIGO-P1900072-v1

9

Squeezing and QM Techniques in GW Interferometers Fiodor Sorrentino and Jean-Pierre Zendri

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Fluctuations of the Electromagnetic Field and States of Light . . . . . . . . . . . . . . . . . Expectation Values of Quantum Fluctuations of the Electromagnetic Field . . . . . . . . . . . . States of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrature Noise Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of Nonclassical States of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Noise in the Interferometric Gravitational Wave Detectors . . . . . . . . . . . . . . . . . . . Quantum Noise in a Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Noise in Power Recycled Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Noise in Dual Recycled Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Noise in Presence of Squeezed Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Noise in Real Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Methods for Quantum Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EPR Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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F. Sorrentino () INFN, Genova, Italy e-mail: [email protected] J.-P. Zendri () INFN, Padova, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_9

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Abstract

Interferometric gravitational wave detectors are limited by quantum noise over a large fraction of the observational band. Therefore any method to mitigate quantum noise would significantly improve their sensitivity. In this section we introduce the interferometer standard quantum limit, and we give an overview of the theoretical models and experimental methods so far developed to surpass it. Keywords

Interferometers · Gravitational wave detectors · Quantum noise · Squeezed states · Quantum non demolition · Back-action evasion

Introduction Quantum noise represents one of the main limitations to the sensitivity of gravitational waves (GW) interferometric detectors. In the standard configuration of an interferometer, the equivalent strain noise hn (the equivalent strain noise hn (t) is the gravitational wave signal which would produce at the detector output the total strain noise actually measured.) it wave power spectral density Shn hn (Ω) is larger than the quantum mechanical Cramér–Rao (CR) fundamental limit [1, 2]: Shn hn (Ω) =

h¯ 2 c2 SP P (Ω)L2

(1)

where L is the length of interferometer arms, SP P (Ω) is the spectral density of circulating optical power, h¯ is the reduced Plank constant, and c is the speed of light in vacuum. In the last few decades several methods have been proposed, and in part implemented, to approach and possibly surpass the CR limit, thus resulting in an overall improvement of detector sensitivity. This manuscript provides an introduction to the theoretical framework and to the experimental implementation of such methodologies. In the first section we summarize the quantum properties of light that ultimately generate quantum noise; in the second section we show how the interplay of such properties determines the standard quantum limit, and how squeezed light can be used to overcome it. The last section illustrates some advanced techniques, which have not yet been implemented in large-scale optical interferometers, but which could potentially approach the CR limit in the next generation detectors.

Quantum Fluctuations of the Electromagnetic Field and States of Light In the initial part of this section, we give a brief review of the quantum electromagnetic field theory, limited to the concepts which are functional to the understanding

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of quantum noise in interferometric GW detectors (a detailed quantum treatment of light is beyond the scope of this manuscript; interested readers may learn more about this subject on many monographs in the literature, see, for instance, [3]). Squeezed states of light, which can help to surpass the standard quantum limit, are the topic of the second part of the section along with the methods for their production and characterization.

Expectation Values of Quantum Fluctuations of the Electromagnetic Field In canonical quantization the electromagnetic field is expanded in terms of creation aˆ ω† and annihilation aˆ ω operators of photons at frequency ω [3]. In the case of interferometric GW detectors it is more convenient to expand the electromagnetic field in terms of the sidebands photon creation operator aˆ ω† 0 ±Ω around the carrier laser frequency ω0 which can be directly related to the sidebands generated by the gravitational wave. In this framework the quantum electric field operator ˆ E(x, y, z, t) for a laser beam traveling in the “z” direction can be written as [4] 

h¯ ω0 ˆ ˆ E(x, y, z) = u(x, y, z) [X1 (z, t) cos ω0 t + Xˆ 2 (z, t) sin ω0 t] cε0

(2)

where ε0 is the vacuum permittivity, the mode shape function u(x, y, z) is normalized as dxdyu(x, y, z) = 1, and the quadrature operators X1,2 (z, t) can be written in the Fourier representation as [5] Xˆ 1,2 (z, t) =

 0

∞

 dΩ  ˆ † X1,2 (Ω) exp(−iΩt + ikz) + Xˆ 1,2 (Ω) exp(iΩt − ikz) . 2π (3)

The hermitian operators Xˆ 1 (Ω) =

aˆ ω0 +Ω + aˆ ω† 0 −Ω √ 2

Xˆ 2 (Ω) =

aˆ ω0 +Ω − aˆ ω† 0 −Ω √ i 2

(4)

coherently create a photon at the sideband −Ω and annihilate a photon at the sideband +Ω. This so-called two photons formalism is particularly suitable for dealing with two-photon systems as in the case of squeezed light [4]. It is straightforward to show that the two operators Xˆ 1 (Ω) and Xˆ 2† (Ω) do not commute, namely (here and below with the notation “δ(x)” we mean the Dirac delta function with the argument “x”), [Xˆ 1 (Ω), Xˆ 2† (Ω  )] = 2iδ(Ω − Ω  )

(5)

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which implies that X1 and X2 cannot be estimated simultaneously with infinite 2 must fulfill the uncertainty principle: accuracy, as their variances ΔX1,2 ΔX12 (Ω)ΔX22 (Ω) ≥ 1

(6)

In other words, the phase and amplitude fluctuations of the electromagnetic field can never cancel out simultaneously without contradicting quantum mechanics. If the field amplitude is much larger than the fluctuations, one can decompose the quadrature operator into a steady-state operator Xˆ 1,2 plus a time-dependent fluctuation operator δ Xˆ 1,2 (t) such that [6] < sl|Xˆ 1,2 |sl >= X1,2

< sl|δ Xˆ 1,2 (t)|sl >= 0

(7)

where X1,2 (t) is the average value of the quadrature amplitude and |sl > the state of light. The quadrature fluctuations associated to δ Xˆ 1,2 are described by the doublesided power spectral density: SδXa δXb (Ω) =

1 < sl|δ Xˆ a (Ω)δ Xˆ b† (Ω) + δ Xˆ b (Ω)δ Xˆ a† (Ω)|sl > π

(8)

States of Light There are many ways to represent a quantum state of light. The most popular is to specify the number nω of photons for each mode at frequency ω of the electromagnetic field. These are called Fock states. Due to the uncertainty principle, if the number of photons is exactly known, the phase of the electric field is completely undetermined. Moreover, for the Fock states the expectation value of the electric field vanishes: < n1 , n2 , ..|E(x, y, z, t)|n1 , n2 , .. >≡< n|E(x, ¯ y, z, t)|n¯ >= 0

(9)

and this holds even in the classical limit n¯ → ∞. For these reasons Fock states are inappropriate to describe a laser field, which is actually used in experiments. Other possible representations of the quantum states of light, which are relevant in the following, are the coherent states and the squeezed states. Coherent states The coherent states |α >, where α = |α| exp[iφ] is a complex number [7–9], are the most appropriate choice to describe the classical field of a laser beam. Indeed for such states the expectation value of the electric field  < α|E(x, y, z, t)|α >= u(x, y, z)

hω ¯ 0 [|α| cos φ cos(ωt) + |α| sin φ sin(ωt)] cε0 (10)

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takes the same form of a classical field with amplitude |α|  |α| =

2Plas hω ¯

(11)

phase φ and power Plas . For coherent states the quantum fluctuation is equally distributed on the two quadrature X1,2 : ΔX12 =< α|δ Xˆ 12 |α >=< α|δ Xˆ 22 |α >= ΔX22 = 1

(12)

which shows that a coherent state satisfies the lower limit of the uncertainty condition (6). For this reason they belong to the class of the minimum uncertainty states. Due to the uncertainty principle, a quantum state cannot be represented in the phase space (X1 , X2 ) with a single point as for a classical state but rather with a probability distribution as shown in Fig. 1 (a more rigorous way to obtain Fig. 1 is to make the projection on the (X1 , X2 ) plane of the Wigner quasi-probabilistic distribution function [10, 11].). Coherent states can be formally generated by applying the unitary displacement operator D(αω ) [3] ˆ ω ) = exp(α aˆ ω† − α ∗ aˆ ω ) D(α

(13)

to the Fock vacuum state |0 >, i.e., the state with zero photons. 'Y1

' X1

X2

X2 ' Y2 T

' X2 _D_ I

_D_

a X1

I

b

X1

Fig. 1 (a): Pictorial view of the coherent state in the phase space X1 , X2 . The uncertainty area is a circle with equal standard deviation ΔX1,2 . (b): Squeezed state representation in the phase space. In this case the quadrature variances are unbalanced. The squeezing ellipse axis ΔY1,2 is squeezed by a factor exp (∓ξ ) with respect to the coherent case (ΔX1,2 ). The color is more intense where the √ Wigner quasi-probability function is higher, while the dashed line corresponds to the points at 2 of the of the maximum probability

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ˆ ω )|0 > |αω >= D(α

(14)

ˆ ω ), the expectation values (7) Given the unitarity of the quantum operator D(α and (8) can be calculated using the the Fock vacuum state |0 > and replacing the operators Xˆ 1,2 with ˆ ω) Xˆ 1,2 → Dˆ † (αω )Xˆ 1,2 D(α

(15)

This operation leads to the double-sided power spectral densities: SδX1 δX1 (Ω) = SδX2 δX2 (Ω) = 1

SδX1 δX2 (Ω) = 0

(16)

Squeezed states Squeezed states are minimum uncertainty states with unbalanced quadrature variance. They can be generated from the vacuum sate |0 > as ˆ ˆ θ )|0 > |α, ξ, φ >= D(α) S(ξ,

(17)

ˆ θ ) is the squeezing unitary operator [3]: where S(ξ,   1 −2iθ 2 1 2iθ †2 ˆ S(ξ, θ ) = exp ξ e aˆ − ξ e aˆ 2 2

(18)

The uncertainty area of a squeezed state appears as an ellipse rotated by an angle θ , with a major axis variance enhanced by factor exp(ξ ) and minor axis variance reduced by factor exp(−ξ ) with respect to the corresponding coherent state (see Fig. 1b). As for the case of the coherent states, the expectation values (7) and (8) can be estimated using the Fock vacuum state after replacing the classical quadratures Xˆ 1,2 with the squeezed quadratures: Xˆ 1,2 → Sˆ † Xˆ 1,2 Sˆ

(19)

which using the Baker-Hausdorff theorem leads for vacuum squeezing (|α| = 0) to the expressions ΔX12 = exp(2ξ )Sin2 (θ ) + exp(−2ξ )Cos2 (θ )

(20)

ΔX22 = exp(2ξ )Cos2 (θ ) + exp(−2ξ )Sin2 (θ ) In the particular case θ = 0 the formula (20) reads ΔX1 = exp(−ξ )

ΔX2 = exp(+ξ )

(21)

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where we see that the variance of the quadrature X1 is “squeezed” by factor exp(−ξ ) with respect the coherent state, while the quadrature X2 is “antisqueezed” by a factor exp(ξ ). The opposite occurs for a squeezing angle θ = π/2. In the general case, with an appropriate rotation of the quadrature axis, it is always possible to define a new set of quadratures Y1,2 for which the conditions (21) hold. In the literature it is customary to define the squeezing factor of a quadrature in decibels as dBsqz = 20 log10

ΔXsqz ΔXcoh

(22)

where ΔXsqz and ΔXcoh are the standard deviations of the squeezed and the coherent state quadratures, respectively.

Quadrature Noise Estimation Detecting quantum states of light requires to measure the variance of a given quadrature of the field. The most common method to estimate the expectation value of the electric field quadratures is the balanced homodyne detection initially ˆ proposed in Ref. [17] and depicted in Fig. 2. The vacuum electric field E(t) (cf. LO ˆ Eq. (2)) and a bright local oscillator (LO) field E (t):     

hω ¯ 0 LO ˆ |α| cos ζ + δ Xˆ 1LO cos(ω0 t) + |α| sin ζ + δ Xˆ 2LO sin(ω0 t) = u(x, y, z) E cε0 (23)

 with power PLO , classical amplitude |α| = 2P LO /hω ¯ 0 , and selectable phase ζ , combine at a beam splitter with 50% splitting ratio, generating the two fields Eˆ P D1,2 (t) at the photodiodes: ˆ Eˆ LO (t) ± E(t) Eˆ P D1,2 (t) = √ 2

(24)

LO and assuming |α| much higher than the E(t) ˆ To leading order in Xˆ 1,2 and δ Xˆ 1,2 ˆ amplitude, the power PP D1,2 impinging the detection photodiodes is

1 PˆP D1,2 = {|α|2 + 2|α|[(δ Xˆ 1LO ± Xˆ 1 ) cos ζ + (δ Xˆ 2LO ± Xˆ 2 ) sin ζ ]} 4

(25)

Therefore the induced differential photocurrent IP D2 − IP D2 ∝ |α|[cos ζ · X1 + sin ζ · X2 ]

(26)

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IPD1

Out

G1

PD1 G2

EPD1

IPD2

EPD2

ELO

PD2

E Fig. 2 Scheme of a balanced homodyne detector: the vacuum field E vac and the LO oscillator field E LO combine in the beam splitter producing the two fields EP D1,2 which generate the two photocurrents IP D1,2 . The difference IP D1 − IP D2 is proportional to a linear combination of the vacuum field quadratures. The linear coefficients depend on the selectable relative phase φ between the two input fields

can be used as an estimator of the quadratures X1,2 amplitudes. In particular X1 is obtained for the LO phase set to ζ = 0 while X2 for ζ = π/2. It is worth to note that the fluctuations δX1,2 of the LO do not appear in Equation (26). This remarkable result is valid only if the beam splitting ratio is exactly 50:50 so that Equation (24) holds. In some experimental setup, the beam splitter imbalance is compensated by properly changing the amplifier gain G1, 2 (see Fig. 2). However, as discussed, for example, in [19], this procedure leads to other problems. In the real world the properties of measured squeezing are affected by practical issues such as noises and optical losses. Effect of losses Any source of optical loss on the path of the squeezed light has the effect of mixing the squeezed vacuum with classical vacuum; this can be seen by modeling the optical loss as a √ beam splitter [6]; a pictorial view of such model is shown in Fig. 3 left. A fraction 1 − η of the incoming squeezed light field Esqz √ is reflected by the beam √ splitter; at the same time, the η transmitted fraction combines with a fraction 1 − η of the uncorrelated coherent vacuum field Ecoh entering the beam splitter from the unused port. Therefore the output quadrature T ot results δ Xˆ 1,2 √ sqz T ot δ Xˆ 1,2 = ηδ Xˆ 1,2 +



coh 1 − ηXˆ 1,2

(27)

The actual quadrature variance in presence of losses can thus be calculated with the previous expression together with Equations (15), (16), and (19). It is interesting to consider the case of phase squeezing (θ = 0) for which the quadrature variances become

369

X2

9 Squeezing and QM Techniques in GW Interferometers

Esqz

X2

Ecoh

n

Esqz + +

1−

Ecoh

X1 X1

Fig. 3 Left: A generic power loss 1 − η sensed by the squeezed beam can be modelled with a √ η √: (1 − η) beam splitter. The fraction η of the incoming squeezed field sum up with the fraction 1 − η of the coherent vacuum field entering from the loss port. This results in a degradation of the squeezing level at the beam splitter output. Right: The jittering θn of the squeezing ellipse produces an effective level of the X1 quadrature variance (ΔX1 ) higher than the actual width of the original squeezing ellipse. For a given θn the squeezing degradation, i.e., the broadening of X1 variance, increases with the anti-squeezing level, i.e., ΔX2

2 ΔX1,2 = 1 + η[exp(∓2ξ ) − 1].

(28)

From Equation (28) one can see that, in the presence of losses, the resulting field is not in a minimum uncertainty state, i.e., ΔX1 ΔX2 > 1. Moreover, as shown in the Fig. 3 left the squeezed quadrature is more affected by losses than the antisqueezed quadrature.

Squeezing angular jitter According to Equation (20) the quadrature variance 2 depends on the squeezing angle θ . In real experiments, as a consequence, ΔX1,2 for instance, of the environmental acoustic, seismic, and temperature noise, θ is not constant in time but fluctuates around an average value θ0 . Therefore the actual 2 is the mean value of Eq. (20) averaged over the rapid fluctuations of value of ΔX1,2 θn (t). Expanding around the angle of phase squeezing θ0 = 0 for small fluctuations θn (t)  1, and combining Equations (21) and (28), one gets ΔX12 ≈ [1 + η(e2ξ − 1)]θn2 + (1 − θn2 )[1 + η(e−2ξ − 1)]

(29)

ΔX22 ≈ [1 + η(e2ξ − 1)](1 − θn2 ) + θn2 [1 + η(e−2ξ − 1)] where θn2 is the mean value of θn (t)2 . Equation (29) shows that the jitter of squeezing angle has larger impact on the quadrature with lower variance, i.e., ΔX1 ; the higher is the anti-squeezing level e2ξ , the higher is the squeezing degradation; see Fig. 3 right. The combined effect of optical losses and jitter of the squeezing ellipse angle is illustrated in Fig. 4 where the measurable squeezing level is plotted as a function of the overall optical losses and of the rms jitter of the squeezing ellipse angle, for a given value of the level of generated squeezing.

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Fig. 4 Effective squeezing level for different values of the optical losses and the rms jitter of the squeezing ellipse angle, assuming 10 dB of generated squeezing

A plot of the measured quadratures variance as a function of the level of generated squeezing is shown in Fig. 5, assuming an rms angular jitter θn = 0.2 rad and an optical efficiency η = 0.8. Equation (29) is commonly used to estimate η and θn by interpolating the experimental data ΔX1,2 as a function of the input squeezing degree ξ . Nonideal detector In general the accuracy of the squeezing level estimation with a balanced homodyne detector is limited by technical noise; therefore, a careful design of the experimental setup is required. • Unbalanced arms. An imperfect balance of the two homodyne detector arms LO onto the output produces a coupling of the local oscillator technical noise δX1,2 signal, thus affecting the squeezing degree estimation. The unbalance can be either opto-mechanical (nonideal splitting ratio, angular jitter of laser beams on the photodiodes sensitive area affecting the detection efficiency, light scattering in the arms) or electronical (photodiodes with different quantum efficiency, unbalanced electronics for the two channels) and in general is more effective at low frequency [18–21]. The parameter used to quantify this effect is the common mode rejection CMMR which gives the fraction of the local oscillator power present in the differential signal output. For the best detectors so far developed, the CMMR is about −80 dB.

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Fig. 5 Measured squeezing (red curve) and anti-squeezing (blue curve) expressed in dB versus the original degree of squeezing ξ . The actual squeezing level is strongly affected by the optical losses 1 − η (η = 0.8 in this plot), and for high squeezing level it is limited by the squeezing ellipse angular jitter (θn = 0.2 rad in this plot)

• Electronic noise. For large squeezing values the electronics noise can compete with the quantum noise signal and therefore must be kept as low as possible. In the audio frequency band, the best so far developed photodetectors show up to 30 dB of electronic noise clearance with respect the quantum shot noise of the light in the mW region [22]. • Detection efficiency. Any limitation in the detection efficiency is equivalent to an optical loss on the squeezed light field. The main loss sources in homodyne detectors derive from the antireflective coatings of the optical elements and from the non-unitary quantum efficiency of the photodetectors. At the wavelength of 1.06 μm, quantum efficiencies greater than 99.5% have been reported [23], while for the 2.2 μm wavelength, which is of interest for the next detectors generation [24], so far 75 ÷ 80% [25] were not exceeded. Although up to now never used in the interferometric GW detectors, the homodyne readout scheme [26] is considered among the main upgrades for the incoming scientific runs [27]. Presently the GW detectors adopt the DC-readout scheme [28] which is a sort of zero-angle homodyne detector in which the carrier suppression is done in the beam splitter.

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Generation of Nonclassical States of Light The statistics of classical (coherent) states of light is determined by the fact that photons are uncorrelated both in phase and in amplitude. A peculiar feature of quantum (squeezed) states of light is the presence of some correlations between photons, which are absent in classical states. For instance, phase correlations would produce amplitude squeezing and phase anti-squeezing, while amplitude correlations would generate phase squeezing and amplitude anti-squeezing. Experimental efforts for the generation of squeezed states of light aim to create correlations between the phase or the amplitude of noise sidebands on an optical carrier and to protect them from the mixing with uncorrelated, classical fields. Several methods have been pursued to such goal using different physical systems, resulting in various kinds and degrees of light squeezing. In the following we give a brief overview of the main results demonstrated so far, and we then concentrate on those more significant for the application to GW detectors, i.e., the CW generation of single mode squeezed vacuum at suitable wavelength with high degree of squeezing and high reliability. The first experimental demonstration of squeezed light was obtained in 1985 with four-wave mixing in a vapor of sodium atoms [29]. Soon after, in 1986 squeezed light was produced with three different methods: by four-wave mixing within an optical fiber at cryogenic temperatures [30]; by parametric down-conversion in a magnesium doped LiNb crystal inside an optical resonator pumped at the second harmonic of degenerate signal and idler fields [32]; and by suppressing the photocurrents driving a semiconductor laser [33]. In those pioneering experiments the level of produced squeezing was lower than 1 dB, with the exception of the experiment on parametric down-conversion which achieved up to 3.5 dB of squeezing. In the last three decades the research on squeezed light did substantial progress, yielding to much higher squeezing degrees; recent experiments with atomic ensembles demonstrated squeezing degrees up to 9 dB; 7 dB of squeezing were obtained with optical fibers, and 13 dB using ferroelectric crystals; the highest squeezing levels, up to 15 dB, were obtained with degenerate parametric down-conversion in an optical resonator. Those results were made possible through methods and technologies to reduce the mixing with classical states and to mitigate noise sources, and included the choice and development optical components with lower losses, detectors with higher efficiency, low-noise electronics. Interaction with dilute atomic ensembles The key concept in the generation of squeezed light is to exploit nonlinear effects in light-matter interaction to produce correlations among photons. Early experiments employed the highly nonlinear interaction of a laser beam with an atomic ensemble. For instance, with a Λshaped configuration of atomic energy levels, a process called four-wave mixing is possible, where two ground states are coupled to a single excited state with two optical pump fields. The nonlinear coupling can be enhanced by embedding

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the atomic system in a doubly resonant optical cavity and allows the emission of two correlated photons in the Stokes and Anti-Stokes optical modes, forming a two-mode squeezed state; squeezing is detected with a LO at optical frequency between the two resonant modes, which in principle can be either degenerate or nondegenerate. A main limitation to the squeezing generated via four-wave mixing is the presence of concurring nonlinear processes such as fluorescence or Raman scattering, yielding the emission of uncoherent photons in the squeezed modes. Such processes can cancel out in the intensity-difference squeezing of a double-Λ system [34], and squeezing levels up to 9 dB were demonstrated. Nonlinear interactions in optical fibers Similarly to second-order nonlinear processes, third-order nonlinearities may generate squeezed light. Though much weaker than second-order processes, they can however be implemented over long interaction lengths in amorphous materials, i.e., inside optical fibers. Squeezing in optical fibers relies on four-wave mixing and on the nonlinear optical Kerr effect. The key aspect is the dependence of the fiber refractive index on the intensity of the light inside it. The propagating optical field experiences a larger phase shift where the intensity is more concentrated. This effect generates correlations between photons, turning the coherent states from an input laser field into squeezed states. Some drawbacks arise from the necessarily high optical intensity inside the medium, possibly driving additional nonlinear processes in the fiber, e.g., from photonphonon coupling. Noise peaks extending from the audio to the RF band may be generated by the combination of Brillouin scattering and the acoustic waveguide effect inside the fiber [35]; such excess noise can be mitigated by cooling the fiber at cryogenic temperatures. Other detrimental effects of the high optical intensities are the stimulated Brillouin scattering, which may retro-reflect a large portion of the input laser field, and the lack of feasibility of homodyne detection. The former effect can be overcome by broadening the pump field spectrum with phase modulation techniques. The measurement of squeezed light quadratures can be achieved instead by means of a phase shifting optical cavity [36]: reflection from a cavity resonance shifts the phase of the bright carrier from that of RF sidebands. With a detuning from cavity resonance, one can project the squeezing ellipse in phase space. A different approach to get rid of many technical noises is the use of a pulsed pump laser [37], which allows high peak intensities in combination with moderate average optical power, thus reducing parasitic processes such as the stimulated Brillouin backscattering. Hybrid methods Another interesting method is to combine the lossless confinement of light over a long path, given by optical fibers, with the high nonlinearity of atomic vapors; this can be achieved with hollow core PCF filled with gaseous or liquid materials. With respect to the use of ordinary fibers, such method offers the advantage of efficient nonlinear interaction with suppression of Brioullin and Raman scattering. Some groups have already demonstrated squeezing in fibers filled with high pressure gas [38] and atomic vapor [39].

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Current control in semiconductor lasers Sub-shot noise emission from semiconductor lasers can be achieved by active control of the injection current down to the space charge-limited regime, similarly to the case of vacuum tubes [40]. The non-Poissonian distribution of electrons in such regime is readily converted into sub-shot noise photons through light-emitting laser diodes. A first demonstration of such method resulted in a modest squeezing level of 0.3 dB [33], which was later improved by placing the detector right in front of the laser diode [41]. By collimating the diode laser emission, 3 ÷ 4 dB of squeezing were achieved at most [42]. The main limitation was understood to arise from the strong correlation between different spatial modes [43]. Parametric down-conversion and optical parametric oscillator The most effective method for squeezed light generation is parametric down-conversion (PDC), which was experimentally realized in 1986 for the first time [32]. A pump photon with frequency ωp impinges on a dielectric with a χ (2) second-order nonlinear susceptibility, generating two new photons: a signal photon of frequency ωs and an idler photon at frequency ωi with ωp = ωi + ωs for energy conservation. In degenerate PDC, the idler and signal photons have the same frequency and polarization. In most cases the χ (2) nonlinearity is weak; in order to produce efficient squeezing, a significant polarization of the medium is required, and several conditions should be fulfilled to this purpose. High values of the optical pump energy density on the crystal are possible by either employing high-power pulsed lasers or by embedding the crystal into an optical resonator. Efficient nonlinear processes require momentum conservation of the involved fields, that is, kp = ks + ki , where kp is the wave vector for the incident pump beam while ks and ki are the wave vectors of signal and idler beams, respectively. This is also known as phase matching condition and can be achieved by temperature and wavelength tuning and/or periodically poling the nonlinear crystal. With type I phase matching, where signal and idler are degenerate both in frequency and polarization, the timedependent Hamiltonian of the PDC process can be written as [31] † † (2) ˆ Hˆ = hω (aˆ aˆ bˆ † − aˆ † aˆ † b) ¯ s aˆ aˆ + hω ¯ p bˆ bˆ + i hχ ¯

(30)

where bˆ is the pump field and aˆ is the signal field. Assuming the pump to be a strong classical field, bˆ and bˆ † can be approximated by the classical operators βe−iωp t and β ∗ eiωp t ; thus, in the interaction picture, the Hamiltonian of the parametric process can be approximated as (2) ∗ Hˆ (t) = i hχ [β aˆ ae ˆ i(ωp −2ωs )t − β aˆ † aˆ † e−i(ωp −2ωs )t ]. ¯

(31)

Considering that ωp = 2ωs , this simplifies into (2) ∗ Hˆ (t) = i hχ (β aˆ aˆ − β aˆ † aˆ † ). ¯

(32)

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The unitary evolution of the input signal under this Hamiltonian is i

ˆ

U = e− h¯ H t = eχ

(2) (β ∗ aˆ a−β ˆ aˆ † aˆ † )

(33)

and has the same form of the squeezing operator, compare with eq. (18) for θ = 0. An optical parametric oscillator (OPO, see Fig. 6), where the PDC process takes place inside a cavity, has been so far the most efficient source of squeezed light. In many of the possible configurations of the optical resonator, signal and idler fields are simultaneously resonant, thus enhancing the nonlinear process. On the other hand, in the presence of a cavity, a critical point appears in the degenerate system: the optical parametric oscillation occurs only above a threshold value of the pump optical power. In an ideal, lossless system, the produced squeezing at threshold would be infinite. With a bright beam seeding, the signal, and idler modes, the critical point disappears, and the OPO becomes a squeezing, phase-sensitive amplifier. The first demonstration of PDC [32] was done with a subthreshold OPO with a MgO:LiNbO3 crystal in a linear optical cavity. The system was pumped to threshold by a frequency doubled laser beam at 532 nm, and squeezed vacuum was produced in the down-converted field at 1064 nm. A different configuration for the OPO, with a bowtie-shaped ring cavity and a KNbO3 crystal, was successfully demonstrated to produce 3.8 dB of frequency tunable squeezed light [44]. A few years later, 6 ÷ 7 dB squeezing was obtained with a monolithic OPO cavity made of a MgO:LiNbO3 crystal with spherical, highly reflective end-facets [45, 46]. More stable squeezing up to the 6 dB level was operated for several hours with a semi-monolithic OPO cavity, seeded with a displacement beam [47]. For some time the maximum degree of produced squeezing was limited to about 6 dB due to intracavity losses, detector losses, and phase noise. By carefully addressing all of these degradation mechanisms, several groups were able to improve the level of produced squeezing. Since 2006 the limit of 6 dB was surpassed first with a periodically poled PPKTP crystal in a ring cavity, reaching 7.2 dB in 2006 [48] and 9 dB in 2007 [49]; shortly later, up to 10 dB squeezing was

Fig. 6 Illustration of the principle of an optical parametric oscillator; left: a nonlinear medium is embedded in an optical cavity resonant on pump, signal, and idler fields; right: two-level system with relevant optical transitions

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generated with a LiNbO3 crystal in a linear cavity [50]. In recent years, further improvements of losses, phase noise, and stay light mitigation allowed squeezing levels above 10 dB with both monolithic [51], semi-monolithic [52], and bowtie OPA cavities [53, 54]. In addition, squeezed light has been finally observed in the audio band which is important for applications in GW detection, after careful mitigation of technical noises at very low frequencies. Up to 15 dB of squeezing was recently achieved [55] by optimizing intracavity losses, detection losses, and phase noise. The former were reduced with the use of PPKTP crystals, which exhibit low absorption at the signal (and idler) wavelength, as well as very small values of pump-induced absorption. Intracavity losses were further reduced with the help of low-loss coatings; these advances allowed to obtain extremely high values, up to more than 97%, for the escape efficiency, that is the ratio of coupling rate and intracavity loss rate. Detection losses are given by the combination of limited quantum efficiency of the photodetectors and the contrast mismatch between LO and squeezed fields. InGaAs photodiodes with ∼99% efficiency at infrared wavelengths became available only recently. Mitigation of phase noise was achieved by suppressing high-order modes from the pump beam by means of an optical resonator and by means of coherent control methods as described below. Pondero-motive squeezing In systems where radiation pressure from a laser field interacts with a soft mechanical oscillator such as a mesoscopic micro-cantilever mirror of a macroscopic suspended mirror, the strong opto-mechanical couplings produce correlations between the phase and the amplitude of the light field. Such correlations may generate squeezed states of light. The interest in pondero-motive squeezing is due to several reasons: application to MOEMS (Micro Opto Electro Mechanical Systems) might allow cheaper and more compact solutions with respect to an OPO; the study of couplings between macroscopic opto-mechanic objects and their quantum behavior has both theoretical and practical interest; the dynamical response of the mechanical system may allow in principle a frequency-dependent squeezing with ellipse rotation in interesting spectral regions for the application to GW detectors. The latter point is briefly discussed in the last section of the chapter.

Quantum Noise in the Interferometric Gravitational Wave Detectors This section deals with the quantum noise in the interferometric GW detectors. In the first paragraph we derive an expression for the standard quantum noise in a Michelson interferometer, and we later extend it to the general configuration of advanced interferometers. The second part describes how squeezed states of light can be used to manipulate and overcome the standard quantum limit. Finally the chapter ends with the state-of-the-art review of the squeezing-enhanced interferometers and the expected improvements foreseen in the short term.

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Quantum Noise in a Michelson Interferometer The optical design of the current interferometric GW detectors is based on the Michelson configuration shown in Fig. 7. Here, for the sake of simplicity, we assumed that the two test masses have the same value (MN A = MEA ≡ M), while the unperturbed North arm length LN A and East arm length LEA ≡ LN A + ΔL are deliberately kept slightly different (ΔL  LN A,EA ) for the implementation of the DC-readout scheme. An incident gravitational wave would produce a thin modification xs  L of the two arm lengths in counterphase to each other. The electric field operator 

hω ¯ 0 ˆ ˆ E(x, y, z, t) = u(x, y, z) E(z, t) ε0 c

(34)

in and the input vacuum field E ˆ in can be written in for the input laser field Eˆ Las V ac terms of Amplitude (‘A’) and Phase (‘P ’) quadratures as [5]

 in A P Eˆ Las (z, t) = [ 2PLas /hω ¯ 0 + δ Xˆ Las (z, t)] cos(ω0 t) + δ Xˆ Las (z, t) sin(ω0 t)] (35) and Eˆ Vinac (z, t) = δ Xˆ VAac (z, t) cos(ω0 t) + δ Xˆ VP ac (z, t) sin(ω0 t)

(36)

North

Input Laser

Input Laser

M

M

East Detection photodiode

PRM

SRM Detection photodiode

Fig. 7 Left: Michelson interferometer configuration. The beam splitter input/output fields are marked with red arrows. The two arms have unperturbed length LN A and LEA respectively. A gravitational wave would produce a length modification xs of opposite sign in the two arms. Without loss of generality, the two arms are assumed to be oriented along the north and east directions. Right: Dual Recycled configuration

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where the amplitude quadrature is defined as the one in phase with the input laser beam whose power and angular frequency are PLas and ω0 , respectively. The amplitude and the phase fluctuations are supposed to be dominated by quantum A fluctuations and thus are represented by the quadrature operators δ Xˆ Las,V ac (t) and P ˆ δ XLas,V ac (t) with power spectral densities expressed in Eq. (16): The two input fields (35,36) combine in the 50:50 beam splitter generating the two fields at the arms input: in EN A,EA (t) =

in (t) ∓ E in (t) ELas √ V ac 2

(37)

which after reflection on the test masses return to the beam splitter input in the form out in EN A,EA (t) = EN A,EA (t − 2τN A,EA ∓ 2xs /c)

(38)

where τN A,EA ≡ LN A,EA /c is the time required for light to cover the unperturbed ASP (t) at the arm length. After recombination these two beams produce the field Eout output of the beam splitter (antisymmetric port (ASP) is the output port of the interferometer after the beams have recombined in the beam splitter). To leading order in xs /c and ΔL/c and δ Xˆ ASP (t) = Eˆ out

 =

out (t) − E out (t) EN A = √ EA 2

 2PLas ω0 P ˆ [2xs (t − τ ) + ΔL] + δ XV ac (t − 2τ ) sin(ω0 t) hω ¯ 0 c

(39)

− δ Xˆ VAac (t − 2τ ) cos(ω0 t) where without loss of generality we made the assumption ω0 LN A /c = nπ (for the case ω0 LN A /c = nπ , we would obtain the same expression of Equation (39) by substituting in the argument of both the trigonometric functions ω0 t + 2ω0 LN A /c in place of ω0 t which is not relevant for the discussion that follows). It is worth A and noticing that in the expression (39) the dependence from the amplitude δ Xˆ Las P phase δ Xˆ Las fluctuations of the input laser field has completely disappeared. This is a direct consequence of having chosen the unperturbed length almost equal for the two arms (ΔL/LN A  1). The detection photodiode measures the expectation value of the power operator  ASP (x, y, z, t)|2 of the ASP beam (39). At leading order in Pˆ (t) = ε0 c dxdy|Eˆ out δ Xˆ and xs and averaging over time the rapid oscillations at frequency ω0 and 2ω0 , we obtain    PLas ω02 ΔL 2hω ¯ 0 c ˆP ˆ ΔL + 4xs (t − τ ) + δ X (t − 2τ ) (40) P (t) = PLas ω0 V ac c2

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From this expression it comes evident that the phase fluctuations δ Xˆ VP ac of the vacuum field generate a sensing noise called quantum shot noise which compete with the displacement signal xs contribution. In addition to the signal displacement xs , the mirrors are subject to a displacement generated by the radiation pressure force FNRP A,EA which acts on the mirrors. To leading order on the quantum fluctuation operators: FˆNRP A,EA (t) = 2ε0



dxdy|Eˆ N A,EA (t − τ, x, y, z)|2 =

(41)

  

Plas 2hω ¯ 0 ˆA δ XLas (t − τ )) ∓ δ Xˆ VAac (t − τ )) ≈ 1+ c Plas For mirrors with equal mass M, the displacement induced by the first two terms in Equation (41) produces an undetectable common motion of mirrors, while the last term generates a differential displacement noise which adds to the signal contribution. The detected power (40) in the Fourier space then becomes

 ω0 ΔL 2 iΩτ e xs (Ω)+ Pˆ (Ω) = 4PLas c  −e

2iΩτ

(42)

  δ Xˆ VP ac (Ω) h¯ A ˆ KMi δ XV ac (Ω) + √ 2MΩ 2 KMi

where we have omitted the unrelevant static term and KMi =

4ω0 PLas c2 MΩ 2

(43)

Equation (42) shows that for a perfectly balanced interferometer (same arm A,P do not length and same mirror mass), the input laser quantum fluctuations δ Xˆ Las contribute to quantum noise which instead is entirely generated from the vacuum fluctuations δ Xˆ VP ac (shot noise) and δ Xˆ VAac (quantum radiation pressure noise) entering from the interferometer dark port [56]. Using Equations (16) (42) and the relationship xs (Ω) = Lh(Ω)/2 between the gravitational wave amplitude h and the induced displacement xs , the quantum noise equivalent strain power spectral density becomes Shn hn (Ω) =

  2h¯ 1 K + Mi KMi ML2 Ω 2

SQL

(44)

which takes the minimum value Shn hn (Ω) called standard quantum limit (SQL):

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SQL

Shn hn (Ω) =

4h¯ ≡ h2SQL−Mi ML2 Ω 2

(45)

when KMi = 1.

Quantum Noise in Power Recycled Interferometers Ground based interferometers are designed to maximize the sensitivity in the audio band, i.e., in the spectral region of a few hundreds of Hz. For these frequencies the theoretically optimal sensitivity of a Michelson interferometer (45) would be approached for input laser powers in the inaccessible range of hundreds of kW (cf. Eq. (43)). In order to overcome this limitation, the optical layout of the advanced detectors incorporate Fabry-Pérot resonators as shown in Fig. 7 Right. First of all an additional input mirror is inserted on each arm, to realize an optical cavity with the corresponding end mirror, resonant with the interferometer carrier field. Consequently within the cavity bandwidth, the effective power sensed by the test masses is enhanced by a factor 4/TI M where TI M is the input mirror power transmittivity. A further optical resonator is realized by inserting a power recycling mirror (PRM) with transmissivity TP R between the laser source and the beam splitter [57]. Indeed, since the interferometer works in the dark fringe condition, all the input power is reflected toward the input port, and the PRM can be used to coherently recycle it into the detector, thus producing a power enhancement by a factor 4/TP R at the BS input. Considering that the symmetric port back-reflected light does not contain any gravitational wave-induced signal or any quantum noise generator δ Xˆ V1,2ac , the only difference expected with respect to the simple Michelson case is that Plas should be replaced by the enhanced value taking into account the cavity transfer functions. Indeed it can be shown that for a power-recycled interferometer, the quantum noise power spectral density takes the form [5]     h2SQL 4h¯ 1 1 KP R + ≡ KP R + Shn hn (Ω) = KP R 2 KP R ML2 Ω 2

(46)

where KP R =

8ω0 2PBS 2γarm 2 + γarm ) MLc TI M

Ω 2 (Ω 2

(47)

γarm the half-width-half-maximum bandwidth of the arm cavity and PBS = 4PLas /TP R the enhanced power at the BS input. The multiplicative factor “2” that appears in the equations (46) with respect equations (44) derives from the presence of two masses in each arm, and therefore the mass of the mirror M must be replaced by the reduced mass M/2.

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Quantum Noise in Dual Recycled Interferometers Adding a Signal Recycling Mirror (SRM) at the interferometer AS port (see Fig. 7 Right) produces a frequency-dependent enhancement of the signal sidebands [58, 59]. Depending on the resonance frequency of the resulting signal recycling cavity (SRC), we can have two different operating regimes: • Tuned configuration in this case the SRC is kept resonant with the interferometer carrier frequency ω0 . This leads to a broadening of the arms cavity bandwidth, preventing destructive interference of the sidebands in the arm [60–62]. • Detuned configuration the carrier frequency is no longer resonant in the SRC, and at least one of the two signal sidebands is resonant in the arm cavity. In this condition during the free propagation the quadratures rotate of an angle related to the cavity detuning Δ = ω0 LSR /c|mod2π [63, 64] where LSR is the signal recycling cavity length. Therefore the shot and the back-action noise generators becomes a combination of δ Xˆ VAac (t) and δ Xˆ VP ac (t) giving rise to a nonvanishing cross-correlation spectral density that can be used to overcome the SQL in a narrow frequency range. The configuration that simultaneously incorporates the power and the signal recycling mirrors is called dual recycling (DR) configuration The first experimental demonstration of a dual recycled interferometer was made in the Garching prototype [65], and it is routinely used in the long-arm interferometers GEO 600 and LIGO [66, 67] while is foreseen in the next Virgo upgrades [68]. The quantum noise derivation for a DR interferometer is rather complex and beyond the scope of this manuscript. The interested reader can refer to the original articles [63, 64]. Here we just report the expected quantum noise power spectral density for a lossless dual recycled interferometer in the most general case in which an homodyne readout scheme with angle ζ is used [69]: Shn hn (Ω) =

(cos ζ, sin ζ )TTT (cos ζ, sin ζ )T (cos ζ, sin ζ )ssT (cos ζ, sin ζ )T

(48)

where the elements of the 2 × 2 matrix T and the vector s are √

√ 2KP R 2KP R 2iΘ t (1+r e ) sin(Δ) s =− tsr (−1 + rsr e2iΘ ) cos(Δ) sr sr 2 R R hPSQL hPSQL (49) 

 1 2 ) cos(2Δ)+ · KP R sin(2Δ) −2rsr cos(2Θ) (50) T11 =T22 =e2iΘ (1+rsr 2 s1 =−

2 2 T12 =−rsr ·[sin(2Δ)+KP R sin2 (Δ)]e2iΘ T21 =rsr ·[sin(2Δ)−KP R cos2 (Δ)]e2iΘ (51)

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where Θ=

  ΩLsr Ω + arctan c γarms

(52)

and rsr tsr , respectively, are the amplitude reflectivity and transmissivity of the SR recycling mirror.

Quantum Noise in Presence of Squeezed Light As shown in the previous section, the quantum shot and back-action noise are ,A . generated by a linear combination of the vacuum field quadrature operators δ Xˆ VP ac By injecting into the dark port of the interferometer a vacuum-squeezed field in place of the coherent vacuum field as shown in Fig. 8 (Left), it is possible to manipulate the quantum noise. Indeed in presence of squeezing, the operators δ Xˆ VP ac and δ Xˆ VAac are formally replaced with the new operators (cf. Eq. (19)): δ Xˆ VP ac → Sˆ † δ Xˆ VP ac Sˆ = δ Xˆ VP ac [cosh ξ − sinh ξ · cos 2θ ] − δ Xˆ VAac sinh ξ · sin 2θ (53) δ Xˆ VAac → Sˆ † δ Xˆ VAac Sˆ = δ Xˆ VAac [cosh ξ + sinh ξ · cos 2θ ] − δ Xˆ VP ac sinh ξ · sin 2θ leading to a modification of the back-action and shot noise value which also become partially correlated for θ = 0, π/2. Putting together equations (42) and (53), the quantum noise power spectral density for a Michelson interferometer in the presence of squeezing takes the form Shn hn (Ω) =

  2h¯ 1 K [cosh 2ξ + cos 2(Φ + θ ) sinh 2ξ ] + Mi KMi ML2 Ω 2

(54)

where Φ(Ω) = arccot[KMi (Ω)]

(55)

A similar expression is obtained for a tuned dual recycled configuration providing that in Equation (54) the mass M is replaced by the reduced mass M/2 and KMi by KT DR [70]: KT DR =

8ω0 2PBS 2γI T F Ω 2 (Ω 2 + γI2T F ) MLc TI M

where the detector bandwidth γI T F is

(56)

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a

Laser Input

383

b

PRM

Laser Input

PRM

SRM

SRM FDS PBS1

M2 OPO

OPO

FIS M1

FIS

FC1

FC2 PBS2

Fig. 8 Dual recycled interferometer with squeezed vacuum injection. (a) Frequency-independent squeezing (FIS). (b) Frequency-dependent squeezing (FDS). For the tuned SRC only one filter cavity (FC) is required to attain the optimal angular dependence for the squeezing ellipse angle. In this case the polarizing beam splitter P BS2 and the filter cavity “F C2” are replaced by a mirror with an incidence angle of 45 degree (like M1 in the left side)

γI T F = −

rI M + rsr c log 2L 1 + rI M rsr

(57)

with rsr and rI M representing the amplitude reflectivity of the signal recycling and the arm input mirror, respectively. For the more general expression of quantum noise in the detuned signal recycling configuration, the reader can refer to [69, 71]. Frequency-Independent Squeezing (FIS) The squeezed vacuum sources so far implemented in long arm interferometers are characterized by a squeezing ellipse rotation angle θ constant over the whole frequency band. In particular, the selected angle is about θ ≈ 0. With this choice according to Eq. (54), the shot noise is decreased by a factor exp(−2ξ ) at the expense of an increment by a factor exp (2ξ ) of the radiation pressure contribution. The latter competes with the low-frequency technical noise that is usually dominant. Therefore the net effect is an increment of the detector sensitivity in the high-frequency region where the shot noise dominates, leaving approximately unaffected the low-frequency sensitivity. In the next section the results so far achieved using this technique are reviewed. It is interesting to note that for θ ≈ 3π/4 the correlations between the shot and back-action noise induced by squeezed light allows to surpass the standard quantum limit by a factor exp[−2ξ ]. However this value of θ would produce a reduction of the quantum noise power spectral density only within a narrow band; for a full-band reduction of the quantum noise, a frequency-dependent squeezing angle would be required as shown in the next paragraph.

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Frequency-Dependent Squeezing (FDS) Assuming a squeezing angle θ (Ω) with the following frequency dependence [72] θ (Ω) = arctan(KMi (Ω))

(58)

the quantum noise expression (54) for a Michelson interferometer takes the form Shn hn (Ω) =

  2h¯ 1 K e−2ξ + Mi KMi ML2 Ω 2

(59)

which corresponds to a suppression of the standard quantum noise (44) by a factor e−2ξ over the whole frequency band. A method for generating squeezed light with a frequency dependent angle (58) was originally proposed by Kimble et al. [72]. In this scheme an FIS beam points to a detuned overcoupled Fabry-Perot (FP) cavity. In the case of a lossless linear cavity, the reflected field acquires a frequency-dependent phase shift: α(Ω). α(Ω) = arg

rin − rout e2i(Ω−Ωf c )L/c 1 − rin rout e2i(Ω−Ωf c )L/c

(60)

where ω0 + Ωf c and Lf c are the cavity resonant frequency and length, respectively, and rin and rout are the input and output cavity mirrors amplitude, reflectivity. Therefore after reflection the sideband annihilation operators aˆ ω0 ±Ω become ref

aˆ ω0 ±Ω = aˆ ω0 ±Ω eiα(±Ω)

(61)

and according to Equation (4) the quadrature Xˆ 1,2 (Ω) changes into Xˆ 1,2 (Ω): ref

Ref Xˆ 1 (Ω) =eiαm [Xˆ 1 (Ω) cos αp (Ω) − Xˆ 2 (Ω) sin αp (Ω)]

(62)

Ref Xˆ 2 (Ω) =eiαm [Xˆ 2 (Ω) cos αp (Ω) − Xˆ 1 (Ω) sin αp (Ω)]

where αp,m (ω) =

α(Ω) ± α(−Ω) 2

(63)

Apart from an irrelevant phase factor, Equation (62) shows that the reflected quadratures are rotated by the frequency-dependent angle αp (Ω):  αp (Ω) = arctan

2γf c Ωf c γf2c − Ωf2 c + Ω 2

 (64)

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where the last equality derives from Equations (60) and (63) and γf c is the halfwidth half-maximum power cavity linewidth. If the parameters of the filter cavity are set such that  Ωf c = γf c = Ω

ΩSQL KMi ≡ √ 2 2

(65)

the condition (58) is fulfilled. Therefore by injecting the FIS beam reflected from the FC into the interferometer dark port (see Fig. 8b), a broadband quantum noise reduction (cf. Eq. 59) is achieved. The previous result applies to the case of a Michelson interferometer and a lossless filter cavity. However it can be shown that the same quantum noise suppression factor (e−2ξ ) can also be achieved for tuned dual recycled interferometers with arm cavities providing the following approximated parameters (equation (66) is well founded in the limit in which the interferometer bandwidth γI T F is much larger than the frequencies in the range γf c around Ωf c where most of the squeezing ellipse rotation is accomplished. For the long arm interferometers Ωf c ≈ γf c is typically few dozens of Hz, while γI T F is several hundreds of Hz, so this limit holds [73].) for the filter cavity [73]:  γf c =

√

Ωf c ΩSQL √ =√ 1 − ε(2 − ε) 1−ε 2 2

(66)

where  ε=

 2+

4

 2 + 2 1 + (2ΩSQL /νf c Δ2rtl )4

ΩSQL

tsr 8 = 1 + rsr c

Parm ω0 MTI M (67)

Here νf c and Δf c are the filter cavity free spectral range and round trip losses, tsr and rsr are the amplitude transmittivity and reflectivity of the signal recycling mirror, Parm is the circulating power at the beam splitter, and M the mirror mass and TI M are the power transmittivity of the arms input mirrors. In the most general case, the expression (58) for the optimal squeezing angle is replaced by [69]  θ (Ω) = − arctan

T11 Cosζ + T21 Sinζ T12 Cos + T22 Sinζ

 (68)

where the coefficients Tij are defined in Eqs. (50), (51) and ζ is the detection homodyne angle. For the detuned configuration the optimal angle (68) can be implemented using two detuned filter cavities in series as shown in Fig. 8b. In

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this circumstance a numerical optimization of the lossy cavity parameters is made starting from the approximated solution [74] for the lossless case.

Quantum Noise in Real Interferometers Optical losses Optical losses affect the interferometers quantum noise. They are generated both inside the interferometer, as, for instance, due to scattering at the mirrors, and in the detection line after the recombination in the beam splitter (output mode cleaner spatial and angular mismatch, absorption and scattering inside the Faraday isolator, limited quantum efficiency of detection photodiodes, etc.). The former can be modeled assuming that the vacuum fluctuations δ Xˆ VA,P ac entering in the dark port are reduced by the interferometer optical efficiency √ ηI T F and simultaneously an “external” vacuum field with quadrature fluctuations √ 1 − ηI T F δ Xˆ VA,P ac−ext “enters” from the lossy port (cf. Eq. (27)). Differently from δ Xˆ VA,P , the fluctuations Xˆ VA,P ac ac−ext are uncorrelated in the two arms, thus producing an increase of the shot noise. This mechanism also applies for the back-action contribution; however, interferometer losses reduce the field amplitudes leading to a reduction of the back-action force amplitude. Detection losses (1 − ηdet ) are accounted in the same way, but in this case no effect on the back-action force is expected as the external vacuum field propagates in the direction of the detection photodiode without entering into the interferometer arms. Taking this into account, the quantum noise power spectral density (46) for a power-recycled interferometer in the presence of losses and in the limit Ω  γarm results [72] Shn hn (Ω) =

h2SQL 2



 

εrtl 1 [1 + εrtl + εDet ] + KP R 1 − KP R 2

(69)

where εrtl = 2Δrtl /TI M , Δrtl is the arm cavities round trip losses, and εDet = 1 − ηDet is the detection losses. As expected, the interferometer internal losses (εrtl ) decrease the back-action contribution and increase the shot noise, while the detection losses act only on the shot noise contribution. Similar considerations can be made for the case of dual recycled interferometers with the additional complications arising from the presence of correlations between the noise generators and by a more complex dynamics in general [71]. In the case of squeezed light injection, we proceed in the same way taking care to replace the vacuum quadrature fluctuations δ Xˆ VA,P ac with the squeezed quadrature in the presence of injection losses 1 − ηinj (cf. Eq. (27)). The value of ηinj is limited by many loss mechanisms, such as misalignment and mismatch of the squeezed beam with respect to the interferometer, absorption from the optical elements of the injection line, etc. For a detailed list of the loss mechanisms affecting the interferometers, the reader should refer to [81, 82]. It is interesting to note that, in contrast to the lossless case, in presence of losses, the shot noise enhancement

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produced by phase squeezing injection is less than the corresponding increasing factor in the back action. This is because the shot noise is limited by the total losses (1 − ηDet ηI T F ηinj ), while the back-action enhancement is not affected by the detection losses 1 − ηDet [81]. The general formula of the quantum noise in the presence of losses and squeezing injection can be found in Ref. [71] for DR interferometers while in Ref. [72] for power recycled interferometers. Squeezing ellipse angular jitter As explained in a previous section, the squeezing ellipse angular jitter may limit the actual sensitivity enhancement achieved by squeezed light injection. Squeezing angle fluctuations may arise from various mechanisms [12] and can exceed the radians scale. Different stabilization methods have been therefore developed [14, 15]. With χ (2) crystal squeezers, the most effective technique is the so-called coherent control (CC) initially proposed and implemented at the Albert Einstein Institute of Hannover [16]. Here a bright field with frequency ω0 + ΩCC is injected in the OPO cavity where, through the nonlinear frequency sum generation process, an additional field with frequency ω0 − ΩCC is generated. The phase φ of the latter depends on the phase of the pump field which also fixes the squeezing angle. An experimental estimate of φ is obtained by demodulating at 2ΩCC the beat note of these two coherent control fields. The squeezing angle is then locked to the CC beam stabilizing the φ value, for instance, by acting on the path length of the pump field. The last step consists in phase locking the CC and the LO beams using the error signal coming from the synchronous demodulation of their beat note. It is worth noticing that, choosing the ΩCC frequency offset in the RF domain, the technical noises of the control fields do not affect the squeezing degree in the audio band around Ω0 . Low-frequency fluctuations deriving from seismic and/or acoustic noise are in some case passively suppressed by suspending the OPO cavity in vacuum [80, 82] or actively via the CC stabilization loop. It should be noted that the latter’s performance can be degraded by squeezed beam pointing jitter which produces squeezed angle locking point fluctuations [12]. More difficult to suppress are the residual high-frequency fluctuations related to the presence of the interferometer control sidebands in case they are unbalanced in amplitude or in presence of the contrast defect field (the contrast defect field is generated when the two arms are unbalanced and exhibits a 90-degree phase shift with respect to the carrier in the detection port). At the state of the art this mechanism contributes for over 10 mrad to the total residual angular noise budget [13, 83] which for the present detectors stand in the range of 20 ÷ 50 mrad [81, 82, 84]. Stray light The scattered light from vibrating surfaces can recombine with squeezing beam generating a spurious signal with a phase noise which can spoil the overall sensitivity of the detector, thus reducing or cancelling the benefits of squeezed light [85–87]. In the case of interferometers with DC-readout, the most relevant contribution comes from the small fraction of the carrier field reflected toward the squeezer source and partially backreflected to the interferometer with a random phase coming from their relative displacement noise xn [83, 85] (the relative phase

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noise φn = 2π xn /λ generally exceeds 2π . Therefore to properly calculate the impact of the backscattered light on the interferometer sensitivity, the nonlinear behavior of the interference process must be considered [88, 89]). The most direct way to reduce this unwanted effect is to minimize xn . This was recently realized by the LIGO collaboration that installed the OPO on the detection bench, which is suspended under vacuum [82], thus drastically reducing the effect of relative seismic motion noise and acoustic noise. An alternative approach is to minimize the squeezer backscattered light which arises from two different sources: • Backscattered light from the squeezing injection line. Imperfections in the optics (finite roughness, surface contamination, bulk index fluctuation, etc.) scatter an incident beam in all directions [90]. The spatial distribution of the scattered light is described by the bidirectional scatter distribution function (BRDF) [90, 91]: BRDF (θinc , θSc , λ) =

π w02 PSc λ2 Pinc cos[θinc ]

(70)

where θinc and θSc , are the incident and the scattered angle, Pinc and PSc are the incident and the scattered power, and λ and w0 are the wavelength and the waist of the incoming beam. In order to minimize backreflected light (and the optical losses), the squeezing injection line must be realized with the lowest possible number of optics, using reflective instead of refractive optics having the lowest possible BRDF. Depending on the type of optics considered, the lowest BRDF so far achieved range between 5 · 10−5 /str. and 5 · 10−7 /str. [92–94]. • OPO backreflection. The direct reflection of the OPO cavity provides the most relevant source of backscattered light [83]. To mitigate this effect, a chain of low loss Faraday isolators [95] is usually inserted between the OPO cavity and the interferometer. In order to reduce the optical losses deriving from the use of Faraday isolators in certain cases, it is preferred to use traveling wave OPO cavities which have up to 40 dB of intrinsic rejection to the back scattered light [85]. Finally, in recent times an active method has been developed for the suppression of stray light coming from the reflection of the OPO [96]. The use of filter cavities adds a potential source of losses and other mechanism that can affect the overall detector quantum noise. The losses produce both an unwanted spurious mix of the quadratures (coherent dephasing) and a frequencydependent squeezing degradation due to the coherent vacuum fluctuations entering from the loss port as discussed above (decoherence) [73, 143]. Optical losses in the filter cavity can arise from several mechanisms: • Intracavity losses. The best reported value of the cavity round trip loss Δrt for a 100-meter scale optical cavity is about 60 ppm [97]. This value is mainly limited by the scattering losses arising from the finite roughness of the cavity mirrors [98, 142]. Therefore in order to minimize this detrimental effect, linear cavities, which minimize the scattering surfaces, are usually preferred [99]. A significant

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decrease in Δrt is expected for the next generation of cavities that are planned to install mirrors of the same quality as those developed for the LIGO/Virgo test masses [100]. Indeed a calculation based on the method presented in Ref. [98] estimates that for these cavities less than 10 ppm of round-trip scattering losses is expected [68]. The effect of the intracavity losses in the total loss budget scale as Δf c /Lf c . Indeed for a long cavity a lower finesse F is required to meet condition (66), and therefore less “bounces” on the scattering surfaces are accomplished. Typically Δf c /Lf c is required to be lower than one ppm/m which constrain to operate with 100-meter scale filter cavities [101]. • Mode matching. An imperfect spatial overlap between the squeezed beam and FC mode and between the FC and the interferometer modes induces optical losses independent from the cavity length. In recent times mode matching sensors [68, 102, 103] and actuators [104, 105] have been developed with the target to keep this kind of losses below (1 ÷ 2)% [106]. Another squeezing degradation mechanism derives from the residual fluctuations δLf c in the length of the filter cavity that induces a change δΩf c in the detuning frequency Ωf c : δΩf c =

Ωf c δLf c Lf c

(71)

and consequently a frequency dependent phase jitter (cf. Eq. (64)). This effect scale is the inverse of the cavity length and can be estimated considering that picometric scale residual displacement δLf c has been demonstrated [77]. Taking in consideration all these effects, the filter cavity design is done by increasing Lf c until the length-independent squeezing degradation sources become dominant. With the current technological limits, this happens for Lf c of the order of a few hundred meters (see, for instance, [68]).

State of the Art While laboratory experiments already demonstrated up to 14 dB of squeezing down to audio sideband frequencies (10 Hz to 10 kHz) [107], the effective reduction of quantum noise in large scale interferometers via the injection of squeezed vacuum represents a technical challenge. As discussed in the previous paragraph, the effect of injected squeezing on quantum noise is limited by optical losses, phase noise, and scattered light. The first demonstration of squeezed state enhancement of a power recycled GW configuration dates back to 2002 [108]. Two years later, squeezed light generation in the GW detection band was demonstrated for the first time [86]. Control of squeezing degradation mechanisms Besides the absorption and scattering of optical elements along the path of the squeezed vacuum field, a

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major source of optical injection losses is given by relative misalignment and mode mismatch between the squeezed light beam and the interferometer. While the mode mismatch is minimized by simply pre-tuning a matching telescope to maximize the coupling of a bright alignment beam from the OPA to the output mode cleaner of the interferometer, automatic control methods are applied to compensate for the relative alignment during science data taking. For instance, during O3 AdV applied angular dithers to the squeezed vacuum beam, demodulating the amplitude of the CC beat note on the interferometer detection photodiodes provided alignment error signals to automatically maximize the coupling of CC field on the interferometer; see Fig. 9. In aLIGO the automatic alignment was based on differential RF wave front sensing applied in reflection from the output mode cleaner. Detection losses are mainly generated by nonperfect transmission of the Faraday isolators, mode mismatch between the interferometer and the output mode cleaner, and the nonunitary quantum efficiency of the detection photodiodes. In addition technical noises at the interferometer detector reduce the amount of observed squeezing and are thus equivalent to an effective optical loss. Dark noise was particularly relevant in AdV during O3, where it was at a variable level around 9 dB below the unsqueezed shot noise, corresponding to a loss term of (7 ÷ 15)%.

Advanced VIRGO

= vacuum system 3km arm cavity

suspended input mode cleaner

Power recycling mirror

High-power laser system

3km arm cavity

1064nm 18Watts External PLL

output mode cleaning stages

SHG 1W squeezing main laser

532nm

h(t)

In-air bench

Mach Zehnder

50/50

PD

PDMZ

PZT auto-alignment actuators

Internal PLL

PDOPA

OPA

D u m p

DBS

Mode matching telescope

Phase shifter

Faraday Isolator

0.5W squeezing auxiliary laser

Squeezed vacuum & coherent control field

50/50 Pre-alignment detectors

Fig. 9 Scheme of the control system for the phase and alignment of the squeezed vacuum in AdV during the observation run O3. (From [81])

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Frequency-independent squeezing Among the long arm interferometers, GEO600, where squeezing enhancement is routinely employed since 2010 [109], demonstrated almost 6 dB of detected squeezing [84], the best level so far achieved. Concerning interferometers with long arm cavities, moderate noise reduction by squeezed light injection was already demonstrated in the first generation LIGO [109]. In the following detector generation, Advanced Virgo (AdV) and Advanced LIGO (aLIGO) achieved a quantum noise reduction by about 3 dB during their third observing run (O3) [81, 82]. Such result is consistent with the expected degradation of injected squeezing level considering the budget of optical losses and path length noise. In particular, the total optical losses on the squeezed light path, including technical noise equivalent loss, amount to (36 ± 5)% and to (25 ± 6)%, respectively, in AdV and in aLIGO, while the rms jitter of squeezing ellipse angle was estimated to be 45 ± 40 mrad in AdV and ∼15 mrad in aLIGO. The injected squeezing level was ∼7 dB in AdV and ∼4 dB in aLIGO. Figure 10 shows the effect of phasesqueezed light injection on the strain sensitivity of AdV and aLIGO. Figure 11 shows the long-term stability of shot noise reduction in AdV during the first half of the O3 science run.

Evidence of quantum back action With further control over loss terms and squeezing ellipse noise, higher squeezing levels will be probably observed in GW detectors in the next future. On the other hand, shot noise reduction by the injection of a frequency-independent phase-squeezed vacuum field in the dark port implies anti-squeezing of amplitude noise with corresponding increase of quantum radiation pressure noise (QRPN). Current GW detectors are not limited by QRPN, as the low-frequency strain sensitivity is still dominated by technical noises. However, for AdV and aLIGO the estimated level of QRPN is close enough to the current sensitivity that strong amplitude anti-squeezing can have an observable impact on

Fig. 10 Spectral strain sensitivity of the GW detectors during the O3 science run. Left: AdV; black curve: without squeezing injection; blue curve: with phase-squeezed light injection. (Adapted from [81]). Right: aLIGO; black curve: without squeezing injection; green curve: with phase-squeezed light injection. (From [82]). During the observational run, O3 AdV operated in the power recycled configuration, while aLIGO operated in the tuned signal recycling configuration

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Fig. 11 (From [81]) Left: time plot of the measured squeezing level in AdV during the first 5 months of the O3 science run; right: histogram of measured squeezing level

the sensitivity curve. Evidence of QRPN driving the displacement of the mirror test masses was found both in AdV [110] and in aLIGO [111]. Analyzing the effect of QRPN on low-frequency sensitivity, where technical noises are dominant, is quite subtle. In AdV, phase-squeezed and amplitudesqueezed light was alternatively injected for different values of the OPA parametric gain. To disentangle QRPN from technical noise at low frequency, the strain sensitivity with and without squeezing injection was compared. Amplitude antisqueezing with large OPA-gain enhances QRPN to observable levels, and the otherwise dominant technical noise is subtracted in differential measurements. At any given OPA-gain, with phase-squeezed light injection the strain sensitivity at high frequency, where shot noise is dominant, provides a precise measurement of the anti-squeezing level, that is, the expected enhancement factor for QRPN. The QRPN enhancement, measured at low frequency with phase-squeezed light injection, was found to equal the shot noise enhancement, measured at high frequency with amplitude-squeezed light injection, thus confirming the validity of the quantum noise model and the evidence for QRPN. A similar method was applied to show QRPN manipulation in aLIGO. An experimental measurement of the quantum noise is derived as the difference of strain sensitivity with and without squeezed light injection. By tuning the squeezing ellipse phase at an angle of 35◦ , a local minimum of the inferred quantum noise is found, consistently with the quantum noise model. Quite remarkably, as a consequence of the quantum correlations between the shot and the radiation pressure noise, such minimum lays below the SQL. Figure 12 shows the experimental evidence for QRPN in AdV and in aLIGO.

Frequency-dependent squeezing The cavity-assisted squeezing rotation was first demonstrated experimentally in the MHz [75] and subsequently in the kHz [76] regions. More recently, some proof-of-principle experiments have shown the

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Fig. 12 Evidence of radiation pressure noise in GW detectors. Left (from [110]): spectral strain sensitivity in AdV with the injection of phase-squeezed light (blue), with amplitude-squeezed light (red), and without squeezing (black); solid lines are experimental data, and dashed lines represent the quantum noise model; between 30 and 70 Hz, strain noise increases with amplitude squeezing due to QRPN enhancement. Right (from [111]): spectral strain sensitivity in aLIGO without squeezing injection (black trace), inferred quantum noise with squeezing injected at 35◦ (green trace), quantum noise model in the two conditions (blue and purple respectively); red dashed lines represent the SQL and 3 dB below SQL

feasibility of implementing FSD using long, low-loss optical cavities with rotation frequency down to tens of Hz, as required for GW detectors. In particular, the TAMA experiment injected a squeezed vacuum field from an OPO into a 300 m long linear Fabry-Pérot cavity with suspended mirrors [113]. An auxiliary laser beam at the second harmonic of the squeezed carrier or alternatively the Coherent Control beam [141] were employed for the longitudinal and angular control of the filter cavity mirrors. A similar experiment was performed at LIGO using a 16 m long linear filter cavity with suspended mirrors [114]. The experimental evidence of frequencydependent squeezing is shown in Fig. 13 for the TAMA and LIGO experiments. Both the interferometric GW detectors AdV and aLIGO are currently upgrading their setups for squeezed light injection by adding ∼300 m long filter cavities with suspended mirrors [27,68]; the goal is to implement frequency-dependent squeezing during the next observing run, O4. The integration of the FCs is also foreseen in the GW detectors under construction and in the baseline design of the future, thirdgeneration detectors. In this case it is also planned to operate with the detuned signal recycling configuration, and therefore two filter cavities are required [78] or as recently proposed a single filter cavity with a semitransparent mirror in the middle [79].

Advanced Methods for Quantum Noise Reduction In this section, a partial list of alternative techniques for quantum noise reduction is presented. Some of these are particularly promising, but in general none have yet been tested on large-scale interferometers. Nevertheless for the third-generation

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Fig. 13 Results of proof-of-principle experiments for frequency-dependent squeezing. Data represent the ratio of measured noise and shot noise versus frequency. Upper plot: results from the experiment at TAMA [113]. Lower plot: results from the experiment at LIGO [114]

detectors, they could represent an alternative or could be used in conjunction with the squeezed light techniques described in the previous section.

Variational Readout This measurement scheme aims to reduce quantum noise and exploits the quadratures correlations in the homodyne detector output signal [72, 112, 115, 116]. The balanced homodyne readout requires equal interferometer arms length (ΔL = 0).

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In this condition using Equations (39) and (41), the antisymmetric port output field becomes ASP (t) = Eˆ out

√

 2KMi 2xs (t − τ ) + KMi δ Xˆ VAac (t − 2τ ) + δ Xˆ VP ac (t − 2τ ) cos(ω0 t)+ hSQL L (72)

+ Xˆ VAac (t − 2τ ) sin(ω0 t)

Therefore according to Equation (25), the homodyne detection output ΔPˆ = PP D1 − PP D2 is ΔPˆ (Ω) = |α| cos ζ

√

 2kMi 2xs (Ω) A P ˆ ˆ + (KMi − tan(ζ ))δ XV ac (Ω) + δ XV ac (Ω) hSQL L (73)

where |α| and ζ are the amplitude and the phase of the local oscillator. Comparing Equation (73) with (42), it becomes evident that the DC-readout is equivalent to a homodyne detection with phase ζ = 0. Assuming the frequency-dependent homodyne angle ζ (Ω) = arctan[KMi ]

(74)

the back-action contribution in Equation (73) vanishes, leading to a drastic reduction of the quantum noise in the low-frequency region and allowing to take full advantage by injecting phase squeezing. This detection scheme is called variational readout and can be implemented by filtering the detector output before the homodyne readout using detuned filter cavities in analogy with the method used for FDS production [72]. Despite the great potential of this scheme, it has been demonstrated [72] that the presence of detection losses εdet severely limits the expected benefits of back-action cancellation. Indeed at each frequency the ratio of the minimum power spectral 1/4 density expected with and without the variational readout scheme scales is εdet 1/2 without FIS injection and εdet in the strong squeezing regime [117].

Speed Meters The speed meter was originally proposed on the basis of the thought that the momentum of the test masses is a quantum non-demolition (QND) observable, which is not affected by the readout back action that limits the sensitivity of mass position sensing [118]. In later times it was shown that also the probe field contributes to the generalized moment [119]; therefore the mass speed can’t be considered a pure QNR observable. Nevertheless speed meters are characterized by a lower back-action contribution with respect to the Michelson interferometers. The

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Fig. 14 Zero area Sagnac speed meter. The beam injected in one direction before returning to the beam splitter it also propagates in the second arm

North

xN

MN

East Input Laser

xE

Detection photodiode

ME

first design of a speed meter interferometer makes use of an external “sloshing” cavity coupled to the output of a traditional Michelson interferometer [74]. Subsequently a zero area Sagnac interferometer, which does not require additional external cavities, has been proposed [120, 121]. With reference to Fig. 14 a beam injected in the “east” arm first senses the position xEast (t) of mirror “ME ” and after the propagation time τ the position xN orth (t + τ ) of mirror MN . Applying the same considerations for the beam injected in the “north” direction, the measured phase difference δφ of the fields recombined in the beam splitter is proportional to δφ ∝ xEast (t) + xN orth (t + τ ) − xN orth (t) − xEast (t + τ ) ≈ 2υτ

(75)

and therefore to the mirror velocity υ. Moreover the radiation pressure on the mass ME resulting from the field injected in the East direction is reproduced after a time τ on the mass MN . This leads to a suppression of the back-action signal as big as the measurement time 1/Ω is higher than τ . Indeed the speed meter strain power spectral density [121]   h2SQL 1 + (KSM (Ω) − tan ζ )2 Shn hn (Ω) = 2 KSM (Ω)

(76)

is formally equal to the Michelson interferometer case as derived using Eq. (73) providing that KMi (Ω) is substituted with KSM (Ω) [121] KSM (Ω) = Ω 2

γ4 KP R (Ω) ∝ 2 arm 2 2 Ω + γarm

(77)

which at low frequency is almost constant on the contrary to the case of Michelson interferometer where KP R (Ω) and thus the back-action contribution grows as Ω −2 . Therefore by choosing the frequency-independent homodyne detection angle ζ equal to arctan[KSM (0)], an almost complete back-action evasion is achieved.

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This property together with the fact that in general speed meters do not require the use of additional filter cavities and are less subject to optical losses with respect to the variational readout detectors led to an intense activity on the optimization of their design (for an extensive and updated review, see [122]). However, a large-scale experimental demonstration on the effectiveness of the method still remains to be completed [123].

EPR Squeezing In the parametric down-conversion process, the pump field at frequency ωp generates the signal and the idler photons with frequency ωs and ωi , respectively. Energy conservation requires that the relations ωp = ωi + ω0 hold. Conditional squeezing can be observed with a nondegenerate OPA, i.e., when the frequency of the pump field ωp = 2ω0 + Δ is kept different from twice the interferometer carrier frequency ω0 . Therefore pairs of correlated photons at frequency ω0 ± Ω and ω0 + Δ ∓ Ω are generated around the carrier ω0 and idler ω0 + Δ field frequencies. The pump frequency offset Δ is chosen such that when the OPO output field is injected in the interferometer dark port, the idler sidebands are completely reflected. On the other hand, the signal sidebands enter in the interferometer and generate quantum noise that competes with the GW signal sideband. In the EPR scheme the idler and the signal sidebands coming from the interferometer are spatially separated and independently detected on separate homodyne detectors (see Fig. 15). The key point is that the quantum fluctuations of the idler and the signal field are correlated. By combining with a proper Wiener filter the idler output, which contains noise alone, and the signal output, which contains the gravitational signal plus noise correlated to the idler channel, one can obtain a squeezed quantum noise in which the squeezing angle coincides with that of the idler’s homodyne detector. Moreover, in reflection from the interferometer, the idler sidebands acquire a frequency dependence that is transferred to the squeezed phase. As demonstrated in Ref. [124], by properly choosing Δ, it is possible to create an FDS similar to that generated by a filter cavity with the advantage of a considerable simplification of the set-up. The drawback is that the maximum achievable squeezing level is 3 dB less than what is achievable with a degenerate OPO cavity. Furthermore despite the losses associated with the filter cavities that are not present, the double readout implies that detection losses must be counted twice. An experimental demonstration of the EPR squeezing was recently carried out by two independent groups [125, 126], while a systematic investigation with large arm interferometer has yet not be undertaken.

Other Methods • A method to avoid the radiation pressure noise increase when phase squeezing is injected consists in using a tuned triangular filter cavity as an high pass optical filter for the squeezed light. In this way the reduction of the shot noise is obtained

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Fig. 15 The EPR squeezing scheme: the signal and the idler after being reflected by the interferometer are separated by the output mode cleaner (OMC). The respective homodyne detector output quadratures are linearly combined with the frequency dependent coefficient K to produce the detector output (“Out”)

Input Laser

LO Signal

OPO

Idler

OMC LO Out

K

without being accompanied by an increase in the radiation pressure which would be generated by the amplitude quadrature fluctuations of the standard coherent vacuum field [127, 128]. In recent times a variant of this scheme that uses two squeezed beams has been proposed to obtain simultaneous reduction of shot noise and radiation pressure noise [5]. • An alternative method to overcome the standard quantum limit is to act on the test mass force susceptibility χ (Ω) leaving unaffected the noise sources generators. Indeed the most general expression for the standard quantum limit is h2SQL =

1 2h¯ 2 2 2 ML Ω Ω |χ (Ω)|

(78)

For a free mass χ = −1/Ω 2 , while for a harmonic oscillator close to the resonant frequency, the susceptibility could be significantly higher. The passage of the mirror dynamics from that of free mass to that of a harmonic oscillator can be achieved in detuned signal recycled interferometers using the restoring force generated by the dependence of the radiation pressure on the mass position [64]. The main advantage of this method is that its effectiveness does not depend on the optical losses while the main limitations come from the narrow band gain and intrinsic instability. One way to fix these problems is to use two carrier fields at different frequencies, each one inducing an optical spring on the test masses [129]. Furthermore enhanced optical spring effect can be achieved inserting parametric amplifiers in the interferometer [130, 131]. A detailed analysis of this method can be found in Refs. [5,119,122] and references therein. • The use of optical resonators in interferometers allows to decrease the Cramér– Rao (QR) limit in the frequency band limited by the linewidth of the Fabry Pérot cavities. Therefore the larger is the gain, the lower is the enhancement bandwidth. A method to overcome this limitation consists in placing an element inside the

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SR cavity for compensating the phase shift accumulated by the light in the cavity round trip [132]. This produces a widening of the cavity resonance without affecting its amplitude gain, leading to an increase of the detector’s bandwidth preserving the peak sensitivity. The schemes proposed are distinguished from the method used to achieve the negative dispersion, namely, using atomic dispersive media [133] or unstable opto-mechanical devices [136]. Finally the detector bandwidth can be expanded by using nonlinear crystals [134, 135]. • Referring the test mass position to a “negative” mass, the radiation pressure effect is cancelled in the differential motion leading to a back action evading detector. The equivalent of a negative mass can be obtained using spin systems [137, 138] or alternatively using nonlinear media inside the cavities [139, 140].

Cross-Reference  Research and Development for Third-Generation Gravitational Wave Detectors  Terrestrial Laser Interferometers

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Environmental Noise in Gravitational-Wave Interferometers

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Irene Fiori, Anamaria Effler, Philippe Nguyen, Federico Paoletti, Robert M. S. Schofield, and Maria C. Tringali

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational-Wave Interferometer at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environment Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensors Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods for Investigating Environmental Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise Hunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of Gravitational-Wave Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Earth Crust Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sea and Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthropogenic Seismic and Acoustic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Fiori () European Gravitational Observatory, Cascina, Pisa, Italy e-mail: [email protected]; [email protected] A. Effler LIGO Livingston Observatory, Livingston, LA, USA e-mail: [email protected]; [email protected] P. Nguyen · R. M. S. Schofield University of Oregon, Eugene, OR, USA e-mail: [email protected]; [email protected] F. Paoletti INFN, Sezione di Pisa, Pisa, Italy e-mail: [email protected] M. C. Tringali European Gravitational Observatory, Pisa, Italy © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_10

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Sound and Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound and Vibration Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam Jitter Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattered Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattered Light Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morphology of Scattered Light Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattered Light Hunting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattered Light Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EM Noise Coupling to Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic and Electric Fields Coupling to Test Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric Field Influences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravity Gradient Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newtonian Noise from Ground Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newtonian Noise from Air Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Noise Considerations in Site Selection and Site Facilities . . . . . . . . . . . . . . . Site Selection Considerations for Minimizing Environmental Noise . . . . . . . . . . . . . . . . . . Site Facilities Considerations for Minimizing Self-Inflicted Environmental Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The first 20 years of operation of gravitational-wave interferometers have shown that, despite the high degree of isolation, detectors are affected by influences from the surrounding environment. Seismic, acoustic and electromagnetic disturbances of natural or human origin may limit the interferometer sensitivity or potentially generate transients of non-astrophysical origin. The study and reduction of environmental influences has been part of the effort that eventually led to the detection of gravitational waves. In this paper, we present a review of environmental noise sources and coupling paths, investigation and mitigation methods. We refer to the experience gained during the commissioning and operation of the existing gravitational-wave interferometers and the most recent documentation on the subject. We wish to share indications useful for the design and commissioning of future terrestrial gravitational-wave detectors.

Keywords

Environmental noise · Environmental monitoring · Gravitational wave · Interferometer · Seismic noise · Acoustic noise · Vibration noise · Scattered light · Magnetic noise · Radio frequency noise · Electronics noise · Noise hunting · Noise mitigation · Site selection · Site facilities · Low-noise design

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Introduction Kilometre scale laser interferometers measure the extraordinarily small space-time strain produced by gravitational waves. On September 14, 2015, the two LIGO detectors made the first direct observation of gravitational waves [1] and opened a new window for the study of the universe. Operative interferometers today are the two LIGO detectors at Hanford (LIGO H1) and Livingston (LIGO L1) [2] in the USA, the Virgo [3] detector in Italy, the KAGRA [4] detector in Japan and the GEO600 [5] detector in Germany. LIGO, Virgo and KAGRA √ have reached, or aim to reach, peak strain sensitivities of a fraction of 10−23 / H z, which means measuring differential changes of the order of 10−19 m in the distance between mirror test masses placed a few kilometres away. These detectors are immersed in the environment and to some degree are influenced by natural, anthropogenic and self-produced noise. The LIGO, Virgo and GEO detectors are located in the countryside, but their environment is not very quiet. The Virgo detector is 1 km away from highways, and a few kilometres from the Pisa city and airport, it is surrounded by an intense agricultural area and some high-voltage power lines run across its two arms. Train transits and tree cutting occur near the LIGO interferometers, and GEO has a wind farm just 1 km away. In addition, all detectors, including the cryogenic underground detector KAGRA, situated in a much quieter place, have to face the self-inflicted noise produced by their infrastructure (e.g. ventilation, water and vacuum pumps, electronics). Detectors are influenced by geophysical phenomena, such as wind, lightning and sea-wave activity and fluctuations in the Earth electromagnetic and gravity fields. Apart from extreme rare events like earthquakes or thunderstorms that “shake” abruptly the interferometer and may cause it to lose the operating position (in jargon, unlock), the effect of environment is adding further non-Gaussian disturbances (i.e. noise) to the detector strain data. In the search for gravitational waves (GW), scientists identify four main categories [6, 7]: (i) continuous waves (CW) produced by isolated spinning massive objects like neutron stars with asymmetric shape or by binary systems long before merger; (ii) chirp-shaped waves generated by the coalescence of compact binary objects (CBC) like double black holes, double neutron stars or neutron star-black hole; (iii) short-duration bursts (from ∼10−3 to ∼103 s) produced, for example, by supernova core-collapse; (iv) and a stochastic GW background hypothesized to be created by a large number of uncorrelated astrophysical and cosmological sources. Environmental noise can impact on all kind of GW searches [8]. For example: broadband noise in the frequency range 10–500 Hz, such as noise from scattered light, mostly affects searches of gravitational waves from coalescence of compact objects; narrow spectral peaks (named lines) impact on continuous wave searches; short-duration transients (named glitches) can mimic supernovae and coalescing binaries. Finally, noise that correlates among distant detectors, such as Schumann magnetic fields [9,10] or noise from clock-disciplined electronics, contaminates the observation of the stochastic GW background [11].

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Interferometer realization alternates periods of construction and commissioning, aimed at improving sensitivity, and periods of astrophysical observation. During the commissioning phases, scheduled upgrades are implemented and residual noise is studied and mitigated to reach the strain sensitivity target. For achieving the sensitivity goal, precautions have been adopted in the design of the detectors and their infrastructure (i.e. detector equipment and site service facilities) in order to drastically reduce the detector coupling to external disturbances and reduce the infrastructure noise. Nevertheless, this is not sufficient to ensure complete immunity from these disturbances. The experience of interferometer commissioning is a long but exciting story of hunting for noise sources and noise coupling paths, identifying and curing unexpected weaknesses. The scope of this paper is to communicate this experience. The dissertation is far for being complete but aims to provide most relevant and updated information to interested readers. This chapter is organized as follows. Section “Gravitational-Wave Interferometer at a Glance” introduces the main characteristics of a gravitational-wave interferometer. Section “Environment Monitoring” illustrates the network of sensors monitoring the interferometer environment. Section “Methods for Investigating Environmental Noise” describes data mining and experimental methods for the investigation of environmental noise. Then, single categories of environmental disturbances are described: seism (Section “Seismic Noise”), sound and vibration (Section “Sound and Vibrations”) and electromagnetic fields (Section “Electromagnetic Noise”). In Section “Scattered Light” we examine the subject of scattered light, one of the most troublesome noise affecting current interferometers. For each category, we illustrate sources, coupling paths and mitigation techniques. A special mention is dedicated to gravity gradient noise (Section “Gravity Gradient Noise”) which, although not yet experienced in detectors at the time of this writing, is considered a serious issue for the next generation of ground-based detectors. Finally, in Section “Environmental Noise Considerations in Site Selection and Site Facilities” we present considerations based on lessons learned that may be helpful in the selection process of new detectors site and in minimizing self-inflicted noise from future facilities. We summarize and conclude in Section “Conclusions”.

Gravitational-Wave Interferometer at a Glance This section illustrates the principle of operation of a gravitational wave interferometer (ITF) and the main components of concern in environmental noise studies. For a thorough description of interferometers and fundamental noise sources, see, for example, [12, 13]. Gravitational-wave detectors today in operation are power-recycled laser Michelson interferometers with 4 km (LIGO), 3 km (Virgo and KAGRA) and 600 m (GEO) long optical cavities in the arms (folded optical cavities for GEO, Fabry-Perot resonators for LIGO, Virgo and KAGRA). To detect a gravitational wave, the distance change ΔL in the horizontal dimensions between pairs of free-falling test masses (TMs) has to be measured.

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Fig. 1 Simplified optical layout of the Advanced Virgo interferometer. (Courtesy of Julia Casanueva, [email protected])

The associated strain is h = ΔL/L, where L is the separation distance between two masses. In GW interferometers, the test masses are mirrors positioned at either end of two perpendicular arms, each forming a resonant optical cavity. The simplified optical layout of a GW interferometer is illustrated in Fig. 1. A laser beam from the input port is equally split by a beam splitter mirror and sent in the two arm cavities. Exiting from the arms, the two beams return at the splitter mirror and superpose along the common output path towards the detection photodiode. The ITF operation point is set such that the two beams interfere almost destructively along the output path (dark fringe condition) and the photodiode measures the instrumental residual noise. A passing gravitational wave produces a differential oscillatory stretching and compression of the arms, causing a slight phase shift of the two beams emerging from the arms towards the beam splitter and as a result a tiny light intensity fluctuation in the photodiode output. The photodiode signal is then digitized and calibrated as dimensionless strain. The ITF optical scheme also includes the following: two additional mirrors named power recycling (PR) and signal recycling (SR) placed before the beam splitter and before the output port, respectively, in order to enhance the power

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Fig. 2 Representative residual strain noise amplitude spectral density of LIGO H1 (red) LIGO L1 (blue) and Virgo (magenta) detectors during their joined third observing run (April 2019 to March 2020). The large peak at about 48 Hz in the LIGO Hanford spectrum was identified using the techniques described in Section “Noise Injections” and was eliminated for the second half of the run

circulating in the interferometer and suppress the shot noise; input and output mode cleaners optical cavities; a thermal compensation system shining CO2 laser light onto silica plates (compensation plates – CP) to recover deformations of the mirror test masses heated by the power stored into the interferometer resonant arm cavities [14]; and squeezed light injection at the output port [15, 16]. The residual instrumental noise of present interferometers is shown in Fig. 2 [17]. The detector bandwidth extends from 10 Hz to 10 kHz with peak sensitivity of a √ fraction of 10−23 / Hz between a few tens and a few hundreds of Hz. Most intense GW signals from coalescence of binary stellar systems observed so far produce strain signals of a few 10−23 m/m, corresponding to arm length changes of the order of 10−19 m. To succeed in measuring such incredibly tiny number, interferometers must be isolated from the external environment as much as possible. Typical displacements due to natural and anthropogenic activities on Earth are of the order of 10−6 m; thus, optics must be seismically decoupled from the ground. The primary test mass mirrors and most optical components are suspended to multistage seismic isolators that provide up to 280 dB of attenuation above 10 Hz. The Virgo and KAGRA seismic isolation systems consist of a cascade of pendula and vertical springs, which perform a passive attenuation in all six (angular and longitudinal) degrees of freedom [18,19]. The LIGO and GEO seismic isolation platforms are based on a similar concept [20, 21]. Isolation systems are actively controlled using sets of position sensors (linear variable differential transformer, LVDT), inertial (accelerometers) sensors and coil-magnet actuators

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positioned along the isolation chain. Their purpose is twofold: (i) to actively damp the mechanical modes of the system (30 mHz to few Hz, for Virgo) and (ii) to control the longitudinal and angular position of each suspended optical element to bring and keep the interferometer in its operational condition [22]. The entire beam path is enclosed in an ultra-high vacuum (UHV) volume of about 10,000 m3 with a residual gas pressure of 10−9 mbar. The vacuum system prevents air density fluctuations, for example, from sound pressure waves, to produce changes in the laser beam path length, which would mimic or mask those produced by a gravitational wave. A few components, which do not require a vacuum, remain in air and not seismically isolated. They are the infrared YAG laser source (λ = 1064 nm, input power ≈100 W) and a few auxiliary optical benches looking at pick-off beams for monitoring purposes. To be sensitive to GW, the interferometer must first be locked, meaning to bring the detector in a regime where maximum power builds up in the arm cavities. This is achieved by controlling the position and alignment of all suspended optics with extreme accuracy [23]. The Pound-Drever-Hall control technique is used. Electrooptical modulators (EOM) modulate the laser light at a few frequencies in the MHz range (in Virgo: 6, 8 and 56 MHz) to produce sideband signals that selectively resonate inside the different optical cavities. Radio frequency photodiodes (PD) and quadrant split photodetector (QPD) placed on auxiliary benches look at cavities reflection, pick-off or transmitted beams. PD and QPD signals are demodulated to extract the carrier sideband information, upon which correction signals are generated and used to adjust the position of each test mass with coil-magnet (Virgo) or electrostatic (LIGO and GEO) actuators. An ultra-stable GPS-locked atomic clock is used to synchronise all interferometer devices, as needed for the implementation of the real-time control, and also to synchronise the different detectors to permit the sky localization of GW sources. In addition to the gravitational-wave strain data, thousands of auxiliary channels are recorded that witness a broad spectrum of potential coupling mechanisms, useful for diagnosing detector faults and identifying noise correlations. Auxiliary channels include measured angular drift of optics, photodiode signals, actuation signals used to control optics position as well as environmental monitors.

Environment Monitoring The external environment can influence a GW detector in multiple ways: through physical contact (via vibrations or temperature fluctuations), electromagnetic waves, electrostatic and magnetic fields and possibly also high-energy radiation. Understanding these influences requires comprehensive monitoring of the environment surrounding the detector. This is achieved with a system of auxiliary sensors carefully selected and positioned. The system includes physical monitors (e.g. seismometers, microphones, magnetometers), positioned in the detector proximity or externally of experimental areas, as well as monitors of the operating status of the

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Table 1 Specifications for important environmental sensor types: the model, the frequency range in which they are used and their typical self-noise. The self-noise is the larger of the quote from the manufacturer for that sensor Type Sensor model Seismometer Trillium 120 or CMG-3T Accelerometer Wilcoxon 731–207 or PCB 393B12 Microphone Brüel Kjaer 4190 or 4193 Magnetometer Metronix MFS-06e or LEMI120 Magnetometer Mayer FL3-100 or Bartington 03CES100 RF receiver AOR AR5000A or ELAD FDMS1

Operating frequency Self-noise √ 0.01–50 Hz 10−8 m/ Hz at 0.1 Hz √ 1–1000 Hz 1 μm/s2 / Hz at 10 Hz 0.1 Hz–10 kHz 0.1 mHz–10 kHz

√ 0.1 mPa / Hz at 10 Hz √ 0.1 pT/ Hz at 1 Hz

0–1000 Hz

√ 10 pT/ Hz at 1 Hz

10 kHz–100 MHz

√ 10 nV/ Hz at 10 MHz

interferometer infrastructure, intending all the equipment needed for the detector operation, like cooling and climatization plants, vacuum system and electronic devices. Hereafter, we outline sensor typologies, positioning choices and installation criteria. We refer mostly to the environmental monitoring sensor network of Virgo; however, the implementation is similar in all GW interferometers [24–27].

Environmental Sensors A few hundred environment monitors are in use at the Virgo site. They are accelerometers for high-frequency (∼10 Hz–10 kHz) vibrations, seismometers for low-frequency (∼0.1–10 Hz) vibrations, microphones, magnetometers that measure audio frequency magnetic fields, voltage and current monitors of the electric power supplied at the detector site, radio frequency (RF) receivers and wind, temperature, humidity and atmospheric pressure probes. Another few hundred sensors are used to monitor the detector infrastructure. Monitored quantities are ambient temperature and overpressure, air and water fluxes, electric power, vacuum residual pressure and dust particle contamination. These sensors have multiple uses: (i) they are used in correlation studies with other signals for tracking offending noise sources; (ii) they are used to measure the coupling of the GW detector to the environment with the purpose to assess its level of immunity or spot possible weaknesses; (iii) they allow for the flagging of noisy conditions to be considered in GW analyses; and (iv) they help in validating astrophysical events by excluding an accidental environmental origin. In order to monitor environmental influences on the interferometer, environmental probes should be much more sensitive to physical disturbances than the interferometer is. Consequently, sensors model and positioning are strategically chosen. Table 1 lists some of the sensor models in use, and Fig. 3 shows the layout

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Fig. 3 Layout of environmental probes in Virgo and LIGO experimental areas. (a) Virgo environmental probes. (b) LIGO environmental probes

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of sensors at Virgo and LIGO Livingston Observatory. The website pem.ligo.org provides detailed information on LIGO monitors. Selected models have enhanced low-frequency response and low self-noise. The sensors operative frequency range mainly provides coverage of the interferometer detection band from ≈10 Hz to a few kHz. However, some sensors also monitor beyond these frequencies to cope with coupling mechanisms that convert low- or high-frequency signals up or down into the detection band.

Physical Environment Monitors Most sensors are located inside the experimental buildings nearby the detector, as illustrated in Fig. 3. Sensors are positioned as close as possible to potential noise coupling sites, specifically: • External optical benches, which are sensitive to acoustically driven vibrations and thermal and humidity fluctuations, are monitored with one 3-axial accelerometer (Episensor Es-T by Kinemetrics), one microphone, one temperature and one humidity probe. • Accelerometers are positioned to monitor vibration of locations which are prone to scattered light coupling (Section “Scattered Light”), such as vacuum chamber walls surrounding optical benches and test masses; in-vacuum separating windows, which are crossed by the beam; and restrictions of the beam pipe, where beam clipping is more plausible to occur. • Electronics rooms, hosting readout and control electronics are instrumented with floor accelerometers (LIGO) to detect vibrational coupling to the electronic boards and to monitor the rooms as seismic sources. Magnetometers are positioned inside electronic racks hosting photodiode demodulation and control electronics to monitor EM disturbances [28]. Temperature and humidity of the room are also monitored. • Each experimental building is equipped with the following: one seismometer and one tiltmeter [29, 30] to monitor the low-frequency motion of the concrete slab, on which all vacuum chambers and seismic isolation systems are placed; an array of geophones as prototype network for Newtonian noise cancellation (Section “Gravity Gradient Noise”); three orthogonally oriented magnetic sensors and one RF antenna; power grid voltage; and current monitors installed at the main electrical switch-boards. Current monitors detect more efficiently than voltage monitors currents from variable loads. LIGO also implements floor accelerometers to aid localizing sources of vibration through propagation delays and amplitude differences. • Temperature and humidity are monitored inside the buildings and clean areas. Temperature is also monitored in-vacuum along seismic isolation chains and in proximity of core optics. • Beam tubes: Vibrations of vacuum beam tubes are not monitored at Virgo, after tubes immunity had been verified with extensive shaking tests. LIGO implemented monitors in the mid-arm stations, which is especially important at

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LHO, where significant coupling has been measured, likely because they contain the smallest aperture between vertex and end stations.

Geophysical Monitors GW detectors can be influenced by meteorological and geophysical phenomena. Effects of sea and wind activity are discussed in Section “Seismic Noise”, while lightning strikes and cosmic rays are discussed in Section “Electromagnetic Noise”. Electromagnetic (EM) waves also travel at the speed of light and, due to their strength, could affect multiple detectors with the same timing as a GW. This is the case of Schumann magnetic fields [9], large-current lightning strikes and solar activity. To monitor wind and lightning, GW detector sites have anemometers and magnetometers. Wind monitors are advisable at the corner and the two end stations to disentangle wind gusts. The lightning sensor is preferably connected to a sensor network (e.g. blitzortung.org), or lightning data from external providers (e.g. www. vaisala.com) are used for specific studies. Extremely low-frequency induction coil magnetometers (Table 1) are set up at one magnetically quiet on-site location to witness anthropogenic or global EM disturbances. LIGO and Virgo also have one detector to witness possible influences from cosmic ray showers. Data from external EM observatories help to improve efficiency and reliability of detecting natural or anthropogenic electromagnetic signals [25]. One such observatory is www.vlf.it. These monitors play a fundamental role in the validation process of GW events, as described in Section “Validation of Gravitational-Wave Events”.

Infrastructure Monitors The detector infrastructure consists of all the equipment needed for the detector operation, like cooling and climatization plants, vacuum system and electronic devices. Infrastructure devices are the principal source of seismic, acoustic and electromagnetic noise nearby the detector. The physical sensors described in Section “Physical Environment Monitors” are often a good witness of the noise but cannot alone track down the specific source. Monitors of the operational status of a single infrastructure device can be used to correlate the device operation or malfunctioning with non-stationary disturbances in the GW signal and identify the culprit. Infrastructure monitors can either be digital signals, indicating the device operating condition (on/off status), or analogue signals from physical probes. For example, hydraulic circuits are monitored by temperature probes, flow and pressure metres. Temperature of sensitive electronics modules is monitored as well. For systems with feedback control, like air conditioning, it is advisable to monitor all sensors (e.g. temperature, airflow, fan speed) and actuators (e.g. valves and grids opening). Often several monitors are included in the commercial product, and it is a matter of integrating them in the experimental data flow. Some custom implementations might however be needed, such as current probes, to monitor the operation status of crude devices, like drain pumps, water coolers, etc.

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One way to monitor the on/off status of devices is plugging them on remote controllable power sockets (“smart plugs”). As a bonus, single devices can be switched on in automated way and kept off when not needed. Virgo makes extensive use of smart plugs, in particular, for noisy devices that are seldom needed, like the following: digital cameras, piezoelectric drivers for optics alignment and stepping motors for suspension static positioning and balancing. One caveat is making sure that both hot and neutral wires are disconnected when the socket is in the open state. Some smart plug models indeed do not disconnect one of the two mains wires and may not eliminate noise from the connected equipment as a result [26].

Sensors Integration Sensors installation usually follows vendor’s instructions, but sometimes a few custom parts are needed, like holders or connector adapters. One example of homebuilt sensors are the voltage monitors. They are simple step-down transformers that adapt the mains voltage to the input of analogue to digital converter (ADC) boards. Other examples are the electric field metre and RF sensors at LIGO with long, halfwavelength wire antennas (Fig. 3). Despite the not adverse environment inside the experimental buildings, the installation of low-frequency seismic sensors requires special care to guarantee good contact with the building floor surface and thermal stability and avoid influences of air turbulence. Good practice is well known to seismologists [31]. The LIGO seismometer isolation kit is described by Martynov [32]. LIGO positioned a few seismometers in vaults placed roughly 1 km from experimental areas in order to distinguish vibration generated at the buildings from vibration generated off-site. Accelerometer installation requires a rigid coupling to the surface to be monitored. For permanent installations, the vendor’s kit and tips shall be adopted. Yet, for a temporary accelerometer installation, a double-sided tape on horizontal surfaces provides a good mechanical coupling up to 1 kHz. Microphones should be protected from rain and wind-induced noise when used outdoor [33]. Indoor magnetometers require rigid supports to avoid noise induced by the sensor motion within the Earth’s magnetic field. Installation of outdoor magnetometers requires a particular care [34]. A quiet location is selected sufficiently far from local sources as result of a measuring campaign but near enough to the site so that magnetometers can observe the same Earth global magnetic field that the interferometer does. Magnetometers are buried at half a metre depth to avoid wind influences. Protection against water and humidity is realized with concentric plastic tubes, filled with hygroscopic material and sealed with silicon glue. Proximity to metallic objects, like fences, that can induce wind-driven signals is avoided, and precaution is taken against overvoltages caused by lightnings. As a general rule for all types of sensors, their output is preferable to be routed to a differential conditioning/acquisition system to reduce any noise, including those due to ground connections. Eventually, probe signals are integrated in the experimental data acquisition system, which performs digitization and anti-aliasing

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filtering. Analogue gain and sampling rates are set to fit the sensor dynamic range and useful bandwidth, as well as the ADC noise constraints. The typical sampling frequency of physical monitors (e.g. accelerometers, microphones) ranges between 1 and 20 kHz. Slowly changing ambient signals, like temperature and humidity, as well as infrastructure monitoring signals have typically 1 Hz sampling. In order to have a prompt monitor of on-site noise and environmental conditions, it proved useful to produce an under-sampled data stream (e.g. 1 Hz sampling) containing the root mean square (RMS) value (full or in-bands) of each environmental monitor. These data have different uses: (i) long-term trend analysis, (ii) within online monitors to flag detector malfunctioning (e.g. anomalous temperature rise) or noisy periods (e.g. a machine forgotten on) to be accounted for in the analysis of GW data and also (iii) inform about a malfunctioning of the environmental sensor itself [35, 36].

Methods for Investigating Environmental Noise Studying environmental noise serves three purposes. The first purpose is to maintain the ambient contribution of environmental noise below all fundamental noise sources of the interferometer. This task is performed by noise hunting, intended as the process of identifying noise sources and their paths to the interferometer readout, and then pursuing a mitigation of their effects. The second purpose is to determine the level at which the existing environmental noise contaminates the interferometer readout. This task is achieved by measuring the interferometer-to-ambient coupling functions. The third purpose is the validation of gravitational-wave events. This means understanding the origin of environmental disturbances that can be correlated between detectors and evaluating their impact on the interferometer readout when they occur around the same time as a gravitational-wave signal. In the following, we elaborate on the three subjects.

Noise Hunting Noise hunting is intended as the process of identifying sources and paths of environmental noise that ultimately contaminate the GW signal. The search usually begins with an attentive examination of the detector signals. This step, called data mining, is supported by ad hoc software tools. Following the clues provided by the environmental sensors and other auxiliary signals, the hunting proceeds with experimental methods. Generally speaking, experimental methods consist in performing controlled actions on the interferometer or the putative sources and interpret the results to eventually track down the culprits. The effort of GW detector collaborations in environmental noise hunting is documented in several articles by LIGO [28,37–40], KAGRA [27] and Virgo [26,41–43]. A collection of public reports tackling a large variety of topics is [44]. A vast supply

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Table 2 Software tools in use for the characterization of environmental noise coupling to Virgo and LIGO detectors Tool name Function VIM spectrograms [49] and Produces daily normalized time-frequency maps to monitor specFscan [39] tral artifacts in strain and auxiliary channels Omicron scan [50] Generates spectrograms optimized for visualization of transient GW and noise events in thousands of signals processed in parallel BruCo [51] Computes coherence between GW strain signal and thousands of auxiliary signals, and produces daily tables of ranked significantly coherent channels NoEMi [52] Finds spectral lines and outputs selected lists in a line database FineTooth [28] Finds families of equally spaced spectral lines (line combs) NonNA [53] and Lasso [54] Identify a ranked list of slow auxiliary monitors capable of recondata regression tools structing the variations of the GW signal in a specific frequency range Buffalo [55] Identifies a ranked list of slow auxiliary monitors capable of reproducing the evolution of a noise feature cropped from a spectrogram graph, e.g. wandering lines MONET [56] Investigates sideband noise due to coupling of a carrier signal with the low-frequency part of several auxiliary signals SILeNTe and other non- Based on data regression methods, model and eventually subtract linear noise tools [57, 58] non-linear noise components Scattered light noise tools Correlate instantaneous amplitude oscillations in the strain signal [59, 60] with predictors computed from the auxiliary channels monitoring the position of relevant optics in the detector

of not-reviewed information can be found in the public electronic logbook of the detectors [45–48].

Data Mining Techniques Careful inspection of the data is usually the first noise hunting step. The search for environmental noise coupling to the GW strain signal makes use of data processing methods and data mining tools dealing with both stationary noise and shortduration transients. Several data mining tools have been developed by the LIGO and Virgo collaborations for the noise characterization of detectors data [8, 28]. Table 2 summarizes the tools used to investigate interferometer noise, including environmental noise. Amplitude spectral density (ASD) plots are useful to examine stationary or slowchanging noise features, like narrow frequency components (lines), wide-frequency structures (bumps) or broadband noise. Examining the coherence between the strain signal and each environmental probe helps sorting out signals that witness linearly coupling noise and give clues about the location and the nature (e.g. seismic or electromagnetic) of the noise source. Time-frequency maps, also known as

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spectrograms, allow for quick spotting of the sudden appearance of a noise source and the determination of its onset time. Figure 4 illustrates two examples. Electronic logbooks [61] with public web access are used for sharing information with the GW community. Interferometer crew adds to e-logbook hourly reports of any activity performed on the detector or the laboratory infrastructure. Upon spotting a new noise, typically the first subsequent step is to query the detector e-logbook for an action coincident or causally related to the onset of the noise, like the switch on of a device or a change in the interferometer operation condition. Furthermore, spectrogram inspection can reveal an intermittent noise whose on/off pattern suggests correlations with other ambient parameters (e.g. temperatures) or the operation cycle of a device (Fig. 5). Given the large number of signals to scrutinize (on the order of a few thousand), often a more efficient approach is to apply the noise search exhaustively to all “safe” auxiliary channels (i.e. channels which do not contain GW signal information) and output a ranked list. For example, the BruCo tool (Table 2) looks for frequency bins with significant coherence between one master channel (typically the GW strain signal) and all safe channels, ranking them by the scored coherence at each bin. Quite useful are exhaustive search tools uncovering correlations between the strain signal and slow ambient monitors, infrastructure monitors and RMS monitors. Some of these tools, like Lasso, NonNA and Buffalo, are described in Table 2. One example of correlation analysis with Buffalo is shown in Fig. 5. Exhaustive search tools also deal with non-linear noise (e.g. sidebands, up-conversion, families of equally spaced lines) that can arise because of bilinear coupling mechanisms as those described in Sections “Beam Jitter Noise” and “Digital Devices”). These tools, also in Table 2, are based on data regression methods that identify and eventually subtract non-linear noise components or up-conversion noise from scattered light processes (Section “Morphology of Scattered Light Noise”).

Experimental Techniques Following clues collected in the data mining step, a set of experimental actions is usually needed to better understand each noise source and pursue its mitigation. The most used experimental investigation methods are sniffing, switch-off tests and noise injections. In the sniffing process, a set of portable magnetic, seismic and acoustic probes is used to inspect the experimental areas and scrutinize potential devices looking for the specific time-frequency signature of the noise evidenced in the data mining step. This could be, for example, a given monochromatic tone (frequency line) or a set of lines. The probe has to be positioned in proximity of the putative source and its signal examined, most conveniently by looking at its ASD. Mobile phone compatible USB sensors are very handy to this purpose. Table 3 lists a few models. The number of noisy devices is so large that sometimes the best strategy for finding a source is to perform selective switch-off of suspected candidates and observe if the noise disappears from the strain signal. One example is illustrated in Section “Switch-Off Tests”.

Fig. 4 Examples of spectrograms of gravitational signal and environmental sensors displaying non-stationary noise features. Left: glitches are present both in the strain signal (bottom) and in one magnetometer (top). Right: a wobbling structure around 80 Hz disappears during the power off of one motor driver of the suspension of the external bench in transmission of Virgo West arm. A residual noisy current in the driver was shaking the bench and enhance noise from scattered light

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Fig. 5 The brute force correlation search tool Buffalo [55] is applied to a non-stationary noise in the Virgo strain signal (left plot, red dots). The identified correlated signal is the temperature of the demodulation electronics (right plot, blue line). The disturbance was then tracked to the thermally susceptible local quartz oscillator and solved by phase locking it to the Virgo master clock [62]. (Courtesy of Bas Swinkels, [email protected]) Table 3 The hunter’s toolkit: a set of tools for investigating environmental noise Type

Description

Portable sensors Seismic Digital accelerometer Acoustic Mobile phone microphone Magnetic 3-axis magnetometer Tools for noise injections Magnetic Small coil Magnetic Large wall coil Seismic Small impact hammer Seismic Small Bluetooth shaker Seismic Medium shaker Seismic Long stroke shaker Acoustic Loudspeaker

Operating frequency

Model

1 Hz–10 kHz 10 Hz–20 kHz 0–900 Hz

Digiducer 333D01 Integrated sensor Bartington 03CES100

DC to few kHz DC – 1 kHz – 50–1000 Hz 10–1000 Hz 1 – Hz 30–3000 Hz

Round, 37 mH, 5Ω Square, 70 mH, 7Ω PCB 086C01 Vibe-Tribe Troll Plus TIRA TV-51110 ELECTRO-SEIS APS 113 18-inch cone

Another way to modify ambient conditions is to generate artificial stimuli. The method is conventionally called noise injections. The concept is that of producing a controlled and localized increase of the detector ambient noise up to the point that a sizeable effect is produced in the strain signal. Excitation tools can be as simple as finger tapping to stimulate a mechanical excitation of a suspicious component. Gentle tapping is commonly performed on in-air optical mounts to search for scattered light, as discussed in Section “Scattered Light Hunting Methods”. More often synthetic stimuli are used. These are generated with commercial or custom devices like loudspeakers (sound injection), coils (magnetic injection), transmitting antennas (RF injection) and shakers (seismic injection). Table 3 lists a few models. There are different uses of synthetic stimuli. One use is to evidence non-linear noise coupling paths. If a pure sinusoidal stimulus is generated and its harmonics

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Fig. 6 Spectral peaks in the quiet strain signal (blue curve) are excited by shaking the vacuum enclosure of the Virgo output bench (red curve)

show up in the strain signal, it is evidence of non-linear coupling. Examples of non-linear noise paths are: (i) Jitter of the beam entering an optical cavity (Section “Beam Jitter Noise”) (ii) Beam clipping (iii) The Barkhausen effect in magnets (Section “Magnetic Field Influences”) (iv) Light backscattered from a vibrating surface (Section “Scattered Light Hunting Methods”) Injection of random noise (e.g. white noise) or frequency sweeps allows to probe the response of the interferometer at all frequencies without blind spots, thus evidencing if the strain signal gets particularly excited at some specific frequency. This could be the case of sound injections that enhance vibration modes of backscattering optics. Figure 6 illustrates one example. Sometimes it is convenient to concentrate the excitation power in a limited frequency region, for example, in the attempt to excite a specific frequency structure in the strain signal. In this case, the injection device is fed with a bandpass-filtered random noise.

Coupling Functions A principal use of noise injections is measuring the coupling of the interferometer, or parts of it, to the surrounding physical environment. The measured coupling function can then be used to estimate the contribution of the ambient noise to the strain signal and evaluate how critical it actually is. The method has been applied in several study contests concerning seismic, acoustic or EM noise [24–26, 63–66]. The experimental method consists in generating a set of stimuli that, for the duration of the measurement – typically one minute – overcome the noise in the undisturbed condition (also called quiet condition) both in the ambient and in the strain signals. The coupling function CF (f ) is estimated taking the ratio of the excess spectral noise in the strain signal and the designated environmental witness sensor (e.g. accelerometer, microphone, magnetometer). If no excess strain noise is generated, an upper limit is measured. Therefore:

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Fig. 7 Measurement of vibrational coupling function. The vibration is generated with a loudspeaker producing sound between 60 and 200 Hz near the interferometer output port. The left plot shows the LIGO LHO laser bench displacement measured by an accelerometer during quiet time (black) and injection time (orange). The middle plot shows the interferometer readout during quiet time (black) and injection time (orange). Estimated ambient levels for the accelerometer are also shown as dark blue dots, with upper limits shown as light blue crosses; they are produced from the coupling function in the right plot. The right plot shows the measured coupling function in metres of test mass displacement per metre of sensor displacement, hence the units of m/m

 CF (f ) =

|Yinj (f )|2 − |Ybkg (f )|2 |Xinj (f )|2 − |Xbkg (f )|2

(1)

where Xbkg (f ) and Xinj (f ) are the ASDs of the witness sensor at quiet and injection times, respectively, and Ybkg (f ) and Yinj (f ) are the ASDs of the strain. Then, the ambient noise contribution to the strain signal, named noise projection, is estimated as follows: Yproj (f ) = CF (f ) · Xbkg (f ). An example of measured coupling function and noise projection is shown in Fig. 7. Note that no coherence is requested between the two channels. For example, in the case of sound injections, a poor coherence is typically measured between the strain and the witness microphone because of the reverberating acoustic environment. Equation 1 relies on two assumptions about the coupling mechanism. First, coupling is assumed to be linear, that is, doubling the amplitude of the measured injected stimulus doubles the amplitude of the response in the GW strain signal. This can be verified by repeating injections with different amplitudes. Second, the sensor is assumed to be a good witness of the coupling. Different factors lead to an imperfect estimate of the coupling functions. One is the limited density on the sensor net; another is the practical difficulty to produce a homogeneous irradiation. To limit these effects, some precautions are taken: (i) Injection locations are chosen to best mimic disturbances from the ambient external to the detector. (ii) Injections are repeated from different locations, possibly at some distance from the detector (i.e. far-field condition) and anyhow at a distance greater than the average distance between witness sensors (typically some metres). (iii) As an estimate of coupling function, the average or conservatively the maximum shall be taken, while differences in measured coupling functions give an estimate of the associated uncertainty. Figure 8 illustrates a coupling function measurement set. A

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Fig. 8 Ambient noise projection for the LIGO HAM6 Y-axis accelerometer estimated from a composite coupling function, using acoustic and seismic injections near the output port

recent work [66] extends the method to estimate coupling functions in the more general case of non-linear noise paths. The method has been practiced with different kinds of physical stimuli (magnetic, radio frequency, acoustic, vibration) and using either sets of sine waves or bandlimited random noise or sweep sinus signals, with the intention of covering the interferometer full sensitivity range or some large portion of it [67–69]. Coupling functions are remeasured periodically or right before and after a noise mitigation action or major modifications of the detector, in order to validate the noise reduction or promptly spot unexpected worsening [26]. Up-to-date projections of estimated environmental noise and other detector technical noise components (e.g. laser frequency noise, control noise, photodiode shot noise, etc.) are combined to produce a noise budget, which is compared to the strain ASD. The noise budget is an essential tool during the detector commissioning in order to assess the consistency between the measured and the design detector sensitivity and identify and monitor any excess or unexplained noise [17, 70, 71].

Validation of Gravitational-Wave Events Gravitational-wave search techniques are optimized for the different types of GW signals mentioned in Section “Introduction”. Two techniques are used: modelled searches, for transient and continuous GW sources that match the data to template waveforms [72–74] and unmodelled searches that identify excess energy coherent between multiple detectors [75]. Environmental disturbances occur frequently around the detector and can occasionally be correlated between detectors, such as through electromagnetic signals from distant sources or glitches in GPSdisciplined electronics (Section “Electromagnetic Noise”). It is therefore plausible that environmental disturbances contaminate candidate astrophysical signals.

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The first observation of gravitational waves occurred on 14 September 2015 [1]. The event, a short-duration binary black hole merger, named GW150914, required a number of follow-up investigations to find potential noise sources around the time of the event [25]. This included an attentive examination of the status of all environmental sensors and any significant signal they observed [76]. The signal-tonoise ratios (SNR) of these signals were also compared to that of the event, showing that even if there were overlapping time-frequency paths, none of the environmental signals were large enough to influence the gravitational-wave channel at the SNR level of the event, based on the estimated ambient levels of the sensors. Among the scrutinized environmental disturbances were geophysical phenomena, such as solar activity, huge cosmic ray showers, Schumann resonances and one large lightning strike over Burkina Faso (at ≈10,000 km distance from LIGO detectors), which was coincident with the GW event [77]. A similar technique was employed for subsequent GW observations [78], but it has been partly automated in order to handle the increase detection rate. When an event is detected by the astrophysical search pipelines, a script automatically analyses environmental signals overlapping with the event duration window, producing estimates of their contribution to the gravitational-wave channel. These results are then used by human reviewers who advise on the data quality of the candidate GW event.

Seismic Noise Even at the quietest sites on Earth, the ground moves several orders of magnitude more than a test mass under the effect of a gravitational wave. Figure 9 shows average ground seismic spectra recorded at GW detector sites. Above 1 Hz displacement spectra approximately decay as the frequency squared: x(f ) ≈ A/f 2 . For Virgo and LIGO, A ∼ 5 · 10−8 m Hz3/2 , while it is about 100 times smaller for the underground site of KAGRA that is surrounded by at least 200 m of rocks. This occurs because seismic noise above 1 Hz is dominated by shallow waves generated by anthropogenic sources on the surface (mostly road traffic), whose amplitude is largely attenuated with depth. Seismic spectra of LIGO and Virgo are similar, but with a few interesting differences. Virgo seismic noise is more intense between 0.3 and 5 Hz. This is partly (0.3–1 Hz) due to the sea-induced microseism (Section “Sea and Wind”) and partly (1–5 Hz) to the presence of elevated highways at 1–2 km distance. Virgo seism is instead quieter than LIGO in the region 5–20 Hz. This is possibly explained by the deeper foundations of Virgo buildings, which sit on compact and quieter soil layers at 15 to 50 m depths. Above approximately 10 Hz, seismic noise is dominated by local sources, like on-site vehicles and laboratory facilities (Section “Sound and Vibrations”). Hereafter, we describe dominant sources in the seismic spectrum from very low frequencies up to 10 Hz. For each category, we discuss noise characteristics, influences on detectors and possible mitigation.

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Fig. 9 Amplitude spectral density of soil horizontal displacement at GW detector sites: Virgo, LIGO Livingston and KAGRA. Curves are 50th percentiles computed over 1 year. Data are recorded with seismometers deployed in experimental buildings. The KAGRA seismometer is positioned at the 2nd floor of X-arm end station. Data are compared with envelopes of seismic noise measured at the quietest sites on Earth [79]. Below 0.1 Hz, the Virgo seismic spectrum is dominated by tilt noise in high wind conditions, and electronic noise otherwise

Earth Crust Deformations At very low-frequency (1 mHz), Earth tides due to the gravity force from the Moon and Sun produce diurnal and semi-diurnal length changes of 150 − 200 μm between plumb lines at the two extremities of each 3 km arm. In Virgo, a feedback control loop is implemented at the level of the end TM suspensions which seeks to keep the arm optical cavities on resonance, by compensating for the Earth crust deformations [22]. The LVDT position sensors placed at the top stage of the suspension measure its displacement with respect to the soil, thus providing a monitor of the crust deformation. Consequently, as a matter of fact, GW interferometers also work as a pair of long base very accurate strain meters monitoring crust deformations along parallel and meridian directions. Reference [80] describes the use of Virgo as an extremely accurate deformograph (δL/L ≈ 10−14 /3000, where δL is the accuracy of interferometer arm length measurement and L is the arm length) for measuring geophysical signals, like oscillations of the Earth’s core at 10−6 to 10−4 Hz frequencies.

Earthquakes Distant earthquakes with a magnitude of 6.5 or greater occur at a rate of approximately one per week. Primary (P) longitudinal body waves (speed ≈6 km/s) arrive

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first followed by secondary (S) shear body waves (speed ≈3 km/s) and finally Rayleigh and Love surface waves [81]. The characteristic frequency of surface waves is in the range 10–50 mHz, and soil tremors larger than a few μm/s can last for more than 1 hour. The suspension control system can usually cope with P and S waves but hardly with surface waves, which require a too large correction and eventually saturate the actuators. In this occurrence, the position control of the test masses is lost and the ITF promptly loses the resonance condition. The controls cannot be re-engaged until the entire earthquake seismic wave has passed by and the oscillation of suspended optics cooled down. This can result in hours of ITF downtime. The adopted strategy relies on algorithms providing low-latency alerts of earthquake arrivals [82]. Upon the alert, a smooth transition is performed of the suspensions system control to a more robust version capable of surviving the seismic wave, although at the price of a slightly worse detector sensitivity [83, 84]. Figure 10 illustrates the effect of the impact of one earthquake on the Virgo detector. Earthquake shakes typically trigger scattered light noise paths in the interferometer.

Fig. 10 Earthquake of mag.7.2 from Papua New Guinea, 6 May 2019 21:19:35 UTC detected at Virgo. Between 2000 and 9000 s, before the arrival of the seismic wave, a more robust control of core optic suspensions is engaged, and the Virgo interferometer remains in operation. Yet, the soil motion induces scattered light noise in the interferometer causing a drop of sensitivity and observation range. (a) Vertical ground velocity signal. (b) Observation range of GW from binary neutron star coalescences. (c) Suspension control correction signal. (d) Spectrogram of GW strain

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Due to the non-linear nature of scattered light (Section “Scattered Light”), seismic wave frequencies are upconverted to frequencies in the 10–100 Hz range that are relevant for GW detection. Earthquakes of magnitude 3 to 5 with epicentre within some hundred km from the site also can affect interferometers [85]. These events can be frequent in proximity of fault lines or in areas subject to hydraulic fracking [85]. A possibility is to implement a fast alert notification within a few seconds from the event, using dedicated seismic arrays strategically located in order to intercept the EQ wave before reaching the detector site [86].

Sea and Wind In the absence of earthquakes, the next largest ground displacement noise is between approximately 50 mHz and 0.5 Hz, or periods between 20 s and 2 s. This Rayleigh surface seismic field consists of the so-called primary, secondary and tertiary microseismic peaks in the approximate range 0.05–0.1 Hz, 0.1–0.2 Hz and 0.2–0.4 Hz respectively, which arise from the interaction of ocean waves with the seabed [87, 88]. Microseism is ubiquitous, but its peak frequency and amplitude depend on the location on Earth and the sea state – storms or extreme meteorological conditions. In the same√location, microseismic peak amplitude can vary a lot, typically from 0.1 to 10 μm/ Hz with a timescale of a day or so. Also, microseismic peak amplitude does not reduce significantly within the first 200 m from surface. Indeed, as Fig. 9 shows, the microseismic peak is as intense at KAGRA as at LIGO and Virgo. The frequency of the most persistent peak is approximately 0.15 Hz at LIGO, 0.2 Hz at KAGRA and 0.4 Hz at Virgo. Microseismic spectral composition varies with season and sea conditions. At Virgo primary, secondary and even forth-order (0.8 Hz) microseismic peaks get excited in winter season and during swells [89]. It is also worth noting that, being the microseism wave speed ∼1–3 km/s (depending on soil rigidity), the differential ground motion over the ITF arms reduces significantly below 0.5 Hz due to the fact that the microseism wavelengths at these frequencies are 10 km or larger. This has implications in the definition of the seismic suspension control strategy. From approximately 1 mHz to 50 mHz seismic noise is dominated by the action of atmospheric pressure deforming the ground. Wind is the major cause. Air turbulence propagating at the wind velocity, v = 5 − 30 m/s, generates surface seismic waves, whose amplitude decays promptly with depth [90]. During calms, the main source of pressure fluctuations is infrasound [91]. To keep the interferometer locked and operating at its optimum point, the net differential motion of the suspended test masses in the two arm cavities needs to be very small, less than 10−14 m RMS. During periods of strong winds (e.g.

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v  15 m/s), keeping the interferometer locked becomes problematic [84, 92]. The reason is that wind gusts pushing onto walls of experimental buildings cause tilting of the floor cement slab, in a way uncorrelated among distant buildings. Moreover, LIGO found that the floor cement slab does not tilt as a unit. Instead, the tilting is mainly within metres of the beams that support the walls, consistent with an elastic dimpling of the ground around the support [93, 94]. The implemented solution at Virgo is named the global inverted pendulum control scheme – GIPC. This technique uses, instead of local position sensors, global virtual sensors that sense the differential motion among each pair of suspended optics [84, 95, 96]. On the other end, the strategy adopted by LIGO focuses on facing the issue related with ground tilt [92]. Tilt noise contaminates accelerometers readings and fools the control of seismic isolation platforms. In case of wind speeds above 10 m/s, induced tilts of the order √ of 10 nrad/ Hz completely contaminate seismometer readings below 100 mHz. LIGO demonstrated a significant improvement by adopting ground rotation sensors to subtract tilting noise from seismometers. The use of tilt-free inertial sensors in the seismic suspensions controls improved robustness for winds up to 20 m/s, significantly improving the duty cycle of the detectors [92]. In addition, LIGO H1 has recently constructed wind fences to divert the wind from LHO buildings [97]. Finally, wind can impact interferometers also as gravity gradient noise, as discussed in Section “Newtonian Noise from Air Density Fluctuations”.

Anthropogenic Seismic and Acoustic Sources Above 1 Hz human-driven seismic noise contributes significantly. This noise is mostly transient and non-stationary and exhibits a working-day cycle with quieter times during nights and festivities [99, 100]. LIGO Hanford experienced noise from water flowing over a dam [101, 102] and cooling fans from a nearby nuclear power plant [103]. LIGO Livingston experienced noise from train transits and tree cutting. At Virgo, seismic bursts peaked at 2–3 Hz correlate to heavy vehicle transits on viaducts exciting mechanical resonances of the structure [104, 105]. A comprehensive study of low-frequency (0.1 to 10 Hz) seismic sources external to the Virgo site is presented in [65]. Wind turbines generate seismic and acoustic low-frequency fields, which might travel considerable distances. This is the case of the wind park at 6 km from Virgo, where the 1.7 Hz vibration mode of the turbine tower couples to subsurface less attenuating soil layers [106]. LIGO has also been impacted by significant infrasound noise emissions at wind turbine’s rotor blade crossing frequency and harmonics, which are poorly attenuated in air [107]. Noise propagation models to evaluate the impact of proposed wind farm installations have been developed [106, 108, 109]. Anthropogenic seismic sources mainly generate surface Rayleigh waves. A rough model of amplitude attenuation of Rayleigh waves as a function of wave

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frequency (f ) and distance (r) from a source in a homogeneous half-space is [110]: A(r, f ) =

r A(0, f ) − πf e Qv . √ r

(2)

The wave phase velocity (v) and the seismic quality factor (Q) vary with soil composition. Typically, wave velocity is highly dependent on frequency, while the Q is not. This occurs because different wavelengths penetrate to different depths in the subsurface and travel through unequal soil types. For example, measured seismic wave velocity at the LIGO Hanford site is 500 m/s at about 5 Hz, but it is 150 m/s at 60 Hz. For the Virgo mud-clay composite surface soil, the values of Q = 30 and v = 200 m/s have been measured for wave frequencies of a few Hz [65]. These soil parameters can be extracted with geophysics exploration techniques, like conventional multichannel analysis of surface waves (MASW) methods [111] or passive methods using seismometer arrays [65, 105, 112]. Particularly large soil excitation at frequencies matching resonant modes of the seismic isolation systems can impair their performance. Most problematic are test mass suspensions, which have mechanical modes with high-quality factors (a few hundreds) in the range from 10 mHz to a few Hz. These modes are actively damped by means of feedback controls using a combination of LVDT relative position sensor below 100 mHz and accelerometer inertial sensors above [22]. Anthropogenic sources that are external to the detector site usually cannot be silenced. It is therefore important to carefully plan new detector site locations and adopt measurements to preserve the site noise climate. In Section “Site Selection Considerations for Minimizing Environmental Noise”, we discuss site selection criteria to minimize environmental noise. Extensive seismic characterization studies are being performed for site selection of the Einstein Telescope detector [113, 114].

Sound and Vibrations Suspension and vacuum systems remarkably insulate most part of the interferometer from the vibration and acoustic noise. Nevertheless, sound and vibrations have been so far the most insidious noise source in GW detectors. Vibration of optics that are in the path of the main beam or auxiliary beams can cause beam jitter, beam clipping or backscattering noise ending up in the strain signal. Vibration and sound are very closely related. They are usually co-generated, since a vibrating body induces pressure waves in the air medium. On the other hand, sound pressure waves induce vibrations of exposed mechanical surfaces, such as optics on in-air benches. The induced vibration can be considerably large in case of extended exposed surfaces (e.g. optical benches or vacuum enclosures) or wherever sound frequencies match mechanical modes of the object.

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In the following, we first describe the major sources of sound and vibration; second, we discuss vibration noise issues; and third, we illustrate how mitigation has been tackled in present detectors. We treat separately issues regarding different detector areas. Finally, we introduce the subject of beam jitter noise. The issue of backscattering is discussed in Section “Scattered Light”.

Sound and Vibration Sources At GW laboratories and close to experimental areas, the majority of sound and vibration disturbance is self-inflicted noise, being produced by the detector infrastructure apparatus. Figure 11 shows typical sound spectra. Most impacting sources are large ventilation fans (whose typical rotation frequency is in the range 10–30 Hz), water circulation pumps, air compressors, refrigerating systems for water and laser cooling, vacuum pumps. Motors driving these devices are typically of asynchronous type, and their operating frequency is at slightly less than the mains frequency. For example, in Virgo (50 Hz mains) the typical frequency of electric motors is ≈48 Hz, or an integer fraction scaling with the inverse of the number of motor poles. Ventilation systems produce 1–100 Hz broadband noise, from turbulent air flux inside the fan enclosure and ducts, and whistles or rattling sounds from air grids. Small cooling fans of electronic modules or vacuum pumps produce vibrations at the rotor’s frequency f and acoustic noise at pass-blade frequency nb f , with nb the number of fan blades [103]. Magnetostriction in power transformer ferromagnetic

Fig. 11 Amplitude spectral density of one microphone in the Virgo laser clean room. The red curve is the measured sound during a temporary switch off of the air conditioning system

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core produces vibration and sound at double the mains frequency. This noise usually has some harmonic content and it is also known as mains hum. Most of the sound and vibration noise generated outside of the experimental buildings are not audible inside. Above 10 Hz typically only sources within a few tens of metres from the ITF experimental buildings do matter. This is the case, for example, of on-site vehicles, which are required to stop during ITF operation. LIGO benefited from repaving near site roads eliminating bumps and holes [115]. However, loud and low-frequency sounds are poorly shielded by building walls and might be relevant. This is the case of aircraft transits, in particular helicopters and propeller airplanes. They generate pressure waves, which induce vibration of the soil and of the building structure. Aircraft events are of easy identification in microphones and vibration sensors because of the typical Doppler-sweep signature peaked at the rotor frequency and harmonics (10–100 Hz) [116, 117]. To reduce the impact of aircraft noise, the Virgo laboratory stipulated a no-fly agreement (NOTAM) within cylindrical volumes of 600 m radius and height around the three ITF vertexes.

Vibration Noise Reduction With the aim of giving useful indications for the design of future detector infrastructures, here we outline the main vibration noise problems experienced and the mitigation solutions implemented. We address issues related to different parts of the detector, namely, the vacuum pumping system, the cooling and air conditioning system, the clean room areas and the optical benches.

Vacuum Pumping System Current interferometers operate within ≈10,000 m3 volume of ultra-high vacuum. A residual gas pressure level of ≤10−9 mbar is necessary in order to minimize pressure noise on test masses from impacting gas molecules and phase noise associated with gas density fluctuations along the beam path. More stringent requirements apply to other gas components, such as hydrocarbons, which contaminate optical surfaces and need to maintain a partial pressure ≤10−13 mbar. To reach and maintain the required residual gas levels, a complex system of vacuum pumps, valves and monitoring gauges is operated [3, 118]. Dry mechanical pumps (e.g. scroll type) backed up by turbo-molecular pumps are used for achieving the initial high vacuum and for single chamber evacuation operations, while cryogenic pumps, ion pumps and getter pumps are continuously operated. Virgo has a higher rate of contamination than LIGO in the suspension chambers and needs also a few scroll pumps and turbo-molecular pumps in continuous operation. All vacuum devices that have moving mechanical parts must be carefully evaluated and seismically decoupled from vacuum chambers and sensitive apparatus. The Virgo scroll pumps are remotely located inside soundproofed rooms, seismically isolated and connected to vacuum chambers through ≈30 m long flexible steel pipes. Virgo turbo-molecular pumps, which need to stay in proximity of the chamber

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to maximize geometrical acceptance, are seismically isolated from the floor and connected to the vacuum chamber though soft edge-welded steel bellows. On the other hand, ion pumps are intrinsically vibration-free but have been a concern for causing electrostatic charging of test masses; see Section “Charging and Discharging Processes”. Passive cryogenic pumps, also known as cryogenic traps, are implemented at Virgo and LIGO to protect the beam tube from the hydrocarbon loads introduced by the suspended optics [3, 118]. A cryogenic trap consists of a donut-shaped tank (cryostat) filled with liquid nitrogen inserted in a few metre-long sections at the extremities of the arm tubes. The cold inner surface (≈10 m2 at 77 K) allows condensation of water and hydrocarbon molecules. The boiling of the liquid inside the cryostat or along the LN2 refill pipeline is a potential source of micromechanical vibrations which might offer a path to scattered light noise. The issue was observed in the initial Virgo [119]. To face this problem, the design of Virgo and LIGO cryogenic pumps featured the following: (i) Precautions to avoid heat concentration spots, like use of heat conductive material (aluminium to be preferred over steel) and large LN2 surface area inside the cryostat; (ii) Seismic isolation of the cryostat; (iii) Absorbing baffles to cover all beam-exposed surfaces. A downside of current cryogenic trap design is the need of periodic refill of the LN2 reservoirs which costs downtime and inflicts noise to the detector. A more practical solution are pulse tube cryocoolers, whose compressor units are intrinsically noisy. Cryocoolers are in use at KAGRA to refrigerate the core optics at 4 K. The implemented solution adopts a vibration reduction system [120].

Cooling and Climatization System Cooling fans of electronic modules emit sound and vibrations and should be avoided, preferring radiative or water cooling instead. In case cooling fans have to be adopted, vibrations are easily cut-off by putting racks on damped spring feet and inserting rubber spacers at fan modules. Mitigating acoustic noise from fans is also crucial, since it can efficiently convert into vibrations of vacuum enclosures or other large exposed surfaces. Noisy electronics should be confined in soundproofed rooms cooled down with mini-split units. Large heating, ventilation and air conditioning (HVAC) systems are remarkable sources of seismic and acoustic noise. Small ductless systems, like mini-splits, should be preferred where possible, for example, for cooling electronic rooms. Acoustic noise mainly arises from air turbulence generated at the fan outlet or inside ducts. An inlet calm plenum helps to slow down air flux and make it laminar. This solution is implemented at LIGO. Low airspeed ( λ/4π ≈ 10−7 m, being λ = 1064 nm the laser wavelength.

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Morphology of Scattered Light Noise Scattered light noise has distinctive spectral features that help to spot its presence and track the source. Often the scatterer surface has a dominant oscillation frequency fsc , for example, a high-Q mechanical mode, or a pick-up vibration from the environment, such as δx = Asin(2πfsc ). Depending on A and fsc , two scenarios arise. In the low-frequency and large-amplitude scenario, the scatterer motion is large, A λ/4π , and the oscillation frequency is 10 Hz. This is the case, for example, of light scattering off a suspended auxiliary bench or a TM mirror reaction mass [143,144]. In this case, as deducible from Eq. 5, the noise is strongly non-linear and can spoil the sensitivity at frequencies much larger than the scatterer frequency. A noise spectrogram shows the characteristic arch-shaped pattern of Fig. 13. To understand the scattered light arches shape, consider that the noise frequency is the rate of constructive superposition of the scattered beam with the main beam, which occurs every time the path length δx changes (increases or decreases) by λ/2. Therefore, the noise frequency as function of time scales with the instantaneous scatterer velocity, υ(t), as fnoise (t) = 2·|υ(t)| λ . An application of this model is illustrated in Fig. 13. The arch spacing is half of the scatterer oscillation period, while the arch height relates to the scatterer maximum velocity: fmax = 4Aπ λ fsc . Sometimes also a second or even a third order of arches is observed. This is explained by scattered light that, because of the presence of a second scattering (or reflective) surface along the beam path, covers the 2δx length two (or more) times before recoupling to the ITF beam. The same arches will appear in the ASD spectrum as a sort of noise shelves or “shoulders”. An example is illustrated in Section “Scattered Light Coupling Fig. 13 Spectrogram of strain signal with arches structures due to scattered light. The scattering source are optics placed on the external auxiliary bench looking at the beam transmitted from the West arm during the Virgo second science run. Superposed curves are computed from the velocity of the bench measured with a seismic c sensor. IOP Publishing. (Reproduced with permission from reference [134]. All rights reserved)

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Measurement”. This mechanism is manifested in the LIGO data, due to lowfrequency relative motion between the test mass and the reaction mass behind it. A small fraction of transmitted light was reflecting directly back into the arm cavity from the gold traces of the electrostatic drive on the reaction mass, or after one to several cycles of reflection between the traces and the back side of the highreflection coating on the test mass [143]. In the low-amplitude and high-frequency scenario, the scatterer oscillatory motion is much smaller than a wavelength, A  λ/4π , and its frequency falls in the detector bandwidth, fs ≥ 10 Hz. In this case a linear coupling would be expected. Instead, as shown in Fig. 14, the noise usually shows up as a broad spectral structure, namely a noise bump, centred at fsc . This broadening is the result of the low-frequency microseism adding to the scatter oscillatory motion. The more intense the sea microseism the broader will be the bump and the lesser the coherence between the scatterer displacement and the noise. Weak or no coherence is indeed a typical signature of non-linear noise, and this is also true for SL noise. In the intermediate case, as the amplitude of the scatterer motion increases, the noise shows up also at multiples of the scatterer motion, as illustrated in Fig. 15. As a general remark, weather conditions are usually a good way to discriminate scattered light from other sources of noise in the strain signal. In fact, the enhancement of microseismic peak during periods of more intense sea activity typically correlates with an increased frequency cut-off of scattering arches or with the broadening of SL noise bumps. As a matter of fact, slow modulations in the amplitude of SL noise can also be associated slow drifts in the ITF alignment, which can enhance scattering due to clipping of beam miscentred on optics.

Fig. 14 Example of scattered light bumps. An air fan produces vibrations at 33 and 41 Hz of the vacuum chamber enclosing the optical bench in transmission of the Virgo North arm. Light scattering off the chamber walls recouples to the main beam and gives rise to noise bumps centred at the same frequencies, which are detected by the photodiode placed on the bench. The noise reduces when the air fan responsible for the vibration is switched off (red curve). (a) Angular motion of auxiliary bench with respect to vacuum chamber. (b) Photodiode signal

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Fig. 15 Sinusoidal shaking of an optical bench. Measurement of scattered light coupling of the auxiliary bench in transmission of the West arm of the initial Virgo. Plots show the bench displacement spectrum (left) and the interferometer strain spectrum in the undisturbed case (cyan curves) and when one sinusoidal excitation is applied to the bench (black curves). The excess noise c produced in the strain signal is well matched by the scattered light model (red curve). (IOP Publishing. Reproduced with permission from reference [134]. All rights reserved)

Scattered Light Hunting Methods The hunt for SL has the double goal of identifying the scatterer and quantifying the noise impact on the interferometer. Investigation techniques are described in several documents [24, 27, 134, 145–147]. The general idea is to perform controlled actions to enhance or reduce the vibration noise of the scattering surface while observing the effect in the strain signal. Below we review the most common techniques.

Inspection and Tapping Visual inspection of stray beams with infrared (IR) viewers or view cards is the first necessary step, but it is limited to accessible volumes, like optical benches not enclosed in vacuum. To this purpose, it is convenient to instrument vacuum chambers with external IR movable cameras for wide-angle inspection. An effective technique to identify critical scatterers on accessible optical benches consists in softly physically tapping the individual optics or optical components, thus enhancing their motion and possible scattering while observing for a correlated noise increase in the strain signal or, more effectively, listening to the ITF output signal through a headphone set. However, note that tapping is poorly controlled and is best used in conjunction with an accelerometer to monitor motion. This is because the motion can be increased so dramatically by tapping or pushing that there appears to be a scattering problem when the nominal noise is actually far below the strain noise floor.

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Contextually, it might prove useful to exclude/attenuate sections of the beam path interposing black glass absorbers or neutral density filters and look for a reduction of noise [145, 146].

Noise Injections If the scatterer is located inside an inaccessible volume (e.g. scattering off vacuum chamber walls, residual scattering from in-vacuum baffles or viewports), the investigation is more troublesome. The applied technique is one instance of the noise injection described in Section “Experimental Techniques”. A controlled external stimulus is generated attempting to selectively stimulate a vibration of the scatter surface and to observe an enhancement of the target noise in the strain signal. Various locations are tested in this way till the target noise gets enhanced. One example is shown in Fig. 6. Different tools are used; some are listed in Table 3. Small electromagnetic or piezoelectric shakers, weighted carts (for very low frequency) and impulse hammers are used to generate substantial noise, continuous or impulsive, in a wide-frequency range (1–1000 Hz) mimicking the noise characteristics of ambient sources. Large weighted carts resting on the floor are used for excitation of wide areas or vacuum chambers. Small shakers can perform more localized injections when attached to benches, beam-pipe sections or viewports. Some parts of vacuum enclosures, for example, viewports, restrictions in pipes and bellow-shaped structures used to join vacuum pipe sections, if not properly protected are prone to scattering and should be investigated more thoroughly, for example, by shaking or softly tapping them with an impact hammer. Usually, localizing scattering sources connected to the vacuum enclosures is the most troublesome. In fact, a mechanical excitation spreads widely over the entire rigidly interconnected system. Injection techniques appropriate to this contest are described in [24]. They rely on the slow propagation speeds (few hundred of metres per second) of vibrations in the steel vacuum enclosure walls, or for acoustic injections, in air. Methods consist in exciting vibrations of the chamber steel wall. One excitation technique consists in directly striking the chamber with impulses and looking for a match between arrival times in the strain signal and in accelerometers. A second technique consists in generating a beating of two sinusoidal excitation slightly shifted in frequency respect to the target noise, with a pair of speakers or shakers. Subsequently, the noise produced in the strain signal is compared with the seismic disturbance produced in a net of seismic sensors (accelerometers) positioned over the chamber. Locations with sensor beat envelopes that do not match the phase of the beat envelope in the strain signal can be excluded being dominant coupling sites. Figure 16 shows one application of this technique. Switch-Off Tests Scattered light is often amplified by machines that produce intense low-frequency sound and vibrations, like air ventilation and conditioning systems (see Section “Sound and Vibration Sources”). Temporarily switching them off, while

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Fig. 16 Spectrograms of a beat injection using two shakers to localize the coupling site responsible for a 48 Hz peak in the differential arm length signal (DARM) of LIGO Hanford (visible in Fig. 2). This set of spectrograms suggests that the accelerometers on the input test mass (ITM) chambers and the Y-axis HAM2 accelerometer are not close to the true coupling location, since the beat envelopes are the furthest offset from the beat envelope in the DARM response. Multiple other injections were made (not shown here) with varying shaker locations in order to rule out other sensors until the most likely candidate remaining was the HAM3 Y-axis accelerometer. Black glass was used to block scattered light at this location, and the peak was eliminated for the second half of the O3 observation run.

observing a reduction of residual strain noise, provides a first rough localization of the scattering source and then to be pursued with the more specific methods illustrated before [26]. Figure 14 shows one switch-off experiment.

Scattered Light Coupling Measurement Coupling functions have been introduced in Section “Coupling Functions” together with experimental methods. Similarly for SL noise, techniques have been devised to measure the amount of scattered light coupling from a specific scatterer to the strain signal. The measured coupling is used to estimate the contribution of the ambient noise to the strain signal (also referred to as noise projection) and evaluate how critical it actually is. The method relies on the model in Eq. 5. The coupling function Tφ0 (f ) depends from the source position inside the interferometer, and it is evaluated with an optical model of the interferometer, like Optickle [148]. The only free parameter in Eq. 5 is the fraction of recoupling backscattered light. To measure it, a sinusoidal excitation is applied to the scatterer while measuring its motion and the induced strain noise, respectively, x(t) and hSL (t) in Eq. 5. The free parameter is derived from a least

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Fig. 17 Example of bench SL noise coupling measurement and projection. The left plot illustrates the coupling function measurement for the auxiliary optical bench in transmission of Advanced Virgo North arm. The undisturbed strain ASD (blue curve) is compared with the strain ASD, while a 0.1 Hz sinusoidal excitation is applied to the bench and noise shelves arise (red curve). The yellow curve is the sum of the reference and the scattered light model fitted to the injection data. The right plot shows the undisturbed strain ASD (blue curve) and the noise contribution from the bench scattered light (green curve). This projection is computed by applying the fitted model to the undisturbed bench motion. (Courtesy of Michal Was, ˛ [email protected])

squares fit to the model. The measurement requires the following: (i) a good sensing of the source motion and (ii) a sufficiently large excitation able of increasing the scattered light noise to a level detectable in the strain signal. Figure 15 shows an example of this technique applied to the Virgo laser bench [134]. The actuation/sensing is performed with a shaker/seismometer placed in contact with the bench. The frequency of the sinusoidal excitation is chosen such that the bench and the optics on it move as a whole. A good choice is matching the frequency of one horizontal mechanical mode of the table. The technique has been applied also to vacuum chambers [147]. In this case, however, the excitation uniformity and sensing of the effective source motion are a possible limitation. The method works quite well for suspended optical benches [144]. In this case, actuators and sensors of the bench position control system are conveniently used to perform the bench excitation and sense its motion. Figure 17 shows one example of the noise injection, the fit model and the subsequent noise projection, which gives an estimate of the contribution of the light scattered off the bench to the residual strain noise. In Virgo the measurement is repeated routinely for all suspended benches or performed after each SL mitigation interventions on benches, in order to monitor improvements.

Scattered Light Mitigation Mitigating SL noise requires either reducing the number of scattered photons recoupling to the interferometer or reducing the amplitude of the scatterer motion, or both (Eq. 5 in section “Scattered Light Noise Model”).

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To reduce vibrations, most of the benches carrying input, output or auxiliary optics of the current generation of interferometers have been seismically isolated [123] and enclosed in light vacuum (∼10−3 mbar, Virgo) or full vacuum (LIGO). In Virgo an active tracking control is implemented for each suspended bench in order to minimize the residual low-frequency relative motion between the bench and the interferometer beam [149]. In LIGO, a similar tracking control was implemented between test masses and reaction masses to reduce scattered light involving multiple reflections between the gold electrostatic drivers on the reaction mass and the highreflectivity coating of the TM [17, 143]. Light might also scatter off vacuum enclosure walls, which vibrate considerably (Section “Sound and Vibrations”). A system of photoabsorbing panels, referred to as baffles, is strategically placed inside the interferometer vacuum enclosure with the purpose of blocking the view of the walls to all light in the main beam and secondary beams. Circular baffles are positioned along the arm tubes, at the inner walls around the suspended optics, around the suspended mirrors to shield the payload structure and in particular in front of any metallic surface/edge perpendicular to the beam (e.g. bellow joints). Baffles are usually made of silicon carbide (SiC, expensive but with higher damage threshold – ∼0.5 kW/cm2 ) or stainless steel plates coated with diamond-like carbon (DLC) topped with an AR layer [150]. The baffle geometrical design and positioning requires an accurate computation aided by optical simulation tools to trace and estimate the power of all secondary beams [136]. Residual reflections from imperfect baffle surface can be problematic because of the large light power usually impinging on it. In order to reduce baffle velocity, implemented solutions are [151]: (i) damped baffle support frames featuring UHV compatible elastomeric spacers to damp baffle internal modes, (ii) suspended baffles with eddy current dampers and (iii) voice-coil actuators to reduce scatterer velocity. Optical benches need extreme attention, because of the presence of multiple beams in a usually crowded space. A list of good practice rules to reduce vibration and the amount of scattered light follows [133, 137, 145]. • Adopt stiff optical mounts, with the goal to move mechanical modes well above 500 Hz and out of the interferometer best sensitivity region. Also, the lower is the height of the optical plane, the shorter and stiffer are the mounts. If necessary, tuned dampers, described in Section “Tuned Damper”, can be used to reduce the quality factor of particularly nasty mechanical resonances. • Avoid beam clipping on optical mounts by adopting large aperture optics whenever possible, or if needed use diaphragm baffles to hide the mount frame. • Use large beam waist and put optics as far as possible from the waist location. • Avoid back reflections by removing windows from photodiodes and set lenses at an angle. • Identify and properly block reflections and ghost beams using diaphragms and beam dumps. • Use high-quality optical surfaces: low roughness, low scratch and dig specifications, high-quality coatings. • Avoid dust deposition on the optics by working in a clean environment.

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• Use Faraday insolators to reduce back-reflected light. • Reduce bench vibration noise (Section “Optical Benches”). • Implement monitoring of vibration of critical parts, such as optical benches, viewports and vacuum chamber walls. This is discussed in Section “Environment Monitoring”.

Electromagnetic Noise The ambient magnetic noise measured at the Virgo site is illustrated in Fig. 18. LIGO and KAGRA are similar. Magnetic spectra measured inside experimental areas consist of a broadband noise, topped by several spectral peaks. The most prominent are persistent narrow peaks at the mains frequency, fmains = 50 Hz (60 Hz at LIGO and KAGRA) and harmonics. This noise is the sum of stray fields radiated by countless devices and cables that are part of the detector apparatus. Noise measured outside experimental buildings is about a hundred times quieter, to the point that the magnetic noise associated with Schumann’s resonances of the Earth magnetic field can be detected. There are two main ways in which electromagnetic (EM) noise affects a GW detector: (i) EM noise generated and coupling to the electronic apparatus and (ii) ambient magnetic and electric fields acting directly onto the TM mirrors through the

Fig. 18 Amplitude spectral densities of indoor (blue curve) and outdoor (red curve) magnetometers at the Virgo site and at Sos Enattos mine in Sardinia (black curve). They are sampled at 20 kHz, 2 kHz and 250 Hz, respectively. The quiet Sos Enattos location shows evidence of the Schumann resonant modes peaked at approximately 8, 14, 21, 27 and 33 Hz. Magnetic noise inside experimental areas (indoor) is mainly due to stray fields radiated from electric loads and cables circulating large currents, for example, HVAC and power supplies. The intense peak at 50 Hz is the magnetic noise at the mains frequency

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magnet coil or electrostatic actuators. In the following, we address in turn the two categories. We conclude this section with a brief discussion on possible influences of cosmic radiation.

EM Noise Coupling to Electronics In this section, we discuss EM disturbances that couple to the interferometer electronic infrastructure. Most of this noise is the so-called self-inflicted noise, meaning that it is generated within the detector laboratory itself. We illustrate the most relevant sources and coupling paths we have encountered, typical noise footprints observed in the detector and implemented or desirable mitigation solutions. Low-noise electronics used in GW experiments are prone to electromagnetic interference. Particularly sensitive is the photodiode readout electronics. RF noise within 10 kHz from the laser modulation frequencies (in Virgo: 6, 8 and 56 MHz) might couple to RF photodiode analogue readout. Upon demodulation, the noise gets downconverted into the detection band and reintroduced through alignment and longitudinal controls. RF injections at LIGO have also shown some sensitivity at 10 MHz, the frequency of the signal used to synchronize electronic equipment via a phase-locked loop technique. Intense RF disturbances might cause saturation of some active component, like integrated circuits in low-noise and high-gain electronic boards. Saturated signals manifest as upconverted noise. Beating effects are also possible, for example, if noise adds to the control signal of resonant optical cavities (e.g. OMC), which have quadratic response [28, 152]. EM noise coupling can occur: • By radiation: cables acting as radio antennae emitters and receivers. This coupling is more efficient in the 100 kHz–100 MHz range, that is, for wavelengths comparable to cable length. • By capacitive or inductive coupling between close-by cables (crosstalk). • By conduction, for example, when two or more currents share a common path. This common path is often a high-impedance ground connection. If two circuits share this path, noise currents from one will produce noise voltages in the other. A vast knowledge exists on the subject of improving the level of EM immunity of sensitive devices against external offending signals, though the focus is often on radio frequency (RF) noise rather than the audio frequency noise that is also important to GW detectors [153–156]. Hereafter, we briefly introduce the most used techniques. A good level of radio frequency immunity of sensitive devices can be achieved with the use of shielding enclosures. Good shielding enclosures have the longest aperture shorter than λ/20 of the highest frequency used by the contained electronics. Even the best shielding enclosure is useless if the noise propagates through

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cables. The use of appropriate filters (e.g. feed-through capacitors) in all connections (power, signals) is mandatory on all inlet and outlet cables at the shield level. Capacitive coupling can be reduced by adopting a cable segregation policy: that is, using separate cable trays for different cable categories, like digital communication cables, power cables and cables carrying low-noise analogue signals. Inductive coupling can be reduced not only by separating conductors carrying incompatible signals but also by twisting pairs of wires instead of running them in parallel. To reduce crosstalk between electronics devices, one technique consists in using differential transmitters and receivers along shielded twisted pair (STP) cables. A differential line is by definition insensitive to the ground problems often encountered with the single connection, in particular when the electronics to be connected are located in different places in the GW experimental area. Furthermore, actions can be applied to the single source or category of sources in order to reduce EM emission or coupling. Hereafter, we examine the main categories of EM sources, which are part of the electronic apparatus of a GW detector.

Noise from the Utility Mains GW sites are powered through the utility mains provided by the local power supplier that in Europe means the general purpose alternating current (AC) 50 Hz power supply (60 Hz in the USA and at KAGRA in Japan). External AC medium voltage (MV) supply is converted on site to three-phase low voltage (LV) AC and distributed to the experimental areas. This is the experiment power grid. The electrical ground of the experimental grid connection provides an equipotential reference to all infrastructure devices and electronics. At Virgo the external supplied voltage is generally clean, with low harmonic distortion and low inter-harmonic noise. At LIGO, the mains voltage and especially the magnetic fields from the currents in the mains can be quite glitchy as external equipment turns on and off, or if the power company inserts or removes the huge capacitors that regulate frequency changes during high use periods [157,158]. Noise arises in the experiment power grid because of reactive or non-linear loads, such as personal computer power supplies or inverter motor controllers. The noisy current eventually results in magnetic fields radiated by cables and devices. Feedforward (FF) noise cancellation schemes have been successfully implemented to tackle mains noise, for example, to minimize residual 50 or 60 Hz noise leaking into Virgo and LIGO strain signals [26]. Successful FF requires a stable noise coupling function and a reliable witness sensor. For example, in Virgo the 50 Hz cancellation uses as sensor the voltage monitor of the UPS of the central building and actuates correcting the longitudinal position of one test mass. FF technique has had several successful applications in GW experiments to reduce contamination of magnetic and seismic noise. For example, feedforward on actuators of one OMC steering mirror using a local magnetic probe removed 60 Hz sideband noise that worsened the Crab Pulsar sensitivity in LIGO during the 6th science run [131] (see also Section “Beam Jitter Noise”).

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Uninterruptible Power Supplies As it is normally done in critical plants where no interruptions or blackouts can be tolerated (e.g. hospitals), in Virgo the AC power supply is generated locally to power the sensitive electronic equipment of the experiment through the use of uninterruptible power supplies (UPS). LIGO and KAGRA, instead, are directly connected to the local electricity grid. UPS units guarantee electrical continuity to devices in case of interruptions of the utility mains. Online double-conversion UPS provides the highest level of protection by isolating the equipment from the raw utility power. UPS devices are not intrinsically clean. Depending on the adopted technology, the synthesized sine wave can have a larger harmonic distortion and higher interharmonic noise than the standard external AC power grid. In this occurrence, the noisy current spreads in the experiment power grid resulting, because of cable EM radiation, in a noisier ambient magnetic field [159]. An issue with the experimental AC power supply is that, if the UPS (or local electricity grid in case of LIGO) cannot keep a constant voltage as currents to electric equipment turn on and off, the voltage fluctuations may couple to the interferometer through sensitive electronics that share the same power grid. For example, LIGO found voltage noise from other electronic equipment coupling into the test mass electronic drivers through the common power supply [28]. The solution is to use smaller size dedicated UPS units to supply power to sensitive devices only.

Switching Devices Often, the mains signal is paired by symmetric spectral components (sidebands). Figure 19 illustrates sidebands of the 50 Hz mains detected by one magnetometer. Sidebands originate from the operation of devices with a large variable electric load. They produce an amplitude modulation of the electric power voltage of the kind: V (t) = V0 g(t) sin(2π fmains ), where g(t) describes the change due to the electric load. Devices of this type are, for example, electric heaters, which use the pulse width modulation (PWM) technique to regulate the delivered power. The mains supply is “cut” with electronic on/off switches at a constant rate, fP W M , and the duty cycle is modulated to guarantee the required average output power. In this case g(t) is a square function. The beating with the mains sine wave produces spectral sidebands at ±nfP W M from fmains , where n is an integer number. One way to mitigate disturbances from PWM devices is to reduce the amount of regulated power to the controlled device by supplying it a constant lower voltage varying only the remaining needed power with a PWM technique. Further mitigation actions are as follows: (i) use of mains twisted cable that help minimizing both the susceptibility to and emission of magnetic fields and (ii) power both the offending and the sensitive devices with separate dedicated UPS. Another type of switching devices are engines with on/off cycle, like air compressors and water chillers. At each switch on, an inrush current of the order of hundreds of A is produced, which gives rise to loud magnetic glitches. Inrush current

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Fig. 19 A week long time-frequency plot of one magnetometer in the Virgo central experimental hall. Sidebands of the 50 Hz mains are visible at many frequencies, with prominent line pairs at 40.5 and 59.5 Hz, 42 and 58 Hz, 47.6 and 52.4 Hz. Some disappear because of the switch off of offending devices. (Courtesy of Didier Verkindt, [email protected])

limiter devices (e.g. negative coefficient resistors or, better, “smart inverters”) help to dilute the current surge reducing instant peak values.

Power Supplies AC-DC power converters, commonly named power supplies, are widely used inside laboratories. There are two categories of AC-DC converters: (i) voltage step-down linear power supply and (ii) switching mode power supplies. Linear step-down power supplies are known to be quieter by an electromagnetic compatibility (EMC) point of view, especially if a toroidal-shaped core for the reduction of stray magnetic field is used. Yet, they are heavy and not efficient for applications needing large amounts of power. They also produce vibration by magnetostriction at double the mains frequency. Switching mode power supplies (SMPS) instead are small, very efficient and widely used in today’s commercial electronic use. But they are also noisy from the point of view of conducted and radiated noise [160]. SMPS devices can induce common-mode noise, extending up to hundreds of kHz, over the entire length of the cable, thus being very efficient emitters [160]. Typically, a large number of SMPS devices is present within experimental areas. Probably less known is that SMPS are used in neon tubes and LED illumination systems. The suggested approach is to switch the lights on/off while observing for appearing/disappearing noise in the RF ambient probe monitors: noisy devices are usually well identifiable.

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Tips to reduce influences from power supplies are: • Feed the front-end electronics with a DC power line from a remote power supply, equipped with remote sensing to guarantee supplied voltage levels. • Avoid placing power supplies (both transformer or switching type) nearby sensitive parts of the experiment whenever possible. • Ferrite beads clamped at both ends of the cables connected to the emitting device help reducing both radiated and recoupling EM fields. • Avoid sharing power lines between the experimental equipment and heavy and “dirty” loads, such as electric heaters, chillers and air conditioners. Instead, consider implementing a “clean” UPS power line dedicated to sensitive devices. A major source of peaks in the LIGO sensitivity spectra has been small but periodic fluctuations in the voltages from regulated DC power supplies, due to periodic variations in the current needed by the equipment that they power. The most common sources of the periodic current variations that produce these voltage fluctuations have been electronics cooling fans [103] and periodically flashing LEDs [28]. The length of an optical cavity may vary with the flashing LED or fan frequencies, if electronic equipment that is controlling the optical cavity length shares a power supply, or sometimes just shares a ground, with the equipment that has the fan or flashing LED. The solution at LIGO has been to improve internal on-board voltage regulation in the sensitive electronics or to power them with their own dedicated DC power supplies. For example, LIGO has moved towards having individual dedicated power supplies for the TM electrostatic drives. LIGO has also eliminated many unnecessary cooling fans and flashing LEDs in order to reduce the voltage regulation requirements.

Digital Devices Digital devices, like microprocessors, programmable communication devices (PLC), ethernet cables, wireless repeaters and mobile telephones, are widely used inside experimental buildings. These devices communicate by exchanging sequences of high-low voltage transition levels timed by clocks. Magnetic noise radiated from devices themselves or from interconnecting signal wires has the spectral footprint of a comb, made of a sequence of frequency peaks equally spaced by 1/T , being T the clock period (Fig. 20). Depending on the stability of the clock, that can span from the poor level of a simple RC oscillator to the impressive level of a GPS disciplined oscillator, comb teeth can be broad or extremely narrow [161]. The ultra-stable disciplined combs, if matched in frequency and leaking in the strain data, can become a coherent noise among different interferometers and an issue for continuous wave searches [28]. Moreover, digital systems’ logic levels (square waves) can produce high-order harmonics, up to hundreds of kHz, of the fundamental communication frequency. One of these harmonics may fall close to laser phase-modulation frequencies and leak into the RF photodiodes and eventually into the strain signal [62].

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Fig. 20 Narrow spectral lines resulting from the superposition (beating) of two square waves with very close frequencies (f1 = 27.778 Hz and |f2 − f1 | = 6.4 · 10−3 Hz). This simulation reproduces the time-frequency pattern of lines observed in the Virgo strain signal, equally spaced by 55.56 Hz. The source was identified in the closed-loop picomotor drivers of the dark fringe mode matching telescope, which was then kept off during data taking. (Courtesy of Francesco Di Renzo, [email protected])

A common practice to reduce emission and recoupling of EM low-frequency (up to a few hundred kHz) fields is using shielded twisted pair (STP) cables to connect devices. At higher frequencies, an effective solution are common-mode filters (e.g. ferrite beads) placed at both cable ends [162]. Where applicable, optical fibres are the optimal interconnection media for digital communications being intrinsically noise emission-free.

External Sources of RF Noise Experimental buildings construction with well-grounded rebar (LIGO) or electrically connected metal cladding (Virgo) allow to screen external RF fields up to some MHz. A 40 dB attenuation at 6 MHz has been measured for Virgo buildings. However, occasional intense external sources might leak through. Solar EM emissions (solar radio flares and currents of charged particles associated with the solar wind) and radio broadcasts are potential sources, but no effect has been reported so far. In particular, solar emissions were examined in the validation process of the GW150914 event [25]. A significant occasional source is lightning. Lightning flashes have a broad EM spectrum, which extends in the MHz range [163]. EM disturbances at these frequencies could get demodulated into the detection band for optic cavity control and are monitored by magnetometers and RF receivers. No effect on present detectors has been reported from lightnings that are more than a few hundred kilometres far (with the exception of the gigantic jet events discussed in Section “Global Magnetic

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Noise”). The noted impact from closer lightnings, but farther than a few tens of km, is the vibration induced by the sonic and seismic shock wave. Overall, no disturbances from external RF emissions have been reported so far. However, LIGO and Virgo periodically measure the coupling function of the ITF to ambient RF fields around the modulation frequencies of the laser in use, in order to monitor the level of immunity [67] (Section “Methods for Investigating Environmental Noise”).

Magnetic and Electric Fields Coupling to Test Masses In this section, we examine magnetic and electric fields that couple noise to GW detectors by acting directly onto the TM mirrors through the built-in magnet coil (Virgo) or electrostatic (LIGO) actuators. This coupling path bypasses the mirror seismic isolation chain and can be effective for frequencies below ≈100 Hz, the so-called extremely low-frequency (ELF) region of the EM spectrum.

Global Magnetic Noise A specificity of EM disturbances is that they can produce correlated noise among distant interferometers. This correlated noise can be an important background for searching for GW signals. There are two types of disturbances to consider. The first type are intense EM pulses of few ms duration, produced, for example, by energetic electric discharges in the upper atmosphere, such as sprites and gigantic jets. These pulses, travelling at the speed of light like gravitational waves, can be detected by multiple interferometers with time delays compatible with gravitationalwave transients and limit the sensitivity to GW signals correlated over multiple detectors, like signals from coalescing binary systems. One such event was detected by LIGO and Virgo [164]. Witness magnetometers are used for vetting these EM noise transients (Section “Validation of Gravitational-Wave Events”). The second type of disturbance are persistent magnetic fields that extend over the entire planet. Geomagnetic phenomena, such as the Schumann resonances [9, 10] are a potential contamination of searches for a stochastic gravitational-wave background. These searches reach high strain sensitivities looking for a signal correlated between sites, integrating over multi-month periods. Schumann resonances are damped modes of the Earth-ionosphere waveguide excited by worldwide lightning activity. They are detected by magnetometers placed in anthropically quiet locations as broad peaks at approximately 8, 14, 21, 27, 33 and 39 Hz dictated by the light travel time around the Earth [9, 163]. As illustrated in Fig. 18, their typical intensity is in the tenths of pico-Tesla and follows daily and seasonal variations related to changes in the ionosphere height and the worldwide lightning activity. Pioneering measurements performed at KAGRA indicate that the Schumann field is not attenuated, and might even be enhanced, underground as consequence of the rocks magnetic permeability [165]. Coherent magnetic noise among distant GW sites has been measured at frequencies corresponding to Schumann resonances [166]. Techniques based on Wiener

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filtering are proposed to subtract it using magnetic sensors as local witnesses [11, 167]. Recent studies demonstrated that coherent magnetic noise between distant detectors extends well above Schumann frequencies, up to some hundreds of Hz. This higher-frequency component is due to correlated noise between energetic lightnings.

Magnetic Field Influences Ambient magnetic fields can exert a force on TM mirrors by acting on the actuation magnets attached onto the mirrors surface or at upper stages of their suspension chain. These forces displace the test masses and thus produce a noise in the strain signal. Both magnetic fields and magnetic gradients contribute, by producing torques (thus mirror rotations) and forces (thus mirror translations), respectively [168]. The magnetic coupling model predicts a strain noise that, in case of the action of μz z field gradients, behaves as h˜ mag ≈ δB δz (L Mf 3 ) , where f is frequency, M the mirror mass, L the arm length and Bz and μz are the ambient magnetic field and magnetic moment of the magnet, respectively, along the beam direction. This model includes a 1/f factor describing the shielding action of the vacuum chamber, which is due to the frequency-dependent eddy currents in the chamber steel walls (skin effect). The magnetic transfer function of Virgo 304L-steel vacuum chambers was measured to behave as a single pole with a few Hz cut-off frequency [64]. Because of the steep frequency decay, the magnetic noise contribution to the interferometer sensitivity becomes relevant for frequencies below ≈100 Hz. This noise affected the sensitivity of both the initial LIGO and initial Virgo detectors, despite the fact that the four actuation magnets attached to each TM were arranged in opposite direction configuration to minimize the total magnetic moment. Virgo mitigated the noise by using smaller magnets and by reducing the ambient magnetic noise level in the proximity of TM vacuum chambers. At frequency below 100 Hz or so, magnetic field inside experimental areas is mainly radiated from electric loads and cables circulating large currents, for example, HVAC and power supplies. The mitigation action consisted in moving devices and rerouting electrical cables at further distance from test masses. The coupling function between the ambient magnetic noise and the interferometer is evaluated periodically at each detector site with the method described in Section “Coupling Functions”. The most recent measured coupling function is similar, within a factor of a few, LIGO and Virgo [25, 64]. Its extrapolation to the Einstein Telescope [169] (shown in Fig. 21) indicates that ambient magnetic noise is a plausible limiting factor for the future interferometers. To reduce coupling to ambient magnetic fields, LIGO replaced TM magnetic actuators with electrostatic actuators (Section “Electric Field Influences”). The residual magnetic coupling measured at LIGO is probably occurring on actuation magnets placed at higher stages of the TM suspension. Further couplings can occur to actuation magnets used for the control of auxiliary benches [26], or possibly

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Fig. 21 Sensitivity model of the Einstein Telescope detector and estimated noise from the environmental sources. Dashed lines indicate the noise levels without the required additional noise mitigation (factor 3 in all three cases), for example, by noise cancellation. It is assumed that the detector depth is 300 m. (Reproduced from Reference [169], with the permission of AIP Publishing)

to permanent magnets used in the Faraday isolators sitting onto input and output benches. Strategies for suppressing this noise include: • Reduce ambient magnetic fields especially in proximity of vacuum chambers. • Reduce the number and size of permanent magnets sitting onto the last stages of suspended optics and benches. • Avoid the use of high conductivity materials (e.g. aluminium and metals in general) for structural parts in proximity of actuation magnets. In fact, the interaction between the time-varying external field and electrically conductive materials generates eddy currents in the metal, which in turn warp the field and produce a gradient. • Passive shielding using materials with high magnetic permeability is feasible for small volumes. Instead, for the large volumes of TM vacuum chambers noise control based on Helmholtz coils [64] looks promising. • Techniques similar to those adopted for Newtonian Noise cancellation (Section “Gravity Gradient Noise”) are being considered for subtracting the Schumann noise correlated among different detectors [11, 170]. At frequencies above 100 Hz, the dominant magnetic coupling is to electronics, especially to cables and connectors. LIGO has found that this coupling can vary in time as electronic equipment is replaced and serviced. For this reason, magnetic coupling was measured weekly during the last part of the third observing run.

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Note on Barkhausen Noise Time-varying magnetic fields like those generated by TM coil actuators can cause flipping of loose magnetic domains in the actuation magnets or in ferromagnetic objects close to the coils, known as the Barkhausen effect. This can result in fluctuating forces on TM and upconverted noise in the GW strain signal. An issue with Barkhausen noise was experienced by the initial LIGO. The culprit was eventually associated with 303-steel fasteners, which became ferromagnetic because they were cold-worked [37,171]. Moreover, use of ferromagnetic materials close to sensitive components like test masses should be avoided. Some kind of magnets, like Sm2 Co17 , are less susceptible to the Barkhausen effect.

Electric Field Influences Test masses are made of non-conductive material (fused silica, sapphire); therefore, electric charge can build up on the surface. LIGO measured surface charge densities of σ ≈ 10−11 C/cm2 . This charge can generate undesired forces on the mirror because of the interaction with voltage fluctuations of nearby conductive surfaces (e.g. vacuum chamber, wires) or stray electric fields [172]. Electric field metres were installed inside one end chamber of both LIGO detectors [24] to witness any time-varying electric field that might induce large enough force on the test mass to impact the interferometer sensitivity. Virgo experienced noise due to the interaction of the charged TM mirrors with residual common-mode voltage on coil actuators [26]. The presence of charge on TM mirrors influences the operation of electrostatic actuators (or electrostatic drivers, ESD) that are used by GEO and LIGO. ESD consist of segmented patterns of pair of conductive strips deposited onto the reference mass facing the test mass mirror back surface. By applying a voltage between the strips, a fringing field is generated which penetrates the dielectric of the TM inducing a force onto the TM dielectric [173, 174]. Occasional variations in the charge level on the mirror means changes in the gain of electrostatic actuators with consequences on the gain of the longitudinal servos and the absolute calibration of the GW signal [175]. Additionally, noise can also arise from the random fluctuation of the charge, although current estimates and measurements are not yet accurate enough to make the case [176].

Charging and Discharging Processes Possible mechanisms that cause charge build up on the TM mirrors are currently debated [17]. Charges may build up through the friction induced by the movements of dust during the evacuation of the chambers, or accidental friction due to contact with non-conductive surfaces like safety stops, or in the action of removing the protective polymer film before a vacuum chamber is closed. According to [177], a low-energy electron flux from cosmic rays could qualitatively explain the monotonic rise of charge observed in [178]. Ion pumps, relied on primarily for non-condensable gas removal, were found to emit UV light and

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X-rays sufficient to cause electrostatic charging of detector components. A mitigation consisted in sets of blackened baffles that block UV/X-ray emission yet allowing the gas to enter the pump. For safe operation, LIGO performs mirrors periodic discharges by occasionally introducing small amounts of an ultra-pure ionized gas into the vacuum chambers [179]. A discharging system through irradiation of the mirror surface with UV light is also under study, but it needs a careful design to avoid potential damaging or contamination of the mirror coating [175].

Cosmic Rays Cosmic ray muons interacting with the test masses can contribute as transient noise or as a continuous background. However, these effects are estimated to be either very rare or beyond the reach of current interferometers. Very energetic secondary muons and hadronic cores of air showers, which deposit >1 TeV of energy in the mirror bulk, can heat-up and deform the TM mirror surface or kick the mirror because of momentum transfer. These effects are expected to produce mirror displacements ∼10−19 m [177, 180]. The rate of these events is estimated in the order of a few per year. Additionally, even for the most energetic showers, the secondary particle flux drops effectively to zero within roughly 10 km of the axis of motion of the primary particle [181], making the coincident observation of a shower among distant detectors highly unlikely. These events can be tagged and vetoed using charged particle detectors to collect a fraction of the shower’s energy. As discussed in [182], the background flux of low-energy muons (100 GeV) can excite and sustain mirror bulk modes leading √ to a stationary strain noise spectrum: hcosmic (f ) = [10−27 − 10−26 ] · 100Hz 1/ Hz, which is a factor 100 f to 1000 below design sensitivities of the current and future planned detectors. At present, no noise correlated to cosmic rays has been evidenced in existing interferometers. One cosmic ray sensor is installed at LIGO H1 and recently also at Virgo. These sensors are intended to be used to detect wide energetic showers, and they are examined for the GW event validation (Section “Validation of Gravitational-Wave Events”).

Gravity Gradient Noise Density fluctuations in the soil or the atmosphere around test masses produce a variable net gravitational force acting directly on the mirror that circumvents any seismic isolation system. This contribution to the detector strain signal is known as gravity gradient noise or Newtonian noise (NN) [183]. Although not yet directly observed in existing detectors, NN is predicted as one of the most insidious limitations of low-frequency sensitivity of the next generation of GW detectors. The issue is seriously considered in the design of both the Cosmic Explorer [184] (CE) and the Einstein Telescope [185] (ET) detectors, which aim for a tenfold improved sensitivity. Unlike mechanically coupled seismic and acoustic

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noise, which can by strongly attenuated by suspending the test masses inside a vacuum chambers (Section “Seismic Noise”), the Newtonian effect of seismic and acoustic fluctuations cannot be attenuated except by reducing the fluctuation amplitude or using auxiliary sensors to estimate and subtract the noise contribution to the detector strain signal. Below we illustrate the main characteristics of gravity gradient noise and briefly overview the noise reduction strategies currently considered.

Newtonian Noise from Ground Density Fluctuations The strain spectral noise associated with soil density fluctuations goes as h˜ N N ∼ x(f ˜ )/f 2 [186], where x(f ˜ ) is the amplitude spectral density of vertical surface displacement of the ground surrounding test masses. Given the typical 1/f 2 ground seism frequency dependence, ultimately the noise decays approximately as the fourth power of frequency. The dominant contribution to Newtonian noise comes from seismic surface Rayleigh waves, P-waves and vertically polarized S-waves. These waves cause either a vertical displacement of the ground or density fluctuations of the bulk. As reported by the most recent estimates and shown in Fig. 21, if not mitigated, this noise is likely to limit the low-frequency sensitivity of the next generation of ground-based interferometers, like the Einstein Telescope (ET) [169], Cosmic Explorer (CE) [187] and Voyager [188]. Newtonian noise cannot be shielded; therefore, it has to be faced either by reducing the soil density fluctuations in the TMs proximity or through the noise cancellation techniques. Low local ground seismicity is essential. At present, evaluation studies are ongoing for the site selection of the ET underground detector [169] and of the surface CE detector [187]. Parameters under evaluation include the following: proximity to antropogenic sources (urban areas, airports, traffic), geophysical soil characteristics (e.g. stability, homogeneity) and soil attenuation properties. Longterm monitoring permits to carefully evaluate daily and seasonal seism variations, seismic wave-field composition (relative abundance of body, surface, and Rayleigh wave types) and correlation lengths. Soil seismicity is mostly due to surface waves, which attenuate exponentially with depth. A significant seismic noise reduction is obtained in underground sites, as demonstrated in KAGRA [189] and in recent studies for the ET detector site [113]. Soil density fluctuations nearby test masses can be reduced simply by removing the earth from underneath them, optionally filling the hollow volume by lightweight material if necessary [190, 191]. Also, seismic cloaking through metamaterials is being investigated by the LIGOIndia [192], Voyager and CE teams [193]. Metamaterials exploit the properties of common materials when organized in periodic patterns to manipulate seismic, acoustic or EM waves by deflecting, blocking or absorbing them [194]. Properly designed seismic metamaterials could deflect or absorb seismic waves before they arrive at the test mass, potentially suppressing surface wave amplitudes by

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a factor of a few. Finally, low seismicity of selected underground and surface laboratories must not be spoiled by self-inflicted noise. A major challenge will be building a silent detector infrastructure, minimizing noise emissions from onsite machinery (Section “Environmental Noise Considerations in Site Selection and Site Facilities”). Mitigation does not have to rely only on reducing ambient vibrations. A promising approach is noise cancellation [195]. The proposed cancellation techniques use an array of seismometers optimally positioned on the soil surface around the test masses and Wiener filtering to reconstruct the seismic wave-field and estimate the instantaneous gravity perturbation. Eventually, the cancellation is achieved either by applying a correction force on the mirror or by performing noise regression from the data stream. First tests [196, 197] indicate that relatively small arrays (≈10 sensors) can achieve cancellation factors of five or larger also in the presence of inhomogeneities (e.g. from local sources) and anisotropies (e.g. from reflecting or scattering surfaces). Other kind of sensors are also being considered such as tiltmeters [30, 199] and distributed vibration sensing with optical fibres. In case of highly non-stationary seismic field (like for Virgo [197]), optimal subtraction may need to relocate the sensors during the day. Machine learning optimization algorithm are proposed [198] to be implemented within mobile robotized sensors. Cancellation schemes using Wiener filters have been evaluated as well for Newtonian noise from P-wave seismic fields [198].

Newtonian Noise from Air Density Fluctuations Gravity gradient noise is also produced by the density fluctuation of air volumes [200]. The relevant mechanism that is likely to impact a surface detector like Cosmic Explorer is the propagation of infrasound (sound frequency f  20 Hz) in the vicinity of test masses [169, 187, 201]. Other mechanisms include moving air volumes driven by wind and temperature gradients [202], but these would have typical timescales longer than a second or occur at large distance from test masses and are unlikely to be significant above a few Hz [203]. The impact of loud infrasound transients, like shock waves from airplanes, explosions, snow avalanches or dam water discharges, should be considered, given the fact that atmospheric damping is pretty low at these frequencies. Underground detectors are shielded from external sources of infrasound noise, which decays exponentially with depth d and frequency f as: e−df/cs , where cs is the speed of sound [203]. On the other hand, a serious potential issue is infrasound noise self-generated by laboratory facility and infrastructure [201]. A challenge for new detector facilities is to keep this noise at a level below the outdoor value. A careful design shall avoid, for example, wind-induced turbulence in surface laboratories, baffling of the interiors or caverns in order to absorb sound and reduce reverberation, proper interior design to break low-frequency standing waves.

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As for seismic NN, microphone arrays around TMs are under study for subtracting infrasound noise [203]. However, the implementation is more challenging than for seismic arrays. Generally, sound fields are largely incoherent and have short spatial correlation distance due to the presence of multiple sources and sound scattering surfaces. While on one hand this has the advantage of reducing the net gravity gradient contribution, on the other hand, it requires a dense sensor net to properly reconstruct the acoustic field and perform NN cancellation. The use of microphones to cancel external sound fields is even more challenging, because of the difficulty to efficiently shield wind-induced turbulence noise in microphones. Low-frequency windshields are under study [33].

Environmental Noise Considerations in Site Selection and Site Facilities The experience gained from the first and second generations of gravitational-wave detectors may be useful for the future generations of detectors. While some coupling mechanisms may differ, for example, Newtonian noise, may couple vibrations that, in earlier detectors, coupled through optic suspensions, the minimization of environmental noise will likely always be prudent. Below we briefly list some considerations, based on lessons learned, that may be helpful in site selection and some considerations that may be helpful in minimizing self-inflicted noise from future facilities. Of course, not all of these considerations apply to underground facilities.

Site Selection Considerations for Minimizing Environmental Noise Environmental noise measurements for evaluating proposed sites should likely include seismic, acoustic (including infrasound), magnetic and RF backgrounds. Measurements of frequency-dependent seismic propagation velocities and quality factors are also important, especially for surface waves. Long-term measurements of background levels can be important, as sources of noise can be seasonal, such as wind, crop harvesting (and trucking) and the ocean-storm-driven microseismic peak. In addition, the background measurements should focus on the transient levels as well as the average levels. Investigations of transients may require examination of noise amplitude spectral density on multiple timescales, so that short transients, such as 0.1 s transients from blasting, are not averaged with transient-free periods. Prior to actual site measurements, candidate sites can be evaluated using information available on the internet and Google Earth. This includes most of the considerations in the list below: 1. Geology. 2. Microseismic peak level (minimize relative motion of detector stations, Section “Sea and Wind”).

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3. Wind speed distribution (minimize wind greater than about 10 m/s). 4. Expected weather changes associated with climate change. 5. Nearest train tracks and density of train traffic on those tracks (trains can produce 1–3 Hz seismic noise, especially within 20 km, and, if electric, can produce magnetic noise transients below 10 Hz especially less than 10 km away. 6. Nearest highway, and truck traffic density (especially within 10 km; frequency, given by axle spacing and speed, is generally in the 4–15 Hz band; transient vibrations from trucks can be an order of magnitude worse than from cars and at lower frequencies due to greater axle spacing [204]). 7. Nearest bridges and viaducts (including those for trains), which may have structural resonances that are excited by vehicles (see also Section “Anthropogenic Seismic and Acoustic Sources”). 8. Conditions of nearest roads (the largest 4–15 Hz seismic signal at a LIGO Hanford Observatory (LHO) end station was reduced by a factor of 2 to 3 when highway 240, 2 km away, was resurfaced [115]). 9. Nearest gravel or dirt road and traffic density (gravel and dirt roads typically produce much more vibration noise per vehicle than other roads). 10. Off-highway recreational vehicle activity within 10 km (the signal from variations in force against the ground is typically in the 3–15 Hz band). 11. Nearest activity involving heavy equipment, such as mining and quarrying (varying forces from massive vehicle motions can be more important than higher-frequency activities such as rock crushing, because of greater attenuation of high frequencies). 12. Potential logging activities (also mainly due to variations in force from massive vehicles). 13. Nearest likely construction (again the worst sources tend to be a massive Earthmoving equipment in motion, mainly within 20 km). 14. Nearest town (within 20 km). 15. Population density. 16. Nearest dam. LHO astrophysical range was limited at night during the spring season high-water flow by 1–3 Hz vibration from water flow over McNary Dam, 60 km away) [101, 102]. 17. Nearest power substation (mains current returning in the ground to a substation near the LIGO Livingston Observatory (LLO) travels on the grounded beam tube and produces magnetic glitches inside the buildings as stoves, heaters, etc. in Livingston turn on and off, Section “Noise from the Utility Mains”). 18. Nearest trunk power line (some of the largest magnetic transients at LHO are produced by current fluctuations in a 500 kV, 700–3000 A transmission line about 2 km away [205]). 19. Does it share an electrical grid with another GW detector (unlikely, but may be a source of correlation between detectors)? 20. Is it on a heavily used flight path of a nearby airport (does it line up with a runway within about 50 km)? 21. Proximity to military training areas (LHO experiences 1–12 Hz seismic and acoustic transients from tank and artillery practice at the Yakima Training Center, about 50 km away [206]).

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22. Proximity to potential wind farms (infrasound coupling to the ground may be important, Section “Anthropogenic Seismic and Acoustic Sources”). 23. Proximity to other facilities with large rotating equipment (cooling fans at a nuclear power plant 6 km from LHO produce seismic signals that are at least an order of magnitude above background at about 2 Hz [103]).

Site Facilities Considerations for Minimizing Self-Inflicted Environmental Noise The considerations in this section are associated with the non-technical facilities. Considerations associated with technical systems, such as vacuum, clean room and electronics systems, are discussed elsewhere in this chapter.

Buildings 1. Consider the most energy efficient, best insulated buildings possible in order to minimize global HVAC requirements: lesser energy usage is likely associated with lesser vibrational noise. 2. To minimize wind-induced floor tilt at wind gust frequencies, vacuum chamber supports should be far from the building walls (>5 m). Varying pressures on a building cause elastic dimpling of the ground near the wall supports (Section “Sea and Wind”). 3. Metal clad buildings can be good at reducing external EM fields. But concrete buildings with well-grounded rebar can also be good Faraday cages at frequencies in the detection band and up to frequencies given by the rebar spacing. 4. Concrete buildings are usually better at blocking sound and infrasound than metal clad structural steel buildings. Also, consider concrete construction around acoustic noise sources such as HVAC chillers. 5. Another advantage of concrete buildings over metal clad structural steel buildings is that the concrete buildings are likely to be more damped. Structural steel buildings, particularly in wind, can increase ground motion inside by more than an order of magnitude at their resonances, typically in the 2–10 Hz band. 6. Siding, flashing and roofing, where present on sensitive buildings, should be rigid, well-supported and damped to minimize acoustic noise in wind and rain, especially in any large regions that could buckle or “breathe” and produce infrasound inside when it is windy. 7. Low-frequency acoustic modes may be damped by suspended ceilings and other techniques. 8. Consider designs that distance human activity (such as in control rooms, offices, labs and shops) from the vicinity of sensitive regions (i.e. experimental areas at interferometer vertexes). 9. Outlying buildings that do not house the interferometer should nevertheless be subjected to the vibrational considerations in Sections “Mains Electrical Considerations” and “HVAC and Other Equipment” because low-frequency vibrations attenuate little with distance (discussed below).

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Site Roads and Parking Areas 1. Vehicle movements on gravel and dirt generate much larger amplitude vibrations than on asphalt or concrete. There should be no gravel/dirt roads or parking lots near the buildings. 2. Gravel and sand located near the roads can contaminate the roads, making them generate larger seismic signals. 3. Shallow pipes under roads, road expansion joints, cattle guards, etc. can make bumps that vehicles drive over, producing vibrational noise. Consider burying pipes and conduits at least 0.5 metres beneath the roads. 4. Consider selecting materials and construction methods that minimize cracking in the road surfaces with age. Mains Electrical Considerations 1. Consider having mains building wiring twisted near sensitive regions in order to minimize magnetic fields at the mains frequency and magnetic field transients. 2. While it is a general practice for electricians to run supply and return wiring in the same conduits, there have been occasions when they were run in separate conduits, leading to large loop areas and unnecessarily large magnetic fields. 3. It may be prudent to inspect high-current equipment installations for unnecessary current loops in the wiring, typically where an electrician connects the equipment to the mains. 4. Consider vibration isolation of the mains transformers that come with the building when they are near sensitive regions. This is occasionally done without special requests, though not as commonly as for equipment with moving parts. But mains transformers can be a strong source of vibrations (at even harmonics of the mains frequency). Acoustic isolation should also be considered. HVAC and Other Equipment 1. Spring isolation of HVAC and other equipment (Section “Cooling and Climatization System”) can be very effective but was sometimes shorted, especially by connected pipes and ducts or simply installed improperly. 2. Vibration from turbulence in HVAC air and coolant lines can be worse than vibration from the actual fans or pumps, which are not broadband and are easy to seismically isolate on springs. a. Coolant lines and air ducts with larger diameters reduce flow speed and turbulence. b. Variable frequency drives can be used to reduce chilled water flows and associated turbulence when demand is lower, if their EM noise is acceptable. c. HVAC fans, like turbines, with high speed airflows, produce turbulence at the output where speed changes rapidly. Screens over the turbine output can reduce low-frequency turbulence [207]. But in general, lower air speeds, such as produced by “squirrel cage” fans, are preferable. If fans are in fan boxes, the boxes themselves should be isolated by placing them on springs. d. To reduce distance-associated resistance, and thus total energy in air and chilled water circuits (a fraction of which will inevitably drive vibrations),

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it may be preferable to locate HVACs nearby. LIGO and Virgo have been using banks of ductless small HVAC systems where extra cooling is needed, such as for electronics rooms (section “Cooling and Climatization System”). The coolant pipes should be isolated. A drawback is that the external chiller units can be strong acoustic sources at fan frequencies. The electrical and vibrational disturbances at startup can be reduced with slow starts. 3. Underground equipment can couple strongly to the ground. a. Underground pipes with the potential to produce strong transients or turbulence, like HVAC chiller pipes, should not be in direct contact with the ground (elastic insulation may reduce coupling), or they should run above ground with vibration isolation. b. Underground motors (e.g. sump pumps and well pumps) should not be in direct contact with the ground and should be on springs like equipment on the surface. 4. The benefits of increasing the distance from a source depend strongly on the seismic wave frequency. Surface wave amplitude only drops geometrically as the square root of distance, until the frequency-dependent exponential attenuation term becomes important:  Afar = Anear

πf Rnear − Qv (Rfar −Rnear ) e Rfar

where f , v and A are the seismic wave frequency, velocity and amplitude; Rnear and Rfar are distances to near and far source options, respectively; Q is the seismic quality factor. For the sandy desert at LHO, the measured wave velocity is about 500 m/s at 5 Hz and 150 m/s at 60 Hz; Q is about 50 at 5 Hz. For the Virgo mud-clay surface soil, values of v = 200 m/s and Q = 30 have been measured at a few Hz. In general, seismic quality factors are between 20 and 200 and change slowly with frequency. The 1/e distance at LHO is about 40 m at 60 Hz, 150 m at 20 Hz, and 1600 m at 5 Hz. 5. Acoustic considerations. a. Ducts should usually be large diameter, double-walled, acoustic ducting. Watch for vibrating single-wall connector segments in double-walled systems, which can be very noisy. b. It is likely advantageous to place equipment with fans or other moving parts in separate rooms (possibly acoustically treated) from the vacuum chambers, in order to reduce acoustic coupling. 6. Considerations associated with heating and with temperature control. a. Electric heaters can be the largest current users in a building and produce the largest magnetic fields. Consider avoiding pulsed building heating (many systems use thyristors to pulse heater currents at about 1 Hz, and control temperature by varying the “on” time in each cycle) and avoiding other pulsed heating currents (in many chiller systems used for lasers). The pulsing transients produce combs in the magnetic spectrum around the mains frequency,

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with the individual peaks spaced by the pulse frequency (section “Switching Devices”). b. Building heating elements can often be current loops with significant areas, and, because the heating currents are large, heating elements can be the most significant generators of magnetic field transients. It is possible to obtain special two-pass heating elements that minimize the loop area. c. Building temperature control sensors should be away from the walls and near (but not directly on) the vacuum chambers where temperature control is most important. 7. Considerations associated with motor and fan frequencies. a. In general, it helps to use higher-frequency motors (2 pole instead of 4 or 6 pole motors) to take advantage of the fact that seismic attenuation is typically greater at higher frequencies. b. Reciprocating compressors should probably be avoided in favor of scroll or other designs. c. If possible, designed mechanical resonances of sensitive components should be away from motor frequencies (for 60 Hz mains, 58–60, 28–30, 17–20, and 14–15 Hz) in order to minimize problems with 2, 4, 6 and the occasional 8 pole motors, respectively. d. Variable speed fans and motors, in HVACs and in other systems, can be problematic because they affect a larger portion of the spectrum than constant frequency systems, and are thus more likely to excite resonances. Variable frequency systems are now almost ubiquitous because of energy savings. However, some variable frequency equipment can be set to maintain constant frequency because frequency variation can be annoying to nearby people. These more versatile systems may be preferable. 8. Considerations associated with control and monitoring. a. Consider HVAC control systems that can be easily modified, both LIGO and Virgo have needed to modify their HVAC control systems. b. The ability to easily monitor the state of heaters, chillers, fans, airflows, etc. and correlate them with detector channels has been expanded at both Virgo and LIGO and is probably essential (Section “Infrastructure Monitors”).

Conclusions Over the next decade, the network of ground-based GW detectors will go through phases of upgrades, in which the sensitivity will be improved, followed by observing runs, in which the detection of numerous gravitational-wave signals is expected. On a longer timescale, the Einstein Telescope will be the first underground GW detector in Europe, while Cosmic Explorer will be a 40 km long interferometer in the USA. They will be at least one order of magnitude more sensitive than the current detectors and will extend the observation band down to approximately 1 Hz. Environmental disturbances will continue to be a main subject of investigation. It will be needed to ensure comprehensive noise monitoring through a widespread

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array of well-positioned sensors and to have an accurate monitoring of the coupling of ambient noise to the detector. This information is essential for the GW event validation and to produce an online budget of the contribution of environmental noise to the interferometer, in order to promptly identify the appearance of new noise or worsening in the noise coupling. A useful effort is the further automation of these measurements. Also important is the development of additional tools for the noise analysis and experimental investigation, particularly for the study of nonlinear processes and scattered light. The experience collected in facing known and unexpected noise taught us the importance of addressing environmental noise issues at the early detector design stage. In short: minimize noise emissions, cut down transmission paths and reduce couplings. A few challenges for next-generation detectors are: • Make a careful choice of the detector site, and plan actions to preserve the site low-noise conditions. • Implement a low-noise and monitored detector infrastructure. • Improve the detector immunity to vibro-acoustic disturbances to suppress scattered light noise. • Increase the EM compatibility of the interferometer electronic apparatus, reduce device emissions and crosstalk, eliminate noisy device models and have a clean experimental power grid. • Improve the interferometer robustness against adverse weather conditions and earthquakes. In these pages, we hope to have provided a useful guidance in addressing some of these issues.

References 1. Abbott BP et al (2016) Observation of gravitational waves from a binary black hole merger. Phys Rev Lett 116:061102 2. Aasi J et al (2015) Advanced LIGO. Class Quantum Gravity 32(7):074001 3. Acernese F et al (2015) Advanced Virgo: a second-generation interferometric gravitational wave detector. Class Quantum Gravity 32(2):024001 4. Akutsu T et al (2019) KAGRA: 2.5 generation interferometric gravitational wave detector. Nat Astron 3:35–40 5. Dooley KL et al (2016) GEO 600 and the GEO-HF upgrade program: successes and challenges. Class Quantum Gravity 33(7):075009 6. Sathyaprakash BS et al (2009) Physics, astrophysics and cosmology with gravitational waves. Living Rev Relativ 12(1):1–141 7. Pitkin M et al (2011) Gravitational wave detection by interferometry (ground and space). Living Rev Relativ 14:5 8. Abbott BP et al (2020) A guide to LIGO-Virgo detector noise and extraction of transient gravitational-wave signals. Class Quantum Gravity 37:5 9. Schumann W (1952) Ückber die strahlungslosen Eigenschwingungen einer leitenden Kugel die von einer Luftschicht und einer Ionosphärenhülle umgeben ist. Zeitschrift Naturforschung Teil A 7:149

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Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum

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Kiwamu Izumi and Karan Jani

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DECIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-DECIGO and Technology Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lunar-Based Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Direct observations of gravitational waves at frequencies around deci-Hertz will play a crucial role in fully exploiting the potential of multi-messenger astronomy. In this chapter, we discuss the detection landscape for the next several decades of the deci-Hertz gravitational-wave spectrum. We provide an overview of the experimental frontiers being considered to probe this challenging regime and the astrophysics and fundamental goals accessible toward them. This includes interferometric observatories in space with heliocentric and geocentric satellites, cubesats, lunar-based experiments, and atom interferometry. A major focus of this chapter is toward the technology behind DECi-hertz Interferometer

K. Izumi () Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa, Japan e-mail: [email protected] K. Jani Department Physics and Astronomy, Vanderbilt University, Nashville, TN, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_50

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Gravitational wave Observatory (DECIGO) and its scientific pathfinder mission concept B-DECIGO. Keywords

Instruments · Laser interferometry · Space mission · Lunar science · Cubesats · Atomic clock · Gravitational-wave astronomy

Introduction The current generation of ground-based gravitational-wave detectors such as Laser Interferometer Gravitational-Wave Observatory (LIGO) [1], Virgo [2], and KAGRA [3] are sensitive at frequencies above 10 Hz. These L-shaped detectors with a few km arm length are suited to probe gravitational waves from the coalescing binaries of neutron stars and stellar black holes (102 M ) [4]. The next generation of ground-based gravitational-wave detectors like Einstein Telescope [5] and Cosmic Explorer [6] are expected to be sensitive as low as 5 Hz. The arm length of these detectors will be tens of km and includes several technological upgrades from the current generation. With these next-generation detectors, the sensitive volume for stellar compact binaries will reach unprecedented cosmological scale and open the prospects for direct detections of lower-range to mid-range intermediate-mass black hole (103 M ) [7]. The upcoming space-based gravitational-wave detector Laser Interferometer Space Antenna (LISA) [8] will have a few million km long arm length, thus measuring low-frequency gravitational waves near the milli-Hertz range. This debut space-based gravitational-wave detector will open a new window into the astrophysics of supermassive black hole binaries of ∼106 − 107 M . Furthermore, the ongoing network of pulsar timing arrays [9] is probing the nanoHertz regime. This can potentially measure gravitational waves from supermassive black hole binaries as heavy as ∼109 M [10]. A particularly challenging regime to measure gravitational waves for all the detectors mentioned above is near the deci-Hertz frequencies (∼0.1 Hz). For a terrestrial laser interferometric setup, it is difficult to go below 1 Hz barrier due to the Earth’s seismic noise. On the other hand, space missions like LISA cannot go higher than 0.1 Hz due to laser shot noise. The universe is filled with rich astrophysical sources in and around deci-Hertz (10−2 ∼ 1 Hz) (see [11, 12] for a recent review). The chief among them are the elusive intermediate-mass black holes (IMBHs: 102 − 105 M ), which are crucial clues in understanding the first population of stars in the universe, as well as the seed formation of the supermassive black holes found at the centers of galaxies (see [13, 14] for a recent review). There are a few promising candidates for IMBHs across its broad mass range from electromagnetic observations, though the first confirmed evidence is attributed to the remnant of binary black hole merger GW190521, observed by the LIGO and Virgo detectors [15].

11 Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum Space missions

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Fig. 1 Gravitational-wave frequencies corresponding to the evolution of some of the prominent astrophysical binaries in the deci-Hertz spectrum. The inspiral of binary neutron star (blue line) and stellar binary black holes (maroon line) will be observed in the deci-Hertz for years. For intermediate-mass black holes, IMBHs (gray lines), depending on their mass ranges, the deci-Hertz access would measure their inspiral, merger, and ringdown

For a coalescing binary, the instantaneous gravitational-wave frequency fGW at a time τ before the merger scales inversely with the chirp mass M [16]. In Fig. 1, we show the evolution of various astrophysical binaries around the deciHz range. Binaries with IMBHs would generally undergo all the three phases of the coalescence – inspiral, merger, and ringdown – within the deci-Hertz band. A binary with an upper-range (∼105 M ) and mid-range (∼104 M ) IMBHs would undergo inspiral for a few minutes to hours in the deci-Hertz band, followed by merger and ringdown. Binaries with lower-range (∼103 M ) IMBHs would inspiral in deci-Hertz spectrum for a few days, though their merger and ringdown may be better accessible with next-generation ground-based detectors. The expected binary population of IMBHs remain fairly unknown, and the constraints on their merger rates differ dramatically from equal-mass systems to intermediate-mass ratios (m1 /m2 ∼ 100). The deci-Hertz spectrum opens a rare observational window into testing a variety of formation scenarios of IMBHs to cosmological distances. Within the frequency band of the current generation detectors such as LIGO, Virgo, and KAGRA (fGW = [10, 1000] Hz), we can only measure about a second of evolution of binaries of stellar black holes (M ∼ 10 M ). In the deci-Hertz band, these binary black holes stay for months (maroon line). The binary neutron star sources would evolve within deci-Hertz band up to a few years (blue line). This extended observation is important to measure the orbital eccentricity and spin

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orientation of the binaries. Both of these measurements are key to distinguish among binary formation channels [17, 18]. A deci-Hertz range detector also acts as an early-alert system. All the binaries that will be seen by next-generation groundbased detectors such as Einstein Telescope and Cosmic Explorer would be seen at the lower frequency many months ahead by a deci-Hertz detector. This opens an interesting possibility of multiband gravitational-wave astronomy, wherein the same astrophysical source is studied in different frequency bands [7, 19–21]. An equivalent analogy of such in astronomy is that of multiwavelength electromagnetic observations across radio, X-ray, and other bands. The precise knowledge of sky localization and distance from deci-Hertz observation will permit high-latency electromagnetic follow-ups and also stronger constraints on the Hubble constant with “dark sirens” [22]. The deci-Hertz spectrum allows a unique peak into the progenitors of Type Ia supernovae (such as double white dwarfs [23]), whose observations will help for better calibration of standard candles [22, 24]. The low-frequency gravitational waves by the asymmetric ejection of neutrino during core-collapse supernovae will also be in the deci-Hertz regime [25]. In context of fundamental physics, the deci-Hertz sources open a wide range of tests for general relativity, including any presence of scalar fields and dipole radiation. The multiband observations between deci-Hertz with LIGO-like ground detectors and LISA-like space missions will allow an independent check on the pre- and post-merger properties of black holes, thus aiding the tests of the no-hair theorem. For probing Beyond Standard Model physics, deci-Hertz will particularly be effective for probing bosonic fields around intermediate-mass black holes merger, subsolar dark matter candidates, and stochastic gravitational-wave background from electroweak symmetry breaking (see [11] for a recent review). In this chapter, we will discuss the various experimental frontiers that are being considered to access this challenging yet rich deci-Hertz gravitational-wave spectrum. We will briefly discuss the astrophysics and fundamental physics goals that can be advanced with each of these frontiers and comment on their feasibility by the next decade. A major part of this chapter focuses on the Japanese space mission, DECIGO [26], and its planned technological development.

Experimental Frontiers For an ideal deci-Hertz gravitational-wave detector, we expect its peak sensitivity around ∼0.1 Hz to be somewhere between that of LISA near the milli-Hz range (characteristic strain, h = δL/L ∼ 10−21 ) or that of LIGO around 100 Hz (h ∼ 10−22 ). While that certainly is not a strict criterion, it sets a target sensitivity for detecting expected gravitational-wave sources. Figure 2 highlights the four broad categories of experimental frontiers that are being considered to achieve the desired sensitivity in and around the deci-Hertz regime. This includes a space-based mission with (i) heliocentric or (ii) geocentric orbits, (iii) an experiment on the lunar surface, and (iv) atomic interferometry. Under each category, we list some of the prominent

11 Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum

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Fig. 2 The four broad categories of experimental frontiers being considered for measuring gravitational-wave frequencies in the deci-Hertz regime. For each category, we list the prominent detector designs and their salient features

detector concepts suggested in the literature, along with their notable advantages and challenges. We plot the sensitivity curves for some of the proposed deci-Hertz detectors in Fig. 3. The detector sensitivity is measured as the dimensionless characteristic strain h = δL/L. The line styles refer to the four categories of experimental frontiers: heliocentric (straight lines), geocentric (dotted-dashed lines), lunar-based (dotted line), and atom interferometry (dashed line). In Fig. 4, we show the horizon distance, i.e., the maximum detection radius, of each of these detectors toward coalescing binaries. Notice that other than the cubesat mission SAGE, all the proposed deciHertz detector can reach up to redshift z ∼ 100 for a certain mass range of binaries. A majority of the proposed efforts are geared toward a large-scale heliocentric space mission similar to that of LISA. With three satellites, each separated by 2.5 million kms arm length forming the interferometer, the LISA mission can, in principle, access gravitational waves with frequencies as low as ∼0.1 mHz and as high as ∼0.1 Hz. At this higher frequency end, LISA’s sensitivity is expected to be at most h ∼ 10−20 . A post-LISA mission with a similar three-satellite configuration but an overall higher sensitivity could achieve the target for deci-Hertz observations. Few of such design ideas include the Advanced Millihertz Gravitational-wave Observatory (AMIGO) [27], which was proposed to ESA’s Voyage 2050 call, and the Chinese-based mission Taiji [28]. To achieve the most of the deci-Hertz science, a detector should have high sensitivity across a broad frequency coverage of 0.01 − −1 Hz. For space missions with LISA-like concepts, it will require shorter arm length and technologies to reduce

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Fig. 3 Sensitivity of the proposed gravitational-wave experiments that can measure frequencies near the deci-Hz spectrum (0.1 Hz). The thick black line refers to the most sensitive heliocentric space mission in this region, DECIGO. The maroon dotted-dashed line refers to its pathfinder space mission in geocentric orbit, B-DECIGO. Other heliocentric space missions shown are ALIA (sky blue curve) and DeciHertz Observatory (blue curve). The lunar-based detector, GLOC, is shown with dotted gray lines. More unconventional approaches such as cubesat mission SAGE (orange dotted-dashed curve) and a detector based on atomic interferometer (dashed green lines) are also highlighted

the acceleration noise. Two of such prominent design ideas are the Advanced Laser Interferometer Antenna (ALIA) [29, 30] and DeciHertz Observatories (DO) [11], proposed to ESA’s Voyage 2050 call as well. Their sensitivities and detection radii are shown in Figs. 3 and 4 (sky-blue and dark-blue curves), respectively. Both these missions could detect IMBH binaries across its three orders of mass range (102 − 105 M ) to an unprecedented cosmological distance (redshift z ∼ 100). Furthermore, they would also detect stellar (∼10 M ) and supermassive (∼106 M ) binary black hole binaries to z  10, thus complementing with the science goals from LISA and LIGO. More recently, there has been modest proposals for relatively low-cost space missions in geocentric orbits that can access deci-Hertz region, albeit not with the same sensitivity requirement. This includes a triangular interferometric constellation with cubesats, called SagnAc interferometer for Gravitational wavE (SAGE) [31]. The sensitivity and detection radius for SAGE is shown in Figs. 3 and 4 (orange dasheddotted curve). The detection sensitivity of SAGE allows a survey of upper-range

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Fig. 4 The detection radius measured in the units of cosmological redshift z for the proposed deciHertz gravitational-wave experiments. The colors refer to the detector sensitivities shown in Fig. 3. The y-axis is equivalent to the horizon distance for an optimally placed, equal-mass, non-spinning binary. We compute the horizon for SNR=8. In the case of space-based detectors (all detectors except GLOC), we put a detector lifetime of 5 years. The x-axis refers to the total mass of the binary as measured in its source frame, i.e., with the redshift correction 1/(1 + z). We compute redshift assuming the Planck 2018 cosmological parameters

IMBH binaries (∼105 M ) within redshift z  1. Other proposals of such geocentric mission in the literature include geosynchronous Laser Interferometer Space Antenna (gLISA) [32] and the Geostationary Antenna for Disturbance-Free Laser Interferometry (GADFLI) [33]. With the advent of the return to Moon, it is now finally possible to think of a gravitational-wave detector on the surface of the Moon that would take advantage of the natural conditions. A few recent proposals that are being studied to access deci-Hertz regime from the Moon are the Gravitational-wave Lunar Observatory for Cosmology (GLOC) [22], Lunar Gravitational-Wave Antenna (LGWA) [24], and Lunar Seismic Gravitational-wave Antenna (LSGA) [34]. We discuss these proposals in more details in the later sections. A promising detector technology is also emerging from atom-based interferometry, which could access gravitational waves in 0.01–1 Hz of frequency range. Some of the prominent design proposals are Mid-band Atomic Gravitational Wave Interferometric Sensor (MAGIS) [35] and the Atomic Experiment for Dark Matter and Gravity Exploration in Space (AEDGE) [36]. The sensitivity and detection radius for a concept atom interferometer (adapted from [11], with parameters based on [37]), is shown in Figs. 3 and 4 (green dashed curve). For more details regarding the science goals and design

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Fig. 5 Orbit of DECIGO. Four clusters of observatories are distributed in the heliocentric orbit with two of them placed at the same position

proposals for atom interferometer, see the chapter “Quantum Sensors with Matter Waves for GW Observation” in this edition by Bertoldi, Bouyer, and Canuel. The most ideal deci-Hertz detector, which has been extensively studied in the literature, is the proposed Japanese space mission DECi-hertz Interferometer Gravitational wave Observatory ( DECIGO). The sensitivity and detection radius of DECIGO is shown in Figs. 3 and 4 (black curve). The detector possesses a rare advantage of accessing gravitational waves to cosmological distances from stellar to IMBHs to supermassive black holes. The mission design and the associated technologies for DECIGO are discussed in the next section.

DECIGO DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) is the future Japanese space gravitational-wave antenna concept observing the frequency band from ∼10−1 to 10 Hz. DECIGO was originally proposed in 2001 [38] to fill the frequency gap between the mHz frequency band covered by LISA and the audio band by the terrestrial interferometers. The conceptual mission design calls for four identical clusters of observatories deployed in heliocentric orbits, as sketched in Fig. 5. Each of the clusters consists of three drag-free spacecraft, 1000 km apart from each other in almost equilateral triangular formation, forming six laser links [39]. Each spacecraft houses two floating test masses serving as proof masses (see Fig. 6). The target sensitivity of a single cluster is set to 4 × 10−24 Hz−1/2 at around 1 Hz [40]. A unique design choice in DECIGO is the utilization of the Fabry-Perot optical cavities consisting of a pair of two test masses apart by 1000 km to improve the

11 Detection Landscape in the deci-Hertz Gravitational-Wave Spectrum

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Fig. 6 Conceptual design of the interferometer for a single DECIGO cluster

Table 1 Design parameters for DECIGO and B-DECIGO [41]

Design parameters Laser wavelength Laser power Arm Length Finesse of the Fabry-Perot cavities Diameter of mirror (test mass) Mirror mass Number of observatory clusters

DECIGO 515 nm 10 W 1000 km 10 1m 100 kg 4

B-DECIGO 515 nm 1W 100 km 100 30 cm 30 kg 1

sensitivity without demanding a significant laser power. However, this means that the optical lengths of the Fabry-Perot cavities must be kept at resonances at a precision level of 10−9 m or less. Therefore, achieving a resonance of a Fabry-Perot cavity in orbit has been identified as one of the key challenges. The characteristic design parameters are summarized in Table 1. One of the main scientific objectives of DECIGO is to conduct searches for primordial gravitational-wave backgrounds such as those amplified during the inflation period [42, 43]. Besides, DECIGO will be capable of delivering new insight into astrophysics and fundamental physics through the observations of various compact star binary systems [38, 44] and other sources [45–48]. In addition, heterogeneous observations of merger events with other gravitational-wave detectors in different frequency bands should improve the performance of the astrophysical parameter estimation.

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B-DECIGO and Technology Developments In order to demonstrate the technologies relevant to DECIGO and simultaneously conduct astrophysically important observations in the deci-Hz band, the pathfinder mission concept called B-DECIGO was recently proposed [49]. The sensitivity and detection radius of B-DECIGO are shown in Figs. 3 and 4 (maroon dasheddotted line). One of the major design drivers for B-DECIGO was the observatory performance being able to observe the stellar mass binary black holes up to a redshift of z ∼ 30. The main scientific objective is to study the origin of the stellar mass black holes by mapping out their mass spectrum and event rate as a function of redshift. B-DECIGO is also capable of contributing to multi-messenger astronomy by issuing the forecasts for merger events with a sky localization error of ∼0.3 deg2 and a merger time accuracy of ∼1 s for a binary black hole with the masses similar to GW150914 at z = 0.1, a day before the merger. Based on the scientific motivations as well as the consideration for making the developments less challenging, B-DECIGO is designed to have a shorter arm length of 100 km and reduced mass of 30 kg for the test masses with only a single observatory cluster deployed. Geocentric orbits with a high altitude of 2000 km has been considered. The design parameters of B-DECIGO are also listed in Table 1. The target sensitivity is set to 2 × 10−23 Hz−1/2 at around 1 Hz. The launch date is currently foreseen to be some time in the 2030s. The recent activities for the technology developments relevant to both DECIGO and B-DECIGO encompass a number of key functionalities and aspects. These include the laser source and its stabilization systems, the relative angular sensor for spacecraft [50], the interferometer control schemes [51, 52], the orbital design, and further improvement in the sensitivity [53].

Lunar-Based Experiments The Moon offers a generous environment for gravitational-wave observations. There is no atmosphere, thus providing a high-quality vacuum just above the lunar surface. There are no ocean tides or significant ground motion to impact the differential displacement of the test masses. There are no large-scale human activities interfering with the detector. Therefore, in principle, one can construct a large-scale interferometer like that of LIGO on lunar soil. The conceptual idea has been long around [54], though the halt of crewed missions to the Moon since the 1970s dissipated any further interest from the community. It is only recently that in initiatives such as the NASA Artemis, Commercial Crew, and ESA’s European Large Logistics Lander project there is a strong possibility of returning to the Moon this decade and building a permanent base. One of the science priorities for NASA Artemis is in utilizing the unique environment of the Moon to study the universe. Unlike space-based deci-Hertz detectors, a lunar-based detector can be in observational mode for decades. This is particularly important in following up rarer

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Fig. 7 Conceptual designs of gravitational-wave detector on the lunar surface. Left: Gravitationalwave Lunar Observatory (GLOC) [22]. Right: Lunar Gravitational-Wave Antenna (LGWA) [24]. (Figures have been reproduced by permission of the corresponding authors)

astrophysical processes in the deci-Hertz spectrum. Hardware failures or upgrading detector technologies is also possible on the Moon, thus providing a better value for long-term investment. Below we discuss two distinct experimental techniques for lunar-based gravitational-wave detection, which in particular opens the access to deci-Hertz gravitational-wave frequencies. The first technique is adapted for Gravitational-Wave Lunar Observatory (GLOC), a proposed interferometer on the Moon [22]. The conceptual design for the GLOC is expected to be similar to that of the next-generation Earth-based detector’s Cosmic Explorer (CE) and Einstein Telescope (ET). The arm length of the interferometer can be set to few tens of kms, and the L-shaped LIGIO-like interferometer could be replaced by a triangular geometry such as that of ET (see Fig. 7 – left panel). The end station could be covered in a temperature-controlled dorm to host the required optics. Ideal sites for GLOC could be inside a crater or within one of lava caves. The tentative sensitivity curve and detection radius of GLOC are shown in Figs. 3 and 4 (gray dotted line). The lowest frequency we expect for GLOC is around ∼0.3 Hz. Below that frequency, the sensitivity of GLOC is limited by the seismic and thermal noises. However, compared to other deci-Hertz detectors, GLOC would remain sensitive at higher frequencies, thus enhancing the overall signal-to-noise ratio. For stellar mass binaries, GLOC could detect them practically across the observable universe. The geocentric orbit of the Moon helps tremendously in constraining the sky localization to few arc-seconds in GLOC, thus opening new dark sirens for measuring Hubble constant. GLOC could measure subsolar dark matter candidates upto redshift z ∼ 10. For IMBH binaries, GLOC could survey lower- to mid-range (∼103 − 104 M ) up to redshift z ∼ 10, while it can find the upper-range (∼105 M ) within redshift z  1. The rare access to ∼1 Hz in GLOC further opens avenues to test progenitor mechanisms to Type Ia, thus providing new calibrations for the standard candles. A second technique has been shown in the Lunar Gravitational-Wave Antenna (LGWA) [24] and the Lunar Seismic and Gravitational-Wave Antenna (LSGA)

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[34]. An extension of the resonant-bar detector idea by Weber [55], in these concept designs, the entire Moon acts as a large spherical detector receiving the incoming gravitational waves. The measurement of the signal is then essentially the vibrational eigenmodes. For LGWA, the proposal is to place an array of highend seismometers in km-long array on the Moon (see Fig. 7-right panel). They will monitor the normal modes of the Moon in the frequency spectrum of 1 mHz – 1 Hz. The predicted sensitivity of LGWA near the deci-Hertz regime would be better than that of LISA, thus serving an important science driver. For LSGA, the proposal is to deploy 10 km optical cables in an L-shaped geometry. The laser light passing through the cables would permit the measurement of the displacement strain from incoming gravitational waves. Both these designs will be able to achieve the science goals of deci-Hertz gravitational-wave astronomy discussed earlier in the chapter.

Summary Continuing experimental efforts have been seen over the world for designing gravitational-wave detectors sensitive at the frequencies around deci-Hertz. Observations in the deci-Hertz band will provide us with unique opportunities to enhance the ability of the detector network and to study the dark and relativistic aspects of our universe. Among the large-scale space missions, DECIGO, its scientific pathfinder mission concept B-DECIGO, ALIA, and DO are expected to open the new gravitational window in the deci-Hertz band up to cosmological distances. For modest geocentric proposals, SAGE will be able to survey the deci-Hertz spectrum in the local universe. The lunar-based experiments GLOC, LGWA, and LSGA offer a new avenue to measure frequencies near deci-Hertz and substantially advance the scope of gravitational-wave astronomy.

Cross-References  Quantum Sensors with Matter Waves for GW Observation

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Part III Gravitational Wave Sources

Binary Neutron Stars

12

Luca Baiotti

Contents Introduction: General Description of Neutron-Star—Neutron-Star Mergers . . . . . . . . . . . . . . Gravitational Waves from the Pre-merger Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal Effects and Their Relation with the Neutron-Star Equation of State . . . . . . . . . . . . . Other Finite-Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detectability and Detection of Pre-merger Gravitational Waves . . . . . . . . . . . . . . . . . . . . . Gravitational Waves from the Merger and Post-merger Phases . . . . . . . . . . . . . . . . . . . . . . . . Spectral Properties and Their Relation with the Neutron-Star Equation of State . . . . . . . . Detectability of Post-merger Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investigating Phase Transitions with Post-merger Waveforms . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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In this chapter, I describe the inspiral, merger, and post-merger phases of binaryneutron-star systems, focusing on the gravitational radiation they emit. After a general introduction to the formation of these systems and to the dynamics of mergers and after some comments on the state of the art of numerical simulations thereof, I descend into some details about how to link gravitationalwave measurements with the equation of state of neutron stars, whose cores have the highest density in the visible universe. This is done in two parts, based on inspiral gravitational waves and post-merger gravitational waves, respectively. The tidal deformability plays a prominent role in the former, while spectral properties of the gravitational-wave signal are important for the latter. I also present current observational capabilities and estimates for future detections and

L. Baiotti () International College and Graduate School of Science, Osaka University, Toyonaka, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_11

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comment on the detectability of equations of state that include deconfined quark matter and possibly phase transitions. Keywords

Neutron stars · Binary-neutron-star mergers · Relativity and gravitation · Numerical relativity · Gravitational-wave sources · Gravitational-wave observations · Equations of state · Phase transitions · Compact stars · Black holes

Introduction: General Description of Neutron-Star—Neutron-Star Mergers The composition of neutron stars is still largely unknown, but they are definitively not made of neutrons only, and perhaps their cores are made of matter in a form rather different from that of neutrons. A better term would be compact stars, because it does not contain any hints on their composition: It just differentiates them from black holes or less dense stars. However, in most of the literature, the term neutron stars is employed, without definite assumptions on their composition. Exceptions to this routine use are works that explicitly study non-nucleonic equations of state for compact stars (see the end of Sections “Detectability and Detection of Pre-merger Gravitational Waves” and “Investigating Phase Transitions with Post-merger Waveforms”). These works often distinguish compact stars into neutron stars, quark stars, strange stars (quark stars that have strange-quark matter), hybrid stars (with a core of deconfined quark matter and an outer region of nucleonic matter, with possibly phase transitions separating them), and so on. Furthermore, the term compact stars may be used to refer to exotic objects that are not made of ordinary matter [25], such as boson stars. I will not write about these here, and I will actually mostly use the term neutron stars to refer to compact stars made of ordinary matter in general. Only where specification may be necessary, or at least useful, I will employ terms with further particularization. Neutron stars are possible outcomes of the evolution of some massive stars, after they collapse and undergo a supernova explosion. Many astronomical observations have revealed that binary neutron stars (BNSs) indeed exist. However, the formation mechanisms of BNS systems are not known in detail. The general picture is that in a binary system made of two massive main-sequence stars of masses between approximately 8 and 25 M , the more massive one undergoes a supernova explosion and becomes a neutron star. This is followed by a phase (whose details are largely unknown) in which the neutron star and the main-sequence star evolve in a common envelope, that is, with the neutron star orbiting inside the extended outer layers of the companion star [53]. At the end of this stage, also the second main-sequence star undergoes a supernova explosion, and a BNS system is formed, if the stars are still bound after the explosions and if neither of them has further collapsed to a black hole. The common-envelope phase, though brief, is crucial because in that phase

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the distance between the stars becomes much smaller as a result of drag, and this allows the birth of BNS systems that are compact enough to merge within a Hubble time, following the dissipation of their angular momentum through the emission of gravitational radiation. Another possible channel for the formation of BNS systems may be the interaction of two isolated neutron stars in dense stellar regions, such as globular clusters and galactic nuclei, in processes like dynamical scattering, dynamical capture, and, in general, multibody interactions (see, e.g., Ref. [46]). Dynamically formed binary systems are different from the others because they have higher eccentricities. In fact, BNS systems that have evolved without close interaction with other stars are thought to have lost, by the time their gravitational-wave signals enter the detectability range, any significant initial eccentricity they may have had through the emission of gravitational radiation and therefore to have very small orbital eccentricities, i.e.,  10−3 (see, e.g., Ref. [48] and references therein). Although the estimates of their event rates are very uncertain (see, e.g., Refs. [46, 93]), it is expected that these eccentric binaries are only a small part of the BNS population. Because of emission of gravitational waves, the neutron stars in a binary are not on closed orbits but inspiral and, if given enough time, eventually merge in a single object, which may be a black hole or a neutron star, surrounded temporarily by a variable amount of matter, mostly in the shape of a disc. Gravitational radiation from the last part of the inspiral before the merger of two BNS systems (GW170817 [87] and GW190425 [90]) has been detected, and gravitational waves form the merger itself and the post-merger part are also thought to be measurable in the near future. Mergers of BNS systems may also be observable in the electromagnetic spectrum if some (even relatively small) amount of matter is ejected during and after the process. Such ejecta, which in numerical simulations have masses of the order of ∼10−4 − 10−2 M and can be powered by different mechanisms like tidal torques, shock waves, neutrino-driven winds, magnetically driven winds, or viscosity-driven winds [7, 84], may give rise to a short gamma-ray burst [18], to radio emission [84], and/or a macronova/kilonova. Note that, the terms macronova and kilonova are often used interchangeably to indicate an astronomical source of electromagnetic radiation about one to three orders of magnitudes brighter than a regular nova, which is caused by hydrogen fusion explosions on a white dwarf accreting from a larger companion star. In the standard scenario, macronovae/kilonovae shine hours to days after the merger in ultraviolet, optical, and infrared bands, and their power source is thought to be the radioactive decay of r-process elements [66]. Macronova emission was observed in association with GW170817 [66], but not with GW190425 [90]. As prime observational targets among strong-gravity systems, BNS mergers have been and are being subject to intense theoretical study, often with the goal to obtain information on the equation of state of ultrahigh-density matter [6], on the origin of the abundance of heavy elements in the Universe [91], on formation mechanisms for magnetars [56], and with the goal to test the theory of gravity [89]. An overview of the dynamics of a BNS merger is given in Figs. 1, 2 and 3, which are taken from Ref. [8]. The images were produced from the data of numerical simulations and show some representative isodensity contours (i.e. contours of

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Fig. 1 Isodensity contours on the (x, y) (equatorial) plane for the evolution of a binary composed of equal stars each with rest mass 1.625M and described by a polytropic equation of state. The time stamp in ms is shown on the top of each panel, and the color-coding bar is shown on the right in units of g cm−3 . The dashed line represents the apparent horizon. (From Ref. [8])

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Fig. 2 Left: Evolution of the maximum rest-mass density normalized to its initial value for the binary of Fig. 1. Right: Evolution of the maximum rest-mass density normalized to its initial value for the binary of Fig. 1, but in which an ideal-fluid equation of state was used. In each panel, indicated with a dotted vertical line is the time at which the binary merges, while a vertical dashed line shows the time at which an apparent horizon is found. After this time, the maximum rest-mass density is computed in a region outside the apparent horizon (in the disc). The non-normalized value of the maximum rest-mass density at t = 0 is 5.91 × 1014 g cm−3 . (From Ref. [8])

equal rest-mass density) on the equatorial plane and a meridional plane. The initial coordinate separation between the maxima in the rest-mass density (defining the stellar centres) is 45 km. The stars inspiral emitting gravitational radiation, and each star is tidally deformed by the gravity of its companion. This leads to a further increase in the inspiral rate (the absolute value of the time derivative of the separation between the centers of the stars) [44], which also depends, of course, on the initial total angular momentum of the system, the orbital one, and the spins of the stars. During the merger, when regions of the two stars with density around a factor of a few less than their maximum density come into contact, a noticeable vortex sheet (or shear interface) develops, where the tangential components of the velocity exhibit a discontinuity. This condition is known to be unstable to very small perturbations, and it can develop a Kelvin-Helmholtz instability, which curls the interface forming a series of vortices. This is indeed what is observed in all simulations of this kind, with features that are not much dependent on the mass or on the equation of state used. Even if this instability is purely hydrodynamical, it can have strong consequences when studying the dynamics of BNS systems in the presence of magnetic fields, because it leads to an exponential growth of the toroidal component of the magnetic field, independently of the original configuration of the magnetic field [8], which is not well-known [31]. Through this mechanism, the instability can lead to an overall amplification of the magnetic field of about three orders

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of magnitude [31]. Such high magnetic fields are presumed to be behind the phenomenology of magnetars [56] and short hard gamma-ray bursts [18]. After the merger, the cores of the two neutron stars coalesce. During their rapid infall, they experience a considerable decompression of ≈15%, for about 1 ms. This is visible in Fig. 2, roughly between 6 and 7 milliseconds. The long-term outcome of the merger can be either a stable neutron star or a black hole. The current estimate of minimum and maximum masses for neutron stars leads to think that most BNS merger would eventually end in a collapse to a black hole (a Kerr black hole, namely, a rotating black hole). Depending on the initial masses of the stars and on the equation of state (which is still largely unknown for rest-mass densities above twice the nuclear saturation density, which is  2.7 × 1014 g cm−3 or, equivalently,  0.16 nucleons per fm−3 ), this can happen promptly after the merger (namely, without any bounce of the stellar cores) or after an interval of time. The merger of BNSs with higher masses and/or softer equations of state ends in a prompt collapse to a black hole, while binaries with lower rest masses and/or stiffer equations of state produce a merged object that does not collapse immediately. In the latter case, the remnant would initially be an unstable, differentially rotating stellar object undergoing violent oscillations. Differential rotation (see, e.g., Ref. [50] for a description of the difficulties of measuring the degree of differential rotation in numerical-relativistic simulations) and thermal gradients [11] prevent prompt collapse also for remnants with masses above the upper limit for static neutron stars (or, more precisely, for uniformly rotating neutron stars (see below), whose mass limit is slightly higher than that of nonrotating neutron stars). Such configurations resulting from BNS mergers are often called hypermassive neutron stars (HMNS), even if the original term refers to equilibrium models [11], while the merger remnant is not in equilibrium at this stage. Supramassive neutron star (SMNS) is used to refer to a uniformly (also said rigidly) rotating neutron star with rest mass larger than the upper limit for static neutron stars in equilibrium. Strictly speaking, also the term SMNS refers only to equilibrium configurations [32]. Remnants supported against collapse by differential rotation and thermal gradients may last for up to ∼1 s, during which time cooling through neutrino emission, angular-momentum transport associated with magnetic-field effects (such as the magneto-rotational instability and magnetic braking), and the gravitational torque resulting from its non-axisymmetric structure act to drive the remnant toward collapse or toward a uniformly rotating neutron star (see, e.g., Ref. [52]). This would likely be a SMNS, which would then collapse to a black hole on timescales of ∼10 − 104 s, after losing angular momentum through secular mechanisms likely related to electromagnetic emission [77]. If, instead, the total mass of the remnant is smaller than the upper limit for a static neutron star, the merged object may never collapse, namely, it may become a stable neutron star. This is probably a marginal case. See, for example, Fig. 1 of Ref. [7] for a schematic view of what was said in this paragraph about the possible outcomes of the merger. Given the choice of masses of the initial configuration and of the equation of state, in the simulation shown in Fig. 1, a black hole forms immediately after the merger. This is evident in the rapid drop of the maximum rest-mass density marked

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at around 8 ms as a vertical dashed line in the left panel of Fig. 2. Note that the drop in the maximum value of the rest-mass density appears in this type of graphs because the region in which such maximum is searched is restricted to the outside of the apparent horizon, whose location is shown as a dashed line in two lower panels of 198 Fig. 1. Note that black holes are defined through their event horizons, which are surfaces bounding a region in spacetime inside which events cannot affect an outside observer. In numerical computations, it is possible but not easy to find event horizons, because information from all times are necessary for this, and so the event horizon can be computed only at the end of the simulation. Instead, apparent horizons are usually searched for in numerical simulations. These are surfaces that are the boundary between light rays that are directed outward and move outward and those that are directed outward but move inward. Apparent horizons are a local concept, and thus they can be computed immediately for each time step of a simulation, with efficient techniques (see Ref. [92] and references therein). The same initial masses but a different equation of state were adopted in the simulation that was used to produce the right panel of Fig. 2. Here, the merged object undergoes a few violent non-axisymmetric oscillations before collapsing. In order to describe stellar deformations, we employ the widely known decomposition of the linear perturbations of the energy (or rest-mass density) as a sum of quasinormal modes that are characterized by the indices (l,m) of the spherical harmonic functions. m here is the dominant term of such expansion: m = 0 is a spherical perturbation, m = 1 is a one-lobed (or one-armed) perturbation, and m = 2 is a bar-shaped perturbation (see Ref. [71] for a review on these topics). In the case of BNS mergers, the dominant overall deformation has m = 2, i.e., it is a bar-like deformation. As the bar rotates, it further loses energy and large amounts of angular momentum through gravitational radiation. m = 1 deformations have also been reported [71]. Together with these oscillations, a secular increase of the central rest-mass density is observed (cf. right panel of Fig. 2). Then, about 14 ms after the merger, the maximum rest-mass density is seen again to increase rapidly, and the object collapses to a rotating black hole. This is because the stellar object has lost enough of its differential rotation that supported it against collapse. The collapse is marked by the appearance of an apparent horizon. When an apparent horizon is born, a large amount of high—angular-momentum matter remains outside of it in the form of an accretion disc. On average such accretion discs have a peak rest-mass density around 1012 − 1013 g cm−3 , a horizontal extension of a few tens of km, and an aspect ratio of 1/3. The initial rest mass of the disc varies in the range 0.001 − 0.2M and is an important quantity to be determined in simulations, since it is related to the ejecta and to electromagnetic emission in general. In particular, the existence of a massive disc around the newly formed rotating black hole is a key ingredient in the modelling of short gamma-ray bursts [18]. Snapshots of the evolution of the disc in the simulation leading to prompt collapse mentioned above are presented in Fig. 3, which shows isodensity contours on a meridional plane. Note that the panels refer to times between the times of the last two panels of Fig. 1. Overall, the disc has a dominant m = 0 (axisymmetric)

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structure, but because of its violent birth, it is far from an equilibrium and undergoes large oscillations, mostly in the radial direction. In order to help and interpret observations, we need solutions of the generalrelativistic equations describing at least the dynamics of spacetime, matter, and magnetic fields, and possibly additional ingredients like neutrino radiation. As everyone knows, analytic solutions of astrophysical relevance for BNS systems are not available; therefore, a lot of effort has been put into the field of numerical relativity – the science of simulating (solving) general-relativistic dynamics on computers – and a sizeable amount of reliable results have appeared. Numerical relativity is now mature, but at the beginning, a couple of decades ago, it has struggled to get decent results because straightforward discretization of the Einstein equations just does not work. In addition to the standard problems of any numerical simulation, there are multiple reasons for the increased difficulty inherent to generalrelativistic simulations: (i) The formulation of the equations is not self-evident; e.g., time is not simply defined, and very careful definitions of variables are needed to obtain a system that is strongly hyperbolic; (ii) Physical singularities may be present and need special treatment; (iii) While not carrying physical information, gauges play an important role in numerical stability, for example, in countering grid stretching. For explanations on all these issues, well-written textbooks are nowadays available [4, 10, 49, 82, 83]. Despite such difficulties, nowadays, the number of research groups with their own independent codes capable of performing (at least in some respects) state-ofthe-art BNS simulations is of the order ten. All codes can robustly compute the matter and spacetime dynamics (including tens of orbits before the merger and longterm evolutions of the formed black hole and accretion disc), and improvements are being constantly made. The selection of appropriate gauges and the extraction of gravitational-wave signals from the dynamics are nowadays routinely done by everyone in the field. The equations of state used in simulations are based on published work from specialists in nuclear theory and so on and are either piecewise polytropes [78] (often with the addition of a thermal part) or tabulated. Also the first general-relativistic simulations of merging neutron stars including quarks at finite temperatures have been performed recently [15, 16, 47, 67]. Some robust results have been obtained also in computations of ejecta from BNS mergers (especially for estimates of heavy-element production and macronovae light curves) [84] and in the general-relativistic treatment of physical viscosity for BNSs [38]. Some other issues, instead, are still very open, in particular the effects of magnetic fields and of radiation transport. Simulating magnetic fields is challenging because of physical instabilities that require very high resolutions to be resolved and because of limitations in the modelling of electromagnetic interactions. Most simulations, in fact, are carried out in the magnetohydrodynamics approximation, which does not capture all the physical effects, like upper limits to the growth of instabilities. Resistive-magnetohydrodynamics simulations exist in small numbers [7], but they are limited by our lack of knowledge about the resistivity of matter in and around neutron stars. The open problems with magnetic fields in BNS mergers apply especially to the post-merger phase, where magnetic fields may have a huge

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importance for the dynamics, for the ejecta, and for electromagnetic emissions from the vicinity of the merged object (like those thought to produce short gamma-ray bursts). Before the merger, magnetic fields are not relevant for the global dynamics, but may produce observable electromagnetic radiation, as found in works employing resistive magnetohydrodynamics [7]. Effects of neutrinos and in general of radiation transport are also important for the production of ejecta and direct electromagnetic emissions, and advances are progressively being made (see, e.g., the introduction of Ref. [95]). Representative waveforms from the merger of different BNS systems are shown in Fig. 4, taken from Ref. [58]. Reflecting the dynamics of matter as described above, gravitational waveforms increase in amplitude and in frequency during the inspiral (the so-called chirp signal), while the waveforms after the merger, reflecting the oscillations of the merged object, are much more varied and in many cases terminate with the ringdown signal of the black hole, during which the distortions with respect to a Kerr black hole are damped in a characteristic gravitational-wave signal. The ringdown signal for black holes formed in BNS mergers is at frequencies of several kHz and so not easily measurable by current detectors. The post-merger signal from before the collapse is at lower frequencies ( 2 − 4kHz) than the ringdown, but still so high that detection is probably limited to close sources (see, e.g., Refs. [40, 94, 99] and references therein). This is in spite of the fact that more energy may be emitted in gravitational waves in this phase than in the inspiral. The inspiral signal can be better measured in current detectors, because of its duration

4

×10−22 IF_q10

IF_q08

H4_q10

H4_q08

3

h+ 22 at 100 MPc

2 1 0 −1 −2 −3

f [kHz]

−4 6 5 4 3 2 1 0

−5

0 5 (t − r) [ms]

10

−15

−10 −5 0 (t − r) [ms]

5

−10

−5 0 5 (t − r) [ms]

10

15

−10

0

10 (t − r) [ms]

20

Fig. 4 Some representative plots of the gravitational-wave strain (l = 2; m = 2 mode only) taken from Ref. [58]. Signals are for (from left to right) an equal-mass binary evolved with an idealfluid equation of state, a binary with a mass ratio of the components of q = 0.8 evolved with an ideal-fluid equation of state, an equal-mass binary evolved with the H4 equation of state (see Ref. [58] for details), and a binary with a mass ratio of the components of q = 0.8 evolved with the H4 equation of state. The top panels show the strain at nominal distance of 100 Mpc. The lower panels show the instantaneous frequency. (From Ref. [58])

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(tens of seconds or even more) and its frequency range of up to ∼1 kHz. Detectors that are currently active (Advanced LIGO [1], Advanced Virgo [2], KAGRA [55], under construction (Indigo/LIGO India [42]) and most of those that are planned (LIGO Voyager [80], Einstein Telescope [74], Cosmic Explorer [80]), in fact, are usually projected to have maximum sensitivity around 1 kHz. Before concluding this introductory section, it is useful to introduce the socalled universal relations, which, in this field, are empirical relations (found through numerical-relativity simulations) between physical quantities of neutron stars and quantities observables in gravitational-wave measurements. Universal here means approximately independent of the equation of state (and of mass); thus, a better naming for them would be quasiuniversal relations, and indeed this term has started being used recently. There are two types of such relations: relations that connect different physical quantities of a neutron star isolated or in a binary system (like radius, mass, moment of inertia, and tidal deformability) among themselves and relations that connect physical quantities of BNS systems (like the compactness C ≡ M/R of the component stars) with quantities that can be measured directly (like the main frequency of the post-merger gravitational-wave spectrum) in gravitationalwave detectors. Famous relations of the former type are the I -Love-Q relations [98], which connect the moment of inertia I , the Love number [which is related to the tidal deformability; see Eq. (3)], and the (spin-induced) quadrupole moment Q of a neutron star. Such universality may originate from the fact that these relations depend most sensitively on the internal structure far from the core, where all realistic equations of state are rather similar. The universality is found to hold at the 1% level, except in exotic cases, like bare solid quark stars (but even in this example, they hold at the 20% level), but it may be affected by strong magnetic fields and large differential rotation [98]. Also, theories of gravity different from general relativity predict different relations and this fact can actually be taken advantage of for testing general relativity versus other theories of gravity [98], if one can obtain for a given neutron star independent measurements of two of the quantities involved in the relations, e.g., the Love number through gravitational-wave observations and the moment of inertia or the quadrupole moment through observations of binary pulsars or x-ray binaries. Other relations of this type are the C-Love relations [98] between the compactness and the Love number of a neutron star. They hold at the few percent level for all equations of state on which they were tried, including those with phase transitions [26]. Still another is the I − C (moment-of-inertia—compactness) relation, which was actually the first to be found, but possesses a lesser degree of equation-of-state insensitiveness (accurate at the 10% level) [98]. More general relations between the lowest few multipole moments of neutron stars have also been found [98]. The binary Love relations will be introduced in Section “Tidal Effects and Their Relation with the Neutron-Star Equation of State.” Belonging to the other type of universal relations is, for example, the relation between the tidal deformability and the frequency of merger of a BNS system, defined as the instantaneous gravitational-wave frequency at the time when the

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amplitude reaches its first peak [19, 59, 79, 81, 86, 94]. This instant is often defined as the time of merger. A similar relation was found between the dimensionless tidal deformability and the gravitational-wave amplitude at its first peak [59,79], between the dimensionless tidal deformability and the time between the merger and the first amplitude minimum of the waveform [94], and between the binary mass ratio and the first amplitude maximum of the waveform after the merger [94]. Another set of universal relations, with different degrees of reliability, has been found to connect the frequencies of the main peaks of the power spectral density of the post-merger gravitational-wave signal with properties (radius at a fiducial mass, compactness, etc.) of a spherical star in equilibrium (see Section “Gravitational Waves from the Merger and Post-merger Phases”). Other relations have been found between the threshold mass for prompt collapse, the maximum mass for a nonrotating neutron star and its radius [13, 14, 61] (see Section “Gravitational Waves from the Merger and Post-merger Phases”), between the total gravitational radiation emitted in a merger event and the angular momentum of the remnant [59] and between the total mass of the binary and the angular momentum in the remnant [13]. Still other universal relations have been found between the frequencies of the fundamental modes (f-modes) of oscillation of stars and certain combinations of the stellar mass and radius (see, e.g., Refs. [5, 30, 37, 51] and references therein). The best of these relations hold at the 1% level [30]. Combining these with the I -Love relation, it is possible to find a universal relation between the f-mode oscillation frequency and the tidal deformability [98]. As mentioned above, these universal relations are in fact not literally universal, since they have some (small) levels of dependence on the equation of state. Hence, if, on one side, using these relations in data analysis allows to perform estimates impossible otherwise, on the other side, they do include further uncertainties in the analyses. In some cases, the issue has been addressed by marginalizing over the equation-of-state variability, for example, when inferring the neutron-star radii of GW170817 employing binary Love and C-Love relations [28]. Since in current detectors statistical uncertainties in parameter estimation are much larger than systematic errors added by using these universal relations, such a marginalization procedure is not a noticeable handicap, but, in order to be useful with the higher sensitivity of (near) future detectors and the large number of expected observations, more accurate universal relations will be necessary [26]. Furthermore, doubts have been raised on the applicability of universal relations by extrapolation to regions where they have not been directly tested through numerical simulations [57]. On the other hand, the observations themselves will impose more and more stringent constraints on the equation of state, and the reduced allowed space of equations of state will allow to decrease the number of equations of state used for computing the uncertainties in the universal relations and thus decrease their variability [26]. In the following sections, I will focus on gravitational waves emitted in BNS mergers and their relation with the neutron-star equation of state. The two main methods to link the observed gravitational waves to the neutron-star equation of

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state involve either tidal deformations during the last orbits before merger or the spectra of the gravitational radiation emitted from the post-merger object (if it does not collapse to a black hole too soon). Tidal deformations have been measured from GW170817 [87] and GW190425 [90], but post-merger gravitational waves have not been observed yet.

Gravitational Waves from the Pre-merger Phase In this section, I will focus on inspiral gravitational waves and estimates on tidal deformations and the equation of state that can be inferred from them. The quantity central to this discussion is the dimensionless tidal deformability Λ (sometimes also called tidal polarizability): an equation-of-state—dependent, dimensionless function of the neutron-star mass that correlates with the pressure gradients inside the star, namely, with the stiffness of the equation of state. In particular, tidal deformations cause a measurable phase shift in the waveform relative to the merger of point particles. Conversely, it has been shown that differences between equations of state of neutron stars that do not appear as differences in the tidal deformability are difficult to measure through inspiral gravitational-wave signals [79]. Note that the phasing of the gravitational-wave signal is significantly more important for parameter estimation than its amplitude [43]. Under our current knowledge, the dimensionless tidal deformability for neutron stars may take values between a few and ten thousands, depending on the stellar mass and equation of state. A larger tidal deformability corresponds to a larger, less compact star that is easily deformable, and vice versa. For a given neutron-star mass, the values of the dimensionless tidal deformability vary up to a factor of ten according to the equation of state assumed.

Tidal Effects and Their Relation with the Neutron-Star Equation of State Stars in a binary system undergo tidal deformations that become larger as the stellar separation decreases. These deformations affect the orbital trajectory of the binary and thus the emitted gravitational waves, encoding in the latter the neutron-star equation of state. A full general-relativistic formalism for tidal deformations has been developed [20,34,63,69], and higher-order tidal terms may have an impact for measurements with next-generation detectors [54], but for use in observations with currently available detectors, it is enough to describe tidal deformations through a single tidal deformability coefficient, defined as the proportionality constant λ between the quadrupole moment of the star Qij and the external tidal field Eij (the field generated by the companion star) Qij = −λ Eij

(1)

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(the same definition applies to Newtonian and general-relativistic theories; see also Ref. [73] for an introduction to tidal deformations in Newtonian theory). However, tidal deformations are more usefully described through the dimensionless tidal deformability Λ≡

λ , M5

(2)

where M is the stellar mass. Note that geometric units are used, in which G = c = 1. Equation (2) can equivalently be written as  5 R 2 , Λ ≡ κ2 3 M

(3)

where κ2 is the quadrupole Love number (whose values for different hadronic equations of state and masses are in the range 0.02 − 0.15) and R the stellar radius. This gives the quadrupole component of Λ, which can be calculated via the following expression (see, e.g., Refs. [34, 98] and references therein) Λ=

16 (1 − 2C)2 [2 + 2C(y(R) − 1) − y(R)] · 15  2C[6 − 3y(R) + 3C(5y(R) − 8)] + 4C 3 [13 − 11y(R) + C(3y(R) − 2) + 2C 2 (1 + y(R))] + 3(1 − 2C)2 [2 − y(R) + 2C(y(R) − 1)] ln (1 − 2C)

−1 (4) ,

where C ≡ M/R is the stellar compactness and y(r) satisfies the differential equation dy 6 4(m + 4π r 3 p)2 y2 r + 4π r 3 (p − ρ) + = − − y− dr r − 2m r r(r − 2m) r(r − 2m)2   ρ+p 4π r 2 5ρ + 9p + , r − 2m (dp/dρ)

(5)

where p and ρ represent pressure and mass density, respectively, and m(r) ≡

 1 − e−γ (r) r , 2

(6)

where γ (r) is in the definition of the metric coefficients ds02 = −eν(r) dt 2 + eγ (r) dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 ) .

(7)

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Equation (5) can be solved for a given equation of state together with the TolmanOppenheimer-Volkoff (TOV) equations [68], which describe spherically symmetric stars in static equilibrium in general relativity. Appropriate boundary conditions need to be imposed. In general, the extraction of higher-order gravitational wave parameters (like tidal deformabilities) from the gravitational-wave signal is difficult because these parameters can be efficiently estimated only in the late part of the inspiral, which may not be very long, and because there exist degeneracies between different higherorder parameters, like the individual neutron-star spins and the tidal deformability. This is why, for example, the tidal deformabilities estimated by the LIGO-Virgo Collaborations were separated in a low-spin and a high-spin scenario [87]. Moreover, because of the strong correlation between the two tidal deformabilities, ΛA and ΛB , associated with each star in a binary, it is challenging to extract them separately from the gravitational waveform, unless one assumes some empirical universal relations connecting ΛA and ΛB [26,97,98], which may add uncertainties to the estimates and which do not necessarily hold for equations of state with phase transitions (see, e.g., Refs. [26, 85] and references therein) or in the intervals where they are employed in the analyses [57]. What can be more easily directly measured from BNS waveforms is the dominant tidal parameter in the post-Newtonian expansions of the waveform. Post-Newtonian expansions are approximate solutions, valid for weaker fields, of the Einstein field equations. The expansion is made in parameters that are small when the approximation is valid, like the velocity v of the objects (for the binary systems considered here, it would be the relative velocity of the binary constituents) with respect to the speed of light or deviations from a background metric. A postNewtonian term of order n is proportional to v 2n relative to the leading-order term in the expression (see Ref. [21] for an introduction). In post-Newtonian expansions, the gravitational-wave signal can be calculated approximately by imposing that the power radiated by a binary system in gravitational waves is equal to the change in the energy of the binary. Then, the lowest-order tidal effects appear as a term of the fifth post-Newtonian order, whose coefficient is given the name of mass-weighted average tidal deformability (also called effective tidal deformability) and has the expression [44] Λ˜ =

16 (mA + 12mB )m4A ΛA + (12mA + mB )m4B ΛB 13 (mA + mB )5

(8)

16 (1 + 12q)ΛA + (12 + q)q 4 ΛB , 13 (1 + q)5

(9)

or, equivalently, Λ˜ =

where q ≡ mB /mA (≤ 1) is the ratio of the masses of the two stars in the binary.

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An alternative way to gain information on the tidal deformability from detector data would be to extract the (mass-independent) coefficients of a Taylor expansion of the tidal deformability about some fiducial mass (see, e.g., Refs. [97, 98] and references therein), usually 1.4M or 1.35M . This parameterization, however, can be efficiently applied only to systems whose neutron-star masses are close to the fiducial mass; otherwise, the systematic error on the leading tidal coefficient can dominate the statistical one. This drawback, on the other hand, may be compensated and overcome by combining the information from multiple events (also with different masses), which can be done easily with this method. However, a sensitivity higher than that of current detectors would be necessary to accurately measure any of these coefficients. In the following, I will focus on the use of the mass-weighted average tidal ˜ allows for better statistical deformability. Reducing the tidal parameters to one (Λ) estimations, but of course some physical information about the two stars is lost. As mentioned in Section “Introduction: General Description of Neutron-Star—Neutron-Star Mergers,” a way around this problem was proposed in Ref. [97], which showed the existence of an equation-of-state-insensitive relation (with variations of at most 20%) between symmetric and antisymmetric combinations of the tidal deformabilities

Λs ≡

ΛA + ΛB , 2

Λa ≡

ΛA − ΛB . 2

(10)

These binary Love relations [e.g., Λa (Λs )] allow to compute the individual tidal deformabilities from the mass-weighted average tidal deformability, but only for stars described by nucleonic equations of state, since these relations do not hold for equations of state with phase transitions when one star in the binary is a neutron star and the other a hybrid star (see, e.g., Refs. [26, 85] and references therein). These relations also depend on the mass ratio of the stars in the binary [57, 97]. Simple Fisher analyses have shown that the binary Love relations improve parameter estimation of the individual tidal deformabilities by up to an order of magnitude with respect to estimations done by extracting ΛA and ΛB from the data directly [97, 98]. Modified binary Love relations for specific purposes have also been proposed (see, e.g., Ref. [26] and references therein). It is important to note that the relationship between the tidal deformability Λ and the radius R cannot be simply derived from Eq. (3), namely, Λ is not necessarily proportional to R 5 , because the quadrupole Love number κ2 also depends on the radius in a complicated manner, different for each equation of state and determined by differential equation (5) coupled to the Tolman-Oppenheimer-Volkov equations. Empirical model-dependent estimates for such a relation found exponents between ≈5.3 and ≈7.5. Even if the correlation between radii and tidal deformability for different equations of state is tight, these two quantities provide complementary information, since for a given tidal deformability, different equations of state may lead to somewhat different radii.

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Although the mass-weighted tidal deformability depends strongly on stellar radius, if the chirp mass is specified (the chirp mass of a binary system is defined as Mchirp = (mA mB )3/5 )(mA + mB )−1/5 and is measured very accurately in observations; see, e.g., Ref. [22]), the mass-weighted tidal deformability is approximately independent of the component masses for a BNS merger [76, 96]. This is an empirical result, obtained by comparing values of the mass-weighted tidal deformabilities of a BNS system for various equations of state and for various combinations of individual masses such that the chirp mass is constant. This fact may be justified by using the I -Love-Q relations between stellar compactness and tidal deformability [98] (see section “Introduction: General Description of Neutron-Star—Neutron-Star Mergers”) to express the mass-weighted tidal deformability of the binary as a function of component masses and stellar radii.

Other Finite-Size Effects In addition to measurements based on the tidal deformability, other ways to gain information on the interiors of neutron stars from gravitational-wave observations have been proposed, even if they require higher sensitivities and thus may be applicable only when third-generation detectors like Einstein Telescope [74] and Cosmic Explorer [80] become operational. Some of these studies involve tidal excitations of resonant modes (see, e.g., Ref. [65] and references therein), gravitomagnetic excitations of resonant modes [45], resonant shattering of the neutron-star crust by tides [93], and nonlinear tidal effects [41, 72]. The measurement of stellar resonant modes in binary mergers may be more easily performed in the (rarer) case of eccentric binaries (cf. section “Introduction: General Description of Neutron-Star—Neutron-Star Mergers”), because tidal perturbations during pericentre passage can excite the stellar fundamental modes of oscillation, which have a time-varying quadrupole moment and therefore act as sources of gravitational radiation, on top of the inspiral emission. The f-mode signal depends on the equation of state (in general, stiff equations of state store more energy in the oscillations compared to soft equations of state [29]) and can significantly affect the phase of inspiral gravitational radiation by enhancing the loss of orbital energy by up to tens of percent. Part of the orbital angular momentum may also be transferred to the stars. Signals from eccentric systems are more difficult to detect with instruments like Advanced LIGO and Virgo, mostly because the power at frequencies around ∼100Hz, where the detectors perform best, is smaller (see, e.g., Ref. [48]). However, if accurate enough templates for eccentric BNS systems are used, parameters such as the chirp mass and sky localization may be estimated more accurately for these signals than for non-eccentric binaries [48], mostly because of the their richer structure that breaks parameter degeneracies. Detection of a few highly eccentric BNS mergers per year might be possible with third-generation detectors [29, 70] or even with the LIGO-Virgo-KAGRA detector network at design sensitivity [48]. Measurements of the frequency, damping time, and amplitude of the tidally excited

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f-modes could yield simultaneous measurements of their masses, moments of inertia, and tidal Love numbers and thus present a prime opportunity to test the I -Love-Q relations observationally. The equation of state also influences other post-Newtonian terms related to the deformations of a neutron star due to its spin (see, e.g., Ref. [36] and references therein). However, some of these terms can be approximated as functions of the tidal deformability of the star through the Love-Q relation (cf. section “Introduction: General Description of Neutron-Star—Neutron-Star Mergers”) and thus need not be necessarily treated as additional parameters.

Detectability and Detection of Pre-merger Gravitational Waves One needs to treat carefully many aspects of the basic idea delineated in the previous subsection when applying it concretely to gravitational-wave observational data. Estimating the parameters of BNS systems during the inspiral phase is based on matched filtering: the gravitational-wave data stream is cross-correlated with theoretically predicted template waveforms (approximants) for different possible physical parameters. These trial waveforms need to be accurate to allow for correct estimates of the stellar masses and spins, and of the internal structure of the stars. This is especially true for the very last orbits before the merger, where instead approximants become increasingly inaccurate. As it can be easily imagined, approximants that do not consider tidal effects are not sufficiently good, especially for spinning BNS systems or stiff equations of state. Furthermore, for spinning systems (spin parameter χ ≡ J /M 2  0.1, where J is the neutron-star angular momentum and M its mass), in order to reduce mismatches to an acceptable level, it is crucial to include spin-induced and equation-of-state—dependent higherorder terms in the waveform approximants (see, e.g., Refs. [36, 54] and references therein). Another issue is the fact that the computation of trial waveforms needs to be efficient and fast, because source properties are generally inferred via a coherent Bayesian analysis that involves repeated cross-correlation of the measured gravitational-wave strain with predicted waveforms. Computational efficiency is crucial also because long waveforms need to be computed, since BNS systems are visible by gravitational-wave detectors for a long time. A few types of waveform models have become the standard in gravitational-wave analysis: post-Newtonian, effective-one-body (EOB), [24] and the Phenom models [3]. Post-Newtonian expansions were introduced in Section “Tidal Effects and Their Relation with the Neutron-Star Equation of State.” Post-Newtonian models of gravitational waves suffer from a lack of complete information on the terms of the necessary orders of the expansion [43, 96]. In order to solve this problem, two classes of effective or phenomenological models for point-mass binaries that to some extent include all post-Newtonian orders in an approximate way have been developed: the effective-one-body model [24] and the so-called Phenom models [3]. Both approaches combine analytical

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results with information from numerical-relativity simulations, and while addressing the problems of the systematic errors of the post-Newtonian expansion, they may introduce other, smaller, systematic errors (which can be assessed by further comparison with numerical-relativity simulations). The Phenom models are phenomenological waveforms that approximate a set of hybrid waveforms constructed by matching numerical-relativity waveforms at higher frequencies with analytical post-Newtonian waveforms at lower frequency [3]. In general, waveforms that are constructed by matching numerical waveforms with some approximant of the general-relativistic equations are called hybrid waveforms. The fundamental idea of the effective-one-body method consists in representing the two-body dynamics by those of a single effective particle in an effective potential. In practice, the dynamics and gravitational waves from a binary are computed by solving the coupled system of ordinary differential equations for the orbital motion, gravitational-wave generation, and radiation backreaction in the time domain. Tidal effects have been fully incorporated in the effective-one-body model, including for (nonprecessing) spinning binaries (see, e.g., [54] and references therein). Refinements and calibrations of the models are performed by comparison with numerical-relativity simulations, which suggest that further improvements of the tidal effective-one-body models are still necessary for a satisfactory description of the signal [36]. Furthermore, effective-one-body waveforms still have a rather high computational cost [36]. Solutions to the latter problem may come from reduced-order-modelling techniques [62], which, however, also add further complexity, from additional inclusion of numerical-relativity results, or from other modelling techniques complementary to the effective-one-body model (see, e.g., Ref. [9]). The actual feasibility of measuring the tidal deformability (and the radius) of compact stars from BNS inspirals with current detectors has been shown in a long series of works, often based on Bayesian analysis and on the use of hybrid waveforms. Current estimates are that for neutron-star binaries with individual masses around 1.4M the dimensionless tidal deformability Λ could be realistically determined with about 10% accuracy by combining information from about 20−100 sources, depending on assumptions about the BNS population parameters (e.g., if one considers also nonzero spins for the initial neutron stars, the necessary number of sources is higher). These estimates are valid under the assumption of hadronic equation of state, while it would be more difficult to make statements about hybrid stars (see below in this section), because they differ from hadronic stars only at higher densities, which contribute less to the tidal deformability. On the other hand, if deviations from this assumption were found in the density regime of neutron-star coalescences, the transition mass could be estimated [27]. Estimates of the tidal deformability have indeed been possible by analyzing GW170817 and GW190425. In these two first detections, Bayesian analyses indicated that the systematic uncertainties due to the modelling of matter effects (choice of representation of the equation of state, of approximant) are smaller than the statistical errors in the measurement [87, 90]. However, further improved

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waveform approximants will be necessary to avoid related systematic errors in future higher signal-to-noise—ratio events (see, e.g., Ref. [35] and references therein). More comprehensive Bayesian analyses have been performed that include not only gravitational emission from GW170817 but also the related electromagnetic emission and other observational and experimental data (see, e.g., Refs. [33, 64] and references therein). The rest of this section focuses on how we might be able to infer knowledge on phase transitions in the equation of state, if they occur. Up to densities as high as twice the nuclear saturation density, the neutron-matter equation of state obtained by theoretical arguments and corroborated by laboratory experiments is expected to be reliable (see Ref. [17] for a review). Beyond that not much is known for sure, and as mentioned previously, matter could also undergo one or more phase transitions or a smooth crossover [17] to quark matter, possibly giving rise to a hybrid hadronquark star, in which the equation of state is purely nucleonic for lower densities and contains deconfined quark matter at higher densities. Phase transitions to hyperonic matter have also been widely investigated, and a short comment on them is at the end of Section “Investigating Phase Transitions with Post-merger Waveforms.” The possibility of more than one phase transition is also envisaged [17]. Such phase transitions would change drastically the properties of the star, in particular because they would produce drastic softening (or perhaps stiffening) of the equation of state. Therefore, neutron stars with similar masses but rather different tidal deformabilities may be observed and yield information about transitions to quark matter. Tens of works have analyzed GW170817 without assuming the absence of phase transitions and found that it is also consistent with the coalescence of a BNS system containing at least one hybrid star (for a review, see, e.g., Ref. [6]). The first numerical-relativity simulations of BNS mergers in which matter undergoes a hadron-quark phase transition have been performed recently [15, 16, 47, 67] (see Section “Investigating Phase Transitions with Post-merger Waveforms”). Most of these simulations employ equations of state in which deconfined quark matter appears only at higher densities, after the merger. Only one work has presented simulations of the inspiral of binaries composed of hybrid stars, pointing out that – at least in their first attempt – the numerical treatment of phase transitions in the inspiral requires higher resolutions than those sufficient for purely nucleonic binaries [47].

Gravitational Waves from the Merger and Post-merger Phases Of course, also gravitational-wave signals from the merger and post-merger phases of GW170817 (and GW190425) have been searched in the data from Advanced LIGO and Advanced Virgo, but no signal was found [87, 88, 90]. In fact, the strain upper limits set by the detector were found to be about one order of magnitude above the numerical-relativity expectations for post-merger emission from a hypermassive neutron star at the distance of GW170817.

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Strong theoretical and observational interest in the post-merger phase originate mainly from their connection with ejecta powering macronovae and from the fact that such observations would probe densities higher (several times the nuclear density) than typical densities in inspiralling neutron stars and would also probe effects of temperature, which becomes much higher (up to ∼50MeV) after the merger. However, even though the energy emitted in gravitational waves after the merger may be higher than that emitted during the inspiral (if there is not a prompt collapse), since post-merger gravitational-wave frequencies are higher (from 1 to several kHz [71]), their signal-to-noise ratio in current detectors is smaller than that of the inspiral, and they are probably only marginally measurable by detectors like Advanced LIGO or Advanced Virgo, namely, only for extremely rare sources within ∼30 Mpc (see, e.g., Refs. [40, 94, 99] and references therein). Also on the theoretical side, the post-merger phase presents more difficulties. Numerical simulations of the merger and post-merger dynamics are more difficult than for the inspiral part, because of strong shocks, turbulence, large magnetic fields, various physical instabilities, neutrino emission, phase transitions, viscosity, and other microphysical effects. Therefore, their accuracy is not as good as for the inspiral. For example, currently, there exist no reliable determinations of the phase of post-merger gravitational radiation, but only of its spectrum (see, e.g., Ref. [23]). For these reasons, it is challenging to construct accurate templates of post-merger signals. The basic idea for obtaining information on the supranuclear equation of state from gravitational waves emitted after the merger is that the main peak frequencies of the post-merger power spectrum correlate, in a rather equation-ofstate—insensitive way, with properties (radius at a fiducial mass, compactness, etc.) of a zero-temperature spherical equilibrium star. These are one type of universal relations, introduced in Section “Introduction: General Description of Neutron-Star—Neutron-Star Mergers,” and actually depend on the spin of the stars in the inspiral and hold only approximately even for irrotational binaries [59]. These relations are being employed to estimate constraints on the neutron-star radius – and thus on the equation of state – from the post-merger signal of future observations (see Refs. [23, 94] and references therein).

Spectral Properties and Their Relation with the Neutron-Star Equation of State As shown in Fig. 5, the general structure of gravitational-wave spectra of BNS mergers consists of a rather featureless part at lower frequencies, stemming from the inspiral, and a part at higher frequencies where some peaks can be discerned. Through numerical simulations of BNS mergers, the frequency of the main peak, fmain peak , which corresponds to the fundamental mode of oscillation of the star and is around 3.5kHz in the case shown in Fig. 5, has been found to correlate with the tidal deformability [81, 86], or with the radius of a 1.6M neutron star [12] with the given equation of state. These relations are affected by the spin of the

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Fig. 5 An example of a gravitational-wave spectrum of an equal-mass BNS merger, with the SFHo

equation of state (see Ref. [59] for details). The quantity on the vertical axis is heff ≡ f (|h˜ + (f )|2 + |h˜ × (f )|2 )/2, where h˜ +,× (f ) are the (2,2) modes of the respective polarization of the gravitational waves extrapolated to infinity (see Ref. [59] for details). Different line types refer to data from simulations with different base-grid spacing Δx. The main peak frequency fmain peak is around 3.5kHz. The vertical dashed line points out the location of the secondary peak (see text) according to Ref. [81]. (From Ref. [59])

neutron stars in the inspiral (see, e.g., Refs. [19, 35]) but otherwise hold at the ≈10% level for equations of state without phase transitions and for both equal-mass and unequal-mass binaries [59]. The likely physical explanation of these empirical relations is that at a given separation, the tidal interaction is more attractive for larger values of the tidal deformability, and thus, such binaries merge at lower frequencies (larger separations). As a consequence, their remnants are less bound and have larger angular-momentum support and extension at formation and thus lower frequency of the fundamental mode [19]. For similar reasons, in eccentric binaries (see Sections “Introduction: General Description of Neutron-Star—Neutron-Star Mergers” and “Other Finite-Size Effects”), post-merger peak frequencies do not follow the abovementioned approximate universal relations with stellar properties like compactness [29, 70]. For eccentric orbits, in fact, the position of the main peak in the power spectrum has been found to vary with the eccentricity of the orbit, as a result of the different angular momentum of the merger remnant [70]. Figure 5 shows with different line types spectra obtained with different grid resolutions. The location of the main peak is estimated to have an error of about

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5% in numerical simulations at the employed resolutions [59]. The location of most other peaks depends, instead, more strongly on numerical parameters, as evident in the figure, and thus cannot be used for estimating physical properties. The only other exception may be the secondary peak marked by the vertical dashed line. This lower-frequency peak, however, appears only in (nearly) equal-mass BNS mergers and for softer equations of state [59]. When physically present and observed, its frequency might be used to gain information on the equation of state, according to a relation with the average compactness of the original stars in the binary [86] or with the tidal deformability [81]. Another peak in the spectrum may be present in some cases because of an m = 1 (see note on page 502) deformation of the merged object [71], especially in mergers of eccentric binaries [39]. This deformation, due to the so-called one-armed spiral instability, is found to be present generically in BNS simulations and to carry information about the equation of state, but its peak in the spectrum, located at about half the frequency of fmain peak , has a much smaller amplitude, and the prospects of observations in gravitational waves appear unlikely in current detectors. Thirdgeneration detectors may be able to target these signals. Note that the merger remnant rapidly evolves toward a more stable configuration, and thus, especially in the first milliseconds after the merger, post-merger frequencies change in time (see, e.g., spectrograms in Ref. [81]), albeit only slightly. In particular, the frequency of the main peak increases up to the formation of the black hole, while its amplitude decreases, as physically expected from the increase of axisymmetry and compactness of the merged object as it loses angular momentum. Hence, the spectral properties of the gravitational-wave signal can only be asserted reliably when the signal-to-noise ratio is sufficiently strong so that even these changes in time can be measured in the evolution of the power spectral densities. In light of these considerations, as mentioned earlier, the prospects for high-frequency searches for the post-merger signal are limited to rare nearby events. Another relevant frequency (but not visible as a peak in the spectra) is the frequency of the merger, already defined above as the instantaneous gravitationalwave frequency at the time when the gravitational-wave amplitude reaches its first maximum at the end of the inspiral. A relation has been found between the tidal deformability and the frequency of the merger [19, 59, 79, 81, 86, 94]. It holds at the 3% level [59], but only for equal-mass or very nearly equal-mass binaries [59, 81], and it has not been tested yet for magnetized and/or highly spinning binaries. A similar relation [59, 79] was found between the tidal deformability and the gravitational-wave amplitude at the time of merger. It has been shown to hold at the 4% level for equal-mass binaries [59]. These same qualitative relations are valid for eccentric binaries [29, 39]. Another tight relation holding at the 3% level has been found between the angular momentum of the remnant and the dimensionless tidal deformability [59], while the relations between the energy emitted in gravitational waves and tidal parameters are more loose [59]. A different way to gain knowledge on the mass-radius relation of neutron stars consists in using some empirical relations obtained from numerical simulations

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of BNS mergers between threshold mass for prompt collapse, maximum mass for a nonrotating neutron star, and its radius [13, 14, 61]. This method is said to be more robust because whether a merger ended in prompt collapse or not may be assessed not only through gravitational radiation but also through associated electromagnetic emissions, which may be observed more easily. The only other quantity this method relies on is the chirp mass, which can be measured precisely from inspiral gravitational waves.

Detectability of Post-merger Gravitational Waves For data-analysis standards and in comparison to the inspiral, numerical simulations of the post-merger phase are still sparse and not accurate enough (as explained above), and this contributes to the difficulty of creating analytical, physically parameterized waveform templates, and this in turn reduces the feasibility of matched filtering. However, methods based on numerical-relativity simulations for generating reliable post-merger spectra or entire inspiral-to-post-merger waveforms rapidly enough to be useful for templated data analysis have begun to appear (see Refs. [23, 40, 94] and references therein). Generic analyses that target signals of unknown morphology might be less efficient than matched filtering, but they have been shown to be able to extract the main features of post-merger signals, such as its main frequency components (see, e.g., Ref. [94] and references therein). The full LIGO-Virgo network operating at design sensitivity may provide in the future reasonable estimates of the dominant post-merger oscillation frequency [87]. In about 1 year of operation of third-generation detectors [74, 80], the dominantpeak frequency may be measured to a statistical error of the order of 10Hz for certain equations of state, corresponding, through universal relations, to a radius measurement to within tens of meters [99]. With further improvements, it will probably also be possible to extract subdominant frequencies. It is to be noted that in this kind of analyses, the error on the radius is currently dominated by the scatter in the universal relations used, rather than the statistical error of the signal reconstruction itself for signal-to-noise ratios larger than ∼10 (see, e.g., Refs. [23, 40, 94, 99]).

Investigating Phase Transitions with Post-merger Waveforms Since the densities reached after the merger are larger than those in the original stars in the binary, it is possible that phase transitions occur only after the merger. In this case, a measurement of the tidal deformabilities, of course, cannot contain information on phase transitions. The case in which such phase transitions occur in both or one of the stars in the binary already before the merger has been briefly treated at the end of Section “Detectability and Detection of Pre-merger Gravitational Waves.”

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The easiest way to identify a phase transition that has a sufficiently strong discontinuity in the density, like one from nucleonic to deconfined quark matter, consists in inspecting the main frequency of the power spectral density of the postmerger phase, fmain peak , which may be different from the one inferred from the measurement of the tidal deformability in the inspiral (namely, before the phase transition occurs) or may change rapidly during the post-merger phase because of the abrupt speedup of the rotation of the differentially rotating core of the remnant [16]. This happens because the formation of a quark-matter core makes the remnant more compact. Such a shift of the dominant post-merger gravitationalwave frequency might be revealed by future gravitational-wave observations using second- and third-generation gravitational-wave detectors and would also allow to constrain the density at which the phase transition occurs [16]. The lifetime of the merged object before collapse and the black-hole ringdown waveforms may also be rather different from the ones expected from purely nucleonic equations of state [67], and, in addition to changes in gravitational waves, the rearrangement of the angular momentum in the remnant resulting from the formation of a quark core could be accompanied by a prompt burst of neutrinos followed by a gamma-ray burst. However, a preliminary study found that there would be no significant qualitative differences in the electromagnetic counterpart of neutron star mergers between a system undergoing a phase transition to quark matter and purely hadronic mergers [15]. All these indications discussed above come from simulations of merging neutron stars described by equations of state that include a phase transition [15, 16, 67]. Up to this point, I have focused on equations of state with phase transitions to deconfined quark matter. Another type of transitions that has received a lot of attention is the one to hyperonic matter. However, numerical-relativity simulations indicate that for BNS systems described with hyperonic equations of state, both the inspiral and the post-merger gravitational waveforms are, in the foreseeable future, indistinguishable from those obtained with the corresponding purely nucleonic equation of state [16, 75]. Amplitude, phase modulation, and total luminosity (but not frequency) in the post-merger gravitational-wave signal are different for hyperonic equations of state, but these quantities are both more difficult to measure and more difficult to obtain accurate estimates of from numerical simulations [75].

Cross-References  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range  Introduction to Gravitational Wave Astronomy  Isolated Neutron Stars  Numerical Relativity for Gravitational Wave Source Modeling  Post-Newtonian Templates for Gravitational Waves from Compact Binary

Inspirals  Principles of Gravitational-Wave Data Analysis  Repeated Bursts

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Acknowledgments Partial support has come from JSPS Grant-in-Aid for Scientific Research (C) No. T18K036220.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Gravitational Wave Emission from Rotating Sources . . . . . . . . . . . . . . . . . . . . . . Multipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition and Material Properties of Compact Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phases of Dense Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigidity and Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Transport and Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Waves Due to “Mountains” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crustal Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic Matter and Core Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Seismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Oscillation Modes and Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f- and r-Modes in Newborn and Young Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r-Modes in Recycled Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-messenger Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave-Driven Spin Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact of Oscillation Modes on the Thermal Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous GW Searches and Current Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Present Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B. Haskell () Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] K. Schwenzer () Science Faculty, Department of Astronomy and Space Sciences, Istanbul University, Istanbul, Turkey e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_12

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Electromagnetic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Neutron star interiors are a fantastic laboratory for high-density physics in extreme environments. Probing this system with standard electromagnetic observations is, however, a challenging endeavor, as the radiation is only emitted by the outermost layers and is scattered by the interstellar medium. Gravitational waves, on the other hand, while challenging to detect, interact weakly with matter and are likely to carry a clean imprint of the high-density interior of the star. In particular, long-lived, i.e., “continuous” signals from isolated neutron stars can carry a signature of deformations, possibly in crystalline exotic layers of the core, or allow to study modes of oscillation, thus performing gravitational wave asteroseismology of neutron star interiors. In this article, we will review current theoretical models for continuous gravitational wave emission, and observational constraints, both electromagnetic and gravitational. Finally, we will discuss future observational possibilities. Keywords

Dense matter · Neutron stars · Asteroseismology · Multi-messeneger astrophysics

Introduction Once detected, continuous gravitational wave (GW) radiation could provide a novel window to the cosmos that would perfectly extend our present electromagnetic spectrum to sb-kHz frequencies. In contrast to observed transient gravitational wave events [1], these would more closely resemble ordinary astronomical sources in that they would allow us to perform precision long-term observations that could probe the presently inaccessible interior of dense compact objects [2]. In this chapter, we give a short overview of this promising subject, ranging from their theoretical description, the properties of the compact sources that emit them, and possible emission mechanisms, as deformations and oscillation modes, to the indirect impact of such an emission on electromagnetic multi-messenger observations, as well as current searches and the direct constraints from their present non-detection.

Continuous Gravitational Wave Emission from Rotating Sources The creation of gravitational waves, of a size that could potentially be detected, in general requires huge masses moving with large velocities in an asymmetric way. These requirements are presently only fulfilled for compact sources, like black

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holes and neutron stars. Whereas featureless black holes cannot produce a sustained emission over longer time intervals, in neutron stars, there are known processes that could power a continuous gravitational wave emission. Since compact sources are very degenerate systems close to the ground state of matter, dynamical processes that could cause a gravitational wave emission are mostly absent. There are then basically two possibilities for continuous gravitational wave emission, namely, that the star is statically, asymmetrically deformed and the rotation causes a time-dependent quadrupole moment, or there are oscillation modes of the star [3] that are unstable to gravitational wave emission. However, dramatic events like the core bounce at birth, special events during the evolution of the source, e.g., phase transitions, crust quakes and glitches, accretion, or tidal deformations before merger, can excite transient oscillations that could be detectable. The main modes that can become unstable in compact sources are the fundamental (f-) modes and the Rosby (r-) modes. F-modes are polar modes, whereas r-modes are axial modes, which are only present in rotating stars. They are obtained from a solution of the perturbed hydrostatic equation in a slow rotation expansion. R-modes to leading order in a Newtonian approximation do not involve compressions of the star and are given by a simple analytic expression for the velocity perturbation, independent of the detailed structure of the source [4] δv = αRΩ

 r m R

iωt YB mm (θ, ϕ) e

(1)

where R is the radius of the source, YB are magnetic vector spherical harmonics, and α is a dimensionless amplitude parameter. Moreover, they have in this approximation a fixed canonical relation between oscillation and rotation angular velocity ω = 4/3Ω, although note that there will be corrections to this relation due to rotation and general relativity.

Multipole Radiation Since there is no way to observationally resolve compact objects, they present perfect point sources. Correspondingly, the gravitational wave emission can be treated within a multipole expansion in terms of the leading coefficients. Gravitational waves are described by the deviation of the metric from its flat Minkowski form. The general expression for the transverse traceless deviation reads [5]

hTijT

 l  ∞ dl G  dl E2 B2 = 4 Ilm (t − r)Tlm,ij + Slm (t − r)Tlm,ij (dt)l (dt)l c r

(2)

i=2 j =−l

where T E2 and T B2 are electric and magnetic tensor spherical harmonics and I and S are generalized mass and current multipole moments.

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The continuous gravitational wave radiation from compact sources typically has a quasi-periodic form with time dependence exp((iω − 1/τ )t), where ω is the angular velocity and τ the driving/damping time. Generally, 2π/ω  τ , so that a harmonic form for the gravitational wave is sufficient, and the evolution of the signal on a secular time scale τ only affects the amplitude of the gravitational wave signal and can be included subsequently, so that the multipole coefficients reduce to conventional time-independent multipole moments. The lowest of these modes generally dominate the gravitational wave emission. The relevant mass quadrupole moments for gravitational wave emission due to deformations (or “mountains”) of the star and f-modes as well as the current quadrupole moments for r-mode oscillations are for sufficiently slowly moving sources [5]

I2m S22

√  16 3π ∗ 2 3 = r d x τ00 Y2m 15 √  32 2π B∗ 2 3 = r d x (−τ0j )Y22,j 15

(3) (4)

respectively, where τ is the stress-energy tensor and Y are scalar and Y B magnetic vector spherical harmonics.

Gravitational Wave Strain As a measure for the detectability of gravitational wave signals, usually, the intrinsic gravitational wave strain amplitude h0 is used. It represents the amplitude that would be measured in the idealized case that a detector is positioned at one of the earth’s poles and observes a source vertically above it, whose rotation axis is parallel to that of the earth. For gravitational waves due to deformations of the star, the intrinsic strain amplitude reads [6]

h0 =

 m=−2,2

G 4π 2 4 c



5 f2 |I2m | 8π D

(5)

although note that in observational gravitational wave papers, a slightly different definition of the mass quadrupole Q22 = (δρ22 )r 2 dr is commonly used, with δρ22 the l = m = 2 component of the density distribution, such that h0 = 4π 2

G c4



8π f 2 Q22 15 D

(6)

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where f is the frequency of the wave (for a pulsar rotating at a frequency ν, f = 2ν), D the distance to the source, and Izz the relevant component of the moment of inertia tensor. Often, for “mountains”, this result is presented in terms of an ellipticity , defined as Ixx − Iyy Q22 = = Izz I



8π , 15

(7)

where Iii are the principal moments of inertia and Ixx ≈ Iyy ≈ Izz ≈ I . One can then simply write h0 = 4π 2

G f2 c4 D

(8)

For gravitational waves due to classical r-modes (Eq. (1)), the current quadrupole moment reduces to S22

√ 32 2π ˜ = J MR 3 Ωα 15c

(9)

where the dimensionless constant J˜ characterizes the source [7] via an integral over the energy density ρ J˜ ≡

1 MR 4



R

dr r 6 ρ

(10)

0

The relation between the oscillation and rotation angular velocity can be written as 2χ 4 ω = (κ(Ω) − 2) Ω ≡ − χ (Ω)Ω = − f 3 3π

(11)

where the function χ (Ω) weakly depends on Ω and the constant χ = χ (0) ≈ 1 parametrizes the leading deviation from the canonical relation due to rotation and general-relativistic corrections. The intrinsic strain amplitude takes then the form [8]

h0 =

2 3 29 G ˜ 3χ f α J MR D 36 5π 5 c5

(12)

The likelihood of detecting continuous gravitational waves in a particular search obviously increases with the observation time Δt. This is usually taken into account by giving the equivalent intrinsic strain amplitude that would be observed within a given search at 95% confidence level. For a coherent search, it is given by

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 h95% 0

=Θ

Sh (f ) . Δt

(13)

in terms of the power spectral density Sh of the noise and a factor Θ that depends on the particular search (Θ ≈ 11.4 for known pulsars and Θ ≈ 35 for sources without timing data [9]).

Composition and Material Properties of Compact Stars Phases of Dense Matter Continuous gravitational waves are very promising as probes for the otherwise inaccessible interior of compact sources [2], whose composition is still largely unknown. Even though the theory of matter, quantum chromodynamics (QCD), is known and intensely studied [10], the particular phase(s) realized at compact star densities is not. There are dozens of proposed potential phases and countless further uncharted possibilities [11]. Yet, they form two qualitatively distinct classes, namely, forms of hadronic matter composed of the familiar hadronic particles (neutrons, protons, etc.), which also make up ordinary matter, and deconfined quark matter, whose long-distance carriers of conserved quantum numbers are the underlying colored quark constituents. That is, in the latter case, quarks are not confined, and dynamical mass generation due to non-perturbative QCD effects, which is responsible for 95% of the mass of atomic matter, could be dramatically reduced. The simplest and prototypical case for the composition of the interior, responsible for the term “neutron star,” is npeμ-matter, i.e., a liquid of nearly 90% neutrons and around 10% protons as well as an equal number of leptons to electrically neutralize the star [2]. Similarly, on the quark side, the simplest option is strange quark matter consisting of u, d, and s quarks neutralized by a small fraction of electrons. The first form of matter is expected at sufficiently low densities, whereas forms of quark matter are theoretically predicted to be realized at asymptotically large densities that are significantly above the density range accessible in the core of compact stars. In this regime, the novel phenomenon of color superconductivity [11] should strongly determine the properties of the corresponding matter. In both limits, there are controlled approximation schemes, namely, chiral perturbation theory and the weak coupling expansion owing to the asymptotic freedom of QCD [10], and these can to some extent constrain bulk properties in the intermediate regime. Nevertheless, the potential options for the composition and symmetry structure at realistic densities are staggering due to the large number of hadrons on the hadronic side (n, p, hyperons, resonances, pions, etc.) and the potential symmetry breaking patterns on the quark side. Both the presence of hadrons and the symmetry of the ground state are determined by the strong interactions in a dense medium which are still only poorly understood due to the inherent complexity to solve QCD at finite

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density [10]. Gravitational wave astrophysics could therefore offer a unique way to probe the dense interior and learn about fundamental properties of matter and the realized phase(s) at neutron star densities.

Rigidity and Shear Modulus The exterior layers of the neutron star, at densities below ρ ≈ 1014 g/cm3 , constitute the crust of the star. Once the star has cooled to temperatures below a few times 109 K, after approximately a minute from birth, the crust solidifies and forms a crystal structure. The composition of the nuclei in the crystal depends on density, with increasingly neutron-rich nuclei appearing at higher densities (see [2] for an in-depth discussion). For the present discussion, we are mainly interested in the crust’s rigidity, as this determines to what extent it can support a deformation, and eventually a GW emitting “mountain.” This information is encoded in the shear modulus, which can be estimated with Monte Carlo simulations of deformations, assuming the crust forms a body-centered cubic (BCC) lattice, leading to [12]  μ ≈ 1020

ρ 1014 g cm−3



g cm−1 s−2 .

(14)

Compared to the pressure, the shear modulus is weak, so the crust behaves more like a jelly than a traditional solid. To understand its relevance for gravitational wave emission, however, another quantity is important, and this is the breaking strain σb . The breaking strain quantifies the level of strain, defined, for example, by the von  Mises criteria as σ = tab t ab with tab the strain tensor, above which the crust will yield. For terrestrial materials, the breaking strain lies roughly in the range 10−5 < σbr < 10−2 , but molecular dynamics simulations suggest σbr ≈ 0.1 for a NS crust [13], which allows for a substantial “mountain” to build up and emit gravitational waves, as we shall see. Before moving on, let us remark that many equations of state predict “pasta” phases, in which nuclei form rods and plates and other structures, at higher densities close to the base of the crust. In this case, one would have a structure similar to a liquid crystal, with a much reduced shear modulus, and possibly breaking strain [12]. Similarly, there may be crystalline phases of quark matter in the deep core, which stem from a spatially periodic fluctuation of the color superconducting condensate [11]. Since the critical energy of color superconductivity is much larger than typical nuclear binding energies in the hadronic crust, this “solid” is much more rigid, can resist shear forces, and could support large “mountains” that would be detectable by ground-based GW detectors.

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Viscous Damping Dynamic material properties, like dissipation, can differ dramatically depending on the form of matter that is present inside a neutron star. The reason for this is that neutron stars are highly degenerate systems where low-energy processes are parametrically suppressed and depend sensitively on the degrees of freedom and the interactions between them. The dissipation of large-scale motion is described by the viscosity coefficients in a hydrodynamic description. In case the different microscopic degrees of freedom form approximately a single nonrelativistic fluid, the dissipated energy density is to leading order described by the shear viscosity η and the bulk viscosity ζ  

2 d 1 ≈ −η ∂i vj − ζ + η (∂i vi )2 dt 3

(15)

while more coefficients arise in relativistic, asymmetric, or multi-fluid cases [14]. The viscosity coefficients describe the dissipative momentum transfer in the fluid and are obtained from the underlying kinetic theory [14]. The shear viscosity stems from friction due to shear motion in the fluid itself or along a boundary [15]. Like most transport properties, it is dominated by the particle with the largest mean free path (i.e., being the lightest and/or having the weakest interactions), and the larger it is, the longer the latter is. In npeμ-matter, it is at neutron star temperatures generally dominated by leptons with long-ranged electromagnetic interactions [16]. In a quark star, quark scattering due to the strong interaction leads to a similar but smaller result. However, if fermionic degrees of freedom are absent due to Cooper pairing, like in the CFL phase [11], the shear viscosity can be drastically different. Bulk viscosity describes the dissipation in compression and rarefaction processes that push the system out of statistical equilibrium. It is a resonant processes and becomes large when the time scale of the underlying interactions matches the time scale of dynamic processes, like oscillations of neutron stars. In neutron stars, bulk viscosity typically stems from slow weak interaction processes, causing chemical equilibration. For small deviation δμ of the chemical potential from chemical equilibrium δμ  T , it has the general form [17] ζ =

δ C 2 Γ˜ T 2δ ω2 + B 2 Γ˜ 2 T

(16)

Here Γ = Γ˜ T δ δμ is the weak equilibration rate and B and C are susceptibilities determining the flavor dependence of the equation of state of dense matter. As seen in Eq. (16), the resonant maximum is tuned by the equilibration rate and therefore for a given angular frequency ω reached at rather different temperatures for different forms of matter. In npeμ-matter chemical equilibration proceeds via semi-leptonic Urca-processes. Yet, except for extremely heavy stars, direct processes are generally not kinematically allowed, and only slow modified Urca reactions are possible,

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where additional strong interactions guarantee momentum conservation. At neutron star temperatures, the dissipation is therefore far from the resonant maximum. In strange hadronic or quark matter in contrast also non-leptonic processes are possible which have a much larger rate, leading to a resonant maximum of the dissipation at temperatures present in neutron stars. The absence of low-energy fermionic modes again strongly alters the dissipation [11]. In phases separated by a sharp phase boundary, there is likewise an analogous boundary version of bulk viscosity stemming from phase conversion at the boundary.

Heat Transport and Cooling The dissipation of large-scale fluid motion can strongly heat the star. This heat which is generally nonuniformly created throughout the star has to be transported to ensure thermal equilibrium and eventually has to be radiatively lost again [2]. Therefore, also thermal properties of dense matter are important to assess the evolution and gravitational wave emission of continuous sources. The thermal conductivity, dominated by weakly interacting, relativistic fermionic particles, is typically very large in neutron stars unless degenerate fermions are absent. In npeμmatter, it is the electrons that dominate heat transport and in strange quark matter the quarks. Correspondingly, the core of such stars is typically in thermal equilibrium. Unless the temperature becomes very low, the cooling of the star is dominated by neutrino emission from the bulk of the star. If fermions are present, these are dominated by the same Urca processes that also cause bulk dissipation. In neutron matter, the modified Urca processes lead to slow cooling of the star [18]. In addition, the neutrino emissivity in superfluid and superconducting phases is enhanced close to the critical temperature. Finally, direct Urca processes at very high density present the fastest known cooling mechanism [18]. The latter are always present in quark matter so stars containing quarks would typically be significantly cooler than hadronic stars.

Gravitational Waves Due to “Mountains” The rigidity of the crust, discussed in the previous sections, is a key ingredient for one of the first proposed mechanisms to create a time-varying quadrupole on a neutron star. The idea is that the crust can sustain a “mountain” on the neutron star in much the same way as the terrestrial crust sustains mountains on Earth. Of course, due to the much larger gravitational pull, neutron star mountains are much smaller, with typical estimates of the heights of the order of δr ≈ 1 mm [12]. In a gravitational wave context, however, we have seen that the size of the mountain is usually discussed in terms of its ellipticity, defined in (7), assuming that the l = m = 2 multipole dominates the emission. In this case, the GW emission will be at twice the rotation frequency of the star, and the GW amplitude is

536

B. Haskell and K. Schwenzer −24

h0 ≈ 4 × 10

 ms 2  kpc   P

r

I 1045 g cm2



 10−6

(17)

with r the distance to the source and P the period of the star. Furthermore, the gravitational radiation carries away angular momentum and spins down the star at a rate ν˙ ≈ −2.67 × 10−8

 ms 5  P

I 1045 g cm2



2 , 10−6

(18)

which leads to a braking index n = ν¨ ν/˙ν 2 = 5. Note, however, that in the more general case where the star is a triaxial ellipsoid that is not rotating around one of its principal axis of inertia, one expects emission also in the l = 2, m = 1 harmonic, which leads to emission at the same frequency as the stellar rotation. The main theoretical models for such a scenario are precessing neutron stars and, as we shall see in the following, deformed magnetized neutron stars. Mountains can also be supported by the strong magnetic field of the NSs, which deforms the spherical shape of the star and, in fact, provides a lower limit to the size of the quadrupole. In accreting neutron stars, the flow of matter can bury the field and locally enhance its strength and confine a larger mountain, as well as source reactions deeper in the crust that can lead to deformations, as we will see in the following.

Crustal Mountains As already mentioned, the neutron star crust can support shear strains and thus a “mountain,” where in this section we will use this term generally to indicate a quadrupolar deformation of the star (naturally the overall angular structure of the mountain may be more complex, but the quadrupolar part will give the strongest contribution to the GW emission). In general, mature NSs are expected to be almost perfectly spherical, as during the life of the star, processes such as plastic flow and crustquakes are expected to release the strain in the crust and erase any mountain. Things are, however, different for neutron stars in Low Mass X-ray Binaries (LMXBs) that are accreting from a less evolved companion and being spun up in the process. In this case, light elements are being accreted onto the NS from the companion and slowly compressed to higher densities, where they undergo a number of reactions such as electron captures and pycnonuclear reactions [2] which release energy locally and heat up the crust, in a process known as “deep crustal heating.” When accretion outbursts end, the neutron star crust can be observed to cool, by observing the spectrum of LMXBs entering quiescence (see [18] for a review). If the heating is not entirely symmetric, due to asymmetries in the accretion process, and there are thermal or compositional gradients, this can source deformations

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of the crust and lead to a quadrupole, which can be estimated, for a single reaction layer, as [19]  Q22 = 3 × 10

35

4 R12

δTq 105 K



Eth 30MeV

3 g cm2 ,

(19)

where δTq is the l = m = 2 part of the temperature increase due to deep crustal heating, R12 the radius of the star normalized to 12 km, and Eth the threshold energy for the reaction layer we are considering. For the whole crust, one can estimate deformations of up to ≈ 10−6 [19], which is within the theoretical limits of what the crust can sustain before breaking, which is of the order of max ≈ 10−5 (σm /0.1), with σm the breaking stain of the NS crust, and from molecular dynamics simulations found to be of the order σm ≈ 0.1, as previously discussed [13]. It has, in fact, been suggested that gravitational wave torques, possibly due to mountains, are setting the spin limit of neutron stars in LMXBs [19]. All the NSs in these systems are spinning well below their Keplerian breakup limit, independently of the equation of state [20]. As the accretion torques in the system spin the stars up, additional torques must necessarily be at work, either due to the interaction between the accretion disc and the magnetic field of the star or due to gravitational waves, from mountains or r-modes, as we will discuss in detail in the following sections. Some systems show, in fact, tantalizing evidence of a mountain, as we will discuss in the following [21]. Recent models have estimated the height of a mountain that can be created in the outer layers of an accreting NS, due to asymmetric accretion and consequently asymmetric heating [22]. In general, the sizes are too small to be detected by ground-based interferometers such as LIGO and Virgo, but if additional shallow heating reactions are present, as suggested also by observations of the cooling of X-ray transients [18], they may be sizable enough to explain the observed phenomenology.

Magnetic Deformations It is well-known that magnetic stars are not spherical, as the Lorentz force leads to deformations of the stellar structure. If the magnetic axis of the star is not aligned with the rotation axis, this leads to a time-varying quadrupole and gravitational wave emission. In general, given an inclination angle α between the magnetic and rotational axis of the star, we expect emission both in the l = m = 2 harmonic at twice the rotation frequency and in the l = 2, m = 1 harmonic, at the rotation frequency Ω. The amplitude of the GWs scales approximately as [23] h0 ≈ h21 sin α cos α + h22 sin2 α ,

(20)

so we see that for large inclination angle, emission at twice the rotation rate in the l = m = 2 harmonic dominates, while for small angles, emission at the rotation

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frequency is dominant, and naturally, there is no emission if α = 0 and the magnetic and rotational axes coincide, so that the deformation is axisymmetric. For the sake of definiteness, we will focus on the Q22 harmonic in the following and discuss the related ellipticytes. However, the order of magnitude of Q21 is expected to be similar, and the discussion would be qualitatively equivalent. In fact, we shall see that in practice, many models simply calculate the axisymmetric deformation Q20 and assume that if the magnetic axis is inclined with respect to the rotation axis, one has Q22 ≈ Q20 . A large amount of work has been devoted to calculating the quadrupolar distortions of magnetized neutron stars (see [24] for a review), and the general result is that poloidal magnetic fields make the star oblate, with ≈ 10−12



2 B¯ , 1012 G

(21)

with B¯ the volume averaged magnetic field strength, while a toroidal field leads to a prolate star with ≈ −10−11



2 B¯ . 1012 G

(22)

The field in a neutron star is expected to be a combination of poloidal and toroidal components, a so-called twisted torus in which the poloidal component that threads the star and extends to the exterior is stabilized by a weaker internal toroidal component of the field, which in most models can account for up to 10% of the magnetic energy [24]. If the toroidal field is strong enough, the overall shape of the star is prolate. In a young neutron star, this leads to an instability which tends to “flip” the star, rapidly leading to inclination angles of α = π/2, given an initial angle α = 0, which is the optimal configuration for GW emission in the l = m = 2 harmonic [25]. In fact, if the outcome of a core collapse or a merger is a meta-stable magnetar, rotating with millisecond periods, such an instability will lead to it being a detectable source of GWs for next-generation detectors [26]. The effect of viscosity on secular timescales counteracts the GW torques and pushes the star back to alignment on longer timescales of hundreds of years. As the star ages, the crust also plays a role, as the field evolves due to the Hall effect, leading to locally enhanced magnetic patches which may lead to detectable deformations in young pulsars [27]. Additionally, as the temperature drops, the interior protons are also expected to become superconducting, leading to an ellipticity of [25]: ≈ 10−8



 ¯  B¯ Hc , 12 10 G 1015 G

where Hc is the lower critical field for superconductivity.

(23)

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Fig. 1 Observed periods and period derivatives of millisecond pulsars (the bottom left corner of the standard P − P˙ diagram. The solid lines represent gravitational wave spindown models with a given ellipticity, and the dashed lines also include magnetic dipole spindown. As shown in [28], the observed cutoff is consistent with sources being spun down by GW due to a mountain with an ellipticity of ≈ 10−9 and a weak magnetic field of B ≈ 107 G. This could signal the presence of a residual ellipticity due to a stronger buried magnetic field in the superconducting core of the stars

Furthermore, if the star has undergone episodes of accretion, the field may be buried, leading to a lower external dipolar field (inferred from measurements of spindown rate, if the NS is visible as a pulsar), but allowing for a substantial buried mountain [24], which may be detectable by next-generation GW detectors. In fact, the presence of a “minimal” ellipticity due to a buried magnetic field may explain the observed clustering and cutoff, shown in Fig. 1, in the P − P˙ diagram of millisecond radio pulsars, which are generally thought to be old NS spun up to millisecond periods by accretion in a binary [28].

Exotic Matter and Core Deformations The crust of the NS may not be the only region that can sustain shear deformations and a “mountain.” As we have seen in Section “Phases of Dense Matter,” at high densities, matter may undergo a number of phase transitions in which standard hadronic matter gives way to phases in which quarks are deconfined and possibly form diquark condensates that depend in a complicated way on the color and flavor

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degrees of freedom [11]. The exact nature of the pairing at NS densities is unknown, as discussed in the previous sections, but in several cases, crystalline phases may exist which allow for a sizable mountain. Early studies showed that such a “solid” core could harbor mountains of up to ≈ 10−4 , which are within reach of current detectors [24]. A more recent analysis [29] has shown that the presence of strange quark matter and type II superconductivity in the core leads to color-magnetic mountains, which for a purely poloidal magnetic field have ellipticities of 2SC ≈ 4 × 10−6 B¯ 14 ,

(24)

if only the u and d quarks are paired in a two flavor phase, and B¯ 14 is the average field strength in units of 1014 G, and CF L ≈ 1 × 10−5 B¯ 14 ,

(25)

if all three species of quarks are paired in a color-flavor-locked (CFL) phase. A detection of such a large ellipticity by the next generation of gravitational wave detectors may thus give us a direct indication of the state of matter at high densities in the interior of the NS.

Gravitational Wave Seismology Orthogonal Oscillation Modes and Instabilities Out of the various oscillation modes of a compact source [3], f-modes and r-modes can according to relativistic hydrodynamics become unstable to gravitational wave emission due to the Friedmann-Schutz mechanism [30]. This instability increases strongly with the oscillation frequency of these modes, which in turn is roughly proportional to the rotation frequency of the source. R-modes are special since they are in the absence of dissipation unstable at any frequency [31], while viscous dissipation can damp them at sufficiently low frequencies. F-modes in contrast generically only become unstable at frequencies close to the Kepler frequency of the corresponding source. Well-established dissipation mechanisms in standard neutron stars, e.g., the Ekman rubbing of the fluid along the solid crust [15], have been shown to be insufficient to damp r-modes in most millisecond sources. Therefore, r-modes could be present in many astrophysical sources, while f-modes could likely only be present in newborn neutron stars. Exotic phases with enhanced forms of dissipation, like the resonant behavior Eq. (16) of bulk viscosity in quark matter [32], or structurally more complicated stars, could damp r-modes in observed sources. In case a mode becomes unstable, its amplitude, conveniently described by a dimensionless amplitude parameter α (see Eq. (1)), grows exponentially. Whereas these modes are orthogonal at infinitesimal amplitude, they mix and couple to other (“daughter”-)modes at finite amplitudes [33]. This mechanism can dissipate an

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increasing amount of energy from the mode as its amplitude grows and can therefore in principle stop the growth and saturate the mode at a finite amplitude αsat . Other such saturation mechanisms include the nonlinear enhancement of bulk viscosity at large amplitude [17], or the resonant coupling to modes in the crust of the star [34]. Sources with saturated modes could be emitters of continuous gravitational waves over long time intervals – in the case of r-modes in millisecond pulsars potentially for billions of years.

f- and r-Modes in Newborn and Young Sources The evolution of the gravitational wave signal due to unstable modes has generally three phases [7]: (a) a steep rise during the exponential mode growth, (b) a plateau phase as the amplitude is saturated but the source does not appreciably spin down yet, and (c) a (potentially slow) power law decay as the source spins down. The gravitational wave emission ends as soon as the source spins out of the instability region. At high saturation amplitudes, the plateau phase might not be realized, resulting in a short but strong gravitational wave burst, whereas at low amplitude, the signal is very stable and hardly changes over long time intervals. F-modes can very likely only be present in newborn sources originating from a core bounce supernova or neutron star merger, in the case when the product spins initially close  to its Kepler frequency, which is given by the empirical formula fK ≈ 0.11 GM/R 3 in terms of the mass M and radius R of the nonrotating configuration [35] and is typically around a kHz. While in Newtonian gravity and for moderate mass sources the f-mode instability region typically extends only down to 0.9fK , in general-relativistic simulations of heavy sources, it might, depending on the composition, extend to frequencies significantly below 0.8fK [35]. The size and length of the gravitational wave signal depend both on the f-mode growth time and saturation amplitude. In supermassive mergers, the typically short live time of the merger product might prevent unstable f-modes to reach sizable amplitudes. The same holds for r-modes which typically have growth times of order seconds. Using mode coupling as saturation mechanism allowed to follow the evolution and predict that f-modes might lead to an observable signal for a sufficiently nearby source [36]. In the case of newborn sources in supernovae, the signal can last until the source spins out of the instability region. While this should still be reasonably quick for f-modes, depending on the saturation amplitude, r-modes could be present in young sources for long times [7]. At sufficiently late times (phase (c) above), the gravitational wave strain of a known source has the interesting property that it is independent of the spin frequency and exclusively determined by its age and its distance [37]  h0 ≈

2.3+3.5 −0.8

−27

× 10

1000 y 1 Mpc , t D

where the uncertainty stems mainly from the unknown saturation mechanism.

(26)

542

B. Haskell and K. Schwenzer

Based on the spindown evolution in the presence of r-modes (discussed below) and based on the observed timing behavior of observed young pulsars, it has been shown that nearly all of them should be stable to r-mode emission. A potential exception is the fastest spinning young pulsar J0537-6910 with a spin frequency of 62 Hz, which might just be in the last stage of its r-mode spindown. Correspondingly, r-mode spindown provides a quantitative explanation for the strikingly low spin frequencies of observed young pulsars [37]. Therefore, sources younger than the youngest known pulsars, which have ages around a thousand years, like the neutron star in Cas A or undetected sources associated with recent supernova remnants are very promising for r-mode gravitational wave searches, as shown in Fig. 2. In particular, the most recent nearby supernova SN1987A is interesting, since there are first indications for thermal emission from the corresponding neutron star [38].

r-Modes in Recycled Sources The progenitors of the millisecond radio pulsars, i.e., the neutron stars in low mass X-ray binaries (LMXBs) that are accreting matter and being spun up, are also an interesting target. In these systems, the neutron stars all appear to be spinning well below their theoretical breakup frequency, and gravitational wave emission due to unstable r-modes may provide additional spindown torques that set in once the system enters the unstable region. Figure 3 shows the instability region for a “minimal” neutron star composition, while the band estimates the uncertainties. However, from the figure, it is also clear that such a minimal model is problematic, as several millisecond sources are well inside the instability window. In such a scenario, the neutron stars would emit gravitational waves and, unless there is a very strong amplitude-dependent saturation mechanism that can saturate r-modes at low amplitudes, would be rapidly spun down out of the region, making it statistically very unlikely to observe so many [39]. The most likely option is thus that additional physics must be at work to damp the r-modes and modify the instability window. The presence of recycled radio pulsars spinning at around 700 Hz also confirms this conclusion, as these systems could not be spun up to such high frequencies if the r-mode would be unstable and reach a sizable amplitude, since the gravitational wave emission would be spinning them down. Several additional effects, including superfluid mutual friction, hyperon bulk viscosity, and coupling to crustal modes, may be at play and lead to a significantly different instability window at low temperatures, which could be probed by future gravitational wave observations. An intriguing possibility is exotic forms of matter, like quark matter [32], where the resonant enhancement of the bulk viscosity at neutron star temperatures could provide the dissipation to completely damp these modes in LMXBs. Taking into account that exotic forms of matter or structural complexity increase the damping compared to a minimal neutron star composition and considering the

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Fig. 2 Expected intrinsic gravitational wave strain amplitude due to r-modes (Eq. (26)) for various young sources [37], where the error bands stem from the unknown source properties. The frequency range is obtained from an r-mode evolution and reflects the range of the unknown saturation amplitudes, where the dots denote the corresponding values for different orders of magnitude from αsat = 0.1 to 10−3 . These estimates are compared to detector sensitivities of advanced LIGO in the standard mode (dashed) and the neutron star optimized configuration (dot dashed), as well as (dotted) to the Einstein Telescope (ET configuration B from http://www.etgw.eu/index.php/etsensitivities). These sensitivities are given both for a known pulsar search with 1 year of data (thick) and a search for potential sources without timing information using 1 month of data (thin). The uppermost curve shows the sensitivity of a previous Cas A search

various uncertainties, a general condition for the presence of r-modes has been derived. For a source with an X-ray and/or UV observation or upper bound for the (un-redshifted) surface temperature Ts , r-modes are absent if it spins with a frequency [40]    Ts Hz f ≤ 389+165 −116 105 K

(27)

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B. Haskell and K. Schwenzer

Fig. 3 Boundary of the r-mode instability region for a standard hadronic neutron star, taking into account the uncertainties from the micro-physics and unknown source properties, compared to thermal data of LMXBs taken from [39, 41]. The main source of damping at high temperatures is bulk viscosity and at low temperatures shear viscosity, but they are insufficient to damp r-modes at high frequencies in the enclosed instability region. Dissipation in an Ekman layer between the crust and the core increases the damping [15], but, using the appropriate shear viscosity [16], it can even in the shown, unrealistic best-case scenario not damp r-modes in the fastest sources [32]. Despite the large errors in temperature (due to the unknown composition of the outer envelope, which must be modelled to infer the core temperature), it is clear that many systems lie in the unstable region for such a basic NS model

While for most millisecond pulsars r-mode emission cannot be excluded based on present X-ray data, the only two millisecond pulsars PSR J0437−4715 and PSR J2124−3358 with actual temperature measurements are very likely r-mode stable. Another possibility is that the observed sources are indeed within the instability region and emit continuous gravitational waves over very long time intervals, but the saturation amplitude for the r-mode is so small that gravitational wave emission does not impact on either the thermal or spin evolution of the stars. However, both the observed spindown behavior and bounds on the thermal X-ray emission, discussed above, set stringent bounds on the r-mode amplitude. These bounds [40] are by

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now several orders of magnitude below theoretical predictions for well-constrained saturation mechanisms, like mode coupling [33]. As discussed, this requires a very strong but yet unidentified source of dissipation within these stars to saturate or even completely damp r-modes in these sources.

Multi-messenger Observations Gravitational Wave-Driven Spin Evolution As discussed in more detail, below, the detectability of gravitational waves from compact sources is strongly increased for known pulsars, where timing data is available. Since gravitational waves from compact sources generally carry angular momentum, the emission spins down the source. In case sizable deformations or unstable modes are present in observed sources, they could easily dominate the spindown of these sources. Therefore, it is important to understand the spindown due to gravitational wave emission. The spindown of a compact source with angular velocity Ω has the canonical power law form Ω˙ ∼ Ω n

(28)

In the case where stellar deformations (“mountains”) provide the main contribution to the spindown, the power law exponent is nd = 5, as seen in Section “Gravitational Waves Due to “Mountains”.” This is sufficiently different from magnetic dipole emission which typically dominates in the absence of gravitational wave emission and has a canonical exponent of nem = 3. In the case of r-modes, the situation is more complicated since due to the instability, the r-mode amplitude has to be saturated by an amplitude-dependent dissipation mechanism, which in addition generally also depends on oscillation frequency and the temperature of the source. This dissipation heats the star so that the evolution of the amplitude, the spin frequency, and the thermal evolution, depending also on the cooling mechanism in the source, are coupled. However, except for newborn sources, the spindown evolution is much slower than the amplitude and thermal evolution. Parametrizing the cooling power by P = Pˆ T θ and saturation amplitude αsat = αˆ sat T β Ω γ by general power laws (as realized for proposed mechanisms), this results in an effective spindown law of the above form with an effective power law exponent [37] nr =

(7+2γ ) θ +2β θ −2β

(29)

In the prototypical but unrealistic case of a constant saturation amplitude, this leads to an exponent of 7, but depending on the saturation mechanism, the spindown law can be rather different. For instance, mode coupling leads to an exponent of 4, and

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for a different mechanism, it could in principle lie anywhere between 1 and 10 making r-mode searches rather involved. The detailed r-mode spindown evolution allows us in addition to the above r-mode analysis for LMXBs also to compare timing data of millisecond pulsars, for which there is no thermal data, to theoretical predictions for the boundary of the instability region in timing parameter space [32]. The corresponding boundary ˙ curves have a very similar form in Ω-Ω-space as in T -Ω-space, showing that sources with a sufficiently low spindown rate are r-mode stable. However, the observed fast-spinning sources have spindown rates that are high enough that they could all be inside the r-mode instability region, just as in the case of LMXBs. A definite conclusion is in this case so far not possible since other mechanisms could be responsible or dominate the observed spindown.

Impact of Oscillation Modes on the Thermal Evolution As discussed, the r-mode evolution strongly affects the thermal evolution as well as electromagnetic observables and therefore indirectly reveals the gravitational emission, even if it is not directly observed. The analysis of r-mode instability regions, discussed before, is an example for this. In turn, observed temperatures or bounds on temperature impose bounds on the r-mode amplitude in observed sources, since they would otherwise be hotter than what is observed. The bound on the r-mode amplitude for a source with an observed surface temperature Ts takes the form [40] −9

αsat < 1.0 × 10



Ts 10 eV

2 

500 Hz f

4 

20π J˜M 1.3 M

−1 

R 10 km

−2 (30)

In LMXBs where direct temperature measurements are available, this leads to bounds as low as 10−8 [42]. In millisecond pulsars where at present only temperature bounds can be obtained, these nevertheless impose rigorous bounds as low as α < 3 × 10−9 [40] which impose significantly tighter constraints than the spindown of these sources and show that a detection of a potential r-mode gravitational wave emission from these sources will require third-generation detectors.

Continuous GW Searches and Current Bounds To date, the only detected GW signals involving neutron stars have been inspirals; however, the sensitivity of the detectors has been steadily improving, and a number of astrophysically significant constraints have been set in continuous wave searches (see [43] for a recent review), and in the next observational runs of the LIGO/Virgo/KAGRA network, there is a very real possibility of detecting these subtle signals.

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Present Searches As we have seen, continuous GWs are expected to be weak, but as they are longlasting signals, longer integration times T can allow to build up signal-to-noise ratio (SNR) as  SNR ∝ h0

T , S

(31)

with h0 the characteristic strain amplitude of the signal and S the spectral density of the noise in the detectors and the signal frequency. Consider one of the sources we have discussed in the previous sections, e.g., a galactic NS with a “mountain,” rotating with a 1 ms period at 10 kpcs and with = 10−7 . The GW amplitude, from Equation (17), is h0 ≈ 10−26 , and from Equation (31), we see that to claim a detection with SNR = 5, given the typical noise spectral density in LIGO for √ O3, S ≈ 10−23 Hz−1/2 , one needs T ≈ 1 year. This is a very promising limit, as the observation time needed is comparable to the duration of the O3 run, and as sensitivity increases during the next runs, observations will begin to dig into astrophysically significant parameter space and confirm, or falsify, GW emission models. In practice, the sensitivity of real-world GW searches depends on a tradeoff between computational costs and accuracy. For isolated NSs, the parameters describing the signal are the two angles which describe the sky position of the source, right ascension α and declination δ, the rotation frequency of the star ν (which we have seen, determines, depending on the emission model, the frequency fGW of the GW signal), and the frequency derivative ν˙ which describes its evolution over the observational window (in some cases, for large variations of ν˙ and long observational windows, it may be necessary to consider also ν¨ ). If the source’s position is known, and ν and ν˙ are accurately determined from EM observations, a targeted search can be performed. These searches are computationally light and allow for a larger accuracy and to probe for weaker signals. In the case where the parameters are not accurately known, a narrowband search is possible, around the expected values, and while similar in principle is slightly more computationally expensive. The situation changes if ν and ν˙ are not known, but the position in the sky of the source is, e.g., if one is searching for CW signals from young neutron stars in supernova remnants, which are not observed as EM pulsars. In this case, one can perform a so-called directed search, scanning over a range of frequencies and frequency derivatives, thus increasing the computational load and decreasing the sensitivity. Finally, the most computationally expensive (and least sensitive) searches are allsky blind searches, where one has to scan the sky for completely unknown signals. Several methods have been developed to deal with this problem and search the data for CW signals from isolated neutron stars, including the F-statistic method, the Hough transform, the 5-vector method, the Band Sampled Data method, and the

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time domain heterodyne method. Methods have also been implemented to search for signals from neutron stars in binaries, where the orbital modulation must be accounted for, e.g., the TwoSpect methods, CrossCorr method, Viterbi/J -statistic method, and the Rome narrowband method, and methods have been devised to detect shorter-duration “continuous” signals (see [43] and references therein for a detailed review of the methods).

Electromagnetic Constraints As mentioned in the previous section, electromagnetic observations can be used to aid GW detections by providing constraints on parameters such as sky position and frequency evolution. However, EM astronomy can reveal valuable information on the emission itself, by constraining the amplitude of the signal and providing indirect evidence that some systems are emitting GW. In the case of CW emission from mountains, measurements of the spindown rate ν˙ of a pulsar, from timing in the radio, X-ray, or other EM bands, can readily provide us with a useful upper limit on the ellipticity. If, in fact, we assume as an upper limit the case in which the spindown is entirely due to GW emission (which is clearly an upper limit, as we observe the EM emission from the pulsar, thus know it must contribute to the energy budget), i.e., that the star is a gravitar, from equation (18), we see that the ellipticity is constrained to be

−5

≤ 1.9 × 10



ν −5/2 100 Hz



|˙ν | 10−10 Hz/s

1/2 

I 1045 g cm2

−1/2 .

(32)

For example, for the Crab pulsar, the spindown upper limit above gives ≤ 7.5 × 10−4 , corresponding to a strain of h0 = 1.4 × 10−24 (see, e.g., [12] for a more detailed derivation). In Fig. 5, one can see a plot of the ellipticity spindown limits for the known pulsars, using data from the ATNF pulsar catalogue (https://www.atnf. csiro.au/research/pulsar/psrcat/) [44]. As we see from the figure, and will discuss in the next section, the spindown limit is the benchmark against which to test the sensitivity of detectors. Analogous spindown limits are available for r-modes [6]. However, taking into account that the same r-mode saturation mechanism should be present in different sources provides in this case even stronger limits [37]. As discussed before, for sources with X-ray bounds, these can impose even much tighter limits showing that r-mode emission from millisecond pulsars is not in reach of present detectors. EM observations can, however, provide more than upper limits. We have already seen that in the case of accreting systems, observations of the spin-distribution of the NSs suggest that additional GW torques may be active, as all systems are spinning well below their theoretical maximum frequency, as set by the Keplerian breakup frequency of the star.

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If we assume that the observed spin-period of these systems is set by the equilibrium between GW torques (see Eq. 18) and an accretion torque Na of the form  Na = M˙ (GMR) ,

(33)

where G is the gravitational constant, M and R are the mass and radius of the star, and M˙ is the accretion rate, which we can infer from the bolometric X-ray flux L of a source at a distance d, according to 4π Rd 2 F M˙ ≈ . GM

(34)

We can thus obtain a torque balance upper limit:

tb

1/2  −1/4 3/4  F R M = 2.4 × 10 10 km 1.4M 10−8 erg cm2 s−1     −1  ν −5/2 d I . (35) 100 Hz 1 kpc 1045 g cm2 −7



as shown for various accreting pulsars in Fig. 4. This is the benchmark for the sensitivity that detector must reach to be able to investigate these sources. For transitional pulsars that are not in torque balance, the limit could be weaker, though. Furthermore, observations of spins and temperatures in LMXBs allow to reconstruct observationally the r-mode instability window and provide strong evidence that exotic phases, such as quark matter, hyperons, or dynamically significant magnetic fields in superconductors, must be present in the interior of the star, given the lack of detections [39]. Individual systems also show evidence for gravitational wave emission, for example, PSR J1023+0038, which transitions from accretion phases where it emits X-rays to quiescent phases where radio emission is present. This system has an enhanced spindown rate during the accretion phase, which suggests that a gravitational mountain is being built [21] and may be present also in the quiescent phase [45]. Another interesting system is the young X-ray pulsar PSR J0537-6910. This is the most prolific glitcher currently known, and measurements of the breaking index between glitches suggest an underlying value of n = 7, consistent with gravitational wave emission from an unstable r-mode driving the spindown [46]. This scenario is theoretically plausible, given our knowledge of the equation of state of dense matter [46], making this system also a target for ongoing GW searches [47]. Millisecond radio pulsars, of which the accreting NSs in LMXBs are thought to be the progenitors, also reveal valuable information, and an observed cutoff in the P-Pdot diagram (period vs period derivative) can be well modelled in terms of a

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Fig. 4 Torque balance upper limits on the ellipticity for known accreting sources in LMXBs from [48]. For reference, we include also the sensitivity curves for Advanced LIGO, Advanced Virgo, and KAGRA at design sensitivity, assuming a distance of 1 kpc for the source and an integration time of 1 year

residual deformation of the stars, due to a buried magnetic field in the stellar interior, which would provide a lower limit for the ellipticity [28].

Gravitational Wave Constraints At the time of writing, no continuous gravitational wave signal has been detected, but, as already mentioned, several pipelines have been developed to search for these signals and now run on data from all the second observational run of LIGO and Virgo (O2) [43], and for some pipelines on parts of data from O3 [49]. The sensitivity of these searches has now reached and beaten the spindown limit in targeted and narrowband searches for a number of pulsars [49], as can be seen in Fig. 5, one of which is the Crab pulsar and others, quite notably, millisecond pulsars. In the next observational runs, it will thus be possible to probe some of the emission scenarios for millisecond pulsars presented in the previous sections. The same is true for searches for GWs from accreting systems, in particular the most luminous X-ray pulsar SCO X-1 [50]. In this case, matters are complicated by the fact that the source is not detected as a pulsar, so the frequency is not known. Nevertheless, at low frequencies, the upper limits are now better than the torque

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Fig. 5 Spindown upper limits on the ellipticity for known pulsars, from the ATNF pulsar catalogue [44]. For reference, we include also the sensitivity curves for Advanced LIGO, Advanced Virgo, and KAGRA at design sensitivity, assuming a distance of 5 kpc for the source and an integration time of 1 year. We also show the upper limits obtained by LIGO and Virgo on selected pulsars using O3 data, from [49]. Note that some of the pulsars are at a distance of approximately 0.1 kpc, which allows for more stringent upper limits than the sensitivity curves plotted for 5 kpc

balance limit in (35), so that again astrophysically significant portions of parameter space are being probed. There is thus a very real possibility that during the next observational runs of LIGO/Virgo and KAGRA, a continuous wave signal will be detected, opening a new window on neutron star interiors.

Conclusions Even though continuous gravitational wave sources have not been observed so far, the future detection of gravitational radiation from isolated neutron stars could provide a wealth of novel information about these extreme objects [8]. Being emitted by the bulk of the source, they have the potential to probe their otherwise opaque interior. For instance, in case the gravitational waves are emitted by a deformed, rigid quark matter core, the observed radiation would literally allow us to see right through the dense hadronic mantle. Similarly, it is generally the dissipation in the bulk of the star that tames unstable oscillation modes and, therefore, shapes the corresponding gravitational wave signal. As discussed, gravitational waves

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could be emitted by various different classes of isolated neutron stars at various stages in their evolution. Taking into account that in addition to the few thousand isolated neutron stars known from electromagnetic observations substantially more are expected from galactic evolutionary models, there could be literally millions of sources once the sensitivity of gravitational wave detectors is sufficient to observe them. Whereas standard hadronic phases very likely cannot sustain deformations that would result in presently detectable gravitational wave emission, exotic crystalline phases should be rigid enough that we should see these sources with next-generation detectors if they have been sufficiently deformed during their evolution. And while the electromagnetic bounds on the r-mode emission of many recycled sources are by now so tight that they are out of reach of current gravitational wave detectors, several young sources are very promising. Present gravitational wave searches already set important bounds on the continuous emission, and several of them already beat the spindown limit for various sources. Therefore, they start to probe the composition of neutron stars, and even a non-detection can significantly limit the presence of exotic forms of matter inside. Combined with multi-messenger electromagnetic observations, such searches have the potential to markedly improve our understanding of neutron stars in the near future.

Cross-References  Binary Neutron Stars Acknowledgments K.S. has been supported by the Turkish Research Council (TÜBITAK) via projects 117F312 and 119F073. B.H. has been supported by the National Science Center, Poland (NCN) via grants OPUS 2019/33/B/ST9/00942, OPUS 2018/29/B/ST9/02013, and SONATA BIS 2015/18/E/ST9/00577.

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r-Process Nucleosynthesis from Compact Binary Mergers

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Albino Perego, F.-K. Thielemann, and G. Cescutti

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matter Ejection from Compact Binary Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejecta from Binary Neutron Star Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejecta from Neutron Star-Black Hole Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejecta Expansion and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r-Process Nucleosynthesis in Compact Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Binary Mergers as r-Process Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Working of the r-Process in Compact Binary Mergers . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Physics Input and Detailed Network Calculations . . . . . . . . . . . . . . . . . . . . . . . . . Detailed Network Calculations and Nucleosynthesis Yields from Compact Binary Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observables of Compact Binary Merger Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Signatures of r-Process Nucleosynthesis in Compact Binary Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Binary Mergers and the Chemical Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Perego () Department of Physics, University of Trento, Trento, Italy INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy e-mail: [email protected] F.-K. Thielemann Department of Physics, University of Basel, Basel, Switzerland GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany e-mail: [email protected] G. Cescutti INAF, Trieste Astronomical Observatory, Trieste, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_13

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Abstract

The merger of two neutron stars or of a neutron star and a black hole often results in the ejection of a few percents of a solar mass of matter expanding at high speed in space. Being matter coming from the violent disruption of a neutron star, these ejecta are initially very dense, hot, and extremely rich in neutrons. The few available protons form heavy nuclei (“seeds”) that absorb the more abundant free neutrons, increasing their size. The neutron density is so high that a substantial number of neutron captures occur before the resulting unstable nuclei can decay toward more stable configurations, converting neutrons into protons. Depending mostly on the initial neutron richness, this mechanism leads to the formation of up to half of the heavy elements that we observe in nature, and it is called rapid neutron capture process (“r-process”). The prediction of the precise composition of the ejecta requires a detailed knowledge of the properties of very exotic nuclei that have never been produced in a laboratory. Despite having long been a speculative scenario, nowadays several observational evidences point to compact binary mergers as one of the major sites where heavy elements are formed in the universe. The most striking one was the detection of a kilonova following the merger of a neutron star binary: the light emitted by this astronomical transient is indeed powered by the radioactive decay of freshly synthesized neutron-rich nuclei and testifies the actual nature of compact binary mergers as cosmic forges. Keywords

r-process nucleosynthesis · Neutron captures · Beta decays · Nuclear fission · Dynamical ejecta · Disk outflows · Kilonovae · AT2017gfo · Heavy elements · Metal-poor stars

Introduction After the discovery of the first stellar system formed by two neutron stars (NSs) [26], it was immediately realized that the decompression of NS matter following the merger of a binary neutron star (BNS) or of a black hole (BH)-neutron star (BHNS) system produces an ideal environment where the rapid neutron capture process (rprocess) nucleosynthesis can take place. The very first calculations were carried out for BHNS binaries [34,35]: in this scenario, during the last orbits of the gravitational wave (GW)-driven inspiral, the NS is tidally disrupted by the gravity of the more massive BH, and a fraction of it is ejected into space. The coalescence of two NSs could also eject neutron-rich matter into the interstellar medium (ISM) through an even richer dynamics [12]. The first modeling of BNS merger nucleosynthesis was indeed accomplished a few years later [17, 62]. Since then, major progress has been made in understanding the mechanisms behind the ejection of matter and the properties of these ejecta. r-process nucleosynthesis is one of the fundamental processes responsible for the production of the heaviest elements in the universe (e.g., [8, 9]; see also [11]

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for a recent review). The binding energy per nucleon in nuclei increases almost steadily from lithium up to 56 Fe and 56 Ni. This allows the production of nuclei starting from H and He (mostly produced during the big bang) up to iron inside massive stars for increasing plasma temperatures in hydrostatic conditions (see, e.g., [10]). The production of heavier elements through reactions involving charged nuclei would require even larger temperatures to overcome Coulomb repulsion. The kinetic energy necessary to synthesize very heavy elements through fusion reactions becomes soon prohibitive for all plausible astrophysical scenarios. Moreover, at such high temperatures, disintegration reactions would even dominate due to highly energetic photons. This strongly limits the nucleosynthesis yields produced through this path. The capture of a free neutron on a nucleus has instead the clear advantage of not having any Coulomb barrier to overcome, and it is indeed the key process to produce the heaviest elements. However, free neutrons in non-degenerate conditions are unstable against β − decay. Thus, neutron capture nucleosynthesis requires a source of neutrons lasting for the relevant timescale over which the nucleosynthesis takes place. Moreover, the neutron-rich nuclei produced by neutron captures are unstable against β − decay. While decay rates are constants, neutron capture rates crucially depend on the neutron density: if the neutron density is high enough (nn  1020 cm−3 ), at least for a short timescale (∼1s), neutrons are rapidly captured (increasing the mass number) before β decays increase the charge number and therefore produce isotopes of the next heavier element. Due to its rapidity, this process can happen in explosive environments. This is the basic idea behind r-process nucleosynthesis, and matter ejected from compact binary mergers provides precisely the conditions necessary for the r-process nucleosynthesis to occur. Over the past few years, several observational evidences have accumulated pointing to compact binary mergers as one of the main sites in the universe where r-process nucleosynthesis takes place, including the observation of elements synthesized through the r-process in the atmosphere of very old metal-poor stars (see, e.g., [61]). The first unambiguous detection of a kilonova (also called macronova), AT2017gfo, as one of the electromagnetic counterparts of the GW signal GW170817 (compatible with a BNS merger) represented the strongest evidence, so far, of this picture (e.g., [3, 31, 60]). In this chapter, we present the most relevant aspects of the r-process nucleosynthesis in compact binary mergers. We start by reviewing the conditions of matter expelled during a BNS or a BHNS merger. After that, we present how r-process nucleosynthesis proceeds in these ejecta. Finally, we overview the main observational evidences supporting compact binary mergers as major astrophysical sites for r-process nucleosynthesis. In most of our calculations, we use cgs units. The physical constants employed through the text are the speed of light c; the reduced Planck constant h; ¯ the Boltzmann constant kB ; the gravitational constant G; the Stefan-Boltzmann constant σSB ; the solar mass M ; the masses of the electron, proton, and neutron me , mp , and mn ; and a generic baryon mass mb , which for our purposes can be assumed mb ≈ mn . Temperatures are expressed both in Kelvin and in MeV (i.e., as kB T ), depending on the context. We recall here that the conversion factor between GK and MeV is roughly one tenth, i.e., kB ×1 GK ≈ 0.086 MeV. The

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distribution of a generic nuclear species i can be expressed in terms of either its mass fraction Xi or number abundance Yi . The former is defined as the ratio between the mass (density) of the species i and the total mass (density), Xi = mi /mtot = ρi /ρ, while the latter as the ratio between the number (density) of the i species and the total baryon number (density), Yi = Ni /Nb = ni /nb . Clearly, nb ≈ ρ/mb and Yi = Xi /Ai , where Ai is the atomic number of the species i. The electron abundance is defined as Ye = ne /nb , where ne is the net density of electrons (i.e., the density of electrons minus the one of positrons). Due to charge neutrality, Ye = Yp,free + Yp,nuclei , where Yp,free and Yp,bound are the abundance of free protons and of protons bound in nuclei, respectively.

Matter Ejection from Compact Binary Mergers Matter ejection in compact binary mergers happens through different channels. These channels are characterized by specific ejection mechanisms, which operate on different timescales and leave an imprint on the ejecta properties and, ultimately, on the nucleosynthesis. In the following, we review the properties of the ejecta from BNS and BHNS mergers by directly relating them to their merger dynamics. Before doing that, we briefly present some of the most relevant features that characterize the modeling of compact binary mergers and some of the more fundamental processes that influence the ejecta properties. For more extended and complete information about these topics, we refer to the dedicated chapters and to a few recent reviews, e.g., [53, 59], where detailed references to the original works can be found. The dynamics of the merger and of the ejecta expulsion depends on several intrinsic parameters of the binary: first of all, on the nature of the coalescing objects (i.e., if it is a BNS or a BHNS system), but also on their masses and spins. Another relevant ingredient is the still uncertain nuclear equation of state (EOS) for matter at supranuclear density [46]. Any quantitative statement (and even a robust qualitative understanding) about the merger dynamics relies on detailed numerical simulations. The latter solve the equations of relativistic neutrino-radiation (ν-radiation) hydrodynamics coupled with dynamical space-time evolution. The hydrodynamics equations are closed by a finite temperature, composition-dependent EOS describing the microphysical properties of matter for a rest mass density that varies between stellar densities (a few g cm−3 ) up to several times 1015 g cm−3 , corresponding to more than ten times nuclear saturation density. Both in BNS and in BHNS mergers, an accretion disk around a central remnant is expected to form after the merger. In the case of BHNS systems, the remnant is always represented by a BH, while in the case of BNS systems, a massive NS, possibly collapsing to a BH on a variable timescale, usually forms. While the inspiraling NSs are in cold neutrino-less weak equilibrium, hot matter inside the merger remnant is out of equilibrium, and its neutron-to-proton content changes due to neutrinomatter interactions. Several hydrodynamics processes increase matter temperature during the merger, and the relevant temperatures range between 0 and ∼150 MeV. Neutrino production is strongly boosted in hot and dense matter. Due to their low

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opacity, neutrinos become the dominant cooling source, and their luminosity is of the order of a few 1053 erg s−1 , at least at merger and during the early aftermath. The decompression and heating of neutron-rich matter favors initially ν¯ e luminosity through the reaction n+e+ → p+ν¯ e on the thermally produced positrons and Lν¯e  2Lνe . The resulting net effect is to increase the electron fraction (leptonization), balancing the proton-to-neutron ratio such that at later times Lνe ≈ Lν¯e . In the densest part of the remnant, where matter density is above ρ ∼ 1012 g cm−3 , and the temperature Trem is of the order of 10 MeV, the neutrino mean free path ν = 1/(nB σν ) is smaller than the size of the system (∼10–100 km) for thermal neutrinos −1  of energy Eν ∼ 3.15Trem : ν ∼ 250 m ρ/1012 g cm−3 (Trem /10 MeV)−2 . For this estimate, we have used an approximated expression for the cross section of neutrino scattering off free nucleons, σν ≈ σ0 (Eν /me c2 )2 with σ0 = 1.76 × 10−44 cm2 , to evaluate the neutrino-matter cross section. Absorption cross sections on free baryons have similar magnitudes and dependences. Then deep inside the remnant, neutrinos equilibrate with matter and diffuse out on the diffusion timescale (∼seconds). Due to the presence of an accretion disk, neutrinos are emitted preferentially along the polar direction rather than along the equator, such that the polar flux can be a few times the equatorial one. The most relevant neutrino decoupling surfaces are located in the density interval 1011−12 g cm−3 , with νe ’s decoupling at lower densities and temperatures than ν¯ e s and νμ,τ s, since the neutron richness favors n+νe → p +e− as absorption process over other inelastic neutrinomatter processes. Typical neutrino mean energies in the decoupling region are Eνe ∼ 10 MeV, Eν¯e ∼ 15 MeV, and Eνμ,τ ∼ 20 MeV. Matter at even lower density, usually located at larger distances from the center, is irradiated by the neutrinos emitted at the inner decoupling surfaces, and this irradiation can change the neutronto-proton content (i.e., Ye ) through neutrino absorption on neutrons, protons, and nuclei. Finally, we recall that NSs are magnetized objects. During the merger and the subsequent remnant evolution, several mechanisms (e.g., dynamo amplification, magneto-rotational instabilities, Kelvin-Helmholtz instabilities) amplify the field strength. Even for initially low magnetic field (B ∼ 109−10 G), the field can be amplified up to several ∼1015 G, and it becomes dynamically relevant during the merger aftermath, also for matter ejection.

Ejecta from Binary Neutron Star Mergers The first kind of ejecta emerging from a BNS merger are the dynamical ejecta. The dynamical timescale of the merger is set by the orbital (angular) velocity at the last orbit before merger, vdyn (Ωdyn ), as tdyn ∼ 2π/Ωdyn ∼ (4π √ RNS ) /vdyn where RNS is the NS radius. Assuming a Keplerian behavior, vdyn ∼ GM/(2RNS ), we obtain: 

vdyn

M ∼ 0.4 c 2.7M

1/2 

RNS 12 km

−1/2 ,

(1)

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and  tdyn ∼ 1.23 ms

M 2.7M

−1/2 

RNS 12 km

3/2 ,

(2)

for a typical total binary mass M of 2.7 M and a NS radius of 12 km [2, 4]. During the last orbits, when the two NSs approach each other, each of them gets deformed by the tidal field of the companion. As soon as the NSs touch, a large fraction of their kinetic orbital energy is converted into internal energy, while the tidal tails retain their orbital speed, Eq. (1). Since this is larger than the radial escape velocity, matter in the tails is ballistically expelled with velocity vej ≈ 0.1–0.3c  vdyn . This is the tidal component of the dynamical ejecta, and it mainly develops across the equatorial plane of the binary. The temperature in these ejecta is only marginally increased by the tidal compression that precedes the merger (T  1 MeV), and the emission of neutrinos is too weak to change Ye significantly. Thus, these ejecta mostly retain their original Ye , 0.05  Ye  0.15, and they increase their specific entropy only up to a few kB per baryon, s  5kB baryon−1 . If the total mass of the binary is too large for the matter pressure and for the rotational support to sustain the forming remnant, the latter collapses immediately to a BH. Otherwise, a (possibly metastable) massive NS forms in the center. This object is far from equilibrium, and it bounces as the result of the gravitational pull and matter pressure response. As the sound waves generated by these oscillations travel through the remnant and reach lower density regions, they convert into shock waves, triggering the ejection of matter heated by compression and shocks from the outer edge of the remnant. These ejecta are called shock-heated ejecta and present a broad distribution of expansion velocity, usually peaked around 0.2–0.3c, but with a possible high-velocity tail extending up to 0.6–0.8c. Due to the action of shocks and compression, the ejecta entropy increases, typically reaching values around 10–20 kB baryon−1 , but with a low-mass, high-entropy, high-speed tail extending up to ∼100 kB baryon−1 . The corresponding increase in temperature (initially, up to several tens of MeV before dropping due to matter expansion) produces a large density of electron-positron pairs and determines an increase in Ye in the ejecta due to positron captures on neutrons. Moreover, neutrino irradiation coming from the forming remnant can further increase Ye through νe absorption on neutrons [70]. Since neutrino emission is more efficient along the polar direction, the effect of irradiation is more evident at high latitudes. The combined effect of tidal tail interactions, hydrodynamics shocks, and weak processes is the expansion of the dynamical ejecta over the entire solid angle, still with a preference along the equatorial plane (the mass distribution retains a sin2 θ dependence on the polar angle θ ), with a clear gradient in the Ye distribution moving from the equator (Ye ≈ 0.1) to the poles (Ye  0.4). In particular, matter above θ ∼ 45◦ is expected to have Ye  0.25. The ejection of dynamical ejecta lasts for a few ms after merger (see Eq. (2)), and its amount ranges between ∼10−4 and ∼10−2 M , depending on the binary properties and on the nuclear EOS (see, e.g., [54]). The tidal component is more relevant if the high-density part of the EOS is rather stiff or if the two NSs in the binary have very different masses. In

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Fig. 1 Color-coded two-dimensional histograms of the conditions of the dynamical ejecta, as obtained by an equal mass BNS merger simulation (BLh q = 1 run presented in [7]). On the left panel, the ejecta are characterized in terms of their specific entropy and electron fraction, while on the right panels their specific entropy and expansion timescale. The top panels refer to an angular slice close to the rotational axis of the binary, while the bottom one to a slice close to the equator

this case, at least one of the two NSs is not very compact (i.e., RNS is larger), and the tidal disruption is very effective. The presence of shock-heated ejecta is instead more relevant in the case of equal mass mergers and for a soft nuclear EOS. In these cases, the NSs are more compact (i.e., RNS is smaller), the collision velocity at merger is larger (see Eq. 1), and the shocks are more violent. Typical conditions of the dynamical ejecta are presented in Figs. 1 and 2 for both an equal and a very unequal mass BNS merger, as obtained by detailed merger simulations in numerical relativity, at two different polar angles. While the massive NS forms in the center, matter compressed and heated up at the contact interface between the two NS cores is expelled outward. Conservation of angular momentum drives the formation of a rotationally supported, thick accretion disk of radial and vertical extension Rdisk and Hdisk , such that its aspect ratio is (H /R)disk ∼ 1/3. The typical disk mass Mdisk (where the disk is usually defined as the part of the remnant with density below 1013 g cm−3 or, in the presence of a BH, outside of the horizon) ranges between 10−3 and 0.3 M . A prompt collapse to BH stops the disk formation, leading to lighter disks. An interesting exception is

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Fig. 2 Same as in Fig. 1, but for a very unequal mass BNS merger (BLh q = 1.8 run presented in [7]). These conditions are qualitatively similar to the ones observed also in BHNS mergers

represented by very asymmetric binaries: in this case, the sudden BH formation is accompanied by a very efficient tidal disruption of the secondary NS, such that a significant fraction of it settles in Keplerian orbital motion outside the BH horizon [7]. The evolution of the rotating disk is governed by hydrodynamics, magnetic, and weak processes. Several mechanisms are responsible for the local amplification of the magnetic field. The resulting viscosity of turbulent origin drives matter accretion onto the central object on the accretion timescale: tacc ∼

1 α



H R

−2

−1 Ωdisk ≈0.76 s

−1/2    α −1 H /R−2  M Rdisk 3/2 rem 0.02 1/3 2.6M 100 km (3)

where α is an effective viscosity parameter and Ωdisk the Keplerian angular velocity, and for the remnant mass Mrem and disk radial scale Rdisk , we set as characteristic values 2.6 M and 100 km, respectively. During its secular evolution, the processes that determine the disk evolution can produce mass outflows, known as disk wind ejecta. Neutrino absorption, for

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example, redistributes energy and momentum inside the remnant, from hot and dense regions (ρ > 1012 g cm−3 ) to regions where the density decreases to ρ ∼ 1010−11 g cm−3 . This process inflates the disk, mainly in the vertical direction (neutrino-driven winds). At the same time, the accretion process implies an angular momentum redistribution inside the remnant: while mass has a net inflow and the bulk of the disk is accreted, a fraction of it expands radially (viscosity-driven winds). In both cases, the disk expansion determines a drop in temperature, and when T  5–6 GK, free neutrons and protons recombine first in α particles and then in heavier nuclei. The energy released is on average nucl ≈ 8.6 MeV baryon−1 , and the corresponding expansion velocity can be estimated by equating the kinetic energy at infinity with the of the gravitational and nuclear energy released by  sum   recombination: v/c  2 nucl /mb c2 − GMrem /(Rc2 ) , where Rrec is the radial scale where recombination occurs. Assuming Mrem = 2.6M , and Rrec ≈ 450– 600 km for disk winds, one obtains v ≈ 0.034–0.074 c. Neutrino-driven winds can emerge on a few tens of millisecond timescale, while viscosity-driven winds on the longer accretion timescale. The earlier the wind develops, the faster the ejecta travel. Indeed, at earlier time, the disk is hotter, and the recombination radius is larger. Other processes happening inside the disk can also drive disk winds. For example, if the central remnant does not collapse to a BH, non-axisymmetric bars extending inside the disk (mainly m = 1 and m = 2 spiral modes) act continuously on its innermost part, producing a net outflow of angular momentum that crosses the disk and expels matter from its edge (spiral-wave winds). Spiral-wave winds develop immediately after disk formation and possibly last up to the point where the central massive NS collapses or the bars are dissipated by GW emission. Moreover, if, in addition to disordered local fields, large-scale magnetic fields develop inside the remnant, magnetic pressure and the Lorentz force can further accelerate matter producing magnetically driven wind disks. Both in the magnetic and in the spiralwave wind cases, expansion velocities are intermediate between the fast dynamical ejecta and the slower recombination disk winds, v ∼ 0.1 − 0.2c. The many processes taking place inside the disk are very effective in unbinding mass from it. The scale that sets the ejecta amount is the mass of the disk itself. While neutrinos alone are able to unbind only a few percents of the disk (and even less in absence of the very luminous massive NS), the other mechanisms unbind between 0.1 and 0.4 Mdisk . It is worth mentioning that these mechanisms can work at the same time, with disk consumption being the only really competing factor. Disk-wind ejecta are thus likely the most relevant source of ejecta for BNS mergers, as visible in Fig. 3. The ejection timescale of disk winds is comparable to the weak reaction timescale. As for the dynamical ejecta, while the initial Ye in the disk is set by the cold weak equilibrium of the merging NSs, the hot temperature increases Ye inside the expanding winds due to positron and neutrino absorption on neutrons. Assuming that neutrino irradiation is effective and long enough to reach equilibrium, Ye tends progressively toward [42, 52]:

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Mej;sec [M ]

10−1 10−2 10−3 10

BHBΛφ DD2 LS220 SFHo

−4

10−5 −5 10

10−4

10−3 10−2 Mej [M ]

10−1

Fig. 3 Comparison between the dynamical and the disk-wind ejecta, as obtained by a large set of BNS merger models, employing several nuclear EOSs. While the amount of dynamical ejecta is computed within the simulations, the mass of the wind-disk is assumed to be 20% of the mass of the accretion disk at the end of the simulations. (Figure taken from [54])

−1  Lν¯ Wν¯e ν¯e − 2Δ + 1.2Δ2 / ν¯e Ye,eq ≈ 1 + e , Lνe Wνe νe + 2Δ + 1.2Δ2 / νe

(4)

where Δ = (mn − mp )c2 ≈ 1.29 MeV, ν is the ratio between the average squared neutrino energy and the average neutrino energy (which for relevant spectral distribution gives ν ≈ 1.2 Eν ), and Wi ≈ 1 + ηEi /mb c2 with ηνe = 1.01 and ην¯e = −7.22 is the weak magnetism correction factor. For typical neutrino luminosities and mean energies, Ye,eq ≈ 0.45 < 0.5. Thus, depending on the ejection time and on the strength of the neutrino irradiation, the electron fraction in the wind ejecta shows a broad distribution between 0.1 and 0.45. In the case of short-lived massive NS, the bulk of the disk wind ejecta have Ye between 0.2 and 0.3, while in the case of long-lived remnant, the higher neutrino luminosity drives the Ye distribution toward Ye  0.3. Matter in the disk is shocked by waves produced by the central remnant, and it is heated by viscous dissipation. Then its entropy increases to an average value of 15–25 kB baryon−1 , with possibly high-entropy tails extending also in this case up to ∼100 kB baryon−1 .

Ejecta from Neutron Star-Black Hole Mergers The ejection of matter from BHNS systems shares many similarities with the one from BNS mergers, but it also presents crucial differences. Astrophysical BHs are

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characterized by their mass, MBH , and spin, JBH . In the following, we assume that the orbital angular momentum of the binary points toward the positive z direction and that the BH is significantly more massive than the NS (MBH  5 M , while MNS  2 M ). Assuming JBH to be also along the z axis, we characterize it through 2 ), where 0 ≤ a < 1 if the the dimensionless spin parameter aBH = ±JBH c/(GMBH spin points toward the positive z direction and −1 < a < 0 otherwise. When at the end of the GW-driven inspiral phase the two compact objects approach each other, the merger fate depends on the location of the BH last stable circular orbit, RISCO (i.e., the radius inside of which circular time-like test-mass orbits in the equatorial plane become unstable to small perturbations), with respect to the tidal distance, dtidal (i.e., the BH-NS distance at which the gravitational force on a test mass at the NS surface equals the tidal force pulling the mass toward the BH). For Kerr BH RISCO = MBH f (a), where f (a) is a monotonically decreasing function such that 1 ≤ f (a) ≤ 9 and f (0) = 6, while dtidal ≈ (2MBH /MNS )1/3 RNS . If dtidal  RISCO , the NS is swallowed by the BH before a significant tidal disruption can occur. No mass is practically left outside the horizon to form a disk or to become unbound. Otherwise the NS experiences a partial tidal disruption before most of its mass gets inside the BH horizon. Mass coming from the farther NS edge can become unbound in the form of tidal dynamical ejecta inside a crescent. The rest of the unswallowed mass sets into a Keplerian accretion torus around a spinning BH. From the above relations, it is clear that the probability of leaving mass outside the BH horizon increases for stiffer nuclear EOS, smaller mass ratios, and larger aBH . Numerical simulations show that 0 ≤ Mdyn  0.15M for the dynamical ejecta, while the mass in the torus is such that 0 ≤ Mdisk  0.5M . While the dynamical ejecta mass decreases for larger NS masses, the most massive torii are observed for very massive NSs (and for smaller mass ratios and larger spins). The NS gets compressed during the inspiral. However, the tidal nature of the merger keeps the entropy in the dynamical ejecta low (s ∼ a few kB ) such that kB T  1 MeV always. The electron fraction of the ejecta stays also very close to its cold neutrino-less weak equilibrium value, i.e., Ye ∼ 0.05. Inside the torus, even in absence of shocks produced by the central remnant, accretion and disk dynamics heat up matter up to kB T ∼ 10 MeV, producing ∼1052 erg s−1 of accretion-powered neutrino luminosities. The subsequent production of disk wind ejecta is similar to the one observed in BNS mergers. Due to the lower neutrino luminosity, the electron fraction of the disk wind ejecta shows a broad distribution, 0.1  Ye  0.35, with a lower average value, compared with the wind disk ejecta produced by long-lived NS remnants.

Ejecta Expansion and Thermodynamics As we have seen both for BNS and for BHNS systems, during the merger, cold and neutron-rich nuclear matter is heated up and leptonized by several processes, including matter compression, hydrodynamics shocks, and neutrino irradiation. The bulk of the dynamical ejecta and of the remnant disk originate from inside the outer

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NS core, where ρ  1014 g cm−3 and Ye  0.1. In these conditions, nuclei are fully dissociated in free neutrons and protons (homogeneous nuclear matter), and due to charge neutrality, their initial abundances are related to the electron fraction by Yp = Ye  0.1 and Yn = (1−Ye )  0.9. After having reached its peak temperature, matter is expelled, and the density and temperature drop, while neutrons and protons start to form nuclei. In the following, we will always assume that the peak temperature of the ejecta exceeds 4 GK. Matter and radiation in the expanding ejecta can be considered as a neutrally charged plasma consisting of nuclei (often distinguished in free neutrons ns, free protons ps, α particles, and a distribution of all other nuclei), electrons, photons, and neutrinos. During most of the relevant nucleosynthesis timescale, photons, electrons, and nuclei are in thermodynamics equilibrium. It is thus useful to consider an expanding fluid element as a Lagrangian particle characterized by an evolving density ρ = ρ(t) and temperature T = T (t). Nuclei and nucleons can be treated as an ideal, non-relativistic classical gas, obeying Maxwell-Boltzmann statistics. Electrons are degenerate in the early phase of the expansion before entering a classical gas phase, and they can be described by an ideal Fermi gas of arbitrary degeneracy. Photons are always characterized by a black body spectrum, typical of massless bosons in thermal equilibrium with matter. Once the density, temperature, and composition are given, the entropy of the system can also be computed through the resulting equation of state, s = s(t). In the first phase of the ejecta expansion, when hydrodynamics processes are still active, the fluid density decreases approximately in an exponential way: ρ(t) ≈ ρ0 exp (−(t − t0 )/τ ), where ρ0 is the density at the onset of the expansion, t = t0 . The expansion timescale, τ , quantifies how fast the density drops during the first phase. Faster dynamical ejecta have smaller τ s (1  τ [ms]  10) than slower disk winds (10  τ [ms]  100). The expansion timescale can be related to the expansion velocity vej through:

 v −1 R˜ R˜ ej ≈ 16.7 ms τ≈ vej 500 km 0.1c

(5)

where R˜ is the lengthscale where matter becomes unbound, depending on the kind of ejecta and ejection mechanism. For the dynamical ejecta, the more impulsive expulsion set a lower R˜ ∼ 300 km than for the disk wind ejecta, R˜ ∼ 600 km. After a timescale thom ∼ 3τ , internal dynamics and momentum redistribution cease and fluid elements start to expand with an approximately constant velocity: the fastest fluid elements at the front and the slowest at the bottom. Such expansion profile is said homologous since v ∝ r and in this phase the density evolves as:  ρ(t) = ρhom

thom t

3

≈ 100 g cm−3



ρhom 3 × 106 g cm−3



τ 3 10 ms



t 1s

−3 (6)

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with ρhom varying inside the range 104 –108 g cm−3 , and the larger values are for low-entropy, fast-expanding ejecta. This density evolution is a very relevant input for the r-process [58]. After the nucleosynthesis, the internal energy of the ejecta is subdominant with respect to the kinetic energy, and the expansion proceeds adiabatically. Moreover, for the relevant density and temperature conditions, the EOS is dominated by the relativistic photons and electrons. For such a gas, an adiabatic expansion satisfies ρT 3 ≈ const and thus T (t) ∝ t −1 . While this temperature evolution profile is not accurate during the nucleosynthesis epoch, it provides a good approximation after it.

r-Process Nucleosynthesis in Compact Mergers We now move to the study of the nucleosynthesis happening inside the ejecta. Before speaking about the r-process, we introduce a few basic, although necessary, concepts in nuclear reaction theory. We address the reader to Refs. [10, 27, 41] for general introductions and to [11, 67] for more specific reviews on the r-process. Nuclear abundances inside the ejecta evolve in time as a result of nuclear (both strong and weak) and electromagnetic reactions. The usage of abundances in nucleosynthesis calculations is very sensible because, being the ratio of densities, it allows to decouple the effects of reactions on the composition from the effects due to matter expansion. Strong and electromagnetic reactions conserve separately the number of protons and neutrons. Strong nuclear reactions include fusion reactions among nuclei. In the case of transfer reactions, they are often indicated as B(i, o)C to emphasize the transfer particles i and o. For example, in a (n, α) reaction, a free neutron is absorbed by a nucleus B = (A, Z), and a nucleus C = (A − 3, Z − 2) and an α particle are produced in the final state. Strong nuclear interactions include also α decays, (A, Z) → (A − 4, Z − 2) + α, and spontaneous or induced fission processes. The most relevant electromagnetic reactions involving nuclei are the photodisintegration reactions, B(γ , o)C, and their inverse absorption processes, B(i, γ )C. For example, in the case of a (γ , n) reaction, the absorption of a photon on a nucleus B = (A, Z) produces a nucleus C = (A − 1, Z) together with a free neutron. The opposite reaction (n, γ ), called neutron capture, consists in the absorption of a free neutron on a nucleus B = (A, Z), producing a nucleus C = (A + 1, Z) and a photon. Weak nuclear reactions include β ± decays, electron, and neutrino captures, and they convert neutrons into protons and vice versa, changing the electron fraction of matter. In the expanding neutron-rich ejecta, β − decays, (A, Z) → (A, Z + 1) + e− + ν¯ e , are the most important weak reactions, while in the initial expansion phase, neutrino irradiation acts through neutrino absorption, as previously discussed. Any of the transfer and absorption reactions B(i, o)C is characterized by its cross section, σB(i,o)C , which in general depends on the energy of the colliding particles. In an astrophysical plasma, the distribution of the colliding energy depends on the local thermodynamics properties, and the reaction rate, rB(i,o)C (defined as

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the number of reactions occurring per unit time, per unit volume, and per reactant pair), is given by rB(i,o),C = σ v B(i,o)C nB ni where σ v B(i,o)C is the product of the reaction cross section times the relative velocity between B and i, averaged over their thermal distributions. In the case of photon or neutrino absorptions on a nucleus B, the reaction rate can be expressed in terms of an effective destruction/decay rate, λB(γ /ν,o)C , as rB(γ /ν,o)C = λB(γ /ν,o)C nB , where λB,γ is a function of T only (since the photon gas properties depend only on temperature), while λB,ν depends on the neutrino spectrum, which is in general not in equilibrium with the plasma. Then, strong and electromagnetic reactions depend only on the local plasma properties. Alpha and beta decays are instead characterized by a constant decay rate λα/β , related to the nucleus half-life t1/2 by λα/β = ln 2/t1/2 . The lifetime of a nucleus against a certain reaction can be defined as the inverse of the corresponding rate, τi = 1/λi . For large temperatures, strong and photodisintegration reactions are characterized by large reaction rates. For T  4–5 GK, the resulting fusion and photodisintegration timescales become much shorter than the weak and dynamical timescales in the ejecta. Thus, these reactions can be considered in equilibrium among them and with their inverse reactions. This condition is called nuclear statistical equilibrium (NSE), and the nuclear abundances in NSE are fully determined by the local thermodynamical conditions, i.e., by ρ, T , and Ye . While ρ and T vary due to the expansion, Ye changes due to weak processes, but in both cases on much longer timescales than the nuclear NSE timescale. When the temperature decreases below ∼4–5 GK, some reactions characterized by small Q-values become slow enough that NSE is no more guaranteed across the entire nuclear distribution and especially for nuclei characterized by magic nuclear numbers, i.e., close to shell closure conditions. This transition is called NSE freeze-out. Since a large fraction of direct and inverse nuclear reactions are still very fast, the nuclear distribution splits into areas of quasi-statistical equilibrium (QSE), where equilibrium conditions still apply on sub-sets of nuclei. When the temperature decreases even further, no equilibrium arguments apply, and fully out-of-equilibrium nucleosynthesis occurs. While in NSE conditions accurate abundances can be computed even without precise information on the reaction rates, a detailed knowledge of the properties of the ejecta and of nuclei all over the nuclear chart are requested to predict accurate abundances after NSE freeze-out. Nevertheless, equilibrium arguments still provide a useful tool to understand the basic feature of nucleosynthesis in QSE conditions.

Compact Binary Mergers as r-Process Site Before entering the details of the r-process, we first motivate why the ejecta of compact binary mergers represent a suitable environment for r-process nucleosynthesis. The ejecta come from high-density conditions and usually experience high enough temperatures such that matter is mostly dissociated into free neutrons and protons under NSE conditions at their peak temperature. As temperature drops, neutrons

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and protons recombine first into α particles. The subsequent building of the most tightly bound iron group nuclei depends on the three body reactions responsible for the assembly of heavier nuclei, namely, 2α + n → 9 Be + γ and 3α → 12 C + γ . The first one is the most relevant in neutron-rich conditions. Triple reactions are in competition with their inverse photodestruction reactions. Due to their triple nature, the former are favored by larger densities, while the latter are strongly enhanced by higher temperatures since for the photon density and mean energy one has nγ ∝ T 3 and Eγ ∝ T , respectively. If the plasma is radiation-dominated, the value of the specific entropy ultimately determines whether iron group nuclei can form. Indeed, if s  sγ + se± ≈ 2sγ , the density can be computed as:  ρ2

4π 2 mb



45 (h¯ c)3

kB3 T 3 ≈ 3.02 × 106 g cm−3 s/kB



T 5 GK

3 

s 10 kB

−1 .

(7)

In very neutron-rich conditions (Ye ∼ 0.1), when temperature decreases between 5 and 2.5 GK, ααn reactions occur more efficiently than their inverse photodestruction reactions only for ρ  3 × 105 g cm−3 , i.e., for s  100 kB baryon−1 . This implies that in low- and moderate-entropy conditions (typical of the merger ejecta), iron group nuclei are formed in NSE conditions, while in the high-entropy tail, α particles are mostly produced when NSE equilibrium is no more guaranteed (α-rich freeze-out). Almost all protons are bound inside nuclei, and a distribution of heavy nuclei around the iron group (called seed nuclei) is present at the NSE freeze-out. For the seed nuclei, one can introduce a representative average nucleus (A seed , Z seed ) defined such that: ⎛ ⎛ ⎞ ⎞

A seed ≡ ⎝ A Y(A,Z)⎠ /Yseed , Z seed ≡ ⎝ Z Y(A,Z)⎠ /Yseed , (A,Z),A>4

(A,Z),A>4

(8)  where we have defined Yseed ≡ (A,Z),A>4 Y(A,Z) . While for Ye ≈ 0.5 the nuclear abundance distribution has its peak around 56 Fe and 56 Ni (the most bound, symmetric nuclei, characterized by the presence of a Z = 28 proton shell closure), for Ye < 0.5, the distribution moves toward more exotic, neutron-rich iron group nuclei. However, there is a limit for the amount of neutrons that can be bound inside a nucleus: for 20  Z  40, nuclei located at the neutron drip line have (Z/A)min ∼ 0.3. Moreover, due to the neutron richness, it becomes energetically favorable to have a certain fraction of free neutrons even for Ye > 0.3. Starting from baryon number conservation written as Xn +Xn,seed +Xp,seed ≈ 1 (where Xn/p,seed indicate the fractions of neutrons or protons bound   in seed nuclei), from the charge  neutrality condition Ye ≈ (A,Z),A>4 Z Y(A,Z) , and using the above definitions, the free neutron fraction can be evaluated as: Yn ≈ 1 − (A seed /Z seed ) Ye .

(9)

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Assuming for simplicity that Z seed ≈ 28 and that the dependence of A seed on Ye is approximately linear between A/Z = 0.5 and A/Z = (A/Z)min : A seed ≈ 56 + 70 (0.5 − Ye ) ,

(10)

we obtain an approximation for Yn : Yn ≈ 1 − 3.25Ye + 2.5Ye2 ,

(11)

that varies between 0 for Ye = 0.5 and 1 for Ye = 0. Since almost all protons are contained inside the seed nuclei, we can also estimate the seed abundance as: Yseed ≈ Ye /Z seed ≈ Ye /28 .

(12)

Neutron capture is the key reaction to produce heavy nuclei beyond the iron group, once NSE freeze-out has occurred. In accordance with the definition of reaction rate and in analogy to the effective rate definition, the lifetime of a generic seed nucleus against a neutron capture reaction can be estimated as:   τ(n,γ ) = 1/ nn σ v (n,γ ) ,

(13)

where nn is the free neutron density. The Q-value of an (n, γ ) reaction is the energygained by the nucleus (A, Z)  by acquiring a neutron, and it is computed as Q = m(A,Z) + mn c2 − m(A+1,Z) c2 . It is also equal to the energy required to remove a neutron from a (A + 1, Z) nucleus, called neutron separation energy of the (A + 1, Z) nucleus, Sn,(A+1,Z) . Neutron captures have Q-values that range from 0 for exotic neutron-rich nuclei at the neutron drip line up to ∼15 MeV close to the valley of stability. The leading contribution to σ(n,γ  ) is provided by the swave term of the partial wave expansion: σn ≈ π/k 2 Ts , where Ts ≈ 4k/k is the transmission coefficient obtained by considering a neutron moving against the potential barrier of the nucleus and k, k are the wave numbers of the particles in the initial and final state, respectively. If μ ≈ mn is the reduced mass of the parent state and E ∼ kB T the thermal energy scale, for non-relativistic energies 2 ), the relative speed is v = √2E/μ, and the wave Q  m c (kB T ∼ 0.1 MeV  n √ √ √ numbers are k ≈ 2μE/h¯ and k ≈ 2μ(Sn + E)/h¯ ≈ 2μSn /h, ¯ so that the typical cross section is: 

σ(n,γ )

kB T ∼ 3.70 barn 0.1 MeV

−1/2 

Sn 5 MeV

−1/2 .

(14)

√ √ Since σn ∝ 1/ T and v ∝ T , σ v n,γ is approximately constant for thermal, non-relativistic neutrons. Thus, the lifetime against neutron capture, Eq. (13), depends mainly on nn , which for a plasma of density ρ and free neutron fraction Yn is simply nn = ρYn /mB ≈ 2.99 × 1024 cm−3 (ρ/10 g cm−3 ) (Yn /0.5), and τ(n,γ ) becomes:

14 r-Process Nucleosynthesis from Compact Binary Mergers

 τ(n,γ ) ∼ 0.21 ns

ρ 10 g cm−3

−1 

Yn 0.5

−1 

571

Sn 5 MeV

1/2 (15)

.

The density in the ejecta changes considerably with time as a result of the homologous expansion, Eq. (6). If we further consider that ρhom ∼ 106 g cm−3 at NSE freeze-out for s ∼ 10 kB baryon−1 (see Eq. (7)), then the free neutron density at time t is: nn ≈ 8.08 × 10 cm 24

−3



ρhom 6 10 g cm−3



   Yn  τ 3 t −3 , 0.5 10 ms 1s

(16)

and τ(n,γ ) can be also expressed as a function of time after merger as:  τ(n,γ ) ∼ 1.18 μ s

Yn ρhom 0.5 × 106 g cm−3

−1 

τ −3 10 ms



Sn 5 MeV

1/2 

t 1s

3 (17) .

The capture of one or more neutrons increases the mass number by one or more units without increasing the atomic number. Then, neutron captures move the nuclear abundances toward the neutron-rich side of the nuclear chart. Two kinds of reactions compete with neutron capture in producing heavier and heavier neutronrich nuclei: (γ , n) reactions and the β − decays. For the (γ , n) reactions, while the high neutron density guarantees high (n, γ ) rates, high-energy photons are required to knock a neutron off a nucleus, overcoming the neutron separation energy Sn and boosting the photodestruction rates. For T  4 GK, the two sets of reactions are in NSE, meaning that the temperature is large enough to bring (n, γ )–(n, γ ) at equilibrium everywhere, also close to the valley of stability where Sn ∼ 8–10 MeV. In neutron-rich conditions, when the nuclear distribution is shifted toward the neutron drip line, the relevant Sn can be as low as ∼1–2 MeV. Thus, temperatures lower than 4 GK are enough to preserve the (n, γ )–(n, γ ) equilibrium after NSE freeze-out on the neutron-rich side of the nuclear chart. This is indeed the QSE typical of neutron-rich conditions. Since (γ , n) reactions are the inverse of (n, γ ) reactions, their rates are related by detailed balance conditions: n(A,Z) nn σ v (n,γ ) = n(A+1,Z) λ(γ ,n) , where λ(γ ,n) is the photodisintegration rate of (γ , n) reactions. Assuming these reactions to be in equilibrium, i.e., (A, Z) + n ↔ (A + 1, Z) + γ , the chemical potentials of the different nuclear species involved are related by μ(A,Z) + μn = μ(A+1,Z) (we recall that μγ = 0). Using the expression of the chemical potentials for an ideal MaxwellBoltzmann gas in its relativistic version (i.e., including the rest mass contribution), one obtains: n(A+1,Z) = n(A,Z) nn



2π h¯ 2 mb kB T

3/2

G(A+1,Z) (T ) 2 G(A,Z) (T )



A+1 A

3/2



Sn,(A+1,Z) exp kB T

 , (18)

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where G(A,Z) (T ) is the nuclear partition function, dependent on the matter temperature. Assuming G(A+1,Z) /G(A,Z) ∼ 1 and (A + 1)/A ∼ 1, we finally obtain an expression for τ(γ ,n) ≡ 1/λ(γ ,n) , the lifetime of a seed nucleus against photodestruction: 

τ(γ ,n)

kB T ≈ 0.19 μ s 0.167 MeV

−3/2 

Sn 5 MeV

1/2

exp (Sn /kB T ) , exp (30)

(19)

where the reference temperature (corresponding to T ≈ 2 GK) and neutron separation energy are chosen such that Sn /kB T ≈ 30. This timescale depends heavily on the temperature due to the presence of the exponential factor that rises by orders of magnitudes as the temperature decreases. Additionally, it depends also on the nuclear masses through Sn , and the latter sets the scale in the exponential argument, meaning that for a given temperature τ(γ ,n) changes dramatically between the valley of stability and the drip line. Initially, at NSE freeze-out (T ∼ 4 GK), the temperature and the density are large enough such that both τn,γ and τγ ,n are much smaller than the dynamical timescale. The exponential term in Eq. (19) sets the typical Sn necessary to guarantee the equilibrium between the two reaction sets. As T and ρ decrease, τ(γ ,n) changes much more sensibly, and (n, γ )–(γ , n) equilibrium can establish only down to T ≈ 1 GK, assuming typical Sn ≈ 1–3 MeV. For the β − decays, using the low-energy limit of the weak interaction theory at leading order, the transition matrix element M is proportional to the Fermi coupling constant, GF . Remembering that in natural units GF is the reciprocal of an energy squared, that the time is the reciprocal of an energy, and that the only relevant energy scale in the process is the Q-value of the decay, the lifetime of a nucleus against β decay, τβ , must be proportional to Q−5 . This dependence can be seen also as a consequence of the three-body nature of the final state. The Q-value of β − decays involving neutron-rich nuclei is roughly proportional to the neutron excess, D = N − Z, and varies between a fraction of MeV close to the valley of stability and ∼5–15 MeV at the neutron drip line (with larger values at lower mass numbers, where the neutron excess can also be much larger). Using the decay of the free n as representative β − reaction (for which Q = Δ and D = 1), we estimate the typical β-decay lifetime as:  τβ ∼ τn

Δ Q

5



Q ≈ 3.19 ms 10 MeV

−5



D ≈ 8.82 ms 10

−5 ,

(20)

where in the last step we have further assumed that Q ∼ DΔ. Then, β − decays act on much longer timescale than neutron captures at NSE freeze-out and during the (n, γ ) − (γ , n) equilibrium and become competitive only when the temperature and density have significantly dropped. This difference in the neutron capture and in the β − decay timescale qualifies the ejecta from compact binary mergers as one of the astrophysical sites for r-process nucleosynthesis in the universe.

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Starting from a certain seed nuclei distribution, the fast neutron and photon captures move abundances within the same isotopic chain (i.e., the sequence of nuclei characterized by the same Z and by an increasing N ). However, nuclear decay is a stochastic process and a fraction of nuclei decay from one isotopic chain to the next one even if τβ is significantly larger than τ(n,γ ) , especially if the (n, γ )–(γ , n) equilibrium is maintained on a timescale comparable to or longer than τβ . Moreover, Sn is not a smooth, monotonic function inside the nuclear chart. The closure of neutron shells inside the nucleus at the magic numbers N = 28, 50, 82, 126 determines an increase of Sn around those values. The corresponding nuclei become waiting points, where τ(n,γ ) increases and matter tends to accumulate. For these nuclei, the β − decay starts earlier to be competitive, and matter flows through them from an isotopic chain to the next one. The net results of the combination of neutron captures, photodestructions, and β − -decays on the whole distribution of nuclei emerging from NSE during the (n, γ )–(γ , n) equilibrium are a characteristic nuclear distribution that proceeds as a river inside the neutron-rich side of the nuclear chart. This is called the rprocess path. The final point of the r-process nucleosynthesis depends on how many free neutrons are available to be captured by the seed nuclei. This number is called the neutron-to-seed ratio, Yn /Yseed , and the end point of the r-process can be estimated as: A final ∼ A seed + Yn /Yseed ,

(21)

where A seed is the average mass number of the seed nuclei. For example, assuming A seed ∼ 80–90 (see Eq. 10), to produce element with A ≈ 130, it is necessary to have Yn /Yseed ≈ 40–50; for A ≈ 195, Yn /Yseed ≈ 95–115 (we will see later that these are the mass numbers of the so-called second and third r-process peaks), while for uranium and thorium (A ≈ 235), Yn /Yseed ≈ 145 − 155. For low- or moderate-entropy ejecta, the value of Ye primary determines the neutron-to-seed ratio and, from that, how far the r-process nucleosynthesis proceeds in producing heavy elements starting from iron group seed nuclei. Using Eqs. (11) and (12), we can obtain a simple estimate for the neutron-to-seed ratio as a function of Ye in low-entropy conditions: Yn /Yseed ∼ Z seed /Ye − A seed ≈ 28/Ye − 70 Ye − 21 .

(22)

This expression diverges for Ye → 0, as there are no seeds, while it goes to 0 for Ye = 0.5. For intermediate values Ye = 0.1, 0.2, 0.3, 0.4, we obtain Yn /Yseed ≈ 250, 105, 51, 21, respectively. These formulae, relying on the simple assumption of a linear dependence of A seed on Ye , Eq. (10), must be understood as very rough estimates that nevertheless catch the most relevant trends for typical entropy (s ∼ 10 kB baryon−1 ) and expansion timescale (τ ∼ 10 ms) in the ejecta. More detailed and physically motivated calculations (e.g., [21, 38]) extending to

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broader ranges of possible conditions show that the neutron-to-seed ratio has a more complex dependence on the (thermo)dynamics conditions at NSE freeze-out, in particular, Yn /Yseed is larger for lower Ye , smaller τ , and larger s. For example, for Ye  0.4, the neutron-to-seed ratio is such that Yn /Yseed ∝ s 3 /Ye3 τ .

The Working of the r-Process in Compact Binary Mergers The evolution of the nuclear abundances and the calculation of the nuclear energy released during the nucleosynthesis are computed by nuclear reaction networks. A nuclear network is a large system of coupled ordinary differential equations. For each nucleus i ≡ (Ai , Zi ), its abundance Yi ≡ Y( Ai , Zi ) evolves according to:

dYi ρ2 i ρ i = Nji λj Yj + Nj,k σ v j,k Yj Yk + Nj,k,l σ v j,k,l Yj Yk Yl , dt mb m2b j

j,k

j,k,l

(23) where the sums run over all possible reactions that include (Ai , Zi ) in the initial or in the final state: the first sum contains decays, photodisintegrations, and semi-leptonic processes, as electron, positron, or neutrino captures; the second and the third ones include nuclear fusions with two and three reactants, respectively. In this context, a three-body reaction is a sequence of two-body reactions with an intermediate state with an extremely short lifetime. The factors N...i account for multiplicity effects in the case of identical particles: if Nm represents the number of m nuclei involved in a specific reaction with sign (i.e., Nm > 0 for creation and Nm < 0 for destruction), i = N /(|N |!|N |!) and N i then Nji = Ni , Nj,k i j k j,k,l = Ni /(|Nj |!|Nk |!|Nl |!) (for identical reactants, double counting must be avoided so that Nj + Nk = 2 and Nj + Nk + Nl = 3 for two- and three-body reactions, respectively. For example, for α + α + α → 12 C, Nj = 3 and Nk = Nl = 0). The calculations of the reaction rates and of the effective decay constants require the knowledge of the evolution of the fluid density and temperature, as well as information about the neutrino irradiation fluxes. In actual computations, the evolution of the matter density is usually prescribed or extracted from hydrodynamics simulations. The subsequent evolution of the temperature and Ye are then self-consistently determined from the detailed abundances, from the EOS of the plasma, and assuming that the expansion proceeds adiabatically, unless for nuclear energy generation and neutrino leakage or irradiation. We will now describe in more details the most relevant features of the nucleosynthesis happening in a fluid element expanding after a compact binary merger, according to the outcome of nuclear network calculations. For the bulk of the neutron-rich ejecta, characterized by low specific entropy, we can identify four phases: • the initial NSE phase; • the r-process nucleosynthesis phase;

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Fig. 4 Evolution of a few selected abundances computed for a fluid element expanding in space with homologous expansion (see Eq. (6)) and initially characterized by a specific entropy s = 10 kB baryon−1 , an initial Ye = 0.10, and expansion timescale τ = 10 ms, corresponding to typical compact binary merger ejecta conditions (left panel). Heavy nuclei are defined as nuclei for which A ≥ 120. In addition, relevant timescales as computed according to Eqs. (33), (34), (35), (36), and (37) are also shown (right panel). Calculations were performed using the SkyNet nuclear network [39]. (Courtesy of D. Vescovi)

• the neutron freeze-out phase; • the decay phase. The different phases and their properties can be observed in Figs. 4, 5, and 6.

The NSE Phase The NSE phase (top panel of Fig. 5) lasts as long as the temperature in the plasma stays above T ≈ 4–5 GK. It is important to notice that a fluid element can enter and exit the NSE phase several times before eventually decreasing its temperature below the NSE threshold due to hydrodynamics processes and intense nuclear heating. In this case, only the conditions at the last NSE freeze-out influence the subsequent evolution. During NSE, since (p, γ )–(γ , p) and (n, γ )–(γ , n) reactions are all in equilibrium, the recursive application of the corresponding equilibrium relation among the relativistic chemical potentials, namely, μ(A,Z) = μp + μ(A−1,Z−1) and μ(A,Z) = μn + μ(A−1,Z) , yields to μ(A,Z) = Zμp + Nμn . For particles described by the Maxwell-Boltzmann statistics, once ρ, T , and the abundances of free ns and ps (Yn,p ) are provided, the abundance Y(A,Z) of any nucleus (A, Z) in NSE is given by [10]: Y(A,Z)

G(A,Z) (T )A3/2 = YpZ Yn(A−Z) 2A



ρ mb

A−1

2π h¯ 2 mb kB T

3(A−1)/2 eB(A,Z)/kB T , (24)

where B(A,Z) is the nucleus binding energy. Yn,p are  ultimately set by requiring baryon conservation and charge neutrality, i.e., 1 = (A,Z) Y(A,Z) A and Ye =

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Fig. 5 For the same trajectory used for Fig. 4, we also present detailed nuclear compositions on the nuclear chart. In this two panels, we show the abundances at the end of the NSE phase (top panel) and the end of the (n, γ )–(γ , n)) equilibrium (bottom panel). In the latter case, the r-process path is clearly visible. These pictures were produced using the SkyNet nuclear network [39] and the dedicated visualization software. (Courtesy of D. Vescovi)



(A,Z) Y(A,Z) Z, respectively. Very high densities and not too large temperatures favor large nuclei (as it happens in the crust of cold NSs), while photodestruction in hot environments produces light nuclei and ultimately free protons and neutrons (as in the mantle above proto-neutron stars in core-collapse supernovae (CCSNe) or in the cores of merging NSs). For intermediate regimes, the nuclear binding energy favors the most tightly bound nuclei, i.e., iron group nuclei or α particles among the light nuclei, and the temperature regulates the width of the distribution. For a given plasma configuration (i.e., for a given set of ρ, T , and Ye ), the NSE condition determines the nuclear abundances according to Eq. (24) without the need

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577

Fig. 6 Same as in Fig. 5, but during the decay phase, just after neutron freeze-out (top panel) and at 109 s after merger (bottom panel). These pictures were produced using the SkyNet nuclear network [39] and the dedicated visualization software. (Courtesy of D. Vescovi)

of solving Eqs. (23). However, the abundances change as a function of time due to the temporal evolution of the expanding and cooling plasma, ρ = ρ(t) and T = T (t). Moreover, weak interactions are out of equilibrium, and Eq. (23) and the charge neutrality conditions reduce to an equation for Ye :

  dYe = λe+ ,i − λe− ,i + λνe ,i − λν¯e ,i + λβ − ,i − λβ + ,i Yi , dt

(25)

i

where the sum runs over all nuclei and the rates span all possible (if any) semileptonic reactions involving each nucleus i.

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As visible in the top panel of Fig. 5, at NSE freeze-out, the composition of the expanding plasma is characterized by a distribution of seed nuclei, peaking around the iron group, α particles, and free neutrons. Depending mainly on the initial Ye  0.4, the seed nuclei are possibly very or extremely neutron-rich, with A ∼ 60 − 100. A high fraction of free neutron is also expected. If the fluid is radiation dominated, its specific entropy can be approximated by the entropy of the photon gas; see Eq. (7). It is very insightful to substitute this expression inside the expression of the abundances in NSE conditions, Eq. (24), to express it also in terms of the entropy: Y(A,Z) ∝

A3/2 YpZ Yn(A−Z) A A−1 2 s



kB T mb c 2

3(A−1)/2 eB(A,Z)/kB T .

(26)

Since Y(A,Z) ∝ s −(A−1) , we recover the result that in the case of matter with high specific entropy, an α-rich freeze-out is obtained.

The r-Process Nucleosynthesis Phase Once T  4 GK, the first reactions that run out of equilibrium are charged nuclear reactions involving the less abundant nuclei. Under the assumption that neutron captures, photodestructions, and β − decays (possibly emitting j delayed neutrons, with j = 0 being the classical β − decay) are the most relevant reactions, Eq. (23) becomes: dY(A,Z) ≈ nn σ v (A−1,Z)(n,γ )(A,Z) Y(A−1,Z) + λ(A+1,Z)(γ ,n)(A,Z) Y(A+1,Z) + dt   − nn σ v (A,Z)(n,γ )(A+1,Z) + λ(A,Z)(γ ,n)(A−1,Z) Y(A,Z) + +

J

λ(A+j,Z−1)→(A,Z)+e− +¯νe +j

n

Y(A+j,Z−1) +

λ(A,Z)→(A−j,Z+1)+e− +¯νe +j

n

Y(A,Z) .

j =0

−

J

(27)

j =0

In the first line, we have considered the creation of (A, Z) nuclei through (n, γ ) reactions on (A − 1, Z) nuclei and (γ , n) reactions on (A + 1, Z) nuclei. In the second line, we have considered the destruction of (A, Z) nuclei through (n, γ ) and (γ , n) reactions. In the third and fourth lines, we have taken into account β − decays that can include a (A, Z) nucleus in the final or initial state, respectively, possibly through the additional delayed emission of 0 < j ≤ J neutrons. As we have seen in the previous section, if the temperature and density are large enough, the timescales of β − decays are much longer than the timescales of (n, γ ) and (γ , n) reactions. The latter absorption processes connect nuclei among the same isotopic chain (Z is fixed), and their equilibrium ensures an almost steady free neutron fraction, while the former decays connect nuclei of contiguous isotopic

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chains (Z−1 and Z and Z to Z+1). Then, the evolution predicted by Eq. (27) can be split into two separate problems: the much faster (n, γ )–(n, γ ) equilibrium inside each isotopic chain and the slower flow through different isotopic chains driven by β − decays. In the following, we will closely analyze each of the two problems separately. We start by consider an isotopic chain characterized by a specific Z. The equilibrium condition inside the chain (i.e., μ(A+1,Z) = μ(A,Z) +μn ) allows to write an equation for the abundances of two adjacent nuclei in the chain, starting from Eq. (18) and simply noticing that Y(A+1,Z) /Y(A,Z) = n(A+1,Z) /n(A,Z) . Evaluating the resulting expression for typical magnitudes, we obtain: Y(A+1,Z) ≈ 5.71 × 103 Y(A,Z)



nn 8 × 1024 cm−3



kB T 0.1 MeV

−3/2 

exp (Sn /kB T ) exp (30)

 . (28)

We notice that the abundance ratio depends only on nn , on T , and on Sn . The latter introduces a dependence on the nuclear masses. Also in this case, the numerical value of this ratio is dominated by the exponential factor whose argument compares the neutron separation energy with the plasma temperature. Close to the valley of stability Sn ≈ 15 MeV  kB T and Y(A+1,Z)  Y(A,Z) , i.e., abundances increase steeply moving toward neutron-richer nuclei. Assuming kB T = 0.1 MeV and Sn = 3 MeV, the above estimate still implies Y(A+1,Z)  Y(A,Z) . For Sn = 2 MeV, the numerical prefactor decreases to 0.26, meaning that Y(A+1,Z) < Y(A,Z) . Since Sn → 0 at the neutron drip line, there exists always a turning point in the isotopic chain, where Y(A+1,Z) stops to increase with respect to Y(A,Z) before decreasing when approaching the neutron drip line. We estimate this point by requiring Y(A+1,Z) /Y(A,Z) = 1. Assuming a certain nn and temperature T , this condition translates in a reference neutron separation energy Sn0 :

2 Sn0 = kB T ln nn



mB kB T 2π h¯ 2

  1 − 0.047 ln

3/2



kB T ≈ 2.14 MeV 0.1 MeV

nn 8 × 1024 cm−3





 ×

kB T + 0.070 ln 0.1 MeV

 .

(29)

The value of Sn0 depends on nn and T , but not on Z. So, at any given time, all populated isotopic chains have their abundance peaks at nuclei characterized by the same Sn0 , and the conditions Sn (A, Z)  Sn0 define the r-process path. A better approximation can be done by considering that, due to nuclear paring effect, even neutron numbers are favored. Indeed, while Sn decreases for increasing D, but with an even-odd modulation, S2n defined as S2n (A + 2, Z) = (mA,Z + 2mn − m(A+2,Z) )c2 and called the two neutron separation energy decreases in a smoother way, with a sudden decrease at the magic neutron numbers. By considering the effective equilibrium (A + 2, Z) + γ ↔ (A, Z) + 2n and the corresponding

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relation on the chemical potentials, μ(A+2,Z) = μ(A,Z) + 2μn , the r-process path can be defined as the set of nuclei belonging to different isotopic chains for which S2n (A, Z) is closer to 2Sn0 (nn , T ) along each specific chain. Since the relative distribution of abundances inside an isotopic chain is set by the fast (n, γ )–(γ , n) equilibrium and determined  by Eq. (28), it is useful to consider the total abundance along the chain, YZ ≡ A Y(A,Z) . Starting from Eq. (27) and considering only β − decays without delayed neutron emission (J = 0), we obtain an evolution equation for YZ : dYZ ≈ λ˜ β,(Z−1) YZ−1 − λ˜ β,Z YZ dt

(30)

where λ˜ β,Z s are the effective, abundance-weighted β decay rates of the whole chains:





˜λβ,Z ≡ λβ,(Z,A) Y(A,Z) / Y(A,Z) . (31) A

A

Under the assumption that the amount of free neutrons and the temperature are high enough, the duration of the r-process becomes comparable or even larger than the β decay lifetime (Eq. 20) over the most relevant part of the nuclear chart, and the chain abundances YZ tend toward an equilibrium configuration, i.e., dYZ /dt ≈ 0, which translates into: λ˜ Z YZ = λ˜ Z−1 YZ−1 ≈ const .

(32)

This condition is known as steady β-flow approximation [16, 33], and it implies that the abundance of a chain is proportional to its effective β decay lifetime. Since the nuclei with magic neutron numbers N = 50, 82, 126 (or just above them) and closer to the valley of stability have the longest β decay lifetime, it is expected that maxima in the abundances will occur at the top end of the kinks in the r-process path corresponding to the neutron shell closures. This is what quantitatively defines the location and the relative importance of the waiting points inside the r-process path. The duration of the r-process nucleosynthesis phase crucially depends on the availability of free neutrons. Under the assumption that neutron captures and photodestructions are the most relevant reactions, the timescale over which the neutron abundance changes, τn , can be evaluated from Eq. 27 as:     1 1 1  dYn  1 1 ≈ . ≡ − τn Yn  dt  Yn /Yseed τ(n,γ ) τ(γ ,n)

(33)

The relevant timescales τ(n,γ ) and τ(γ ,n) can be estimated through Eqs. (15) and (19) as we did in the previous section, but they can also be computed more rigorously from the reaction rates that enter Eq. (23):

14 r-Process Nucleosynthesis from Compact Binary Mergers

⎛ 1 τ(n,γ )

≡⎝

(A,Z)

1 τ(γ ,n)

581

⎞

YA,Z nn σ v (A,Z)(n,γ )(A+1,Z) ⎠ / ⎛

≡⎝

⎞

YA,Z λ(A,Z)(γ ,n)(A−1,Z) ⎠ /

(A,Z)

Y(A,Z) ,

(34)

Y(A,Z) .

(35)

(A,Z)

(A,Z)

As long as the neutron-to-seed ratio is large, τn is relatively long, and there is enough time for many isotopic chain (with increasing Z) to be reached and for the β-flow equilibrium to establish. Detailed calculation show that, for typical conditions in compact binary merger ejecta, this r-process phase can last up to ∼1 s after the merger (see, e.g., the bottom panel of Fig. 5). Eventually, the r-process path can reach the neutron magic number N = 184. Above that, fission becomes the dominant nuclear process. Typical super-heavy, neutron-rich nuclei have (A, Z) ∼ (250, 100), while daughter nuclei can be approximated by distribution around the double-magic number nucleus (A1,2 , Z1,2 ) ∼ (132, 50). Thanks to fission and depending on the lighter fission fragment distribution, the abundances of heavy elements double and specific region of the nuclear chart start to populate. If the initial neutron richness is high enough (in particular, when the initial Ye  0.1), fission becomes relevant, while (n, γ ) and (γ , n) reactions are still in equilibrium. The lighter nuclei resulting from the fission still capture many neutrons, reaching again N = 184, and many fission cycles occur [32, 38, 43]. The possibility of reaching (n, γ )–(γ , n) equilibrium and its duration are critically related with the temperature evolution. For a radiation-dominated plasma, if the expansion proceeded adiabatically, the temperature would simply evolve as T (t) ≈ Thom (thom /t), with Thom of the order of a few GK. Then, within ∼0.1 s, the temperature would drop below 1 GK. While this relation describes in good approximation the evolution of matter temperature before and after the r-process, it is rather inaccurate during it. Indeed, during the r-process, the energy released by nuclear reactions and the matter density are often large enough such that matter is significantly re-heated and temperature becomes again larger than 1 GK for a much longer time (up to ∼1 s), before dropping again as predicted by the adiabatic expansion law, T ∝ t −1 . To estimate if the specific nuclear heating e˙nucl (nuclear energy per unit mass, per unit time) can affect significantly the matter temperature, we compare the density of energy of the radiation field with the energy released by the nuclear decays during a time comparable to the expansion timescale τ , i.e., 4 /15(hc)3 ∼ e˙ π 2 kB4 Tmax ¯ nucl ρτ , and we estimate e˙nucl ∼ Sn Yn /(mb Δtr−proc ) [43]. By solving for Tmax , we can estimate the maximum temperature reached by matter due to the nuclear heating. Assuming that around t ≈ 0.1 s the density has decreased down to 105 g cm−3 and the (n, γ )–(γ , n) equilibrium lasts for Δtr−proc ∼ 1 s: 

Yn ρ Tmax ≈0.75 GK 0.5×105 g cm−3

1/4 

Sn 5 MeV

1/4 

  τ 1/4 Δtr−proc −1/4 . 10 ms 1s (36)

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When Tmax  0.7 GK, the temperature is large enough to guarantee (n, γ )– (γ , n) equilibrium during most of the neutron captures. This is defined as the hot r-process. If, on the other hand, Tmax  0.3 GK, the density is large enough to sustain the neutron captures, but not (n, γ )–(γ , n) equilibrium. In this latter case, the neutron density decreases much faster due to neutron consumption, and this is referred to as cold r-process. It is important to stress again that the relevant r-process regime (hot or cold) does not depend on the absolute peak temperature, but on the maximum temperature during the re-heating phase occurring when seed nuclei capture neutrons. Other factors, as, for example, the expansion timescale, determined if the r-process happens in hot or cold conditions.

The Neutron Freeze-Out and the Decay Phases As the r-process proceeds, and matter expands and cools below ∼1 GK, τ(γ ,n)  τ(n,γ ) , and photodestruction becomes inefficient in keeping a high neutron density outside the heavy nuclei, while the still effective neutron captures and β − decays produce more and more heavy nuclei. Once Yn /Yseed ∼ 1, according to Eqs. (16) and (33), it happens that τn ∼ Yn /Yseed τ(n,γ )  1s: the free neutron lifetime suddenly decreases because the seed nuclei start to compete for the few available neutrons. This phase is called neutron freeze-out, and it is characterized by a sudden drop of the neutron density and of the neutron-to-seed ratio, as visible both in Fig. 4 and in the top panel of Fig. 6. After this drop, the β − decays (often followed by the emission of free neutrons) start to compete with the neutron capture, since τ(n,γ ) becomes comparable to τβ . The latter can be estimated as Eq. (20), but also as: ⎛ 1 ⎝ ≡ τβ



(A,Z), A>4 j =1,J

⎞ ⎛ Y(A,Z) λ(A,Z)→(A−j,Z+1)+e− +¯νe +j n ⎠ /⎝

⎞ Y(A,Z) ⎠ .

(A,Z), A>4

(37) The presence of freshly emitted free neutrons provides a new source of neutrons available to be captured also at later time. The result is that while the nuclei that were located along the r-process path during the (n, γ )–(γ , n) equilibrium decay collectively toward the valley of stability, the competition between the neutron captures and the β decays smooths the r-process abundances. In particular, it removes the strong oscillations in the mass number that characterize the abundances just before neutron freeze-out, due to pairing and collective effects in the nuclear properties. After a timescale ranging from a few up to a few tens of seconds (depending on the environment and increasing for decreasing initial Ye ), the neutron density has decreased such that also the neutron captures become negligible. Most of the matter is still in the form of unstable nuclei, with neutron excess D ∼ 5–10. At this stage, β − decays and, depending on how extended the r-process path is, α decays are the most relevant nuclear processes that bring the abundances toward the valley of stability. While most of the nuclei have reached stable configurations within a few

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tens of days, a few heavy isotopes have longer half-lives, extending above 107 yrs for 14 isotopes and above 109 yrs for 7 isotopes. Since all these reactions liberate nuclear energy, they can heat up matter or become a distinct source of radiation. The detection of the various, distinct signals is thus a clear indication of the r-process nucleosynthesis occurring inside the ejecta.

The r-Process Peaks and the s-Process Nucleosynthesis During the (n, γ )–(γ , n) equilibrium, the r-process path is characterized by the presence of waiting points, where matter accumulates due to the longer lifetimes occurring in correspondence of the neutron magic numbers N = 50, 82, 126. Waiting points are identified by neutron close shells, but they can span a large range of Zs, and they can also be influenced by the presence of the proton magic numbers Z = 28, 50 (see the bottom panel of Fig. 5). Since at neutron freezeout nuclei are still very neutron-rich (typically Z/A ∼ 0.5), these waiting point nuclei are characterized by A ∼ 80, 130, 195. When decaying back to stability, nuclear abundances proceed along A ≈ const paths, i.e., they keep their mass number approximately constant while increasing their Z. However, the residual delayed neutron emissions and neutron captures can further change A by a few units during the process. Due to the decreasing neutron density, the effect due to β decays eventually takes over. The result is that the final abundance pattern is characterized by three peaks (known as the r-process peaks): the first peak is located around A ≈ 80 and covers the range A ≈ 72−−94, the second around A ≈ 130 with range A ≈ 111 − −145, and the third around A ≈ 195 with range A ≈ 188 − −206. Due to the β decays, the nuclei neutron content has globally decreased, and the Ye has increased toward 0.4. As a function of Z, the peaks are located around Z ≈ 35 (e.g., selenium, bromine, krypton) for the first peak, Z ≈ 53 (e.g., tellurium, iodine, xenon) for the second peak, and Z ≈ 77 (e.g., osmium, iridium, platinum) for the third peak. Thus, due to the combination of multiple neutron captures followed by many β decays, the final peaks are shifted toward the left by several units with respect to the nuclei characterized by magic neutron numbers along the valley of stability. However, when considering the element above Zn, the abundances that we observe inside the solar system reveal, close to the r-process peaks, other peaks shifted toward the right by a few units in the mass number and happening precisely in correspondence of the magic neutron numbers. Then, different from what happens in the r-process, the production of these peaks must take place close to the valley of stability, in such a way that a neutron capture on a stable or long-lived nucleus is followed by a β decay before another neutron can be captured, meaning that t(n,γ )  tβ . This is possible if the neutron densities are in the range 106−11 cm−3 , so several orders of magnitudes lower than the ones required by the r-process. This kind of nucleosynthesis is called slow neutron capture process (s-process), and it is thought to happen inside low and intermediate mass stars (starting their life with a mass between 0.6 and 10 M ) during their asymptotic giant branch (AGB) phase. Starting from iron group nuclei produced in a previous SN explosion and

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already present inside the star, the s-process proceeds through the s-process path up to lead on a timescale of several thousands of years. The abundances of the s-process nucleosynthesis are well understood since they rely on the well-known properties of nuclei at or very close to the valley of stability. Thus, the solar r-process abundances are defined as the residual of the solar abundances, once the s-process contribution has been removed; see, e.g., [51]. Most of the heavy elements receive a contribution from both the s- and the r-process. It is important here to stress that information about the isotopic composition is in general harder to obtain, while astrophysical observations often provide information on the elemental composition. So, an element usually receives contributions from different isotopes, and these isotopes can be synthesized in different environments. Nevertheless the structure of the valley of stability is such that some stable isotopes can only be produced in the s-process or in the r-process. In the first case, this happens when a stable isotope is shielded by another stable isotope along a β decay line. In the second case, this happens when a stable nucleus can be reached by a neutron capture and a subsequent β decay sequence only from a nucleus that has a very short lifetime or that is outside the s-process path. When this happens for all or for the most relevant isotopes of an element, this element is called s-only or r-only element, respectively. A r-only isotope has the advantage of allowing to study the contribution of the r-process independent from possible contamination from the s-process. Europium and elements above lead (including uranium, thorium, and plutonium) are among the r-only elements.

Nucleosynthesis in High-Entropy and Fast-Expanding Ejecta Not all the ejecta are characterized by low or moderate entropy or by expansion timescales ranging from a few ms up to several tens of ms. In particular, detailed merger models show the presence of high-entropy and/or extremely fast-expanding tails in the ejecta distribution. Moreover, it is not always guaranteed that the peak temperature will be above 4 GK or that the temperature will be high enough during the r-process phase to ensure (n, γ )–(γ , n) equilibrium. The nucleosynthesis in these conditions can be significantly different than the one described above. For example, if the expansion timescale τ becomes of the order of (or even smaller than) 1 ms, the re-heating during the r-process phase could become inefficient, and (n, γ )–(γ , n) equilibrium cannot establish; see Eq. (36). In this case, a cold r-process happens. Moreover, despite the possible large abundance of free neutrons and the availability of seed nuclei, the decrease in the neutron density is so fast that neutron captures become inefficient too early and the r-process does not proceed much. A large fraction of the free neutrons are not captured by nuclei and decay into protons. The resulting abundances are very different from the one observed in a full r-process. According to Eq. (5), for the dynamical ejecta emerging from BNS mergers, this is the case for matter expanding at v ∼ 0.6–0.8c. Such ejecta are observed in some models in the high-speed tail, especially if the EOS of nuclear matter is rather soft and the shocks produced by the NS collision are very violent.

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Finally, if the specific entropy is large enough (s  80 kB baryon−1 ), the NSE ends with a α-rich freeze-out (see, e.g., [29] and reference therein). If, additionally, Ye  0.4, there is a significant abundance of free neutrons that are not bound inside the very abundant α particles and inside the fewer seed nuclei. Thus, in these conditions, the neutron-to-seed ratio is still very high, and the r-process nucleosynthesis can still occur to produce heavy r-process nuclei. The fundamental difference is that the final abundances are overall dominated by the α particles emerging from the NSE freeze-out; see, e.g., [21, 52].

Nuclear Physics Input and Detailed Network Calculations Equilibrium arguments are very useful to get a qualitative understanding of the main features of the r-process nucleosynthesis. However, only the numerical solution of Eqs. (23) provides detailed nucleosynthesis yields. In nuclear network calculations, abundances are typically initialized in NSE conditions at high temperatures (T ∼ 6– 8 GK) and then evolved consistently following the chemical evolution of the ejecta during their expansion. For r-process nucleosynthesis calculations, more than 7000 nuclei are necessary, ranging from free neutrons and protons up to very neutron-rich transuranic elements (e.g., curium) and covering the whole neutron-rich side of the valley of stability. The reaction rates needed by the network require a vast amount of nuclear physics knowledge. In the following, we briefly discuss the most relevant inputs, namely, the nuclear masses, the β-decay half-lives, the neutron capture rates, and the fission physics. For more detailed information, we address the reader to Ref. [11]. Nuclear masses. The most basic nuclear property are the masses of all nuclei, since they determine the threshold energies for all relevant reactions and, in particular, the neutron separation energies. While the masses of stable nuclei and of nuclei close to the valley of stability are experimentally well known, only theoretical values predicted by nuclear mass models are available for exotic neutron-rich nuclei. Nuclear mass models are tuned on experimentally known masses and then extrapolated, so that the uncertainties grow moving toward the neutron drip line and just above the shell closure points, where correlations and deformation may be very relevant, but difficult to be taken into account. Beta decay rates. Also the values of β decay half-lives are experimentally unknown far from the valley of stability, and theoretical calculations are necessary. Their values are crucial, since they determine the matter flow between different isotopic chains. In particular, the most relevant decay rates are those of nuclei at the neutron magic numbers, since they are the waiting points of the r-process path. The calculation of these decay rates depends on the reaction Q-value (and thus on the mass model), on the transition strength, and, in particular, on its energy distribution. If the transition leads to a final state whose energy is above the neutron separation

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energy, the emission of one or more neutrons in the final state (β-delayed neutron emission) is possible. This emission is relevant after the neutron freeze-out phase. Also in this case, shell effects and nucleon correlations are possibly very relevant but difficult to model. Neutron capture rates. While neutron captures are essential to establish (n, γ )– (γ , n) equilibrium, the corresponding QSE condition makes the detailed knowledge of their rates not very relevant in modeling the r-process. It is after neutron freezeout, when the temperature drops below 1 GK and the photodisintegration becomes less and less relevant, that a detailed knowledge of the relevant neutron capture rates becomes more important. At this stage, nuclei are still very exotic, and theoretical calculations are once again necessary. They are usually computed within the statistical model of nuclei, which is done using the Hauser-Feshbach approach and modeling the nuclear energy density, the γ -strength function for the decay of the compound nucleus, and the potentials of several light particles. Nuclear fission. The fission properties of super-heavy nuclei (A  280) are very relevant in calculations involving very neutron-rich matter (at least for initial Ye  0.10), but they are very uncertain. The theoretical description of fission is highly non-trivial, and it mainly depends on the fission barriers, defined as the energy required for a nucleus to undergo fission. This energy is necessary to deform the nucleus such that the transition to the fission fragments states becomes energetically favored. If the transition to the final state does not require any additional energy, the fission is spontaneous. If additional energy is provided by the interaction or the absorption of another particle, then the fission is induced. All possible fission channels (e.g., spontaneous, neutron-induced, γ -induced) and all the other competing reactions (e.g., α and β decays) need to be included in the model to provide reliable predictions. In the case of a very neutron-rich environment, as the one typically expected in the ejecta of compact binary mergers, neutroninduced fission is usually the most relevant channel. The very uncertain fission fragment distributions are also very important as they influence the abundances around A  140. While the basic features of the r-process nucleosynthesis in the ejecta of compact binary mergers are robust and, especially for low-entropy ejecta, mainly influenced by the initial Ye , the nuclear input physics can significantly change the fine structure of the abundance pattern. For example, the β decay half-lives of nuclei with Z  80 regulate the mass flow toward the magic neutron number N = 126 (possibly enhancing the amount of nuclei that go through fission cycles) and possibly affect the position of the third r-process peak [13, 43]. Moreover, since fission fragments are located around A  140, fission physics can influence the width of the second r-process peak, as well as the abundances of the rear-earth elements, between lanthanum and lutetium [19].

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Detailed Network Calculations and Nucleosynthesis Yields from Compact Binary Mergers Once all the relevant nuclear input physics has been considered, nuclear network calculations can predict with accuracy the distribution of the yields synthesized by a specific fluid element. Nuclear network calculations for fluid elements expanding adiabatically show that the nucleosynthesis outcome depends mainly on the three parameters that we have used to described the properties of the ejecta at the onset of the homologous expansion phase, namely, the electron fraction, the specific entropy, and the expansion timescale (see, e.g., [38] for a systematic study). As representative cases for the abundances obtained at ∼30y after merger, in Fig. 7 we present the abundance patterns obtained by detailed nuclear network calculations. All trajectories have the same specific entropy and expansion timescales (s ≈ 11 kB baryon−1 and τ ≈ 11 ms), but differ because of their initial electron fraction. In all cases, we compare the calculation outcome with the solar r-process residual. For Ye  0.2, all elements between the second and the third r-process peaks are synthesized with a pattern that well reproduces the solar one. Here the observed differences in the shape of the rare-earth peak could be mostly ascribed to the nuclear physics input. This nucleosynthesis outcome is often referred to as the “strong r-process.” For very low electron fractions (Ye  10), the nucleosynthesis proceeds up to Ur and Th, and the abundance pattern is very robust, due to the extremely long r-process path and to the effect of several fission cycles, while for 0.1  Ye  0.2, the production of actinides decreases. Around Ye,crit ≈ 0.23–0.24, the neutron-to-seed ratio at NSE freeze-out is only of a few, the production of nuclei above the second r-process peak is inhibited, and the abundances move toward the first r-process peak for increasing Ye . This nucleosynthesis outcome is sometimes called “weak r-process.”

log Final abundances at 109s

Ye = 0.25 Ye = 0.30

−2

Ye = 0.40 Ye = 0.50 Lanthanides Solar r-process

−3 −4 −5 −6 −7 −8

40

60

80

100 120 140 160 180 200 220 240 260

Mass number A

Ye = 0.10 Ye = 0.20

−1

log Final abundances at 109s

Ye = 0.10 Ye = 0.20

−1

Ye = 0.25 Ye = 0.30

−2

Ye = 0.40 Ye = 0.50 Solar r-process

−3 −4 −5 −6 −7 −8

20

40

60

Atomic number Z

80

100

Fig. 7 Abundances as a function of the mass number (A, left panel) and of the atomic number (Z, right panel) at 109 s after merger for trajectories characterized by s ≈ 11 kB baryon−1 and τ ≈ 11 ms, but for different initial Ye s, computed using the SkyNet nuclear network [39]. Black dots represent the solar r-process residual, as reported by [51]. (Courtesy of D. Vescovi)

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Nucleosynthesis Yields from Compact Binary Mergers Detailed abundances in the ejecta from compact binary mergers reflect the ranges and the relative relevance of the distributions of entropy, expansion timescale, and electron fraction that emerge from the merger dynamics. Different abundance patterns characterize the different ejection channels and keep an imprint of the merger nature, dynamics, and aftermath. Dynamical ejecta. The low-entropy, equatorial dynamical ejecta have Ye  0.2 and produce robustly all elements between the second and the third r-process peaks. In addition, if Ye  0.1, significant abundances of translead nuclei and actinides are produced in relative abundances that can be comparable or even larger than the solar ones. This is the case especially for BHNS mergers and BNS mergers characterized by very different NS masses. In the case of BNS mergers of comparable NS masses, at polar latitudes, the entropy increases but stays on average below 20 kB baryon−1 , while Ye  0.25, and the production of heavy r-process elements is suppressed. In this case, the r-process nucleosynthesis does not proceed up to very high mass numbers, but produces elements from the first up to the beginning of the second peak. The relative importance of the two contributions (polar and equatorial) to the dynamical ejecta depends on the binary mass ratio and on the properties of the nuclear EOS. However, the equatorial component is expected to be always present and overall dominant. Representative results for mass-integrated abundances in the dynamical ejecta of BNS and BHNS simulations are presented in Figs. 8 and 9, and they show that the dynamical ejecta can produce a significant fraction of all r-process nuclei from the first to the third peaks; see, e.g., [54, 70].

Disk wind ejecta from BH-torus systems. For a BHNS merger or for a BNS merger whose the central remnant collapses to a BH within a few dynamical timescales, the ejecta from disk winds are dominated by the viscous component,

100 Relative abundances

Relative abundances

100 10−1 10−2 10−3 LS220_M135135_LK LS220_M135135_M0_LTE LS220_M135135_M0

10−4 50

75

100

125 A

150

Solar

175

200

10−1 10−2 10−3 LS220_M135135_M0 LS220_M140120_M0

10−4 50

75

100

125 A

Solar

150

175

200

Fig. 8 Mass-integrated abundances in the dynamical ejecta obtained by simulations of BNS merger models. In the left panel, the different curves represent different neutrino treatment, with the blue curve not including neutrino irradiation, while the other two including it. In the right panel, neutrino irradiation is always accounted for, but the different curves show the difference between an equal (blue) and an unequal (yellow) mass mergers. (Figures taken from [54])

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Fig. 9 Mass-integrated yields in the dynamical ejecta obtained by simulations of BHNS merger. In the left panel, mass fractions are presented for different combinations of NS and BH masses. In the right panel, the different curves represent abundances obtained for different intensities of the neutrino luminosities. (Figures taken from [29] (left panel) and [55] (right panel))

solar r abundances S-def

-3

Final Mej YA (arbitrary scale)

abundances at 1 Gyr

10-2 10

-4

10

10-5 10-6 -7

10

0

50

100

150

mass number, A

200

250

2nd peak rareearth 3rd peak peak

100 10−1 10−2 1st peak

10−3 10−4 10−5

solar r-process

10−6 10

H000 H010 H030

−7

10−8 10−9

0

H100 H300 Hinf

B070 B090 BF15

25 50 75 100 125 150 175 200 225 250 Mass number A

Fig. 10 Mass-integrated nucleosynthesis results (as a function of the mass number A) for the diskwind ejecta obtained in the case of a BH-disk systems (left panel) and of a massive NS remnant (right panel, with different colors corresponding to different NS lifetimes). (Figures taken from [76] (left panel) and [37] (right panel))

have low entropy, and cover a broad range of electron fractions across the critical value Ye,crit ≈ 0.23–0.25. In these conditions, the production of all heavy elements between the first and the third r-process peaks is expected (see, e.g., [29, 76]). Due to the smaller effect of neutrino irradiation, the synthesis of elements between the second and the third r-process peaks is significant, while the production of the first peak elements is below the solar ratio. The angular distribution of the abundances is in this case rather insensitive to the latitude: high-Ye matter at polar latitude possibly synthesizes only elements below the second r-process peak, but this ν-driven component is very subdominant for BH-torus systems, due to the lower neutrino luminosities. Nucleosynthesis results for the viscous ejecta from BH-torus systems are presented in the left panel of Fig. 10.

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Disk wind ejecta from systems hosting a massive NS. If the massive NS resulting from the merger of two NSs is not a short-lived one, a larger variability is expected, reflecting the presence of several components in the disk winds and the impact of neutrino irradiation [37]. If the remnant survives for a timescale comparable to the disk lifetime (tacc ; see Eq. (3)), neutrino irradiation progressively shift the Ye distribution of the viscous ejecta toward larger values  Ye,crit , compared to the BH-torus system. While the production of both light and heavy r-process elements is foreseen, the relative importance of the former with respect to the latter increases as a function of the remnant lifetime. The presence of a very longlived massive NS (collapsing on a timescale  tacc ) could possibly prevent the formation of the second peak and of all the elements above it in the bulk of the ejecta, producing a significant amount of light r-process elements. The presence of a massive NS causes also disk wind ejecta emerging from distinct portions of the solid angle to be characterized by different nucleosynthesis patterns. As in the case of the dynamical ejecta, matter expelled at high (polar) latitudes is more efficiently irradiated by neutrinos, and for it, the production of heavy r-process elements at and beyond the second r-process peak becomes soon inefficient, while elements characterized by 75  A  120 and 33  Z  55 are eventually produced in solar proportions even for ejecta expelled within the first few tens of milliseconds [40, 47]. Nucleosynthesis yields obtained for the neutrino- and the viscosity-driven ejecta from merger remnant hosting a massive NS in the center are presented in the right panel of Fig. 10. High-entropy, high-velocity tails. In the case of shock-heated ejecta and ejecta expelled from strong magnetic fields in torii around BHs, a high-entropy tail is observed in the ejecta distribution. In the first case, the ejecta also expand with a high velocity, such that τ  2 ms. In this case, assuming Ye  0.4 (but often Ye < 0.3), the r-process nucleosynthesis proceeds in α-rich freeze-out conditions, and it synthesizes a significant fraction of nuclei up to lanthanides (Ye  0.4) or even actinides (Ye  0.2). Different from the low-entropy, neutron-rich ejecta, these matter present also a significant amount of H and He in the final composition.

Observables of Compact Binary Merger Nucleosynthesis Binary compact mergers have long been thought as promising astrophysical sites for r-process nucleosynthesis. While the expected conditions of the ejecta and the outcome of detailed numerical models indicate that the production of heavy r-process elements above the iron group is robust, observational evidences are crucial to validate our models and to discriminate between the many theoretical uncertainties that still affect our theoretical understanding. In the following, we will discuss two major observables and their relation with the outcome of r-process nucleosynthesis in compact binary mergers: the electromagnetic transient called kilonova and the evolution of the chemical abundances of r-process elements in the stars of our galaxy and in its satellites.

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Electromagnetic Signatures of r-Process Nucleosynthesis in Compact Binary Mergers What Is a Kilonova? Starting from a few seconds and continuing for several hundreds of days after merger, a large amount of nuclear energy is released by the combination of β decays, α decays, and fission processes that follow the r-process nucleosynthesis in the ejecta of a compact binary merger. This energy heats up the expanding matter and produces a nuclear-powered transient called kilonova [36]. Depending on the ejecta properties, a kilonova is expected to have its peak luminosity between a few hours and several days after merger in the UV/optical/near-IR frequencies, with a fast declining luminosity. For a detailed treatment of this transient, we refer to the dedicated chapter and to recent reviews, e.g., [11, 14, 45]. The nuclear energy associated with each decay is distributed among the daughter nuclei and the other particles in the final state (electrons, photons, and neutrinos), in a way that primarily depends on the nature of the decay. Typical Q-values are of the order of 1–100 MeV, much larger than the decreasing matter temperature in the expanding ejecta. Indeed, if the temperature at the end of the r-process nucleosynthesis (tr−proc ∼ 1 s) is kB Tr−proc ∼ 0.1 MeV, then:  kB T (t) ∼ kB Tr−proc

t tr−proc

−1

 ≈ 1.16 eV

kB Tr−proc 0.1 MeV



tr−proc 1s



t 1d

−1 . (38)

While the energy emitted in neutrinos is always lost, several processes can thermalize at least a fraction of the released nuclear energy, making it available for the kilonova. The intensity of the nuclear heat and the efficiency of the thermalization processes are maximal immediately after the r-process nucleosynthesis. However, matter is initially very opaque to photons, and the timescale for photon diffusion is much larger than the dynamical timescale over which the ejecta expand. The photon optical depth depends on the matter density profile and on the photon opacity. It is thus necessary to wait for the density to drop such that thermal photons can efficiently diffuse and be emitted at the photosphere.

r-Process Nucleosynthesis and Kilonovae If this is the mechanism behind kilonovae, r-process nucleosynthesis can influence these transients mainly through three aspects: the nuclear heating, the thermalization efficiency, and the photon opacity. In the following, we will analyze the main features of each of them. Nuclear heating. On the kilonova timescale, nuclear abundances in the ejecta change in time according to a set of decays and fission reactions. Each reaction is characterized by an exponential behavior, Ni (t) = Ni,0 exp(−λi t), where Ni (t) is the number of parent nuclei at time t and λi the reaction rate. The specific nuclear heating rate as a function of time can be computed as:

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 e˙nucl (t) =

i

Qi λi Ni (t) ≈ e˙e− (t) + e˙ν (t) + e˙α (t) + e˙γ (t) + e˙fission (t) , mej

(39)

where Qi is the Q-value of each reaction i and mej is the ejecta mass. In the last step, we have explicitly indicated the decay particles whose kinetic energy provides the available nuclear energy, neglecting the kinetic energy of the daughter nuclei in the case of α and β decays. Detailed network calculations (e.g., [32, 38, 44]) show that the dominant contribution to e˙nucl can be approximately described by a power law dependence in time, with possible corrections in the form of exponential terms: e˙nucl (t) ≈ e˙nucl,0 (t/t0 )

−α

+

N

βi exp(−γi t) .

(40)

i=1

with t0 being a reference timescale. The precise values of e˙nucl,0 , α, βi , and γi depend on the initial ejecta properties, i.e., on (Ye , s, τ ) of the specific trajectory, and on the nuclear input physics. For low-entropy ejecta, Ye is the dominant parameter. For Ye  0.25, α ≈ 1.3 with a possible variation interval 1.1–1.4, while different mass models give e˙nucl,0 ∼ 1016 –1017 erg s−1 g−1 (assuming t0 = 1 s) [57]. This is mainly due to the fact that, especially for very low Ye , translead nuclei tend to decay through α decays and fission, which are very sensitive to the specific mass model [6, 57, 75]. The dominant power law term in Eq. (40) can be understood by considering that in most of the cases (and especially for Ye  0.2, when strong r-process nucleosynthesis occurs), a large statistical ensemble of nuclei is produced [44]. Assuming β − decay to be the dominant decay channel, along an isotopic chain (i.e., for fixed Z), the Q-value is roughly proportional to the neutron excess D. Since λβ ∝ Q5 , then λβ ∝ D 5 . The nuclei at the end of the r-process distribute along the r-process path, characterized by Sn ≈ Sn0 ; see Eq. (29). According to the Sn distribution on the nuclear chart, at high N, a larger D is required to fulfill the r-process path condition, and, for a given time t, the nuclear distribution is such that the number of nuclei per interval of D is approximately constant within the relevant interval of neutron excess: dN dN dN dD ≈ const → = = Cλ−4/5 , dD dλ dD dλ

(41)

with C being a constant. Due to the large number of reactions and nuclei involved, Eq. (39) can be converted in an integral over λ: 

+∞

e˙nucl ∝

λ1/5 λ exp (−λt)λ−4/5 dλ ∝ t −7/5 .

(42)

0

While this argument explains well the presence of the power law and provides a good estimate of its slope, many details can affect the precise calculation, accounting for the lower exponent obtained in detailed calculations.

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The power law term in Eq. (40) is very robust for ejecta characterized by Ye  0.25 and undergoing strong r-process nucleosynthesis. For Ye  0.25, the decay of a few nuclei dominates the heating rate, and deviations from the power law behavior are accounted by the exponential terms in Eq. (40). These nuclei (including neutronrich isotopes of Kr, Rb, Br, Sr, Ce, As, and Ge; see, e.g., [40]) are usually below the second r-process peak and are characterized by large Q-values and half-lives of a few hours. Thus the heating rate is enhanced by a factor of a few within the first day and decreased by a factor ∼2 at later time with respect to the Ye  0.25 case. However, since the exponential corrections depend significantly on the ejecta distribution in the (Ye , s, τ ) space and the latter is usually rather broad in Ye , a power law behavior is partially recovered, and e˙nucl varies only by a factor of a few for relevant ejecta conditions during the first week after merger (e.g., [38]). At late time (between 10 and several hundreds of days), independent from the initial Ye , the nuclear distribution is very close to the valley of stability, and only very few nuclei have the right lifetime to decay within this time window. Thus, the heating rate becomes possibly very sensitive to the details of the abundance distribution through emerging bumps in its time evolution. Due to the temporal evolution of the thermalization efficiency (see below) and depending on the detailed yields, heavy nuclei with β decay half-lives around 14 days can be relevant for the light curve behavior at a few weeks, while a few actinides and transuranium elements with α-decay half-lives of several tens of days can affect the kilonova emission up to hundreds of day after the merger [75, 77]. Thermalization efficiency. Thermalization efficiency is defined as the ratio of the total energy released by all radioactive processes, e˙nucl , to the energy effectively transferred to the ejecta, e˙th , i.e., fth ≡ e˙th /e˙nucl . Since both these quantities changes with time, also fth is a function of time. Detailed studies on fth can be found, e.g., in Refs. [6, 23]. First of all, the thermalization efficiency depends on how relevant each decay channel is and on how energetic the particles in the final state are. These two points ultimately relate with the physics of the decays and with the actual abundances in the ejecta. Beta decay is the dominant decay channel for nuclei A  200, and it is relevant for all Ye conditions, especially for Ye  0.25. The Q-values of the β decays relevant for the kilonova emission are of the order a 1–2 MeV, and most of this energy is emitted in the form of γ rays (∼45%, emitted from the excited daughter nucleus), while e− s take usually 20%, the rest being lost in neutrinos. The Q-values and the e− energies are larger for larger Ye . Alpha decay is the dominant decay channel for A  200. Thus, it affects significantly the heating rate if actinides are produced, i.e., if Ye  0.20. For these decays, typical Q-values are in the range 5–9 MeV. Most of this energy is taken by the α particle, while the daughter nucleus de-excitation is negligible. Fission is very effective for super-heavy nuclei (A  250) produced for Ye  0.10. The Q-value of these reactions can be approximated by the kinetic energy of the fission fragments, and the latter can be estimated as

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the Coulomb repulsion energy of the two fragments at their formation, Qfission ∼ 1/3 1/3 ECoul = Z1 Z2 e2 /(r0 (A1 + A2 )), where we have approximated the radius of a nucleus of mass number A as r0 A1/3 with r0 ≈ 1.8 fm. For typical super-heavy nuclei with (A, Z) ∼ (250, 100) and daughter nuclei with (A1,2 , Z1,2 ) ∼ (132, 50), Q is ∼ 100 MeV. Additionally, fth depends on how efficiently the different final state particles thermalize in the plasma. This in turn depends on the physical processes providing thermalization and on the density of the medium, since fth is larger in high-density ejecta (e.g., in more massive and slower ejecta). High-energy charged particles (e− s, αs, and fission fragments) lose their energy in the ambient plasma through Coulomb interactions with the free and the atomic e− s. In the latter case, they can excite or even ionize atoms, and these are the most efficient thermalization mechanisms. Each of these distant interactions transfers a relatively small amount of energy to other electrons. Then many interactions are required to thermalize a single projectile particle, but due to the low transferred energy, the target electrons in the final state thermalize very rapidly. Due to its Z1 Z2 dependence, Coulomb interactions with ions are relevant only for fission fragments, while strong nuclear interactions with other nuclei are negligible. The cumulative nature of the Coulomb processes that affect charged particles allows to thermalize efficiently at least a fraction of the available energy. Also γ -rays lose their energy in the plasma by interacting with electrons through photoionization and Compton scattering. Due to the relatively high ionization thresholds of heavy elements (∼100 keV), photoionization is the most relevant process up to ∼1 MeV, while for higher photon energies, the Compton scattering becomes dominant. Any thermalization process is efficient as long the timescale over which it acts is smaller than the ejecta expansion timescale. Due to the relatively low opacity provided by Compton scattering and photoionization, high-energy photons stop to thermalize when matter becomes transparent to them, and for typical kilonova conditions, this happens within the first day after merger (tineff,γ ∼ 1 day, where tineff is the timescale when thermalization becomes inefficient). For supra-thermal electrons and α particles, an efficient thermalization can occur up to several days after the merger (tineff ∼ 8 day), while fission fragments (again due to the ∝ Z1 Z2 dependence of the Coulomb interaction) thermalize efficiently up to a few weeks (tineff ∼ 16 day). Detailed calculations (e.g., [6]) show that fth (t) decreases from 0.5–0.6 during the first day down to 0.1 around 10 days after merger. Atomic opacity. The photon opacity, κγ , quantifies the degree of transparency of matter to electromagnetic radiation. In particular, it can be understood as the cross section (σ ) per unit mass of a fluid element to radiation: κγ = nσ/ρ, where ρ is the matter density and n the target particle density. In general, κγ depends on the energy of the incident photon and on its complex interactions with the electron structure of the atom, in all its possible ionization states. Thus information about the composition and the ionization degree of each species is crucial to take properly into account the most relevant atomic opacities. While the ejecta expand and cool,

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electrons recombine with atoms to form ions and neutral atoms. First ionization energies vary between 3 and 25 eV, while the innermost electrons of heavy elements have ionization energy of ∼100 keV. Given the expected temperature, Eq. (38), most of the electrons have recombined at the time of the kilonova emission, and the abundance of free electron is Ye,free ∼ a few 0.01, decreasing as a function of time due to electron recombination. Despite the fact that plasma collisions are not effective enough to maintain thermodynamics equilibrium during the ejecta expansion, due to the high opacity expected in the ejecta of compact binary mergers, radiation can effectively drive the ion abundances towards local thermodynamical equilibrium (LTE), at least during the first days after merger (e.g., [31] and references therein). For the photon energy interval relevant for kilonovae, bound-bound transitions are the most important atomic processes, followed by bound-free and free-free opacities, usually smaller by several orders of magnitudes. Detailed and exhaustive experimental values of κγ for heavy elements, for a broad range of photon energies and in relevant thermodynamics conditions, are mostly missing, and the recourse to sophisticated, but still uncertain, atomic physics calculations is necessary (e.g., [30, 64]). The energy-independent electron scattering opacity (Thomson’s opacity),     2 2n −3 cm2 g−1 Y κγ ,Th = 8π/3 α hc/m ¯ ec e,free /ρ ≈ 3.97 × 10 e,free /0.01 , where α = 1/137 is the electromagnetic coupling constant, sets a lower limit that becomes relevant only for photon wavelengths λγ  104 Å. Energy- and compositionaveraged values of the opacity (sometimes called gray opacity) can be used to roughly characterize the global plasma behavior. For example, for the Planck mean opacity, κγ (λγ ) is averaged over the Planck distribution function, while for the Rossland mean opacity, 1/κγ (λγ ) is averaged over the temperature derivative of the Planck distribution function. When the ejecta contain light r-process elements (A  140), bound-bound transitions involve d-shell valence electrons. For λγ  103 Å, the opacity strongly decreases with the photon wavelength (κγ (λγ = 103 Å) ∼ 103 cm2 g−1 , while κγ (γ = 104 Å) ∼ 10−3 cm2 g−1 ), and typical gray opacities are s 1 cm2 g−1 . If lanthanides and/or actinides are also present, the opening of the electron f -shell increases enormously the number of possible transitions, and the bound-bound opacity is characterized by a forest of lines, Doppler-broadened by the large expansion velocity. The spectral opacity still decreases with the photon wavelength, but much mildly (κγ (103 Å) ∼ 103 cm2 g−1 , but κγ (2.5 × 104 Å) ∼ 10−3 cm2 g−1 ), and typical gray opacities are in this case  10 cm2 g−1 . We stress that even a small amount of lanthanides or actinides (XLa s + XAc s  10−4 ) can already change the global matter opacity.

Modeling Kilonovae Kilonova modeling is an extremely challenge task. State-of-the-art models require the solution of the photon radiative transfer equation in the expanding ejecta, possibly in multiple dimensions, e.g., [31, 63, 74]. In addition to physically

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motivated density and temperature profiles, these models also require detailed information on the nuclear composition, on the ion abundances, and on the spectral opacities at every time and everywhere in the ejecta. These models are able to provide light curves and spectra from a given configuration, at different epochs after merger. If the measurement or computation of all relevant transition lines is one of the major theoretical uncertainties, also their treatment in radiative transfer codes and in particular their translation in an effective opacity (“effective” because it refers to a discretization procedure involving finite wavelength interval, where many single lines are present) is not obvious. In the following, we will derive fundamental scaling relations, based on a simplified analytical model, which are in qualitative agreement with more detailed models. In addition to provide the fundamental scales of the problem, they also highlight the impact of nuclear physics input and the variety implied by the different ejecta conditions expected in compact binary ejecta. To model the ejecta, we consider a spherically symmetric distribution of total mass mej and average speed vej , characterized by a gray opacity κγ and expanding homologously. Matter at the outer edge is moving at velocity vej,max , and at each time t, its radial position is Rmax = vej,max t. Any internal shell of mass δm is expanding at a speed vej < vej,max , constant in time and proportional to the radius, such that its radial position evolves according to R = vej t. We further define menv the mass of the envelope above δm and ΔR = Rmax − R its radial thickness. The expansion timescale of this envelope can be computed as texp ∼ ΔR/vej . Thermal photons produced inside the envelope will contribute to the kilonova if texp ∼ tdiff , where tdiff is the photon diffusion timescale. The latter can be determined from random-walkarguments starting from  the photon optical depth (τγ ) and mean free path (γ = 1/ κγ ρ ) as tdiff ∼ τγ2 γ /c. The optical depth is defined as the integral of the photon inverse mean path, −1 γ , along an outgoing radial path. In words, τγ counts the average number of interactions that a photon experiences before being emitted at the photosphere, and it can be approximated by τγ ∼ ρ κΔr, where ρ is the average density experienced by the outgoing photon, ρ ∼ menv /(4π R 2 ΔR). By equating the expansion and the diffusion timescale, we can determine the time t˜(menv ) at which the photons emitted by menv will contribute to the kilonova, and we can estimate the peak time of the kilonova emission tpeak by taking menv ∼ mej and vej ∼ vej :  tpeak ∼

1/2  1/2    mej vej −1/2 mej κγ κγ ≈ 4.6 days . 4π vej c 0.01 M 0.1 c 10 cm2 g−1 (43)

The energy available to power the kilonova at t˜ is the nuclear energy released by mej and thermalized by the plasma, Lγ (t˜) ≈ e˙0 (t˜/1 sec)−α fth (t˜) mrad (t˜), with α = 1.3 as a typical value. Once again, we can estimate the peak luminosity by taking t˜ = tpeak and menv ∼ mej to obtain:

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−α/2 

597

mej 0.01 M   α/2   vej e˙0 fth . 0.1 c 0.5 5 × 1016 erg s−1 g−1

Lpeak ∼ 2.4 × 1040 erg s−1

κγ 10 cm2 g−1

1−α/2

(44)

The radius of the photosphere at t˜ is Rph (t˜) ≈ vej t˜ and then at the luminosity peak:  Rph,peak ∼ 1.26 × 1015 cm

κγ 10 cm2 g−1

1/2 

mej 0.01 M

1/2 

vej 0.1 c

1/2 . (45)

Finally, assuming black body emission and using the Stefan-Boltzmann law, the effective photospheric temperature, Tγ ,eff , can be determined as Tγ4 =   2 σ ) . This expression can be evaluated at the peak: Lγ /(4π Rph SB 



vej 0.1 c

−α/8 mej 0.01 M 1/4   e˙0 fth 1/4 , 0.5 5 × 1016 erg s−1 g−1

κγ 10 cm2 g−1 (α−2)/8 

Tγ ,peak ∼ 2.15 × 103 K

−(α+2)/8 

(46)

and translated in a peak wavelength λγ ,peak = 1.35 × 103 nm (Tγ ,peak /2.15 × 103 K)−1 . The above estimates have been done assuming κγ ∼ 10 cm2 g−1 , i.e., considering ejecta that contain a significant fraction of lanthanides and actinides (meaning that the ejecta contains some matter with initial Ye  0.25). In this case, the luminosity peak is expected to occur around 1 week and at near-IR wavelengths. The luminosity is more than 100 times larger than the one of a typical nova, but several orders of magnitudes lower than a supernova (SN). Since SNe are also powered by radioactive decays in expelled matter, the mean reason for such a 1−α/2 large difference is in the amount of ejecta (Lγ ∝ mej ), much lower in the case of compact binary mergers. If the wind ejecta or the dynamical ejecta have been significantly irradiated by neutrinos, the initial Ye could have increased such that the production of lanthanides is prevented. In that case, κ  1 cm2 g−1 , and the luminosity peak is expected to happen earlier (around 1 day), with a higher luminosity (more than 103 times the one of a nova) and at bluer peak frequency (λγ ∼ 500 nm).

GW170817 and Its Kilonova On August 17, 2017, and in the subsequent weeks, the first unambiguous kilonova resulting from a compact binary merger was detected; see, e.g., [3]. This kilonova

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(called AT2017gfo) followed GW170817, the first GW signal compatible with the late inspiral of two NSs [2]. The total mass of the system was 2.74 M , while the mass ratio, assuming slowly spinning NSs, was measured to be between 0.7 and 1.0. Light curves in different photometric bands of this unprecedented UV/visible/IR emission showed an early peak (around 1 day after merger) in the visible frequencies, followed by a later peak (around 5–7 days after merger) in the near-IR; see [68] and references therein. The bolometric luminosity of the event 1.5 days after merger was ∼3.2 × 1041 erg s−1 , while it decayed approximately following a t −1.3 power law during the first week so that around 7 days Lγ ≈ 6 × 1040 erg s−1 . The spectrum at 1.5 days was very close to a black-body of 5 × 103 K, while around 7 days, it was broadly compatible with a 2.2 × 103 K black-body spectrum with a forest of absorption features, e.g., [49, 60, 65]. The identification of elements in the spectrum is very challenging due to the high density of lines and to their broadening due to the high expansion velocities. An analysis of the spectra recorded during the first days has revealed features compatible with the presence of Sr, an element of the first r-process peak [72]. Theoretical modeling of AT2017gfo requires the presence of more than one component of the ejecta. The different components are characterized by different masses, velocities, and opacities and possibly a non-trivial dependence from the geometry of the ejection. This is indeed necessary to explain the observed blue and red peaks [48, 68]. For example, one can consider the presence of two distinct components to explain the color evolution of AT2917gfo. Using Eqs. (43), (44), (45), and (46), assuming a α ∼ 1.3 and a thermalization efficiency of 0.8 and 0.4 for the peaks at 1.5 and 7 days, respectively, the above peak properties imply that the blue peak was characterized by mblue ≈ 0.019 M , κγ ,blue ≈ 1 cm2 g−1 , and vej,blue ≈ 0.20c, while for the red peak, mred ≈ 0.058 M , κγ ,red ≈ 4.2 cm2 g−1 , and vej,red ≈ 0.09c. The estimated total amount of ejecta is thus of the order of several percents of a solar mass. Results from one of these models are presented in Fig. 11. According to more detailed models, e.g., [31, 63, 74], the amount of ejecta in this event was ∼0.02–0.05 M . Nuclear physics input (e.g., the nuclear mass model) can introduce an additional uncertainty factor of a few, possibly reducing the total ejecta mass. The inferred opacities suggest a negligible amount of lanthanides in the blue component and a lanthanide mass fraction between 10−3 and 10−2 in the red one. The emerging picture is certainly very compatible with results of compact binary merger simulations and nucleosynthesis calculations. In particular, the decline rate of the bolometric light curve and the inferred opacities are compatible with what expected from the collective decay of freshly synthetized r-process elements (see, e.g., Fig. 11).

Compact Binary Mergers and the Chemical Evolution Due to the production and ejection of heavy nuclei, compact binary mergers are a possible key player in shaping the evolution of chemical abundances in the universe. A first relevant question is whether they are primary astrophysical sites

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Fig. 11 Modeling of the bolometric light curve of AT2017gfo, the kilonova associated with the GW170817 BNS merger event. The slope of the light curved, fitted with four different models characterized by different ejecta mass and opacity, is consistent with a power law decay, t −β such that β ≈ 1.0–2.0 (α in the text), compatible with the decay of r-process material in the ejecta of compact binary mergers. (Figures taken from [60])

for the production of r-process elements. Two main ingredients are necessary to answer this question: detailed abundance predictions and reliable merger rates. The former are obtained by combining the amount of ejecta predicted by merger simulations with the distributions of the nucleosynthesis yields corresponding to the properties of the ejecta. The latter are related to the binary formation channels and evolution. In particular, they depend on the probability that a compact binary forms and on the time that it takes for it to reach coalescence due to GW emission. To provide a simple answer, we compare the amount of r-process elements in the Milky Way (MW, Mr,MW ) to the whole amount of r-process material ejected by compact binary mergers during the galactic evolution (Mr,CBM ). Their ratio, fCBM ≡ Mr,CBM /Mr,MW , quantifies the relevance of compact binary mergers in accounting for the observed r-process elements. The enumerator can be evaluated as the product of the average merger rate in the galaxy, RCBM , times the average mass of r-process elements expelled by a single event, mr,CBM , times the age of the galaxy, tMW ≈ 13.5 Gyrs. The mass per event can be estimated from the lower bound obtained from GW170817 ejecta, i.e., mr,CBM ∼ 0.02M . The merger rate is still very uncertain, but it can be measured in different ways. From the GW events detected so far in the first two observing runs, the LIGO-Virgo collaborations have provided a rate of 110–3840 Gpc−3 yrs−1 for BNS mergers and a robust upper bound of 660 Gpc−3 yrs−1 for BHNS mergers [5]. Assuming that a significant fraction of mergers produce also a short gamma-ray burst (SGRB), the merger rate can be inferred also from the SGRB rate. The latter is bracket by RSGRB = 0.6–6 Gpc−3 yrs−1 [18, 71], and since GRBs are collimated emissions with a beaming correction factor fbeam ∼ 100 (fbeam = 1 − cos θjet and typical opening angles are θjet ≈ 5 − 10◦ , [15]), the resulting lower bound on RCBM rate is ∼80–600 Gpc−3 yrs−1 . More theoretical bounds can be obtained from population

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synthesis studies and from extrapolations of the observed populations of galactic BNS system (see, e.g., [1] and references therein). These rates broadly agree with the measured rates, even if with larger uncertainties. Assuming a galaxy density of ∼0.01 galaxy Mpc−3 , a conservative merger rate inferred from theoretical models and observations is RCBM ≈ 30 Myr−1 galaxy−1 . For the total mass of r-process elements in the MW, we consider the Sun as representative of the averaged enrichment of the galaxy, a common assumption in galactic chemical evolution models. We also assume that all this mass is produced by the one single type of events and with a relative constant yield and rate for the entire evolution of our galaxy. These hypothesis are also grounded in the observations of the abundances of r-process elements in metal-poor stars. Indeed the spectroscopic analysis of the light emitted by a star reveals the chemical composition of the ISM from which the star has formed. Low-mass stars can live longer than the galaxy; thus, some of them are among the oldest objects in the MW. These old stars have an extremely low metallicity since they formed very early in the galactic history when only a few SNe had exploded and polluted the ISM. They can be considered fossils of the early chemical enrichment of our galaxy, and the chemical abundances measured in their spectra can be studied to infer the characteristics of the first stellar nucleosynthesis events. Noteworthy, in the spectra of these old stars, the lines of neutron capture elements are identified and their abundances measured. Among these fossil stars, there is a group that appears particularly rich in r-process elements (so-called r-process-rich stars; see, e.g., [61]). For this group, the derived abundance pattern reproduces very closely the solar residual r-process pattern, at least beyond the second peak, while the first peak presents a significant dispersion. Since the low metallicity implies that these stars have been polluted by very few (if not one single) r-process sources, the common patterns observed in the solar r-process residual, in this group of metal-poor stars, and in compact binary merger nucleosynthesis calculations reveal the presence of robust features that characterize the r-process nucleosynthesis in very general terms. Nevertheless, the emerging picture is more complicated than that: other metal-poor stars present a substantial amount of first peak elements and a lower enrichment in the heavy ones [22]. This is still compatible with the large variability expected in the yields of compact mergers (see Fig. 12), but it leaves also space for other production sites. To account for these uncertainties, starting from the solar r-process abundances, we compute the cumulative mass fraction of the r-process materials above a certain mass number Amin , X> (A > Amin ); see Fig. 13. To estimate the typical total mass fraction of r-process elements in the MW, we assume Xr−proc ≈ X> (A > Amin ), and we consider Amin = 68, 89, 124 such that Xr−proc (A > Amin ) ≈ 40, 9, 6 × 10−8 , respectively. Since the mass in stars and ISM of the MW is MMW ≈ 6 × 1010 M , fCMB can be finally estimated as:  fCBM ≈ 1.35

RCBM 30 Myr−1



mr,CBM 0.02M



tMW 13.5 Gyr



Xr−proc 10−7

and fCBM ≈ 0.34, 1.50, 2.25 for Amin = 68, 89, 124, respectively.

 ,

(47)

Fig. 12 Elemental abundance comparison between BNS merger models and metal-poor star observations. The violet curve represents nucleosynthesis yields for tidally dominated dynamical ejecta, while the other curves yield from neutrino-driven wind ejecta for different massive NS lifetimes. Theoretical abundances are compared with two classes of metal-poor stars. In the right panel, dynamical ejecta have been diluted by a factor of 50 with respect to the neutrino-driven wind one. (Figures taken from [40])

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10−6

X> (Amin )

X> (Zmin )

10−10

100 150 200 minimum mass number Amin

Actinides

III peak

10−9

II peak

10−8

Lanthanides

10−7

I peak

Solar r-process residual cumulated fraction

602

40 60 80 minimum atomic number Zmin

Fig. 13 Cumulative mass fraction obtained by the solar residual r-process abundances from [51] as a function of both the mass numbers (left panel) and the atomic numbers (right panel)

These estimates are rather crude in many ways. For example, they assume that the present-day merger rate is representative of the average merger rate, while we know that it has significantly changed during the galaxy history. Chemical evolution models can be used to better consider the merger rate and set more stringent constrain on the impact of the enrichment of compact mergers to the total balance of r-process in the galaxy; see [11]. Moreover, it is not obvious that the reference amount of ejecta is representative for the merger population. However, it is worth noticing that an IR excess, observed in the afterglow light curves of a few SGRB, is broadly compatible with the emission expected from a kilonova and it usually points to the presence of large ejecta masses (10−2 –10−1 M ), e.g., [66]. In addition, assuming that a significant fraction of the accretion disk is expelled in disk winds, our reference value is also well compatible with the results reported in Fig. 3. Some astrophysical processes are also neglected for simplicity. For example, when compact objects emerge from CCSNe, they receive a kick at birth. This kick, in addition to being a threat for the survival of the binary, can put the binary in a wide orbit inside the galactic potential, such that the merger could happen with a significant offset from the stellar and gas distributions inside the galaxy (see [24] for a more detailed discussion). Despite the large uncertainties, fCBM ∼ 1, and this testifies that compact binary mergers are primary astrophysical site where r-process elements are produced. This is especially true for elements above the second r-process peak, while the explanation of the first peak could be a clear confirmation that different mergers can produce different yields or could require additional production sites. A related question is whether other astrophysical sites are able to synthesize heavy elements through the r-process. Historically, proto-neutron star winds emerging after a successful CCSN explosion have long been thought to be a possible

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r-process nucleosynthesis site [52]. The argument applied above to relate the rates and the ejecta from compact binary mergers to the observed amount of rprocess elements in the MW could be easily adapted to regular CCSNe, and it is not able to distinguish alone between these two scenarios. However, CCSNe are significantly more frequent than compact binary mergers, and their rate is well constrained, RCCSN ≈ 2.8 × 104 Myr−1 (this rate can be related to RCBM by considering that approximately half of the massive stars are in binaries, two CCSNe are required to form a compact binary, and only ∼1% of the stellar binaries survives the two CCSN explosions). From Eq. (47), it is evident that CCSNe could explain the bulk of the galactic r-process nucleosynthesis if every SN would eject ∼10−5 M of r-process elements. Thus, the competition here is between rare events that expel large amounts of r-process material (e.g., compact binary mergers) or frequent events with much smaller amounts (e.g., regular CCSN). Over the past years, several observational evidences (in addition to GW170817) have accumulated pointing to the fact that the r-process elements come from rare events that produce a significantly large amount of r-process elements. In the following, we will briefly review them: Eu abundance in galactic metal-poor star. The analysis of the abundance of r-process elements, and especially of Eu, as a function of metallicity (and in particular of iron and of α elements, which are good tracers of the early evolution of the ISM composition due to the explosions of the very first massive stars) has revealed that the average abundance of Eu correlates with the one of α elements produced in CCSNe through the entire metallicity evolution (it correlates also with the one of iron, but only before type Ia SN start to explode, i.e., for low metallicity). However, different from the α elements case, the distribution of the single observations changes considerably: while at present metallicity the cumulative effect of several nucleosynthesis episodes and the efficient mixing of gas inside the galaxy has homogenized the ISM composition and reduced the observed spread, at early times, when the amount of iron was 102 –103 times smaller than the present one, the ratio of the Eu over iron abundances shows a two orders of magnitude scatter, ranging from metal-poor stars where Eu is underrepresented to cases where this ratio is more than ten times the one observed in the present solar system (see the left panel of Fig. 14 and [61]). This large scatter suggests that the possible r-process elements pollution comes from a single rare event that ejects large amount of r-process material in ISM clouds that have seen only a few SNe. Considering that ISM mixing over the entire galaxy requires a timescale much larger than the star formation timescale, this naturally introduces the inhomogeneous character to the early chemical evolution necessary to explain the observations. Sophisticated galactic chemical evolution models have been specifically developed to mimic this observational spread (see [11] and references therein). It shall be underlined that this is a peculiar characteristic of the r-process elements, not present in other chemical elements, such as α−elements produced together with iron by the more frequent SNe.

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Galactic rate [Myr−1 ]

103

Rp

roc

ess

CBM simulations

ma

ss

102

in

MW

GW BNS G Galacti c BNS S

101

sGRB Dwarf

100

Amin = 90

10−3

244

Pu

10−2

Mass per event [M ]

10−1

Fig. 14 Left: Scatter plot of the relative abundance ratio of iron over hydrogen versus europium over iron, for a large sample of metal-poor stars. The quantity [A/B] is relative with respect to the solar one, i.e., it is defined as [A/B] ≡ log10 (YA /YB ) − log10 (YA /YB ) . (Data have been taken from [56].) Right: Summary of the observational evidences for the production of r-process elements expressed on the r-process mass (mr ) versus galactic event rate (RMW ) plane. For concreteness, r-process abundances are assumed to be equal to the solar residual for Amin = 90, while the BNS rate is inferred from galactic BNS observations [50]. (Courtesy of K. Hotokezaka; see also [24])

r-process abundance in ultra-faint dwarf galaxies. In addition to galactic observation, Eu can be observed also in classical dwarf and ultra-faint dwarf (UFD) galaxies and satellites of the MW and formed by very old stars (interestingly, these observations are possibly related with the ones of galactic metal-poor stars, because the latter are often located in the so-called galactic halo, that is thought to be formed by accretion episodes of dwarf galaxies on the MW). In particular, in the case of UFD galaxies, while Fe is observed in all cases, Eu has been detected so far only in a couple of cases (Reticulum II and Tucana III), corresponding to ∼30% of the available sample [20, 28]. Since UFD galaxies are formed by 103 –105 stars, such a low detection rate points again to a single, rare event (no more than one per 104 –105 stars). Moreover, it is interesting to compare the ratio between the amount of iron and of europium observed in Reticulum II (MFe,Ret II ≈ 0.7 M and MEu,Ret II ≈ 10−5 M ), with the theoretical value predicted by assuming that Fe is produced by CCSNe (typically, mFe,CCSN ≈ 0.07 M ) and Eu by merger events (for which we conservatively assume again mr,CBM ∼ 0.02 M ) with an abundance equal to the solar one (XEu ≈ 4.1 × 10−10 ). Since (MFe /MEu )theory ∼ mFe,CCSN RCCSN /mEu,CBM RCBM and mEu,CBM ∼ mr,CBM XEu /Xr−proc , we finally obtain: (MFe /MEu )theory (MFe /MEu )Ret II

   (MFe /MEu )Ret II −1 RCCSN /RCBM ≈ 5.69 7 × 104 4.7 × 102 −1     X> XEu mFe,CCSN /mr,CBM . (48) 7/2 4.1 × 10−10 10−7 

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Thus, despite the crude approximations, the large uncertainties, and the possible peculiar character of Reticulum II, the relative amount of Fe over Eu is broadly compatible with what expected from compact binary enrichment. Radioactive elements in Earth sediments and in meteorites. The analysis of the abundances of long-lived r-process isotopes (t1/2 > 107 yr) and of their daughter nuclei in meteorites (formed when the solar system was formed) and in sediments (e.g., deep sea floor) reveals information on the local isotopes production at specific times (see [24] and references therein). Due to the paucity of the r-process nuclides 129 I and 247 Cm, whose lifetimes are 20 Myr, and to the relative large abundance ratio (0.008) between 244 Pt (t1/2 ∼ 80 Myr) and its stable daughter nucleus 238 U in meteorites, it is possible to set the delay time between the solar system formation and its last r-process pollution to ∼100–120 Myr. In addition, the analysis of the deposition rates in deep sea sediments of radioactive nuclides like 60 Fe (very shortlived nuclide usually produced in SNe) and the r-only 244 Pt over the past 25 Myr shows a large fluctuation, with the present rate ∼10−2 times smaller than the rate 25 Myr ago. This indicates that the enrichment of r-process elements is uncorrelated with the CCSNe enrichment and the former is due to rarer events that produce large yields [25]. A summary of the above observational constraints is provided in the right panel of Fig. 14. All of them work against regular CCSNe and in favor of compact binary merger as major site for the r-process production. A final interesting question is whether compact binary mergers are the only sources of r-process elements. The large abundances of the first peak elements in the solar r-process pattern and in some metal-poor stars, with respect to the elements beyond it, could be an indication of the occurrence of mergers in which the production of heavy r-process elements is disfavored (e.g., due to a strong neutrino irradiation). However, it leaves also space for other sources able to provide weak r-process nucleosynthesis (e.g., electron capture SN, [69]). Moreover, it is non-trivial for compact binary mergers to explain the abundances at very low metallicity. Indeed, a compact binary merger requires the successful explosion of at least one CCSN to form it (two for a BNS system), and, in addition to the stellar evolution timescale, it is necessary to wait for the GW-driven inspiral timescale. The latter can be estimated from the semi-major axis, a; the eccentricity, e; and the total and reduced mass, M and μ, of the binary at formation as: tGW ∼ 0.66 Gyr



a 4 0.01AU



M 2.7 M

−2 

μ 0.68M

−1  7/2 1 − e2 .

(49)

This is a possibly rather long timescale to be reconciled with fast mergers happening at extremely low metallicity. However, the strong dependence of tGW on a and e does not exclude them, but it requires a population of tight, possibly eccentric, compact binaries. Possible alternative sites are represented by rare classes of SNe characterized, for example, by very intense magnetic fields and fast rotating cores

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(e.g., [73]). It is important to stress that the relation between time and metallicity is not obvious: yields expelled by stars, SNe, and compact binary mergers need to mix with the ISM before entering in the composition of the next stellar generations. Moreover, due to kicks and fast traveling ejecta, the places where SNs explode, compact binaries merge, and new stars form could be distinct. All these effects are amplified in low metallicity conditions, where only a few enrichment episodes have happened.

Summary and Outlook Compact binary mergers involving at least one neutron star represent ideal environments where the production of heavy elements through the r-process nucleosynthesis takes place. The prediction of the precise composition of the ejecta and a clear understanding of its origin depend both on the modeling of the astrophysical sites and on detailed nuclear physics knowledge. A lot of progress has been achieved in the past few years, and the predicted abundances are able to explain many independent observations. Compact binary mergers are also possibly very relevant in explaining the nuclear evolution of the universe, in terms of the abundances that we observe at different epochs and in different astrophysical environments. Still, many open questions remain. On the one hand, more realistic and sophisticated compact binary merger models are required to predict the properties and the amount of ejecta with better accuracy and properly taking into account all the relevant physics. On the other hand, a more robust knowledge of the properties of exotic neutron-rich nuclei is key to reduce present nuclear uncertainties. In this respect, existing and upcoming worldwide nuclear facilities (including FAIR, FRIB, HIAF, RAON, RIKEN, and SPIRAL) will finally produce some of the neutronrich nuclei relevant for the r-process and measure their properties. Multimessenger observations, as well as the study of the composition of matter in different astrophysical and terrestrial contexts, will also sharpen our understanding and test our models, helping reducing our ignorance and forcing us to look at the problem from many viewing angles. Although a conclusive answer about the presence and the role of other possible r-process nucleosynthesis sites still needs deeper investigations and clear evidences, it is nowadays certain that compact binary mergers are one of the major sources of r-process elements in the universe.

Cross-References  Binary Neutron Stars  Black Hole-Neutron Star Mergers  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black Hole-Neutron Star Binary Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of BHNS Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disk Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Post-merger Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing BHNS Mergers Through Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . Waveform Models and Their Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detectability and Detection Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current BHNS Merger Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R-Process and Kilonovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nucleosynthesis in BHNS Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radioactively Powered Transients: Kilonovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kilonova Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UV/Optical/IR Follow-Up of BHNS Merger Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . Short Gamma-Ray Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other EM Counterparts to BHNS Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radio Emission from Mildly Relativistic Outflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended X-Ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pre-merger Electromagnetic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Pushed by the rapid rise of gravitational wave astronomy, the study of compact binary mergers has made significant progress over the last 5 years. Multiple binary black hole and binary neutron star systems have been detected, and while we are still waiting for the first confirmed observation of a mixed black hole-neutron star (BHNS) merger, a number of candidate events have been announced over the last observing run of the LIGO-Virgo Collaboration. BHNS mergers have many of the same advantages as neutron star mergers, including their potential to help us constrain the properties of dense nuclear matter and their ability to power bright electromagnetic counterparts and produce heavy r-process elements. There are however important differences between BHNS and binary neutron star mergers that significantly impact what we can hope to accomplish from their observation. In this chapter, we will review the basic physics of BHNS mergers and the properties of the observable gravitational wave and electromagnetic signals that they power. We will also discuss the current status of theoretical and numerical models of these signals, their successes, limitations, and likely future areas of improvements. Finally, we will consider the role that these models play in the robust interpretation of BHNS observations and how current model limitations impact what we can learn from multi-messenger observations of BHNS binaries. Keywords

Black hole · Neutron star · Gravitational waves · Kilonovae · R-process nucleosynthesis · Gamma-ray bursts

Introduction Since 2015, multiple detections of gravitational wave (GWs) signals by the LIGO and Virgo Collaboration (LVC) have opened an entirely new way for us to observe the universe. This first observation of the merger of two black holes (GW150914) [1] was followed by nine more black hole mergers during the first and second observing runs of the LVC (O1,O2) [2] and by the first detection of merging neutron stars (GW170817) [3]. The first half of the third observing run of the LVC with advanced detectors (O3a) included many more events released as part of the GWTC-2 catalogue [4], while the second half (O3b) is still being analyzed. Current GW detectors are mainly sensitive to the late inspiral and merger of solar mass objects, with masses M ∼ (1 − 100)M : binary neutron star (BNS) mergers, binary black hole (BBH) mergers, and mixed black hole-neutron star (BHNS) mergers. Of those, BHNS binaries remain the most elusive. Three events from the GWTC-2 catalogue (O3a) could plausibly be BHNS mergers. GW190425 [5] is a system of total mass 3.4M that is most likely a BNS merger but could involve a low-mass black hole. GW190814 [6] is the merger of a 23M black hole with a 2.6M object that could be either the most massive neutron star or the least massive black hole observed to date. Finally, GW190426 is most likely a BHNS merger, but

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is a marginal event with a non-negligible probability of being terrestrial in nature [4]. None of these systems can unequivocally be associated with a BHNS merger. This is not a coincidence: we will see that confirming the BHNS nature of a GW source is a difficult task. Some O3b candidates also have a non-zero probability of being BHNS mergers, but are not yet confirmed as GW detections at the time of this writing. BHNS binaries, like their BNS counterparts, can be multi-messenger sources whose observation informs our understanding not only of the properties of compact objects and gravity but also of the equation of state of dense matter, neutrino physics, cosmology, the origin of many heavy atomic nuclei, the production of gamma-ray bursts, high-energy neutrinos and cosmic rays, and a host of other electromagnetic signals observable from X-rays to radio waves. The power of multi-messenger observations of compact binary mergers was clearly demonstrated by GW170817 and the associated EM signals. The wide range of open questions in physics and astronomy that even a few good observations of BHNS and BNS merger can help us answer makes these systems prime targets for modern astronomy. There is however an important distinction that needs to be made between BHNS and BNS binaries. Most BNS systems are expected to power at least some post-merger EM emission, and any BNS system that is not massive enough to immediately collapse to a black hole most likely powers bright EM signals [7, 8]. The same cannot be said of BHNS binaries. The main driver of post-merger EM emission is the ejection of matter, either during the merger or through disk winds and relativistic jets in the post-merger remnant. For BHNS binaries, many mergers result in the absorption of the entire neutron star by the black hole, with little to no mass ejection and thus no significant post-merger EM emission. On the other hand, when the neutron star is tidally disrupted by its black hole companion, BHNS mergers tend to eject more matter than BNS systems [9–12]. These differences have important consequences for follow-up strategies, for the properties of postmerger EM signals, and for the respective impact of BHNS and BNS mergers on astrophysical nucleosynthesis. In this chapter, we will focus first on what we already know about the event rate of BHNS mergers and the expected properties of the merging objects. We will then discuss the main physical processes at play in BHNS mergers and how the physical parameters of BHNS binaries impact the outcome of the merger. We will then discuss the associated GW and EM signals (with particular emphasis on GWs and radioactively powered kilonovae), what we can learn from them, and the status of current efforts to model these signals. Unless otherwise specified, we will work in geometrical units G = c = 1.

Black Hole-Neutron Star Binary Population Event Rates In the absence of any confirmed detection, the event rate of BHNS mergers and the properties of the merging objects remain very uncertain. As for BBH and BNS systems, the main source of uncertainty is not which massive stars in binary systems

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end their life as BHs or NSs, but rather how often these systems can form the very compact binaries that are required for GW emission to drive the system to merger fast enough for us to observe them today. For field binaries, this generally requires the system to go through a common envelope phase leading to a significant decrease of the orbital separation. Large velocity kicks in core-collapse supernovae can also disrupt the binary when the black hole or neutron star forms, preventing the formation of a sufficiently compact system. Unfortunately, the details of common envelope evolutions and magnitude of supernova kicks remain very uncertain. When other astrophysical uncertainties such as the mass function of stars in binaries, stellar metallicities, winds, mass transfer events, etc. are also taken into account, we are left with theoretical event rates predicted by population synthesis models that span multiple orders of magnitude [13–15]. Observations of kilonovae [16] and short gamma-ray bursts (SGRBs) [17–19] and the inferred abundances of r-process elements [20] can also provide some constraints on the rate of BHNS mergers – or at least upper bounds obtained under the conservative assumption that all observed events are due to BHNS mergers. Finally, the lack of GW observations provides an upper bound on the rate of BHNS mergers. At the end of O1, predictions for the event rate of BHNS mergers ranged between 0.02and3000 Gpc−3 yr−1 [21]. To put this into context, we note that an optimally oriented BHNS binary containing a 10M black hole is expected to be observable by the advanced LIGO detector at design sensitivity up to distances of ∼1 Gpc [22]. Once we account for orientation effects, an event rate of 1000 Gpc−3 yr−1 corresponds to ∼300 events per year at design sensitivity [22]. The O1 rates thus covered scenarios going from likely detections in O2 to no detections even by the end of the lifetime of current instruments. The upper bound from non-detection of GW was decreased to 610 Gpc−3 yr−1 after O2 [2], now a meaningful constraint in tension with the most optimistic astrophysically motivated event rates (i.e., incompatible with the assumption that all short GRBs are BHNS events [18] or that all of the r-process elements are produced in BHNS mergers [20]). Event rate calculations post-O3 will certainly be more difficult to make given the uncertain nature of many recent GW triggers: at this point, one could plausibly argue for anywhere between ∼0and4 BHNS events in O3a-O3b. This range will hopefully narrow with the release of more detailed information about O3b events and their statistical significance, yet we are unlikely to obtain a definite number of BHNS events in O3 given the uncertain nature of already published events (e.g., GW190426). Theoretically, BHNS mergers can also form from dynamical interactions in globular clusters (GCs). However, while there is a lively discussion regarding the relative contributions of field binaries and globular clusters for BNS and BBH systems, compact BHNS binaries are probably formed at a very low rate in GCs [23, 24]. Recent estimates also predict a very low rate of BHNS mergers in galactic nuclei, (0.05 − 0.6) Gpc−3 yr−1 [25]. The formation of compact binariesin dense environment can have an important impact on the properties of GW sources, for example, increasing the eccentricity of observed binaries. For BHNS binaries, it seems that this channel can only dominate the event rate if that event rate is itself negligible for existing GW detectors.

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Binary Parameters Before discussing in more detail the physics of BHNS mergers, it is useful to stop and consider the parameter space of possible BHNS binary mergers and what our current priors on these parameters are. We will see however that this section will serve more to highlight how much we do not know about the properties of BHNS binaries, than to provide us with a clear picture of what a BHNS system will look like. Let’s start with the properties of the black hole. Assuming that astrophysical black holes have negligible electric charges, the black hole is entirely defined by 2 , with χ¯ its mass MBH and angular momentum vector J¯BH = χ¯ BH MBH BH the dimensionless spin of the black hole. Population synthesis models predict that for field binaries, mass accretion from a common-envelope phase and/or mass transfer from the black hole’s companion has a relatively small impact on the mass and spin of the black hole [26], so that the mass and spin distributions of black holes in BHNS mergers are mostly set by the properties of black holes at the time of their formation. At this point, that does not necessarily tell us that much about the properties of black holes in BHNS mergers: the observation of solar mass black holes in the Milky Way predicts a mass distribution peaking at (7 − 10)M [27], in reasonable agreement with core-collapse simulation results [28]. Yet, higher mass black holes are common in observed GW events [2] and expected in low-metallicity environments [29]. The lower bound of the BH mass distribution is also uncertain. In particular, whether a “mass gap” exists between the most massive neutron stars and least massive black holes is a crucial question to understand the population of BHNS binaries and to determine the nature of GW sources like GW190814 [6]. As for the black hole’s spin, while some observations of galactic black holes favor near-extremal spins [30, 31], black holes observed through GWs seem to prefer low spins [2]. The main exception is the high-spin black hole inferred for the most marginal of the six GW candidates detected with the alternative pipeline of Venumadhav et al. [32]. The existence of at least a small number of moderately spinning black holes in O2 events was also confirmed through a reanalysis of the LVC results with a modified parametrization of the black holes’ masses and spins [33]. If we account for the fact that there is no guarantee that the population of black holes in merging BHNS systems matches either the galactic black hole population, the black hole population within the part of the universe observable through GWs, or the black hole population in BBH mergers, we see that there remain large uncertainties regarding the masses and spins of black holes in BHNS mergers. Moving on to neutron stars, the galactic BNS systems that are compact enough to merge within a Hubble time have a narrow mass distribution MNS ≈ (1.33 ± 0.05)M [34]. However, the wider population of galactic neutron stars has a broader range of masses, with indications of a bimodal distribution possibly separating recycled pulsars [MNS ≈ (1.81 ± 0.18)M ] from other neutron stars [MNS ≈ (1.39 ± 0.06)M ] [35]. Multiple pulsars with masses ∼2M have now also been confirmed [36, 37]. Predictions from recent core-collapse simulations are broadly

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consistent with the population of non-recycled neutron stars [28]. We are thus left with a narrower range of potential masses than for black holes, but no clear picture of the mass distribution of neutron stars within that range. It is also worth noting that the two observed neutron star mergers already involved neutron stars of very different masses: while GW170817 is consistent with expectations derived from galactic binaries [3], GW190425 is a significantly more massive system [5]. A more constraining astrophysical prior is generally assumed for neutron star spins. The fastest rotating neutron star observed so far is the 716 Hz radio pulsar PSR J1748-2446ad [38]. However, while millisecond-period pulsars are observed in binary systems, it is relatively unlikely that a neutron star would remain rapidly spinning by the time of merger (at least for field binaries). As a result, many studies take the reasonable prior that neutron stars observed through GWs have a low dimensionless spin (χNS  0.05) that minimally impacts the GW and EM signals. In most of this review, we will make the same assumption and limit ourselves to neutron stars with relatively low spins. GW and EM observations of neutron star mergers may also help us put meaningful constraints on the mass-radius relationship of neutron stars. As there is a one-to-one relationship between the equation of state of dense matter and the massradius relationship, measuring the size of neutron stars allows us to put constraints on the strength of nuclear interactions in dense neutron-rich matter and on the possible existence of phase transitions in the core of neutron stars. We note that the equation of state of neutron stars is a very different free parameter than the masses and spins of the compact objects: while we naturally expect a distribution of masses and spins for astrophysical objects, the equation of state is most likely unique. From a modeling point of view, however, the fact that we do not know the equation of state of dense matter and want to use observations to constrain that equation of state means that we need to construct a range of models covering physically acceptable equations of state and thus treat it as freely specifiable in models and simulations. Before GW observations and for MNS ∼ 1.4M , some estimates from quiescent X-ray binaries preferred relatively compact stars RNS ∼ (7.6−10.4) M [39], while combining these observations with nuclear physics constraints led to higher radius estimates RNS ∼ (10.4 − 12.9) km [40], as did, e.g., estimates from millisecond pulsars (RNS > 11.1 km) [41]. Nowadays, GW170817 provides its own radius constraints RNS ∼ (10.5 − 13.3) km [42], as does the NICER observatory (RNS ∼ (11.5−14.3) km) [43–45]. Slightly narrower predictions have been obtained by also including EM observations of GW170817 [8,46–48]. Their result shows a relatively broad agreement, preferring the range RNS ∼ (11 − 13) km. Further improvements to these constraints would have a significant impact on nuclear physics. Another way to study the properties of dense matter is to measure the maximum mass of isolated, non-rotating neutron stars. This has already proven to be an extremely useful observable in nuclear astrophysics: the detection of a 2M neutron star remains one of the main constraints on the equation of state of dense matter today. Joint GW and EM observations of merging compact objects can help tighten these constraints. For example, the observation of a bright EM counterpart to GW170917 has led to a range of new upper bounds for the maximum mass

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of neutron stars [49–51] that remain however more model-dependent than the observation of ∼2M pulsars. We note that all of these upper bounds predict that the 2.6M object in GW190814 should be a black hole, possibly an important hint regarding the nature of that event. Finally, while the masses and spins of the merging compact objects and the size of neutron stars are probably the most important parameters determining the outcome of BHNS mergers, there are at least two additional parameters that are worth discussing here: the eccentricity of the binary and the initial magnetic field configuration within and outside the neutron star. Eccentricity is expected to be small in field binaries, at least by the time the emitted GW signal becomes detectable by current gravitational wave observatories. For binaries formed in globular clusters or galactic nuclei, however, this is not necessarily the case. While here we generally assume circular orbits, eccentric BHNS mergers have unique observational signatures that are worth keeping in mind, including the potential for partial disruption of the neutron star and for an excitation of the fundamental mode of the neutron star leading to monochromatic GW emission [52,53]. As for magnetic fields, observed neutron stars have a wide range of inferred surface field strength, from B ∼ 108−14 G rotation-powered pulsars to B ∼ 1014−15 G magnetars [54]. At the low end of that range, the initial field strength most likely only impacts the production of (very dim) pre-merger electromagnetic signals, while at the very high end, it may contribute to the large-scale structure of the magnetic field post-mergers. In most cases, we expect the magnetic field to grow from instabilities in the postmerger remnant, largely erasing the effects of initial conditions on the strength and geometry of the post-merger field. Realistic initial magnetic fields are also expected to be irrelevant to the dynamics of BHNS binaries up to merger [55]. However, current simulations of BHNS mergers including magnetic fields often use magnetarlevel initial field strengths, which impacts the outcome of simulations. Overall, we thus note that there are large uncertainties on both the event rate and the most likely parameters of merging BHNS binaries. We have some astrophysical priors on the masses of black holes and neutron stars and information about the equation of state of dense matter; yet even so the parameter space to explore when studying BHNS mergers remains vast and poorly constrained.

Dynamics of BHNS Mergers Tidal Disruption From the formation of a compact BHNS binary to the emission of detectable gravitational waves, millions of years will be spent in a slow gravitational wavedriven inspiral. During that time, the binary is expected to be largely invisible to us. As the binary separation decreases, the orbital frequency of the system increases, and so does the amplitude of the GW signal, up to the point when, seconds to minutes before merger, the system enters the LIGO-Virgo frequency bands. We will discuss the GW signal and the information that can be extracted from it in

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more detail later in this chapter. For now, let us focus on the qualitative features of the merger itself, which will affect not only the GW signal but also most EM counterparts to BHNS mergers. To first order, the main question that we need to ask ourselves is whether the neutron star can be tidally disrupted by the black hole or if it is entirely absorbed by the black hole. The main parameters determining that outcome were first described by Lattimer & Schramm [56] and largely parallel the physics of the tidal disruption of main sequence stars by supermassive black holes. We can estimate a binary separation Rdisrupt where tidal forces overcome the self-gravity of the neutron star. We also know the radius of the innermost stable circular orbit (ISCO) of the black hole RISCO , below which no stable circular orbits exist and an orbiting test mass has to plunge into the black hole. Roughly speaking, if Rdisrupt > RISCO , the neutron star is tidally disrupted while orbiting the black hole. This leads to unstable mass transfer between the two objects and the formation of a partially unbound tidal tail (Fig. 1). The bound portions of the tidal tail then circularize and form an accretion torus around the black hole. If on the other hand Rdisrupt < RISCO , the neutron star plunges into the black hole without being disrupted. No matter is ejected; no accretion torus is formed. In Newtonian physics, we have  Rdisrupt ∼ k

Fig. 1 Disruption of a neutron star by a black hole for a mass ratio Q = 4 system, with χBH = 0.9. The disrupted neutron star forms a tidal tail, part of which is unbound. The bound material will later circularize into an accretion disk

MBH MNS

1/3 RNS

(1)

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with k ∼ 2 − 3 a parameter with a mild dependence in the equation of state of the neutron star and the spin of the black hole [57, 58]. On the other hand, for pointparticles orbiting in the equatorial plane of a spinning black hole, we have [59] RISCO = f (χBH )

GMBH c2

(2)

with  (3 − Z1 )(3 + Z1 + 2Z2 )   2 1/3 (1 + χBH )1/3 + (1 − χBH )1/3 Z1 = 1 + (1 − χBH )  2 + Z2, Z2 = 3χBH 1

f (χBH ) = 3 + Z2 ∓

(3) (4) (5)

with the minus sign for prograde orbits and the plus sign for retrograde orbits. The function f (χBH ) is 6 for non-spinning black holes, 1 for maximally spinning black holes and prograde orbits, and 9 for maximally spinning black holes and retrograde orbits. We can then estimate the ratio 1 RISCO ∼ Q2/3 CNS f (χBH ) Rdisrupt k

(6)

that only depends on the dimensionless numbers Q = MBH /MNS , CNS = MNS /RNS , χBH , and k. Clearly, disruption will be favored for low mass ratios Q, small neutron star compaction CNS (i.e., large neutron stars), and large prograde spins. From the range of binary parameters estimated in the previous section, we see that this ratio can easily be smaller or greater than one and thus that disruption and plunge are both possible outcomes of a BHNS merger. At the order of magnitude level, the disruption condition RISCO < Rdisrupt works reasonably well. Consider, for example, a non-spinning black hole and a neutron star with MNS = 1.4M , RNS = 12 km (CNS = 0.17): we would predict a critical mass ratio Qcrit ∼ (3 − 5) below which tidal disruption occurs. Our best current estimate is slightly more pessimistic, with Qcrit ∼ 2.5 (see below) [12]. For this order-of-magnitude estimate, we assumed that the orbit is circular and that the neutron star is non-spinning and orbits in the equatorial plane of the black hole. A similar argument can however be made for orbits that are not in the equatorial plane of the black hole, replacing the ISCO with the innermost stable spherical orbit (ISSO) [60]. Treating eccentric trajectories and spinning neutron stars is more complex. Qualitatively, we expect a high eccentricity and a rapidly spinning neutron star to facilitate disruption. Bound eccentric orbits can get closer to the black hole than the ISCO [59], while centrifugal forces work against self-gravity within the neutron star. Yet, this will only impact the tidal disruption of the neutron star if the binary has a significant residual eccentricity at the time of merger, which requires dynamical capture on a highly eccentric orbit, or if the rotation period of

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the neutron star is comparable to the orbital period at merger, i.e., if the neutron star is a millisecond pulsar. The frequency of the GW signal emitted when the neutron star is disrupted can also be estimated reasonably well from Newtonian physics and the fact that the GW frequency is twice the orbital frequency. Using the Newtonian formula for the orbital frequency and our previous estimate of Rdisrupt , we get

fGW,disrupt

1 ∼ π



MBH + MNS 3 Rdisrupt

  1 1 MNS ∼ 1+ 3 Q RNS π k 3/2

(7)

which for typical neutron stars is fdisrupt ∼ (0.5−1.5) kHz and only weakly depends on the mass of the black hole. This indicates that if a neutron star is tidally disrupted, the corresponding merger signal will be in the ∼1 kHz frequency range. If the ISCO frequency is lower than this, which indicates that there is no tidal disruption, we naturally expect less of an imprint of the finite size of the neutron star on the GW signal. As current GW observatories are most sensitive at a few hundreds Hz, they are more sensitive to the late inspiral of disrupting BHNS mergers than to the disruption signal itself. To go beyond these qualitative statements, one needs numerical simulations of BHNS mergers. These were first performed using Newtonian or pseudo-Newtonian potentials [61–65] and then with codes evolving both Einstein’s equations and the relativistic fluid equations [66–68]. Over the years, general relativistic codes have improved to include effects of the black hole spin magnitude [9,69] and orientation [70, 71], more realistic binary mass ratios [72, 73], nuclear theory-based equations of state [74–76], magnetic fields [55, 77–80], neutrino cooling [81] and neutrino transport [82, 83], and eccentricity [52]. The development of high-order numerical methods [84] has also allowed simulations to provide predictions for the GW signal accurate to a fraction of radian [85]. In the rest of this section, we draw upon these results to discuss in more detail the physics of BHNS mergers and the limits of current simulations. Early general relativistic simulations of BHNS mergers largely focused on moderate mass ratios Q ∼ 3−5 and black hole spins χBH  0.75. These simulations rapidly confirmed that disruption of the neutron star is more likely for low-mass black holes, rapidly spinning black holes, and large neutron stars and that when a neutron star is tidally disrupted, mass transfer onto the black hole is unstable. The neutron star rapidly expands into an elongated tidal tail, with the vast majority of the neutron star mass accreting onto the black hole within ∼1 ms of the onset of mass accretion. The disruption of the neutron star can occur well outside the ISCO, or as the neutron star plunges into the black hole, with distinct impacts on the gravitational wave signal. Shibata et al. [88] classify BHNS waveforms into three types: nondisrupting binaries, for which the GW signal shows the traditional inspiral-mergerringdown phases observed in BBH systems; systems with disruption well outside of the ISCO, for which the GW signal abruptly shuts off when the star disrupts; and

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systems that disrupt as the neutron star plunges into the black hole, which show a typical inspiral-merger signal, but no ringdown. This classification is still used to guide the construction of modern BHNS waveform models (see Fig. 2). The second and third types typically leave significant amounts of matter (tenths of solar masses) outside the black hole after merger, some bound and some unbound. The unbound material is particularly interesting: as it comes directly from the tidal disruption of the neutron star, is typically cold and neutron rich, is confined to a crescent extending ∼(20 − 30)◦ away from the orbital plane of the binary, and has a significant velocity v ∼ (0.2 − 0.4)c. Similar outflows are produced in BNS mergers, but in much smaller quantity: where a favorable BNS merger will produce 0.01M of cold tidal ejecta, BHNS mergers can eject ∼(0.01 − 0.1)M . This is not due to any difference in the physics of tidal disruption between BHNS and BNS binaries, but rather to the mass asymmetry expected from BHNS systems. A nearequal-mass BHNS merger does not produce more tidal ejecta than a BNS merger [89]. Nevertheless, the production of a large amount of rapidly moving neutronrich ejecta is a notable property of most disrupting BHNS mergers, which can be leveraged to design appropriate EM follow-up strategies. The tidal disruption of the neutron star by its black hole companion is probably the best understood phase of BHNS mergers. Fitting formulae inspired by numerical simulations were rapidly developed to attempt to predict the amount of matter remaining outside of the black hole after merger and available to power post-merger EM emission [90,91]. The most recent [12] is expected to be valid for most systems with MBH > MNS and χBH < 0.9. Models calibrated to numerical simulations also exist for the mass and velocity of the tidal ejecta [92, 93], the mass and spin of the post-merger black hole remnant [94, 95], and the frequency of GWs at the time of disruption [96]. For low-eccentricity systems with small neutron star spins, the latest models are expected to predict the outcome of BHNS mergers within uncertainties that are negligible compared to those associated with later phases of the evolution of the system. Finally, we note that the clear distinction between disrupting and non-disrupting binaries is particularly important if a bright EM counterpart is observed after a BHNS merger: the presence of an EM signal indicates that the neutron star was disrupted, immediately providing us with constraints on the properties of the merging compact objects. In particular, a joint EM-GW observation of a BHNS merger can provide a robust lower bound on the radius of the neutron star, as a function of the binary mass ratio and black hole spin. Even if BHNS mergers are more rare than BNS mergers, a single multi-messenger observation has the potential to provide important constraints on the equation of state of neutron stars.

Disk Formation After merger, the bound material circularizes into an accretion disk as it falls back onto the black hole. The circularization time scale is relatively short ∼(10 − 20) ms, though fallback of marginally bound material onto the disk continues for a few

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Fig. 2 Gravitational wave signals for BHNS mergers in the frequency domain (top) and time domain (bottom), according to the IMRPhenomNSBH model [86]. All waveforms are for optimally oriented binaries at 150 Mpc, with MNS = 1.35M , ΛNS = 400. The black hole mass is chosen to obtain representative examples of non-disrupting binaries (MBH = 6.8M ), mildly disrupting binaries (MBH = 3.8M ), and disrupting binaries (MBH = 2M ). On the frequency-domain plot, we also show the LIGO noise power spectrum at design sensitivity (zero-detuned high-power). The data is generated using the open-source pyCBC library [87]

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seconds after the merger. This is when the physics of BHNS mergers becomes significantly more complex. So far, general relativity and hydrodynamics were largely sufficient to describe the evolution of the system. From now on, magnetic fields and neutrinos are going to play an important role. As a consequence, predictions from numerical simulations become more uncertain, as no code evolves all of the physical processes relevant to these systems to sufficient accuracy. Magnetic field lines inside the neutron star are stretched during tidal disruption, leading to a largely toroidal field structure at the beginning of disk formation. Unless the initial neutron star was a magnetar, this field should remain dynamically unimportant. The growth of the magnetic field to strengths that can affect the merger and post-merger evolution most likely comes from two small-scale instabilities: the Kelvin-Helmholtz shear instability (KHI) that develops at the interface between the forming disk and bound material falling back onto it [79] and the magnetorotational instability (MRI) in the post-merger accretion disk [97]. These instabilities are expected to grow the magnetic field up to the point when the magnetic pressure is of the same order as the fluid pressure, with rapid variations in the orientation of the magnetic field on scales comparable to the growth length scales of the KHI and MRI. The MRI and KHI drive turbulence in the remnant disk, with energy cascading down to the viscous scale where it is dissipated – in theory at least, as current numerical simulations barely resolve the growth length scale of these instabilities [79] and do not come close to resolving the turbulent cascade or dissipation scale. These instabilities are the main source of heating and angular momentum transport in the post-merger remnant once the disk’s circularization is completed and are likely responsible for the ejection of a significant fraction of the disk mass in a relativistic wind. Whether there also is a dynamo mechanism acting in the system that can lead to the formation of a large-scale magnetic field remains an important open question today, as we will see later in this chapter. Magnetic field lines outside the neutron star interact with the black hole before and during merger and may thus be important for pre-merger EM emission [98] and for the eventual formation of relativistic jets and gamma-ray bursts [78]. As for neutrinos, they are the main source of cooling for the post-merger accretion disk. The balance between MRI-driven heating and neutrino cooling is expected to set the temperature, thickness, and accretion rate of the remnant at least on time scales τ  100 ms. Additionally, emission and absorption of electron (anti)neutrinos drive changes in the electron fraction Ye of the disk and its outflows – where Ye = np /(np + nn ) and np,n are the number density of protons/neutrons. The electron fraction is a critical parameter in determining the outcome of nucleosynthesis in the outflows and the properties of kilonovae. Unfortunately, there has not yet been a BHNS merger simulation including both magnetic fields and neutrino transport. As a result, while we understand the overall dynamics of disk formation, we are left with large uncertainties regarding a particularly important aspect of this process: the amount of mass ejected during disk formation and the properties of that ejecta. Kiuchi et al. [79] demonstrated that turbulent heating during disk formation in the absence of neutrino cooling can lead to the ejection of about half the remnant disk within ∼50 ms of the merger. Nouri

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et al. [80] find comparably negligible outflows in a simulation that includes neutrino cooling but does not capture the KHI or neutrino absorption and initializes magnetic fields ∼15 ms post-merger. Qualitatively, it is reasonable to assume that with all the physics properly included, some outflows would be produced and that the unbound matter will be hotter and less neutron-rich than the dynamical ejecta, yet faster than later disk outflows (v/c ∼ 0.1) – as seen, for example, in simulations by Miller et al. [99] that include both magnetic fields and state-of-the-art neutrino transport but start from idealized initial conditions. Whether such outflows represent 1%, 10%, or 50% of the initial disk mass can make a huge difference in the properties of kilonovae and the outcome of r-process nucleosynthesis in BHNS mergers and is thus an important open question for numerical relativists to answer and an important source of uncertainty for modelers to take into account.

Post-merger Evolution Once a nearly axisymmetric accretion disk forms around the remnant black hole, the time scale for the evolution of the system slows down. Angular momentum transport in the disk is driven by the MRI, and MRI-driven turbulence is initially the main source of heating in the disk. The initial density of the disk remains ∼1010−12 g/cm3 , and photons are thus trapped in the fluid. The most massive disks formed in BHNS mergers have initial neutrino optical depths τ ∼ 1 − 10, while less massive disks are optically thin to neutrinos [81]. Either way, neutrinos are the main source of cooling in the disk. For a few tenths of seconds, neutrino cooling allows the disk to remain moderately thick (H /R ∼ 0.2, with H, R the height and radius of the disk), and accretion proceeds efficiently [100, 101]. At later times, as the density of the disk decreases, weak interactions freeze out [102]. The disk becomes radiatively inefficient and geometrically thick. During the first phase, 3D MHD simulations indicate that magnetically driven winds can eject a significant portion of the disk mass [99, 103, 104]. During the second phase, a combination of recombination of nucleons into α-particle and viscous heating due to MRI-driven turbulence can unbind even more material, over time scales of ∼10 s [105–107]. Finally, at some point in the lifetime of the disk, for at least some post-merger remnant, we expect the remnant to produce collimated, ultra-relativistic outflows that may power short gamma-ray bursts [78, 104]. These disk winds and relativistic jets are the main observable that we want to characterize in this long post-merger evolution. When studying this phase of the evolution, the difficulty of including all relevant physical processes (as during disk formation) is compounded by the time scale that simulations need to capture. The disruption of a neutron star by its black hole companion takes ∼1 ms. The formation of a largely axisymmetric accretion disk around the remnant black hole takes ∼10 ms. The evolution of that disk to the point where most of the matter is either accreted into the black hole or unbound from the system, however, lasts ∼10 s. Evolving the remnant for long enough to capture the lifetime of the disk is a significant challenge: the codes currently used to simulate

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BHNS mergers have typically not been used for simulations longer than ∼100 ms. Accordingly, we have to reconstruct the evolution of the post-merger remnant from (relatively) short simulations that directly evolve the outcome of merger simulations [78, 79], a few 3D simulations starting from idealized equilibrium accretion disks [99, 103, 104, 108], 2D simulations starting from axisymmetrized merger results [109, 110], and a much larger number of 2D simulations starting from idealized initial conditions [105–107, 111]. The 2D simulations allow for parameter space exploration and longer simulations, but cannot capture the turbulent cascade due to the MRI and instead rely on subgrid, α-viscosity model for angular momentum transport and heating. These simulations also have very different level of complexity in their treatment of neutrino transport and gravity. Only Miller et al. [99] evolve Boltzmann’s equations; Just et al. [106] use an approximate twomoment transport scheme, while other simulations use leakage schemes, or do not evolve neutrinos at all. All of that missing physics leads to predictions with sometimes poorly constrained error bars: the impact of initial conditions and neutrino transport algorithms remains uncertain, but is probably not negligible [99,107]; evolving magnetic fields is known to be crucial to capture outflows early in the disk evolution, when neutrino cooling is efficient [103,104]; and finally, the initial magnetic field configuration can itself have a huge impact on these early outflows and the production of relativistic jets [78, 108]. Given these uncertainties, what do we know about this phase of the evolution? Let’s start with jet formation. At the moment, there are two proposed mechanisms for the production of a jet and gamma-ray burst through magnetic effects and one due to neutrinos. The first mechanism relies on the existence of a magnetic field exterior to the neutron star before merger, which interacts with the black hole, is twisted and amplified, and eventually leads to the production of a magnetically dominated outflow region along the spin axis of the black hole, an incipient jet (Current ideal MHD simulations are unable to accurately evolve regions with Lorentz factor Γ  10 − 50 and can thus only produce magnetically dominated outflows with energetics comparable to SGRBs, not ultrarelativistic jets.) [78]. Similar simulations without external magnetic fields do not produce incipient jets [55, 112]. The main open question in this scenario is whether the magnetic field can be amplified from realistic initial strengths, as simulations have so far used magnetar-level initial field strengths. The second mechanism relies on the amplification of the magnetic field through a dynamo mechanism in the post-merger remnant disk. Disk simulations initialized with a large-scale poloidal field robustly produce jet-like outflows [103, 104]. However, this is neither surprising nor sufficient to explain the production of a jet from more realistic initial conditions. Indeed, jets are naturally created in black holedisk systems when the black hole accumulates a sufficient net magnetic flux [113], which automatically occurs when a black hole accretes matter from an accretion disk seeded with a large poloidal field. The production of a weaker, fluctuating jet in a simulation initialized with a large-scale toroidal magnetic field is a more

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promising result [108], as it required the creation of a sufficiently large-scale poloidal field from an initially toroidal field. What has not been shown so far, however, is a dynamo mechanism that is active for the type of large-amplitude, small-scale magnetic fields that might be expected if the field grows due to the KHI during the circularization of the disk. Numerical simulations produce jets with energy and duration compatible with observed short gamma-ray bursts [114], yet a lot of work still has to be done to understand the growth of large-scale magnetic fields and the connection between the properties of the jet and those of the postmerger remnant and pre-merger compact objects. The last mechanism is for neutrino-antineutrino pair annihilation to deposit enough energy in the polar region to drive a SGRB. We discuss this mechanism in more detail in the SGRB section of this chapter, but already note here that while it may play an important part in the development of the jet, it has some difficulties explaining the most energetic SGRBs observed so far. Let us now move away from the jet and toward the characterization of the mildly relativistic outflows. There are broadly two types of outflows known to be produced during the post-merger evolution: magnetically driven outflows on a timescale 1 s and viscous outflows ejected over (1 − 10) s. Magnetically driven outflows produced while the disk is efficiently cooled by neutrinos are the most difficult to model. They are typically hotter, faster, and less neutron-rich than the tidal ejecta, but their mass and velocity depend heavily on the chosen initial conditions. 3D GRMHD simulations starting from equilibrium tori have found that ∼5 − 25% of the disk mass is ejected in that process [103, 104, 108], depending on the magnetic field strength and geometry. In addition to the uncertainty due to the magnetic field configuration, the very meaning of “disk mass” here may differ from the mass measured in merger simulations: we should remember that the circularization of the disk itself may drive significant outflows [79], while accretion onto the black hole is initially driven by hydrodynamical shocks, not the MRI [79, 80]. By the time a system reaches the kind of steady-state configuration evolved in long post-merger simulations, a significant fraction of the matter left outside of the black hole after tidal disruption may have been accreted or ejected from the system. As a result, translating the outcome of merger simulations into initial conditions for post-merger simulations is non-trivial and a significant source of uncertainty when modeling post-merger outflows. Simulations treating neutrinos with a leakage scheme find that these outflows have Ye  0.25. The one simulation performing neutrino transport, however, predicts a wider range of Ye in the outflows, indicating that the composition of these outflows is far from being settled [99]. The viscous outflows produced while the disk is radiatively inefficient are comparatively better understood: (5−30)% of the disk is unbound in these outflows, with the exact fraction depending mostly on the compactness of the accretion disk (or, equivalently, its binding energy), which is itself correlated with the mass ratio of the binary [107]. While the black hole spin also is an important determinant in the amount of unbound matter, black holes formed as a result of a disrupting BHNS merger have a fairly narrow range of post-merger spins χBH ∼ 0.8 − 0.9 [95, 107]. This is because at low mass ratio, the black hole spin mainly depends on the angular

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momentum of the system just before disruption, while at high mass ratios, only rapidly spinning black holes lead to tidal disruption of the neutron star. The viscous outflows are typically slow, at least compared to other merger outflows: v ∼ 0.03c. Finally, they are relatively isotropic, with a broad distribution of electron fractions peaking at Ye ∼ 0.3. Our confidence in the value of Ye is still limited by the fact that long numerical simulations of the post-merger remnant use approximate leakage schemes for neutrino transport, but this may be less of an issue here than at earlier times as viscous outflows have had time to reach an equilibrium composition within the remnant accretion disk before weak interaction freeze out, fixing the value of Ye . Finally, it is worth mentioning that most of the results discussed so far assume a black hole-accretion disk system in isolation. In reality, matter from the bound tidal tail continuously falls back onto the remnant and interacts with the disk outflows. Besides the fact that fallback material may affect the mass and properties of the outflows [109, 110], fallback material has also been suggested as a potential reason for late-time X-ray emission following gamma-ray bursts [115, 116]. The modeling of post-merger remnants has made a lot of progress in recent years: all of the long 3D GRMHD simulations discussed in this section have been published in the last 3 years. The first incipient jet observed in a shorter BHNS merger simulation was published in 2014. Finally, 2D simulations of viscous outflows in the context of neutron star mergers are less than a decade old. Yet we can see that, in order to reach reliable models of jets and relativistic outflows, much remains to be done: understanding the growth of magnetic fields during merger and their large-scale structure post-merger is crucial to the production of both jets and slower outflows; improved neutrino transport in simulations is necessary to properly capture the electron fraction of the ejecta; and a better connection between the output of merger simulations and the initial conditions of post-merger simulations will be needed in order to connect the properties of observed jets and kilonovae to the mass and spin of merging compact objects and the equation of state of neutron stars. We will come back to this when discussing in more detail the properties of EM signals powered by BHNS mergers but already illustrate current uncertainties in Table 1, which provides rough estimates for the mass of the three main types of outflows known to be produced by BHNS mergers.

Gravitational Wave Signals Observing BHNS Mergers Through Gravitational Waves Having described the main features of BHNS mergers, we can now move to a more detailed study of the observable signals that they power. We will begin in this section with gravitational wave emission. As we saw earlier in this chapter, disrupting BHNS systems merge at a time when their gravitational wave frequency is fGW,dis ∼ 1 kHz, at which point gravitational wave emission largely stops. A non-disrupting BHNS binary will merge at the ISCO frequency

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Table 1 Estimated outflows for a few BHNS configurations. All systems assume RNS = 12 km, a black hole spin aligned with the orbital angular momentum, and no neutron star spin. We show estimated masses for the dynamical ejecta, post-merger accretion disk, early post-merger magnetically driven outflows, and late post-merger viscous outflows. We take the dynamical ejecta from [93] and the sum of disk mass and dynamical ejecta from [12] and estimate the MHD disk outflows using MMHD = (0.05 − 0.25)Mdisk . The range of viscous disk outflows is estimated from the disk mass and the predictions from [107]. We note that the error bars here are estimates themselves – the table is meant to provide an idea of potential merger outcomes and existing uncertainties, rather than to provide a rigorous model MBH 3M 3M 7M 7M 7M 10M 10M

χBH 0 0.5 0.0 0.7 0.9 0.7 0.9

fGW,ISCO

MNS 1.35M 1.35M 1.35M 1.35M 1.35M 1.35M 1.35M

1 ∼ π



Mdyn (M ) 0 0.003–0.011 0 0.013–0.021 0.046–0.054 0–0.004 0.046–0.054

MBH + MNS 3 RISCO

Mdisk (M ) 0–0.02 0.09–0.13 0 0.06–0.08 0.18–0.24 0.01–0.03 0.11–0.15

MMHD (M ) 0–0.005 0.005–0.03 0 0.003–0.02 0.01–0.06 0–0.01 0.005–0.04



10M ∼ 450Hz MBH + MNS



Mvis (M ) 0–0.005 0.01–0.03 0 0.003–0.01 0.01-0.025 0–0.003 0.005–0.015

6(MBH + MNS ) RISCO

3/2 , (8)

before emitting a higher-frequency merger-ringdown signal similar to that of a BBH merger. Current ground-based gravitational wave detectors are sensitive from ∼(10 − 40) Hz to a few kHz, with their best sensitivity in the few ∼100 Hz range. Most of the information gathered from gravitational wave observations thus come from the late inspiral (for low-mass BHNS mergers) or merger-ringdown (for high-mass systems). The disruption phase, which if it exists is probably the most interesting part of the signal as far as studies of the properties of neutron stars are concerned, is in a frequency range where the sensitivity of existing detectors is rapidly dropping. To understand what information we can extract from gravitational wave observations of BHNS binaries, we first go back to the main parameters driving the evolution of BBH systems. At large separation, the gravitational wave signal is nearly entirely determined by the chirp mass Mchirp =

(MBH MNS )0.6 (MBH + MNS )0.2

(9)

and the geometry of the system (distance, relative orientation of the binary and the detector). As the system gets closer to merger, the impact of the mass ratio and spins becomes more significant. Finally, the merger-ringdown signal is mostly determined by the mass and spin of the post-merger remnant. For low-mass systems, current detectors are mostly sensitive to the inspiral signal, while the merger happens at

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higher frequencies, where detectors are less sensitive. The chirp mass is then known very accurately, but the mass ratio is uncertain. This is the case, for example, for GW190425 [5], a system with a ∼3.4M total mass: the chirp mass of that system is measured to ∼1%, but individual masses are only known to ∼10% if low spins can be assumed (as is reasonable for BNS systems) and to ∼40% if high spins are possible (as in BHNS binaries). This uncertainty in the individual masses of the objects is in fact part of the reason we are uncertain of the exact nature of GW190425. For higher-mass systems, the chirp mass may be less accurately measured, yet a higher accuracy in measurements of the total mass and/or mass ratio can lead to smaller errors for the masses of individual objects. For example, for GW190814 [6], a system of total mass ∼26M , the chirp mass is measured to ∼1%, and individual masses are known within ∼5%. In particular, the lower-mass object has a mass m2 = (2.58 ± 0.10)M . In this case, we are uncertain of the exact nature of GW190814 because we do not know whether neutron stars and/or black holes of that mass exist, more than because m2 is poorly measured. We note that GW190814 has a higher signal-to-noise ratio than GW190425 (22 vs 12), so these numbers should not be considered a rigorous comparison. They do however provide us with practical examples of the impact of the mass of a system on the accuracy of parameter estimation – and in particular of the fact that a disrupting BHNS system is likely to have poorly measured component masses. More detailed studies of our ability to distinguish between BBH, BHNS, and BNS systems, performed using analytical waveforms, also indicate that most disrupting BHNS binaries cannot be confidently identified as such through gravitational wave measurements only [117], even if we trust our astrophysical priors for the mass range of black holes and neutron stars. If we go beyond the point-mass limit, gravitational waves from BHNS binaries also contain information about the size of neutron stars. Finite size effects in BHNS mergers mainly impact the gravitational wave signal in two ways: a dephasing due to the tidal distortion of the neutron star during the inspiral and an abrupt cutoff in gravitational wave emission if the neutron star is disrupted by the black hole. During inspiral, the evolution of a binary can be derived from the equation

dEbin = −PGW dt

(10)

with Ebin the energy of the binary system and PGW the power in the emitted gravitational waves. The tidal distortion of a neutron star due to the gravitational field of a black hole impacts both sides of this equation, with energy going into the distortion of the neutron star and additional gravitational wave emission associated (to lowest order) to the coupling of the quadrupole moment of the distorted neutron star to the quadrupole moment of the binary. To lowest post-Newtonian order, this leads to a correction to the evolution of the phase Φ of the gravitational wave signal [118]

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dΦ 195 3/2 |tides = − x dx 256



M2 MNS MBH



Λ˜

(11)

with x = (ΩM)2/3 , Ω the orbital frequency, M the total mass of the system, Λ˜ =

4 16 (MNS + 12MBH )MNS ΛNS 13 M5

(12)

the effective tidal deformability of the system (expressed in a form adapted to BHNS systems), and ΛNS =

2 −5 k2 CNS 3

(13)

the tidal deformability of a neutron star. Here, k2 ∼ 0.1 is the tidal Love number, which depends on the equation of state of the neutron star. The tidal deformability has a very strong dependence on the compaction of the neutron star and thus on 5 ). From these expressions, we can see clear differences in the its radius (Λ ∝ RNS magnitude of tidal effects between BHNS and BNS systems: tidal effects in BHNS systems are of course smaller due to the presence of only one neutron star, but additionally the mass dependence of the tidal effects means that dΦ dx |tides decreases rapidly for more asymmetric binaries (MBH ∼ M  MNS ). Closer to merger, the low-order expression Eq. 11 becomes inaccurate. Higherorder post-Newtonian terms become significant, and the assumption of static tides made in this derivation breaks down [119]. We should also note that dΦ dx |tides is not, in itself, a particularly good measure of the detectability of tidal effects. BHNS binaries merge at lower frequencies than BNS systems, where detectors are more sensitive. BHNS systems with spinning black holes may also reach much higher values of x before merging than BNS systems or BHNS binaries without spinning black holes. Yet more detailed studies of the detectability of finite size effects in BHNS mergers largely confirm that finite size effects are only detectable for loud signals and low-mass binaries [69, 120]. For non-disrupting neutron stars, tidal dephasing is the main impact of the finite size of the neutron on the gravitational wave signal – and it is a small effect, considering that non-disrupting BHNS systems have higher black hole masses than disrupting systems. Numerical simulations [10], analytical studies [120], and injections into LIGO noise [121] all indicate that non-disrupting BHNS systems are indistinguishable from BBH systems in most of our observable volume. For disrupting systems, on the other hand, we have already argued that the gravitational wave signal is cut off at a frequency fGW,disrupt ∼ (0.5 − 1.5) kHz that depends on the radius of the neutron star. Large neutron stars disrupt earlier and can thus more easily be distinguished from black holes. Both tidal dephasing and tidal disruption should then be taken into account when using gravitational waves to extract information about the size of neutron stars. Rather interestingly, even for disrupting systems, the tidal deformability Λ apparently remains the best

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measured parameter related to the finite size of the neutron star [69, 120]. It does however remain a difficult measurement to make: from Lackey et al. [69], we can infer that to measure RNS to ∼1 km with a single BHNS event and the aLIGO detector, we need an optimally oriented merger event within 100 Mpc, even for very favorable mass ratios (MBH = 2MNS ), and 50 Mpc for a more likely configuration (MBH = 5MNS , χBH = 0.75). The latter would correspond to a signal-to-noise ratio > 100, i.e., less than one in a thousand events. A direct measurement of Λ from the observation of BHNS mergers thus appears unlikely to be competitive with BNS measurements or with NICER observations. However, that does not necessarily mean that finite size effects should be ignored in waveform models. One of the reason Λ is difficult to measure is due to degeneracies with other binary parameters (e.g., the black hole’s spin) [69]. Ignoring tidal effects in merger waveforms may lead to biases in the inferred parameters of observed systems even if the tidal effects themselves cannot be directly measured [121], and low-mass BHNS mergers may still provide useful information about the size of neutron stars if some BHNS mergers that include black holes within the mass gap are observed. Furthermore, any increase in the high-frequency sensitivity of the detectors can help with the observation of the disruption of a neutron star. The A+ update to the LIGO detectors, expected to be operational in 2024, provides about a factor of 2 increase in sensitivity over the aLIGO design sensitivity at ∼1 kHz (Technical note LIGO-T1800042-v4), while third-generation detectors such as the Einstein telescope would allow us to measure tidal deformabilities with a few percent accuracy [69].

Waveform Models and Their Accuracy Gravitational wave observatories require accurate signal templates to both detect and analyze observed merger events. Early gravitational wave models were generally derived from post-Newtonian theory, while most modern waveform templates come from one out of two broad classes of models: inspiral-merger-ringdown phenomenological waveforms (IMR-Phenom) or the effective one-body (EOB) formalism. These models come with a range of free parameters that are typically calibrated to the result of numerical simulations in order to retain high accuracy during the late inspiral and non-linear merger phase. The IMR-Phenom models are frequency domain waveforms that smoothly interpolate between the known post-Newtonian and ringdown portions of a merger signal, with the merger part of the waveform fitted to a large number of numerical simulations. The IMR-PhenomD model used during recent LIGO-Virgo runs [122, 123] was recently updated by the IMR-PhenomXAS model [124], which provides better results for unequal spins and asymmetric mass ratios. The base models, built for detecting merger events, are aligned-spin waveforms: they do not include precession, tides, or disruption and are additionally limited to the dominant (2, ±2) modes of the expansion of the gravitational wave signal in spinweighted spherical harmonic. More advanced models including precession [125],

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higher-order modes [126,127], tides [86,128,129], and the disruption of the neutron star [69, 86, 130] can be used to perform more accurate parameter estimation. The EOB maps the general relativistic two-body problem onto that of a test particle moving in an effective metric [131]. It remains the main analytical technique to directly generate analytical waveforms that remain reasonably accurate through mergers. Recent models are based on ideas developed over the last two decades [132–140] and are partially calibrated to the result of numerical simulations. As for IMR-Phenom waveforms, models with different levels of complexity are currently available: aligned-spin waveforms with or without higher-order modes [141–143], precession [144, 145], static and/or dynamic tides [69, 146–149], and tidal disruption [150]. These models have rapidly progressed over the last few years yet still have a few important limitations when applied to BHNS mergers. This is largely because BHNS mergers are particularly complex systems to model. We expect BHNS mergers to have significant mass asymmetries, in a part of parameter space where numerical relativity simulations used to calibrate models are less accurate and subdominant modes are important. The black hole may have a large spin misaligned with respect to the orbital angular momentum, leading to significant precession of the orbital plane. And of course we would like models to include tidal effects during the inspiral and take into account the disruption of the neutron star during merger. So far, models constructed to capture the disruption of the neutron star do not include precession nor subdominant modes [69, 86, 130, 150]. The lack of more advanced BHNS-specific waveform models can also in part be blamed on the limited availability of accurate numerical waveforms. Numerical simulations are particularly critical to calibrate tidal effects during the late inspiral and, even more, the disruption of the neutron star. There are however only two sets of BHNS simulations that have been used for model calibration so far, and both have important limitations. First, a group of 134 waveforms generated with the SACRA code [69, 151] provides (by far) the best coverage of the BHNS parameter space available to date. These are however older simulations that only simulated the last few orbits before merger and do not have the high phase accuracy of the most recent SACRA simulations of BNS mergers [152]. Second, five SpEC BHNS waveforms [68] are publicly available as part of the SxS catalogue [85] or have been used for model calibration and will soon be released [150]. While these waveforms are significantly longer and accurate and have been useful to calibrate the phase evolution of BHNS models [86, 150], they clearly provide nearly no parameter space coverage. None of the systems used to calibrate BHNS models (and none of the publicly available waveforms) include any precession. It will only be possible to develop more robust models for the tidal disruption of the neutron star if a sufficiently large number of numerical simulations of precessing systems become available in the future. Existing numerical simulations do however allow us to test the accuracy of the models currently used for gravitational wave data analysis. Lackey et al. [69] find that their model for the amplitude of the signal during tidal disruption has ∼30% relative errors, reduced to ∼10% in the model of Pannarale et al. [130]. This latter

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model forms the basis of the implementation of tidal disruption in the IMR-Phenom model [86] and the EOB model [150]. The latest IMR-Phenom and EOB models also agree with the phase of the waveforms in numerical simulations to better than 0.5 rad over ∼20 − 30 cycles (∼0.4% relative error), except during the last orbit before merger, when errors can rise to (1 − 2) rad. This is true both for simulations used to calibrate the models and for simulations with large black hole spins that are nominally outside of the calibration range of the models [150]. Matas et al. [150] also compute the unfaithfulness between waveforms generated by hybridizing the results of SpEC simulations with the IMR-Phenom or EOB model and the waveforms directly generated with these models. The unfaithfulness is a more useful measure of modeling uncertainties than the phase and amplitude errors, as it takes into account the sensitivity of current detectors as a function of frequency. For the existing five-parameter models [86, 150], a rough rule of thumb is that two waveforms with unfaithfulness F¯ are distinguishable for signal to-noise ratio ρ  2/F¯ . For low-mass (MBH ≤ 3MNS ), non-spinning systems and using the advanced LIGO design sensitivity, F¯  0.002 and the models and simulations are undistinguishable for ρ  32. For systems with spinning black holes and/or higher mass ratios, F¯  0.01 in most cases (and up to F¯ ∼ 0.03 for the most unfavorable case). The highest unfaithfulnesses are obtained when comparing hybrids using one analytical model to analytical waveforms using the other model. This indicates that for spinning black holes, parameter estimation may be biased even for moderate signal-to-noise ratios ρ ∼ 10 − 20. A more in-depth study of parameter biases in BHNS mergers was recently performed by Huang et al. [121] through injection of numerical relativity-analytical waveform hybrids into LIGO noise and their recovery using the full parameter estimation pipeline. That study is limited to non-spinning configurations but considers multiple binary inclination and signal-to-noise ratios, and mass ratios Q = 2, 3, 6. The results confirm that for nonspinning black holes, modeling biases are smaller than statistical errors at ρ ∼ 30, while modeling biases become significant for loud signals (ρ ∼ 70), particularly for models that are not calibrated to BHNS simulations. Overall, BHNS models are thus in very good shape for non-spinning systems, may show some noticeable biases for systems with large black hole spins and no precession, and have unknown errors for precessing systems.

Detectability and Detection Biases The fact that BHNS systems may have very asymmetric masses and/or show significant precession has another important consequence: the use of aligned-spin waveforms in detection pipelines means that we are more likely to completely miss the detection of a BHNS merger than a BNS merger (low spin, near-equal mass) or BBH merger (on average closer to equal mass). Harry et al. [153] find that for 10M black holes and assuming a uniform distribution of black hole spin magnitudes and an isotropic orientation of that spin, ∼30% of BHNS mergers will be missed by

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the advanced LIGO detectors (35% if using non-spinning templates). The recovery fraction of BHNS signals varies significantly with the properties of the binary. Binaries with large black hole masses, binaries with rapidly spinning black holes, and binaries seen edge-on or with significant orbital plane precession are all more likely to be missed by searches using aligned-spin waveform. It is thus important to note that there are significant observational biases involved in the observation of BHNS binaries. This may not be a major limitation in the analysis of a population of BHNS observations, as the observational biases are known or can at least be estimated through the injection of more accurate waveforms in simulated noise and their attempted recovery using spin-aligned models. Yet it is an important effect that should not be neglected. We should also note that the lack of precessing waveforms and its impact on the detection of BHNS systems are more of an issue for pure gravitational wave observations than for multi-messenger observations. This is because the subset of BHNS binaries that experience tidal disruption and can power bright electromagnetic signals is more likely to be recovered by gravitational wave searches using aligned-spin waveforms than the set of all BHNS binaries. Disrupting BHNS binaries have lower masses than non-disrupting systems and require black hole spins that are mostly aligned with the orbital angular momentum of the binary. Additionally, for joint observations of gravitational waves and gamma-ray bursts, we need the binary to be observed close to face-on. Accordingly, joint observations of gravitational waves and gamma-ray bursts are less affected by biases in the detection rate of BHNS binaries than the GW observations themselves.

Current BHNS Merger Candidates No detection of a BHNS merger has been confirmed so far, but a few candidate events currently exist. We can broadly categorize them into two classes: published GW candidates that have a non-zero probability of being a BHNS system and GW triggers that are categorized by the LVC as potential BHNS mergers. For the first category: • GW170914 [3] is nearly certainly the first observed BNS merger. While the presence of EM counterparts only guarantees the presence of one neutron star, the most massive object in this binary has mass M1  1.89M (90% probability, allowing for a spinning BH) and is thus probably a neutron star. • GW190425 [5] is a more massive system and thus more intriguing as a potential BHNS binary. Its most massive component has M1 ∈ [1.61, 2.52] (90% probability, allowing for a spinning BH). Considering the evidence that (a) the maximum mass of neutron star is likely  (2.17 − 2.3)M [49–51] and (b) objects of mass M ∼ 2.6M can be found in merging binaries (see GW190814), this event is a more reasonable BHNS candidate than GW170914, though still more in line with the masses of observed neutron stars than black holes.

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• GW190426 [4] is a lower significance event with a false alarm rate of 1.4 per year (as estimated by GstLAL). If real, it is most likely a BHNS merger, with M1 ∈ [4.4, 9.7]M and M2 ∈ [1.0, 2.3]M . This is the most likely BHNS merger candidate so far, with the main uncertainty being the possibility that the event is not a true gravitational wave signal. • GW190814 [6] is a high mass ratio systems with component masses M1 ∼ 23M , M2 ∈ [2.50, 2.67]M . The lower mass object is either the lowest mass black hole ever observed (well below the expected boundary of a potential mass gap ∼5M ) or the most massive neutron star (in tension with constraints on the maximum mass of neutron stars coming from GW170817). It is one of the most intriguing events reported by the LVC so far, yet its exact nature will most likely remain unknown until we determine whether (2.5−2.6)M objects can be black holes, neutron stars, or both. If GW190814 was a BHNS binary, it would not disrupt [12]. Accordingly, the lack of EM counterparts to this event is not informative. In addition to these published candidates, the S191205ah GW trigger from O3b is currently listed as a likely BHNS merger (More than 50% chances of being a BHNS in https://gracedb.ligo.org/superevents/public/O3/). We note however that this event has a false alarm rate higher than 1 every 10 years; that any object with an inferred mass of less than 3M is categorized as a neutron star by the public alert database (e.g., GW190814 is labeled as a BHNS binary by the alert system); that in the absence of a full parameter estimation, some events can easily be mislabeled; and that even with full parameter estimation, characterizing an event as BHNS, BNS, or BBH can be difficult. On the other hand, a few other interesting BHNS candidates are worth mentioning, despite the fact that their exact nature is unknown and that they were not initially categorized as BHNS mergers in rapid alerts for EM follow-up: • S200105ae was initially categorized as likely terrestrial, yet its significance is underestimated because it is a loud single-detector event. If real, it is most likely a BHNS binary with a small probability of disruption (12%, GCN 26640 [154]). • S200115j is a higher significance event (false alarm rate of 1 per 1513 years) currently associated to the merger of a neutron star with a “mass gap” object (with mass M ∈ [3M , 5M ]), which makes it most likely a BHNS mergers. It also has a non-zero probability of powering an EM counterpart (8% according to the rough estimates provided in rapid alerts by the LVC). A similar story unfolded for GW190426, an event that was not initially categorized as a likely BHNS merger by the alert system. We thus see that some of the most interesting BHNS candidates that exist today were not listed as BHNS events in rapid alerts, another consequence of the difficulty of recognizing that a system is a BHNS binary. Our understanding of the nature of these events is certain to evolve as more complete information is released and as we improve our understanding of the mass

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range of black holes and neutron stars. Given the number of “likely” or “very likely” (e.g., GW190426, S200115j) BHNS mergers in O3, we have probably already detected a BHNS merger, proving that detection beyond a reasonable doubt has not however been possible so far.

R-Process and Kilonovae Nucleosynthesis in BHNS Mergers We previously saw that disrupting BHNS binaries can eject a few percent of a solar mass of neutron-rich ejecta at the time of merger (dynamical ejecta) and at least a comparable amount of matter in disk winds of more uncertain properties (see, e.g., Table 1). Lattimer & Schramm [56] proposed more than 40 years ago that this could result in a non-negligible contribution of BHNS binaries to heavyelement nucleosynthesis. While the fractional contribution of BHNS mergers to the production of heavy elements remains unknown today, BHNS mergers remain viable candidate for the production of at least some heavy nuclei. Elements heavier than iron are expected to be made mostly through the slow and rapid neutron capture processes (s-process and r-process) [155]. During neutroncapture nucleosynthesis, heavy nuclei are formed mainly through a combination of neutron capture events (increases the atomic mass number A at constant atomic number Z) and β-decays (increasing Z at constant A). In the s-process, neutron capture rates are small or comparable to the rate of β-decays for nuclei close to stable isotopes. Nuclei then evolve mostly along the valley of stable isotopes; any neutronrich nucleus formed by neutron capture rapidly β-decays. If the neutron density is high enough, on the other hand, we get r-process nucleosynthesis: rapid neutron captures will create very neutron-rich nuclei, and nuclei will grow along a path far away from the valley of stable isotopes. The higher the neutron density around these nuclei, the farther from the valley of stability this progression will take place. For the dynamical ejecta of a BHNS binary (Ye ∼ 0.05), the r-process evolves very close to the neutron drip line. For less neutron-rich environment (e.g., in a neutrino irradiated wind), the r-process may instead proceed closer to the valley of stability. Rapid neutron capture stops when most neutrons have been captured in nuclei, at which point the very neutron-rich nuclei created during the r-process decay back toward stability. If enough neutrons are initially present in the system, r-process nucleosynthesis may also lead to the production of nuclei heavy enough to undergo fission, creating lower-mass neutron-rich nuclei that will then themselves go through r-process nucleosynthesis – a process called fission cycling. There are clear differences in the relative amounts of each element produced in the s-process and r-process. In particular, nuclear models predict that the r-process mostly produces elements around 3 peaks at A ∼ 70 − 80 (“first peak”), A ∼ 120 − 130 (“second peak”), and A ∼ 190 − 200 (“third peak”), as well as a non-negligible amount of lanthanides (between the second and third peaks) and actinides (above the

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third peak) that will play an important role in the properties of the electromagnetic signals powered by r-process nucleosynthesis. This allows us to use observations to infer the relative contribution of each process to the enrichment of the universe in heavy elements. Quantitative studies of r-process nucleosynthesis require nuclear reaction networks with thousands of isotopes and hundreds of thousands of nuclear reactions. Such networks have been applied to the study of the dynamical ejecta from neutron star mergers [106, 110, 156–159] and from post-merger accretion disks [99, 103, 160–163]. For the purpose of this review, we mostly emphasize a result important to our understanding of the potential role of BHNS mergers in astrophysical nucleosynthesis: the crucial role played by the electron fraction Ye of the ejecta in setting the outcome of r-process nucleosynthesis. A low electron fraction should lead to the r-process reaching a quasi-stationary distribution of nuclei where neutron capture, beta decays, and fission rates mostly balance each other. This is the situation encountered in the dynamical ejecta of a BHNS merger. The outcome of r-process nucleosynthesis should then be fairly insensitive to small changes in Ye . Nucleosynthesis in these very neutron-rich environments produces mostly heavy elements (A  100), including second and third-peak elements as well as lanthanides and actinides. At higher Ye , less neutrons are available and the r-process can be truncated. Elements around the first peak are then produced, while elements beyond the second peak are not (third-peak elements, lanthanides, actinides). Above Ye  0.2 − 0.3 [159, 164], lanthanides, actinides, and third-peak elements are no longer produced in significant amounts. This provides us with a good first-order estimate of the outcome of the r-process. We note however that while a division into two potential types of r-process is sometimes useful to estimate the outcome of nucleosynthesis and the properties of kilonovae, it should not be taken too literally: there is not a sharp transition between two fixed nucleosynthesis outcomes at a given Ye . For example, small variations in Ye can lead to changes in the ratio of actinides and lanthanides produced during the r-process even for values of Ye that avoid production of first-peak elements (e.g., Ye ∼ 0.1 − 0.3) [165]. This is of particular importance when attempting to ascertain the origin of elements around and past the third peak (e.g., gold, uranium): we will see in the next section that the production of lanthanides and/or actinides can reasonably easily be inferred from electromagnetic observations of kilonovae; yet the presence of lanthanides does not by itself guarantee the production of large amounts of gold or uranium. The main determinants of the contribution of a BHNS merger to the enrichment of the universe in various r-process elements are thus the mass of baryonic matter that it ejects and the properties of that ejecta – primarily its electron fraction Ye . We have already seen that due to the limits of current numerical simulations, there are significant uncertainties in the properties of that ejecta, particularly for the postmerger outflows. These are not, however, the only sources of uncertainties when predicting the outcome of the r-process. The relative abundances of various elements produced in the r-process and the energy released during the r-process both depend on the masses of the very neutron-rich, short-lived nuclei that are temporarily

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created during the r-process, as well as the rate of β-decay of these nuclei, and the fission rate and distribution of fission fragments if fission cycling occurs. The exact outcome of the r-process is thus tied to the determination of the properties of very neutron-rich nuclei. These are also of great interest in experimental nuclear physics today, with experiments such as FRIB attempting to measure more accurately the properties of neutron-rich nuclei away from the valley of stability (see [166] for a review of the connection between r-process nucleosynthesis and experimental nuclear physics and [167] for a review of the impact of various nuclear-physics parameters on the outcome of the r-process). With some assumptions regarding the source of r-process elements, the abundances of r-process elements inferred from observations can even be used to reverse-engineer the problem and constrain nuclear physics [168]. Going back to BHNS mergers, the dynamical ejecta is most likely sufficiently neutron rich to avoid production of first-peak elements and to robustly produce third-peak elements, lanthanides, and actinides. Given that first-peak elements are observed in the solar system, the dynamical ejecta from BHNS mergers cannot on its own be responsible for all r-process nucleosynthesis. The same is true for the material ejected in the disruption of a neutron star during a BNS merger. In BNS merger, the first proposed solution to this problem was the discovery that matter ejected during the decompression of the neutron star cores right after they collide is sufficiently hot to emit and absorb large numbers of electron (anti)neutrinos and thus reach values of Ye sufficient to create first-peak elements [159]. HighYe ejecta can also be produced through neutrino-driven winds in the presence of a long-lived neutron star remnant [169] and in spiral density waves generated by that neutron star remnant in the post-merger accretion disk [170]. In BHNS mergers, the situation is more uncertain. The dynamical ejecta represents a larger fraction of the outflow mass (often > 50%) and thus has a much larger impact on nucleosynthesis yields than in BNS mergers. The post-merger outflows have a more uncertain electron fraction. The most recent simulations indicate that most of the viscous outflows produced seconds after the merger may be sufficiently neutronpoor to produce first-peak elements [107]. Earlier magnetically driven outflows are more neutron rich [103, 108], though simulations including more detailed neutrino transport indicate that at least a fraction of the magnetically driven outflows can have Ye  0.2 − 0.3 [99]. Explaining the entire r-process through BHNS mergers may however not be necessary (or even desirable). BNS and/or core-collapse events may contribute at comparable or higher level to the enrichment of the universe in r-process elements. Vangioni et al. [20] estimate that if all of the r-process elements were produced in BHNS mergers, the advanced LIGO detectors should observe (2 − 12) disrupting BHNS binaries a year at design sensitivity. This is already in tension with the latest upper bound on the event rate of all BHNS mergers presented by the LVC [2]. BHNS mergers may contribute significantly to the production of r-process elements (or not), but they seem unlikely to be the sole dominant source of r-process nucleosynthesis.

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Radioactively Powered Transients: Kilonovae The most direct observational consequence of the ejection of neutron-rich matter undergoing r-process nucleosynthesis is the production of kilonovae: radioactively powered transients evolving on time scales of days to weeks after the merger and observable with UV/optical/IR observatories [171–174]. In a kilonova, radioactive decays of neutron-rich nuclei produced during the r-process release energy initially distributed between the decay products: non-thermal β and α particles, gamma rays, neutrinos, and fission fragments. Neutrinos immediately escape the ejecta, while the other decay products partially thermalize in the ejecta [175, 176]. Initially, the ejecta is dense enough to be optically thick to UV/optical/IR photons. As the matter expands and cools down, however, the opacity of the ejecta decreases, up to the point when the energy being released by the ashes of the r-process becomes visible as thermal emission from the radioactively heated ejecta – a kilonova. As a result, the brightness, duration, and color of a kilonova strongly depend on the opacity of the ejecta, which is dominated by absorption due to a very large number of boundbound transitions in heavy nuclei. Early predictions for the properties of kilonovae assumed that the opacity of the ejecta would be similar to that of supernovae remnants [157, 171, 172], i.e., that the elements produced in the r-process have opacities comparable to the iron group elements produced in supernovae. A typical kilonova would then peak on a timescale of a few days after the merger, with a brightness of the order of a thousand novae and a spectrum peaking at optical wavelenghts [177]. However, this ignored the fact that lanthanides and actinides have many more transition lines than iron group elements. As a result, in the presence of lanthanides and/or actinides, the opacity of the ejecta to optical/IR photons is significantly higher [173]. This led to the prediction that kilonovae should instead be dimmer and peak 1 − 2 weeks after merger at infrared wavelengths [178]. Searches for infrared kilonovae following short gamma-ray bursts (GRBs) rapidly led to a viable kilonova canditate associated with GRB130603B [179, 180], with more candidates proposed after reanalysis of archival data [181]. A third type of proposed signal is UV emission from the decay of rapidly moving free-neutrons, if a small amount of fast-moving neutronrich material is ejected during merger or due to emission from the hot surface of the ejecta [182]. That signal peaks on hour timescales and is thus naturally more difficult to catch as follow-up of GW events and GRBs requires searches through relatively large areas of the sky. UV emission can also come from the outer layers of the ejecta, before the ejecta becomes transparent to photons. UV, optical, and IR emissions were all observed in the kilonova following the first detection of a BNS merger, starting ∼11 hrs after merger [183–190]. This remains the only kilonova directly associated with a GW signal. The detection of a kilonova tells us that r-process nucleosynthesis was very likely occurring in a system, and their color informs us about the outcome of the r-process (i.e., the presence of lanthanides and actinides). Differentiating between an r-process that produces elements beyond the third peak and one that produces lanthanides

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but not many heavier nuclei is however difficult using observations in the week following the merger; they may become distinguishable at later times, when energy deposition is dominated by the decays of a small number of elements (weeks to months post-merger) [191], but no such observation is available for GW170817. Kilonovae may also help us determine the properties of merging compact objects. For spherically symmetric mass ejection, there are already important variations in the duration, color, and brightness of kilonovae with the mass, velocity, and opacity of the ejecta [178], all properties that vary with the nature of the merging binary (BNS vs BHNS), and the exact properties of the merging compact objects (mass, spin, nuclear equation of state). A more massive ejecta leads to a brighter, longer duration kilonova; a faster ejecta creates a brighter but shorter-lived kilonova than for lower velocities; and of course Ye impacts the color and duration of the kilonova, as previously discussed. With sufficiently accurate models mapping the properties of a BHNS / BNS system to those of its ejecta, one could in theory obtain a lot of information about the system from kilonovae observations. However, we are not quite there yet: we have already seen that there are significant uncertainties in current predictions for the mass and composition of merger outflows in simulations. Nuclear physics uncertainties also impact potential inferences made from kilonovae: predicted heating rates from the decay of r-process elements vary by factors of a few between nuclear models [175]. Going beyond spherical symmetry, we note that in a BHNS merger, the dynamical ejecta is confined to within 20◦ − 30◦ of the orbital plane (at the time of the disruption of the neutron star), spreading in a crescent covering about half that plane [92]. Matter ejected from the post-merger accretion disk is likely more isotropic and slower. An observer looking at a BHNS merger edge-on may not be able to see optical emission from high-Ye disk outflows if they are obscured by the higher opacity dynamical ejecta, and the properties of kilonovae will clearly depend on the relative orientation of the binary and the observer [110, 192]. This naturally complicates our work when attempting to extract information about merging compact objects from kilonova observations. One possible advantage of these geometrical effects is that kilonovae from BHNS mergers may look different from kilonovae from BNS mergers [193–195]: a larger amount of neutron-rich ejecta mostly confined to the equatorial plane would lead to a redder, longer-duration kilonova with higher integrated bolometric luminosity. One might however need to be careful with this argument: simulations of low-mass BHNS mergers indicate that the properties of their ejecta do not fundamentally differ from those of BNS mergers with similar component masses [89]. So while the properties of a kilonova might allow us to differentiate between typical BHNS / BNS mergers (i.e., systems with very different component masses), it is unclear whether it is possible to do so for systems with well-measured component masses. Finally, we note once more that the most robust results that can be obtained from the observation of a kilonova in a BHNS system comes from the existence of the signal itself – regardless of its detailed properties. A kilonova will only be possible

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Fig. 3 Estimated mass ejected by a BHNS merger as a function of mass ratio Q = MBH /MNS and dimensionless black hole spin χBH . We estimate the dynamical ejecta and disk mass from [89, 93] and assume that a fraction frem of the disk mass is unbound. The six panels vary CNS = GMNS /(RNS c2 ) and frem = 0.1, 0.5. The neutron star properties are chosen to range from optimistic for tidal disruption (top) to pessimistic (bottom). Even more massive neutron stars are nearly never disrupted. The white region in the top right panel has Mej > 0.2M

if the neutron star is disrupted by the black hole, and thus the detection of a kilonova provides us with a lower bound on the neutron star radius as a function of mass ratio and black hole spin [12]. We illustrate this in Fig. 3. We see that the observation of an EM signal itself can strongly constrain the binary parameters. Upper bounds on the ejected mass at the level of a few percents of a solar mass are also interesting. An actual measurement of the total ejected mass would provide meaningful constraints on the binary parameter, though with significant degeneracies between mass ratio, black hole spin, and neutron star compactness. In the following sections, we cover in more details existing kilonova models used for BHNS binaries, as well as recent searches for optical counterparts to potential BHNS mergers and what we learn from them.

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Kilonova Models To go beyond the simple dichotomy between disrupting and non-disrupting BHNS systems, and make detailed inferences about the properties of the merging compact objects, we need reliable models for the kilonova signal as a function of the parameters of the binary. In the previous section, we argued that current models still have significant uncertainties due to numerical errors and incomplete physics in numerical simulations, nuclear physics uncertainties, and the use of approximate geometries for the ejecta. Nevertheless, there has been significant progress in kilonova modeling of both BNS and BHNS binaries in recent years, and we are now, for example, much better at connecting the properties of the merging compact objects to specific features of kilonovae than we are at doing the same for GRBs. The main ingredients going into a kilonova model are a choice of initial conditions for the ejected material, a choice of nuclear heating rate, a model for the opacity of the ejecta, and an algorithm for radiation transport. Initial conditions are typically inspired by or directly taken from the result of merger simulations. The heating rate depends on the assumed nuclear model and the thermalization efficiency of the decay products. The opacity of the ejecta requires a choice of composition (heavy nuclei present in the ejecta) and absorption lines of each nucleus or a simplified model capturing their effect. Existing radiation transport codes typically assume either a spherically symmetric or axisymmetric ejecta. Detailed energy-dependent radiation transport simulations have been performed by a number of groups (see, e.g., [173, 174, 178, 196, 197]) and play an important role in guiding our understanding of kilonova properties and EM follow-up to gravitational waves. Energy-dependent radiation transport simulations are however too costly to be used directly for parameter estimation when observing a kilonova. Simplified models have thus been developed to provide kilonova light curves for a range of ejecta masses and composition. Kasen et al. [198] produced a grid of spherically symmetric kilonova models varying the mass and lanthanide/actinide content of the ejecta that can be used to at least estimate the mass and composition of the ejecta powering a kilonova. The POSSIS code [192] has similarly been used to produce grids of axisymmetric transport simulations with two ejecta components, observed from various orientations and with opacities chosen to represent a neutronrich dynamical ejecta (within an angle θ < θej of the orbital plane) and a higher Ye disk wind (θ > θej ). The grids include models adapted to BNS [47] and BHNS [199] systems. An alternative to these grids of models is to use faster, simplified methods that can generate light curves for a wider range of parameters. Hotokezaka & Nakar [176] developed a spherically symmetric model that can rapidly generate light curves for specific ejecta profiles by assuming an effective gray opacity for the ejecta, calibrated to the result of energy-dependent radiation transport simulations. They also show that the total energy of a kilonova is strongly correlated with the total mass of the ejecta, without much dependence in its composition. If we can measure the bolometric luminosity of a kilonova as a function of time – which is not necessarily easy if observations are sparse – we can thus directly estimate the

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mass of the ejecta, up to variations between the total energy of the kilonova and its estimated isotropic luminosity. Going beyond spherical symmetry, Kawaguchi et al. [92] developed a model specifically adapted to the geometry of the dynamical ejecta of BHNS binaries, predicting the bolometric luminosity of the kilonova before applying color corrections calibrated to more detailed radiation transport simulations. Barbieri et al. [200] expanded on this idea and constructed a semianalytical model including contributions from a dynamical ejecta, wind ejecta, and viscous ejecta (as well as a model for gamma ray and radio emission from BHNS mergers). The most natural use of these models is to attempt to determine the properties of the ejecta, given a set of observations. This has of course not been done for an actual BHNS-powered kilonova, as no such system has been detected so far, but Barbieri et al. [201] show that the brightness of a BHNS kilonova will vary significantly with the binary parameters (mainly the size of the neutron star and mass and spin of the black hole). As they fix some of the parameters that numerical simulations have not managed to reliably predict yet (e.g., the fraction of the disk ejected in magnetically driven and viscous winds), the results are a demonstration of what could be done with more reliable predictions from merger and post-merger simulations. If a kilonova from a BHNS merger was observed today, we would have to account for numerical uncertainties. This thus represents a powerful motivation for more accurately predicting the properties of post-merger outflows. Kilonovae models have also been used to assess whether GW170817 could potentially have been a BHNS binary [201–203], with the overall conclusion that at this point there is no smoking-gun evidence that it cannot be: the merger of a BHNS binary with the component masses of GW170817 and a moderately stiff equation of state can at least explain the overall energy budget of the observed kilonova. Whether it can produce the optical signal observed after GW170817 is more debatable: that signal is inferred to come from a relatively high-velocity (v  0.2c), high-Ye outflow (see, e.g., [183]). Magnetically driven outflows during disk formation and/or in the first 1 s post-merger are likely able to produce such outflows [99], but possibly not in sufficient quantity. The strongest argument in favor of GW170817 being a BNS merger remains of course the low masses of the merging compact objects. Finally, these models can be used to constrain the properties of a BHNS binary given observational upper bounds on the brightness of an associated kilonova. This analysis leads to much weaker constraints than an actual detection but has the advantage of having been performed for actual follow-ups of GW triggers [195, 199, 204–206]. When performing such analysis, one of course has to assume that the event was within the region covered by EM follow-up observations; the resulting constraints are placed in the high-dimensional space of potential binary parameters and sky localizations, with significant degeneracies between the limits on the binary parameters, the distance to the source, and its orientation. Using this method, Andreoni et al. [199] found that for GW190814, the ejected mass had to satisfy Mej  0.04M if observed mostly face-on, while Kawaguchi et al. [195]

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and Viera et al. [207] found Mej  0.05M . This provided new constraints on the spin of a potential low-mass black hole at the time of publication, as could also be inferred from our Fig. 3. However, parameter estimation results from the LVC released long after these EM observations found component masses incompatible with a disrupting BHNS binary (23M + 2.6M ) [6]. More recent events generally had much weaker constraints on Mej (see next section).

UV/Optical/IR Follow-Up of BHNS Merger Candidates Besides our two likely BNS events, the first potential BHNS binaries followed by UV/optical/IR telescopes were S190426c and GW190814. S190426c had a high probability of being terrestrial (58%), a 1260 deg2 90%-credible region, and at a distance of (377 ± 100) Mpc; it was thus far from an ideal candidate for followup, but as the first likely BHNS merger and the only one so far with a high probability of tidal disruption, it generated a lot of interest (see [205] for a summary of observations). While more than half of the credible localization region was observed, constraints on the ejected mass in the observed region remained relatively weak (Mej  0.1M ). GW190814, on the other hand, was rapidly reported to be a likely BHNS binary and had relatively good sky localization (23deg2 at 90% confidence) and reasonably close-by distance (267 ± 52) Mpc. Followup observations of GW190814 with a combination of galaxy-targeted searches and tiling did not uncover any optical/infrared counterparts compatible with a kilonova [199,204,207–209], but they produced useful upper bounds on the amount of mass ejected by the merger if it occurred within the observed region of the sky (see previous section). A summary of other follow-up observations performed during the first half of O3, which included BHNS candidates S190910d, S190923y, S190924h, and S190930t, can be found in [205]; a summary for the second half of O3, which includes S191205ah, S200105ae, and S200115j, can be found in [206]. The only candidate localized to better than 2000deg2 was S200115j (765deg2 ), complicating followup searches. Survey instruments (ZTF, GRANDMA-TAROT, MASTER, ATLAS, SEGUARO) covered (parts of) the localization region of the more poorly localized events, while more teams were involved in follow-up of S200115j due to its better localization and non-zero estimated probability of powering a transient. As for GW190814, none of the transients observed during these follow-up campaigns passed the cuts required to be categorized as a kilonova (typically, a combination of evolution time scale, color, and/or color evolution). An analysis of follow-up observations of GW200105ae and GW200115j by the GROWTH network and ZTF [210] provides more information about what can currently be inferred from the lack of detection of EM counterparts for most of these O3 events. The more recent searches were less constraining than follow-up of GW190814, due to a combination of larger distances, poorer sky localization, suboptimal observing conditions, and exposure times. Useful constraints on the

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ejected mass (less than a few percent of a solar mass) could only be obtained for the deepest observations available at the closest possible distance; and even these only exclude nearly maximally rotating black holes as potential sources for the GW signal. This emphasizes the need for improved observing strategies if we want non-detection to provide meaningful constraints on the parameter of a binary for “average” BHNS candidates. Among the recently proposed improvements to survey strategies, we note a recommendation to modify the exposure times used by wide field surveys [206], a proposition to search for signals that decay rapidly (days) in optical band but over longer time scales (weeks) in infrared bands [211], and the possibility to combine constraints from follow-up of multiple events to better understand the luminosity of kilonovae [212]. While follow-up of the many candidate BHNS mergers in O3 did not result in any multi-messenger detection, the lessons learned from these campaigns should prove useful in optimizing observing strategies in the future.

Short Gamma-Ray Bursts Gamma-ray bursts (GRBs) are among the brightest events observable in the universe, powered by ultra-relativistic outflows that can be observed at cosmological distances. They have long been classified into two main populations: short gammaray bursts (SGRBs) typically defined as GRBs with 90% of the burst energy observed within a time T90 < 2 s and long GRBs with T90 > 2 s [213]. Long GRBs are preferentially observed in star-forming regions of galaxies, which motivated their association to core-collapse supernovae. A more direct confirmation for that theory then came from joint observations of long GRBs and supernovae (see, e.g., [214] for a review of the long GRB/supernova connection). SGRBs, on the other hand, are preferentially found offset from their most likely host galaxy [215]. From a decade of observations of SGRBs and their optical, X-ray, and radio afterglow, Fong et al. [19] find that SGRBs have inferred isotropicequivalent gamma-ray energy of Eγ ,iso ∼ 1050−53 erg. Large uncertainties in the beaming angle of the jet make the true energy budget of the SGRB more difficult to determine; the true gamma-ray energy of the burst is most likely 10−3 − 10−1 of the isotropic equivalent energy, with median beaming-corrected energies of ∼1050 ergs. Finally, the density of the medium in which the jet propagates inferred from these observations is fairly low, with median densities n ∼ (10−3 − 10−2 ) cm−3 . These observations were in general agreement with the theoretical idea that SGRBs are powered by BHNS and/or BNS mergers [216–221]. Compact object mergers most naturally explain the high-energy and short timescales of SGRBs, the time delay between SGRBs and star formation, and the fact that SGRBs are observed out of their presumed host galaxies (as supernova kicks can lead to compact binaries being kicked out of their host galaxy). The joint detection of a SGRB and GW signal produced by a BNS merger provided the final evidence that at least some SGRBs are powered by compact binary mergers [222]. This remains the only joint GWSGRB detection to date.

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There are a few interesting differences between SGRBs in BHNS and BNS mergers that are worth mentioning here. First is the fact that the polar regions in a BHNS mergers are more likely to be free of baryonic matter than in a BNS merger. The tidal ejecta of BHNS binaries is largely confined to the equatorial plane, while BNS merger may eject ∼(0.001 − 0.01)M in a largely isotropic manner at merger and form a neutron star remnant that continually drives winds from its surface and from the surrounding disk. Magnetically driven outflows launched during or immediately after disk formation could still lead to baryon loading of the polar regions after a BHNS merger, but not as much as in a BNS merger. Second, the remnant is guaranteed to be a black hole. Accordingly, one does not need to wonder whether a neutron star-accretion disk system can power a SGRB. And finally, the black hole spin and accretion disk angular momentum may be slightly misaligned. This could lead to observable precession of the jet and modulation of the SGRB signal [60]. Joint observations of GWs, SGRBs, and SGRB afterglows from BHNS mergers can provide us with additional information about the properties of the merging objects and the mechanism powering SGRBs. As usual, the easiest inference from a joint observation may be a lower bound on the size of neutron star, required for tidal disruption. One might however need to be a little bit more careful with this argument for SGRBs than for kilonovae: 1050 ergs ≈ (5 × 10−5 )M c2 , and thus if the SGRB engine is very efficient, it is in theory possible to power a SGRB with a very low-mass remnant. BHNS merger simulations cannot at this point guarantee that a “non-disrupting” BHNS merger is unable to produce a small accretion disk of ∼10−4 M . Multi-messenger observations of GWs and SGRBs from BHNS and BNS mergers should however provide us with information about the type of systems that power SGRBs, the efficiency of the jet production process, and possibly the time delay between merger and jet production. We can obtain additional information from X-ray, optical, and radio observations of SGRB afterglows: with sufficient data, it is possible to obtain accurate sky localizations, as well as information about the opening angle of the jet and the density of gas around the merger [19]. Radio observations of the afterglow of GW170817 even provide tentative evidence for (apparent) superluminal motion of the radio source, which would be a direct confirmation that the observed SGRB is actually powered by an ultra-relativistic collimated outflow [223]. From a theoretical and numerical point of view, there remain a number of important open questions about the physical mechanism that powers SGRBs. The most common model involves the formation of magnetically driven outflows from a black hole-accretion disk system produced as the result of a BHNS or BNS merger, resulting in the production of a jet in the nearly baryon-free environment likely to exist in the polar regions. There have been significant progresses in the numerical modeling of this process over the last decade (see [224] for a review). In the context of the post-merger evolution of BHNS binaries, we have in particular already discussed the existence of two proposed mechanisms for jet formation that have both obtained promising results for simulations initialized with large magnetic fields, but whose robustness to these chosen initial conditions remains uncertain [78, 108].

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Both scenarios lead to estimates for the energy of the jet compatible with SGRB observations when starting with strong poloidal fields [78, 108]. We refer the reader to the section covering the post-merger evolution of BHNS binaries for a more indepth discussion of this topic. Historically, collimated magnetically driven outflows have been just one of the two leading explanations for the production of SGRB, the other being outflows powered by energy deposition from neutrino-antineutrino pair annihilation in the baryon-free regions along the spin axis of the remnant black hole. Recent estimates of neutrino luminosities in mergers and the efficiency of the pair annihilation process indicate that neutrinos could reasonably power SGRBs up to about their median inferred energy, but not explain the full range of inferred energies. Indeed, neutrino luminosities peak at ∼1053 erg/s in BNS [225] and BHNS [226] mergers and decay over a (10 − 100) ms timescale [81]. A few percent of the neutrino energy is likely deposited in the polar regions due to pair annihilations [227]. The resulting energy deposition ∼1049−50 ergs is thus not too far from the median gamma-ray energy of SGRBs, indicating that pair annihilation may play a subdominant yet non-negligible role in SGRB production and/or help cleaning up the polar regions for the production of a magnetically driven jet. In the context of low-mass BNS mergers (with a neutron star merger remnant), Fujibayashi et al. [228, 229] find that pair annihilation plays an important role in heating matter outflows in the polar region, but is not sufficient to accelerate these outflows to the high Lorentz factors expected in a SGRB jet. In a BHNS merger, the neutrino flux immediately above the remnant will be lower, but so will the density of the baryonic matter polluting the polar regions. It is unknown at this point what the energy deposition from pair annihilation and asymptotic Lorentz factor of the associated outflows will be for a BHNS binary once neutrinos, baryonic matter, and magnetic fields are evolved self-consistently.

Other EM Counterparts to BHNS Mergers SGRBs and kilonovae are the most commonly discussed EM counterparts to BHNS and BNS mergers and the most promising for EM follow-up campaings. There are however a number of other potential EM signals associated with BHNS mergers that have been proposed over the years and that have the potential to provide valuable additional information under the right circumstances. We cover them briefly in this section.

Radio Emission from Mildly Relativistic Outflows We have already discussed in detail the mildly relativistic outflows produced in BHNS mergers and their role in r-process nucleosynthesis and the production of kilonovae. Over longer timescales, the acceleration of electrons at shocks created between the expanding outflows and the circummerger material leads to synchrotron

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emission from the ejecta [230]. The energy released by that process is set by the kinetic energy of the outflows and the timescale by their deceleration timescale. Radio emission from a disrupting BHNS merger is thus expected to show two distinct peaks. The first, 1 − 2 weeks after merger, is due to synchrotron emission from the ultra-relativistic jet material; the second, years to decades later, is due to the deceleration of the dynamical ejecta and disk outflows [231]. The amplitude and timescale of this radio signal are particularly sensitive to the density of the circummerger environment. For the densities n ∼ (10−3 − 10−2 ) cm−3 inferred from SGRB afterglows [19] and expected for binaries kicked out of their host galaxy, emission from the dynamical ejecta evolves over ∼10 yrs, while for a binary within a galaxy, it may evolve on a year timescale. Hotokezaka et al. [231] show that follow-up observations with LOFAR, JVLA, MeerKat, or ASKAP can only detect the most optimistic ejecta models in low-density environments (for mergers at ∼300 Mpc), while detection of the same systems is within reach of these detectors at higher circummerger densities (∼1cm−3 ).’ This is clearly a difficult measurement to make, but also one that provides us with relatively easy-to-interpret results (at least when compared to kilonovae and SGRBs): the total energy emitted provides us with a good estimate of the total kinetic energy of the ejecta. In BHNS merger, this is likely dominated by the best-modeled component of the outflows: the dynamical ejecta. Indeed, the dynamical ejecta has v ∼ (0.2 − 0.3)c, while later mass ejection most likely has v  0.1c. This synchrotron emission is thus much less sensitive to the currently large uncertainties in the mass of the post-merger outflows. Additionally, Kyutoku et al. [11] find that the asymmetry of the mass ejection in BHNS merger would lead to motion of the radio image by O(1) milliarcseconds over the time of the observations, allowing us to observe the motion of the ejecta’s center of mass. This could put very useful constraints on the asymmetry of the system and its BHNS / BNS nature.

Extended X-Ray Emission At least ∼20% of SGRBs are followed by ∼(10 − 1000) s of extended X-ray emission [232, 233], with peak emission multiple seconds after the burst. If these SGRBs are the result of compact binary mergers, explaining the timescale of the extended X-ray emission is challenging. One possibility is emission from a magnetar formed as the result of a BNS merger [234]; the lack of detection of late time radio emission from SGRBs with extended X-ray emission is however beginning to place some constraints on the magnetar model [235, 236]. A possible alternative is emission due to late fallback onto the post-merger remnant. In that model, an emission gap between early accretion and the peak of the extended emission is required to explain the time evolution of the observed signal. This could occur due to r-process heating of the ejecta, but only if the post-merger remnant is sufficiently massive, MBH > (6 − 8)M [116]. One requirement for this model to explain observations, however, is a rate of disrupting BHNS mergers comparable

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to the rate of SGRB-producing BNS mergers. With at most two disrupting BHNS mergers (S190426c, S200115j – the first with a high probability of being terrestrial and the second with a high probability for no disruption) and ∼5 BNS mergers observed so far (including O3 triggers), there are now some tensions between the model and observations.

Pre-merger Electromagnetic Signals Most discussions of EM counterparts to BHNS and BNS mergers focus on postmerger signals. There are a number of good reasons for this: SGRBs were already observed before any GW event was detected; the feasibility of performing searches for kilonovae based on GW triggers had been demonstrated before the detection of GW170817; many (but not all) pre-merger signals are significantly dimmer than SGRBs and kilonovae; and triggered searches are currently impossible for premerger signals. That last point could theoretically be alleviated for BNS mergers once current detectors reach their target sensitivity: the feasibility of an early alert system releasing triggers (10 − 60) s before merger has indeed been demonstrated [237]. This alert system however relies on the relatively large amount of time that BNS mergers will spend within the LIGO/Virgo frequency band (10 − 15 min). BHNS mergers are less likely to accumulate a large enough signal-to-noise ratio long enough before merger to provide early alerts. Considering that the expected number of detections of an EM counterpart to a BNS merger within seconds of the merger is already O(1) over the lifetime of current detectors [237], the observation of a pre-merger signal to a BHNS binary would nearly certainly require a signal observable by all-sky observatories, an extremely lucky serendipitous discovery, an alert system based on a BHNS detection by a lower-frequency space-based GW observatory, or third-generation ground-based GW detectors. There is however a very important advantage to pre-merger signals for BHNS mergers: they do not require disruption of the neutron star. We have already argued that there is a high likelihood that most BHNS mergers will be non-disrupting systems; pre-merger signals may provide the only avenue for EM observations of these systems. The brightest potential pre-merger signal proposed to date is the shattering of the neutron star curst following resonant excitation of tide-crust interface modes during the inspiral [238, 239]. This process could release ∼1047 ergs a few seconds before merger and was originally proposed as a potential mechanism for preburst flares observed seconds before a small fraction of SGRBs (∼0.4% [240] or 2.7% [241] of SGRBs, according to recent estimates). The time interval between the pre-merger signal and the merger itself would then provide information about the equation of state of neutron stars, as the frequency of resonant modes is very equation-of-state dependent. The exact properties of these pre-merger bursts remain however difficult to constrain: existing models provide an estimate of the total energy released by crust breaking, but not of the emission mechanism or the energy of the emitted photons. Schnittman et al. [242] take a more agnostic

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approach to the study of these pre-merger flares, assuming that pre-merger gammaray flares are powered by emission from the surface of a neutron star (in either a BHNS or BNS system). They demonstrate that relativistic beaming and Doppler effect lead to a modulation of the signal due to the orbital motion of the binary. The resulting signal should then have the same “chirping” evolution as the GW signal, possibly allowing for matched filtering searches in the future and thus deeper observations. The main alternative to emission from the surface of a neutron star is the interaction of the magnetosphere of the neutron star with the black hole. Using force-free simulations of the magnetic field outside of the neutron star, Paschalidis et al. [98] find emission of ∼6 × 1042 erg/s for a BHNS system a few orbits before merger, assuming a 1013 G magnetic field on the surface of the star (the luminosity scales as B 2 ). As for surface emission, the signal is expected to be modulated by the orbital motion of the binary. From the Poynting flux measured in the simulations, the system is expected to radiate within broad opening angles ∼40◦ − 60◦ . Very similar results have been obtained independently for emission from the interaction of the magnetosphere of two neutron stars [243]. Given the relatively low-energy emission from magnetosphere interactions before merger, pulsar-like coherent radio emission may be a more realistic target for pre-merger observations [244]. Most & Philippov [245] estimate that for BNS systems, coherent radio emission similar to fast radio bursts is at least conceivable given the magnetic field geometry observed in mergers. I am not aware of any detailed calculation of this emission in the case of BHNS binaries. Practically, the development of numerical algorithms for kinetic plasma simulations in neutron star mergers will be required before more accurate predictions can be made for the high-energy signals and coherent radio emission powered by BHNS and BNS mergers.

Conclusions The study of black hole-neutron star mergers is at an interesting crossroad today. The LVC is quite likely to have detected BHNS mergers during its third observational run (O3), yet so far no detection has been robustly confirmed. We saw in this chapter that this is not particularly surprising. Among compact binary mergers, BHNS mergers are the least likely to be unequivocally distinguished from other sources – either surprisingly massive BNS mergers or surprisingly low-mass BBH mergers. BHNS mergers are potentially powerful sources of multi-messenger signals: they can power SGRBs (and their afterglows), kilonovae, synchrotron radio emission, and/or pre-merger EM signals, and these signals can provide us with valuable information about the production site of heavy atomic nuclei, the properties of merging compact objects, and the equation of state of dense matter. As opposed to BNS mergers, which should relatively robustly power at least some EM signals,

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the likelihood that a given BHNS merger will be a multi-messenger source is highly dependent on the parameters of the binary. Post-merger EM emission requires tidal disruption of the neutron star during merger, which only occurs when merging with low-mass and/or rapidly spinning black holes. Additionally, finite size effects in the GW signal are negligible if the neutron star does not disrupt. Accordingly, if future detections of BHNS mergers by GW observatories will certainly provide us with useful information about the mass and spin distribution of black holes and neutron stars, the importance of BHNS mergers for nuclear physics, high-energy astrophysics, and the study of astrophysical nucleosynthesis hangs on the existence of BHNS mergers with specific parameters. Most of all, the existence of black holes within the mass gap 2M < MBH < 5M would be wonderful news for the prospect of multi-messenger astronomy with BHNS mergers. Before O3, the existence of such objects was doubtful. The latest GW triggers from the LVC however provided a surprisingly large number of candidate objects within the mass gap, and GW190814 confirmed the existence of ∼2.6M objects. As a result, the final analysis of O3 events and the next observational run of the LVC will nearly certainly play an important role in our view of BHNS mergers and of their future role for multi-messenger astronomy and nuclear astrophysics. If BHNS mergers turn out to be useful multi-messenger sources, we will soon rely on models of their GW and EM emission to extract information about the properties of the merging compact objects. The observation of GW signals, kilonovae, and radio transients (if detectable) is particularly useful in that respect: over the last decade, theoretical and numerical models have made significant progress toward ab initio modeling of these signals. Nevertheless, these models should be treated with caution: with the exception of GW models for low-SNR events, all models are likely to have important systematic biases. This is due to the extreme complexity of properly modeling the merger and post-merger evolution of BHNS binaries: accurate solutions would require the evolution of Einstein’s equations of general relativity, the equations of relativistic fluid dynamics, magnetic fields, neutrinos, nuclear recombination,... with numerical resolution capable of capturing the growth of MHD instabilities and over timescales sufficient to follow the lifetime of the post-merger remnant. No code is currently capable of performing this feat. Additionally, in the case of kilonovae, nuclear physics uncertainties (nuclei masses, reaction rates,...) add another potentially significant source of modeling errors. Accordingly, careful monitoring of potential modeling errors is crucial to the use of BHNS mergers in multi-messenger astronomy. Finally, we note that even in the absence of confirmed detection so far, O3 has already provided us with a useful practice run for follow-up of BHNS candidates and joint work between observers and modelers. Follow-up of the most promising BHNS candidates provided some bounds on the amount of mass ejected by the merger [195, 199, 204–206], as well as valuable information about the limitations of current follow-up methods and potential improvements for the next LVC observing run.

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Cross-References  Binary Neutron Stars  Effective Field Theory Methods to Model Compact Binaries  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range  Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravita-

tional Waves  Numerical Relativity for Gravitational Wave Source Modeling  Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspi-

rals  r-Process Nucleosynthesis from Compact Binary Mergers

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Dynamical Formation of Merging Stellar-Mass Binary Black Holes

16

Bence Kocsis

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BH Mergers in Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Processes in Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merger Probability in Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merger Rate Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Frequency and Eccentricity Distribution for Globular Clusters . . . . . . . . . . . . . . . . . . BH Mergers in Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mergers Driven by Binary-Single Encounters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Single Gravitational Wave Captures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mergers Triggered by the Kozai-Lidov Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Assisted Mergers in Active Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Observational Diagnostics of the Dynamical Channel . . . . . . . . . . . . . . . . . . . . . . . . . Mass, Spin, and Redshift Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Universal Gravitational Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observed Merger Fraction and Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The astrophysical origin of black hole mergers is one of the most important outstanding questions in gravitational wave astronomy. How do black holes find each other in vast space, form binaries, and get so close to one another that gravitational wave emission can successfully merge them within the present age of the universe? In this chapter we review the dynamical formation channel,

B. Kocsis () Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_15

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where the binary separation is reduced by dynamical processes. These processes are important for mergers in dense stellar clusters, galactic nuclei, mergers in supermassive black hole accretion disks, and mergers in stellar triple and quadruple systems and possibly in the dark matter halo. We introduce a unified framework to interpret the theoretical expectations on the characteristics of these merging binaries such as their gravitational wave frequency, eccentricity, mass, and spins to identify these gravitational wave source populations in merger catalogs. For dynamical merger pathways, we show that GW source catalogs carry information on the escape velocity of the host environment.

Keywords

Sources of gravitational waves · Black holes · Dense stellar systems · Globular clusters · Nuclear star clusters · Galactic nuclei · Astrophysical dynamics · Dynamical encounters

Introduction The recent detection of gravitational waves (GWs) shows evidence that black holes (BHs) merge frequently in the universe. Based on the 85 BH-BH merger events in the first 3 observing runs O1, O2, and O3 of the LIGO and VIRGO Scientific Collaboration, the BH merger rate density is estimated to lie in the range 17−45 Gpc−3 yr−1 [2, 3]. Based on the quickly growing sample of detections, it is clear that it will soon become possible to probe the demographics of BH-BH binaries in different environments in the universe [1]. The astrophysical origin of the detected mergers represents a major open question in gravitational wave astronomy [22]. Most of the observed stellar binaries in the universe are widely separated compared to the scales on which gravitational wave (GW) emission may efficiently reduce the binary separation. GWs reduce the binary separation on a characteristic timescale of tGW = 1018 yr (r/100 AU)4 (m/10 M )−3 , where r is the orbital separation and m is the binary component mass [103]. Clearly tGW is much larger than the age of the universe, unless some physical mechanism other than GW emission delivers the compact object binaries down to sub-AU scales. This is referred to as the final AU problem. Several mechanisms have been proposed to solve the final AU problem. One possibility is the isolated evolution of massive stellar binaries in galactic fields, if the progenitors form at a very small separation and undergo either common envelope evolution where one star completely engulfs the other in the red giant phase [45,71, 81, 82, 92, 129] or via stable mass transfer [62, 93, 141] or chemically homogeneous evolution where the near-contact binaries experience strong internal mixing before turning into black holes [28,79,84]. Another possibility, which is the subject of this chapter, is the dynamical channel, where the merger is facilitated by gravitational

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

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dynamics. Tight BH-BH binaries may form via direct dynamical capture [73, 95, 108], through exchange interactions in dense stellar environments, and may merge after repeated binary-single and binary-binary interactions [106, 127], or as a result of long-term perturbations of a third body or an anisotropic mass distribution if present in the binary’s proximity [4, 10, 27, 58, 88, 142]. In the following sections, we discuss the various environments in which dynamics may play an important role in merging black hole binaries and highlight the expected gravitational wave signatures to possibly distinguish the different source populations.

BH Mergers in Globular Clusters Globular clusters are dense stellar environments of 104 –106 stars where the number density is up to 105 × higher than in the galactic field, where dynamical encounters are sufficiently common that binary formation, hardening, and merger occur frequently [24]. These systems may host abundant sources of gravitational waves. Indeed, recent studies of the LIGO/VIRGO merger rate distributions combined with population synthesis model predictions found that at least 50% and up to 98% of the observed sources may originate dynamically [118, 143], and that 9%–60% may be from globular clusters specifically. In this section, we summarize the relevant dynamical processes, derive the black hole merger rates from first principles, and discuss observational probes of this merger channel. We highlight the observable quantities that may be predicted in a very robust way relatively independent of theoretical uncertainties.

Dynamical Processes in Globular Clusters We start with simple analytical estimates of the dynamical processes which shape the distribution of merging black hole binaries in globular clusters. Binaries may form dynamically during close encounters of three objects, in which one object takes away the initial kinetic energy of the system and is ejected from the system, leaving behind a bound binary [25]. In two-body encounters, the change in the velocity satisfies δv/v ∼ b90 /b where b90 = 2Gm/v 2 is the critical impact parameter of a large-angle (90◦ ) scattering, m is the average binary component mass, and v is the initial relative velocity which we approximate with 21/2 σ where σ is the velocity dispersion. The rate at which one star approaches a 2 given star with impact parameter b ≤ b90 is approximately nσss v, where σss = π b90 is the scattering cross section for single-single encounters, n is the number density. During such an encounter the probability to have a third object in the neighborhood 3 , implying that the rate of three single body encounters in a cluster of N stars is nb90 5 v, i.e. that may lead to binary formation is Γsss = π N n2 b90

664

Γsss =

B. Kocsis

25 π N n2 G5 m5 =3×10−7 yr−1 v9



v 20 km/s

−9

N 106



n 5 10 pc−3

2 

m 10M

5 . (1)

Equation (1) shows that massive stars and/or black holes may form ∼ 300 binaries with typical main sequence stars in 1010 yr. Note however, that the binary formation rate is extremely sensitive to the velocity dispersion of the host environment as σ −9 and the mass of the most massive  object participating in the encounter as m5 . The typical relative velocity is v ∼ 0.5 GN m/Rgc , where m and Rgc ∼ (3N/4π n)1/3 are the average stellar mass and half-mass radius. Thus, v = 0.6(Gm)1/2 N 1/3 n1/6 = 23

km s



Mgc 106 M

1/3 

n 105 pc−3

1/6 

m 0.3M

1/6 . (2)

The globular cluster’s mean density correlates with its mass Mgc = Nm roughly as [46]  ρ = mn = min 105 M pc−3 , 103 M pc−3



Mgc 105 M

2  ,

(3)

so that 6 km/s ≤ v ≤ 28 km/s for 105 M ≤ Mgc ≤ 106 M . The escape velocity may be related to the velocity dispersion σ analytically using the potential-density pair introduced by Stone and Ostriker [132]. In terms of the√typical relative velocity (v = 21/2 σ ) this gives approximately vesc ∼ 2v = 8σ . This implies that 10 km/s ≤ vesc ≤ 60 km/s for globular cluster mass between 105 M and 106 M (Eqs. 2 and 3) [47]. Also note that if the objects reach kinetic energy equipartition then we expect v = 12 (m/m)1/2 vesc , but since the results derived below depend mostly on vesc and are otherwise only weakly sensitive to v, and the observed systems are typically far from equipartition [100, 139], we shall neglect the mass dependence of v for simplicity. Binary formation is further facilitated by dynamical friction and core collapse, which causes the massive components in the cluster to settle to the core increasing the density therein and reducing their velocity dispersion [72, 127]. The dynamical friction timescale is (see Eq. 8.4 in [25]) tfric =

3  −1  −1  ns m 3 v v3 7 = 10 yr . √ 20 km/s 10M 105 pc−3 16 π G2 mmns ln Λ (4)

Here ln Λ is the Coulomb logarithm defined as ln Λ = ln[Rgc /b90 ] ∼ 10 where Rgc ∼ 3 pc is the size of the system. The binaries undergo frequent dynamical encounters with single stars or other binaries as shown next. If the incoming object is more massive than the binary components, such binary-single or binary-binary interactions result in an exchange

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

665

interaction, where the binary components are replaced by the two most massive objects participating in the encounter and the lowest mass object(s) is/are ejected. Thus, if stars are born in binaries in globular clusters, exchange interactions lead to binaries with higher mass stars and compact objects. Ultimately the most massive components pair up to form binaries in the cluster. Binary-single interactions also change the binaries’ separation. If a binary’s binding energy is larger than the mean kinetic energy of stellar objects in the cluster, it is defined to be a “hard binary” which tends to shrink after binary-single interactions [59]. The binary semimajor axis (separation) and mean relative orbital velocity of hard binaries v 2orb = 2Gm/a satisfy [61] 3 Gm2 a ≤ ah = = 2200 AU × 2 mσ 2  v orb ≥ v orb,h =



v 20 km/s

−2 

m 0.3M

−1 

m 10M

2 , (5)

4m σ = 0.14 v 3m



m 0.3M

1/2 

m 10M

−1/2 .

(6)

To approach within a distance a from the binary center of mass, where a is the binary separation, the timescale between successive binary-single interactions for each binary may be estimated as tbs = 1/(nσbs v), where the hard binary-single scattering cross section is √ σbs = 4 π (a 2 + ab90 )

(7)

[25], where the second term arises due to gravitational focusing. For hard binaries, the second term dominates. By substituting b90 =Gm123 /v 2 where m123 =2m+ms , the timescale between encounters per binary becomes −1    m123 −1  a −1 v v ns tbs = √ =4×107 yr 20 km/s 30M 1 AU 105 pc−3 4 π ns Gm123 a −1 −1  −1   2  m123 m v v orb ns 7 = 9.9×10 yr . 20 km/s 10M 30M 200 km/s 105 pc−3 (8) Binary-single interactions are very frequent initially when the separation is of order 103 AU, taking place once every 105 yr for each binary. As the binary shrinks, encounters become less frequent, but the time between encounters is always smaller than a Hubble time as long as a > 0.006 AU. The distribution function of binding energy after binary-single interactions has been determined for comparable mass scatterings [59]. During each binary-single hardening encounter, the binary separation is changed on average by a factor of δ = 7/9 for a single mass population. Numerical scattering experiments [147] show that

666

B. Kocsis +0.10 δbs = 0.84+0.21 −0.12 and δbb = 0.34−0.09

(9)

for binary-single and binary-binary scattering, respectively, see also Refs. [15, 75, 76]. For binary-single interactions a simple approximation for scatterer mass ms ≤ m and binary component mass m is δ =1−

2 ms . 9 m

(10)

Assuming a fixed ms , the binary separation decreases exponentially as a geometric sequence, an = ah δ n . The hardening timescale of such binaries with m ≥ ms is th =

n

max

tbs δ n ≈

n=0

v tbs = 1−δ GH ns ms a

(11)

√ √ where H = 4 π (m123 /ms )(1 − δ) = 89 π m123 /m. For ms m and ms =m, H = 3.2 and 4.7, respectively. Numerical scattering experiments give a faster hardening rate of H = 15–20 for ms m [107]. The backreaction of the change in the binary’s energy increases the kinetic energy of the scattering object. Furthermore, conservation of linear momentum implies that the center of mass of the binary receives a recoil kick velocity. The binary is ejected from the cluster if its recoil kick velocity exceeds the escape velocity from the cluster, vesc . The binary recoil velocity vbin may be estimated from energy and 2 + 1 m v 2 = (δ −1 − 1)Gm2 /a momentum conservation to be ΔEbs = 12 (2m)vbin 2 s s and 2mvbin + ms vs = 0, which gives  mms G 1 −1 2 δ 2(2m + ms ) vesc ⎧ 2  −1  −2  m vesc ms ⎪ −4 AU ⎪ 3 × 10 ⎨ 0.3M 10M 40 km/s =   −2  ⎪ v m esc ⎪ ⎩ 0.26 AU 10M 40 km/s 

aej =

(ms m) (ms = m) (12)

where in the second line we used the Eq. (10) for binary-single interactions. We shall find it useful below to represent the binaries with their relative orbital velocity averaged over an orbit v orb = (2Gm/a)1/2 , for which we get that at ejection  v orb,ej =

8m + 4ms δ vesc = ms 1−δ

    m m 9 2 1+2 − 2 vesc ms ms

(13)

Equation (12) suggests that binaries with m ms ∼ 0.3M will not be ejected only at separations less than a solar radius R = 0.005 AU, where the assumptions

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

667

break. However, massive BH binaries may be ejected by occasional encounters with other compact objects. Comparing Eqs. (2), (3), and Eq. (13), we find that v orb,ej increases with Mgc such that for 105 M ≤ Mgc ≤ 106 M and m ≥ mBH it ranges between 80

km m km m ≤ v orb,ej ≤ 370 . s mBH s mBH

(14)

These results apply for binary-single interactions. For binary-binary interactions v orb,ej is a factor of 2.6 smaller, see Eq. (9). Scattering encounters have five possible outcomes. 1. The binary is ejected, if its recoil kick velocity exceeds the escape velocity of the cluster. Much later the ejected binary merges due to GW emission outside the cluster. These sources are approximately circular when they reach GW frequencies characteristic of LISA and LIGO GW observations [98]. 2. The binary merges in the cluster due to GW emission between subsequent encounters [89]. These sources are typically eccentric for LISA but become circular by the time they reach the LIGO GW frequency band. 3. The binary merges in the cluster during a binary-single scattering encounter during a stable intermediate state when the third object is relatively far [112,120]. These binaries form in between the LISA and LIGO bands and have low eccentricities once they enter the LIGO band. 4. The binary merges in the cluster during the three-body encounter phase of the binary-single interaction. These binaries may form at high GW frequencies just below the LIGO band and remain eccentric in the LIGO band. 5. Objects can also merge following a close approach of two single objects, if the energy radiated in GWs exceeds the initial kinetic energy. These binaries in globular clusters form at a factor of 10 lower frequencies than the three-body mergers and so they have a smaller eccenticity in the LIGO band. We discuss the probabilities of these expected outcomes, and the corresponding GW frequency and eccentricity distribution in turn next.

Merger Probability in Globular Clusters Let us now derive the probabilities of the five outcomes of scattering encounters from first principles and then estimate the GW frequency and eccentricity distributions. We highlight how the results depend on the two most important quantities of the problem: the binaries’ component masses and the mean orbital velocities. Given the geometric sequence starting from ah (Eq. 5), the number of binarysingle interactions required to shrink the binary to aej is ntot = ln(aej /ah )/ ln δ ∼ 30. The ejected binaries are driven by gravitational waves to merger within [103]

668

tGW =

B. Kocsis

−3    a(1 − e) 4 5 c5 [a(1 − e)]4 m 10 g(e) = 1.6 × 10 yr g(e) 512 G3 10M 0.1 AU m3 (15)

where e is the binary eccentricity and g(e) = (1 − e)−1/2 (1 + e)7/2 [1 + (73/24)e2 + (37/96)e4 ]−1 for which g(0) = 1 for circular orbits. This shows that the initial periapsis distance after ejection rp = a(1 − e) must be of order 0.1 AU or less in order for the binary to merge within a Hubble time, and this is not satisfied for binaries in globular clusters with e = 0 and a ≥ aej as long as vesc ≤ 61 (ms /m) km/s as indicated by Eq. (12). A useful approximation for the GW merger time (15) is tGW ≈ tGWc (1 − e2 )7/2 ,

(16)

which is accurate to better than 15% for e < 0.9 and within a factor 2 for e → 1 where tGWc denotes the circular GW inspiral time tGWc =

5 c5 a 4 5 Gm = 512 G3 m3 32 c3



v orb c

−8 (17)

.

Here m denotes the binary component mass, assuming an equal mass binary. The eccentricity is typically far from zero. After each scattering encounter, the 2 eccentricity is drawn from a thermal distribution P (e) = √ 2e, for which e is uniformly distributed, and the median eccentricity is e = 1/ 2 = 0.7, reducing the GW merger time by a factor of 30 compared to circular orbits. Thus, a significant fraction of the ejected BH binaries are expected to merge within a Hubble time. The merger probability among ejected BHs outside the cluster from Eq. (16) is roughly  pout =

2 1 − ecrit,ej

≈

tH tGWc

2/7



m = 0.16 10M

−2/7 

v orb,ej 200 km/s

16/7 (18)

where ecrit,ej is the minimum eccentricity for which tGW ≤ tH = 1010 yr for the ejected binaries. The merger probability among ejected BHs is weakly sensitive to mass but it is very sensitive to the orbital velocity at ejection (Eq. 14), implying that for 105 M ≤ Mgc ≤ 106 M and m = mBH , 0.02 ≤ pout ≤ 0.64. For heavier BHs, pout ∝ m2 since v orb,ej ∝ m. The overall binary merger probability outside the cluster is Pout = pout Pej where Pej is the probability for the BH binary to be ejected, which is determined in Eq. (27). Let us now turn to the second possibility, in which the binary merges already inside the cluster due to GW emission. This happens if tGW ≤ tbs . For a comparable mass scatterer, the binary eccentricity after each binary-single interaction is drawn randomly from the thermal distribution dp/de = 2e, for which e2 is uniformly distributed. The probability that the eccentricity becomes sufficiently large e ≥ ecrit,2b to satisfy tGW (a, e) ≤ tbs after a single encounter such that the encounter

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

669

leads to a GW-driven merger before the subsequent binary-single encounter is from Eq. (16)  p2b =

2 1 − ecrit,2b

=



tbs

2/7

tGWc

m = 0.048 10 M

−6/7 

 =



2/7 4 vv 10 orb = √ 15 π G3 c5 nm3 2/7  20/7 v v orb . 30km/s 200km/s (19)

128 G2 m2 v √ 15 π c5 na 5 −2/7 

nBH 105 pc−3

2/7

Here we used the fact that e2 is uniformly distributed for a thermal eccentricity distribution between 0 and 1, and substituted Equations (8), (12), and (15) in the limit of ecrit,2b ∼ 1 and m = ms . The 2b index labels two-body mergers, to distinguish from three-body mergers discussed below. As the binary separation a changes from ah to aej , the mean orbital velocity changes from v orb,h to v orb,ej and the probability of merger for a single interaction for Mgc = 106 M changes typically from ∼ 10−6 to 0.21, increasing by a factor of δ −10/7 = 1.43 during each scattering event with a comparable mass scatterer. The time between binarysingle scatterings increases with decreasing binary separation or with increasing v orb (Eq. 8), but this is typically still smaller than 10 Gyr, all the way to ejection for m ≥ 1.2 M . In the following we will denote the merger probability during the nth binary-single scattering by p2b,n = δ0−n p2b (v orb,h ) ≈ δ0n p2b (v orb,ej ) where δ0 = δ 10/7 . The probability that the binary merges in any one of ntot = ln(aej /ah )/ ln δ binary-single encounters from an initial separation ah to aej equals 1 minus the probability that the binary avoids merger after each interaction, which we may write as n  ntot tot 

P2b = 1 − [1 − p2b,n ] = 1 − exp ln[1 − p2b,n ] (20) n=0



≈ 1− exp −

ntot

n=0

 p2b,n +

n=0

2 p2b,n

2



   2 −p2b,ej −p2b,ej exp ≈1− exp 1 − δ 10/7 2(1 − δ 20/7 ) 

where −1/7  −2/7    v orb,ej 22/7 p2b,ej nBH m = 0.15 10 M 200km/s 1 − δ 10/7 105 pc−3

(21)

and this expression approximates P2b in the limit P2b 1. In Eq. (20) we have expanded ln(1 − p2b ) in a Taylor series in p2b to second order, substituted the formula for a finite sum of the geometric series and neglected terms of order δ01+ntot and δ02+2ntot , as well as multiplicative factors of δ (10/7)frac(ntot ) if ntot is not an

670

B. Kocsis

integer (here frac(ntot ) = ntot − floor(ntot )). In the last equality we used Eq. (10), 2 so that 1 − δ 10/7 = 10 7 × 9 ms /m for m ≥ ms and assumed ms = 10M and 1 1 v ∼ 2 vesc ∼ 12 (ms /m) v orb,ej (see Eqs. 10 and 13). For 105 M ≤ Mgc ≤ 106 M , we get 0.02 ≤ P2b ≤ 0.53. The large variations with Mgc is caused by the extreme sensitivity on the orbital velocity at ejection which is proportional to the cluster’s escape velocity (Eq. 13). The two-body merger probability depends on mass and the globular cluster properties almost exclusively through v orb,ej ∝ mvesc (Eq. 13); it 3 . increases with BH mass and the cluster escape velocity roughly as P2b ∝ m3 vesc Up to this point, we assumed that a binary-single interaction randomizes the eccentricity once per interaction and neglected the possibility of merger during the three-body interaction. However a strong binary-single interaction consists of multiple scattering and intermediate episodes where two of the three objects form an intermediate state (IMS) binary . Numerical experiments show that the number of such episodes is NIMS = 20 [120]. The eccentricity of the IMSs is drawn from a thermal distribution f (e) after each of these scattering episodes. This leads to the possibility of a GW capture and merger in any one of these NIMS intermediate states. For encounters with three objects of total mass m123 = m1 + m2 + m3 , the 3/2 lifetime of each IMS is tIMS = 2π(Gm123 )−1/2 aout , where aout is the semimajor axis (separation) of the outer orbit during the IMS. The probability of merger during one IMS is  2 pIMS = 1 − ecrit,IMS =

tIMS tGWc

2/7 =

4 π 2/7 1/7  aout 3/7 q a 52/7 out



v orb c

10/7 (22)

where qout = (m1 + m2 )/(m123 ) is the mass ratio of the inner to the outer binary. The probability of a merger during any one of NIMS number of IMSs during the nth binary-single interaction episode, pIMS,n , is one minus the probability of no mergers during all NIMS independent trials, i.e.  pIMS,n =

2 1 − (ecrit,IMS )NIMS

 ≈ 1 − exp −NIMS

=1− 1−



tIMS tGWc



tIMS tGWc

2/7 NIMS

2/7 

(23) 

≈ NIMS

tIMS tGWc

2/7 = NIMS pIMS .

Here the last line is valid if much smaller than unity, which is justified for nonrelativistic binaries, v orb c. Note that the final result is independent of the globular cluster parameters, the mass of the triple, and the details of the initial binary-single encounter other than the binary’s mean orbital velocity, as expected due to the scale-free nature of general relativity. To give a sense of pIMS , let us assume that aout /a ∼ 100. Then for 105 M ≤ Mgc ≤ 106 M , the relative probabilities range between 110 ≤ pIMS,n /p2b ≤ 410 initially at v orb = v orb,h and 0.001 ≤ pIMS,n /p2b ≤ 0.02 at v orb,ej if ejection is due to comparable mass scatterers. This shows that the probability of merger during one of NIMS

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

671

intermediate states with the same triple is initially much higher but ultimately smaller than in between binary-single scattering episodes, due to the different scaling with v orb . The total merger probability during all IMSs among multiple binary-single scattering episodes may be estimated as in Eq. (20): one minus the probability of no mergers during ntot number of independent scattering episodes: PIMS = 1 −

ntot 

[1 − pIMS,n ] = 1 − exp

n=0



≈ 1 − exp −

ntot

 ln[1 − pIMS,n ]

n=0

 pIMS,n +

n=0

n tot

2 pIMS,n



2



  2 −pIMS,ej −pIMS,ej exp ≈ 1 − exp 1 − δ 5/7 2(1 − δ 10/7 ) 

≈

 10/7 v orb,ej pIMS,ej NIMS 1/7 m  aout 3/7 q ≈ 0.08 20 out ms 100a 200 km/s 1 − δ 5/7

(24)

Thus, the overall IMS merger probability is dominated by the conditions of the binary before the final ejection. The result is independent of the binary mass and all other binary and globular cluster parameters other than v orb,ej . For 3 ≤ aout /a ≤ 100 and globular cluster masses between 105 M and 106 M , we get respectively, 0.005 ≤ PIMS ≤ 0.20 and 0.05 ≤ PIMS ≤ 0.20 for the most common BH masses. For heavier BHs, the IMS merger probability increases roughly as PIMS ∝ (m/ms )17/7 . Let us now examine the merger probability during all three-body scattering events. There are NIMS number of such events during each binary-single scattering episode, each lasting for approximately the orbital time of the inner binary torb = 2π(2Gm)−1/2 a 3/2 . Thus three-body merger probabilities are given by the formulae describing two-body IMS mergers with aout /a = 1 and qout = 1, i.e.  v orb 10/7 , p3b,n ≈ NIMS p3b , c    10/7 v orb,ej π 2/7 4NIMS v orb 10/7 NIMS m = 2/7 ≈ 0.01 c 20 ms 200 km/s 5 1 − δ 5/7 (25)

2 p3b ≈ 1 − ecrit,3b =

P3b ≈

p3b,ej 1 − δ 5/7

4 π 2/7 52/7



For 105 M ≤ Mgc ≤ 106 M , this leads to 0.003 ≤ P3b ≤ 0.03 for the most common BH masses. For heavier BHs, the three-body merger probability increases as P3b ∝ m17/7 . Since the in-cluster mergers in the three cases are independent and the probability of in-cluster mergers is one minus the probability of not having any of the three

672

B. Kocsis

cases: Pin = 1−(1−P2b )(1−PIMS )(1−P3b ) ≈ 1−e−(P2b +PIMS +P3b ) ≈ P2b +PIMS +P3b , (26) where the approximations are valid if Pin 1, and the probability of ejection is Pej = 1 − Pin = (1 − P2b )(1 − PIMS )(1 − P3b ) .

(27)

Assuming that aout /a = 100 and Mgc = 106 M , we find that Pin = 59% and 78% for m = ms = 10M and m = 3ms = 30M , respectively, and the relative rate of the three in cluster merger processes are (P2b , PIMS , P3b )/(P2b + PIMS + P3b ) = (70%, 26%, 4%) and (50%, 42%, 8%), respectively. For Mgc = 105 M , the incluster merger probability is much lower: Pin = 4% and 10% for m = ms = 10M and m = 3ms = 30M , respectively. Close approaches during encounters of two single objects in a globular cluster may also lead to GW-driven mergers if the energy radiated in GWs is larger than the initial kinetic energy of the binary [96]. For equal mass objects, the typical time between mergers may be estimated using the GW capture cross section σGW as 1 where bGW = = tss = 2 v nσGW v π nbGW 1



85π 3

1/7

2Gm  v −9/7 . c c2

(28)

Thus, 1 tss = 4π



3 85π

2/7

v 11/7 c10/7 . G2 m2 n

(29)

The relative rate of single-single captures relative to three-body mergers for a given object per unit time may be estimated using Eqs. (8), (25), and (29) as −1 Γss Ns tss 0.3Ns ∼ = −1 Γ3b N Nbin p3b,n tbs bin NIMS



v orb v

4/7 .

(30)

Here Nbin /Ns = fbin is the number of binaries with mean relative orbital velocities v orb compared to the number of singles. The ratio v orb /v varies between 0.14 and 13 as v orb increases from the hard-soft boundary (Eq. 6) to the value at ejection (Eq. 12), but the three-body merger probability is dominated by its value at v orb,ej . Here Γss /Γ3b ∼ 7%/fbin for a binary fraction fbin = Nbin /Ns . The total rate of singlesingle GW capture mergers may be comparable to or exceed the three-body merger rate if the binary fraction is lower than 7%. Observationally, the instantaneous binary fraction of low-mass main sequence stars ranges from less than a percent to 40%. However the binary fraction of black holes is expected to be much higher, due to exchange interactions.

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

673

Furthermore, we note that the initial eccentricity and GW frequency of singlesingle captures are more similar to the two-body IMS mergers, as we show below, and the probability of those is enhanced compared to three-body mergers by a 1/7 factor of qout (aout /a)3/7 , i.e. ∼ 7 for aout ∼ 100a. This suggests that single-single captures will be a subdominant source of GW sources in globular clusters. The simple analytical estimates derived above are in good agreement with stateof-the-art Monte Carlo simulations which include post-Newtonian dynamics. These show that of order 50% of globular cluster mergers happen inside the clusters and the mergers are dominated by the massive Mgc ∼ 106 M clusters [112, 121] (c.f. discussion below Eq. 26). Finally note that black hole mergers may be facilitated in hierarchical and nonhierarchical triples in globular clusters where the binary system is perturbed by a bounded outer third object at moderate to large semimajor axis aout compared to the semimajor axis of the inner binary [7, 15, 85, 142]. In these cases the long-term effect of the third object leads to the so-called Kozai-Lidov effect: an oscillation in inclination and eccentricity while the semimajor axis is fixed. These sources have been found to also produce eccentric mergers in numerical simulations, where the eccentricity remains detectable for LIGO for 20% of the sources [7,38,85,142]. We discuss the characteristics of Kozai-Lidov-triggered mergers in Section “Mergers Triggered by the Kozai-Lidov Effect.”

Merger Rate Density Given the merger probability per BH for a particular channel P , the mean rate of black hole mergers may be estimated directly for a given mean number of black hole binaries per globular cluster NBHBH,bin and given globular cluster number density in the universe ngc as Rgc = NBHBH,bin

ngc Pmerger . tH

(31)

For a simple estimate of NBHBH,bin , let us assume that the initial fraction of stars −2.3 dm for m > 0.5M , between [m, m + dm] is f∗ (m)dm = 0.0795M−1   (m/M ) and stars more massive than 20M turn into black holes. The normalization factor accounts for  ∞the Kroupa [70] initial mass function of stars with mass m ≥ 0.01M , such that 0.01M f∗ (m)dm = 1. This implies that the number fraction of BHs is fBH =

NBH = N∗



∞ 20M

f∗ (m)dm = 1.1 × 10−3

(32)

Assuming that the number of stars per globular cluster is N∗ = 106 and the number density of globular clusters is ngc = 0.77 Mpc−3 [114] and labeling the probability of remnants to become members of a binary with fbin , we get

674

B. Kocsis

Rgc =

ngc 1 fBH N∗ fbin Pmerger = 43 Gpc−3 yr−1 × fbin Pmerger . 2 tH

(33)

Here the factor 1/2 accounts for two components in a binary. In summary, the merger rate depends on the binary fraction and the merger probability. If fbin = 20% of the black holes become binary members and Pmerger = 50% of them merge then Rgc ∼ 4 Gpc−3 yr−1 , and the upper limit for one merger per black hole with fbin = 100% and Pmerger = 100% is Rgc ≤ 43 Gpc−3 yr−1 . Recent state-of-the-art Monte Carlo simulations which include post-Newtonian dynamics show that around 50% of globular cluster mergers happen inside the clusters, and about 10% among those sources may have a measurable eccentricity with LIGO/VIRGO, particularly for three-body mergers as we show below [112, 112, 121]. Detailed Monte Carlo simulations of globular clusters yield black hole merger rates of Rgc ∼ 5 Gpc−3 yr−1 [18, 115]. However these estimates are conservative in that they use the present-day observed globular cluster density in the universe ngc = 0.77 Mpc−3 [114] and neglect the black hole mergers from globular clusters that have evaporated by now. This may increase the rate estimates by a factor 3 [34, 113]. In comparison, based on LIGO-VIRGO measurements, the BH merger rate density at low redshift (z ∼ 0.2) is estimated to lie in the range 17.3−45 Gpc−3 yr−1 [2] showing that the globular cluster channels may contribute significantly to the detected merger rates.

GW Frequency and Eccentricity Distribution for Globular Clusters After each strong binary-single scattering event, the initial eccentricity is drawn randomly from a thermal distribution where e2 is uniformly distributed. Mergers 2 . Thus, the take place if e ≥ ecrit , such that the merger probability is pmerger = 1−ecrit initial eccentricity and maximum initial periapsis distance leading to merger statisfy   pmerger 1 tlife 2/7 =1− (34) 2 2 tGWc    apmerger Gm tlife 2/7 = 2 = a(1 − ecrit ) = a(1 − 1 − pmerger ) ≈ 2 v orb tGWc (35)

ecrit = rp,crit



1 − pmerger ≈ 1 −

where pmerger resembles the merger probability during individual encounters pout , p2b , pIMS , or p3b , respectively, and the approximation is valid if pmerger 1, and the probability of merger was related to the relevant timescales, i.e. the lifetime of the binary in between perturbations (i.e. tH , tbs , tIMS , or torb for the four cases, respectively) compared to the circular GW merger time. These quantities depend on the mean relative orbital velocity v orb of the binary, see Eqs. (18), (19), (22), and (25).

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

675

The initial eccentricity of merging binaries is typically close to unity for the twobody, IMS, and three-body channel mergers (Eq. 22), higher than 2/7 4 vv 10 orb ecrit,2b √ tGWc 15 πG3 c5 nm3 −6/7  −2/7  2/7  20/7  nBH v v orb m = 1−0.024 10 M 30km/s 200km/s 105 pc−3 (36) 1 ≈1− 2

ecrit,IMS ≈ 1 −

ecrit,3b

1 2

1 ≈1− 2







2/7

tbs

tIMS tGWc torb

1 =1− 2

2/7

2 π 2/7 1/7 q 52/7 out

=1− 2/7

tGWc



2 π 2/7 = 1 − 2/7 5





v orb c

v orb c

10/7 

10/7

aout 3/7 , a 

v orb = 1 − 1.8 c

(37) 10/7 . (38)

For single-single GW captures with initial relative velocity v, the orbit becomes bound with e < 1 due to GW dissipation following a close encounter, which may be calculated from the critical initial periapsis distance leading to capture as [95] rp,crit,ss =

ecrit,ss

2 v2 bGW = 4Gm



85π √ 24 2

2/7

rp,crit,ss v 2 =1− =1− 2Gm



2Gm  v −4/7 , c c2 85π √ 24 2

(39)

2/7   v 10/7 v 10/7 = 1 − 1.8 . c c (40)

Interestingly, the minimum initial eccentricities of merging binaries in the threebody and the single-single channel are independent of their mass and the properties of the host environment other than the relative velocity of the particles when they are scattered into merging trajectories. Comparing with Eq. (38), it is reassuring that for practical purposes ecrit,ss = ecrit,3b if v orb is replaced by v despite the different approximations leading to these results. This is expected as the close approach leading to a merger during the three-body encounter also represents a GW capture where the initial relative velocity is v orb . Since in both cases v orb c and v c, this shows that mergers may be triggered only for very high initial eccentricity close to unity for both the three-body channel and for single-single captures. Based on the arguments above, the cumulative probability distribution function of the initial eccentricity of merging binaries is the truncated thermal distribution pth (e0 > e) =

1 − e2 pmerger = 2 1 − ecrit



1 − e2 if e2 ≥ 1 − pmerger , 0 otherwise .

(41)

676

B. Kocsis

Remarkably, the cumulative distribution function of initial eccentricity for a given binary with fixed parameters (m, v orb , n, v) is universal for all of the merger channels in isotropic systems: two-body, IMS, three-body, and single-single mergers, they are all described by the thermal distribution. The differences in the distributions are limited to the eccentricity of truncation. This universality is manifested also in the distribution after multiple interactions. In particular, for IMS and three-body mergers there are Nint trials, so in this case the eccentricity distribution after a full binary-single episode follows from arguments similar to that used for Eq. (23): pIMS,n (e0 >e)=1 − (1 − pth (e0 >e))NIMS ≈1 − e−NIMS pth (e0 >e) ≈NIMS pth (e0 > e) . (42) Now add up the contributions of all binary-single interactions for a given binary for the mergers in between encounters: P2b (e0 > e) = 1 −

ntot 

(1 − p2b,n (e0 > e)) ≈ 1 − e−Nbs (e)(1−e ) . 2

(43)

n=0

Hence ln P2b (e0 < e) ≈ −Nbs (e)(1 − e2 ) .

(44)

Here Nbs (e) is the number of binary-single interactions for which a merger was possible at critical eccentricity lower than e. This function is 0 for e2 < 1 − p2b,ej . Since the process is a geometric series p2b,n = p2b,ej δ0−n , Nbs (e) = ln[(1 − e2 )/p2b,ej ]/| ln δ0 | where δ0 = δ 10/7 . We derive PIMS and P3b similarly where δ0 is replaced by δ 5/7 . Thus, the cumulative distribution functions of initial eccentricity of merging systems in the 2b and IMS channels satisfy ln P2b (e0 < e) ≈ (1 − e2 )

ln(1 − e2 ) − ln(p2b,ej ) 10 7

ln δ

if 0 > e2 > 1 − p2b,ej (45)

ln PIMS (e0 < e) ≈ NIMS (1 − e2 )

ln(1 − e2 ) − ln(pIMS,ej ) 5 7

ln δ

if e2 > 1 − pIMS,ej (46)

These results explain the eccentricity distribution derived using Monte Carlo simulations as we will discuss, see Figure 2 below. The GW spectrum of an eccentric orbit is peaked at angular frequency ωGW = 2πfGW close to the angular velocity at periapsis [95]. This is roughly ωp = vp /rp , where for a highly eccentric orbit the velocity at periapsis is vp ≈ (2Gmtot /rp )1/2 . This follows from energy conservation 12 vp2 = 12 va2 − Gmtot /ra + Gmtot /rp ≈ Gmtot /rp . Therefore the GWs are emitted in bursts during close approaches where

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

677

the peak frequency of the bursts is roughly constant, given by the periapsis distance, until the eccentricity is no longer closer to unity. More precisely, the GW frequency peaks at [142]

fGW

(1 + e)1.1954 1 = 2fp , where fp = 2π (1 + e)1.5



2Gm rp3

1/2 .

(47)

Unlike for circular orbits where fGW is narrowly peaked at 2fp , for e ≈ 1 the waveform is broadband, with significant power within the range 12 fGW ≤ f ≤ 2fGW [140]. In this case, the initial GW frequency may be approximated by fGW0 ≈ 1.7fp0 , where

fp0

1 = 2π



2Gm

1/2

3 rp0

1 = 2π



 v 3orb (1 − e0 )−3/2 . 2Gm

(48)

For all sources with a given v orb , the fGW0 distribution vanishes at a lower cutoff frequency that corresponds to eccentricity ecrit (Eq. 34). For all the processes considered above, the merger probability increases strongly with v orb until a maximum v orb,ej where the binary is ejected from the cluster. We shall assume arbitrarily that the peak of the fGW0 distribution of all binaries in a cluster equals approximately fGW0 ≈ 2fp0 (ecrit , v orb,ej ). We get

fGW0,out

√ 3    −3/7   2Gm −4/7 tH v orb,ej −3/7 tH 8 v orb,ej 1 ≈ =√ π 2Gm tGWc,ej 10 c c3 8π = 4 × 10−5 Hz

fGW0,2b

v orb,ej 200 km/s

−3/7 

m 10M

−4/7 (49)

,

√ 3  3/7   tbs,ej −3/7 8 v orb,ej nBH −4 ≈ = 3 × 10 Hz π 2Gm tGWc,ej 105 pc−3  ×

fGW0,IMS



v 30 km/s

−3/7 

v orb,ej 200 km/s

−9/7 

m 10M

2/7 ,

(50)

√ 3       −3/14 a 9/14 v orb,ej 6/7 c3 8 v orb,ej tIMS,ej −3/7 53/7 qout ≈ = √ π 2Gm tGWc,ej c 8 π 10/7 2Gm aout −1  6/7   v orb,ej m aout −9/14 = 0.15 Hz , (51) 200 km/s 10M 100 a

678

B. Kocsis

fGW0,3b

fGW0,ss

√ 3     v orb,ej 6/7 8 v orb,ej torb,ej −3/7 c3 53/7 ≈ =√ π 2Gm tGWc,ej c 8 π 10/7 2Gm −1 6/7   v orb,ej m = 2.6 Hz , 200 km/s 10M 1 ≈ π



2Gm

3 rp0  = 0.4 Hz

1/2 =

v 20 km/s

(52)

√ c3  v 6/7 33/7 8 853/7 π 10/7 2Gm c

6/7 

m 10M

−1 .

(53)

In each case, we assumed ms = m for the final expression. The results show that the mergers that happen outside the cluster following ejection form near the lower frequency limit of LISA above 0.03 mHz, the in-cluster mergers between scattering episodes form in the LISA band above 0.3 mHz, and the mergers that take place during the scattering episodes either in IMSs, three-body, or single-single encounters form in between the LISA and LIGO band [29, 124, 125]. The GW frequency at formation carries information about the binary characteristic velocity of the host environment. Higher orbital velocities lead to larger initial GW frequencies for mergers during scattering episodes (IMS, 3b, and ss) and fGW0 decreases with v orb,ej for in-cluster two-body mergers in between scattering episodes and for the mergers outside of the cluster. Note that in the limit v orb = v, three-body and single-single encounters are nearly identical in terms of the fGW0 and initial eccentricity (Eqs. 38, 40, 52, and 53). The typical initial GW frequency for single-single encounters is lower because the typical relative velocity v is much smaller than the orbital velocity at ejection v orb,ej typically by a factor of √ 2 42 and 12m/ms for m = ms and m ms , respectively, see Eq. (13). Thus, fGW0,3b /fGW0,ss = (v orb,ej /v)6/7 ∼ 8.4 for m = ms . Since three-body mergers form relatively close to the LIGO band (fLIGO ≥ 10 Hz), the residual eccentricity when the sources reaches the LIGO band may be significantly nonzero, which carries information about the formation process. Let us now estimate the residual eccentricity when a given GW detector detects it at GW frequency fdet . For an isolated binary, its periapsis and eccentricity evolve such that [103] rp (e, e0 , rp0 ) = rp0

e12/19 1 + e0 [1 + (121/304)e2 ]870/2299 ≈ 1.7rp0 e12/19 12/19 1 + e [1 + (121/304)e2 ]870/2299 e0 0 (54)

where rp = a(1 − e) is the periapsis distance and rp0 = a0 (1 − e0 ) is its initial value, and the final approximation assumes e0 ≈ 1 and e 1. This shows that the periapsis distance changes by only 50%, while the eccentricity changes from a value close to 1 down to 0.2, after which it follows rp ∝ e12/19 . This makes the

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

679

peak GW frequency (Eq. 47) to increase by a factor 2.6 in the highly to moderately eccentric phase passing from e = 1 to 0.2, and later it increases further as fGW ∝ −3/2 rp ∝ e−18/19 . Thus, once the GW frequency is lower than a factor ∼ 3 of its initial value fGW0 , the eccentricity scales approximately inversely with fGW as the binary shrinks due to GW emission. The fact that the evolution follows a roughly constant periapsis in the highly eccentric phase, while the semimajor axis changes by orders of magnitude, is due to the fact that energy dissipation rate due to GWs scales very steeply with relative velocity, 12 mv 2 /tGW ∝ (v/c)10 (see Eq. 17), and hence with separation. Thus, for highly eccentric orbits, there is approximately no energy or angular momentum loss due to GW emission along the orbit except for the short duration close encounters near periapsis. There r ∼ rp is fixed, but the particle loses kinetic energy, or orbital speed impulsively. Conservation of angular momentum and energy in between close approaches ensures that the particle comes back to the same rp and vp that it had immediately after the previous close approach. Using this evolutionary sequence (Eq. 54), the residence time at different rp for binaries initially shrinking due to binary-single scatterings and later by GW emission, and the relationship between the fGW and rp , we may calculate the instantaneous GW frequency distribution of sources in the cluster. The distribution −3 function follows P (fGW ) ∝ fGW above fGW0 for the merging population in the −2/3 −34/9 LISA frequency band [124] and a broken power law with fGW and fGW , respectively, at low and intermediate frequencies in the LISA band for the nonmerging populations [124]. Figure 1 shows the expected initial and present-day distribution of GW frequencies in a random sampling of binaries from the various channels, as adopted from [123]. We introduce two quantities that characterize the eccentricity in the detector: er when the binary periapsis distance from the center of mass reaches a given value; we

Fig. 1 The initial and present-day distribution of gravitational wave frequency for merging binaries in globular clusters. (Adopted from Refs. [123, 124]. The eccentricity of the inspiraling binary at a fixed GW frequency fdet fGW0 is roughly ef = fGW0 /fdet ; see Eq. (56))

680

B. Kocsis

will use r10M = 10GM/c2 where M = 2m is the binary’s total mass, and the eccentricity e10Hz at a given frequency for LIGO detections, fdet = 10 Hz. Here, e10M may be derived from Eq. (54) by finding e for which rp (e; e0 , rp0 ) = 10GM/c2 for given e0 and rp0 and e10Hz is given by solving fGW (e, rp (e; e0 , rp0 )) = fdet for e (Eqs. 47). In the limit where e0 ≈ 1 and e < 0.2,  e10M =

17 GM c2 rp0

19/12



17 v 2orb ≈ 2 c (1 − e0 )

19/12

 =266

v orb c

19/6 

tlife

−19/42

tGWc

, (55)

 e10Hz ≈

fGW0 1.7 fdet



19/18 ≈

19/18  √ 3  8 v orb tlife −19/42 . (1.7 fdet )(2π Gm) tGWc

(56)

Here tlife is the lifetime of the binary in the different phases tH , tbs , tIMS , and torb , respectively for the various processes. For practical purposes the eccentricity at entering the LIGO band is ef ≈ 0.6fGW0 /fdet where fGW0 is given by Eqs. (49)– (52). Clearly ef /er = const × m19/18 , where the constant does not depend on the binary parameters. The cumulative distibution function of er and ef may be obtained from the cumulative distribution function of e0 , (Eqs. 45–46). The minimum detectable eccentricity for Advanced LIGO at design sensitivity is e10Hz = 0.02 − 0.08 [48, 77]. The peak distribution of e10Hz and e10M may be obtained for the five merger channels in globular clusters by substituting Eqs. (49)– (53) in (55)–(56). We find that the eccentricity is significantly nonzero for a significant fraction of three-body mergers, since in this case the distribution peaks at −19/18  19/21 v orb,ej m . 10M 200 km/s  19/21   v orb,ej 19/21 v orb,ej = 37 = 0.048 c 200 km/s 

e10Hz,3b = 0.14 [6pt]e10M,3b

(57) (58)

√ Note that since v orb,ej is between 42vesc and 6(m/ms )vesc for m ≥ ms (Eq. 13, thus e10Hz,3b is roughly independent of the mass of the merging binaries: e10M,3b = 0.14(vesc /30 km s−1 )19/21 if the scattering is due to 10M BHs. The measured eccentricity distribution will provide a clean measurement of the escape velocity of the source environment, if v orb ≥ 100 km/s, or escape velocities larger than 17 km/s. Figure 2 shows the eccentricity distribution from Monte Carlo simulations. Both the in-cluster and ejected merging sources have an initially thermal distribution of e0 , but the distributions for frequencies above 10−5 Hz is significantly different. The derivation presented above shows that this is due to the differences in the distribution of pth (e > e0 ) and P2b (e > e0 ) in Eqs. (41) and (45). The eccentricity of in-cluster mergers is skewed toward higher values due to the merger condition requiring

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

681

Fig. 2 The distribution of eccentricity for merging binaries in globular clusters, adopted from [69]. The left, middle, and right panels show, respectively, all binaries, those in the GW band of LISA, and those which have a LISA signal-to-noise larger than 2

an inspiral time within subsequent binary-single interactions and because of the repeated trials during the series of hardening encounters. The highest eccentricity tail corresponding to three-body mergers does not appear in the figure as this forms at higher GW frequencies than shown (Eq. 52). Up to this point, we have focused on binary-single and single-single interactions. For binary-binary interactions, the maximum orbital velocity before ejection is lower than for binary-single interactions by a factor of 2.6 (Eq. 13), implying that fGW0 is lower by a factor of 2.66/7 = 2.3 and e10Hz and e10M by a factor of 2.619/21 = 2.4.

BH Mergers in Galactic Nuclei Nuclear star clusters at the centers of galaxies are the densest stellar environments in the universe [101]. The supermassive black hole at their centers creates a deep potential well where stars and black holes accumulate in a steep density, and the central number density can reach nc ∼ 1010 pc−3 [41]. These regions are expected to host up to 25,000 black holes and even more neutron stars and white dwarfs [90]. There is X-ray observational evidence of 12 compact object binaries within the central parsec of the Galactic center, 6 of which likely contain a stellar-mass black hole [52]. The black hole distribution forms a mass-segregated profile, where the number density for different mass objects depends on radius as [95] nm (r) ∝ r −1.5−p0 m/mmax .

(59)

Here p0 = 0.5–0.6 for a range of reasonable black hole mass functions [95], and p0 may reach 3 in special cases [64].

682

B. Kocsis

Black hole mergers in galactic nuclei may happen in multiple ways. 1. Mergers induced by binary-single interactions as in globular clusters. 2. Single-single GW captures [50, 95, 111]. The probability of scattering is greatly increased by the high number density in these regions compared to globular clusters where the typical velocity is extremely high. These are the most eccentric sources in the universe. 3. Kozai-Lidov-triggered mergers [11, 54, 60, 104]. The gravitational effect of the central supermassive black hole drives the eccentricity of nearby binaries to high values, which may lead to a GW-driven merger. [12, 74]. 4. Gas-assisted mergers in active galactic nuclei [23, 87, 131, 134]. We discuss these possibilities in turn next.

Mergers Driven by Binary-Single Encounters Let us examine how the results derived for globular clusters may be extended to nuclear star clusters. Since the binary formation rate Eq. (1) scales with v −9 , a supermassive black hole may largely inhibit the formation of binaries if present. But several galaxies are known to have a dense nuclear star cluster (NSC) with no supermassive black holes [94]. These systems are similar to massive globular clusters with escape speed increased to peak above vesc ∼ 100 km/s extending to vesc ∼ 500 km/s for some systems. However although the binary formation rate by three-body encounters becomes prohibitively long for many NSCs (Eq. 1), nevertheless, black hole binaries may form through exchange interactions in these regions −1 −1 [12]. The binary-single interaction rate scales with ns tbs,ej ∝ nBH ns v −2 orb,ej v (see Eq. 8), the increase in the mean density mitigates the effect of the increasing binary-single timescale for high orbital velocities, which makes strong binary-single interactions still possible within a Hubble time if v orb ≤ 1,700 km/s if n = 106 pc−3 and v = 300 km/s. Also, as mentioned below Eq. (11), the hardening timescale may be a factor ∼ 5 shorter than our simple estimate for tbs , implying that binary-single interactions may continue to efficiently shrink the binaries also for very large v orb . The merger probability increases strongly as P2b ∝ v 3orb , implying that the merger −1 P2b ∝ nBH ns v orb /v, so it may be dominated by rate is proportional to nBH ns tbs √ sources having an initial relative orbital velocity of v orb,ej = 42vesc in the range of 600–3,200 km/s (Eq. 13). In the following we will consider this range to examine the expected properties of mergers in these systems. In Sec. “Merger Probability in Globular Clusters”–“GW Frequency and Eccentricity Distribution for Globular Clusters,” we have shown that the probabilities and the properties of merging sources in the ejected, in-cluster two-body, IMS, threebody, and single-single channels are primarily set by the binary orbital velocity near ejection. Plugging in the numbers in the analytical expressions for the merger probabilities, fGW0 , the distribution of e0 , and e10Hz and e10M , we may draw

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

683

the following conclusions for v orb,ej = (600, 1000, 3000) km/s and m = ms = 10 M . • The probability for ejection is close to zero even for the high-mass black hole masses, two-body encounters cause mergers within the cluster with nearly 100% probability. • The probability of intermediate state mergers is PIMS = (33%, 56%, 98%), respectively. These binaries enter the GW-driven regime at a peak GW frequency of fGW0,IMS = (0.4, 0.6, 1.5) Hz, and the eccentricity when the binaries reach 10 Hz is e10Hz,IMS = (0.02, 0.03, 0.08), respectively. These values may possibly be detected to be significantly non-zero with LIGO, VIRGO, and KAGRA. More massive binaries with m ≥ 10M have smaller fGW0,IMS , and therefore will be more circular at 10 Hz (Eq. 57). • The probability of three-body mergers is P3b ≤ (6%, 12%, 30%) respectively for the same three cases and the GW frequency at formation is fGW0,3b = (6, 10, 27) Hz. This shows that the eccentricity at 10 Hz is expected to be very high above 0.5 for v orb > 600 km/s for this merger channel. Thus threebody mergers in nuclear stars clusters with no supermassive black holes will be systematically much more eccentric than three-body mergers in globular clusters. The overall probability of eccentric mergers is much higher in nuclear star clusters. This is not surprising since Eqs. (26) and (57) show that P3b ∝ 0.9 1.4 0.9 v 1.4 orb,ej ∝ v esc and e10Hz ∝ v orb,ej ∝ v esc . For more detailed discussions of the properties of mergers in nuclear stars clusters see Refs. [9, 12, 27, 39, 80, 83, 104, 110].

Single-Single Gravitational Wave Captures The rate of single-single captures may be written using Eq. (29) as  Γss =



 n2 σGW v dV =

16π 2

85π 3

2/7

  G2 m2 2 2 v(r) −11/7 r n (r) dr c c4 (60)

√ Here v(r) = 2GMSMBH /r is the typical relative velocity for an encounter that happens at a radial distance r from the SMBH. Higher v leads to a lower Γss . However the number density increases steeply in the galactic center (Eq. 59), which leads to dΓss /d ln r ∝ r 11/14−2p0 m/mmax .

(61)

Thus, single-single captures are uniformly distributed on a logarithmic scale at m ∼ 0.7mmax , it is skewed toward the inner and outer regions for higher and lower masses, respectively. The GW frequency at formation is given by Eq. (53).

684

B. Kocsis

Fig. 3 The gravitational wave frequency at formation for single-single GW captures in galactic nuclei for different binary total masses (left panel, cf. Figure 1 for globular clusters) and the cumulative distribution function of eccentricity when the binary reaches a pericenter distance of 10Gm/c2 . A significant fraction of sources form in the LIGO band above 10 Hz. Note that while fGW0,ss scales with m−1 (Eq. 53) higher mass black holes segregate closer to the supermassive black hole where the velocities are much higher. Adopted from [49]

Since v > 3000km/s in the inner regions, fGW0,ss > 21 Hz(m/10 M )−1 for these sources. This shows that a significant fraction of single-single captures in galactic nuclei form inside the LIGO band [95]. The eccentricity at the start of the GW inspiral is higher than 0.95 (Eq. 40) [49]. Note that e10Hz is undefined for sources forming at higher than 10 Hz. A well defined quantity is e10M > 0.04 see Eq. (40). Figure 3 shows the distribution of the GW frequency at formation and the eccentricity at e10M for single-single GW captures in mass-segregated nuclear star clusters (adopted from [49]). Single-single captures in galactic nuclei represent the highest eccentricity systems in the universe. In these regions lower eccentricities are forbidden for single-single captures since the high velocity dispersion requires a very close passage to result in a bound binary, and a merger happens before circularization may take place. A measurement of such high eccentricity sources without the detection of a larger population of moderately eccentric population would be a smoking gun signature of a source in the vicinity of supermassive black holes. Due to the increasing typical velocity for more massive sources due to mass segregation, e10M , the eccentricity when the binary reaches a pericenter of 10Gm/c2 , is systematically higher for larger masses [49, 50]. The probability of merger is a sensitive function of p0 as shown in [111]. For a BH mass function of dN/dmBH ∝ m−2.35 BH , the merger rate density in this channel is Rss = 0.003 Gpc−3 yr−1 for p0 = 0.5 and 0.008 Gpc−3 yr−1 for p0 = 0.7.

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

685

Mergers Triggered by the Kozai-Lidov Effect Eccentricity excitation of a binary is also possible directly by the supermassive black hole, the Kozai-Lidov effect [11, 54, 60, 104]. The effect arises due to the change in the binary angular momentum when averaging the gravitational effect of the SMBH when averaging over the orbital time of the binary and around the SMBH [63, 91]. The effect may be derived in a multipole expansion, where the quadrupolar effect leads to a periodic oscillation of the eccentricity, and the octupole term leads to a random walk in the peak eccentricity of the quadrupole cycles, while the semimajor axis of the binary and v orb is fixed. The Kozai-Lidov effect may operate efficiently if its characteristic timescale is smaller than the GR precession timescale, but the binary is not too close to the supermassive black hole to be tidally disrupted. This requires that the orbital period of the binary is shorter than the period around the SMBH and that [5] 2

tKL,quad =

t 8q (1 − e2 ) c2 2 3/2 orb,out (1 − eout ) ≤ tGR = torb , 15π torb 6π v 2orb

(62)

where q = (m1 + m2 + mout )/mout and torb,out and eout are the orbital period and eccentricity for the binary’s center of mass motion around the SMBH. For a SMBH, mout m1 + m2 , so q = 1. Thus,  5 (1 − e2 )1/2 c torb,out 1≤ ≤ . (63) 2 )3/4 v torb 16q (1 − eout orb For typical galactic nuclei, this corresponds to the regions very close to the supermassive black hole with a semimajor axis of aout ∼ 102 –104 AU, where the binary semimajor axis is a = 0.1–10 AU so that v orb = 30–300 km/s [11, 60]. Note that the velocity dispersion of the star cluster at these distances from the supermassive black hole is v = 600–20,000 km/s, implying that these binaries are soft and get tidally disrupted or ionized by binary-single interactions (Eq. 6). However, the binary-single encounter timescale is often much larger than the KozaiLidov-triggered merger timescale [11, 130]. During the Kozai-Lidov effect, the semimajor axis and v orb are fixed. The merger is triggered when the eccentricity is excited above the value e ≥ ecrit,KL where tGW ≤ tKL , and the GW merger time (Eq. 15) is less than a Kozai-Lidov timescale (Eq. 62), where  2 = 1−ecrit,KL

tKL,quad tGWc

2/7

 =

1024q 75

2/7 

v orb c



10/7 2 3/7 (1−eout )

torb,out torb

4/7 . (64)

Note that the criteria (63) and (64) for tKL ≤ tGR and tGW ≤ tKL are typically contradictory since tGR ∝ (v orb /c)2 torb is a first post-Newtonian (PN) effect and tGW ∝ (v orb /c)5 torb is a 2.5PN effect, so typically tGR tGW . In other words,

686

B. Kocsis

GR precession quenches the Kozai-Lidov effect when GW emission dissipation becomes significant. This does not happen if the eccentricity jumps within one orbit from the regime satisfying tKL ≤ tGR (Eq. 63) above the critical value of tGW ≤ tKL (Eq. 64). This is possible if the triple system is only moderately hierarchical. Since the change in the eccentricity during one orbit is of order Δe ∼ torb /tKL,quad , so for 2 high eccentricity we require that Δe ≥ 1 − ecrit,GR where ecrit,GR is defined such that tKL,quad = tGR . Using Eq. (62) this yields  torb,out ≤ 1≤ torb

4

 1/2  75π c v orb 90 2 −3/4 2 −3/4 (1 − eout ) = √ (1 − eout ) . v orb q 200 km/s 128q 2 (65)

This criterion is indeed more stringent than Eq. (63). If only Eq. (63) holds and Eq. (65) is violated, GWs gradually decrease the inner binary semimajor axis ultimately leading to more circular mergers [58,109]. In the following we will focus on the case of moderately hierarchical mergers when Eq. (65) holds. Generally since the actual torque of the third body can exceed the average torque by a factor of order unity, the criterion for Kozai-Lidov-induced mergers (Eq. 64 and 65) may be violated in practice by an order unity factor. Combining the two criteria, we get the following range for the critical eccentricity: 

8q 4 75

2/7 

v orb c

10/7 2 3/7 (1 − eout )

≤

2 1 − ecrit,KL

    π 1/7 v orb 8/7 ≤4 . 150 c (66)

The lower bound of this equation is very similar to that for three-body mergers (Eq. 25), as expected, since this represents the case where torb,out /torb = 1 for which tKL,quad ∼ torb . However the equation breaks down near the lower bound, as the binary becomes unstable for intersecting orbits. Based on the upper bound for v orb ∼ (600 − 3000) km/s, 1 − ecrit,KL is up to a factor ∼ 2.5 × −3.9× larger than for the three-body mergers in binary-single interactions 1 − ecrit,3b . Once ecrit,KL is reached, the initial GW frequency follows from Eqs. (47)–(48). If we assume that the peak frequency is twice the lower cutoff of fp0 as in Eqs. (49)– (52), we get that it is 

 −6/7    75 3/7 c3 v orb 6/7 2 −9/14 torb,out fGW0,KL (1 − eout ) 64q 2Gm c torb     −1   −9/14 6/7  2 torb,out −6/7 m v orb 0.3 Hz 1 − eout = 3/7 . 0.5 90 torb 1000 km/s 10M q (67) 1 =√ 8π

The peak eccentricity at 10Hz (Eq. 57) is e10Hz ∼ 0.6(fGW0,KL /10Hz)19/18 ∼ 0.01, just below the LIGO/VIRGO detection limit. However if vesc = 500 km/s and

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

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v orb = 3200 km/s then fGW0,KL = 0.8 Hz and e10Hz = 0.04 for the peak of the distribution. The KL-triggered mergers triggered by the SMBH form in between the LISA and LIGO band with very high eccentricity in nuclear star clusters. For globular triple systems with stellar-mass companions, v orb is lower by a factor 10 − −100, and the peak GW frequency at formation is in the LISA band, and the mergers in the LIGO band do not have any significant eccentricity. These results are consistent with numerical simulations, which have shown that the initial GW frequency peaks at fGW0,KL = 10−3 − 10−1 Hz for KL-driven mergers of hierarchical triples in field triples and in globular clusters depending on the parameters of the triples, and the eccentricity at 10Hz is 10−3.5 –0.03 [36, 85]. The typical time for merger may be estimated analytically as follows. Numerical simulations show that the distribution of periapsis distance for triples undergoing the Kozai-Lidov process is uniform implying that e is drawn from a thermal distribution for which e2 is uniformly distributed [63]. Without dissipation or direct collision, the eccentricity distribution reaches e  1 for orbits that have inclinations sufficiently close to 90◦ that they undergo orbital flips. The fraction of orbits above ecrit,KL is 2 1 − ecrit,KL ≈ 2(1 − ecrit ); thus, the time of merger is exponentially distributed with a mean tmerger,KL =

torb , 2 1 − ecrit,KL

(68)

2 is given by Eq. (64). where 1 − ecrit,KL This argument shows that the initial eccentricity distribution for merging binaries is expected to be close to a truncated thermal distribution pth (e0 > e) Eq. (41) with a cutoff at ecrit,KL . The cumulative distribution of e0 for merging binaries is

⎧ ⎪ ⎨

1 − e2 if e < ecrit,KL 2 pKL (e0 > e) = 1 − ecrit,KL ⎪ ⎩ 0 otherwise .

(69)

The probability of merger is sensitive to the orbital inclination of the binary relative to the orbital plane around the supermassive black hole. Strong eccentricity excitation occurs for initial inclination close to i0 = 90◦ . While a direct collision due to the quadrupole-order Kozai-Lidov effect would require an extreme amount 2 of fine-tuning i within a narrow range of width 1 − ecrit,KL around 90◦ , this is not required due to either the breakdown of the double-orbit averaged approximation or the octupole corrections. In a related problem for white dwarf mergers, Katz and Dong [63] find that the condition for merger requires that Δi0 = |i − 90◦ | satisfies Δi0 =

torb mout (1 − eout )−3/2 m1 + m2 torb,out

(70)

688

B. Kocsis

where the binary masses are m1 and m2 and the perturber mass is mout . While this formula breaks down in the limit of m3 m1 + m2 , the fraction of systems which exhibit an orbital flip and may reach very high eccentricities is of order 1%–3% for m3 m1 + m2 [78]. For an isotropic distribution or binaries in the vicinity of a SMBH, numerical experiments show that the probability of merger is 7%-15% with a median merger time of 4×105 yr [37,60]. Orbital inclination diffusion due to resonant relaxation can help increase the fraction of systems driven to merger by the Kozai-Lidov effect by driving the inclination to near i = 90◦ . This leads to a KozaiLidov merger probability of 1%–30% depending on the mass of the supermassive black hole [54]. These results suggest a very high merger rate and merger fraction in galactic nuclei, where the SMBH triggers the KL effect. Indeed, if the supply rate of binaries to the inner regions was faster than the KL timescale, this merger channel would produce a very high rate of mergers possibly accounting for all of the detections [33, 54, 60]. However, the formation and delivery rate of binaries in the innermost regions of galactic nuclei are currently not well understood. Observations suggest a binary fraction between 0 and 60% [105], and the rate of delivery by inward diffusion due to two-body relaxation may be larger than 100 Myr [95]. Kozai-Lidov-triggered mergers may happen not only in galactic nuclei but in triples in other environments including the galactic field and globular clusters. Indeed, the third object need not be a supermassive black hole to drive the binary to high eccentricity, but it may be another compact object or an ordinary star either to form an isolated triple or a triple in globular clusters [6, 13, 36, 38, 85, 128, 142]. The fraction of mergers driven by the Kozai-Lidov process and its eccentricity distribution are weakly sensitive to q = (m1 + m2 + mout )/mout as shown above as long as mout ≥ m1 + m2 . Even more generally, an analogous process may be driven by the tides of a stellar cluster [27, 57, 58, 110].

Gas-Assisted Mergers in Active Galactic Nuclei In approximately 1% of all galactic nuclei, the supermassive black hole is accreting gas and produces bright electromagnetic active galactic nucleus (AGN) activity. Gas effects may accelerate mergers in multiple ways [23, 87, 131, 134, 136, 138]. Gas helps to: 1. Capture binaries from the surrounding nuclear star cluster from initially inclined orbits. 2. Form massive stars in the unstable outer regions of the disk. 3. Grow stars and black holes via accretion increasing their mass and spin. 4. Catalyze binary formation during dissipative single-single encounters in the ambient medium. 5. Drive the binaries to orbital radii similar to that in planetary migration where they accumulate and undergo multiple dynamical encounters.

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

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6. Reduce the binary separation due to resonant circumbinary torques. 7. Align the binary’s spin angular momentum with its orbital angular momentum around the supermassive black hole. Many of these mechanisms are not well understood at the time of this writing, they represent some of the most difficult problems in theoretical astrophysics. The main reason for the complications is a lack of understanding of the efficient angular momentum transport in accretion disks, i.e., “viscosity”, the nonlinear nature of turbulence over a vast range of scales, a leading order cancellation of large opposing torques exerted by the inner and outer disks leading to uncertainties in their net contribution, nonlinearities and resonances in the torque cutoff phenomenon, and strongly coupled processes involving magnetohydrodynamics, thermodynamics, radiative transport, and plasma physics. Existing studies of compact object mergers are limited to either order-of-magnitude estimates or semi-analytical simulations with many simplifying assumptions. Nevertheless, it is clear that the AGN merger channel has a potential to strongly contribute to the observed GW events. The capture of objects on initially inclined orbits by the gas disk can be estimated using Chandrasekhar’s dynamical friction during disk crossings. The alignment time is [23, 133] 2 sin i sin3 (i/2) m2SMBH talign,CDF = torb,out , (71) ln Λ md,loc m where md,loc = dmd /d ln r = 4πρr 2 h is the local disk mass in a logarithmic annulus integrated over the h disk thickness, mSMBH is the supermassive black hole mass, i is the orbital inclination, ln Λ is the Coulomb logarithm, and torb,out is the orbital time around the supermassive black hole, which is typically hours to 100 yr. The local disk mass ranges from less than m to mSMBH from the inner edge of the disk to its outer edge at 0.01 pc [51]. The alignment time depends sensitively on i. At large inclinations the alignment time is of order 108 –109 yr, but for i ≤ 15◦ , it is typically less than 106 yr. Once in the disk, the objects excite density waves in the disk and are driven inward similar to how planets migrate in proto-planetary disks [17]. The migration timescale in the limit when the object is sufficiently massive to open an annular gap (type-II migration) is approximately [53, 68]  m d,loc tmig ≈ 7 × 104 torb,out × min 1, m

(72)

but it may be much shorter in the limit when the objects do not open a gap (typeI migration). Thus, the objects are rapidly transported to the gap-opening radius, where they accumulate. Once an object (or a number of objects) opens an annular gap, and the local disk mass is relatively small, this may lead to a pressure maximum in the disk near the gap, which may serve to create a migration trap, where low-mass objects accumulate (see also [99, 102, 126]).

690

B. Kocsis

Fig. 4 The dominant mechanisms for reducing the binary separation in the AGN disk, adopted from [134]. Gas effects drive the binary outside of 0.01 pc. The binaries accumulate at 0.01 pc, where binary-single interactions reduce their separations in the LISA band, until GW emission becomes dominant

The objects may form hard binaries efficiently in a gaseous ambient medium during single-single encounters, where the initial kinetic energy is dissipated by the gas [134]. In particular, this may be expected if the gas mass within the annular width of the Hill sphere where the binary can interact strongly gravitationally rh = rout (2m/mSMBH )1/3 is much larger than the binary mass, i.e. md,loc

2krh 1 1/3

m or equivalently if md,loc 1/3 m2/3 mSMBH r 2 k

where k is a constant of order unity.

(73)

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

691

Fig. 5 The eccentricity distribution at 10Hz in various models of mergers in AGN, adopted from [134]. Gravitational wave capture events during binary-singe and single-single interactions lead to a high eccentricity peak. The left panels assume isotropic binary-single interactions, the middle panels assume initially 2D binary-single scattering which leads to a different thermal eccentricity distribution, and the right panels show the case of no binary-single interactions, only single-single captures

The binaries experience binary-single interactions with disk objects and objects from the spherical nuclear star cluster. However the capture cross section is truncated by the Hill sphere and the thickness of the disk. The interaction rate is still high, which reduces the binaries’ separation and the recoil effect leads to their ejection from the thin gas disk (see Eq. 13). The ejected binaries settle back to the disk from low inclination orbits relatively rapidly due to dynamical friction (Eq. 71). During each binary-single interaction, the eccentricity is drawn either from an isotropic thermal distribution (Eq. 41), or its analog for a planar configuration, which is skewed even more toward high eccentricities [122, 137]. Semi-analytical simulations of Tagawa et al. [134, 136] show that the merger probability is very high for a wide range of model assumptions. The merger remnants are retained in the cluster, form new binaries, and may merge multiple times. Figure 4 shows the GW frequency and the radial location of all binaries at two snapshots in the semi-analytical simulation of Tagawa et al. [134], indicating the dominant hardening mechanism. The binaries are hardened mostly by gas effects in the outer regions of the AGN, and by dynamical interactions at 0.01 pc, where they accumulate. Figure 5 shows the eccentricity distribution at 10 Hz, for different models. All models show a high eccentricity peak detectable by LIGO due to GW captures during strong binary-single interactions and single-single GW captures. This is explained by Eqs. (52) and (57), where now v orb is increased given that the ejection following binary-single interactions is inhibited by the SMBH potential.

692

B. Kocsis

The binary-single interactions lead to an especially high probability of eccentric mergers if the scattering takes place in 2D [122].

Further Observational Diagnostics of the Dynamical Channel Mass, Spin, and Redshift Distributions We now turn to the expected signatures in the mass and spin distribution to identify the dynamical origin of mergers in globular clusters. A particularly interesting possibility for the dynamical channel is mergers among merger remnants, often referred to as “hierarchical” mergers, which build up a black hole mass hierarchy with a characteristic spin magnitude distribution [31, 43, 98]. Hierarchical mergers may involve BHs of masses and spins beyond the maximum values for first-generation BHs, mmax (due to the pulsational pair instability mass loss for higher masses) and amax , the so-called mass gap and spin gap for the isolated binary evolutionary channel. Mergers in the corresponding regions of parameter space may have a dynamical origin. Hierarchical mergers are not possible in the ordinary isolated binary evolutionary channel; it is a smoking gun signature of the dynamical channel. In massive globular clusters, second-generation mergers may constitute between 9-15% of first-generation mergers [19, 26, 31, 43, 44, 97, 112]. Second and highergeneration mergers may be even more likely in nuclear star clusters, where the escape speed is higher. Recent population synthesis studies coupled with dynamical models have also determined the branching ratios for hierarchical mergers in globular clusters and nuclear star clusters under different assumptions for the stellar population and the corresponding implications for the merger rate, mass, and spin distributions [16, 20, 40, 80, 83]. Hierarchical mergers in nuclear star clusters with escape velocities exceeding 300 km/s lead to the formation of intermediate mass black holes [9,14,21,39,40]. Second-generation mergers are also possible in young massive clusters and open clusters if the black hole birth spin is close to zero [21]. Semi-analytical simulations of Tagawa et al. [134,136] support expectations that the AGN merger channel is a factory of hierarchical mergers [144]. The fraction of second- and higher-generation mergers in AGN is 33%, but this implies 70% for the observable merger fraction after accounting for the higher signal-to-noise ratio of higher-generation mergers. Despite the fact that the initial black hole mass is possibly limited by the solar metallicity of nuclear star clusters to 15M , the black hole masses quickly increase beyond 50M [137]. This leads to a characteristic value for the spin magnitudes. The spin magnitude may be further increased by accretion in the AGN channel. Furthermore, the spin directions of the merging black holes align due to the gas torques, while the binary-single interactions may randomize the orbital inclination. A smoking gun signature of the AGN channel is the parallel black hole spin configuration for the binary components (more precisely, equal inclination angle between the orbital angular momentum and two spin vectors) but misaligned with respect to the orbital angular momentum [134, 135].

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

693

While hierarchical mergers are obviously impossible in isolated binaries they may possibly take place in triples and quadruples [35, 55, 56, 119]. For 2+2 field quadruples, the rate of second-generation to first-generation mergers is 10−7 [55]. The spin direction distribution in globular clusters is expected to be isotropic with respect to the angular momentum direction, assuming that the binaries formed dynamically by chance encounters, and the orbital planes have been randomized isotropically [30]. The mass-weighted summed spin components projected along the orbital angular momentum χeff are distributed with zero mean and broad scatter for the dynamical merger channel, implying a comparable probability for antialigned systems compared to aligned systems. This is very different from the observed distribution in the three X-ray black hole binaries with high-mass companions [86], where the spin directions have been tidally synchronized [145]. With the further development of the GW detector sensitivities and the commissioning of KAGRA, the redshift evolution may soon be one of the key diagnostics to identify mergers in the dynamical channel. The cosmic history affects the host environments of dynamical sources in a predictable way. Globular clusters evaporate due to ejections following binary-single interactions, and they may be torn apart by the tidal field of the galaxy in the central regions [46]. Many of the globular clusters might have already dissolved by the time their black hole binaries merge [34]. The cosmological redshift distribution carries information on the globular cluster formation rate and evolution [8, 113, 116].

Universal Gravitational Wave Statistics One difficulty in identifying GW source populations in different merger channels is their dependence on quantities like the initial mass function of black holes and binary fraction which are not well understood. There are, however, certain universal properties for the dynamical channels which do not depend on these unknowns. One such universal feature identified in Monte Carlo and direct N-body simulations of globular clusters, is the merger probability relative to the random pairing probability as a function of total binary mass mtot = m1 + m2 marginalized over the distribution of the mass ratio. This quantity P (mtot ) is proportional to m4tot or m5tot for mtot ≤ 2mmax where mmax is the maximum mass of firstgeneration compact object remnants [97]. This is independent of the underlying black hole mass function and binary fraction. The increased merger probability toward heavy masses is a manifestation of the mass dependence of binary formation in triple single scattering (Eq. 1), exchange interactions, mass segregation, and the mass dependence of the maximum orbital velocity before ejection in binary-single scattering encounters, which enhance the likelihood of merger among the heavier objects, see Eqs. (21). Note that the observed distribution is furthered skewed toward higher masses due to observational bias as GWs are more intense from higher mass sources. The combination of these effects implies that the observed rate of mergers from heavy (mh ) vs. light black holes (ml ) in globular clusters is roughly [mh /ml ]n fBH (mh )/fBH (ml ) where n ∼ 5–6 and fBH (m) is the mass distribution of

694

B. Kocsis

black holes. This shows that the heavier black holes dominate the observed merger sample even if there are much fewer of them in total. There is a related universal quantity that may be derived from R(m1 , m2 ), the binary component mass distribution of the rate of mergers for dynamical channels, which does not depend on the unknown black hole mass function [67]. This is α = −(m1 + m2 )2

∂2 ln R ∂m1 ∂m2

(74)

which is universal for different source populations of dynamical origin. In particular, if R(m1 , m2 ) = p(m1 , m2 )fBH (m1 )fBH (m2 ) where fBH (m) is the mass function of black holes and p(m1 , m2 ) is the probability of merger relative to the random pairing probability, then fBH (m1 )fBH (m2 ) drops out from the mixed derivative of the logarithm. Thus α depends only on p(m1 , m2 ) which may be determined by theoretical models for the dynamical merger channels. In particular, for primordial BH binaries formed in the early universe, BH mergers that originate from single-single captures in collisionless systems and in single-single captures in mass-segregated galactic nuclei, these are, respectively [67] αPBH = 1 ,

(75) 10 , 7

(76)

αss,heaviest = −5.39 ,

(77)

αss,collisionless =

αss,lightest =

10 . 7

(78)

We may also derive a universal feature for the initial eccentricity distribution of merging binaries (Eqs. 45–46) which depends on the parameters of the binary and the environment only through the constants p2b,ej and pIMS,ej given by Eqs. (19)– (22). The quantity   (1 − e2 ) d ln P (e0 < e) κ(e) = − 2e de 1 − e2

(79)

is universal, as it does not depend on any of the parameters describing the cluster and the merging binaries, including the eccentricity, wherever the probability density is d nonzero. In terms of 0 = 1 − e02 , it is κ() = − d [ −1 ln P (0 > )]. κ2b =

10 10 = = 2.785 7| ln δ| 7| ln 79 |

κIMS = κ3b =

5 5 = 5.571 = 7| ln δ| 7| ln 79 |

(80) (81)

16 Dynamical Formation of Merging Stellar-Mass Binary Black Holes

695

κss = 0 .

(82)

The universality of this functional of the initial eccentricity distribution is an important prediction of the dynamical merger channel. The corresponding universal quantity for the observables er and ef may be defined by mapping the e0 distributions (see Eqs. 56 and 55). For er and ef much smaller than unity, κ≈−

12 d er 19 der



12/19 ln Per (er ) 12/19 er

⎛

 ≈−

d ⎝ 18 ef 19 def

18/19

ln Pef (ef

18/19

⎞ )

⎠

(83)

ef

where Pef (ef > x) and Per (er > x) are the cumulative distribution functions of ef and er , respectively, e.g. e10Hz and e10M . Note that for heavier BH mergers than average m ≥ ms , the mass depependence of δ (Eq. 10) may break the universality of κ in a predictable way. More generally, mtot

∂κ(e, mtot ) ∂κ(e, m1 , m2 ) or (m1 + m2 )2 ∂mtot ∂m1 ∂m2

(84)

are expected to be universal, independent of e and m for merging BHs with masses higher than the typical scatterer mass.

Observed Merger Fraction and Branching Ratios Hierarchical mergers have been identified in the LIGO third observing run [65, 66]. Five merging sources are likely 1g+2g (GW190517, GW190519, GW190602, GW190620, and GW190706), and one source 2g+2g (GW190521), where ng refers to the number of previous mergers. In fact GW190521 may also be eccentric [42, 117], further supporting its dynamical origin. The fraction of mergers in the 1g+2g and the 2g+2g channel is estimated to be 5 × 10−3 ± 1 dex and 3.1 × 10−5 ± 2 dex, respectively [66]. A number of studies have investigated the fraction of observed GW detections in the different merger channels. The spin distribution suggests that at least 50% of the observed sources may originate dynamically [118]. Wong et al. [143] found that if all mergers originate either through the isolated binary evolution common evelope path or in globular clusters, then 2–52% and 48–98% formed in the former and latter, respectively. Zevin et al. [146] found that the common envelope, chemically homogeneous evolution, and stable mass transfer types of the isolated binary evolutionary channel may represent 1–30%, 1–16%, and 2–49%, respectively, while the dynamical channel in globular clusters and nuclear star clusters correspond to 9%–60% and 6%–47% of the detections. Despite the great progress, these results are preliminary as they are sensitive to assumptions, and the goodness-of-fit is not investigated.

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B. Kocsis

Fragione and Banerjee [32] argue that young massive and open clusters alone can explain the mass and spin distributions of the detected sources in the LIGO observing runs O1, O2, and O3a.

Conclusions This chapter summarized the key characteristics of the dynamical merger channels of gravitational wave source populations. We have shown that the dynamical channel is expected to produce a high rate of mergers in the universe comparable with the observed merger rate by LIGO and VIRGO. Although the dynamical channels span a large variety of physical processes (binary formation, binary-single and binarybinary encounters, exchange interactions, ejections, single-single gravitational wave captures, triple or quadruple interactions, secular triple interactions, etc.) and many different host environments (dense star clusters including open clusters, globular clusters, nuclear star clusters with and without supermassive black holes, active galactic nuclei, primordial black holes binaries formed in the early universe or in dark matter halos), we have shown that the single most important quantity that determines the properties of merging sources (i.e. the merger fractions, GW frequency at formation, and eccentricity distribution) is the mean relative orbital velocity v orb . The mergers are typically dominated by binaries where v orb reaches its maximum, set by the escape velocity of the host environment (Eq. 13). This suggests that the most important characteristic that GW observations will tell us about the host environment is their typical escape velocity. We have also shown that there are several unified observational characteristics with respect to the GW frequency distribution of mergers, their merger fractions, and eccentricities. The unification is possible statistically by the common theme of the thermal eccentricity distribution, which determines the observable eccentricity distribution at 10 Hz or at rp = 10Gmtot /c2 uniquely up to a minimum initial eccentricity cutoff e0,crit . This led to the existence of universal statistics to uniquely classify dynamical merger channels without the knowledge of the underlying unknowns describing the populations’ mass and spin distributions (Eqs. 74 and 79). These universal constants are different for single-single interactions, binary-single interactions, intermediate states and three-body mergers, and for binaries formed in the early universe, collisionless dynamical systems, and mass-segregated dynamical systems. The dynamical environments and the dynamical processes differ in the implied minimum initial eccentricity cutoff e0,crit , which sets the minimum value of the initial GW frequency and the minimum eccentricity at any given separation or frequency approaching merger. However, due to instrumental noise, and the fact that the eccentricity decreases during the inspiral, eccentricity is measurable only for the highest velocity processes with LIGO, VIRGO, and KAGRA. These are the single-single GW captures in galactic nuclei with velocity dispersions over 200km/s and the three-body mergers in binary-single interactions where the binary relative orbital velocity v orb is higher than this value either in massive globular

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clusters, galactic nuclei, or AGN accretion disks. In nuclear star clusters, the more common intermediate state mergers during binary-single interactions also lead to sources with residual eccentricity in the LIGO band. Ultimately LISA will be able to distinguish the GW sources from individual dynamical processes and different types of host environments by measuring the minimum eccentricity truncation of the GW source populations as a function of frequency between 10−4 and 10−1 Hz. The dynamical merger channel also allows for the possibility of hierarchical mergers which has distinctive fingerprints in the mass and spin distribution of GW source populations. These features have the potential to determine the fractions of mergers originating in the known processes, and may reveal the presence of additional source populations if the measured distribution is inconsistent with all of the known source populations.

Cross-References  Formation Channels of Single and Binary Stellar-Mass Black Holes  LISA and the Galactic Population of Compact Binaries  The Gravitational Capture of Compact Objects by Massive Black Holes Acknowledgments This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Program for Research and Innovation ERC-2014-STG under grant agreement no. 638435 (GalNUC).

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Contents Introduction: Observational Facts About Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . The Formation of Compact Remnants from Single Stellar Evolution and Supernova Explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar Winds and Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Core-Collapse Supernova (SN) or Direct Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pair Instability and the Mass Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Mass of Compact Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Object Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natal Kicks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binaries of Stellar Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Envelope (CE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative Evolution to CE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BBH Spins in the Isolated Binary Evolution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the Isolated Binary Formation Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dynamics of Binary Black Holes (BBHs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamically Active Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Body Encounters, Dynamical Friction, and Core Collapse . . . . . . . . . . . . . . . . . . . . . Binary: Single Encounters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar Mergers and BHs in the Pair-Instability Mass Gap . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Three-Body Binary Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Ejections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation of Intermediate-Mass Black Holes by Runaway Collisions . . . . . . . . . . . . . . . . Hierarchical BBH Formation and IMBHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative Models for Massive BHs and IMBH Formation in Galactic Nuclei . . . . . . . . .

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M. Mapelli () Dipartimento di Fisica e Astronomia Galileo Galilei, Padova, Italy INFN, Sezione di Padova, Padova, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_16

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Kozai–Lidov Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Dynamics and Open Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BBHs in the Cosmological Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data-Driven Semi-analytic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmological Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

These are exciting times for binary black holes (BBHs). LIGO and Virgo detections are progressively drawing a spectacular fresco of BBH masses, spins, and merger rates. In this review, we discuss the main formation channels of BBHs from stellar evolution and dynamics. Uncertainties on massive star evolution (e.g., stellar winds, rotation, overshooting, and nuclear reaction rates), corecollapse supernovae, and pair instability still hamper our comprehension of the mass spectrum and spin distribution of black holes (BHs), but substantial progress has been done in the field over the last few years. On top of this, the efficiency of mass transfer in a binary system and the physics of common envelope substantially affect the final BBH demography. Dynamical processes in dense stellar systems can trigger the formation of BHs in the mass gap and intermediate-mass BHs via hierarchical BH mergers and via multiple stellar collisions. Finally, we discuss the importance of reconstructing the cosmic evolution of BBHs. Keywords

Black holes · Gravitational waves · Pair instability · Massive stars · Binary systems · Binary black holes · Intermediate-mass black holes · Stellar dynamics · Star clusters · Merger rates

Introduction: Observational Facts About Gravitational Waves On 2015 September 14, the LIGO interferometers [1] captured the gravitational wave (GW) signal from a binary black hole (BBH) merger [2]. This event, named GW150914, is the first direct detection of GWs, about hundred years after Einstein’s prediction. Over the last 5 years, LIGO and Virgo [19] witnessed a rapidly growing number of GW events: the second gravitational wave transient catalogue (GWTC-2, [14, 335, 336]) consists of 50 binary compact object mergers from the first (O1), the second (O2), and the first part of the third observing run (O3a) of the LIGO–Virgo collaboration (LVC). Based on the results of independent pipelines, [343, 349, 375], and [255] claimed several additional GW candidates from O1 and O2. Furthermore, several dozens of public triggers from the second part of the third observing run (O3b) can be retrieved at https://gracedb.ligo.org/.

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Among the aforementioned detections, GW170817, the first binary neutron star (BNS) merger detected during O2, is the first and the only GW event unquestionably associated with an electromagnetic counterpart to date [10, 11, 22, 73, 82, 83, 145, 235, 253, 269, 302, 309]. Several other exceptional events were observed during O3a: the first unequalmass BBH, GW190412 [15]; (ii) the second BNS, GW190425 [6]; (iii) the first black hole (BH) – neutron star (NS) candidate, GW190814 [16], and (iv) GW190521, which is the most massive system ever observed with GWs [17, 18]. Also, GW190521 is the first BBH event with a possible electromagnetic counterpart [152]. This growing sample represents a “Rosetta stone” to unravel the formation of binary compact objects. Astrophysicists have learned several revolutionary concepts about compact objects from GW detections. Firstly, GW150914 has confirmed the existence of BBHs, i.e., binary systems composed of two BHs. BBHs have been predicted a long time ago (e.g., [48, 51, 80, 187, 273, 306, 307, 337, 342]), but GW150914 is their first observational confirmation [3]. Secondly, GW detections show that a number of BBHs are able to merge within a Hubble time. Thirdly, GW170817 has confirmed the connection between short gamma-ray bursts, kilonovae, and BNS mergers [10]. Finally, most of the LIGO–Virgo BHs observed so far host BHs with mass in excess of 20 M . The very first detection, GW150914, has component masses equal +3.0 to m1 = 35.6+4.7 −3.1 M and m2 = 30.6−4.4 M [4]. This was a genuine surprise for the astrophysicists [3], because the only stellar BHs for which we have a dynamical mass measurement, i.e., about a dozen of BHs in X-ray binaries, have mass ≤20 M [243, 258, 260]. Moreover, most theoretical models available 5 years ago did not predict the existence of BHs with mass mBH > 30 M (but see [43, 128, 159, 220, 227, 228, 313, 370, 378] for a few exceptions). Thus, the first GW detections have urged the astrophysical community to deeply revise the models of BH formation and evolution. The recently published O3a events add complexity to this puzzle. In particular, the secondary component of GW190814, with a mass m2 = 2.59+0.08 −0.09 M [16], is either the lightest BH or the most massive NS ever observed, questioning the proposed existence of a mass gap between 2 and 5 M [111, 260]. The merger product of GW190521 (mf = 142+28 −16 M , [17, 18]) is the first intermediate-mass BH (IMBH) ever detected by LIGO and Virgo and is the first BH with mass in the range ∼100 − 1000 M ever observed not only with GWs but also in the electromagnetic spectrum. Moreover, the mass of the primary component, m1 = 85+21 −14 M , falls inside the predicted pair-instability mass gap, as we will discuss later in this review. According to the re-analysis of [254] (see also [115]), GW190521 could be an intermediate-mass ratio inspiral with primary (secondary) +33 mass 168+15 −61 M (16−3 M ), when assuming a uniform in mass ratio prior. This interpretation avoids a violation of the mass gap but requires the formation of an IMBH to explain the primary component. LIGO and Virgo do not observe only masses; they also allow us to extract information on spins and merger rates. The local BBH merger rate density inferred

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+54 −3 yr−1 within the 90% credible interval, from GWTC-2 is 23.9+14.9 −8.6 (58−29 ) Gpc when GW190814 is included in (excluded from) the sample of BBHs [335]. The LVC also estimated an upper limit for the merger rate density of BH–NS binaries (RBHNS < 610 Gpc−3 yr−1 , [7]) and inferred a BNS merger rate density −3 yr−1 within the 90% credible interval [335]. RBNS = 320+490 −240 Gpc The uncertainties on the spins of each binary component are still too large to draw strong conclusions, apart from very few cases (e.g., the spin of the primary component of GW190814 is close to zero; see Figure 6 of [16]). However, LIGO and Virgo allow to give an estimate of two spin combinations, which are called effective spin (χeff ) and precessing spin (χp ). The effective spin is defined as

χeff =

(m1 χ 1 + m2 χ 2 ) L · , m1 + m2 L

(1)

where m1 and m2 (χ 1 and χ 2 ) are the masses (dimensionless spin parameters) of the primary and secondary component of the binary, respectively, and L is the Newtonian orbital angular momentum vector of the binary. The dimensionless spin parameters are defined as χi ≡ Si c/(G m2i ), where Si is the spin vector, c the speed of light, and G the gravity constant. Hence, the effective spin can take any values between −1 and 1, where χeff = 1 (−1) means that the two BHs are both maximally rotating and perfectly aligned (anti-aligned) with respect to the orbital angular momentum of the BBH, while χeff = 0 indicates either that the two BHs are perfectly non-rotating or that both spins lie in the plane of the BBH orbit. The precessing spin is defined as χp =

c max (B1 S1⊥ , B2 S2 ⊥ ), B1 G m21

(2)

where B1 = 2 + 3 q/2 and B2 = 2 + 3/(2 q), q = m2 /m1 ≤ 1, S1⊥ , and S2⊥ are the components of the spin vectors perpendicular to the orbital angular momentum. Hence, χp can take values between 0 (no spin components in the orbital plane) and 1 (at least one spin being maximal and lying in the orbital plane). χp is called precessing spin because spin components misaligned with respect to the orbital angular momentum of the binary drive precession [304]. In current GW observations, small values of χeff are preferred, and χp is unconstrained [7, 8], with a few exceptions. GW151226, GW170729, GW190412, and GW190425, together with a few candidate events (e.g., GW190517_055101), show support for positive values of χeff [7, 14, 15]. If we look at the overall population of GWTC-2 BBHs [335], ∼12% to 44% of BBH systems have spins tilted by more than 90◦ with respect to their orbital angular momentum, supporting a negative effective spin parameter. The precessing spin of GW190814 has a strong upper bound χp < 0.07 [16]. Finally, GW190521 shows mild evidence for a non-zero precessing spin (χp = 0.68+0.25 −0.37 within the 90% credible interval, [18]). In contrast, spin measurements in X−ray binaries point to a range of spin magnitudes, including high spins [242,243].

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This review discusses the formation channels of BHs and BBHs in light of the challenges posed by recent GW detections. This field has witnessed an exponential growth of publications and models in the last few years. While I will try to give an overview as complete as possible of the main models and connected issues, it would be impractical to mention every interesting study in the span of this review.

The Formation of Compact Remnants from Single Stellar Evolution and Supernova Explosions Black holes (BHs) and neutron stars (NSs) are expected to form as remnants of massive (8 M ) stars. An alternative theory predicts that BHs can also form from gravitational collapse in the early Universe (the so-called primordial BHs, e.g., [53, 66, 67, 168]). In this review, we will focus on BHs of stellar origin. The mass function of BHs is highly uncertain, because it may be affected by a number of barely understood processes. In particular, stellar winds and supernova (SN) explosions both play a major role on the formation of compact remnants. Processes occurring in close binary systems (e.g., mass transfer and common envelope) are a further complication and will be discussed in the next section.

Stellar Winds and Stellar Evolution Stellar winds are outflows of gas from the atmosphere of a star. In cold stars (e.g., red giants and asymptotic giant branch stars), they are mainly induced by radiation pressure on dust, which forms in the cold outer layers (e.g., [347]). In massive hot stars (O and B main sequence stars, luminous blue variables, and Wolf–Rayet stars), stellar winds are powered by the coupling between the momentum of photons and that of metal ions present in the stellar photosphere. A large number of strong and weak resonant metal lines are responsible for this coupling (see, e.g., [186] for a review). Understanding stellar winds is tremendously important for the study of compact objects, because mass loss determines the pre-SN mass of a star (both its total mass and its core mass), which in turn affects the outcome of an SN explosion [43, 127, 129, 220, 227]. Early work on stellar winds (e.g., [13, 185, 195]) highlighted that the mass loss of O and B stars depends on metallicity as m ˙ ∝ Z α (with α ∼ 0.5 − 1.0, depending on the model). However, such early work did not account for multiple scattering, i.e., for the possibility that a photon interacts several times before being absorbed or leaving the photosphere. Vink et al. (2001, [353]) accounted for multiple scatterings p and found a universal metallicity dependence m ˙ ∝ Z 0.85 v∞ , where v∞ is the terminal velocity and p = −1.23 (p = −1.60) for stars with effective temperature Teff  25000 K (12000 K  Teff  25000 K). The situation is more uncertain for post-main sequence stars. For Wolf–Rayet (WR) stars, i.e., naked helium cores, [355] predict a similar trend with metallicity m ˙ ∝ Z 0.86 . With a different numerical approach (which accounts also for wind

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Fig. 1 Evolution of stellar mass as a function of time for a star with ZAMS mass mZAMS = 90 M and seven different metallicities, ranging from 0.005 Z up to Z (we assumed Z = 0.02). These curves were obtained with the SEVN population-synthesis code [313], adopting PARSEC stellar evolution tracks [70]

clumping), [149] find a strong dependence of WR mass loss on metallicity but also on the electron-scattering Eddington factor Γe = κe L /(4 π c G m), where κe is the cross section for electron scattering, L is the stellar luminosity, c is the speed of light, G is the gravity constant, and m is the stellar mass. The importance of Γe has become increasingly clear in the last few years [150, 352, 354], but only few stellar evolution models include this effect. For example, [70, 327] adopt a mass-loss prescriptions for massive hot stars (O and B stars, luminous blue variables, and WR stars) that scale as m ˙ ∝ Z α , where α = 0.85 if Γe < 2/3, α = 2.45 − 2.4 Γe if 2/3 ≤ Γe ≤ 1, and α = 0.05 if Γe > 1. This simple formula accounts for the fact that metallicity dependence tends to vanish when the star is close to be radiation pressure dominated, as clearly shown by Figure 10 of [149]. Figure 1 shows the mass evolution of a star with zero-age main sequence (ZAMS) mass mZAMS = 90 M for seven different metallicities, as obtained with the SEVN code [313]. At the end of its life, a solar metallicity star (here we assume Z = 0.02) has lost more than 2/3 of its initial mass, while the most metal-poor star in the Figure (Z = 0.005 Z ) has retained almost all its initial mass. Other aspects of massive star evolution also affect the pre-SN mass of a star. For example, surface magnetic fields appear to strongly quench stellar winds by magnetic confinement [136, 175, 268]. In particular, [268] show that a nonmagnetic star model with metallicity ∼0.1 Z and a magnetic star model with solar metallicity and Alfvén radius RA ∼ 4 R undergo approximately the same mass loss according to this model. This effect cannot be neglected because surface magnetic fields are detected in ∼10 percent of the hot stars [359] but is currently not included in models of compact object formation. Finally, rotation affects the evolution of a massive star in several ways (e.g., [71, 72, 197, 198, 212]). As a general rule of thumb, rotation increases the stellar luminosity. This implies that mass loss is generally enhanced if rotation is accounted for. On the other hand, rotation also induces chemical mixing, which leads to the formation of larger helium and carbon-oxygen cores. While enhanced mass loss implies smaller pre-SN masses, the formation of bigger cores has strong

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implications for the final fate of a massive star, as we discuss in the following section.

Core-Collapse Supernova (SN) or Direct Collapse Whether a star undergoes a successful core-collapse SN or a failed SN is the first key question to address, in order to assess the properties of the final compact object. A star undergoing a successful core-collapse SN explosion will leave a NS or a light BH, while stars that end their life with a failed SN will become rather massive BHs (>20 M ), because most (if not all) of the final mass of the star collapses into a BH directly. Addressing this question is a challenge for several reasons. The mechanisms triggering iron core-collapse SNe are still highly uncertain. The basic framework and open issues are the following. As the mass of the central degenerate core reaches the Chandrasekhar mass [68], the degeneracy pressure of relativistic electrons becomes insufficient to support it against collapse. Moreover, electrons are increasingly removed, because protons capture them producing neutrons and neutrinos. This takes the core into a new state, where matter is essentially composed of neutrons, which support the core against collapse by their degeneracy pressure. To reach this new equilibrium, the core collapses from a radius of few thousand km down to a radius of few ten km in less than a second. The gravitational energy gained from the collapse is W ∼ 5 × 1053 erg (mPNS /1.4 M )2 (10 km/RPNS ), where mPNS and RPNS are the mass and radius of the proto-neutron star (PNS). The main problem is to explain how this gravitational energy can be – at least partially – transferred to the stellar envelope triggering the SN explosion [52, 79]. Several mechanisms have been proposed, including rotationally driven SNe and/or magnetically driven SNe (see, e.g., [116, 173] and references therein). The most commonly investigated mechanism is the convective SN engine (see, e.g., [128]). According to this model, the collapsing core drives a bounce shock. For the SN explosion to occur, this shock must reverse the supersonic infall of matter from the outer layers of the star. Most of the energy in the shock consists in a flux of neutrinos. As soon as neutrinos are free to leak out (because the shock has become diffuse enough), their energy is lost and the shock stalls. The SN occurs only if the shock is revived by some mechanism. In the convective SN scenario, the region between the PNS surface and the shock stalling radius can become convectively unstable (e.g., because of a Rayleigh–Taylor instability). Such convective instability can convert the energy leaking out of the PNS in the form of neutrinos to kinetic energy pushing the convective region outward. If the convective region overcomes the ram pressure of the infalling material, the shock is revived and an explosion is launched. If not, the SN fails. While this is the general idea of the convective engine, fully self-consistent simulations of core collapse with a state-of-the-art treatment of neutrino transport do not lead to explosions in spherical symmetry except for the lighter SN progenitors (10 M , [105,116]). Simulations which do not require the assumption of spherical

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symmetry (i.e., run at least in 2D) appear to produce successful explosions from first principles for a larger range of progenitor masses (see, e.g., [248, 249]). However, 2D and 3D simulations are still computationally challenging and cannot be used to make a study of the mass distribution of compact remnants. Thus, in order to study compact object masses, SN explosions are artificially induced by injecting in the pre-SN model some amount of kinetic energy (kinetic bomb) or thermal energy (thermal bomb) at an arbitrary mass location. The evolution of the shock is then followed by means of 1D hydrodynamical simulations with some relatively simplified treatment for neutrinos. This allows to simulate hundreds of stellar models. Following this approach, O’Connor & Ott (2011, [256]) propose a criterion to decide whether a SN is successful or not, based on the compactness parameter: ξm =

m/M , R(m)/1000 km

(3)

where R(m) is the radius which encloses a given mass m. Usually, the compactness is defined for m = 2.5 M (ξ2.5 ). [256] measure the compactness at core bounce ( [344] show that ξ2.5 is not significantly different at core bounce or at the onset of collapse.) in their simulations and find that the larger the ξ2.5 is, the shorter is the time to form a BH (as shown in their Figure 6). This means that stars with a larger value of ξ2.5 are more likely to collapse to a BH without SN explosion. The work by Ugliano et al. (2012, [344]) and Horiuchi et al. (2014, [166]) indicates that the best threshold between exploding and non-exploding models is ξ2.5 ∼ 0.2. Ertl et al. (2016, [105]) indicate that a single criterion (e.g., the compactness) cannot capture the complex physics of core-collapse SN explosions. They introduce a two-parameter criterion based on M4 =

m(s = 4) M

 and

μ4 =

dm/M dR/1000 km

 ,

(4)

s=4

where M4 is the mass (at the onset of collapse) where the dimensionless entropy per baryon is s = 4 and μ4 is the spatial derivative at the location of M4 . This choice is motivated by the fact that, in their 1D simulations, the explosion sets shortly after M4 has fallen through the shock and well before the shell enclosing M4 + 0.3 M has collapsed. They show that exploding models can be distinguished from nonexploding models in the μ4 versus M4 μ4 plane (see their Figure 6) by a linear fit y(x) = k1 x + k2

(5)

where y(x) = μ4 , x = M4 μ4 , and k1 and k2 are numerical coefficients which depend on the model (see Table 2 of [105]). The reason of this behavior is that μ4 scales with the rate of mass infall from the outer layers (thus the larger the μ4 is, the lower the chance of the SN to occur), while M4 μ4 scales with the neutrino

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luminosity (thus the larger the M4 μ4 is, the higher the chance of a SN explosion). Finally, [105] find that fallback is quite inefficient (< 0.05 M ) when the SN occur. The models proposed by O’Connor & Ott (2011, [256]) and Ertl et al. (2016, [105]; see also [106, 266, 324, 325]) are sometimes referred to as the “islands of explodability” scenario, because they predict a non-monotonic behavior of SN explosions with the stellar mass. This means, for example, that while a star with a mass m = 25 M and a star with a mass m = 29 M might end their life with a powerful SN explosion, another star with intermediate mass between these two (e.g., with a mass m = 27 M ) is expected to directly collapse to a BH without SN explosion. Thus, these models predict the existence of islands of explodability, i.e., ranges of mass where a star is expected to explode, surrounded by mass intervals in which the star will end its life with a direct collapse. The models discussed so far depend on quantities (ξ2.5 , M4 , μ4 ) which can be evaluated no earlier than the onset of core collapse. Thus, stellar evolution models are required which integrate a massive star till the iron core has formed. This is prohibitive for most stellar evolution models (with few remarkable exceptions, e.g., FRANEC [71] and MESA [265]). Fryer et al. (2012, [128]) propose a simplified approach (see also [46,127,129]). They suggest that the mass of the compact remnant depends mostly on two quantities: the carbon-oxygen core mass mCO and the total final mass of the star mfin . In particular, mCO determines whether the star will undergo a core-collapse SN or will collapse to a BH directly (viz., stars with mCO > 11 M collapse to a BH directly), whereas mfin determines the amount of fallback on the PNS. In this formalism, the only free parameter is the time to launch the shock. The explosion energy is significantly reduced if the shock is launched 250 ms after the onset of the collapse (delayed SN explosion) with respect to an explosion launched in the first ∼250 ms (rapid SN explosion, [128]). While this approach is quite simplified with respect to other prescriptions, [198] and [230] show that there is a strong correlation between the final carbon-oxygen mass and the compactness parameter ξ 2.5 at the onset of collapse, regardless of the rotation velocity of the progenitor star (see Figure 1 of [230]). Thus, we can conclude that the simplified models by [128] can effectively describe the overall trend of a collapsing star, although they do not take into account several details of the stellar structure at the onset of collapse. Recently, [262] propose an interesting alternative approach. They integrate a large grid of naked carbon-oxygen (CO) cores to the onset of core collapse and estimate the explodability of each model with the compactness ξ 2.5 [256] and with the parameter M4 [106]. Naked CO cores are faster and simpler to evolve than full stellar models (with hydrogen and helium) and are less sensitive to metallicity. They made available their grid of simulations for implementation into populationsynthesis codes. Other works (e.g., [77, 216, 217]) highlight the stochasticity of the final direct collapse or core-collapse SN. Finally, even the most advanced formalisms to derive the explodability of massive stars should be taken with a grain of salt, because of the complexity of the processes involved in core-collapse SNe and because of the simplifications still included in the models [63].

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Even if we were in the conditions to tell if a given star undergoes a failed SN instead of a successful SN, this would not mean we can automatically infer the final mass of the compact object. In the case of a failed SN, the main uncertainty on the final compact object mass is represented by the fate of the envelope [230]. In fact, the envelope of a massive giant star is rather loosely bound, and even a small energy injection can unbind a fraction of it. Fernandez et al. (2018, [113]) show that a 0.1−0.5 M neutrino emission during the PNS phase causes a decrease in the gravitational mass of the core, resulting in an outward going pressure wave (sound pulse) that steepens into a shock as it travels out through the star. This might cause the ejection of a fraction of the loosely bound stellar envelope. According to [113], the ejected mass is a monotonically decreasing function of the envelope compactness (Figure 6 of [113]), defined as ξenv ≡

Mcc /M , Rcc /R

(6)

where Mcc and Rcc are the total mass and radius of the star at the onset of core collapse. With this formalism, the ejected mass is up to a few M for red supergiant stars and 1 M for more compact stars like blue super-giant stars and WR stars. The possibility of a direct collapse is supported by observations. A survey conducted with the Large Binocular Telescope to find quietly disappearing stars [137, 178] reported evidence for the disappearance of a ∼ 25 M red super-giant star [21]. In addition, surveys of SNe indicate a dearth of red super-giant progenitors with mass > 20 M associated with Type IIp SNe [177].

Pair Instability and the Mass Gap If the helium core of a star grows above ∼30 M and the core temperature is 7 × 108 K, the process of electron–positron pair production becomes effective. It removes photon pressure from the core producing a sudden contraction of the carbon-oxygen core, before the formation of an iron core [40, 118, 281, 367]. For mHe > 135 M , the contraction cannot be reversed and the star collapses directly into a BH [367]. If 135  mHe  64 M , the collapse triggers an explosive burning of heavier elements, especially oxygen and silicon. This leads to a pair-instability SN (PISN): the star is completely disrupted, leaving no compact remnant [159]. For 64  mHe  32 M , pair production induces a series of pulsations of the core (pulsational pair-instability), which trigger an enhanced mass loss [367]. At the end of this instability phase, the star finds a new equilibrium and evolves toward corecollapse: a compact object with non-zero mass is produced, less massive than we would expect without pulsational mass loss. The main effect of (pulsational) pair instability is to open a gap in the mass spectrum of BHs between approximately ∼40 − 70 M and ∼120 M [44, 140, 142,233,234,312,321,348,367,368]. The large uncertainty on the edges of the mass

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gap is due to our poor understanding of the physics of massive stars. In particular, [109] and [108] integrate pure-He stars with MESA [263–265] and show that the uncertainties on the 12 C(α, γ )16 O reaction rate [87] are responsible for a change in the lower edge of the mass gap of more than ∼15 M . Mapelli et al. (2020, [230]) show that assuming that the hydrogen envelope collapses to the final BH can lead to an increase of the lower edge of the mass gap from ∼40 M to ∼65 M . Finally, [81] combine the uncertainties on the 12 C(α, γ )16 O reaction rate with the uncertainties on the collapse of the hydrogen envelope and find that the mass gap progressively reduces from ∼80 − 150 M (for a rate computed with the standard 12 C(α, γ )16 O reaction rate −1 σ ) to ∼92 − 110 M (for a standard rate − 2 σ ) and even disappears (for a standard rate −3 σ ) as an effect of convection and envelope undershooting. These uncertainties leave open the possibility that the primary mass of GW190521 is the result of stellar evolution [47, 81, 330, 331].

The Mass of Compact Remnants The previous sections suggest that our knowledge of the compact remnant mass is hampered by severe uncertainties, connected with both stellar winds and SNe. Thus, models of the mass spectrum of compact remnants must be taken with a grain of salt. However, few robust features can be drawn. Figure 2 is a simplified version of Figures 2 and 3 of Heger et al. (2003 [158]). The final mass of a star and the mass of the compact remnant are shown as a function of the ZAMS mass. The left- and the right-hand panels show the case of solar metallicity and metal-free stars, respectively. In the case of solar metallicity stars, the final mass of the star is much lower than the initial one, because stellar winds are extremely efficient. The mass of the compact remnant is also much lower than the final mass of the star because a core-collapse SN always takes place. In contrast, a metal-free star (i.e., a population III star) loses a negligible fraction of its mass by stellar winds (the blue and the black line in Fig. 2 overlap). As for the mass of the compact remnant, Fig. 2 shows that there are two regimes: below a given threshold (≈30 − 40 M ), the SN explosion succeeds even at zero metallicity, and the mass of the compact remnant is relatively small. Above this threshold, the mass of the star (in terms of both core mass and envelope mass) is sufficiently large that the SN fails. Most of the final stellar mass collapses to a BH, whose mass is significantly larger than in the case of a SN explosion. The only exception is represented by the pair-instability window: single metal-free stars with ZAMS mass mZAMS ∼ 140 − 260 M undergo a PISN and are completely destroyed, while single metal-free stars with mZAMS ∼ 110 − 140 M undergo pulsational pairinstability and leave smaller compact objects. In this simplified cartoon, we neglect the existence of islands of explodability. What happens at intermediate metallicity between solar and zero? Predicting what happens to a metal-free star is relatively simple, because its evolution does not depend on the interplay between metals and stellar winds. The fate of a solar

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Fig. 2 Final mass of a star (mfin , blue lines) and mass of the compact remnant (mrem , red lines) as a function of the ZAMS mass of the star. The thick black line marks the region where mfin = mZAMS . Left-hand panel: solar metallicity stars. Right-hand panel: metal-free stars. The red arrow on the left-hand panel is an upper limit for the remnant mass. Vertical thin black line in the right-hand panel: approximate separation between successful and failed SNe at Z = 0. This cartoon was inspired by Figures 2 and 3 of Heger et al. (2003 [158])

metallicity star is more problematic, because we must account for line-driven stellar winds, but most observational data about stellar winds are for nearly solar metallicity stars and allow us to calibrate our models for such high metallicity. In contrast, modelling intermediate metallicities is significantly more complicated, because the details depend on the interplay between metals and stellar winds and only limited data are available for calibration (mostly data for the Large and Small Magellanic Clouds, e.g., [298]). As a rule of thumb (see, e.g., [128, 313]), we can draw the following considerations. If the zero-age main sequence (ZAMS) mass of a star is large (mZAMS  30 M ), then the amount of mass lost by stellar winds is the main effect which determines the mass of the compact remnant. At low metallicity (0.1 Z ) and for a low Eddington factor (Γe < 0.6), mass loss by stellar winds is not particularly large. Thus, the final mass mfin and the carbon-oxygen mass mCO of the star may be sufficiently large to avoid a core-collapse SN explosion: the star may form a massive BH (20 M ) by direct collapse, unless a pair-instability or a pulsational pair-instability SN occurs. At high metallicity (≈ Z ) or large Eddington factor (Γe > 0.6), mass loss by stellar winds is particularly efficient and may lead to a small mfin and mCO : the star is expected to undergo a core-collapse SN and to leave a relatively small remnant. If the ZAMS mass of a star is relatively low (8 < mZAMS < 30 M ), then stellar winds are not important (with the exception of asymptotic giant branch stars),

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regardless of the metallicity. In this case, the details of the SN explosion (e.g., energy of the explosion and amount of fallback) are crucial to determine the final mass of the remnant. This general sketch may be affected by several factors, such as pairinstability SNe, pulsational pair-instability SNe (e.g., [367]), and an island scenario for core-collapse SNe (e.g., [105]). The effect of pair-instability and pulsational pair-instability SNe is clearly shown in Fig. 3. The top panel was obtained accounting only for stellar evolution and

Fig. 3 Mass of the compact remnant (mrem ) as a function of the ZAMS mass of the star (mZAMS ). Lower (upper) panel: pulsational pair-instability and pair-instability SNe are (are not) included. In both panels: dash-dotted brown line, Z = 2.0 × 10−2 ; dotted dark orange line, Z = 1.7 × 10−2 ; dashed red line, Z = 1.4 × 10−2 ; solid red line, Z = 1.0 × 10−2 ; short dash-dotted orange line, Z = 8.0 × 10−3 ; short dotted light orange line, Z = 6.0 × 10−3 ; short dashed green line, Z = 4.0 × 10−3 ; dash-double-dotted green line, Z = 2.0 × 10−3 ; dash-dotted light blue line, Z = 1.0 × 10−3 ; dotted blue line, Z = 5.0 × 10−4 ; dashed violet line, Z = 2.0 × 10−4 . A delayed core-collapse SN mechanism has been assumed, following the prescriptions of [128]. (This Figure was adapted from Figures 1 and 2 of Spera & Mapelli (2017, [312]))

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core-collapse SNe. In contrast, the bottom panel also includes pair-instability and pulsational pair-instability SNe. This figure shows that the mass of the compact remnant strongly depends on the metallicity of the progenitor star if mZAMS  30 M . In most cases, the lower the metallicity of the progenitor is, the larger the maximum mass of the compact remnant [43,158,220,227,228,312,313]. However, for metal-poor stars (Z < 10−3 ) with ZAMS mass 230 > mZAMS /M > 110 pairinstability SNe lead to the complete disruption of the star and no compact remnant is left. Only very massive (mZAMS > 230 M ) metal-poor (Z < 10−3 ) stars can collapse to a BH directly, producing intermediate-mass BHs (i.e, BHs with mass 100 M ). If Z < 10−3 and 110 > mZAMS  60 M , the star enters the pulsational pair-instability SN regime: mass loss is enhanced, and the final BH mass is smaller (mBH ∼ 30 − 55 M , bottom panel of Fig. 3) than we would have expected from direct collapse (mBH ∼ 50 − 100 M , top panel of Fig. 3). Finally, the mass spectrum of relatively low-mass stars (8 < mZAMS < 30 M ) is not significantly affected by metallicity. The assumed core-collapse SN model is the most important factor in this mass range [128]. The models presented here do not take into account stellar rotation. Studied in [198, 230, 233], stellar rotation produces larger cores because of chemical mixing. This shifts the minimum ZAMS mass for a star to undergo pulsational pairinstability and PISN to lower values, because of the larger He core masses. The net result is a decrease of the maximum BH mass for fast-rotating stars (≥ 150 km s−1 ) with respect to low-rotating stars [230].

Compact Object Spins In the previous sections, we have seen that the connection between the mass of a BH and the properties of its progenitor star is still highly uncertain. Our knowledge on the origin of BH spins is even more uncertain. It is reasonable to assume that a compact object inherits the spin of its progenitor (or at least of the core of its progenitor) if the progenitor collapses to a BH directly, without any SN explosion. In contrast, mass ejection during a SN explosion can significantly dissipate part of the final spin of the progenitor star. Hence, if the final spin of the progenitor star is not negligible, we would expect large birth spins for the most massive BHs, which form from direct collapse, and low birth spins for NSs and light BHs, which form from successful SN explosions. Observations of Galactic pulsars seem to confirm the idea that NSs are born with relatively low χ . Young pulsars (i.e., pulsars which did not have much time to slow down or to be recycled) have a value of the spin parameter χ  0.01 [180,214]. This indicates that most of the spin of the progenitor was lost either before or during the SN. In contrast, millisecond pulsars are significantly spun up by mass accretion from a companion star: the fastest millisecond pulsar (B1937+21, [161]) has χ  0.4, but this value has almost nothing to do with the initial spin of the NS. On the other hand, the observed spins of BHs make us “scratch our head.” Relatively low-mass BHs in high-mass X-ray binaries seem to show very large spins

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[147, 148, 199, 243]. Considering that these BHs did not have enough time to spin up by mass accretion (for the short lifetime of their companion stars), this suggests large spins at birth. In contrast, LIGO–Virgo results support low values of χeff for most of the BBHs [8], which can be interpreted either as strongly misaligned spins with respect to the binary angular momentum or as very low values of χ . From a theoretical perspective, the key question is then: what is the final spin of the progenitors of BHs and NSs? The final spin of a massive single star depends on mass loss and angular momentum transport. A mildly efficient transport by meridional currents (as adopted in, e.g., [101, 198]) leads to a non-negligible final spin of the star and to a high spin of the BH, if born from direct collapse [47,230]. In contrast, [131] investigate efficient angular momentum transport via the magnetic Tayler instability [132,318] and find extremely slow spins (χ ∼ 10−2 ) for BHs born from single stars. If the magnetic Tayler instability is the dominant process for angular momentum transfer in massive stars, BH spins larger than χ ∼ 10−2 can be produced only by tidal torques in binary stars [190, 278] or by chemically homogeneous evolution [211, 369, 374]. Overall, the initial rotation speed of very massive stars and the process of angular momentum transfer remain uncertain, hampering the predictive power of theoretical models on the spin distribution of BHs.

Natal Kicks Compact objects are expected to receive a natal kick from the parent SN explosion, because of asymmetries in the neutrino flux and/or in the ejecta (see [173] for a review). The natal kick has a crucial effect on the evolution of a binary compact object, because it can either unbind the binary or change its orbital properties. For example, a SN kick can increase the orbital eccentricity or misalign the spins of the two members of the binary. Unfortunately, it is extremely difficult to quantify natal kicks from state-of-theart SN simulations, and measurements of natal kicks are scanty, especially for BHs. As to NSs, indirect observational estimates of SN kicks give contrasting results. Hobbs et al. (2005, [164]) found that a single Maxwellian with root mean square σ CCSN = 265 km s−1 can match the proper motions of 233 single pulsars in the Milky Way. Other works suggest a bimodal velocity distribution, with a first peak at low velocities (e.g., ∼0 km s−1 according to [126] or ∼90 km s−1 according to [32]) and a second peak at high velocities (>600 km s−1 according to [126] or ∼500 km s−1 for [32]). Similarly, Verbunt et al. (2017, [350]) indicate that a double Maxwellian distribution provides a significantly better fit to the observed velocity distribution than a single Maxwellian. Finally, the analysis of Beniamini & Piran (2016, [50]) shows that low kick velocities (30 km s−1 ) are required to match the majority of Galactic BNSs, especially those with low eccentricity. A possible interpretation of these observational results is that natal kicks depend on the SN mechanism (e.g., electron-capture versus core-collapse SN, e.g., [141]) or on the binarity of the NS progenitor. For example, if the NS progenitor evolves in a close binary system (i.e., in a binary system where the two stars have exchanged

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mass with each other, see section “Mass Transfer”), it might undergo an ultrastripped SN (see [332] and references therein for more details). A star can undergo an ultra-stripped SN explosion only if it was heavily stripped by mass transfer to a companion [333, 334]. The natal kick of an ultra-stripped SN should be low [334], because of the small mass of the ejecta (0.1 M ). Low kicks (50 km s−1 ) for ultra-stripped core-collapse SNe are also confirmed by recent hydrodynamical simulations [172, 326]. As to BHs, the only indirect measurements of natal kicks arise from spatial distributions, proper motions, and orbital properties of BHs in X-ray binaries (e.g., [244]). Evidence for a relatively small natal kick has been found for both GRO J1655–40 [364] and Cygnus X-1 [366], whereas H 1705–250 [284, 285] and XTE J1118 + 480 [123, 245] require high kicks (>100 km s−1 ). By analyzing the position of BHs in X-ray binaries with respect to the Galactic plane, Repetto et al. (2012, [284]) suggest that BH natal kicks should be as high as NS kicks. Repetto et al. (2017, [285]) perform a similar analysis but accounting also for binary evolution and find that at least two BHs in X-ray binaries (H 1705–250 and XTE J1118 + 480) require high kicks. Most models of BBH evolution assume that natal kicks of BHs are drawn from the same distribution as NS kicks but reduced by some factor. For example, linear momentum conservation suggests that vBH =

mNS vNS , mBH

(7)

where vBH is the natal kick of a BH with mass mBH and vNS is the natal kick of a NS with mass mNS . Alternatively, the natal kick can be reduced by the amount of fallback, under the reasonable assumption that fallback quenches the initial asymmetries. Following [128], vBH = (1 − ffb ) vNS ,

(8)

where ffb quantifies the fallback (ffb = 0 for no fallback and ffb = 1 for direct collapse). Most studies assume that BHs born from direct collapse receive no kick [128]. The model by [143] can unify BH kicks and NS kicks, naturally accounting for ultra-stripped SNe and electron-capture SNe, which mostly lead to reduced kicks (e.g., [326, 332]). According to this toy model, the kick of a compact object can be described as vk = fH05

mNS  mej , mrem mej 

(9)

where mNS  is the average NS mass, mrem is the mass of the considered compact remnant, mej is the mass of the ejecta, mej  is the ejecta mass of a SN that

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leaves a single NS with mass mNS , and fH05 is a number randomly drawn from a Maxwellian probability density curve with one-dimensional root-meansquare velocity dispersion σ 1D = 265 km s−1 [164]. To derive this model, [143] assume that the Maxwellian distribution fitted by [164] is a good proxy to the distribution of the kicks of NSs born from single stars and that kicks are the effect of asymmetries in mass ejecta (∝ mej ; see also [60, 61, 328]), modulated by linear momentum conservation (∝ m−1 rem ). This new simple formalism seems to solve the tension between the local merger rate of BNSs derived from gravitational wave interferometers [6, 8] and the proper motions of Galactic young pulsars [164]. In summary, natal kicks are one of the most debated issues about compact objects. Their actual amount has dramatic implications on the merger rate and on the properties (spin and mass distribution) of merging compact objects.

Binaries of Stellar Black Holes Naively, one could think that if two massive stars are members of a binary system, they will eventually become a BBH and the mass of each BH will be the same as if its progenitor star was a single star. This is true only if the binary system is sufficiently wide (detached binary) for its entire evolution. If the binary is tight enough, it will evolve through several processes which might significantly change its final fate. The so-called binary population-synthesis codes have been used to investigate the effect of binary evolution processes on the formation of BBHs in isolated binaries (e.g., [41, 46, 102–104, 140–143, 167, 183, 223, 228, 239, 270, 276, 314, 320, 331]). These are Monte Carlo-based codes which combine a description of stellar evolution with prescriptions for SN explosions and with a formalism for binary evolution processes. In the following, we mention some of the most important binary evolution processes, and we briefly discuss their treatment in the most used populationsynthesis codes.

Mass Transfer If two stars exchange matter to each other, it means they undergo a mass transfer episode. This might be driven either by stellar winds or by an episode of Rochelobe filling. When a massive star loses mass by stellar winds, its companion might be able to capture some of this mass. This will depend on the amount of mass which is lost and on the relative velocity of the wind with respect to the companion star. Based on the Bondi & Hoyle (1944, [56]) formalism, Hurley et al. (2002, [167]) describe the mean mass accretion rate by stellar winds as 1

m ˙2 = √ 1 − e2



G m2 2 vw

2

αw 1 |m ˙ 1| , 2 2 a [1 + (vorb /vw )2 ]3/2

(10)

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where e is the binary eccentricity, G is the gravitational constant, m2 is the mass of the accreting star, vw is the velocity of the wind, αw √∼ 3/2 is an efficiency constant, a is the semi-major axis of the binary, vorb = G (m1 + m2 )/a is the ˙ 1 is the mass orbital velocity of the binary (m1 being the mass of the donor), and m loss rate by the donor. Since |m ˙ 1 | is usually quite low (|m ˙ 1 | < 10−3 M yr−1 ) and vw is usually quite high (>1000 km s−1 for a line-driven wind) with respect to the orbital velocity, this kind of mass transfer is usually rather inefficient. However, most of the observed high-mass X-ray binaries [345, 346] and in particular all the systems with a WR star companion [107] are wind-fed systems. Mass transfer by Roche lobe overflow is usually more efficient than wind accretion. The Roche lobe of a star in a binary system is a teardrop-shaped equipotential surface surrounding the star. The Roche lobes of the two members of the binary are connected in just one point, which is the Lagrangian L1 point. A widely used approximate formula for the Roche lobe is [99] RL,1 = a

0.49 q 2/3  , 0.6 q 2/3 + ln 1 + q 1/3

(11)

where a is the semi-major axis of the binary and q = m1 /m2 (m1 and m2 are the masses of the two stars in the binary). This formula describes the Roche lobe of a star with mass m1 , while the corresponding Roche lobe of a star with mass m2 (RL,2 ) is obtained by swapping the subscripts. A star overfills (underfills) its Roche lobe when its radius is larger (smaller) than the Roche lobe. If a star overfills its Roche lobe, a part of its mass flows toward the companion star which can accrete (a part of) it. Mass transfer obviously changes the mass of the two stars in a binary, and thus the final mass of the compact remnants of such stars, but also the orbital properties of the binary. If mass transfer is non conservative, which is the most realistic case, it leads to an angular momentum loss, which in turn affects the semi-major axis. Recently, [58] show that the hypothesis of a highly non-conservative mass transfer (mass accretion efficiency fMT ≤ 0.3) is in tension with LVC data, if we assume that all BBHs observed by the LVC form via isolated binary evolution. A crucial information about Roche lobe overflow is whether it is stable or unstable and on which timescale. The most commonly used approach can be described as follows [100, 167, 276, 341, 361]. Let us assume that the stellar radius and mass are connected by a simple relation R ∝ mζ . Thus, the variation of the donor’s radius during Roche lobe is ∂R1 R1 dm1 dR1 = +ζ . dt ∂t m1 dt

(12)

1 In the above equation, the term ∂R ∂t is due to nuclear burning, while ζ measures the 1 adiabatic or thermal response of the donor star to mass loss. Note that dm dt is the mass loss from the donor; hence it is always negative.

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Similarly, the change of the size of the Roche lobe of the donor RL,1 can be estimated as dRL,1 ∂RL,1 RL,1 dm1 = + ζL , dt ∂t m1 dt

(13)

∂R

where ∂tL,1 depends on tides and GW radiation, while ζL describes the response of the Roche lobe to mass loss: the Roche lobe might shrink or expand depending on m1 , m2 , a, m ˙ 1 , and m ˙ 2 . If ζL > ζ , the Roche lobe shrinks faster than the radius of the star does and the mass transfer is unstable; otherwise, it remains stable until the radius changes significantly by nuclear burning. Mass transfer can be unstable either on a dynamical timescale (if ζ describes the adiabatic response of the donor and ζ < ζ L ) or on a thermal timescale (if ζ describes the thermal response of the donor and ζ < ζ L ). If mass transfer is dynamically unstable or both stars overfill their Roche lobe, then the binary is expected to merge, if the donor lacks a steep density gradient between the core and the envelope, or to enter common envelope (CE), if the donor has a clear distinction between core and envelope.

Common Envelope (CE) If two stars enter CE, their envelope(s) stop co-rotating with their cores. The two stellar cores (or the compact object and the core of the companion star, if the binary is already single degenerate) are embedded in the same non-co-rotating envelope and start spiraling in as an effect of gas drag exerted by the envelope. Part of the orbital energy lost by the cores as an effect of this drag is likely converted into heating of the envelope, making it more loosely bound. If this process leads to the ejection of the envelope, then the binary survives, but the post-CE binary is composed of two naked stellar cores (or of a compact object and a naked stellar core). Moreover, the orbital separation of the two cores (or the orbital separation of the compact object and the stellar core) is considerably smaller than the initial orbital separation before the CE, as an effect of the spiral in. (A short-period (from a few hours to a few days) binary system composed of a naked helium core and BH might be observed as an X-ray binary, typically a WR X-ray binary. In the local Universe, we know a few (∼ 7) WR X-ray binaries, in which a compact object (BH or NS) accretes mass through the wind of the naked stellar companion (see, e.g., [107] for more details). These rare X-ray binaries are thought to be good progenitors of merging compact object binaries.) This circumstance is crucial for the fate of a BBH. In fact, if the binary which survives a CE phase evolves into a BBH, this BBH will have a short semi-major axis (a  100 R ), much shorter than the sum of the maximum radii of the progenitor stars, and may be able to merge by GW emission within a Hubble time. In contrast, if the envelope is not ejected, the two cores (or the compact object and the core) spiral in till they eventually merge. This premature merger of a binary

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Fig. 4 Schematic representation of the evolution of a BBH through CE. The companion of the BH is initially in the main sequence (MS). In the cartoon, the BH is indicated by a black circle, while the MS companion is indicated by the light blue circle. When the companion evolves off the MS, becoming a giant star, it overfills its Roche lobe. The BH and the giant star enter a CE (the CE is indicated in orange, while the core of the giant is represented by the dark blue circle). The core of the giant and the BH spirals in because of the gas drag exerted by the envelope. If the envelope is ejected, we are left with a new binary, composed of the BH and the naked helium core of the giant. The new binary has a much smaller orbital separation than the initial binary. If the naked helium core becomes a BH and its natal kick does not disrupt the binary, then a BBH is born, possibly with a small semi-major axis. In contrast, if the envelope is not ejected, the BH and the helium core spiral in, till they merge together. A single BH is left in this case

during a CE phase prevents the binary from evolving into a BBH. The cartoon in Fig. 4 summarizes these possible outcomes. The α formalism [360] is the most common formalism adopted to describe a common envelope. The basic idea of this formalism is that the energy needed to unbind the envelope comes uniquely from the loss of orbital energy of the two cores during the spiral in. The fraction of the orbital energy of the two cores which goes into unbinding the envelope can be expressed as ΔE = α (Eb,f − Eb,i ) = α

G mc1 mc2 2



1 1 − af ai

 ,

(14)

where Eb,i (Eb,f ) is the orbital binding energy of the two cores before (after) the CE phase, ai (af ) is the semi-major axis before (after) the CE phase, mc1 and mc2 are the masses of the two cores, and α is a dimensionless parameter that measures which fraction of the removed orbital energy is transferred to the envelope. If the primary is already a compact object (as in Fig. 4), mc2 is the mass of the compact object. The binding energy of the envelope is Eenv =

G λ



 menv,1 m1 menv,2 m2 , + R1 R2

(15)

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where m1 and m2 are the masses of the primary and the secondary member of the binary, menv,1 and menv,2 are the masses of the envelope of the primary and the secondary member of the binary, R1 and R2 are the radii of the primary and the secondary member of the binary, and λ is the parameter which measures the concentration of the envelope (the smaller the λ is, the more concentrated is the envelope). By imposing ΔE = Eenv , we can derive the value of the final semi-major axis af for which the envelope is ejected. If the resulting af is lower than the sum of the radii of the two cores (or than the sum of the Roche lobe radii of the cores), then the binary will merge during CE; otherwise the binary survives. This means that the larger (smaller) the α is, the larger (smaller) the final orbital separation. Actually, we have known for a long time (see [170] for a review) that this simple formalism is a poor description of the physics of CE, which is considerably more complicated. A healthy treatment of CE should take into account not only the orbital energy of the cores and the binding energy of the envelope but also (i) the thermal energy of the envelope, which is the sum of radiation energy and kinetic energy of gas particles [156], (ii) the recombination energy (as the envelope expands, it cools down, the plasma recombines, and some atoms even form molecules, releasing binding energy, [184]), (iii) the tidal heating/cooling from stellar spin down/up [170], (iv) the nuclear fusion energy [171], (v) the enthalpy of the envelope [169], and (vi) the accretion energy, which might drive outflows and jets [84,202,207,208, 310, 311]. Moreover, the envelope concentration parameter λ cannot be the same for all stars. It is expected to vary wildly not only from star to star but also during different evolutionary stages of the same star. Several authors [201,371] have estimated Eenv directly from their stellar models, significantly improving this formalism. However, even in this case, we cannot get rid of the α parameter. Thus, it is essential to model the physics of CE with analytic models and numerical simulations. A lot of effort has been put on this in the last few years, but the problem remains largely unconquered. Several recent studies investigate the onset of CE, when an unstable mass transfer prevents the envelope from co-rotating with the core and leads to the plunge-in of the companion inside the envelope [203– 206, 351]. Several hydrodynamical simulations model the fast spiral in phase after plungein [257,261,286,287], when the two cores spiral in on a dynamical timescale (≈100 days). At the end of this dynamical spiral in, only a small fraction of the envelope (∼25%, [257]) appears to be ejected in most simulations. When the two cores are sufficiently close that they are separated only by a small gas mass, the spiral in slows down and the system evolves on the Kelvin–Helmholtz timescale of the envelope (≈103−5 years). Simulating the system for a Kelvin–Helmholtz timescale is prohibitive for current three-dimensional simulations (e.g., [193]). Fragos et al. (2019, [124]) reduce the complexity of the problem by simulating the entire CE evolution in just one dimension, with the hydrodynamic stellar evolution code MESA [263–265]. They evolve a binary system composed of a NS and a 12 M red super-giant star

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for a thermal timescale. In their model, envelope ejection is mostly driven by the thermal energy of the envelope and by the orbital energy, while recombination contributes to 10% of the total energy required to eject the envelope. The final system is a NS – naked helium star system with an orbital separation of a few R , i.e., a good progenitor for a merging BNS. Their simulated system can be reproduced by the α formalism if α ≈ 5. The apparently unphysical value of α > 1 is motivated by the fact that orbital energy is only one of the energy sources that participate in envelope ejection.

Alternative Evolution to CE Massive fast-rotating stars can have a chemically homogeneous evolution (CHE): they do not develop a chemical composition gradient because of the mixing induced by rotation. This is particularly true if the star is metal poor, because stellar winds are not efficient in removing angular momentum. If a binary is very tight, the spins of its members are even increased during stellar life, because of tidal synchronization. The radii of stars following CHE are usually much smaller than the radii of stars developing a chemical composition gradient [86, 215]. This implies that even very tight binaries (few tens of solar radii) can avoid CE. Marchant et al. (2016, [232]) simulate tight binaries whose components are fastrotating massive stars. A number of their simulated binaries evolve into contact binaries where both binary components fill and even overfill their Roche volumes. If metallicity is sufficiently low and rotation sufficiently fast, these binaries may evolve as “over-contact” binaries: the over-contact phase differs from a classical CE phase because co-rotation can, in principle, be maintained as long as material does not overflow the L2 point. This means that spiral-in can be avoided, resulting in a stable system evolving on a nuclear timescale. Such over-contact binaries maintain relatively small stellar radii during their evolution (few ten solar radii) and may evolve into a BBH with a very short orbital period. This scenario predicts the formation of merging BHs with relatively large masses (>20 M ), nearly equal mass (q = 1), and with large aligned spins. The latter prediction suggests that the CHE scenario cannot reproduce the observed χeff distribution of GWTC-2 BBHs: the CHE scenario alone cannot explain all the LIGO–Virgo BBHs.

BBH Spins in the Isolated Binary Evolution Model Most evolutionary processes in an isolated binary star (tides, mass transfer, CE) lead to the alignment of the spins of the components to the orbital angular momentum of the binary [167]. The only evolutionary process that can significantly misalign BH spins with respect to the orbital angular momentum is the SN explosion [47, 138, 139, 167, 290]. However, Stegmann & Antonini (2021, [319]) recently proposed a possible spin flip mechanism during mass transfer.

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Under the assumption that the system has no time to re-align spins between the first and the second SN and that SNe do not change the direction of compact object spins but only the direction of the orbital angular momentum of the binary, we can derive the angle between the direction of the spins of the two compact objects and that of the orbital angular momentum of the binary system as [139, 290] cos θ = cos (ν 1 ) cos (ν 2 ) + sin (ν 1 ) sin (ν 2 ) cos (φ),

(16)

where ν i (with i = 1, 2) is the angle between the new (Lnew ) and the old (L) orbital angular momentum after a SN (i = 1 corresponding to the first SN, i = 2 corresponding to the second SN), so that cos (ν) = Lˆ new · Lˆ old , while φ is the phase of the projection of Lˆ into the orbital plane. As shown by [290], the most commonly adopted SN kick models fail producing a significant misalignment. Hence, we expect that the isolated binary evolution model has a preference for BH spins aligned with the orbital angular momentum of the binary, especially in the case of massive BHs which undergo just a failed SN.

Summary of the Isolated Binary Formation Channel In this section, we have highlighted the most important aspects and the open issues of the isolated binary formation scenario, i.e., the model which predicts the formation of merging BHs through the evolution of isolated binaries. For isolated binaries, we mean stellar binary systems which are not perturbed by other stars or compact objects. To summarize, let us illustrate schematically the evolution of an isolated stellar binary (see, e.g., [45, 142, 223, 320]) which can give birth to merging BHs like GW150914 and the other massive BHs observed by the LVC [2, 4, 5, 7, 8, 12]. In the following discussion, several details of stellar evolution have been simplified or skipped to facilitate the reading for non-specialists. Figure 5 shows the evolution of an isolated binary system composed of two massive stars. These stars are gravitationally bound since their birth. Initially, the two stars are both on the main sequence (MS). When the most massive one leaves the MS, its radius starts inflating and can grow by a factor of a hundreds. The most massive star becomes a giant star with a helium core and a large hydrogen envelope. If its radius equals the Roche lobe (equation 11), the system starts a stable mass transfer episode. Some mass is lost by the system, and some is transferred to the companion. After several additional evolutionary stages, the primary collapses to a BH. A direct collapse is preferred with respect to a SN explosion if we want the BH mass to be >20 M (as most BBHs observed by the LVC) and if we want the binary to remain bound (direct collapse implies almost no kick, apart from neutrino mass loss). At this stage, the semi-major axis of the system is still quite large: hundreds to thousands of solar radii. When the secondary also leaves the MS, growing in radius, the system enters a CE phase: the BH and the helium core spiral in. If the orbital energy and the thermal

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Fig. 5 Schematic evolution of an isolated binary star which can give birth to a BBH merger (see, e.g., [45, 142, 223, 320])

energy of the envelope are not sufficient to unbind the envelope, then the BH merges with the helium core leaving a single BH. In contrast, if the envelope is ejected, we are left with a new binary system, composed of the BH and of a stripped naked helium star. The new binary system has a much smaller orbital separation (tens of solar radii) than the pre-CE binary, because of the spiral in. If this new binary system remains bound after the naked helium star undergoes a SN explosion or if the naked helium star is sufficiently massive to directly collapse to a BH, the system evolves into a tight BBH, which might merge within a Hubble time. The most critical quantities in this scenario are the masses of the two stars and also their initial orbital separation (with respect to the stellar radii): a BBH can merge within a Hubble time only if its initial orbital separation is tremendously small (few tens of solar radii, unless the eccentricity is rather extreme, [267]), but a massive star (>20 M ) can reach a radius of several thousand solar radii during its evolution. Thus, if the initial orbital separation of the binary star is tens of solar radii, the binary system merges before it can become a BBH. On the other hand, if the initial orbital separation is too large, the two BHs will never merge. In this scenario, the two BHs can merge only if the initial orbital separation of the progenitor binary

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star is in the range which allows the binary to enter CE and then to leave a short period BBH. This range of initial orbital separations dramatically depends on CE efficiency and on the details of stellar mass and radius evolution. There is one more interesting comment to add. Some stellar wind models predict a maximum BH mass of ∼65 − 70 M from single stellar evolution and from the collapse of the residual hydrogen envelope at solar metallicity (e.g., [140,142]). This means that the maximum total mass of a BBH is mTOT = m1 +m2 ∼ 130−140 M . However, the maximum total mass of a BBH which merges within a Hubble time is only mTOT ∼ 80−90 M [140], as shown in Figure 6. The reason for this difference is that if the initial orbital separation is sufficiently small (a ∼ 102 − 104 R , [314]) to allow mass transfer and CE, then the two stars lose almost all their hydrogen envelope: they might become a BBH that merges within a Hubble time, but their total mass is at most equal to the total mass of the two naked helium cores. In contrast, if the two stars are metal poor (Z  0.0002) and if the initial orbital separation is too large to induce mass transfer and CE, the binary star becomes a BBH almost without mass loss, with a total mass up to 140 M , but the final semimajor axis is too large to allow coalescence by GW emission.

Fig. 6 Upper panel: total mass (mTOT = m1 + m2 ) of all BBHs formed from a stellar population of 2×107 massive binary stars simulated with MOBSE and with different metallicities (from 0.0002 to 0.02) [140]. Lower panel: total mass of the BBHs merging within a Hubble time from the same simulations. See https://mobse-webpage.netlify.app/ for more details on MOBSE and on these runs. (Courtesy of Nicola Giacobbo)

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The Dynamics of Binary Black Holes (BBHs) In the previous sections of this review, we discussed the formation of BBHs as isolated binaries. There is an alternative channel for BBH formation: the dynamical evolution scenario.

Dynamically Active Environments Collisional dynamics is important for the evolution of binaries only if they are in a dense environment (103 stars pc−3 ), such as a star cluster. On the other hand, astrophysicists believe that the vast majority of massive stars (which are BH progenitors) form in star clusters [191, 275, 362, 363]. There are several different flavors of star clusters. Globular clusters [153] are old stellar systems (∼12 Gyr), mostly very massive (total mass MSC ≥ 104 M ) and dense (central density ρc ≥ 104 M pc−3 ). They are sites of intense dynamical processes, such as the gravothermal catastrophe. Globular clusters represent a small fraction of the baryonic mass in the local Universe (1 percent, [157]). Most studies of dynamical formation of BBHs focus on globular clusters (e.g., [28, 33, 34, 49, 62, 93, 94, 119, 181, 219, 273, 289–292, 294–296, 307, 329]). Young star clusters are young (100 Myr), relatively dense (ρc > 103 M pc−3 ) stellar systems and are thought to be the most common birthplace of massive stars [191, 275]. When they disperse by gas evaporation or are disrupted by the tidal field of their host galaxy, their stellar content is released into the field. Thus, it is reasonable to expect that a large fraction of BBHs which are now in the field may have formed in young star clusters, where they participated in the dynamics of the cluster. A fraction of young star clusters might survive gas evaporation and tidal disruption and evolve into older open clusters, like M67. Figure 7 shows a snapshot of an N-body simulation of a young star cluster. Population-synthesis prescriptions are included in this simulation to follow the evolution of binary stars and the formation of BBHs. Studies of BBHs in young and open clusters include [36–39, 88–90, 130, 176, 188, 189, 218, 273, 378]. Finally, nuclear star clusters lie in the nuclei of galaxies. Nuclear star clusters are rather common in galaxies (e.g., [55, 114, 151, 252]), are usually more massive and denser than globular clusters, and may co-exist with super-massive BHs (SMBHs). Stellar-mass BHs formed in the innermost regions of a galaxy could even be “trapped” in the accretion disc of the central SMBH, triggering their merger (see, e.g., [42, 238, 322]). These features make nuclear star clusters unique among star clusters, for the effects that we will describe in the next sections.

Two-Body Encounters, Dynamical Friction, and Core Collapse The main driver of the dynamics of star clusters is gravity force. Gravitational twobody encounters between stars lead to local fluctuations in the potential of the star

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Fig. 7 Snapshot of an N-body simulation of a young star cluster. Different particle sizes and colors refer to different star luminosities and temperatures, as estimated by population-synthesis calculations. This simulation has been included in [228]

cluster and drive major changes in the internal structure of the star cluster over a two-body relaxation timescale [316, 317]: trlx = 0.34

σ3 , G2 m ρ ln Λ

(17)

where σ is the local velocity dispersion of the star cluster, m is the average stellar mass in the star cluster, ρ is the local mass density, G is the gravity force, and ln Λ ∼ 10 is the Coulomb logarithm. The two-body relaxation timescale is the time needed for a typical star in the stellar system to completely lose memory of its initial velocity due to two-body encounters. In star clusters, trlx is much shorter than the Hubble time (trlx ∼ 10 − 100 Myr in young star clusters, [275]), while in galaxies and large-scale structures, it is much longer than the lifetime of the Universe. Hence, close encounters are common in dense star clusters. Dynamical friction is another consequence of gravity force: a massive body of mass M orbiting in a sea of lighter particles feels a drag force that slows down its motion over a timescale [69]: tDF (M) =

3 4 (2 π )1/2

σ3 . G2 ln Λ M ρ(r)

(18)

It is apparent that two-body relaxation and dynamical friction are driven by the same force and are related by

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tDF (M) ∼

m trlx , M

(19)

i.e., dynamical friction happens over a much shorter timescale than two-body relaxation and leads to mass segregation (or mass stratification) in a star cluster. This process speeds up the collapse of the core of a star cluster and can trigger the so-called Spitzer’s instability [315]. Two-body relaxation, dynamical friction, and their effects play a crucial role in shaping the demography of BBHs in star clusters, as we discuss below.

Binary: Single Encounters We now review what are the main dynamical effects which can affect a BBH, starting from binary–single star encounters. Binaries have an energy reservoir, their internal energy: Eint =

1 G m1 m2 μ v2 − , 2 r

(20)

where μ = m1 m2 /(m1 + m2 ) is the reduced mass of the binary (whose components have mass m1 and m2 ), v is the relative velocity between the two members of the binary, and r is the distance between the two members of the binary. As shown by Kepler’s laws, Eint = −Eb = −G m1 m2 /(2 a), where Eb is the binding energy of the binary system, a being its semi-major axis. The internal energy of a binary can be exchanged with other stars only if the binary undergoes a close encounter with a star, so that its orbital parameters are perturbed by the intruder. This happens only if a single star approaches the binary by few times its orbital separation. We define this close encounter between a binary and a single star as a three-body encounter. For this to happen with a non-negligible frequency, the binary must be in a dense environment, because the rate of threebody encounters scales with the local density of stars. Three-body encounters have crucial effects on BH binaries, such as hardening, exchanges, and ejections.

Hardening If a BBH undergoes a number of three-body encounters during its life, we expect that its semi-major axis will shrink as an effect of the encounters. This process is called dynamical hardening. Following [160], we call hard binaries (soft binaries) those binaries with binding energy larger (smaller) than the average kinetic energy of a star in the star cluster. According to Heggie’s law [160], hard binaries tend to harden (i.e., to become more and more bound) via binary–single encounters. In other words, a fraction of the internal energy of a hard binary can be transferred into kinetic energy of the intruders

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and of the center of mass of the binary during three-body encounters. This means that the binary loses internal energy and its semi-major axis shrinks. Most BBHs are expected to be hard binaries, because BHs are among the most massive bodies in star clusters. Thus, BBHs are expected to harden as a consequence of three-body encounters. The hardening process may be sufficiently effective to shrink a BBH till it enters the regime where GW emission is efficient: a BBH which is initially too loose to merge may then become a GW source thanks to dynamical hardening. The hardening rate for hard binaries with semi-major axis a can be estimated as [160] d dt

  1 ρ = 2π Gξ , a σ

(21)

where ξ ∼ 0.1 − 10 is a dimensionless hardening parameter (which has been estimated through numerical experiments, [162, 279]), ρ is the local mass density of stars, σ is the local velocity dispersion, and G is the gravity constant. Dynamical hardening is the main responsible for the shrinking of a binary, till it reaches a semi-major axis short enough for GW emission to become effective, which can be derived with the following equation [267]: da 64 G3 m1 m2 (m1 + m2 ) −3 =− a . dt 5 c5 (1 − e2 )7/2

(22)

By combining equations 21 and 22, it is possible to make a simple analytic estimate of the evolution of the semi-major axis of a BBH which is affected by three-body encounters and by GW emission: G ρ 2 64 G3 m1 m2 (m1 + m2 ) −3 da = −2 π ξ a − a , dt σ 5 c5 (1 − e2 )7/2

(23)

This equation holds under the assumption that the binary star is hard, that the total mass of the binary star is much larger than the average mass of a star in the star cluster (exchanges are unlikely), and that most three-body encounters have a small impact parameter. The first part of the right-hand term of equation 23 accounts for the effect of three-body hardening on the semi-major axis. It scales as da/dt ∝ −a 2 , indicating that the larger the binary is, the more effective the hardening. This can be easily understood considering that the geometric cross section for three-body interactions with a binary scales as a 2 . The second part of the right-hand term of equation 23 accounts for energy loss by GW emission. It is the first-order approximation of the calculation by Peters (1964, [267]). It scales as da/dt ∝ −a −3 indicating that GW emission becomes efficient only when the two BHs are very close to each other. In Figure 8, we solve equation 23 numerically for three BBHs with different mass. All binaries evolve through (i) a first phase in which hardening by three-body encounters dominates the evolution of the binary; (ii) a second phase in which the

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Fig. 8 Time evolution of the semi-major axis of three BH binaries estimated from equation 23. Blue dashed line, BH binary with masses m1 = 200 M , m2 = 30 M ; red solid line, m1 = 36 M , m2 = 29 M ; black dot-dashed line, m1 = 14 M , m2 = 7.5 M . For all BH binaries: ξ = 1, ρ = 105 M pc−3 , σ = 10 km s−1 , e = 0 (here we assume that ρ, σ , and e do not change during the evolution), initial semi-major axis of the BH binary ai = 10 AU

semi-major axis stalls because three-body encounters become less efficient as the semi-major axis shrink, but the binary is still too large for GW emission to become efficient; and (iii) a third phase in which the semi-major axis drops because the binary enters the regime where GW emission is efficient.

Exchanges Dynamical exchanges are three-body encounters during which one of the members of the binary is replaced by the intruder. Exchanges may lead to the formation of new BBHs: if a binary composed of a BH and a low-mass star undergoes an exchange with a single BH, this leads to the formation of a new BBH. This is a fundamental difference between BHs in the field and in star clusters: a BH which forms as a single object in the field has negligible chances to become member of a binary system, while a single BH in the core of a star cluster has good chances of becoming member of a binary by exchanges. Exchanges are expected to lead to the formation of many more BBHs than they can destroy, because the probability for an intruder to replace one of the members of a binary is ≈0 if the intruder is less massive than both binary members, while it suddenly jumps to ∼1 if the intruder is more massive than one of the members of the binary [163]. Since BHs are among the most massive bodies in a star cluster after their formation, they are very efficient in acquiring companions through dynamical exchanges. Thus, exchanges are a crucial mechanism to form BH binaries dynamically. By means of direct N-body simulations, Ziosi et al. (2014, [378]) show that >90 percent of BBHs in young star clusters form by

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dynamical exchange. Moreover, BBHs formed via dynamical exchange will have some distinctive features with respect to field BBHs (see, e.g., [378]): • BBHs formed by exchanges will be (on average) more massive than isolated BBHs, because more massive intruders have higher chances to acquire companions. • Exchanges trigger the formation of highly eccentric BBHs; eccentricity is then significantly reduced by circularization induced by GW emission, if the binary enters the regime where GW emission is effective. • BBHs born by exchange will likely have misaligned spins: exchanges and other dynamical interactions tend to lead to isotropically distributed spin directions with respect to the binary orbital plane, because dynamical interactions remove any memory of previous alignments. Zevin et al. (2017, [376]) compare a set of simulations of field binaries with a set of simulations of globular cluster binaries, run with the same population-synthesis code. The most striking difference between merging BH binaries in their globular cluster simulations and in their population-synthesis simulations is the dearth of merging BHs with mass 150 M ), blue straggler stars [221], and unusually massive BHs. Figure 10 shows the dynamical formation of a BBH with primary mass m1 = 88 M and secondary mass m2 = 47.5 M [90]. The masses of the components of this BBH are similar to those of GW190521 [17, 18]. In particular, the primary BH has a mass in the pair-instability mass gap. Its formation is possible because its stellar

Fig. 10 Cartoon of the dynamical assembly of a GW190521-like BBH from the simulations of [90]. (Courtesy of Ugo N. Di Carlo)

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progenitor is the result of the merger between a giant star with a well-developed He core and a main sequence companion (MS). The result of this stellar merger is a massive core helium burning (cHeB) star with an over-sized hydrogen envelope with respect to the He core [88–90, 182, 283]. Given the short timescale for He, C, O, Ne, and Si burning with respect to H burning, the star collapses to a BH before the He core grows above the threshold for pair instability: the result is a ∼88 M BH. After its formation, this BH acquires a companion by dynamical exchanges and merges within a Hubble time. This is a viable scenario to produce systems like GW190521 not only because of the good match of the BBH mass but also because a BH born from the direct collapse of a very massive star might form with a large spin (depending on the final spin of the massive progenitor) and because this dynamical formation leads to isotropically oriented spins.

Direct Three-Body Binary Formation In the most massive star clusters (globular clusters, nuclear star clusters), stellar velocities are so large that dynamical encounters can unbind most of the original binary stars (i.e., those binary stars that were already there at the formation of the star cluster). The minimum relative velocity vc between a binary star and an intruder star needed to unbind a binary is in fact [308]:  vc =

G m1 m2 (m1 + m2 + m3 ) , m3 (m1 + m2 ) a

(24)

where m1 , m2 , and m3 are the masses of the two binary members and that of the intruder, while a is the binary semi-major axis. In these extreme environments, most BBHs are expected to form by direct encounters of three single bodies [247, 295], during core collapse. This leads to the formation of extremely hard BBHs, which survive further ionization from intruders. The timescale for binary formation via three single body encounters is [194]:  t3bb = 0.1 Myr

n 106 pc−3

−2

9 σ −1 30 km s



mBH 30 M

−5 ,

(25)

where n is the local stellar density, σ is the local velocity dispersion of the star cluster, and mBH is the typical BH mass in the star cluster. The properties of BBHs born from three single body encounters are similar to those of BBHs born via dynamical exchanges: they tend to be more massive than isolated binaries and have high initial eccentricity and isotropically oriented spins (e.g., [25]). Direct three-body encounters are likely the most common BBH formation channel in globular clusters and nuclear star clusters [247], while binary–single star exchanges are likely the most common formation channel of BBHs in young star

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clusters [88, 378]. In young star clusters, dynamical exchanges affect both already formed BHs and their stellar progenitors.

Dynamical Ejections During three-body encounters, a fraction of the internal energy of a hard binary is transferred into kinetic energy of the intruders and of the center of mass of the binary. As a consequence, the binary and the intruder recoil. The recoil velocity is generally of the order of a few km s−1 but can be up to several hundred km s−1 . Since the escape velocity from a globular cluster is ∼ 30 km s−1 and the escape velocity from a young star cluster or an open cluster is even lower, both the recoiling binary and the intruder can be ejected from the parent star cluster. If the binary and/or the intruder are ejected, they become field objects and cannot participate in the dynamics of the star cluster anymore. Thus, not only the ejected BBH stops hardening but also the ejected intruder, if it is another compact object, loses any chance of entering a new binary by dynamical exchange. A general expression for the recoil velocity of the binary center of mass if (m1 + m2 ) m (where m is the average mass of a star in a star cluster) is

vrec

m ∼ m1 + m2

 2ξ Eb , (m1 + m2 + m)

(26)

where Eb = G m1 m2 /(2 a) is the binary binding energy. The above equation can help us deriving the minimum binding energy above which a binary star is ejected by a binary–single encounter Eb, min [240]: Eb, min ∼

(m1 + m2 )3 2 vesc , 2 ξ m2

(27)

where vesc is the escape velocity from the star cluster. A BBH will merge inside the star cluster only if Eb, min > Eb, GW , where Eb, GW is the minimum binding energy to reach coalescence by GW emission. Most BNSs, BBHs, and BH–NS systems in young star clusters are ejected before they merge [88, 282]. Dynamical ejections of BNS and BH–NS binaries were proposed to be one of the possible explanations for the host-less short gamma-ray bursts, i.e., gamma-ray bursts whose position in the sky appears to be outside any observed galaxy [117]. Host-less bursts may be ∼25% all short gamma-ray bursts. In general, ejections of compact objects and compact object binaries from their parent star cluster can be the result of at least three different processes: • Dynamical ejections (as described above) • SN kicks [128, 164] • GW recoil [64, 146, 200]

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GW recoil is a relativistic kick occurring when a BBH merges. It is the result of asymmetric linear momentum loss by GW emission, when the binary has asymmetric component masses and/or misaligned spins. It results in kick velocities up to thousands of km s−1 and usually of the order of hundreds of km s−1 . Ejections by dynamics, SN kicks, or GW recoil may be the main process at work against mergers of second-generation BHs, where for second-generation BHs we mean BHs which were born from the merger of two BHs rather than from the collapse of a star [138]. In globular clusters, open clusters, and young star clusters, a BH has good chances of being ejected by three-body encounters before it merges [240] and a very high chance of being ejected by GW recoil after it merges [229, 292]. The only place where merging BHs can easily avoid ejection by GW recoil are nuclear star clusters, whose escape velocity can reach hundreds of km s−1 [23, 26, 122, 229].

Formation of Intermediate-Mass Black Holes by Runaway Collisions In section “The Mass of Compact Remnants”, we have mentioned that intermediatemass black holes (IMBHs, i.e., BHs with mass 100  mBH  104 M ) might form from the direct collapse of metal-poor extremely massive stars [312]. Other formation channels have been proposed for IMBHs, and most of them involve dynamics of star clusters. The formation of massive BHs by runaway collisions has been originally proposed about half a century ago [78,299] and was then elaborated by several authors (e.g., [88, 90, 125, 144, 155, 218, 221, 226, 271, 272, 274, 288]). The basic idea is the following (as summarized by the cartoon in Fig. 11). In a dense star cluster, dynamical friction [69] makes massive stars to decelerate because of the drag exerted by lighter bodies, on a timescale tDF (M) ∼ ( m/M) trlx (equation 18). Since, the two-body relaxation timescale in a young star cluster can be as short as trlx ∼ 10 − 100 M [275], for a star with mass M ≥ 40 M , we estimate tDF ≤ 2.5 Myr: dynamical friction is very effective in dense massive young star

Fig. 11 Cartoon of the runaway collision scenario in dense young star clusters (see, e.g.. [274]). From left to right: (1) the massive stars (red big stars) and the low-mass stars (yellow small stars) follow the same initial spatial distribution; (2) dynamical friction leads the massive stars to sink to the core of the cluster, where they start colliding between each other; (3) a very massive star ( 100 M ) forms as a consequence of the runaway collisions; (4) this massive star might be able to directly collapse into a BH

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clusters. Because of dynamical friction, massive stars segregate to the core of the cluster before they become BHs. If the most massive stars in a dense young star cluster sink to the center of the cluster by dynamical friction on a time shorter than their lifetime (i.e., before corecollapse SNe take place, removing a large fraction of their mass), then the density of massive stars in the cluster core becomes extremely high. This makes collisions between massive stars extremely likely. Actually, direct N-body simulations show that collisions between massive stars proceed in a runaway sense, leading to the formation of a very massive ( 100 M ) star [274]. The main open question is: “What is the final mass of the collision product? Is the collision product going to collapse to an IMBH?”. There are essentially two critical issues: (i) How much mass is lost during the collisions? (ii) How much mass does the very massive star lose by stellar winds? Hydrodynamical simulations of colliding stars [133, 134] show that massive star can lose ≈25% of their mass during collisions. Even if we optimistically assume that no mass is lost during and immediately after the collision (when the collision product relaxes to a new equilibrium), the resulting very massive star will be strongly radiation-pressure dominated and is expected to lose a significant fraction of its mass by stellar winds. Recent studies including the effect of the Eddington factor on mass loss [218, 312] show that IMBHs cannot form from runaway collisions at solar metallicity. At lower metallicity (Z  0.1 Z ), approximately 10 − 30% of runaway collision products in young dense star clusters can become IMBHs by direct collapse (they also avoid being disrupted by pair-instability SNe). The majority of runaway collision products do not become IMBHs, but they end up as relatively massive BHs (∼20 − 90 M , [218]). If they remain inside their parent star cluster, such massive BHs are extremely efficient in acquiring companions by dynamical exchanges. Mapelli (2016, [218]) find that all stable binaries formed by the runaway collision product are BBHs and thus are possibly important sources of GWs in the LIGO–Virgo range.

Hierarchical BBH Formation and IMBHs The runaway collision scenario occurs only in the early stages of the evolution of a star cluster, when massive stars are still alive (the lifetime of a ∼ 30 M star is ∼6 Myr). However, it has been proposed that IMBHs form even in old clusters (e.g., globular clusters) by repeated mergers of smaller BHs (e.g., [26,122,144,230,240]). The simple idea is illustrated in Fig. 12. A stellar BBH in a star cluster is usually a rather hard binary. Thus, it shrinks by dynamical hardening till it may enter the regime where GW emission is effective. In this case, the BBH merges leaving a single more massive BH. Given its relatively large mass, the new BH has good chances to acquire a new companion by exchange. Then, the new BBH starts hardening again by three-body encounters, and the story may repeat several times, till the main BH becomes an IMBH.

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Fig. 12 Cartoon of the repeated merger scenario in old star clusters (see, e.g., [144, 240]). From top to bottom and from left to right: (1) a BBH undergoes three-body encounters in a star cluster; (2) three-body encounters harden the BBH, shrinking its semi-major axis; (3) the BBH hardens by three-body encounters till it enters the regime where GW emission is efficient, the BBH semimajor axis decays by GW emission, and the binary merges; (4) a single bigger BH forms as result of the merger, which may acquire a new companion by dynamical exchange (if it is not ejected by GW recoil); (5) the new BBH containing the bigger BH starts shrinking again by three-body encounters (1). This loop may be repeated several times till the main BH becomes an IMBH

This scenario has one big advantage: it does not depend on stellar evolution, so we are confident that the BH will grow in mass by mergers, if it remains inside the cluster. However, there are several issues. First, the BBH may be ejected by dynamical recoil, received as an effect of three-body encounters. Recoils get stronger and stronger, as the orbital separation decreases (equation 27). The BBH will avoid ejection by dynamical recoil only if it is sufficiently massive (50 M for a dense globular cluster, [80]). If the BBH is ejected, the loop breaks and no IMBH is formed. Second and even more important, the merger of two BHs involves a relativistic kick. This kick may be as large as hundreds of km s−1 [200], leading to the ejection of the BH from the parent star cluster [165]. Also in this case, the loop breaks and no IMBH is formed. Finally, even if the BH binary is not ejected, this scenario is relatively inefficient: if the seed BH is ∼50 M , several Gyr are required to form an IMBH with mass ∼500 M [240]. Monte Carlo simulations by Giersz et al. (2015, [144]) show that both the runaway collision scenario and the repeated merger scenario can be at work in star clusters: runaway collision IMBHs form in the first few Myr of the life of a star cluster and grow in mass very efficiently, while repeated-merger IMBHs start forming much later (5 Gyr) and their growth is less efficient.

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Alternative Models for Massive BHs and IMBH Formation in Galactic Nuclei Several additional models predict the formation of IMBHs in galactic nuclei. For example, Miller & Davies (2012, [241]) propose that IMBHs can efficiently grow in galactic nuclei from runaway tidal capture of stars, provided that the velocity dispersion in the nuclear star cluster is 40 km s−1 . Below this critical velocity, binary stars can support the system against core collapse, quenching the growth of the central density and leading to the ejection of the most massive BHs. Above this velocity threshold, the stellar density can grow sufficiently fast to enhance tidal captures and BH–star collisions. Tidal captures are more efficient than BH–star collisions in building up IMBHs, because the mass growth rate of the former scales as m ˙ IMBH ∝ mIMBH 4/3 (where mIMBH is the IMBH seed mass), whereas the mass growth of the latter scales as m ˙ IMBH ∝ mIMBH [323]. Furthermore, McKernan et al. (2012, [237]; see also [42, 236, 238, 372, 373]) suggest that IMBHs could efficiently grow in the accretion disc of a SMBH. Nuclear star cluster members trapped in the accretion disk are subject to two competing effects: orbital excitation due to dynamical heating by encounters with other stars and orbital damping due to gas drag. Gas damping is expected to be more effective than orbital excitation, quenching the relative velocity between nuclear cluster members and enhancing the collision rate. This favors the growth of IMBHs via both gas accretion and multiple stellar collisions. This mechanism might be considered as a gas-aided runaway collision scenario. As a final remark, it is worth mentioning that all the IMBH formation scenarios we have discussed here – (i) runaway collisions of massive stars, (ii) hierarchical merger of BHs, and (iii) BH trapping in AGN discs – can also work as possible formation mechanisms for GW190521-like events: they can lead to the formation of BHs with mass in the pair-instability mass gap and with large spins. These BHs are born in dense stellar environments, where they can acquire companions dynamically and form BBHs with isotropically oriented spins [18].

Kozai–Lidov Resonance Unlike the other dynamical processes discussed so far, Kozai–Lidov (KL) resonance [179, 196] can occur both in the field and in star clusters. KL resonance appears whenever we have a stable hierarchical triple system (i.e., a triple composed of an inner binary and an outer body orbiting the inner binary), in which the orbital plane of the outer body is inclined with respect to the orbital plane of the inner binary. Periodic perturbations induced by the outer body on the inner binary cause (i) the eccentricity of the inner binary and (ii) the inclination between the orbital plane of the inner binary and that of the outer body to oscillate. The semi-major axis of the binary star is not affected, because KL resonance does not imply an energy exchange

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between inner and outer binary. KL oscillations may enhance BBH mergers, because the timescale for merger by GW emission strongly depends on the eccentricity e of the binary tGW ∝ (1 − e2 )7/2 (see equation 22, [267]). It might seem that hierarchical triples are rather exotic systems. This is not the case. In fact, ∼10% of low-mass stars are in triple systems [280, 338, 339]. This fraction gradually increases for more massive stars [95], up to ∼50% for B-type stars [246, 297, 340]. In star clusters, stable hierarchical triple systems may form dynamically, via four-body or multiple-body encounters. Kimpson et al. (2016, [176]) find that KL resonance may enhance the BBH merger rate by ≈40% in young star clusters and open clusters. According to Fragione et al. (2019, [120]), the merger fraction of BBHs in galactic nuclei can be up to ∼5 − 8 times higher for triples than for binaries. On the other hand, Antonini et al. (2017, [24]) find that KL resonance in field triples can account for 3 mergers Gpc−3 yr−1 . The main signature of the merger of a KL system is the non-zero eccentricity until very few seconds before the merger. Eccentricity might be significantly non-zero even when the system enters the LIGO–Virgo frequency range. KL resonances have an intriguing application in nuclear star clusters. If the stellar BH binary is gravitationally bound to the super-massive BH (SMBH) at the center of the galaxy, then we have a peculiar triple system where the inner binary is composed of the stellar BH binary and the outer body is the SMBH [27]. Also in this case, the merging BBH has good chances of retaining a non-zero eccentricity till it emits GWs in the LIGO–Virgo frequency range.

Summary of Dynamics and Open Issues In this section, we have seen that dynamics is a crucial ingredient to understand BBH demography. Dynamical interactions (three- and few-body close encounters) can favor the coalescence of BBHs through dynamical hardening. New BBHs can form via dynamical exchanges and via direct three-body encounters or GW captures. Hierarchical mergers of BHs in dense star clusters and in AGN disks can trigger the formation of unusually massive BHs (like GW190521) and even IMBHs. Stellar mergers might also lead to the formation of unusually massive BHs (like GW190521) and IMBHs in dense star clusters. All these dynamical processes suggest a boost of the BBH merger rate in dynamically active environments. Overall, dynamically formed BBHs are expected to be more massive than BBHs from isolated binary evolution, with higher initial eccentricity and with misaligned spins. Also, KL resonances favor the coalescence of BBHs with extremely high eccentricity, even close to the last stable orbit. On the other hand, three-body encounters might trigger the ejection of binary compact objects from their natal environment, inducing a significant displacement between the birth place of the binary and the location of its merger. Figure 13 summarizes one of the possible evolutionary pathways of merging BBHs which originate from dynamics (the variety of this formation channel is too

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Fig. 13 Schematic evolution of a merging BBH formed by dynamical exchange (see, e.g., [34, 93, 94, 218, 290, 291, 378])

large to account for all dynamical channels mentioned above in a single cartoon). As in the isolated binary case, we start from a binary star. In the dynamical scenario, it is not important that this binary evolves through Roche lobe or CE (although this may happen). After the primary has turned into a BH, the binary undergoes a dynamical exchange: the secondary is replaced by a massive BH and a new BBH forms. The new binary system is not ejected from the star cluster and undergoes further threebody encounters. As an effect of these three-body encounters, the binary system hardens enough to enter the regime in which GW emission is efficient: the BBH merges by GW decay. We expect dynamics to be important for BBHs also because massive stars (which are the progenitors of BHs) form preferentially in young star clusters [275], which are dynamically active places. It is reasonable to expect that most BHs participate in the dynamics of their parent star cluster before being ejected by dynamical recoil or relativistic kicks.

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BBHs in the Cosmological Context At design sensitivity, LIGO and Virgo will observe BBHs up to redshift z ∼ 1. The farthest event observed to date, GW190521, is at z ∼ 0.8, when the Universe was only ∼6.8 Gyr old. Next-generation ground-based GW detectors, such as the Einstein Telescope [277], will observe BBH mergers up to redshift z  10, when the Universe was 500 Myr old. Almost the entire Universe will be transparent to BBH mergers, and it will be possible to investigate the cosmic evolution of stars and galaxies through the observations of BBH mergers. In other words, we will do cosmology through BBH mergers. Accounting for the cosmological context of BBH mergers might appear as a desperate challenge, because of the humongous dynamical range: the orbital separations of BBHs are of the order of tens of solar radii, while cosmic structures are several hundreds of Mpc. Several theoretical studies have addressed this challenge, adopting two different methodologies.

Data-Driven Semi-analytic Models Some authors (e.g., [35, 45, 54, 91, 92, 97, 98, 102, 103, 140, 192, 250, 300, 301, 328, 356]) combine the outputs of population-synthesis codes with analytic prescriptions. The main ingredients are the cosmic star formation rate density and the evolution of metallicity with redshift [74–76,209,210]. In some previous work (e.g., [91,192]), a Press–Schechter-like formalism is adopted, to include the mass of the host galaxy in the general picture. Lamberts et al. (2016, [192]) even include a redshift-dependent description for the mass-metallicity relation (hereafter MZR), to account for the fact that the mass of a galaxy and its observed metallicity are deeply connected. The main advantage of this procedure is that the star formation rate and the metallicity evolution can be derived more straightforwardly from the data. The main drawback is that it is more difficult to trace the evolution of the host galaxy of the BBH, through its galaxy merger tree. In the semi-analytic description, the merger rate density evolution can be described as [300]: R(z) =

d dtlb (z)



z

zmax

ψ(z )

dtlb (z )  dz dz



Zmax

 η(Z)F (z , z, Z) dZ ,

(28)

Zmin

where tlb (z) is the look-back time at redshift z, Zmin and Zmax are the minimum and maximum metallicity, ψ(z ) is the cosmic star formation rate (SFR) at redshift z , F (z , z, Z) is the fraction of compact binaries that form at redshift z from stars with metallicity Z and merge at redshift z, and η(Z) is the merger efficiency, namely, the ratio between the total number NTOT (Z) of compact binaries (formed from a coeval population) that merge within an Hubble time (tH0  14 Gyr) and the total initial mass M∗ (Z) of the simulation with metallicity Z. The value of F (z , z, Z)

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Fig. 14 Cosmic merger rate density of BBHs in the comoving frame (RBBH ) as a function of the look-back time (tlb , lower x−axis) and of the redshift (z, upper x−axis) from some of the models presented in [301], which are based on Equation 28. We use catalogues of population-synthesis simulations run with MOBSE [142, 223]. In both panels: thin gray line and right y− axis, cosmic star formation rate density as a function of redshift from [210]. Gray shaded box: 90% credible interval for the local merger rate density of BBHs, as inferred from the LVC data, considering the union of the rates obtained with model A, B, and C in [8]. The width of the gray-shaded area on the x−axis corresponds to the instrumental horizon obtained by assuming BBHs with total mass of 20 M and O2 sensitivity [9]. Upper panel: BBH merger rate density for different assumptions on the α parameter (i.e., the efficiency of CE ejection). Models for α = 0.5, 1, 2, 3, 5, 7, and 10 are shown. Lower panel: the solid red line shows the median value of the BBH merger rate density for α = 5, while the shaded areas show the 50% credible areas estimated from the uncertainties on metallicity evolution (hatched region) and star formation rate density evolution (shaded region). The cosmic star formation rate density evolution is modelled as in [210] and is obtained by fitting data. The metallicity evolution uses the fitting formula by [85], adapted as described in [300]. (Courtesy of Filippo Santoliquido)

and that of η(Z) can be calculated either from catalogues of population-synthesis simulations or from phenomenological models. This formalism yields an evolution of the merger rate density with redshift as shown in Fig. 14. From the upper panel, we see that the cosmic merger rate density evolution is sensitive to the choice of the CE parameter α. From the lower panel, it is apparent that the BBH merger rate density is tremendously affected

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by the metallicity evolution and by the observational uncertainties on metallicity evolution. Most models agree on this result (e.g., [35, 54, 103, 250, 300, 301, 328]). The reason for this trend is twofold. On the one hand, according to current models of BBH formation, the merger efficiency η is 2−4 orders of magnitude higher at low metallicity (Z < 0.0002) than at high metallicity. Hence, the merger rate is extremely sensitive to the underlying metallicity evolution. On the other hand, the uncertainties on metallicity evolution from observational data are large, as discussed by, e.g., [74–76, 209, 213, 300]. The trend of the cosmic merger rate in Fig. 14 is similar to the trend of the cosmic star formation rate density curve, modulated by both metallicity evolution and time delay (we define as time delay the time elapsed between the formation of the progenitor binary and the merger of the two BHs). The peak of the BBH merger rate density curve (z ∼ 3 − 4, depending on α) is at a higher redshift than the peak of the cosmic star formation rate density (z = 2). This happens because the lower the metallicity is, the higher is the efficiency of BBH mergers, and metal-poor stars are more common at high redshift than at low redshift. With this approach, it will also be possible to put constraints, in a statistical sense, on the fraction of BBH mergers from the dynamical channel with respect to BBH mergers from the isolated evolutionary channel (e.g., [57, 289, 300, 365, 377]).

Cosmological Simulations The alternative approach to reconstruct the BBH merger history feeds the outputs of population-synthesis simulations into cosmological simulations [29–31, 65, 154, 222–225, 231, 259, 305], through a Monte Carlo approach. This has the clear advantage that the properties of the host galaxies can be easily reconstructed across cosmic time. However, the ideal thing would be to have a high-resolution cosmological simulation (sufficient to resolve also small dwarf galaxies) with a box as large as the instrumental horizon of the GW detectors. This is obviously impossible. High-resolution simulations have usually a box of a few comoving Mpc3 , while simulations with a larger box cannot resolve dwarf galaxies. Moreover, this procedure requires to use the cosmic star formation rate density and the redshiftdependent MZR which are intrinsic to the cosmological simulations. While most state-of-the-art cosmological simulations reproduce the cosmic star formation rate density reasonably well, the MZR is an elusive feature, creating more than a trouble even in the most advanced cosmological simulations. For example, [225] investigate the main properties of the host galaxies of merging BHs, by combining their population-synthesis simulations [222, 223] with the ILLUSTRIS cosmological box [251,357,358]. The size of the ILLUSTRIS (length = 106.5 comoving Mpc) is sufficient to satisfy the cosmological principle, but galaxies with stellar mass 108 M are heavily under-resolved. Their results show that BHs merging in the local Universe (z < 0.024) have formed in galaxies with relatively small stellar mass (< 1010 M ) and relatively low metallicity (Z ≤ 0.1 Z ). These BHs reach coalescence either in the galaxy where they formed or in

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larger galaxies (with stellar mass up to ∼1012 M ). In fact, most BHs reaching coalescence in the local Universe appear to have formed in the early Universe (9 Gyr ago), when metal-poor galaxies were more common. A significant fraction of these high-redshift metal-poor galaxies merged within larger galaxies before the BBHs reached coalescence by GWs. Moreover, these models show that the mass spectrum and the other main properties of BBHs do not evolve significantly with redshift [224]. Schneider et al. (2017, [305], see also [154, 231]) adopt a complementary approach to study the importance of dwarf galaxies for GW detections. They use the GAMESH pipeline to produce a high-resolution simulation of the Local Group (length = 4 Mpc comoving). This means that the considered portion of the Universe is strongly biased, but the resolution is sufficient to investigate BH binaries in small (106 M ) dwarf galaxies. One of their main conclusions is that GW150914-like events originate mostly from small metal-poor galaxies. Similarly, Cao et al. (2018, [65]) investigate the host galaxies of BBHs by applying a semi-analytic model to the Millennium-II N-body simulation [59]. The Millennium-II N-body simulation is a large-box (length = 137 comoving Mpc) dark-matter-only simulation. The physics of baryons is implemented through a semi-analytic model. Using a dark-matter-only simulation coupled with a semianalytic approach offers the possibility of improving the resolution significantly, but baryons are added only in post-processing. With this approach, Cao et al. (2018, [65]) find that BBHs merging at redshift z  0.3 are located mostly in massive galaxies (stellar mass 2 × 1010 M ). Finally, [29–31] combine population-synthesis simulations with the EAGLE cosmological simulation [303]. They show that the merger rate per galaxy strongly correlates with the stellar mass of a galaxy for both BBHs, BHNSs, and BNSs. They also find a dependence of the merger rate per galaxy on the star formation rate and a weaker dependence on the metallicity, but the correlation with the stellar mass is definitely the strongest one. This result has been used to optimize electromagnetic follow-up strategies, weighting the galaxies in the LVC uncertainty box by their stellar mass [20, 96]. These studies show that the combination of populationsynthesis tools with cosmological simulations is an effective approach to understand the cosmic evolution of the merger rate and the properties of the host galaxies of BBH mergers.

Summary and Outlook We reviewed our current understanding of the formation channels of BBHs. The last 5 years have witnessed considerable progress in this field, thanks to the groundbreaking results of the LVC and to a renewed interest in BH astrophysics, mostly triggered by GW data. The final stages of massive star evolution deserve particular attention to understand the mass and spins of BHs. GW data have recently challenged the concept of

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pair-instability mass gap, i.e., the existence of a mass range (between ∼65 M and ∼120 M ) in which we do not expect to find BHs as an effect of (pulsational) pair instability. The process of mass transfer between two massive stars in a binary system is another key aspect shaping the demography of BBHs. The efficiency of mass accretion via Roche lobe overflow and the stability of mass transfer are possibly the main unknowns affecting the mass and the delay time of a BBH. In addition, the dynamical formation channel with its richness and multifaceted processes (hardening, exchanges, ejections, runaway collisions, hierarchical mergers, etc.) adds to the complexity of the general picture. Current astrophysical models of BBH formation face a number of essential questions: • Impact of core overshooting, rotation, and possibly magnetic fields on massive star evolution and BH formation • Angular momentum transfer in massive stars and its link to BH spins • Explodability of massive stars • Stability and efficiency of mass transfer • Evolution of common-envelope systems • Physics of stellar collisions and mergers • Impact of the environment (e.g., metallicity, star clusters versus field) on the formation of BBHs across cosmic time Overall, we have a better understanding of the main dynamical processes with respect to both massive star evolution and SN physics. But our understanding of BBH dynamics is mainly hampered by two problems: (i) we cannot properly model the dynamics of BBHs if we do not know BH masses and spins from stellar evolution; (ii) dynamical simulations are still too computationally expensive to allow us to investigate the relevant parameter space. Next-generation ground-based detectors will observe BBH mergers up to redshift z  10, beyond the epoch of cosmic reionization [174]. This will open a completely new scenario for the study of BBHs across cosmic time and for the characterization of their evolutionary pathways.

Cross-References  Dynamical Formation of Merging Stellar-Mass Binary Black Holes  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range  Inferring the Properties of a Population of Compact Binaries in Presence of

Selection Effects  LIGO, VIRGO, and KAGRA as the International Gravitational Wave Network  Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravita-

tional Waves  Primordial Gravitational Waves  Third-Generation Gravitational-Wave Observatories

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Acknowledgments We thank the DEMOBLACK team for useful discussions and for providing us with some essential material for this review. MM acknowledges financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no. 770017.

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The Gravitational Capture of Compact Objects by Massive Black Holes∗

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Contents Introduction: Why Is This Important? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme-Mass Ratio Inspirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Long Story Short . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar Tidal Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relaxation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Loss-Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation of EMRIs via Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation of EMRIs via Tidal Separation of Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geodesic Motion and Relativistic Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kerr Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution in Phase-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulated Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event Rate of Relaxation EMRIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate-Mass Ratio Inspirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate-Mass Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wandering of IMBHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulations of IMRIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, as well as the COST Action GWverse CA16104. This work was supported by the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303). He is indebted to Marta Masini for her support during the lockdown, without which this work would not have been possible. P. Amaro Seoane () Institute of Multidisciplinary Mathematics, Universitat Politècnica de València, València, Spain Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing, China Kavli Institute for Astronomy and Astrophysics, Beijing, China © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_17

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Event Rate of IMRIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-bandwidth IMRIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling IMRIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extremely Large Mass Ratio Inspirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Relativistic Fokker-Planck Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newtonian Motion Around a Newtonian Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations Between the Relativistic and Newtonian Parameters for a Schwarzschild SMBH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations Between the Relativistic and Newtonian Parameters for a Kerr SMBH . . . . . . . A Possible Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The gravitational capture of a stellar-mass compact object (CO) by a supermassive black hole is a unique probe of gravity in the strong field regime. Because of the large mass ratio, we call these sources extreme-mass ratio inspirals (EMRIs). In a similar manner, COs can be captured by intermediate-mass black holes in globular clusters or dwarf galaxies. The mass ratio in this case is lower, and hence we refer to the system as an intermediate-mass ratio inspiral (IMRI). Also, substellar objects such as a brown dwarf, with masses much lighter than our Sun, can inspiral into supermassive black holes such as Sgr A* at our Galactic Centre. In this case, the mass ratio is extremely large, and hence, we call this system extremely large mass ratio inspirals (XMRIs). All of these sources of gravitational waves will provide us with a collection of snapshots of spacetime around a supermassive black hole that will allow us to do a direct mapping of warped spacetime around the supermassive black hole, a live cartography of gravity in this extreme gravity regime. E/I/XMRIs will be detected by the future spaceborne observatories like LISA. There has not been any other probe conceived, planned, or even thought of ever that can do the science that we can do with these inspirals. We will discuss them from a viewpoint of relativistic astrophysics. Keywords

Gravitational captures · Supermassive black holes · Stellar dynamics · LISA · Tests of general relativity

Introduction: Why Is This Important? For many years we have known that at the center of most nearby bright galaxies, a very massive, compact, and dark object must be lurking [e.g., 74, 106, 107]. Recently, the Event Horizon Telescope finally delivered the result that we were expecting: the “shadow” of one of these dark objects at the center of the galaxy M87, with a mass of 6.5 × 107 M [66]. In Fig. 1, we show their result with the position of Voyager1 and Pluto if our Sun was located at the center of the image.

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This is exceptionally to scale. Note however, as the author of the comic told us, that the blurring makes the dark shadow look smaller than it is. But the diameter of the brightest part of the ring should be ∼640 AU. In 2020, the Nobel Prize went to Roger Penrose, Reinhard Genzel, and Andrea Ghez. The latter two have devoted a significant part of their careers to study an interesting phenomenon happening at our own Galactic Centre: a cluster of young stars revolving around a point of small size, a radio source, on observable timescales. After analyzing the orbits, Genzel’s and Ghez’ groups came to the conclusion that these stars, which are called the S- or SO-stars, are orbiting a point mass of about 4 × 106 M enclosed in a volume as small as 1/3 the distance between the Earth and the Sun. We call this “point” mass Sgr A* (see for a review [74], and references therein). The properties of these stars are interesting. For instance, S4714 moves at about 8% the speed of light, and S62 is as close as 16 AU from SgrA* [138]. The team of Genzel presented this year the detection of the Schwarzschild precession in the orbit of S2 after its second periapsis passage. According to their measurements, the predictions of general relativity (GR) and the observations are in agreement by 17% [85]. Therefore, even if we still cannot yet completely exclude other exotic possibilities that nature might have come up with, it looks like SgrA* and the dark object located at the center of M87 are supermassive black holes, and this is what we will assume in this chapter. In that case, the Event Horizon Telescope has measured the event horizon to be of about 0.0013 pc [66]. We will however have a stronger evidence when we detect the gravitational waves (GWs) from a system involving a supermassive black hole (SMBH), in particular an E/I/XMRI. In this chapter, we focus on this particular source of GWs. In the literature, the most studied inspiral has so far been the gravitational capture of a stellar-mass compact object (i.e., a stellar-mass black hole, a neutron star, or a white dwarf) by a SMBH. In a matter of a few years, these sources significantly shrink their semi-major axes, precess and if the SMBH is Kerr (and that is the most likely situation in nature; see, e.g., [157]), the plane of the orbit changes, so that they map spacetime around SMBHs. The number of cycles that they describe around the SMBH is inversely proportional to the mass ratio, so that an EMRI will provide us with a collection of about ∼105 snapshots of spacetime around a SMBH. This translates literally into a cartography of warped spacetime in the strong regime of gravity. In this sense, these sources of GWs are unique. The ideal instrument to look for in this kind of inspiral is the ESA/NASA Laser Interferometer Space Antenna [LISA, see 16] mission, because the total mass is too large for ground-based detectors (with the exception of some particular cases of IMRIs, as we will discuss). LISA consists of three spacecraft arranged in an equilateral triangle of 2.5 × 10 6 km armlength flying on a heliocentric orbit, with a strain sensitivity of less than 1 part in 1020 in frequencies of about a millihertz. Most of this chapter will be devoted to addressing the question of how do you form these sources. We will see that not all phase-space is at our disposition to create one of these inspirals and that the conditions change a lot depending on whether it is an EMRI, an IMRI, or an XMRI.

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Fig. 1 The shadow of the dark object found in M87. From https://xkcd.com/2135/ and with the XKCD team authorization, using the original figure from the EHT team web page, https://eventhorizontelescope.org/

Fundamental Science EMRIs, IMRIs, and, within some limitations, XMRIs are probes of fundamental physics with GWs, mostly due to the fact that these inspirals spend a large number of cycles very close at the verge of the last stable orbit, where precessional are the strongest [see, e.g., 23, 34]. An EMRI has three fundamental frequencies associated with it [13,15] which are encoded in the gravitational wave emitted by the system. The associated timescales depend on the Kerr geometry and will evolve due to gravitational backreaction, which can evolve in very different ways depending on the type of gravity one considers, or whether there are extra dimensions or fields. In this sense, the detection and extraction of this information can be used as a tool to explore these new ideas or might turn into something that we had not previously considered or even thought of. Because we want to extract information from the modelling of a source that will last months to years, the waveforms must be very precise and therefore subject to important (accumulated) deviations if what we are measuring does not

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correspond to the general relativity picture. In some sense, this might be regarded as a double-edged sword. On the one hand, we can extract very accurately gravitational multipole moments describing the SMBH geometry, as well as other parameters that provide us with information about modifications of GR, such as coupling constants and extra dimension length scales. On the other hand, if such modifications are there and are important, we might lose the source altogether. Therefore, we can in principle use EMRIs as they generate GWs [30] as tools to explore the nature and geometry of SMBHs, i.e., the no-hair conjecture [see, e.g., 13, 15], complementary in a sense with the detection of quasinormal models excited in the ringdown of a SMBH binary merger [35], and rule out exotic objects such as, e.g., boson stars [14, 89]. This is particularly true for multi-bandwidth detections, i.e., IMRIs, which can be detected in the early inspiraling phase by LISA and later by groundbased detectors [6, 60], because the combined detection would help break up different degeneracies in the parameter extraction. EMRI GWs are subject to be affected by additional mechanisms that might alter them on their way toward us, such as extra polarizations, gravitational parity violation, and breaking of Lorentz invariance, all of them related to highenergy effects [see 31]. There are also other potential effects that could importantly modify the waves, such as the aberrational and beaming effects and their combination [170, 171], which emerge when the source is moving relative to us. In a more speculative manner, we could also use E/IMRIs to explore the possibility that primordial black holes do exist [45], assuming that we have confidence in our astrophysical modelling of what the mass of the CO is, which might not necessarily be true. This connects to the idea that dark matter could be constituted of such primordial black holes [see the review of 44] For astrophysics, these sources of GWs are important because we can try to reverse-engineer the information that they will provide us with to probe regions of the galaxy which are unaccessible to light because of obscuration or distance.

Extreme-Mass Ratio Inspirals A Long Story Short The problem of how a CO could become an EMRI in a galactic nucleus is, as of writing these lines, a problem which goes back in time three decades to the best of our knowledge [88, 97, 155, 158]. The first reference uses the fundamentals of the theory of loss-cone in the context of relaxation to derive the rates and characteristics. It is natural that this first attempt happened at that time and place, because the study of stellar disruptions was one of the main interests of the second author of the paper. However, as compared to a tidal disruption, the slow, progressive inspiral of a CO is a more challenging problem, as we will see. We will adopt the Milky Way as a typical, reference galaxy host to the kind of sources that we are going to address in this chapter. The mass of the central SMBH

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will have that of SgrA*, 4 × 106 M , or of the order of it. The fiducial CO we will consider is a stellar-mass black hole of mass mBH = 10 M for historical reasons: This was the default assumed mass before the first discoveries of LIGO/Virgo, since we cannot explain how to form stellar-mass black holes with larger masses via (reasonable) stellar evolution. In this chapter, we are going to focus in detail on the most well-understood mechanism (in the sense that the number of free parameters is very small), which is relaxation, more specifically two-body relaxation. Other mechanisms have been proposed, in particular the so-called resonant relaxation [147], which seemed very important a few years ago. Another interesting scenario has been proposed, which involves the tidal separation of a binary in which one of the stars is a CO. This possibility is an interesting one, but we are missing a fundamental piece of information, which is the distribution of initial semi-major axes and eccentricities, as well as the fraction of such binaries harboring a CO. Nonetheless, since the mechanism is simple and robust enough, we will also address it in some detail. To summarize in a few paragraphs what has been the work of decades, the study of EMRI event rate was addressed in the framework of two-body relaxation, since the problem shared similarities to the tidal disruption of a star. The event rate for a typical galaxy (i.e., a Milky Way-like galaxy, as stated before) is very low, of the order of 10−5 − 10−6 yr−1 [see, e.g., 17, and references therein]. Because of the dynamical properties that the COs have when they form a potential EMRI source with the central SMBH, i.e., have very large eccentricities and semi-major axis of about 0.1 − 1 pc, at apocenter, they risk being scattered off the EMRI orbit via the accumulation of gravitational tugs from other stars (we will elaborate this later). If we perturb the apocenter, the pericenter is also perturbed, and the fate of the potential EMRI is twofold: It can either increase more and more the initial eccentricity to the point that it forms almost a straight line which leads it to directly cross the event horizon of the SMBH, or it can simply be reabsorbed by the stellar system. In the first case, it would produce one burst of gravitational radiation and then be lost in terms of GWs [33, 100]. It is important to note here that such orbits have unfortunately been dubbed in the (astrophysical) literature as “plunges” or “direct plunges,” which leads to confusion and should be avoided, since any EMRI will eventually cross the event horizon of the SMBH. This is referred to in other texts as the plunge of the orbit. We will only use the term “plunge” in the sense that the CO crosses the event horizon of the SMBH. In the second, it would never emit (detectable) GWs. A better term would be a “1-burst orbit,” because the system emits one intense burst of GWs, and then it is lost. We note that there are scenarios in which we can have repeated bursts coming parabolic orbits, not bound ones, and the rate is of about one burst per observation year [see references in 7]. We are not referring to these here. The fact is that the assumptions under the classification of orbits which led to consider 1-burst orbits have been highly simplified in the literature. In reality, both for Schwarzschild and Kerr SMBHs, 1-burst orbits are extremely difficult to achieve. The work of [12] illustrates this with an example. The authors take a SMBH with

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Fig. 2 Different separatrices for a Kerr SMBH of a• = 0.999. The separatrices which correspond to retrograde orbits are shown above the Schwarzschild LSO (black, thick line) and prograde below it

a (pc)

no spin, Schwarzschild, and analyze orbits for a stellar-mass black hole with such an eccentricity that would lead to 1-burst scenarios. The authors then calculate the number of periapsis passages before it plunges through the event horizon. From the initial semi-lactus rectum, eccentricity, and inclination p, e, and ι, they calculate the constants of motion E, Lz , and C (energy, z-component of the angular momentum, and Carter’s constant) and the average flux ˙ L˙ z , and C. ˙ This allows of these “constants,” i.e., the average time evolution, E, them to calculate the time to go from apoapsis and back (radial period) and thus the change in E, Lz , and C and so the new constants of motion, pnew , enew , and ιnew . The authors find that the family of separatrices in the Kerr case deviate from the Schwarzschild case depending on (1) the spin of the SMBH, (2) the inclination of the orbit, and (3) whether the orbit is pro- or retrograde. For prograde orbits, the last stable orbit (LSO) is closer to the event horizon, and, hence, the possibility that the CO is on a 1-burst orbit becomes more rare. Retrograde orbits, on the contrary, have LSOs farther away than in the Schwarzschild case. Therefore, it is easier to find 1-burst situations. Nevertheless, the situation is not symmetric. The farthest possible away separatrix for a retrograde orbit is much closer to the Schwarzschild separatrix than the equivalent for a prograde orbit. In Fig. 2, which is using the same method as in [12], we can see this. The green separatrices for retrograde orbits are closer to the Schwarzschild case than the blue ones, which correspond to prograde orbits. The work of [12] shows that the number of cycles in the band of the detector before plunging through the event horizon is as large as 105 for some prograde cases, and at least of a few hundreds, depending on the spin and inclination. Retrograde cases, however, can lead more easily to 1-burst orbits. However, due to the asymmetry, these are much more rare. It must be stressed out that, even in the Schwarzschild case, it is difficult to have 1burst orbits. This is so because the energy radiated away at periapsis is proportional

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to q, so that even for orbits with very small periapsis distances, the EMRI in general cannot radiate away the required amount of energy for the CO to plunge. There are exceptions to this situations which, however, are fine-tuned, such as the zoom-whirl orbits [83]. In this case, even if we only have one radial oscillation, there is an arbitrary number of (different) oscillations along the azimuthal axis, which translate into a very strong precession of the orbit. When a Keplerian orbit extends beyond the potential well in radius, the orbits are non-closed anymore [the Newtonian precession; see, e.g., 7]. Such non-closed orbits have a periapsis advance as a consequence. There are two periods in these orbits: the orbital period is the required time to go from apo- to periapsis, and another one is the time to revolve 2π in the orbit. These two periods are identical in the Keplerian case of closed orbits. In the relativistic case, these two periods always differ, and the more relativistic, the larger the difference, to the point of being arbitrarily different. This leads to a precession of the orbit with an arbitrary number of cycles in a single radial period, if the initial conditions are matched. This means that one needs a high degree of fine-tuning in the Hamiltonian system which describes the initial dynamical parameters. In such a case, the amount of energy radiated is considerable, and it could lead to the CO to indeed plunge through the event horizon after one single periapsis passage. In the Kerr case, the physical picture is even more complex, depending on whether the orbit is pro- or retrograde. If it is prograde, since the orbit gets closer to the last stable orbit (LSO), it will induce a stronger precession of the orbit than in the Schwarzschild case. If it is retrograde, the number of cycles will be less than in the Schwarzschild case. Another possibility would be that the SMBH is Kerr and the orbit of the CO is perfectly parallel and aligned with the axis of the spin of the SMBH. In that case, from the perspective of the CO, the SMBH would be de facto a Schwarzschild SMBH, and this peculiar situation could have a 1-burst orbit. These are nonetheless very peculiar configurations and tuning of the initial conditions of the orbits, so that the probability for these scenarios to happen frequently in the formation of EMRIs is low. The belief that most sources would be lost to 1-burst orbits led a number of researchers to look at alternative scenarios, in particular to resonant relaxation. The idea was that by getting closer and closer to the SMBH, the number of density of stars would decrease, and, hence, the danger of turning a successful EMRI orbit into a 1-burst orbit would decrease. But, at the same time, because of the drop of density, the timescale associated with relaxation would increase more and more, so that the event rate would drastically drop. This is why the concept of (scalar) resonant relaxation as presented by [147] was envisaged as a possible way to enhance the rate. This motivation led to interesting discoveries in the field of theoretical stellar dynamics (see in particular the work of [27, 28, 64, 99, 164]). However, even in the range of radii in which scalar resonant relaxation was thought to be a promising driver of EMRIs, it is now well settled that in the end, two-body relaxation is the main mechanism. Therefore, in this work, we will focus on two-body relaxation

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and loss-cone theory. For more details about resonant and scalar relaxation, and alternative scenarios, we refer the reader to [17].

Stellar Tidal Disruptions The theory on which we address the problem of formation and evolution of EMRIs is two-body relaxation [see e.g. 37, 160] in the context of stellar tidal disruptions (for a recent review of the theory and derivation of the rates, and references therein [165], and for some classical references, see [126, 148, 166, 175]). When an extended star (meaning a star which is not a CO), such as our Sun, approaches “too closely” a SMBH, it will suffer a difference of gravitational forces on points diametrically separated on its structure due to the fact that it is not a point particle. Hence, depending on how close the star, typically on a hyperbolic orbit, gets to the SMBH, it will get disrupted or not, since we have to compare the work exerted over it by the tidal force with its own binding energy. The radius within which this happens is what we call the tidal radius rt and can be derived to be [see, e.g., Sec. 1.3 of 7, but note the small typo: the factor 2 should be outside of the brackets] 

(5 − n) MBH rt = 3 m

1/3 2r,

(1)

with r the radius of the star, m its mass, MBH the mass of the SMBH, and n the polytropic index. For our Sun, considering a n = 3 polytrope, and MBH = 106 M rt  1.83 × 10−6 pc  0.38 AU.

(2)

We show in Fig. 3 a smoothed-particle hydrodynamics simulation of the tidal disruption of a star of mass m = 1 M , which corresponds to Fig. (2) of [7]. We can see that the star adopts a spheroidal shape after it has passed through periapsis on its orbit around a SMBH of mass 106 M . The top-left panel corresponds to the initial time and the rest of the panels to later moments. As we can see in Figs. 3 and 4, the Sun will be able to only describe a close passage around the SMBH to then be tidally torn apart (if the trajectory crosses the tidal radius and the disruption degree depends on how deep the passage is). Contrary to extended stars, COs can revolve around the SMBH many times, as we have already mentioned. This is so because the tidal radius of a neutron star (NS) is located within the event horizon of the SMBH, so that we will never see the NS being disrupted. An interesting situation, however, is systems composed of an IMBH and a white dwarf (WD), because the WD will be tidally disrupted before plunging through the event horizon [see, e.g., 127, 129, 149, 156]. In Fig. 5, we show the relation between the mass of a SMBH and the mass of a star, or sub-stellar object, for it to cross the event horizon without being tidally

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Fig. 3 Tidal disruption of a sun-like star around a SMBH of mass MBH = 106 M , on the frame of the star. The QRC link to the URLs https://youtu.be/Ryc44v4Eb7I and https://youtu. be/uZqXBD8R9Dw. This is Fig. (2) of [7], distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/)

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Fig. 4 Same as Fig. 3 but in the general frame. The SMBH is located at the origin of coordinates. We have selected four snapshots right after the periapsis passage. The length is expressed in solar radii. We can see how the star is torn apart and extends to much longer radii than the initial radius of the star, m = 1 M

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Fig. 5 Adapted figure Fig. (1) from [18]. Minimum mass MBH as a function of the mass of a star, or sub-stellar object, for the latter to cross the event horizon without suffering significant tidal stresses. The arrow points to the mass of the SMBH at the center of our Galaxy and the dashed lines the interval of masses for stars and brown dwarfs (BD) which can directly plunge without disruption

disrupted. For instance, we can see that a red giant with a mass of 50 M can plunge directly the event horizon of a SMBH of mass  4 × 108 M without being disrupted. For this figure we have taken into account the mass-radius relations from the modelling of stars of [46, 49, 130, 152]. These relations reproduce almost identically the more recent data of [47] for brown dwarfs. The problem of an EMRI is conceptually very similar to that of a tidal disruption. However, instead of one periapsis passage, the source will pass through periapsis tens to hundreds of thousand of times. EMRIs form initially with very high eccentricities, as we will see, so that the apocenter can extend well into the bulk of the stellar system. A perturbation at apocenter can lead to (i) the EMRI scattering off the orbit, so that we lose the source of GWs, or to (ii) increase even more its eccentricity.

Relaxation Theory In Newtonian physics we cannot analytically solve systems with more than two bodies (but for very special configurations, in which there is an additional body which has a much smaller mass than the two other bodies, which is the restricted

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three-body problem). However, when we try to understand the formation of EMRIs, IMRIs, or XMRIs, we need to look at dense stellar systems which might have up to 108 M pc−3 . These densities are orders of magnitude above what we find around our solar system, which typically has of the order 10−2 M pc−3 . Since we cannot solve this problem, we try to address it by borrowing ideas from other fields in physics, such as plasma physics, because it also shares the property of inverse square laws. However, plasmas are often nearly uniform, rest and large spatial extent, which is not the case of a dense stellar system. Another possibility is thermodynamics because, in some sense, we can think of a dense stellar system as a gaseous medium. Nonetheless, thermodynamics excludes a description of selfgravitating systems. We cannot have an asymptotic thermodynamic limit because the gravitational forces are long-range and we cannot ignore the effect of a star which is “far away.” This translates into the fact that the thermodynamical equilibrium is ruled out by construction. In spite of this, using thermodynamics in the context of stellar dynamics has proven to be an important asset. Nevertheless, surprises arise in this context, such as the concept of negative heat capacity [“(...) stars act like donkeys slowing down when pulled forwards and speeding up when held back.” 123]. We introduce now the fundamentals of relaxation theory, just as a way to understand in an intuitive way how to ponder the phenomena that lead, in the Newtonian limit, to centrophilic orbits, i.e., those which will approach the SMBH enough so as to stand a chance of forming an EMRI. For a more detailed description, we remit the reader to [37, 93, 150, 160]. In relaxation, the potential of the dense stellar system is described by approximating it in the addition of two separated components. One is given by a dominating smooth contribution, which we will call Φs , plus the contribution of individual stars, δΦ, which are responsible for the phase-space distribution function to follow the collisionless Boltzmann equation. A way to describe the evolution of these dense stellar systems is the FokkerPlanck (or Kolmogorov forward-) equation, which is a partial differential equation that allows one to study the time evolution of the velocity probability density function of a particle (a star) due to drag forces and random forces. When two stars interact with each other, the associated volume typically has dimensions which are small compared to the macroscopic lengths of the system, such as the radius of the galactic nucleus. We can then approximate the evolution by assuming that the entire cumulative effect of all encounters on our star are identical to an ideal situation. We assume that the star is embedded in a homogeneous system with the same local distribution function everywhere [this is usually referred to as the “local approximation”; see, e.g., Sec. 8.5 of 17, for a detailed description]. As time flows, the accumulated variations of δΦ alter E and J and lead to a modification of the distribution function. We then treat the role of the individual changes of δΦ as the sum of uncorrelated hyperbolic encounters between two stars with a small deviation angle. Hence, if we consider a star “1” in a stellar system which is perfectly homogeneous of stars of the type “2,” with the same masses and velocities, after a time δt, the initial trajectory will suffer a deflection θ of

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θ δt = 0,

    bmax G2 (m1 + m2 )2 2 θ  8π N ln δt, 3 δt b0 vrel

(3)

with m1 and m2 the masses of two stars, N the stellar number density, b0 the impact parameter that leads to a deflection angle of π/2, bmax a normalization factor to avoid logarithmic divergence (interpreted as the maximum impact parameter, of the order the size of the stellar system), and vrel the relative speed between star 1 and the background stars 2. Note that we obtain θ δt = 0 because this is a diffusion process, and hence θ 2 δt ∝ δt. We define the velocity dispersion σ of the nucleus or cluster via the statistical concept of root mean square dispersion. The variance σ 2 provides us with a measure of the dispersion of the observations within the statistical population at our disposal (i.e., the observational data). This means that σ2 =

N 1  (Vi − μ)2 . N i=1

where Vi are the individual stellar velocities and μ the arithmetic mean μ≡

N 1  Vi . N i=1

If we take M and R as the total stellar mass and radius of the nucleus, respectively, and define m∗ as the average stellar mass, then the argument of the Coulomb logarithm (remember our discussion about plasma physics) is approximately 2 R vrel M σ 2R bmax γ   = γ N. b0 G (m1 + m2 ) GM m∗

(4)

In the last equation, γ is a dimensionless proportionality constant. Note that this proportionality applies only to self-gravitating, virialized stellar systems. In order to derive the relaxation timescale of the system, we would have to integrate over all possible values of bmax /b0 ; however, in practice, this is again approximated by assuming that γ is constant. This global value can be derived using analytical arguments [94, 161], which has been corroborated with numerical simulations. For systems composed of stars with a single type of mass, [94] derived γ  0.10−0.17. He also proved that this value should be significantly smaller in the case that we consider a mass spectrum. When we analyze relaxation via the local approximation, we are implicitly assuming that the dense stellar system is finite and homogeneous. Real systems however, are quite the contrary, with important density gradients. By adopting the local

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approximation, we are imposing the fact that b R. This allows us, as explained, to neglect Φs – and hence approximate the trajectories as Keplerian orbits – as well as to use the local properties around star 1 as representative ones. This a priori conceptual grotesque approximation is however physically tolerable because we are talking about the argument of a logarithm, ln(b2 /b1 ). This assumption has been tested in a large number of works and seems to be acceptable for pragmatic purposes [77, 78, 143, 162, 163, 167]. Although there is a conceptual problem in the way we have introduced it [see discussion in 17], we can use Eq. (4) to introduce a characteristic timescale associated with relaxation   π 2 δt θ2  . δt 2 Trlx

(5)

The full expression of this timescale is given in, e.g., [48] (see [112]). In this work, we will not derive it. We only introduced the basic notions so that the reader can develop a physical intuition for the process. The expression is Trlx =

9 σ3 , √ 16 π G2 mρ ln(γ N )

(6)

which can be rewritten as Trlx ∼ 2 × 1010 (ln Λ)−1 yrs



σ 100km/s

3 

m 1M

−1 

r 1pc

γ .

(7)

In this expression, we have adopted a power-law distribution in the stellar density, ρ(r) ∼ r −γ (see [32, 72, 136, 154]). Within the influence radius of a SMBH rinfl (which can be loosely defined as the radius within which the potential is dominated by the SMBH), we can approximate as explained before bmax = rinfl , and relaxation leads to a steady-state distribution of orbital energies on, of course, a timescale ≈ Trlx . So far, we have considered that all stars have the same mass in the nucleus. This is obviously not correct. Stellar-mass black holes, in particular, have a higher mass than our Sun, which we adopt to be in this work mbh = 10 M . If we consider that a fraction of all stars are stellar-mass black holes, typically as small as ≈ 10−3 for a standard initial mass fraction[see, e.g., 110], then a limiting form of relaxation leads to an interesting phenomenon. Stars, or COs, with masses larger than the average stellar mass, assumed to be of 1 M will segregate in phase-space in a timescale shorter than the associated Trlx by a factor given approximately by the ratio Q of the average stellar mass divided by the mass of the CO. That is, if mbh = 10 M and m = 1 M , this timescale is 1/10 of Trlx . This limiting form of relaxation was described by Chandrasekar [see, e.g., 37] and is called dynamical friction, with the associated timescale

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TDF =

  2X −X2 −1 vc3 m Trlx erf(X) − , e ∼ Trlx := √ 2 mbh Q 4π G ρmbh ln Λ π

(8)

where vc is the (local, obviously) circular velocity, ρ is the mass density of the stars √ other than COs, and the mass ratio is Q = mbh /m and X = v/( 2σ ). What this means is that we do expect to have stellar-mass black holes very close to the SMBH in a timescale shorter than the Trlx , which is what we need to create the orbit which will ultimately lead to an EMRI. This leads us to the question whether nuclei in the bandwidth of LISA will have Trlx shorter than a Hubble time because otherwise the nuclei will not be relaxed, and hence we cannot expect stellar-mass black holes close to the SMBH. If nuclei which are target of LISA, i.e., with masses approximately ranging between 104 and107 M , are not relaxed, then we cannot expect EMRIs to be detected. If these nuclei follow the so-called mass-sigma relationship (which is not clear, since they are at the low-mass end of the correlation (see, e.g., [105]), MBH ∝ σ 4 , then we can express the influence radius assuming that the nucleus is isothermal (i.e., ρ ∼ r −2 , σ (r) = constant, for r  few × 10−1 rinfl ) as rinfl ∼ GMBH /σ 2 ∝ 1/2 MBH , with G the gravitational constant. On the other hand, the average stellar 3 ) (since we have density at rinfl can be estimated roughly to be ρ ∼ 2MBH /(rinfl the same mass in stars as the mass of the SMBH). The velocity dispersion within rinfl is σ 2 (r) ∼ GMBH /r (as determined via the Jeans equation; see [37]), so that we finally have Trlx ∼ σ 3 /ρ, which means that Trlx =

17 (mΛ)−1 100



MBH G

1/2

3/2

(9)

rinfl

Adopting ln Λ ∼ ln(MBH /mbh ) ∼ 11.5 (see Eq. 13 of [17]), and normalizing to typical values, we find that

Trlx

∼ = 1.1 × 109 yrs



MBH 4 × 10 6 M

1/2 

m 0.4 M

−1 

rinfl 1 pc

3/2 ,

(10)

where we have used the average stellar mass at the Galactic Centre. This can be roughly derived by assuming that m = Mtot /Ntot , with Mtot the total stellar mass and Ntot the total number of stars. In our approximation, we only consider main sequence stars and stellar-mass black holes, which represent a fraction 10−3 of the total number. Using a typical IMF, the average stellar mass is of ∼0.4 M , which is in agreement with the Milky Way models of [70]. Using the fact that stellar distribution follows a power-law expression of the radius, as explained before, we can furthermore derive that [see 18]  rinfl  2.5 pc

MBH 4 × 106 M

3/5 .

(11)

18 The Gravitational Capture of Compact Objects by Massive Black Holes

787

1011

tot (rinfl) (yrs) T rlx

1010 10

T Hubble SgrA*

9

108 107 106 105 103

SgrA* 104

105

106

107

108

109

MBH (M ) Fig. 6 Relaxation time as a function of the SMBH mass MBH . The horizontal, dashed line marks the Hubble time limit and the arrows the position of the SMBH in our Galaxy; in the relation, we have taken into account the influence radius as a function of MBH . Nuclei harboring SMBHs with masses MBH  107 M have a relaxation time longer than the age of the Universe and have not had time to converge into a steady state and develop segregated stellar cusps. Their stellar distribution keeps memory from their formation and (if any) of the last strong perturbation process

Then, we are finally left with tot ∼ Trlx = 4.35 × 109 yrs



MBH 4 × 10 6 M

11/10 

m 0.4 M

−1 .

(12)

In this expression, we have the relaxation time as a function only of the mass of the SMBH while taking into account the scaling of the influence radius with this mass. This allows us to have an idea of what nuclei will be relaxed. We depict this in Fig. 6 for a range of masses and note that in the low end of masses, the results should be taken with skepticism, since SMBHs with light masses will wander off the center. By wandering off, the massive black hole (in this case an IMBH) could potentially explore regions of phase-space with different relaxation times, and since the wandering timescale is much shorter than the relaxation timescale, the system does not have time to re-adjust. Hence, any capture is led by a dynamical process, and not a relaxational one. It is remarkably interesting to note that the relaxation time will exceed a Hubble time for nuclei harboring SMBHs with masses slightly above 107 M (the exact number should not be envisaged as a realistic one in view of the – few – assumptions that we have made in the derivation). Surprisingly, this type of masses fit perfectly well in the kind of nuclei that LISA will probe. If this had not been the case, LISA would never be able to detect an EMRI. To the best of my knowledge, nobody thought of this potential risk before designing the sensitivity curve of the instrument.

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I would appreciate it if somebody could provide me with more details, in case that I am wrong (which is a possibility, obviously).

The Loss-Cone After having conceptually introduced what is the relaxation of a dense stellar system, we now look at the definition of the region of phase-space denoted as the “loss-cone.” Not all stars in a galactic nucleus or a globular cluster will have such an energy and angular momentum that will bring them very close to the SMBH; after all, from the point of view of the stellar system, the SMBH is something microscopic, with a size (i.e., a Schwarzschild radius) of RSchw  3.8 × 10−7 pc if its mass is that of SgrA*. In comparison, its influence radius, as we have seen, is about rinfl  2.5 pc, i.e., seven orders of magnitude bigger in length. Only a very peculiar subset of all stars with a given semi-major axis will have an eccentricity high enough to bring them close at periapsis to interact with the SMBH. Since historically this was studied in the context of stellar disruptions, this subset is dubbed the “loss-cone,” because (1) the orbit around the SMBH can be defined in terms of an angle that will lead the star to get close enough to the SMBH, and (2) if at periapsis it crosses the tidal radius, the star can be tidally disrupted, as we saw before, and is hence lost for the system. Previously, we introduced the relaxation time. It is useful to define another interesting timescale, the dynamical timescale or the “crossing time.” This is the time required for a star to cross the host dense stellar system. For a cluster, Tdyn = rinfl /σinfl , where σinfl is the velocity  dispersion at the influence radius. If

3 /(GM )  (Gρ)−1/2 , with M we impose virial equilibrium, then Tdyn ≈ rinfl infl infl the total stellar mass comprised within rinfl (which is in the order of magnitude equivalent to the mass of the SMBH). Following our previous discussion of (cheating with) thermodynamics, we note that contrary to a gaseous system, in a stellar system, the thermodynamical equilibrium timescale will be much longer than the crossing one, Trlx Tdyn . If we consider a perfectly homogeneous stellar system, we will reach a stationary state in the limit t → ∞. How quickly a virial equilibrium is reached depends on the timescale in which a perturbation in the stellar system is washed out. We have introduced the relaxation time as the time that we need to wait for the system to “mix up,” i.e., more formally, for the perpendicular velocity component of a star to be of the same order than the perpendicular velocity component itself, 2 /v 2  1, as illustrated in Fig. 7. That is, the relaxation time is the required v⊥ ⊥ time to induce a change in the perpendicular velocity component of the same order 2  v 2 , after a number of as the perpendicular velocity component v⊥ itself, v⊥ ⊥ dynamical times (e.g., crossing times) n, which allows us to define the associated 2 /v 2 = T /T , as in, e.g., [37]. This timescale for mixing up the system: v⊥ rlx ⊥ 2 2 2 /v 2 = 1, so that T means that we have that v /v = n δv rlx = n Tdyn = ⊥ ⊥ ⊥ ⊥

2 2 T v⊥ /δv⊥ dyn .

18 The Gravitational Capture of Compact Objects by Massive Black Holes

789

v θD

1 δv ⊥

Fig. 7 The change in the perpendicular component of the velocity of the star is assumed to be small after every single encounter with another star. This leads to the definition of a diffusing angle θD which, ab definiti¯o (see text), must be small. When this component is of the same order than the perpendicular velocity component itself, a relaxation time has passed

The mean deviation of a orbit in a dynamical timescale Tdyn (which is typically defined as the time required for a star to cross the system) can be estimated via a “diffusion angle” θD2 := Tdyn /Trlx , and we furthermore assume that this angle is very small, following the line of thought of, e.g., [9,25,69,115]. Therefore, sin θD  δv⊥ /v  θD , and so θD  Tdyn /Trlx . This angle is very useful to understand the evolution of the system; it will help us quantify how efficiently (or not) the loss-cone can bring stars close to the SMBH. Stars which are in the loss-cone are lost in a dynamical time, i.e., a crossing time. We can define the stars that belong to the loss-cone by evaluating the angular momentum of the star, the influence radius, and the velocity dispersion of the system. We can illustrate this angle in an approximate way as in Fig. 8. If the star of mass m has a velocity vector such that the periapsis distance rp falls within the tidal radius rt , it will suffer important tidal stresses. A total disruption happens when the so-called penetration factor β := rt /rp > 1.85; see [86]. This will happen when rp (E, L) ≤ rt and θ ≤ θlc . One can show that the loss-cone angle is θlc = √ 2rt /(3r), with rt √ the tidal radius and r the distance from the MBH to the star, if r ≤ rinfl , and θlc  2rt rinfl /(3r) for r ≥ rinfl [for details about this derivation, see, e.g., 17]. With these considerations we can define a critical radius which will determine the future of stars in the loss-cone. If we define the ratio ξ := θlc /θD , when ξ = 1, θlc = θD , which can be converted into a distance rcrit . Stars inside this radius are removed on a Tdyn because ξ > 1, θlc > θD . We can therefore make an educated guess for the loss-cone to be replenished. The condition is that ξ 2 × Tdyn , where the square appears because, as illustrated in Fig. 9, the surface which corresponds to the radius b must cover the empty surface of radius a. When a star is outside of the critical radius, even if it is in the loss-cone, it will be diffused out of it before it can reach the central SMBH. This is so because ξ < 1, θlc < θD , and by definition, θD corresponds to the variation of θ in a Tdyn . Inside the critical radius, however, stars will fall on to the SMBH without being perturbed. We have that ξ = a/b because a = v sin θlc ≈ vθlc , and b = v sin θD ≈ vθD .

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P. Amaro Seoane

v

m θlc

MBH

rp

rt

R Fig. 8 Definition of the loss-cone angle for a star of mass m approaching a SMBH of mass MBH

MBH

θD

θlc

b

a

Fig. 9 Definition of the critial radius via the loss-cone- and diffusion angles

18 The Gravitational Capture of Compact Objects by Massive Black Holes

791

v

m• θlc

MBH

rp

R Fig. 10 Same as Fig. 8 but for a CO orbiting around a SMBH. The orbit precesses, losses energy, and thus shrinks over time, and the plane of the orbit basculates due to frame-dragging if the SMBH is spinning, which is not shown in the illustration because it is a two-dimensional projection. At periapsis, there is a strong burst of gravitational radiation, and the semi-major axis of the orbit shrinks a bit (the shrinkage is not to scale, as well as the precession). This figure illustrates the difficulty of defining analytically the relativistic equivalent of the loss-cone treatment for a tidal disruption. The gravitational waves are not to scale; they are just illustrative

While it would be ideal to be able to define a relativistic loss-cone, i.e., a “GWcone,” in practice, this turns out to be challenging, if not impossible. In Fig. 10, we depict the complications of this situation. The orbits are not closed and shrinking over time, and, if the central SMBH is Kerr, there will be a change in the inclination of the orbit when the CO approaches the pericenter.

Formation of EMRIs via Relaxation The evolution of a CO on its way to becoming a source of gravitational waves is determined by two very different types of physics: stellar dynamics, for which we do not have to bother with general relativistic effects, but with two-body relaxation, and gravitational radiation. Since we obviously cannot solve the 106 −−108 problem in Newtonian gravity and not even the two-body problem in general relativity (even less at these mass ratios), we need to work with timescales to derive information about the process. We can define a threshold that separates the evolution via

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P. Amaro Seoane

dynamics from general relativity by equating the two associated timescales, times a factor C of order 1 Trlx, peri = C TGW (a, e)

(13)

In this equation, Trlx, peri is the relaxation time at pericenter, i.e., Trlx, peri := Trlx (a) × (1 − e) [7], and TGW the time derived in the approximation of Keplerian ellipses of [139]. Since the initial eccentricities of EMRIs are typically very large, as we will explain√later, the function F (e) of TGW (a, e) can be estimated to be F (e) = 425/(768 2). We also have to equate 8 GMBH = a (1 − e), c2

(14)

with c the speed of light, and we note that the position of the last-stable orbit (LSO) depends on (i) the spin of the central SMBH a• , (ii)the inclination of the orbit θ , and (iii) whether the orbit is pro- or retrograde, as described in the work of [12]. This is of course only true for Kerr SMBHs. The Schwarzschild case serves as a reference point, which has the LSO at a distance of 4 RS , with RS the Schwarzschild radius. The work of [12] provides us with the function W (θ, a• ) which captures this information and accordingly modifies the position of the LSO. This position is crucial because it determines the upper integration limit when calculating the event rate, as we will discuss. In Fig. 11 we give the full shape of the function. The GW timescale is given by TGW (a, e) ∼

√ 24 c5 a 4 (1 − e)7/2 2 , 2 85 G3 m CO MBH

(15)

where m CO is the mass of the CO in consideration. It could be a neutron star, a white dwarf, a stellar-mass black hole, or even a brown dwarf, which is not a compact object, but in the case of masses in the range of SgrA*, as we have discussed, a powerful source of gravitational radiation. This is important because the timescale is given by two bodies only, the SMBH and the CO. Moreover, we note that stars with different masses will segregate in different fashions. That is, the density profile will follow a different power law, but, still, relaxation will be dominated by stellarmass black holes, as discussed in [18]. This implies that, while relaxation is the driving mechanism dominating the stellar dynamical evolution and is provided by the stellar-mass black holes at the radii of relevance, the CO must not necessarily follow the same density distribution as that of the stellar-mass black holes. This is why we will use two different power indices, γ for the distribution of stellar-mass black holes and β for the distribution of the other species, the COs. Similarly, we will use two separated masses, one for the stellar-mass black holes, mbh , and another for the COs, m CO . If we are interested in EMRIs, then obviously, mbh = m CO , and γ = β.

18 The Gravitational Capture of Compact Objects by Massive Black Holes

793

1.5

0.5

W (a• , θ)

1.0

0.0 −0.5

0.0 0.1

0.2 0.3

0.4 0.5

a•

0.6 0.7

0.8 0.9 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

d)

a θ (r

Fig. 11 Function W (θ, a• ) which gives us the multiplying factor to identify the location of the LSO for a Kerr SMBH relative to the Schwarzschild case for prograde and retrograde orbits

As shown in [18], we can define the threshold between dynamics and general relativity, taking into account the change of relaxation as a function of the radius with 7/2

(1 − e)5/2 =

85 1 G 5/2 Mbh m CO 3−γ γ −11/2 4.26 R a . √ (3 − γ )(1 + γ )3/2 24 2 c5 ln(Λ) NBH mbh 0 (16)

In this equation, R0 is the radius within which relaxation, as discussed before, is dominated by stellar-mass black holes. Accordingly, NBH is how many of them are enclosed within R0 . We choose R0 ≡ rinfl . This allows us to solve for the critical semi-major axis, which is given by  acrit = rinfl

C 6144 (3 − γ )(1 + γ )3/2 4.26 85

 × W (θ, a• )



5/2

MBH NBH ln(Λ) m CO



1 γ −3



MBH mbh

1 −2  γ −3

,

(17)

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P. Amaro Seoane

Fig. 12 Evolution in phase-space (semi-major axis and orbital period as a function of the eccentricity) of a potential source of GWs, represented with a black circle. The mass of the SMBH is of MBH = 4 × 104 M . Green lines show the correlation between a and e as given in [139], and blue, dotted lines are the isochrones for a given TGW , as described in Eq. (15). The standard relaxation scenario is marked with the letter “A”: We have marked a few important points. The first and the second points show the dynamical parameters for which the CO would merge in less than 1010 and 108 , respectively, if it only evolved due to the emission of gravitational radiation, which is not the case, because the source is on the right of the oblique, dashed red line, where dynamics dominate the evolution. Point three represents the crossing of this very line, from which the evolution is dominated uniquely by GWs. Point four is the conjunction of the LSO (for a Schwarzschild SMBH) and the threshold line, which defines in the Y-axis the critical semi-major axis acrit , and finally point five is a rough location of where the LISA bandwidth stars (based on the orbital period, which will change depending on the dynamical properties of the CO). The tidal separation scenario, which will be discussed later, is marked with the letter “B”: We can see that, when entering the LISA band, these sources will have signifincatly lower eccentricities. The values which we have chosen for this plot are shown in the left-bottom corner

With the threshold between dynamics and general relativity and acrit , we can now plot the evolution of a potential source of GWs in phase-space. We represent this in Fig. 12. The source describes a random-walk-like evolution in energy and angular momentum due to the interaction with other stars in the stellar system, since we are on the right hand of the red line, with a larger scatter in eccentricity than in semi-major axis until it crosses the threshold given by Eq. (16). From that moment onward, on the left hand of the red line, the driving mechanism is gravitational

18 The Gravitational Capture of Compact Objects by Massive Black Holes

795

radiation, and the evolution in phase-space follows very closely one of the green lines, which gives the relation between the semi-major axis and the eccentricity in the two-body problem as approximated by [139]. We note, however, that for strongfield and fast-motion orbits, radiation reaction will enhance the eccentricity for very small values of the semi-major axis (i.e., if the semi-lactus rectum p is close to its minimum value 6 + 2e), which can be used as an indicator of the imminent plunge of the orbit [58]. We give an example of this in Fig. 17 and see the discussion in the text of that section. EMRIs with a acrit will have a much lower event rate because in the powerlaw solution for the stellar distribution, we have that the numerical density of stars n ∝ r −γ , so that the total number of stars N per unit log(a) scales as dN/d(log a) ∝ a (3−γ ) . Moreover, as we move to deeper and deeper radii, the value of γ is lowered [10, 115]. This is so because the loss-cone is much more quickly depleted, and in order to re-populate it, we need to wait for several relaxation times. Also, as we get closer to the SMBH, the value of the relaxation time increases, as we can see from Eqs. (6) and (7). Indeed, unless SMBHs in the Universe are Schwarzschild, which is very unlikely [157], EMRIs will originate at large semi-major axes with very high eccentricities [see discussion in 12].

Formation of EMRIs via Tidal Separation of Binaries An interesting idea about how to produce an EMRI has its origins in the work of Hills [96], who estimated that, in the same way a star can be tidally disrupted, as we have seen in section “Stellar Tidal Disruptions”, a binary could be separated, “ionized,” via the same process: The gravitational forces acting onto one of the two companions would be different than on the other one, and depending on the distance to the SMBH, this difference of forces could overcome the binding energy of the star. He predicted that this would lead to the creation of the so-called hyper-velocity stars, stars with a velocity of > 103 km s−1 . The discovery of these stars in the work of [41] led to literally an avalanche of theoretical and observational work. Among these works, Miller and collaborators in [132] presented an interesting idea: If one of the two stars happened to be a CO, more specifically a stellar-mass black hole. Because the separation happens very close to the SMBH, the stellar-mass black hole could eventually become an EMRI. These EMRIs, contrary to those produced by relaxation, would have a much lower eccentricity when entering the LISA band, as we can see in Fig. 12, case “B.” Because the initial semi-major axis is smaller than in the relaxation case “A,” the eccentricity at bandwidth entrance is much lower. As in the tidal disruption problem, we can derive the splitting radius, which obviously is very similar to the former one  Rsplit ∼ a

MBH mbin

1/3 ,

(18)

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P. Amaro Seoane

with, e.g., mbin = m + mCO , if one of the companions is a star and the other one a CO. To get an idea of the ejection√ velocity that the companion will receive, we note that the orbital velocity is vorb ∼ G mbin /a, i.e.,

vorb  312 km s

−1



mbin 11 M

1/2

a −1/2 , 0.1AU

(19)

assuming that m = 1 M and mCO = 10 M . Note that this is just an example, because in a nucleus, it is more likely that the mass of the extended star has a similar mass to that of the CO. Assuming a parabolic encounter, which is the most probable scenario, the velocity of the center-of-mass VCoM can be estimated to be  VCoM 

G MBH ∼ vorb Rsplit



MBH mbin

1/3

 1.4 × 104 km s−1  vorb ,

(20)

for a MBH = 106 M . Following the derivations of [17], we have that the ejection velocity of the companion is veject  (M/mbin )1/6 and the new semi-major axis of the binary acapt ≈ a (MBH /mbin )2/3  104 a  5 × 10−4 pc. One can approximate the separation radius Rsplit to the periapsis distance, and with that information, we can derive the initial eccentricity when the CO is captured by the SMBH which is of ecapt = 1 − (MBH /mbin )−1/3  0.97 for the values that we have adopted here. These values for the initial semi-major axis and eccentricity are responsible for the source to enter the LISA band at much lower eccentricities, as we can see in Fig. 12, case B (which corresponds to a slighter more massive SMBH, of 4 × 104 M ). This signature in the eccentricity as compared to a relaxation EMRI is what will allow us to extract information about the mechanism that produced the source in the first place. This is so because if we set all dynamical parameters of two EMRIs identical but for the eccentricity, we can calculate the mismatch M between them, as done in [17], to find hrlx |hbin  = 99.9971%, M := 1 − √ hrlx |hrlx  hbin |hbin 

(21)

with hrlx and hbin the waveforms of an EMRI produced via relaxation and binary separation, respectively, using the kludge approximation of [71]. The natural scalar product product < | > is introduced by treating the waveforms as vectors in a Hilbert space [92], and we refer to this work as well as to [67, 169] for further details. For this kind of sources, as a rule of thumb, a mismatch M < 0.1 will make detection impossible, and a mismatch of M < 10−3 will make parameter extraction challenging [see, e.g., 53, 54, 59, 120].

18 The Gravitational Capture of Compact Objects by Massive Black Holes

797

Geodesic Motion and Relativistic Precession In this section, which profits from parts of [11], we give a summary of the concepts that we will need for the evolution of an EMRI around a Schwarzschild SMBH, the geodesic motion, and relativistic precession. The Kerr case is much more complex, and we will simply give an illustrative summary. We then give examples that solve the evolution of an EMRI (or an IMRI, XMRI) in phase-space by resorting to approximate, semi-analytical, and numerical techniques.

Geodesic Motion Around a Schwarzschild Black Hole The only spherically symmetric solution of Einstein’s equations for vacuum is the Schwarzschild solution (Birkhoff’s theorem) (Using normal units, we have f =1−

2GMBH c2 r

where G is Newton’s constant and c denotes the speed of light.) ds 2 = −f dt 2 +

dr 2 2 +r dΩ 2 , f

f = 1−

2MBH , r

dΩ 2 = dθ 2 +sin2 θ dϕ 2 .

(22) This solution predicts the existence of a horizon at rg = 2MBH (= 2GMBH /c2 in normal unit conventions) and hence a black hole geometry. We are interested in the motion of a test mass around such black hole (whose geometry is described by Eq. (22)). As in Newtonian gravity, the motion takes place in a plane that we can take to be the equatorial plane θ = π/2. There are two constants of motion, the energy E and the angular momentum J , associated with the time and azimuthal Killing symmetries (We use the letters (E , J ) to distinguish them from the Newtonian definitions, (E, J ). Later we will see what are the relations between them.). Thanks to this, the motion is completely separable and hence integrable. Let us see how this works. The energy and angular momentum, (E , J ), in terms of the Schwarzschild coordinates of Eq. (22), can be written as E =f

dt , dτ

J = r2

dϕ , dτ

(23)

where τ denotes proper time (the time measured by the clocks of an observer moving with the test mass) and t is the coordinate time of Eq. (22), the time measured by the clocks of distance observers, at infinity). Notice that the constants of motion (E , J ) are also specific constants of motion (per unit mass of the test particle), like in the Newtonian situation of Eqs. (106) and (107). In the language of general relativity, we have the four-velocity uμ =

dx μ , dτ

 (uμ ) =

 dt dr dθ dϕ , , , , dτ dτ dτ dτ

(24)

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P. Amaro Seoane

which is a normalized vector with respect to the metric (because it has been defined in terms of proper time) gμν uμ uν = −1 ,

(25)

where gμν are the metric components, which in the case of Schwarzschild can be read off from Eq. (22). Then, Eq. (25) is equivalent to the following expression:  − 1 = −f

dt dτ

2

1 + f



dr dτ



2 +r

2

dθ dτ

2

 + sin θ 2

dϕ dτ

2  .

(26)

If we substitute in this equation dt/dτ and dϕ/dτ from Eq. (23) and use the choice θ = π/2, which implies dθ/dτ = 0, we obtain the following equation for the radial motion: 

dr dτ

2

   2MBH J2 1− . = E2 − 1+ 2 r r

(27)

To summarize, the equations of motion, in terms of proper time, are dt E , = dτ 1 − 2MrBH    2  2MBH dr J2 2 1− , =E − 1+ 2 dτ r r dϕ J = 2. dτ r

(28)

(29) (30)

Equation (28) just gives the relation between coordinate time t and proper time τ . We can use it to rewrite the equations of motion, the geodesics, in terms of coordinate time. For the purposes of this work, it is very convenient to introduce in an appropriate way a three-velocity. This can be done in a natural way by factoring out the time component of the four-velocity:

uμ = ut v μ = ut 1, v i ,

vi ≡

ui . ut

(31)

Here, it is important to notice that this three-velocity, or better spatial velocity, is now defined in terms of the coordinate time t as follows: vi =

dx i . dt

(32)

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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In this way, and by virtue of the normalization property of uμ , we can interpret ut as the relativistic gamma factor associated with the observers that measure the coordinate time t: gμν uμ uν = −1

1 ut =  ≡Γ . −gtt − 2gti v i − gij v i v j

⇒

(33)

√ In special relativity, the gamma factor is just Γ = 1/ 1 − v 2 . We can check that this is what we get from Eq. (33) for the Schwarzschild metric of Eq. (22) when we take the limit of no gravity, i.e., G → 0 (or, in our units, the limit MBH → 0). Going back to the study of Schwarzschild geodesics, let us now focus on bound orbits, since they are the ones we are interested in. By definition, the radial coordinate for these orbits must lie inside a finite interval, [rp , ra ], where rp is the pericenter radial coordinate and ra is the apocenter one. This means that for these particular values of r, dr/dτ must vanish. If we look at Eq. (29), we realize this can only happen if E 2 ≤ 1. Assuming E > 0 (negative values correspond to the timereverse orbits), this means 0 ≤ E ≤ 1. More in detail, Eq. (29) can be rewritten as follows: 

dr dτ

2 =



1 2 1 − E (ra − r) r − rp (r − ro ), 3 r

(34)

only valid for bound geodesics. Here, ro is a third root that satisfies ra > rp > ro , and hence it will not be reached during the motion. By comparing this with Eq. (29), we can obtain (rp , ra , ro ) 2MBH , 1−E2 J rp ra + ro (rp + ra ) = , 1−E2 r p + ra + ro =

rp ra ro =

(35) (36)

2MBH J 2 . 1−E2

(37)

In order to solve them, we can introduce the eccentricity and dimensionless semilatus rectum orbital parameters, (e, p), in the usual way (The general rule to recover proper units is to make the substitution: MBH −→ GMBH /c2 .): rp =

p MBH , 1+e

ra =

p MBH 1−e

⇐⇒

p=

2rp ra , MBH (rp + ra )

e=

ra − rp . ra + rp (38)

Then, we can first find the expressions of (E , J ) and ro in terms only of (e, p) and the black hole mass:

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P. Amaro Seoane

ro = E2 = J2 =

2 p MBH , p−4

(39)

(p − 2 + 2 e)(p − 2 − 2 e) , p(p − 3 − e2 )

(40)

2 p2 MBH . p − 3 − e2

(41)

An interesting relation comes out by imposing that ro < rp as we have assumed at the beginning: p − 6 − 2e > 0,

(42)

which is a separatrix between stable and unstable bound orbits. In practice, in order to numerically integrate the equations of motion for bound orbits, we have to take into account that r is not a good coordinate to use due to the existence of turning points. In order to avoid the numerical problems derived from this, it is very convenient to introduce the following alternative angular variable: r=

pMBH , 1 + e cos ψ

(43)

The orbital motion, in terms of the variables (t, ψ, ϕ), is described by the following set of ODEs: √ (1 + e cos ψ)2 (p − 2 − 2e cos ψ) p − 6 − 2e cos ψ dψ  = , dt p2 MBH (p − 2)2 − 4e2

(44)

(1 + e cos ψ)2 (p − 2 − 2e cos ψ) dϕ  = . dt p3/2 MBH (p − 2)2 − 4e2

(45)

Relativistic Precession In Keplerian motion, the time that a particle takes to go from rp to ra and back to rp (the radial period) is exactly the same as the time it takes to go 2π around the central object, that is, to cover a ϕ-period. In general relativity, this is not the case, and in the case of a non-spinning black hole, these two periods do not coincide. The consequence of this is that the orbit does not close itself and there is precession of the pericenter (it is not located in the same position with respect to Cartesian coordinates associated with (r, ϕ), i.e., (x, y) = (r cos ϕ, r = sin ϕ)), also known as perihelion advance or periastron advance for test masses. Using the equations of motion given in Eqs. (44) and (45), the amount of angle ϕ covered during one single radial period is given by

18 The Gravitational Capture of Compact Objects by Massive Black Holes

 Δϕ = 2

ta

dϕ dt , dt

tp

801

(46)

where tp and tp indicate the coordinate time corresponding to the apocenter and pericenter locations. We can rewrite this integral in terms of ψ to get 

π

Δϕ = 2 0

dϕ dψ = 2p1/2 dψ



π 0

√

dψ , p − 6 − 2e cos ψ

(47)

and using that cos ψ = − cos(π − ψ) = −1 + 2 sin2 (π/2 − ψ/2) and defining x ≡ (π − ψ)/2, we get 4p1/2 Δϕ = √ p − 6 + 2e



π/2 0

4p1/2 K =√ p − 6 + 2e





dx 1−

4e p−6+2e

4e p − 6 + 2e

sin2 x

 ,

(48)

 π/2 where K(k) = 0 dα (1−k sin2 α)−1/2 is the complete elliptic integral of the first kind. It turns out that this integral diverges as p approaches 6 + 2e, that is, when we approach the separatrix of Eq. (42).

The Kerr Case Now that we have addressed the Schwarzschild case, which is our reference point, we will describe the characteristics that distinguish a Kerr SMBH from it. This is a simple summary focused on the main ideas which are important for the evolution on phase-space that we address later. For a rigorous, detailed review, we refer the reader to the reviews of [168, 174]. The most interesting feature is that, while in the Schwarzschild case the geometry is spherically symmetric, in Kerr, it is axisymmetric with respect to the spin axis. This translates into the fact that orbits outside the equatorial plane are not planar. There is no such (Keplerian- or Schwarzschild-like) concept in Kerr. As a matter of fact, the concept of “orbit” is not exactly straightforward in the relativistic case if it is not a bound one, and there is no simple way to compare it with the Keplerian concept. In any case, the inclination of the orbit ι with respect to the spin axis is fundamental in Kerr and decides the dynamics of the system (Fig. 13). Non-equatorial orbits (which are a special case) precess with a given ι at a frequency fθ , with θ the polar Boyer-Lindquist coordinate [40, 133]. We can define the inclination for Kerr geodesic orbits in two different ways. One possibility is via Carter’s constant Q

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P. Amaro Seoane

cos ι = 

Lz

,

(49)

 − θmin ,

(50)

L2z + Q

and another with the minimum value of θ ι = sign (Lz )

π 2

where sign (Lz ) allows us to distinguish between prograde (positive) and retrograde orbits (negative). In Fig. 14, we depict the orbit of a prograde EMRI revolving around a SMBH of spin a• . The minimum value of the polar angle θ determines ι. This figure is to be regarded as an illustration of the process to show the effect of radiation reaction. The orbital plane however precesses on a much faster timescale, so that the orbit does not look like depicted for a static observer. In general, it fills the volume of a torus. For a more realistic illustration, see, e.g., Fig. (4) of [30]. The inclination ι must vary as the EMRI approaches more and more the LSO; it is a constant of motion for geodesics, not to be misinterpreted with the instantaneous orbital colatitude θ of Boyer-Lindquist coordinates. As in the Schwarzschild, the periapsis precesses, so that we need to introduce two additional frequencies. One is related to the radial motion and the time to go from periapsis, rperi , to apoapsis, rapo , and back. The second one is linked to the azimuthal motion around the spin axis and the required time to describe 2π around it, i.e., the

Fig. 13 Tidal separation of a binary of two stars of semi-major axis a. We consider star “1” to be a compact object of mass mBH which is gravitationally bound to the SMBH of mass MBH because the orbit crosses the splitting radius Rsplit . This radius is equivalent to the tidal disruption radius in the tidal disruption problem. After the separation, star 2 is ejected, and star 1 forms a new binary around the SMBH with a new semi-major axis acapt , which typically is much smaller than acrit , as defined in Eq. (17). This leads to much smaller eccentricities in the LISA band

v

1 a

binary 2

CO of mass mBH

ejected companion

Rsplit

MBH

acapt

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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time for the azimuthal angle ϕ to increase 2π radians. We call these two frequencies fr and fϕ , respectively. These three fundamental frequencies and their harmonics are responsible to make so rich in information an EMRI, because they allow the CO, the EMRI, to probe the geometry around the SMBH with gravitational radiation, which encodes this cartography of warped spacetime. The emitted gravitational waves backreact on to the CO itself and lead to a change of its orbital parameters which can be estimated by the energy and angular momentum carried away from the radiation, namely, the semi-lactus rectum – or alternatively the semi-major axis – the eccentricity, and the inclination of the orbit, (p, e, ι). The geodesic motion around a Kerr SMBH has three constants of motion, the energy per unit mass E (which we normalize to mCO ), L z , and the Carter constant per unit mass square. This last is linked to an extra symmetry of the Kerr geometry, in the same way that some axisymmetric Newtonian potentials display [see, e.g., 7, 37]. These three constants are modified due to the emission of gravitational

Fig. 14 The minimum polar angle θ from the spin axis is defined as the inclination ι of the EMRI. We illustrate this with a prograde orbit of a CO revolving around a SMBH of spin a• on a prograde orbit. While θ changes significantly, the inclination does not, as we can see in Fig. 19

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P. Amaro Seoane

radiation. The work of [153] shows that there is a connection between the constants (E, L z , C) and the set of orbital parameters (p, e, ι); more precisely, there is a (bijective, an injective-surjective) mapping between both sets, which is useful to analyze the evolution of the EMRI without having to explicitly solve the evolution of the orbit. This mapping is however complex, and we refer the reader to, e.g., the implementation of [159] for a reference. Again, as a reference, we illustrate this mapping for a simpler case, a Schwarzschild SMBH. It is simpler because ι and C are not required. In this case

E 2 (p − 2 − 2e)(p − 2 + 2e) , = c2 p (p − 3 − e2 ) L2z =

2 p2 G2 MBH . − 3 − e2 )

c2 (p

(51) (52)

Using the symmetries of the geometry of a Kerr SMBH, we can separate the equations for geodesic orbital motion so that the trajectory of a massive body, described in terms of Boyer-Lindquist coordinates {t, r, θ, ϕ}, can be written as follows:   L 1 dt E = Σ 2 − 2a• r• z r dτ Δ c c  2    

E L 2 dr Q ρ4 − a• z − 2 + r 2 Δ ≡ R(r) = r 2 + a•2 dτ c c c  2   L2z dθ E2 C 4 2 2 ρ = 2 − 2 cot θ − a• 1 − 2 cos2 θ dτ c c c   L z Δ − a•2 sin2 θ 1 E 2 dϕ = ρ . 2a• r• r + dτ Δ c c sin2 θ ρ2

(53) (54) (55) (56)

where we have defined the gravitational radius r• ≡ GM• /c2 , Q ≡ C +

2 L z − a• E , ρ 2 ≡ r 2 + a•2 cos2 θ , and Δ ≡ r 2 − 2r• r + a•2 = r 2 f + a•2 , with f ≡ 1−2 r• /r, and Σ 2 ≡ (r 2 +a•2 )2 −a•2 Δ sin2 θ . In this set of equations, Eq. (53) gives us the link between the change of coordinate time t, the time of observers at infinity, and the proper time τ , while Eqs. (54), (55), and (56) give us the link for the spatial trajectory. This set can be combined to derive the spatial trajectory in terms of coordinate time t, (r(t), θ (t), ϕ(t)).

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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Evolution in Phase-Space We can analyze the evolution of the radiation emitted by an EMRI, thanks to the approximation of Keplerian ellipses of [140]. In this approximation, the orbital parameters change slowly because of GWs, which are emitted at every integer multiple of the orbital frequency, ω n = n G MBH /a 3 . At a given distance D, the strain amplitude in the n-th harmonic is G2 MBH mBH D a c4 −1  −1  a D −22  1.6 × 10 g(n, e) 1 Gpc 10−2 pc    MBH mBH . 10 M 4 × 104 M

hn = g(n, e)

(57)

(58)

In this expression, g(n, e) is a function of the harmonic number n and the eccentricity e [see 140]. We consider the RMS amplitude averaged over the two GW polarizations and all directions. While there are more accurate descriptions of the very few last orbits, as, e.g., [29, 71, 81, 141], the scheme of [140] yields a qualitatively correct estimation of the frequency cutoff at the LSO. In Fig. 15, we show the first ten harmonics in the plane of characteristic amplitude as a function of the frequency. We can see that at detector “entrance,” the source is already quite circular, even if the initial eccentricity was set to what we expect from a relaxation EMRI. This is why the second harmonic dominates over the rest of them. Even if initially the periapsis distance was of Rp0 = 26.1 RS , 104 yrs before the plunge (i.e., crossing the event horizon of the SMBH), the EMRI has an eccentricity of e = 0.547 and a periapsis distance of Rp = 21.2 RS . This depicts quite clearly the slow shrinkage and circularization of EMRIs. Ten years before the plunge, we have that e = 1.97 × 10−2 , with Rp = 3.77 RS , and nine years later, e = 1.44 × 10−2 , with Rp = 3.11 RS . This means that the CO, a stellar-mass black hole in this case, is at the verge of falling on to the SMBH but for a small epsilon, and this distance becomes smaller by extremely amounts over a timescale of 9 years. This translates into a very large number of strong bursts from a coherent source at the verge of the abyss, being transported to us in an almost unperturbed way and carrying information about spacetime from a region inaccessible to the photon. Moreover, since the CO is changing the plane of the orbit (for a Kerr SMBH), shrinking the semi-major axis and precessing, we are de facto gathering a cartography of warped spacetime. We now give a few examples for the dynamical parameters of relaxation EMRIs as well as their polarizations. We follow the “numerical kludge scheme,” which conceptually computes the trajectory of the CO using Boyer-Lindquist coordinates [40], to then identify them with flat-space spherical polar ones. This allows us to then derive the waveform from the multipole moments in the context of linearized

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Fig. 15 Cascade of the first 10 harmonics in the approximation of [139]. The Y-axis shows the characteristic, dimensionless amplitude (how much the length of the arms of the detector changes divided by the length) as a function of the frequency in Hz. The CO has been set on an orbit whose dynamical parameters are summarized on the top-left corner, with e0 the initial eccentricity, Rp0 the initial periapsis distance, and Tpl the associated timescale TGW as described in Eq. (15). We have adopted values corresponding to a relaxation EMRI, as in Fig. 12. The upper, red curve corresponds to the second harmonic, which is dominant in this case because, although the CO initially has a large eccentricity, it has circularized. Different points on this curve display information about the system as it evolves with time. The diamond corresponds to 1 year before the plunge on to the SMBH

gravitational perturbation theory in flat spacetime. This waveform is computed with a slow-motion quadrupole formula, a quadrupole/octupole formula, and the weakfield approach for fast motion of [144]. This scheme has evolved over the years and has been significantly improved; see [22, 29, 52, 53]. For the particular calculations we have done, we have used the “EMRI Kludge Suite” of [52, 53]. Figure 16 corresponds to the evolution of the eccentricity for an EMRI with characteristics similar to those of Fig. 15, at detector entrance, when the eccentricity is about e  10−2 . In Fig. 17 we show the same but for a system of a much lower

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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eccentricity, so that the separatrix is located very close to the minimum value of the semi-latus rectum of a Schwarzschild black hole, p = 6 + 2e. In Fig. 18 we depict the evolution of the energy E of the same system as a function of time. We normalize to the initial value of the energy when the system still has 126.23 months (∼10.5 yrs) to go before the plunge. In Fig. 19 we show the evolution of the inclination for the same system. At this point, it is important to note that when we say inclination, we are not referring to the polar angle but to the maximum value reached by the polar angle with respect to the equatorial plane of the orbit if the SMBH was Schwarzschild (or if the direction of the spin was perfectly aligned with the z-component of the angular momentum of the orbital plane, Lz ). Because the minimum value of θ , i.e., the inclination ι (see Fig. 14), changes very slowly [see, e.g., Fig. (2) of 82], it has been assumed to be fixed by the kludge models in the past, as, e.g., in the Teukolsky approach of Fig. (2) of [101]. In that figure, the dashed line is the location of the LSO. We can see that the secondary explores the maximum and the corresponding minimum value during every period. In Fig. 20 we show the evolution of the polarizations only for the few last hours because otherwise the figure is too crowded. The system corresponds to the same one as in the rest of the previous figures.

Fig. 16 Eccentricity evolution as a function of time to plunge in months. The labels correspond (from the top to the bottom, left to right) to the initial values of the semi-latus rectum p/M, the spin of the SMBH a• , the inclination ι, the true anomaly ψ, the source polar angle θS , the azimuthal angle φS , the SMBH spin azimuthal angle φK (all of them in ecliptic coordinates), the azimuthal orientation α as defined in Eq (18) of [29], and the distance to the source D. Note that we do not see an increase in the eccentricity as in Fig. 17 because the separatrix for a• = 0.9 located at a different minimum value of the semi-lactus rectum p

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P. Amaro Seoane

Fig. 17 Evolution of the eccentricity of an EMRI displaying the increase in the last orbits for a SMBH of mass MBH = 4 × 106 M , of spin a• = 0.2, and a CO which initially had e0 = 0.5. The rest of the parameters are identical to those of Fig. 16

Fig. 18 Evolution of the energy of the EMRI normalized to the initial energy at the beginning of the integration. This system corresponds to the one described in Fig. 16

Accumulated Phase Shift We have seen that there are different mechanisms to form EMRIs. In this chapter, we are focusing only on relaxation EMRIs and those formed via tidal separation of binaries. These, but also other scenarios, leave a fingerprint in the dynamical parameters of the EMRIs when it enters the LISA band that allows us to reverseengineer the properties of the host environment, which is interesting from a point of view of astrophysics. The same applies to IMRIs, as we will discuss later.

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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Fig. 19 Same as Fig. 16 but for the evolution of the inclination

Fig. 20 The two waveform polarizations h+ (left panel) and h× (right panel) as a function of time to merger. The system corresponds to that of Fig. 16

In particular, we can look at the impact on how the phase shifts due to any residual eccentricity that the source might have. As an example, we have seen that relaxation EMRIs will typically have higher eccentricities as those produced via the tidal separation of a binary. This eccentricity will induce a difference in the phase evolution of the signal as compared to a circular source. We can estimate the accumulated phase shift to lowest post-Newtonian order and to first order in e2 (via the derivation of [108] of the phase correction) 7065 2 e (π f Mz )−5/3 . ΔΨe (f ) = Ψ last − Ψ i ∼ = −Ψ i = 187136 i

(59)

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P. Amaro Seoane

101

101

acrit Corrected acrit , s=0 Corrected acrit , a• =0.999,θ=0.4 rad

0

10

10−1

10−1

−2

a [pc]

10

a [pc]

acrit Corrected acrit , s=0 Corrected acrit , a• =0.999,θ=-0.4 rad

100

10−3 10−4

TGW [yr]

MBH =4 × 106 M mBH = 10M NBH =5000 γ=1.75 rinfl=2.5 pc

−5

10

10−6

1010 108 106 LSO

10−2 10−3 MBH =4 × 106 M mBH = 10M NBH =5000 γ=1.75 rinfl=2.5 pc

10−4 10−5

TGW [yr] 1010 108 106 LSO

10−6 10−6

10−5

10−4

10−3 1-e

10−2

10−1

100

10−6

10−5

10−4

10−3 1-e

10−2

10−1

100

Fig. 21 Derivation of acrit as in Eq. (17) and in Fig. 12 with the modified TGW , which takes into account both the effects of the first-order post-Newtonian perturbation with a simple fit to account for the Newtonian self-consistent evolution of the eccentricity, and the effect of the spin as well. Note that the correction for the spin is a different one as the location of the LSO via the W (θ, a• ) function presented in Fig. 11, altough we do use this function as well (for details, see the upcoming work of Zwick et al 2020 and Vázquez et al 2020). The left panel corresponds to prograde orbits and the right panel to retrograde ones

Here ei represents the eccentricity at the frequency of the dominant harmonic when it enters the detector, we have introduced the quantity Mz :=

G(1 + z) (MBH mCO )3/5 , c3 (MBH + mCO )1/5

(60)

and f is the frequency for the n = 2 harmonic. We have furthermore assumed that ΔΨe (f ) = Ψ last − Ψ i  −Ψ i ,

(61)

with Ψ last and Ψ i the final and initial phase. As shown in section B.2 of [57], Ψe (f ) has a pronounced falloff with increasing frequency. Hence, to derive the accumulated phase shift in terms of f and the remaining time to merger, we now recall from [102] that the semi-major axis of the binary is a3 =

G (MBH + mCO ) (π f )2

,

(62)

so that we can derive the accumulated phase shift in terms of f and the remaining time to merger, because this last quantity is given by [139]   c5 G(MBH + mCO ) 4/3 5 Tmrg ∼ , = 256 G3 MBH mCO (MBH + mCO ) (π f )2

(63)

and we will elaborate on this in the section about the rates. From the same reference, relation 5.12, assuming 1/(1 − e2 )  1 and Eq. (62), we have that e2 f 19/9 ∼ =

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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constant, which means that a ∝ f −2/3 . Hence, from Eqs. (63) and (62) πf ∼ =



5 256

3/8

−5/8

Mz

−3/8

(64)

Tmrg ,

so that the accumulated phase shift is  ΔΨe (f ) =

5 256

−17/12

7065 25/36 17/12 Tmrg (πfi )19/9 ei2 Mz 187136

25/36 17/12 ∼ Tmrg , = 10 (πfi )19/9 ei2 Mz

(65)

and is detectable if  π .

Event Rate of Relaxation EMRIs In this section, and as mentioned in the introduction, we will focus on relaxation as the mechanism for producing EMRIs. To derive the event rate, we have to solve the following integral: Γ˙ CO 



acrit amin

dn CO (a)

. Trlx (a) ln θlc−2

(66)

where the loss-cone angle can be estimated to be θlc  (Jmax /Jlc )−1 , as can be derived √ from the previous discussion (see [17] for more details). Following [5] θlc2  8 RS /a. The numerator is given by [see 18] CO dn CO (a) = fsub (3 − β)

CO Ninfl MS rinfl



a rinfl

2−β da .

(67)

CO the fraction of the type of CO in consideration in the stellar system (e.g., with fsub for stellar-mass black holes ∼10−3 ) and N infl CO MS the total number of objects (mainsequence stars and COs, or brown-dwarfs) within rinfl . The lower limit in the integral can be estimated by calculating the radius within which we expect to have at least one CO of the type we are considering. Since CO Ninfl ¯ ∗ , with m ¯ ∗ the average stellar mass, we have that amin  MS = MBH /m   β 1.65 × 10−5 pc f CO, sub rinfl /(1 pc) , with f CO, sub the fraction of CO taken into consideration. With these values, we can solve analytically Eq. (66), which yields

Γ˙ CO =

   CO 3 − β Ninfl 1 MS CO λ fsub acrit ln(acrit /8 RS ) − λ 2 λ T0 rinfl λ

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λ −amin

  1 ln(amin /8 RS ) − , λ

(68)

where we have introduced [18]  T0 

4.26 (3 − γ )(1 + γ )3/2

 3 (GM )−1  rinfl MBH 2 BH . ln(Λ) Nbh mbh

(69)

These equations are not easy to interpret in terms of rates, so we will now investigate Eq. (68) for the standard EMRI scenario, i.e., we choose the CO to be a stellar-mass black hole with a mass of 10 M . As for the power indeces, we address two possibilities. One is a classical solution, the so-called Bahcall-Wolf result [BW, 25]. We remark, however, that their results for two-mass components (the stellar-mass black holes and the main-sequence stars, all with the same type of mass, 1 M ) are heuristically derived from their earlier work of [24]. In their paper, they derive γ = 7/4 and β → 3/2. This result of based only on the mass ratio of the two populations and assumes that stellar-mass black holes have a fraction as high as 50% of all stars. A physically realistic solution of the problem must require that stellar-mass black holes have a fraction at most of 10−3 of all stars, as derived from a standard IMF. When using this occupation fraction, [4] and [8, 145] found that diffusion is more efficient and γ = 2 and β = 3/4. We shall call this solution the strong-mass segregation result (SM). For legibility reasons, we introduce the quantities Λˆ := ln(Λ)/13, Nˆ infl := Ninfl /12 × 103 , rˆinfl := rinfl /(1pc), and m ˆ bh := mbh /(10 M ). With these definitions, we derive that for the (mathematically correct but physically unrealistic) Bahcall & Wolf solution −5/2 2 ˆ bh × Γ˙BW, bh ∼ 2.63 × 10−6 yrs−1 Nˆ infl Λˆ rˆinfl m  −4/5 4/5 5 × 10−2 rˆinfl Nˆ infl Λˆ −4/5 m ˆ bh W (θ, a• )−2 ×

  −4/5 4/5 ln 16318 rˆinfl Nˆ infl Λˆ −4/5 m ˆ bh W (θ, a• )−2 − 1 × 

  2 × 10−3 rˆinfl × ln 618 rˆinfl − 1 .

(70)

Setting m ˆ bh = 1 and all of the other quantities with a hat to unity as well, Γ˙BW, bh ∼ −6 −1 10 yr , which is the usual solution [see 7]. As for the strong mass segregation result, which does represent correctly the segregation in a galactic nucleus

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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−2 2 Γ˙SM, bh ∼ 1.92 × 10−6 yrs−1 Nˆ infl Λˆ rˆinfl m ˆ bh ×  1/2 −1/2 1/2 1.6 × 10−1 rˆinfl Nˆ infl Λˆ −1/2 m ˆ bh W (θ, a• )−5/4 ×

  −1 ˆ −1 Λ m ln 9138 rˆinfl Nˆ infl ˆ bh W (θ, a• )−5/2 − 2 − 

  −2 1/2 4 × 10 rˆinfl × ln 618 rˆinfl − 2 ,

(71)

which is then Γ˙SM, bh ∼ 2 × 10−6 yr−1 . Note that for both results, we have adopted a Schwarzschild solution, which means that we have set W (θ, a• ) = 1. The rates for a Kerr SMBH will have a multiplying factor that depends on the inclination of the orbit and spin of the SMBH. A detailed analysis of (i) the relativistic evolution in phase-space and (ii) the rates depending on the mass species and distribution is in preparation and will be submitted elsewhere (Amaro Seoane & Sopuerta in preparation). It is important to note that these results are subject to be revisited, since recently [181] derived an improved timescale TGW which differs from the results of [139] because it includes the effects of the first-order post-Newtonian perturbation and additionally provides a simple fit to account for the Newtonian self-consistent evolution of the eccentricity. These improvements can be captured via relatively trivial modifications to the usual timescale, which must be multiplied by two factors √

R(e0 ) = 81− Qf (p0 ) = exp

1−e0



2.5 RS p0

 (72)

,

where e0 is the initial eccentricity and p0 = a0 (1 − e0 ) the periapsis. The final corrected expression for the GW-induced decay of two orbiting bodies MBH and mCO is TGW

√   24 2 c4 a 5 (1 − e)7/2 1−√1−e0 2.5 RS , = 8 exp 85 p0 G3 mCO MBH

(73)

as derived in [180]. Since we are modifying TGW , this will be propagated into the rates, as we can see from Eq. (13). For high eccentricity orbits and spin, the correction factors are [182] √

R(e0 ) = 8

1−e0



2.8RS Qh Qs → exp + s1 p0



0.3RS p0



 + |s1 |

3/2

1.1RS p0

5/2  (74)

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And accordingly, the corrected timescale is √ 24 2 c4 a 5 (1 − e)7/2 1−√1−e0 8 TGW = 85 G3 mBD MBH     5/2  2.8RS 0.3RS 3/2 1.1RS + |s1 | × exp + s1 , p0 p0 p0

(75)

where s1 is a spin parameter defined as s1 := s cos θ , with s the magnitude of the spin and θ the angle between the SMBH spin vector and the angular momentum vector of the orbit. For a detailed derivation of this result, see [182]. We note that the inclusion of these corrections on the event rate is negligible, as shown by Vázquez et al. 2021 [173]. We see that when we take into account both corrections, namely, the first-order post-Newtonian perturbation with a self-consistent evolution recipe for the evolution of the eccentricity and the spin effects on that timescale (additionally to the location of the LSO, via the W (θ, a• ) function), the results do not differ significantly from the location of acrit in Eq. (17).

Intermediate-Mass Ratio Inspirals Intermediate-Mass Black Holes For an IMRI to exist, we first have to introduce what intermediate-mass black holes are, what motivates this search, and how many of these have been detected. On the one hand, for a long time, we know that stellar-mass black holes must be present, thanks to electromagnetic observations. Ground-based gravitational wave observatories have opened up a new window and have corroborated this. On the other hand, the understanding of galactic nuclei (the innermost cores of galaxies) has advanced rapidly during the past decade, not least due to major advances in high-angular-resolution instrumentation at a variety of wavelengths. As we have previously mentioned, the overwhelming evidence is that supermassive black holes (with masses between a million and ten thousand million Suns) occupy the centers of most galaxies for which such observations can be made. Moreover, an intimate link exists between the central supermassive black hole and its host galaxy [105], as exemplified by the discovery of correlations between the mass of the supermassive black hole and global properties of the surrounding stellar system, e.g., the velocity dispersion σ of the spheroid of the galaxy. Despite much progress in recent decades, many fundamental questions about these relations remain open. These two flavors of black holes have masses that differ by up to nine orders of magnitude. The same way humans grow from babies to teenagers and, later, to adults, black holes must also exist in the intermediate regime. Such “teenager,” intermediate-mass black holes (IMBHs) must have masses typically ranging between ∼102 − 104 M , and in fact we have detected high X-ray luminosities

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not coincident with the nucleus of the host galaxy, which translate into these masses under the assumption that they are black holes. Theoretically, we know that IMBHs form and are located at the center of dense stellar systems such as globular clusters, young clusters, or the cores of dwarf galaxies, and indeed this is what the correlations we mentioned before predict [see 73,75], and the reviews of [122,131]. IMBHs are key to understanding how supermassive black holes gained their titanic masses from stellar-mass black holes, but they are elusive, and we do not have any conclusive evidence of their existence in X-rays or radio, although they should accrete matter. For decades, despite efforts with electromagnetic radiation, we lacked an evidence of their existence. Recently, this has changed, thanks to the observation of GW190521, a gravitational wave signal resulting from the merger of two progenitor black holes of 85 M and 66 M , resulting into an IMBH of mass 142 M [116], which means that a total of 9 M were completely transformed into energy in the form of gravitational waves [118]. These massive black holes must be lurking in star clusters, as we explained before. These clusters are complicated and interesting, not least because they allow us to probe the event horizon. They are, however, key to understanding IMBHs but difficult to simulate due to the many orders of magnitude we have to overcome. If we want to observe IMBHs with light in globular clusters, we need ultraprecise astronomy, because the influence radius of an IMBH of mass 104 M , rh ∼ 5 , assuming a central velocity dispersion of σ = 20 km s−1 and a distance of ∼5 kpc. Within that radius, we only have a few stars. At such a distance and assuming an observational timescale of 10 years, with adaptive optics, we could in the best of the cases have a couple of measurements of velocities. The sensitivity limits correspond to a K-band magnitude of ∼15, like the kind of stars that we observe in our own Galactic Centre, at 8 kpc and of type B- MS. To derive the influence of the mass of the IMBH on the stellar population around it, as it has been done in our galactic centre with SgrA*, one needs instruments such as the VSI or GRAVITY [65, 80]. Until the recent discovery of the LIGO/Virgo team, IMBHs were simply a logical conjecture which has now been corroborated.

Wandering of IMBHs It is relatively simple to estimate in an analytical way how far away an IMBH can wander off the center due to Brownian motion. For simplicity, we assume that the stellar cluster is described by a so-called eta model [61, 172] with enclosed stellar mass  M∗ (r) = Mtot

r/Rcut 1 + r/Rcut

η ,

(76)

where Mtot is the total mass in stars and Rcut the cutoff radius. For r Rcut , we have the usual power-law distribution ρ ∝ r −γ with γ = 3 − η. The massive black hole will wander within a given radius Rw , where it can capture COs by perturbing stellar orbits (i.e., dynamically) for light enough IMBHs. We simplify the

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problem by assuming a constant stellar density, so that the gravitational potential in which the IMBHis located can be described as a harmonic oscillator, with angular 3 . Given an equilibrium point, the maximum speed frequency ω = GM∗ (Rw )/Rw Vosc achieved by the oscillator for an amplitude (the maximum displacement from 2 = ω2 R 2 . This value, commonly referred the equilibrium position) Rosc is Vosc osc to as the root mean square amplitude of the oscillations in velocity, is linked to its equivalent in space 2 2 2 = ω2 Rosc ≈ ωRw = Vosc

GM∗ (Rw ) . Rw

(77)

Assuming equipartition of kinetic energy between the IMBH and the stellar component, we have that [see 62, for the specific case η = 1.5] 2 MBH Vosc  m∗ σ (Rcut )2 ,

(78)

where σ is the stellar velocity dispersion at r = Rcut . We note that σ 2  0.1G Mtot /Rcut . Since we can assume that typically Rw Rcut and, hence, M∗ (Rw )  Mtot (Rw /Rcut )η and combining Eqs. (77) and (78) and the expression for σ 2 , we obtain  Rw ∝ Rcut

m∗ MBH

1/(2−γ ) .

(79)

In Fig. 22 we show the wandering radius of an IMBH with two different masses for different values of the power-law index γ . As we can see, for shallow values, the IMBH can reach significant values. As we have seen in Eq. (12 and in Fig. (6), the system should be relaxed, but this implicitly assumes that the mass-σ relationship holds at the low end of masses, which has not yet been confirmed observationally. This means that we expect a cusp to build around the IMBH, so that for a singlemass population, γ = 1.75, and in the case of a mass spectrum, it will be typically of the order of γ  1.75 for the heavier components and γ  1.5 for the lighter ones. This quick estimate agrees well with the much more detailed work of [119]. The authors identify three main mechanisms for the wandering of the IMBH: (1) Brownian motion, (2) the effect of the segregated stellar cusp, and (3) three-body interactions between stars and the IMBH. The authors however find that Brownian motion is the most important one. Later, [87] ran scattering, numerical experiments of single stars onto a binary formed by the IMBH and another star. Their results are that only IMBHs with masses above 300 M have small wandering radii. Below this limit, the IMBHs wander off the center of the cluster. Hence, if Brownian motion was the only phenomenon to take into account, we could assume that the IMBH is fixed at the center, since for any realistic value of γ , the displacement is negligible, and so the treatment of loss-cone could be applied, but in view of the results of [87], this can only be done for rather massive IMBHs.

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Fig. 22 Wandering radius Rw of an IMBH due to Brownian motion in a dense stellar cluster as a function of the power-law exponent γ . We use a so-called “eta” model with cutoff radius Rcut , and the mass of the IMBH is shown as MBH

Numerical Simulations of IMRIs Because of the challenges related to the analytical approach in the case of an IMBH, in particular the wandering of the massive black hole, we need to resort to numerical simulations to at least win some intuition about the physics at role. For this aim, and as discussed in detail in the relativistic context in [7], the most accurate integration one can do is the so-called direct-summation N−body algorithms. These integrators have been used for decades in astronomy and have been put to test a number of times with observations and analytical techniques. It all boils down at integrating Newton’s equations of motion between all stars in the stellar system at every timestep, with a regularization algorithm for binaries. Since we are integrating directly, all phenomena purely related to dynamics (i.e., a star is a point) emerge naturally [see, e.g., 1–3, and the latter for the concept of regularization]. Nowadays, many of these Newtonian integrators have adapted a post-Newtonian correction, first implemented in [111]. We note, however, that the post-Newtonian expansion assumes the bodies to be completely isolated and without additional perturbations. This has been neglected in all astrophysical integrations with relativistic corrections of this type, but it has been shown that it can induce important errors. Indeed, it has been shown that the so-called cross-terms must be taken into account [179]. These are terms that represent a coupling between the SMBH and the stellar potential. In the case of having a hierarchical triple system, the author shows that the effects of such terms can actually be enhanced to amplitudes of Newtonian order. But, in general, and even for a binary in vacuum, the post-Newtonian approach is not yet fully understood, and as a matter of fact, it has received criticism. In particular, the work of [63] pointed that the derivations either contained inconsistencies or are incomplete. These points have been addressed almost completely, already in the 1990s. There remains however one important open question in my opinion,

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about the nature of the sequence itself, because we do not know yet whether it converges, diverges, or is asymptotic. However, on the calculation side, we can avoid divergences by carefully constructing the hierarchy of the equations [see, e.g., 21]. From a pragmatic point of view, the lack of a formal proof of convergence, divergence, or asymptoticity of the post-Newtonian series might be envisaged as irrelevant because for some reason, it is a very efficient tool to address the evolution of isolated binaries. However, in my opinion, it is crucial to concile the physical effectiveness of this series with its mathematical framework. Why does it work at all? I refer the reader to the elegant work “On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics” presented in [178]. The summary could be the following sentence from the article: “The reasons for this effectiveness are largely unknown.” We note that an interesting alternative is implementing a geodesic solver to the Newtonian algorithm in the regime of interest, i.e., when the CO is detached from the rest of the stellar system, at distances close to periapsis from the SMBH (or IMBH, in this case). The details are given in Sec. (8.8.2) of [7]. In the context of stellar dynamics and IMRIs, the first numerical work that addressed the formation and evolution of an IMRI in a cluster is the work of [104]. They observe the formation of an IMRI with masses MBH = 500 M and mCO = 26 M . After some 50 Myrs, the IMRI merges, and the IMBH receives a relativistic recoil [26,43,84]. The result is that, due to the low escape velocity of the cluster, the IMBH leaves the host stellar system. They noticed in their simulations that the IMBH forms a binary with a (relatively massive) stellar-mass black hole for about 90% of the time of the simulation. The interesting fact is that the stellar-mass black hole that forms the IMRI is the result of an abrupt exchange of companion with the IMBH. In view of the initial semi-major axis and eccentricity, a ∼ 10−5 pc and e = 0.999, this capture seems to follow the “gravitational brake” capture of the parabolic orbits: A sudden loss of energy via gravitational radiation can lead to the spontaneous formation of a relativistic binary. This was first presented in the work of [146], while the energy and angular momentum changes in the case of a hyperbolic orbit presented in [90]. This scenario has been recently explored numerically by [98, 103, 113, 128, 135]. The work of [104] has been confirmed by [114], who finds very similar results but using a different numerical scheme. Also, the work of [91] is basically a reproduction of [104] but with a different integrator, which leads the authors to similar results as well. Later, the detailed analysis of [124] explores lighter IMBHs and also finds that the IMBH spends about 90% of the integration time on a binary with a CO, also with a distribution of semi-major peaking at values of 10−5 pc.

Event Rate of IMRIs As we have discussed previously, we rely on numerical simulations to address the formation and evolution of IMRIs. Therefore, the derivation of the event rate must

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Fig. 23 Sketch of the set of ∼ 3 × 104 numerical simulations of [19]. The three boxes display the different stages in the simulation. The first one corresponds to the initial conditions, which are based on previous works, in particular on [104], which has been confirmed by independent groups. An IMBH (the larger black circle) forms a binary with a stellar-mass black hole (smaller black circle) and a perturber, which is another stellar-mass black hole, but in principle could be another type of star with the proviso that the mass is large enough to have been segregated to small distances from the IMBH (black star). This three-body problem is embedded in an external stellar potential representing the background stellar component, and the evolution can be approximately divided in three different possibilities, as we see in the “dynamical interaction” box. The last box gives us the result of the interaction depending on the details of the initial configuration

accordingly rely on a large sample of numerical simulations that allows us to at least have an educated guess about the free parameters that riddle the problem. In Fig. 23 we show a cartoon of the set of 3 × 104 simulations of [19], which sweep the parameter space based on the initial conditions motivated by the previous work of [91, 104, 114, 124]. We elaborate more on this in the next section. Thanks to the outcome of the numerical study, we can make a guess on the event rate that we expect for different detectors. That is, depending on the horizon distance zhor of the detector we consider for a given type of mass, the rates will be different. The work of [19] derives the rates by estimating the total number of IMRIs within a given cosmological volume of radius zhor . We reproduce here their calculation as a way to exhibit clearly the number of free factors that are at play in the calculation. The number of IMRIs, ΓIMRI , can be obtained via the following integral:  ΓIMRI = Ωs

M2

M1

 0

zhor

d nIMRI dVc d z dMIMBH , dMIMBH d z d z 1 + z

(80)

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where dVc /d z is the comoving cosmological volume element, (1 + z)−1 takes into account the time dilation, and d nIMRI /dMIMBH expresses how many IMRIs we have per unit of mass. This, as pointed by the authors, can be rewritten as d n d nGC dMGC d nIMRI = ξBH fGW pIMBH nrep . dMIMBH dMg d z dMGC dMIMBH

(81)

In this last expression, d n/(dMg d z) is the number of galaxies per z, ξBH is (a priori) a free parameter that gives us the the probability for an IMBH to be in a binary with a stellar-mass black hole, d n/dMGC represents the number of clusters per cluster mass in a certain host galaxy, dMGC /dMIMBH links the mass of the GC to that of the IMBHs, nrep is another unknown factor that expresses how often an IMBH can for a binary with a CO, and fGW is the fraction of successful IMRIs (i.e., those which merger within a Hubble time, an information drawn from their simulations). Finally, pIMBH is the probability that an IMBH is indeed lurking in the considered GC. This is very likely the most unknown parameter. The authors rely on the set of Monte Carlo simulations of [79] and decide to set it to pIMBH = 0.2, i.e., 20% of all GC are supposed to be hosting an IMBH. They assume a power law for d n/dMGC of the following fashion: dn −s = kMGC , dMGC

(82)

with a slope s = 2.2, for consistency reasons with the initial mass function of young and old star clusters in galaxies, as derived in the work of, [e.g., 76], with the normalization factor k defined as δMg (2 − s)

. k= 2−s 2−s MGC2 − MGC1 We need to assume some reference values for this normalization. The authors adopt Galactic values, so that Mg = 6 × 1010 M and MGC1,2 = (5 × 103 − 8 × −s −bs 106 ) M , and derive MIMBH  (30 − 4.6 × 104 ) M , as well as MGC = aMIMBH , which again contain two parameters. As for d n/(dMg d z), they resort to the results of [56]. Hence, the number of IMRIs contained in a given observable volume is  NIMRI = ka

1−s

M2

bpIMBH nrep ξBH M1

with

 0

zhor

(1−s) b−1

fGW MIMBH

dMIMBH

φ(z) dVc d z. 1 + z dz (83)

18 The Gravitational Capture of Compact Objects by Massive Black Holes

φ(z) = −

φ∗ 10(α∗ +1)(M2 −M∗ ) , α∗ + 1

821

(84)

a parametric expression of galaxies number density, where φ∗ , α∗ , M∗ a function of z [see Table 1 in 56], and M2 = 12. The estimation limits zhor to either z = 2 (which corresponds to the peak of GC formation) or z = 6 (formation of the first stars). The probability that the IMBH forms a tight binary with a CO is given by ξBH , and it is assumed that ξBH → 1, based on initial mass function arguments but, more importantly, on the results of the different numerical simulations performed by different groups. The last piece of information required is the timescale for the IMRIs to form, which can be expressed as the accumulative sum of the timescales for cluster formation, IMBH formation, IMRIs formation, and coalescence. Since we are, to put it mildly, at debate about the process that leads to IMBH formation, [19] adopt a weighted timescale for it of 2 Gyr based on the numerical experiments of [20, 79]. They take the mass-segregation timescale for the IMRI formation, which is sensible, of ∼0.1 − 1 Gyr, and they derive from their simulations that tGW = 0.6 − 1.5 Gyr. As [19] point out, an alternative way of deriving the merger rate is via the cosmological GC star formation rate ρSFR (z). This can be used to obtain the total number of GCs as a function of z 

zmax

N(zmax ) = 0

ρSFR (z) dVc d z . < MGC > d z 1 + z

(85)

If one adopts the power-law GCs mass function, the normalization in this case is k=

(1 − s) 1−s MGC,1

1−s − MGC,2

,

(86)

so that the total number of IMRIs within a given volume is  NIMRI =ka 1−s bpIMBH nrep

M2

M1

 0

zhor

(1−s) b−1 MIMBH fGW ρSFR (z)

dVc d z dMIMBH . dz 1 + z (87)

Therefore, so as to obtain the merger rate, we convert Eqs. (83) and (87) at a given z with the relation ΓIMRI = NIMRI /T . Therefore, we can convert the rates to specific detectors, and we have that • • • •

LIGO/Virgo should detect up to 1 − 5 IMRIs per year with MIMBH  100 M LISA is in the position of observing up to ∼6 − 60 events per year ET could detect 100 − 800 IMRIs per year with masses MIMRI < 2000 M Decihertz detectors could detect up to 3800 events per year

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Fig. 24 Evolution of the eccentricity for the three different stellar-mass black holes which form a binary with the IMBH (data from [104]), as a function of time. The masses of the three different COs are shown in the left panel, which is the complete evolution of the three systems. On the right panel, we have a zoom in of the last binary, which lasts 1.25 × 104 yr from the formation to the merger. We can see the extreme eccentricity that it achieves and how quickly it circularizes as an effect of the radiation of energy

In view of the detection of [116] of the system GW190521 by LIGO/Virgo with a mass 142 M these predictions seem to be robust, but we are talking about singlepoint statistics. We need to wait to have more detections to derive any conclusion.

Multi-bandwidth IMRIs Intermediate-mass ratio inspirals are particularly interesting because, as discussed in [6], they cannot only be detected by future space-borne gravitational wave observatories. Depending on their masses and, in general, dynamical parameters, they could already be present in ground-based detectors, such as LIGO/Virgo. Moreover, also depending on their dynamical characteristics, a percentage of them will be detectable by both space-borne- and ground-based facilities. As we have already mentioned in the previous section, we rely on numerical simulations to address this kind of source. We stress out that the findings of the first numerical simulation by [104] have been confirmed by at least another three different groups. This is important, because IMRIs seem to form at very high eccentricities and very small semi-major axes. To illustrate this, in Figs. 24 and 25, we show the eccentricities and semi-major axes of one of the simulations of [104], which led to the formation of an IMRI. This particular example is a representative of what has been found by other groups. The IMBH forms a binary with a compact object, a stellar-mass black hole of mass ∼20 M for most of the time of the simulation. The binary is ionized and the CO replaced with another one, which is also of the same type and mass, while the semi-major axes become smaller and smaller. Eventually, after an abrupt interaction, a final binary forms with an initial

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Fig. 25 Same as Fig. 24 but for the semi-major axes. On the right panel, we only show the evolution of the two last COs in the order of the dynamical interactions with the IMBH

Fig. 26 Evolution of the first harmonics of IMRIs with typical values for the semi-major axis and eccentricity. As in Fig. 15, we show different moments in the evolution, as well as the periapsis distances, distance to the source, and the merger time. Additionally, we also display the Einstein Telescope [95, 151] sensitivity curve on the right, at higher frequencies, and above it, the LIGO sensitivity curve

eccentricity of e ∼ 0.999 and the IMRI forms and merges in a fraction of the total simulation time (∼2.5% of the total). In these figures, we can see how quickly both the semi-major axis and eccentricity decay. Based on these results, and in view of the similarity with other groups, we can take these initial values as representative and ask how these systems would be observed from the space and from the ground, as in the work of [6]. In Fig. 26, we follow the approach of Eq. (58). We can see that the dominant harmonic n = 2 crosses a range of frequencies in a timescale of a few years. This means that if LISA and ground-based observatories are operational at the same time, a LISA detection

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would translate into a forewarn to decihertz observatories (if any) and the groundbased LIGO/Virgo (or ET). This is interesting because the combination of these detections would allow us to impose enhanced constraints on the system’s parameters. Since LISA can observe the inspiral, we can derive parameters such as the chirp mass easily. Ground-based detectors are in the position of observing the merger and ringdown, and hence one can derive additional parameters such as the final mass and spin. This compound detection has also the advantage of splitting various degeneracies and therefore to achieve better measurements of the parameters, as compared to individual detections. In the work of [6], it is shown that IMBHs with light and medium size can be observed by space-borne and ground-based observatories with eccentricity values ranging between e ∈ [0.99, 0.9995]. The most eccentric ones can only be seen from the ground [see discussion in 50]. This is an implication of the characteristic frequency changing as the pericenter distance decreases because eccentricity increases. The peak frequency is therefore shifted [see Eq. 37 of 177]. Heavier IMBHs have their frequency peak receding in frequency as compared to medium-sized ones, and therefore, the harmonics are embedded in the LISA range. We can estimate the signal-to-noise ratio (SNR or ρ) as follows. Assuming that the hc of an IMRI is, at a given frequency f [68] hc =



˙ f˙)/(π D), (2E/

(88)

where E˙ is the power emitted, and f˙ the time derivative of the frequency, the skyand orientation-averaged ρ of a monochromatic source (assuming the the ansatz of ideal signal processing) is [68] (ρ)2 =

4 π D2



E˙ f˙ ShSA (f )

df f2

(89)

where ShSA (f ) ≈ 5Sh (f ) is the sky- and orientation-average noise spectral density. For a source with multiple frequency components, the total value of ρ 2 is obtained by summing the former equation for the different components. For high-eccentricity sources, we use the expression [see 6, for an explanation]  (ρ)2n =

fn (tf in )  h fn (tini )

c, n (fn )

hdet (fn )

2

1 d (ln(fn )), fn

(90)

where fn (t) is the (redshifted) frequency of the n harmonic at time t (fn = n × forb , with forb the orbital frequency), hc, n (fn ) the characteristic amplitude of the n harmonic when the frequency associated to that component is fn , and hdet is the square root of the sensitivity curve of the detectors.

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With this in mind, we can estimate typical SNRs for different detectors and sources. Typically, the values are from tens of LISA and LIGO to thousands in ET, at D = 500 Mpc [6]. We therefore have that IMRIs are (a) sources that might already be present in current ground-based observatories and (b) multiband probes. The fact that no IMRI has been yet detected (and we are excluding here the “mysterious source” [117], because it has q  0.1) is very likely due to the reasons given in the next section. However, there is another particular characteristic of IMRIs that makes them interesting, at least from the point of view of modelling. This is the fact that they are “clean sources.” What this means is that, contrary to EMRIs, which might have their inspiral affected by gas effects (or even dark matter, if one invokes such a possibility), IMRIs can and must be considered as a relativistic two-body problem in vacuum from the moment in which it is formed, and the driving mechanism is the emission of GWs (i.e., once it has detached from the stellar dynamics regime; see Fig. 12 and related discussion). This is so because (a) globular clusters have a negligible amount of gas and (b) the timescale for merger, once the IMRI has formed, is very small as compared to other typical timescales associated to perturbations. Indeed, [6] shows this by analyzing the evolution of a completely isolated IMRI and another one embedded in a dense stellar system with typical densities of ∼105 − 106 M pc−3 . When comparing the evolution of the isolated case with the multibody one, there is no difference. Finally, we note that it is possible to use IMRIs to extract properties of the host environment by following the method of accumulated phase shift. Indeed, [6] finds that in the LISA and ET band some systems can have ΔΨf (e)  104 radians (with a threshold on SNR = 5). In the LIGO/Virgo band, the number ΔΨf (e)  10 radians for some systems, typically the light ones. This means that we can extract information from the host cluster from IMRI detections.

Modelling IMRIs Detecting IMRI systems in GW-data will require us to push data analysis techniques to beyond the state of the art. In particular, we need to include higher signal harmonics and to combine data from multiple detectors coherently. All the gravitationalwave events found so far are from binaries in which the two components have comparable masses, in any case very different from the asymmetric systems we are considering here. This asymmetry has two implications. First, the gravitational wave signal is much more complicated, mostly due to the higher modes and harmonics becoming more important. Second, the higher asymmetry implies that the signals have a lower amplitude (compared to a more symmetric system which is equally far away). Thus, the analysis must be more sensitive by including more physical effects and by combining the data from multiple detectors more optimally.

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All of the searches thus far have only looked for the dominant harmonic of the signal with additional physical restrictions (especially assuming no-precession), and moreover, they have combined the data from different detectors without considering phase coherence. It is thus not a surprise that the highly asymmetric systems have not yet been found. This would require dropping these simplifying assumptions. We would need to include higher signal harmonics and precession and also combine data from the different detectors coherently. However, even assuming that the signal detection problem have been understood and addressed, we will still not be able to find IMRIs. This requires critical advances in general relativity and numerical analysis. To explain this, we need to briefly discuss the existing methods and their range of validity. Comparable mass ratios, as exemplified by the current LIGO/Virgo detections, are modeled by post-Newtonian methods [see 38, for a review] for a review, while the extreme-mass ratio systems relevant for LISA require the self-force calculations in general relativity [see 30, 142, for a review]. The intermediate-mass ratio systems fall in the middle and represent an unexplored part of parameter space. The most reliable calculations of the gravitational signal are by solving the Einstein equations numerically. For comparable mass-ratio systems, it is now routine to solve the equations with astrophysical initial conditions and to calculate the full gravitational wave signal. Current methods do not work when the mass ratio q = mCO /MBH is small; it is computationally far too expensive to simulate even a fraction of an orbit for q ∼ 10−3 . The most extensive publicly available catalogues of gravitational waveforms [117, 134] currently only go until q = 0.1, and q = 0.01 represents the smallest ever mass ratio simulated [121, unfortunately covering too few orbits]. The most extreme-mass ratio found by the LIGO/Virgo collaboration is GW190814, the “mystery object” [117], and it has q  0.1. The presence of the much smaller black hole introduces a small length scale in the problem which, in the standard approach, needs to be resolved. This affects then the time steps in the integration of the Einstein equations and the computational cost increases correspondingly. Going to q ∼ 10−3 or 10−4 is well beyond the capabilities of current methods. At the other end, very small values of q  10−5 can be studied using the self-force calculations, analogous to the self-force due to electromagnetic radiation in classical electrodynamics. Here, q is a small parameter linearly perturbing a background solution [30, 142]. While several difficulties remain, it is reasonable to expect that current methods can address the problem. These methods cannot however be extended for q ∼ 10−3 as this would necessitate going well beyond the linear perturbation approximation. It is worth pointing out here that there are attempts to interpolate between the two regimes, in particular by the “effective-one-body” approach [see, e.g., 39, 42]. In this approach, as the name suggests, the idea is to replace the true 2-body system by an effective description in terms of a 1-body system with a modified Hamiltonian. The results of this approach have proven to be effective in the comparable mass

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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and extreme-mass ratio regimes; however, it is not clear that this will work in the intermediate-mass ratio regime.

Extremely Large Mass Ratio Inspirals We have seen in Fig. 5 the range of masses for a stellar object to plunge directly through the event horizon of a SMBH without suffering significant tidal stresses, as a function of the mass of the object. We note that there is a range of masses which correspond to sub-stellar objects, in particular to brown dwarfs (BDs) which will inspiral and cross the event horizon of SgA* in our Galaxy without being disrupted. The typical mass of a BD translates into a mass ratio, for SgrA* of q ∼ 10−8 . This is particularly interesting because if one of these BDs was to form a relativistic binary with the SMBH in our Galaxy, because of the following points: 1. The number of cycles that it would revolve until plunging into the event horizon is roughly inversely proportional to q [see, e.g., 125]. This means that a BD would describe of the order of ∼108 cycles in the band of the detector (in this case LISA). Because of this value, these sources are called “extremely large mass ratio inspirals” (XMRIs). 2. Due to the previous point, XMRIs would have a much longer life in the band of the detector as compared to an EMRI. 3. Because of the distance to our Galactic Centre, of ∼8 kpc, the signal-to-noise ratio (SNR) of an XMRI would be extremely high, achieving that of binaries of SMBHs or even exceeding it. 4. Since back-reaction is proportional to the mass ratio, the modelling of XMRIs would be much easier than that of EMRIs and, in any case, trivial as compared to IMRIs. The orbit approaches more and more a pure geodesic and is, hence, trivial to calculate. These points were realized by [18]. In this work, the author addresses the possibility that one of these sources exists in our Galactic Centre, and estimates the associated SNR. The mass of a BD ranges approximately between 0.012 M and 0.076 M . We note that this is a lower limit, since they can also have masses in the range 0.07 − −0.15 M through the BD formation process (see [109]), so that the results here are to be regarded as conservative, since we assume masses lower than 0.07 M . By following a similar procedure as the one presented starting from Eq. (88), and in the same approach, [18] calculates the SNR for conservative values of XMRI systems and finds that LISA would be able to detect them with SNRs ranging from 10 millions of years before the final plunge to values as high as SNR  104 thousands of years before it, assuming only 1 year of observation. We can see this in Fig. 27.

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Fig. 27 SNR of an XMRI with typical parameters in our Galactic Centre. About 107 years before the BD plunges through the event horizon, and assuming that we can only follow the XMRI for 1 year, it can be observed with a SNR of 10. Different lines correspond to different harmonics, while the black, upper line is the sum of them. One million years before the plunge, it achieves SNR  60. From a thousand years before the plunge, it is SNR  104

In Fig. 28, we show the evolution in phase-space of a typical XMRI. We can see that the XMRI can spend millions of years in the band of the detector. This is the reason (and the proximity in distance) of the large SNR of Fig. 27. The event rate for an XMRI to plunge into the SMBH of our Galaxy is not drastically higher than what can be expected for an EMRI, given in Eqs. (70) and (71). The procedure to derive the event rate is very similar to that of stellar-mass black holes, i.e., EMRIs, with a crucial subtlety related to the dynamics of BDs, which we summarize here for the convenience of the reader and refer to details to the work of [18]. BDs have their own initial mass function, which is not well-constrained, but it can be approximated by a single power law [Eq. 4.55 of 109], and this is consistent with the observational data of [176]. Moreover, we can assume that relaxation is dominated, driven, by stellar-mass black holes. This implicitly assumes that relaxation can be added up individually from these two mass species, the BDs and

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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Fig. 28 Evolution in phase-space of an XMRI following the same approach and convention as in Fig. 12. However, instead of showing the different harmonics as in that figure, we display the isochrones. This means that a curve depicts the position of the different contribution of the different harmonics at the same time. At the time of formation, the XMRI time to plunge is of Tpl ∼ 4 × 109 years

the stellar-mass black holes. Indeed, close to the central region, energy equipartition is found only among the largest masses, and this equipartition is spread then to velocity at low masses first [36]. This means that for a given distribution f (m, v) with v the velocity and m the mass, the moment of the change of velocities, the mean value, is  2

< dv >=

dv 2 f (m, v) dm dv.

Because we can neglect equipartition among the low-mass object, we can then express this as

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2

< dv >=



 n(m)

 2

dv f (v)dv ,

m

with n(m) the density of stars of mass m. This is crucial, because it implies that (1) BDs must be close to the center and (2) we can ignore the contribution of relaxation provided by them. The event rate, as derived in [18], and following the same nomenclature starting from Eq. (68), is Γ˙X−MRI =

     3 − β N0BD 1 1 MS BD λ λ ln(Λ ln(Λ − a , f × a ) − ) − crit min crit min 2 λ T0 Rhλ sub λ λ

where we have introduced λ = 9/2 − β − γ , Λcrit = acrit /(8 RS ) and Λmin = amin /(8 RS ). As with the EMRI rates, and following the same notation, we give now two examples, the BW- and SM-solution −11/4 Γ˙BW ∼ 1.8 × 10−4 yrs−1 Nˆ 0 Λˆ rˆinfl ×  5/4 1.34 × 10−4 rˆinfl Nˆ 0−1 Λˆ −1 m ˆ BD W (ι, a• )×



4 −4/5 ˆ −4/5 4/5 −2 ˆ ln 262 rˆinfl N0 Λ − − m ˆ BD W (ι, a• ) 5  

4 −25/4 5/4 6.86 × 10 . rˆinfl × ln 15.22 rˆinfl − 5

(91)

where we have introduced m ˜ BD := mBD /(0.05 M ). In the case of SM, we have that −5/2 ˆ Γ˙SM ∼ 2.3 × 10−3 yrs−1 rˆinfl Nˆ 0 Λ×  ˆ BD W (ι, a• )−5/2 × 1.4 × 10−4 rˆinfl Nˆ 0−1 Λˆ −1 m

  ln 46 rˆinfl Nˆ 0−1 Λˆ −1 m ˆ BD W (ι, a• )−5/2 − 1 −  

 −7 4.67 × 10 rˆinfl × ln 15.24 rˆinfl − 1 .

(92)

We reproduce here the results of [18], in Fig. 29. We can see that the event rate is not much higher as for the EMRI case (see Eq. (68) and the examples). The important and interesting point about XMRIs is that, contrary to more massive EMRIs, XMRIs spend a long time on band, as we have seen. They start

18 The Gravitational Capture of Compact Objects by Massive Black Holes

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Fig. 29 Event rate for XMRIs at the galactic centre in the BW- and SM-solution

with a SNR > 10 in LISA T ∼ 106 yr−1 before the final plunge, and the event rate Γ˙ ∼ = 10−5 yr−1 . Therefore, the number of sources in band at any given moment (with the proviso that the nucleus is relaxed) is the multiplication of the event rate times their lifetime, i.e., ∼ a few 10. From the continuity equation of the events, we can derive the relative occupation fractions of the (line) density g = dN/da, with N the number of sources and a their semi-major axes. Taking into account the eccentricity of the sources when integrating N , [18] finds •  15 X-MRIs at low frequencies with high eccentricities and associated SNRs of  a few 100. •  5, at higher frequencies, i.e., at very high SNRs (from a few 100 up to 2×104 ), in circular or almost circular orbits. It is important to stress that this derivation has been done assuming a steady-state distribution; it is a statistical representation of the system. This means that, at any given moment of the Milky Way – under the assumption that our Galactic Centre is in the range of relaxed nuclei (see Fig. 6) – the numbers of sources that we quote above hold. If this requirement is fulfilled, this means that millions of years ago and millions of years in the future, those are the numbers of XMRIs that we can expect to be populating the Galactic Centre. We stress that these are conservative values. These numbers can be multiplied by a factor of a few depending on the eccentricity of the sources when they form. Also, this approach is purely analytical and based on pure power laws. These should decrease as one approaches the innermost radii, which translates into a more

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efficient diffusion of stellar-mass black holes. Hence, we are artificially enhancing the relaxation time and therefore decreasing the event rate. While the SNRs are very high, to the point of being able to bury the SNR of binaries of SMBHs in some cases, the distance plays a crucial role. This means that while we can expect to detect them in our own Galactic Centre, other galaxies are out of question but for, maybe, satellite galaxies of M82 with SMBHs in the range of LISA. Because of their extreme-mass ratios, these systems are closer to a geodesic than EMRIs, which makes it easier to model them. However, because XMRIs evolve so slowly, in the lifetime of LISA, they can be envisaged as monochromatic sources. This, in turn, poses the question of how many of them and in what distribution of orbital parameters would we need to reproduce the exploration of timespace that an EMRI can do. An EMRI probes a vast range of values in the semi-major axis, eccentricity, and change of plane of the orbit in as short as a few months or years. Each burst can be considered like a picture of warped spacetime, and hence it delivers us with thousands of pictures to allow us to do a cartography of that spacetime around the SMBH. An XMRI, on the contrary, is “fixed” in phase-space, and the question arises of how many of them, and in what distribution, would we need to derive that mapping of spacetime. There is an additional problem related to XMRIs. Since we can expect the Milky Way to be a proxy galaxy of the type that LISA will probe, it is very likely that an important fraction of all of them will have a few XMRIs distributed around their SMBHs. As we have noted before, this is a steady-state representation of relaxed nuclei. In other words, we can expect that all relaxed nuclei in the range of masses that LISA will observe will have a few tens of XMRIs distributed around the SMBH moving at a fraction of the speed of light. Any infalling EMRI will therefore inevitably encounter them on its orbit toward the final plunge into the SMBH. Because the XMRIs are moving at such high velocities, their effective masses are similar to those of the EMRI, which means that an interaction EMRI-XMRI can significantly alter the orbit of the EMRI or, even worse, scatter it completely off, cancelling the merger of the system.

A Relativistic Fokker-Planck Algorithm As we have seen in the previous sections, the problem of forming an EMRI (or IMRI/XMRI) involves both astrophysics (in particular stellar dynamics) and general relativity. This is challenging because this translates into addressing the two-body problem in general relativity, which is unsolved at these mass ratios, as we have briefly described before, and the 106 –108 body problem in Newtonian gravity. The host galactic nucleus (in the case of an EMRI or XMRI) or globular cluster (for an IMRI) has that order of stars. While we care about relativistic effects between COs and the central SMBH/IMBH, relativity does not play a significant role in general among the stars that form part of the stellar system in which the inspiral takes place.

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833

In this section, which gives the primary idea of [11], we describe what ought to be done to address this problem with a two-dimensional Fokker-Planck integrator, which has shown to be very robust even compared to direct N−body simulations [see, e.g., 8, 145], with the advantage that (i) one has control over the physics as it is implemented into the integrator scheme and (ii) the Fokker-Planck approach is orders of magnitude faster than any brute force, direct-summation integrator. Before we describe the main idea, we need to introduce a few physical concepts and the connection between the Newtonian and relativistic regimes.

Newtonian Motion Around a Newtonian Potential We start from the Hamiltonian of Eq. (106). In this case, the equation for radial motion can be written as 

dr dt

2 = −2E +

J2 2GMBH 2E − 2 = 2 (ra − r)(r − rp ) . r r r

(93)

From here, and using the definitions of the previous section of the pericenter and apocenter radial coordinates, we find E= J2 =

GMBH G(1 − e2 ) , = rp + ra 2p

(94)

2GMBH rp ra 2 = GMBH p. rp + ra

(95)

Relations Between the Relativistic and Newtonian Parameters for a Schwarzschild SMBH In order to connect the Newtonian and relativistic calculations into a single FokkerPlanck integrator, it is important first to have a clear way of connecting the Newtonian and relativistic parameters describing the orbital motion. Here, we want to relate the Newtonian constants of motion (E, J ) of Eqs. (106) and (107) with the relativistic constants of motion (E , J ) of Eq. (23). Let us start first with the angular momentum. For that, we need to compare the Newtonian definition of the angular momentum in terms of the spherical coordinates of motion J = r 2 ϕ˙ ,

(96)

with the second equation in (23). We realize that, by identifying the radial and azimuthal coordinates (there is a justification for this), we can just identify the angular momentum parameters, i.e.,

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J ≡J.

(97)

The case of the energy is more complex, in part because of the intrinsic four-dimensional character of general relativity. Moreover, the properties of the Einsteinian and Newtonian gravitational fields are different and complicate the comparison since, at finite distance, the energy involves the gravitational potential. Hence, the ideal way of comparing things is by going to spatial infinity, which means analyzing the energy at r → ∞. At infinity, the relativistic specific energy satisfies Γ∞ = ut∞ = E ,

(98)

1 Γ∞ =  , 2 1 − v∞

(99)

where we have used Eqs. (23) and (33) in the case of the Schwarzschild metric. Here 2 is the spatial velocity at infinity, given by v∞ 2 ≡ v∞

i j g∞ ij v∞ v∞

 =

dr dτ



 ∞



+ r2

dϕ dτ

2  .

(100)

∞

Then, combining Eqs. (98) and (99), we obtain the well-known relation in special relativity: 2 v∞ =

2 −1 Γ∞ E2 −1 = . 2 Γ∞ E2

(101)

In the Newtonian case, from Eq. (106), we have 2 = −2 E . v∞

(102)

Hence, by comparing we can relate both energies in the following way: E=

1−E2 , 2E2

E =√

1 1 + 2E

.

(103)

Since we are analyzing the equations at infinity, this is only valid for unbound orbits, which satisfy E < 0. For bound orbits, we have to make a different analysis. A possibility is to identify the (e, p) parameters as they have the same meaning both in Newtonian and Einsteinian gravity, as they determine the turning points of the motion, i.e., rp and ra . However, this produces J = J , which in principle is not a problem by itself. By doing this identification, one can get the following relations: J 2 − J2 4+e , = 2 p J

(104)

18 The Gravitational Capture of Compact Objects by Massive Black Holes

E − 1−E E2 1−E 2

E2

2

=

4(1 − e2 ) , p(p − 4)

835

(105)

where we are comparing the Newtonian energy E with (1 − E 2 )/E 2 [as in Eq. (103)]. We have normalized with respect to the relativistic values as they are supposed to be closer to reality. Notice that for orbits with big p, the identification can be reasonable, as both expressions go to zero as p goes to infinity.

Relations Between the Relativistic and Newtonian Parameters for a Kerr SMBH In this case, the situation is a bit more complicated, as we have an additional constant of motion. In a spherically symmetric gravitational potential, the constants of motion are just two, (E, J ), but in the case of a spinning SMBH (Kerr geometry), there are three: (E, Jz , C), where Jz is the angular momentum component in the spin direction and C, the Carter constant, related to the total angular momentum. In the Newtonian limit J2z +C ≈ J2 . For some triaxial Newtonian potentials, there is an analogous to the Carter constant. An important point is that in the spherically symmetric case, (E, Jz ) is equivalent to (e, rp ), where rp denotes the pericenter radial coordinate. With spin, (E, Jz , C) is equivalent to (e, rp , ι), where ι denotes the inclination of the orbit with respect to the equatorial plane (the plane orthogonal to the spin). This means that the inclination plays a different role in the spherical symmetric case with respect to the axisymmetric one.

A Possible Scheme The first step is to obtain the steady-state distribution, defined by the density profile ρ(r) and the phase-space distribution function (DF) f (E), of stars around the SMBH. This is given by the solution of the Fokker-Planck equation (or by N-body simulations). We have solutions for clusters which are spherically symmetric and which have an isotropic distribution of velocities. Here, we assume that the cluster is spherical and the SMBH spin does not change this appreciably. It is possible that the distribution of inclinations for stars with lowest J will deviate from sphericity due to close interactions with the hole, but not the cluster as a whole. We now define J = (Jx , Jy , Jz ). We note that we use specific energy E and angular momentum J. A E > 0 corresponds to bound orbits, so for an elliptical orbit in the field of the SMBH, the Hamiltonian of the system becomes 1 GMBH H = − v2 + = E > 0,. 2 r

(106)

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We note that in physics literature, usually it is used −H , so that E < 0 corresponds to bound Newtonian orbits and E > 0 for unbound ones. The magnitude of total angular momentum is GMBH . J = √ 2E

(107)

Since the stars originate from a spherical cluster, all components of J are allowed to diffuse independently under two-body relaxation, and, as a result, orbital inclinations diffuse with J. We define Jz  a• , where a• is the spin axis of the SMBH. Stars interact relativistically with the Kerr SMBH, and this leads to deterministic changes in (E, Jz , C); we need (E, J) to evolve them in the cluster. This needs to establish a correspondence between Newtonian and relativistic constants of motion (either for spinning or non-spinning black holes). With this at hand, we can then try to distinguish between changes in different components of the angular momentum, ΔJ. Then, we want to determine the region of phase-space (E, J) from which EMRIs originate. For such calculation, we need to include not only the effect of scattering on the EMRI candidate due to the other stars but also the deterministic loss (ΔEGW , ΔJGW ) that results from the emission of GWs at close passages to the SMBH. We need to then estimate the changes in the constants of motion due to GW ). This means that we do not gravitational wave emission, (ΔE GW , ΔJGW z , ΔC GW for the Fokker-Planck simulations. Nevertheless, in order to work with need ΔC the orbital parameters (p, e, ι), we have to calculate ΔC GW to estimate the changes in the orbital parameters. The idea is to study the phase-space evolution of many individual EMRI candidates in a Monte Carlo fashion. One starts by sampling the stellar population according to the DF given by FP calculation. So, each EMRI candidate has an initial set of phase-space coordinates such as (E0 , Jo ). In principle, we should be interested in orbits which are bound to the SMBH, but one is also interested in the case when they are unbound to the hole, but bound to the cluster. The FP solution ρ(r) and f (E) completely determines the diffusion coefficients for scattering with the cluster stars – assuming that two-body relaxation is the only dynamical mechanism. The EMRI candidate will undergo a random walk-in phasespace plus a determination loss of energy and angular momentum due to radiation reaction. The angular momentum jump δJP (E, J ), per orbital period P (E), is given by

1/2 − ΔJGW (E, J ) = δJP (E, J ) = ΔJ t P (E) + ξJ (ΔJ )2 t P (E) =

Jc2 (E) P (E) + ξJ 2 J Trlx (E)



Jc2 (E) P (E) Trlx (E)

1/2 − ΔJGW (E, J ), (108)

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where ξJ = ±1 and the quantities inside  .  are the first and second diffusion coefficients per unit time. The second equality is obtained from simple scaling arguments. The orbital energy jump, per orbital period, is given by

1/2 δEP (E, J ) = ΔEt P (E) + ξE (ΔE)2 t P (E) + ΔEGW (E, J ),

(109)

In this equation, DEE and DE are the diffusion coefficients, introduced when defining the time-dependent, orbit-averaged, isotropic, Fokker-Planck equation in energy space. For each component, this is [51, 160] p(E)

∂FE,i ∂fi ∂fi =− , FE,i = −DEE,i − DE fi , ∂t ∂E ∂E

(110)

with  DEE,i = 4π 2 G2 m2∗ μ2i ln Λ  +

+∞

μj μi

2  Nc  

 q(E)

j

E −∞

dE  fj (E  )

dE  q(E  )fj (E  ) ,

E

DE,i = −4π 2 G2 m2∗ μ2i ln Λ

 Nc   μj j

μi

+∞

dE  p(E  )fj (E  ).

(111)

E

The indices i, j run from 1 to the number of mass components considered, Nc , and we define μi = mi /m∗ , with m∗ = 1/N. One can obtain gravitational potential Φ from Poisson’s equation ∇ 2 Φ(r) = 4π Gρ(r), which needs to be updated as ρ(r) evolves over time. This can be achieved following the technique of operator splitting by [55] and [51]. One successively updates the distribution function f (E) through the diffusion equation and the gravitation potential Φ(r) via the Poisson equation. In each diffusion step, Φ(r) is kept constant, while f (E) and the diffusion coefficients are updated. Correspondingly, for every Poisson step, Φ(r) is updated, and f (E) is kept constant as a function of the adiabatic invariant. One then needs to take into account the deterministic losses (ΔEGW , ΔJGW ) due to GW emission that needs to be added to the right-hand sides of the equations. These could be obtained from the numerical calculations given some (E, J) and some interval of time Δt. In the simplest case, Δt = P (E), i.e., one orbital period; but, in general, we need to have P (E) Δt Trlx (E). This can be achieved by reading the values from the Fokker-Planck calculations to derive the new values (E, J), after computing (ΔEGW , ΔJGW )Δt over some N orbital periods. Let us consider Nporb as the number of radial periods, i.e., the time to go from rp to ra and back to rp . This has to be distinguished from the other two periods present in generic orbits around a spinning SMBH.

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Given Nporb (or alternatively Δt), the relativistic Fokker-Planck integrator comGW ) and/or the new (p, e, ι). Again, besides the radial putes (ΔE GW , ΔJGW z , ΔC period, Tr , we have the polar and azimuthal periods. The first one, the polar Tθ , is related to the fact that when we have a spinning black hole, the motion, in general, is not confined in a plane, as mentioned previously. Nevertheless, one can define an instantaneous plane of the orbit, which oscillates in a symmetric way around the equatorial plane. In that case, Tθ is the period of such oscillations. The third period, the azimuthal one Tϕ , corresponds to the time that the azimuthal angle ϕ takes to run 2π . For generic orbits, the three periods are different. For a non-spinning black hole, we just have Tr and Tϕ which are different because of the relativistic precision. In Newtonian dynamics, these two are identical. The algorithm could then be summarized with the following steps: 1. Obtain the initial steady-state distribution of stars around the SMBH, described by (ρ(r), f (E)), via a Newtonian, two-dimensional Fokker-Planck integrator. This allows us to have (E, J) for each star in the distribution. That is, {(Ei , Ji )}i=1,...,N∗ , where N∗ is the number of stars. Assume Jz  a• . The orbital inclinations are already encoded in the angular momentum vector. 2. Establish a correspondence between (E, J)N and (E, Jz , C)E (where the subscripts N and E stand for Newtonian and Einsteinian, respectively). We note that Nporb is not the same for all stars. A solution to this is to make Δt equal to a fixed fraction of the (local) relaxation time. For a cusp with f (E) ∝ E p , Trlx ∝ E −p [137], where we expect p to range between ∼0 and ∼1/2. The Newtonian orbital periods are P (E) ∝ E −3/2 , so that the steps will change significantly between different particles and also during the inspiral. 3. Given the relativistic parameters, and given Nporb , we need to compute the changes due to gravitational wave emission on them, (ΔE GW , ΔJzGW , ΔC GW ). The relativistic Fokker-Planck scheme then delivers the new (E, Jz , C)E . 4. Using the new (ΔE GW , ΔJzGW , ΔC GW ) map back to obtain the new (E, J)N . In this way, we compute {(ΔEiGW , ΔJGW )}i=1,...,N∗ i 5. Use the results of Point (4) in the equations for (δEP (E, J ), δJP (E, J )). 6. Iterate from point (2). Here we should pass the information of when one of the EMRIs has plunged. This is decided by the relativistic integrator. Obviously, we can derive the corresponding orbital parameters, so as to have a statistical distribution.

Cross-References  Binary Neutron Stars  Black-hole Superradiance: Searching for Ultralight Bosons with Gravitational

Waves  Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

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 Formation Channels of Single and Binary Stellar-Mass Black Holes  Introduction to Gravitational Wave Astronomy  LISA and the Galactic Population of Compact Binaries  Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravita-

tional Waves  Principles of Gravitational-Wave Data Analysis  Research and Development for Third-Generation Gravitational Wave Detectors

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Enrico Barausse and Andrea Lapi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dark Matter Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation Time, Fast Collapse vs. Slow Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Median and Average Halo Mass Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial Mass Profile and Pseudo-evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halo Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halo Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halo Merger Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halo Merger Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryons and Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black-Hole Mass Function at High Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Delays Between Galaxy and Black-Hole Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predictions for LISA and PTAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E. Barausse () SISSA – Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy IFPU – Institute for Fundamental Physics of the Universe, Trieste, Italy INFN-Sezione di Trieste, Trieste, Italy e-mail: [email protected] A. Lapi SISSA – Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy IFPU – Institute for Fundamental Physics of the Universe, Trieste, Italy INFN-Sezione di Trieste, Trieste, Italy INAF-Osservatorio Astronomico di Trieste, Trieste, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_18

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E. Barausse and A. Lapi

Abstract

At low redshift, massive black holes are found in the centers of almost all large elliptical galaxies, and also in many lower-mass systems. Their evolution is believed to be inextricably entangled with that of their host galaxies. On the one hand, the galactic environment provides gas for the black holes to grow via accretion and shine as active galactic nuclei. On the other hand, massive black holes are expected to backreact on the galactic dynamics, by injecting energy in their surroundings via jets or radiative feedback. Moreover, if galaxies and darkmatter halos form hierarchically, from small systems at high redshift coalescing into larger ones at more recent epochs, massive black holes may also merge, potentially generating gravitational-wave signals detectable by present and future experiments. In this chapter, we discuss the predictions of current astrophysical models for the mergers of massive black holes in the mHz frequency band of the Laser Interferometer Space Antenna (LISA) and in the nHz frequency band of pulsar-timing array experiments. We focus in particular on the astrophysical uncertainties affecting these predictions, including the poorly known dynamical evolution of massive black-hole pairs at separations of hundreds of parsecs, the possible formation of “stalled” binaries at parsec separations (“final-parsec problem”), and the effect of baryonic physics (e.g., SN feedback) on the growth of massive black holes. We show that nHz-band predictions are much more robust than in the mHz band and comment on the implications of this fact for LISA and pulsar-timing arrays. Keywords

Black holes · Gravitational waves · Dark matter halos · Galaxies: formation and evolution · LISA · Pulsar-timing arrays

Introduction In the local universe, massive black holes (MBHs) with masses of 105 –109 M are ubiquitous in the center of large elliptical galaxies [55, 73] and are also present in some low-mass dwarf galaxies [5, 110, 111]. MBHs are also believed to accrete from the nuclear gas and thus power Active Galactic Nuclei (AGNs) and quasars [53], which are in turn expected to exert a feedback (via jets or radiation) on their surroundings and even on their galactic host as a whole [24, 33, 65]. As a result, the evolution of MBHs and galaxies is expected to proceed “synergically,” as suggested also by the observed galaxy-MBH scaling relations [72,91,114,115,117,118], which link, e.g., the MBH mass to the stellar velocity dispersion of the host’s spheroid (Mbh − σ relation) and the MBH mass to the spheroid’s stellar mass (Mbh − M ∗ relation). (See however [8, 61, 123, 141] for recent work on these scaling relations, which shows that their interpretation may be subtler and at least partially related to observational bias.)

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Despite their crucial importance for galaxy formation and evolution, the mechanisms that drive the growth, feedback, and dynamics of MBHs are surprisingly little understood. From a theoretical prospect, this is to be ascribed to the difficulty of resolving the scale of MBHs and their sphere of influence, which is tiny compared to galactic scales. Moreover, processes such as star formation and supernova (SN) explosions, but also MBH accretion, feedback, and generally the interaction of MBHs with the surrounding gas, are not yet fully understood from first principles and have therefore to be modeled via “sub-grid” prescriptions in hydrodynamic simulations (cf., e.g., [40, 46, 100, 105, 112, 116, 131, 140, 143]). From an observational point of view, a crucial limitation consists in the difficulty of observing MBHs at moderate to high redshift, where their AGN/quasar activity is expected to be the brightest and their evolution the most rapid. Compared to electromagnetic probes, gravitational waves (GWs) offer several advantages in this respect (cf.  Chap. 1 “Introduction to Gravitational Wave Astronomy”). GWs interact very weakly with matter, unlike electromagnetic radiation. Moreover, GW detectors are sensitive not to the energy flux (which decays as the inverse square of the luminosity distance dL ) but to the field (which decays as dL−1 ). For these reasons, GWs are in principle observable up to very high redshift. GWs, unlike electromagnetic radiation, are also not produced microscopically, i.e., they are not the collection of a huge number of quanta produced by atoms/plasmas. Instead, they are produced macroscropically by the bulk motion of large masses moving at relativistic speeds. As such, they encode directly information on the dynamics and macroscopic properties of the system that generates them. For GWs resulting from the merger of two black holes, one can, e.g., extract the masses and spins of the two objects from the GW signal alone (cf.  Chap. 41 “Principles of Gravitational-Wave Data Analysis”). Dark-matter (DM) halos and galaxies are believed to form hierarchically in the Λ Cold Dark Matter (ΛCDM) model, from small systems at high redshift, which evolve to larger ones at lower redshift through a combination of major and minor mergers, as well as accretion of DM/hot gas from the intergalactic medium (IGM). It was therefore recognized early on [11] that galaxy mergers are a promising environment for the production of GWs, because they could lead to the coalescence of the MBHs present at the centers of the two merging galaxies. Because of the large MBH masses, these black-hole mergers are not observable with current interferometers such as LIGO and Virgo (cf.  Chap. 1 “Introduction to Gravitational Wave Astronomy”). Indeed, the merger frequency of a blackhole binary system of total mass M scales roughly as ∼1/(GM/c3 ), which lies in the band [10−4 Hz, 0.1 Hz] of the Laser Interferometer Space Antenna (LISA; cf.  Chap. 3 “Space-Based Gravitational Wave Observatories”) for masses 104 – 107 M , and in the band [10−9 Hz, 10−7 Hz] of pulsar-timing arrays (PTAs; cf.  Chap. 4 “Pulsar Timing Array Experiments”) for masses of 108 –109 M . In this chapter, we will review the astrophysics of the co-eval evolution of MBHs and their galactic hosts. We will start from the large scales (i.e.,  100 kpc) of DM halos and the diffuse, chemically pristine intergalactic medium, whose properties and merger history we will derive in detail within the framework of the ΛCDM

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model. We will then focus on the intermediate scales (tens of kpcs down to pc) relevant for the baryonic physics of galaxies (e.g., stellar and gaseous disks and spheroids), down to the sub-pc scales of nuclear objects (nuclear star clusters and MBHs). We will then proceed to discuss the formation and evolution of MBHs (isolated and in binaries), starting from the formation of MBH pairs, which may then give rise to bound binaries and eventually merge. We will discuss the uncertainties of this “pairing” phase of the MBH evolution and present its implications for LISA and PTAs. Throughout this chapter, we adopt the standard flat ΛCDM cosmology [104] with rounded parameter values: fractional matter density at present time ΩM ≈ 0.3, dark-energy density ΩΛ ≈ 0.7, baryon density Ωb ≈ 0.05, Hubble constant H0 = 100 h km s−1 Mpc−1 with h ≈ 0.7, and mass variance σ8 ≈ 0.8 on a scale of 8 h−1 Mpc.

Dark Matter Halos In the standard cosmological framework, the seeds of cosmic structures like quasars, galaxies, and galaxy systems are constituted by DM perturbations of the cosmic density field, originated by quantum effects during the early inflationary universe. The perturbations are amplified by gravitational instabilities and, as the local gravity prevails over the cosmic expansion, are enforced to collapse and virialize into bound “halos.” In turn, these tend to grow hierarchically in mass and sequentially in time, with small clumps forming first and then stochastically merging together into larger and more massive objects. The halos provide the gravitational potential wells where baryonic matter can settle in virial equilibrium, and via several complex astrophysical processes (cooling, star formation, feedback, etc.) originate the luminous structures that populate the visible universe (see textbooks such as [29, 92]). In the following, we recap, with a modern and original viewpoint, the crucial issues concerning the formation and evolution of DM halos, which will provide the backbone for the description of MBH mergers in the rest of this chapter.

Basic Quantities Provided a background cosmology and suitable initial conditions in terms of a power spectrum P (k) of density fluctuations (e.g., [10]), the statistical evolution of the halo populations in mass and redshift can be characterized at leading order via three basic quantities: • The mass variance σ (M), which describes the statistics of the density perturbation field when filtered on different mass scales M; this is defined as σ 2 (M) =

1 (2π )3



2 (k) , d3 k P (k) W˜ M

(1)

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2 (k) is the Fourier transform of a window function whose volume in where W˜ M real space encloses the mass M. For a scale-invariant power spectrum P (k) ∝ k n with effective spectral index n > −3 (to ensure hierarchical clustering), σ (M) ∝ M −(n+3)/6 applies. • The linear threshold for collapse δc (z), which is a measure of the typical amplitude required for a perturbation to collapse efficiently; in fact, the condition σ (M) ∼ δc (z) yields the characteristic mass Mc (z) that, on statistical grounds, is prone to collapse at redshift z. A useful approximated expression for the collapse threshold in a flat ΛCDM cosmology is given by δc (z) = δc0 D(0)/D(z), in terms of the normalization [49]

δc0 

  3 (12π )2/3 1 + 0.0123 log10 ΩM (z) ≈ 1.68 , 20

(2)

and of the growth factor [103] 5 ΩM (z) D(z)  2 1+z



−1 1 209 1 4/7 2 + ΩM (z) − Ω (z) + ΩM (z) 70 140 140 M

(3)

where ΩM (z) = ΩM (1 + z)3 /[ΩΛ + ΩM (1 + z)3 ]; this latter redshift dependence is close to D(z) ∝ (1 + z)−1 at z  1 and then slows down toward the present when the dark energy component kicks in. • The virial overdensity Δvir , which approximately renders the typical density contrast of perturbations at virialization (hence in the nonlinear regime). In a flat ΛCDM cosmology, it can be computed via the approximated formula [26] Δvir ≈ 18 π 2 + 82 [ΩM (z) − 1] − 39 [ΩM (z) − 1]2 ,

(4)

with typical values Δvir ≈ 100 at the present time and increasing toward Δvir ≈ 180 for z  1. Related to this, the virial mass Mvir of a DM halo is defined as the mass contained within a radius Rvir inside which the mean interior density is Δvir times the critical density ρcrit ≡ 3 H02 /8π G ≈ 2.8 × 1011 h2 M Mpc−3 .

Formation Time, Fast Collapse vs. Slow Accretion The formation redshift zf of a halo with mass M at redshift z is routinely defined as the highest redshift at which the mass of its main progenitor is larger than f M, i.e., a fraction f of the current mass. [58] have provided a fitting formula of the formation redshift distribution extracted by N−body simulations, valid for any value of f . It writes 2

dpf αf νf eνf /2 (zf |M, z) = 2 dzf (eνf /2 + αf − 1)2

   dνf     dz  f

(5)

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where αf ≈ 0.815 e−2 f /f 0.707 and 3

δc (zf ) − δc (z) νf ≡  . σ 2 (f M) − σ 2 (M)

(6)

The corresponding cumulative distribution is αf

pf (> zf |M, z) = e

νf2 /2

+ αf − 1

.

(7)

Despite f ≈ 1/2 is often exploited to define the formation redshift of a halo, this choice is rather arbitrary. In fact, intensive N-body simulations and analytic studies (see [77, 148, 149]) have demonstrated that the growth of a halo actually comprises two different regimes: an early fast collapse during which the central gravitational potential well is built up by dynamical relaxation processes and a late slow accretion when mass is slowly added in the outskirts of the halo in the way of an inside out growth. The transition between the fast and slow accretion regime is a more motivated definition of halo formation; this is found to occur at a cosmic time tzFS ≈ 3.75 tz0.04 , proportional to that at which the halo assembled a fraction f ≈ 4% of its current mass. Thus, the generic formation time distribution dp/dzf in Eq. (5) can be exploited to derive the distributions dp/dz0.04 and dp/dzFS . Note that the latter can in principle extend even to redshifts zFS < z, meaning that a halo at redshift z has not yet entered the slow accretion regime. The corresponding distributions for different halo virial masses Mvir ≈ 109 − 1015 M at z ≈ 0 are illustrated in Fig. 1.

Median and Average Halo Mass Growth The median history of the main progenitor for a halo with mass M at redshift z can be derived very easily from the formation redshift distribution Eq. (5). By definition, if dpf /dzf is the formation redshift distribution, and dpMP /dM is the distribution of main progenitor masses, one has the trivial identity 

∞

zf

dpf dz (z |M, z) = dz



M

fM

dM

dpMP (M , zf |M, z) . dM

(8)

From this it is easily recognized that the cumulative distributions of formation redshift and main progenitor masses are equal; thus, both the median main progenitor mass at given redshift and the median formation redshift at given main progenitor mass both satisfy Eq. (6), i.e., [σ 2 (f M) − σ 2 (M)] ν˜ f2 = [δc (z ) − δc (z)]2

(9)

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Fig. 1 Formation time distribution for different halo virial masses Mvir ≈ 1015 M (red), ≈ 1013 M (orchid), ≈ 1011 M (blue), and ≈ 109 M (cyan) at current redshift z ≈ 0. Dotted lines refer to the redshift z0.04 at which the halo has accumulated 4% of its current mass, dashed lines to the redshift z0.5 at which the halo has accumulated half of its current mass, and solid lines to the redshift zFS of the transition between fast collapse and slow accretion

where ν˜ f =



2 ln(1 + αf )

(10)

is the median value that has been computed explicitly from Eq. (7). The above equation can be solved for the median f˜(z ), and hence the main progenitor median mass is f˜(z )M. The average mass growth of the main progenitor f M can be related to the (cumulative) formation time distribution. By definition one has f (z ) M =



M

dM M

0

dpMP (M , z |M, z); dM

(11)

M Now, one can rewrite M = 0 dM , reverse the double (triangular) integral as

M



M M M dM , and use Eq. (8) to obtain 0 dM 0 dM = 0 dM M f (z ) M = M

 0

1

df pf (> z |M, z).

(12)

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Fig. 2 Mass growth history of the main progenitor for halos with different virial masses Mvir ≈ 1015 M (red), ≈ 1013 M (orchid), ≈ 1011 M (blue), and ≈ 109 M (cyan) at current redshift z ≈ 0. Thick solid lines refer to the median growth history, and shaded areas illustrate 5th and 95th percentile around it. Thin solid lines refer to the median history depurated from pseudo-evolution, dashed lines to the average growth history, and dotted lines show the direct simulation fits by [13]. The symbols illustrate on every median growth history the location of the redshift z0.04 (circles) where the main progenitor mass is 4% of the current one, z0.5 (squares) where it is 50% of the current one, and zFS (stars) where transition from fast to slow accretion regimes takes place

Notice that to compute this quantity the formation time distribution for any f is necessary. The corresponding median and average halo growth histories for different halo virial masses Mvir ≈ 109 −1015 M at z ≈ 0 are illustrated in Fig. 2 and found to be in overall good agreement with the tracks directly extracted from simulations [13].

Radial Mass Profile and Pseudo-evolution N-body simulations indicate that the mass profile of a DM halo follows the universal Navarro-Frenk-White (NFW) shape [99]  M(< r) = MΔ g(cΔ ) ln (1 + cΔ s) −

cΔ s 1 + cΔ s

 (13)

where s ≡ r/RΔ is the radius normalized to that RΔ ≡ (MΔ /4π Δρcrit )1/3 within which the average density is Δ times the critical one ρcrit ≈ 2.8 × 1011 ΩM h2 M Mpc−3 ; moreover, cΔ is the concentration parameter and g(x) ≡ [ln(1 + x) − x/

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(1 + x)]−1 . Very often Δ is taken to be the virial threshold for collapse Δvir defined in Eq. (4), so that the related concentration will be indicated by cvir . Simulations indicate (see [44,96,148,149]) that during the fast collapse, the virial concentration cvir stays put to a value around 4, while it progressively increases as mass is added in the outer parts of the halo during the slow accretion phase. A simple fitting formula is given by [149] 1/8  cvir (z |Mvir , z) ≈ 4 1 + (tz /tzFS )8.4

(14)

in terms of the fast/slow transition time tFS (Mvir , z); note that the latter depends on the halo virial mass Mvir at redshift z, and that actually is not a unique value but follows the distribution dp/dzFS defined above. The median concentration corresponding to the halo mass growth histories discussed above for different halo virial masses Mvir ≈ 109 − 1015 M at z ≈ 0 is shown in Fig. 3. It has been pointed out by several authors (e.g., [43, 44, 96] that during the slow accretion phase, the halo growth is mainly driven by a pseudo-evolution in radius and mass due to the lowering of the reference density defining the halo boundary. For example, pseudo-evolution implies that a halo of 1013 M with turning point at redshift zFS ≈ 2 will end up, say, at z ≈ 0 in a halo of several 1014 M ;

Fig. 3 Median virial concentration of halos with different halo virial masses Mvir ≈ 1015 M (red), ≈ 1013 M (orchid), ≈ 1011 M (blue), and ≈ 109 M (cyan) at current redshift z ≈ 0; colored shaded area illustrate 5th and 95th percentile around the median. Solid lines refer to the prescription by [149] and dotted lines to that by [58]. The inset shows the overdensity Δconc of the inner region formed during the fast collapse phase

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clearly, the latter is a halo typical of a galaxy group/cluster, but has little relevance to the physical processes occurring within the galaxy hosted by the original halo of 1013 M at formation. In other words, the “galactic” halo must have evolved in mass much less than predicted by pseudo-evolution. Under the assumption that the universal NFW mass distribution is retained, the mass growth depurated from pseudo-evolution can be estimated as [44, 96] Mnops (z)  Mvir (z) g[cvir (z)]/g[cvir (zFS )]

(15)

This also implies that the overdensity of the central region not subject to pseudoevolution is given by  Δnops (z)  Δvir

cvir (z) cvir (zFS )

3

g[cvir (z)] g[cvir (zFS )]

(16)

We have highlighted in Figs. 2 and 3 the impact of the pseudo-evolution on the mass growth and on the concentration/central region overdensity.

Halo Spin DM halos slowly rotate with a specific (i.e., per unit mass) angular momentum given by jH ≈ 1670

λ 0.035

MH 1012 M

2/3

−1/6

Ez

km s−1 kpc ,

(17)

where Ez ≡ ΩΛ + ΩM (1 + z)3 and λ is the halo spin parameter. Numerical simulations (e.g., [27, 84, 150]) have found that λ has a roughly log-normal distribution with mean value λ ≈ 0.035 and dispersion σlog λ ≈ 0.25 dex, approximately independent of mass and redshift. The small values of λ testify that rotation is largely subdominant with respect to random motions, which indeed are mainly responsible for sustaining gravity and enforcing virial equilibrium. The distribution of DM halo mass MH (< j ) as a function of the specific angular momentum j has been studied in detail via N-body simulations by [27, 125, 133], who found the convenient one-parameter representation MH (< j ) = MH γ [α, αj/jH ]

(18)

∞

x where γ [a, x] ≡ 0 dt t a−1 e−t / 0 dt t a−1 e−t is the normalized incomplete gamma function, and the value α ≈ 0.9 applies. All in all, a representation in spherical mass shells j ∝ M(< r)s with s ∼ 1.1 − 1.3 holds to a very good approximation over an extended range.

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Halo Mass Function The halo mass function is the statistics describing the number of DM halos per unit comoving volume as a function of halo mass and redshift. This is routinely estimated via high-resolution, large-volume N-body simulations (see [15, 66, 126, 130, 144]), although given the natural limits on resolution, computational time, and storing capacity, it can be probed only in limited mass and redshift ranges. We also caveat that the results of simulations depend somewhat on the algorithm used to identify collapsed halos (e.g., FoF=friend-of-friend vs. SO=spherical overdensity), and on specific parameters related to the identification of isolated objects (e.g., the linking length). One of the most accurate determination has been obtained by [144]. Their fit to the simulation outcomes reads   dN ρM (0)  d log σ  Γ (σ, z|Δ) f (σ, z|Δ) (19) = dM M 2  d log M  where 

a  b 2 f (σ, z|Δ) = A 1 + e−c/σ . σ

(20)

For FoF-identified halos, the function Γ (σ, z|Δ) = 1 and the relevant parameters read A = 0.282, α = 2.163, β = 1.406, and γ = 1.210; thus, the function f (σ ) is actually independent of the redshift, and the mass function is said to be universal. For halos identified with a spherical overdensity algorithm, one has

Γ (σ, z|Δ) = C(Δ)

Δ 178

d(z) ep (1−Δ/178)/σ

q

(21)

and redshift-dependent parameters A = ΩM (z) [1.907 (1 + z)−3.216 + 0.074], a = ΩM (z) [3.136 (1 + z)−3.058 + 2.349], b = ΩM (z) [5.907 (1 + z)−3.599 + 2.344], c = 1.318, C(Δ) = 0.947 e0.023 (Δ/178−1) , d(z) = −0.456 ΩM (z) − 0.139, p = 0.072, and q = 2.130. The halo mass function for both FoF and SO algorithms is illustrated in Fig. 4, together with the contribution from halo in the fast and slow accretion regime. The shape of the mass function and other halo statistics (like progenitor mass function and large-scale bias) can be theoretically understood from first principles, on the basis of the celebrated Press & Schechter theory by [107] and its extended version by [19, 76, 93]. Such a framework, modernly dubbed excursion set theory, remaps the issue of counting numbers of halos into finding the first crossing distribution of a random walk that hits a suitable barrier. The random walk is executed by the overdensity field around a given spatial location when considered as a function of the mass variance σ 2 (M) of Eq. (1). The barrier is provided by the linear collapse threshold δc (t) in Eqs. (2–3), with possibly an additional dependence

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Fig. 4 Halo mass function N (log M, z) ≡ dN/d log M per logarithmic mass bins (dex units) at different redshifts z ≈ 0 (red), 1 (orchid), 3 (green), 6 (blue), and 10 (turquoise). Solid line is the total mass function by [144] for a nonlinear threshold Δvir , dot-dashed line the contribution from halos in the fast collapse stage, and dashed line from those in the slow accretion regime. Dotted lines illustrate the mass function for FoF-identified halos

on the mass scale (“moving barrier”). Remarkably, the resulting halo statistics are found to be in overall good agreement with the N-body outcomes, especially when a moving barrier with shape inspired by the ellipsoidal collapse is adopted [126, 146, 147]. Recently, an alternative framework to describe the halo statistics based on stochastic differential equations in real space has been proposed by [78].

Halo Merger Trees Halo merger trees are numerical yet approximate realizations of a halo merging history (see [31, 68, 102, 127]); these constitute the skeleton of many semi-analytic models aimed to describe the properties of galaxies and MBHs (Fig. 5). The basic ingredient to build up the tree is the halo progenitor mass function dN/dM , i.e., the distribution of halo masses M at different redshifts z that will end up in a given descendant halo mass M at a later time z. This quantity is usually represented as 

2 2 dN M Δδc e−(Δδc ) /2Δσ Δσ 2 δc2 (M , z |M, z)  √ G 1+ 2 , 2 dM M σ σ 2π (Δσ 2 )3/2

 2  dσ     dM  M

(22)

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Fig. 5 Example of a merger tree for a descendant halo with final mass 1013 M at z ≈ 0 (top), down to an initial redshift z ∼ 10 (bottom); bluer colors correspond to more massive progenitors

in terms of Δδc = δc (z ) − δc (z) and Δσ 2 = σ 2 (M ) − σ 2 (M); actually this expression is inspired by the excursion set theory (see above), but includes a correction term to reproduce the results of N-body simulations [67, 102], which reads

Δσ 2 δ 2 G 1 + 2 , c2 σ σ



γ γ Δσ 2 1 δc2 2  G0 1 + 2 σ σ2

(23)

with G0 ≈ 0.57, γ1 ≈ 0.19, and γ2 ≈ −0.005. There are various possible algorithms to exploit the above and build up the tree, but one of the most effective in reproducing N -body outcomes is the [31] binary method with accretion, which is briefly recalled next. First, provided a mass resolution Mres , one computes the smooth mass accretion integrating the progenitor mass function 

Mres

Macc 

dM M

0

dN (M , z |M, z) . dM

(24)

Second, one computes the mean number of progenitors in the range [Mres , M/2] as  P

M/2

Mres

dM

dN (M , z |M, z) , dM

(25)

and chooses the merger time step so that P  1 to ensure that multiple fragmentation is unlikely. A uniform random number R generated in the interval [0, 1] determines whether the descendant has one (R > P) or two progenitors (R ≤ P). In the former case, one prescribes M = M − Macc ; in the latter case,

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one of the masses M1 is drawn from the progenitor mass function in the range [Mres , M/2], and the second mass is just M2 = M − M1 − Mres .

Halo Merger Rates After formation, a DM halo is expected to grow because of mergers and smooth accretion from the surrounding field and the cosmic web. The merger rates per descendant halo, per unit cosmic time t (corresponding to redshift z), and per halo mass ratio μH can be described with the fitting formula originally proposed by [50, 79] dNH,merg −b−2 (μH /μ˜ H )c dδc a = NH MH,12 μH e dt dμH dt

(26)

in terms of the descendant halo mass MH,12 = MH /1012 M and of the linear threshold for collapse δc . [56] have determined the parameters entering the above expression from the Illustris-Dark simulations, finding NH = 0.065, a = 0.15, b = −0.3, c = 0.5, and μ˜ H = 0.4. Major mergers are typically identified with the events featuring μH > 1/4, minor mergers with those featuring 1/10 < μH < 1/4, and smooth accretion with those having μH < 1/100. The resulting halo merger rates as a function of the mass ratio for different redshifts are illustrated in Fig. 6. We stress that these are often exploited to derive

Fig. 6 Halo merger rate as a function of redshift for different halo masses MH ≈ 1014.5 M (red), 1012.5 M (green), 1010.5 M (blue), and different thresholds in halo mass ratios μH > 1/4 (solid lines), μH > 10−2 (dashed lines), and μH > 10−4 (dotted lines)

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the MBH merger rates, by taking into account possible time delays due to a variety of dynamical processes; these will be discussed to some extent in the rest of this chapter.

Baryons and Black Holes The DM merger trees described above provide the backbone on top of which the hierarchical evolution of the baryonic structures of galaxies, as well as the MBHs, can be computed. In the following, we briefly describe the physics of baryons, before focusing on MBHs and presenting results for the MBH merger rates. We refer, e.g., to [7] for a more extensive description of the physics. A schematic view of the various baryonic structures involved and their relations is presented in Fig. 7. On the largest scales, baryons are mostly in the form of a chemically unprocessed gas that accretes onto DM halos from the IGM. While accreting, the gas is shockheated to the virial temperature of the halo in low-redshift systems of sufficiently large mass. However, in smaller systems and/or at high redshift, the IGM may simply stream into the DM halo along cold flows/filaments [28,36,37]. (In practice, in simulations, a smooth transition is observed between these “hot” and “cold” accretion modes of the IGM; see, e.g., [32].) Cooling of the shock-heated gas and/or inflow of the IGM along cold filaments is then expected to give rise to a cold-gas phase (“interstellar medium,” ISM). Because of conservation of angular momentum, this gaseous component may be disk-like [94], although its geometry can be significantly altered and rendered spheroidal by bar instabilities, compaction, and (major) mergers. Moreover, the ISM (and particularly its molecular clouds) will also undergo star formation. The latter will take place both in disks and in spheroids (or “bulges”), although possibly more efficiently in the latter [35, 57], since the influx of gas toward the galaxy’s center as a result of major mergers and disk instabilities may trigger starbursts. Star formation will also exert a feedback on the stellar surroundings via SN explosions [54, 109, 128], whose winds are expected to expel gas from the ISM, thus quenching (or self-regulating) star formation. SN ejecta also enrich the metal content of the ISM and contribute to its chemical evolution. In their central regions, small-to-intermediate local galaxies with bulge velocity dispersion σ  150 km/s typically contain dense (∼10 pc) stellar clusters with mass up to 108 M [52]. These “nuclear star clusters” might form from local (“in situ”) star formation episodes and/or infall of globular clusters from the galactic disk/bulge as a result of mass segregation (i.e., dynamical friction from the gas and field stars) [1, 2]. Their evolution is intimately connected with that of MBHs, which also dwell in the same nuclear region [59, 60]. The merger of two galaxies, and the ensuing formation of a MBH pair or bound binary, is likely responsible for the relative scarcity of nuclear star clusters in systems with σ  150 km/s [1, 2, 14]. Indeed, because of the galaxy-MBH scaling relations, those large galaxies tend to host large MBHs. When the latter form pairs/binaries, they erode nuclear star clusters by ejecting stars through the slingshot effect. Moreover, MBH accretion

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Fig. 7 Figure adapted from [71]. Schematic description of the semi-analytic model of [7], with the improvements of [1, 9, 22, 23, 121]. Highlighted in red are the processes (black-hole seeding, delays and SN feedback on the MBH growth) that most significantly affect MBH merger rates

occurs from the same nuclear gas (mainly resulting from inflows due to major mergers and/or disk instabilities) that feeds the growth of nuclear star clusters via in situ star formation episodes. It is therefore natural to expect SN explosions, which tend to deplete/heat up the ISM, to impact also the amount of gas that can accrete onto the MBH. This effect is indeed observed in hydrodynamic simulations, though it depends sensitively on the details of how the energy released by SNae couples to the ISM. For instance, weak thermal and kinetic SN feedbacks do not dramatically impact BH growth, but if the shocks caused by the explosions delay cooling of the gas, SN feedback may hamper BH growth in low-mass galaxies [62].

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Besides being passive actors in galaxy formation, with their mass growth and spin evolution proceeding through gas accretion and coalescences triggered by galaxy mergers, MBHs can also backreact on their galactic hosts. This process is known as AGN feedback [24, 33, 65]. When accreting and shining as quasars/AGNs, MBHs emit photons in a variety of bands. A fraction of this radiation may interact with the (optically thick) material surrounding MBHs, heating it up and thus quenching its cooling and star formation. Moreover, AGN feedback may be provided by outflows/jets from the accretion disk (“disk winds”) [16], or by jets launched by the MBHs when they are highly spinning and immersed in an external magnetic field (anchored to a circum-nuclear disk). The latter process is known as BlandfordZnajek effect [17], is linked to the presence of a black-hole ergoregion, and is believed to be the mechanism underpinning radio-loud AGNs. The baryonic physics that we have described is typically implemented in both semi-analytic galaxy-formation models (e.g., [7,30,31,69,95]) and in hydrodynamic simulations (e.g., [40,46,62,100,105,112,116,131,140,143]). We will focus now on the two main uncertainties that affect the predictions for the merger rates of MBHs: the initial mass function of the MBH population at high redshift and the time delays between the merger of two DM halos and the merger of the hosted MBHs.

Black-Hole Mass Function at High Redshift Several formation mechanisms for the initial population of black-hole seeds at high redshift have been proposed (see, e.g., [80] for a review). “Light” seeds (LSs) may form from the first generation (Population III) of stars [86]. These metal-free stars are believed to form at high redshift (peaking around z ∼ 10) in the deepest potential wells (i.e., in the most massive DM halos at a given epoch) and to be very massive (up to several hundred solar masses). When they explode as SNae, they may leave behind black holes of a few hundred M , which may provide the seeds for the subsequent growth (via mergers and accretion) of the MBH population. A problem with this scenario is that it is difficult to reconcile it with the AGN luminosity function at high redshift, and particularly with the discovery of active MBHs with masses  109 M in gas- and dust-rich galaxies at z  7 [6, 51, 97, 137–139]. At those redshifts, the age of the universe was  0.77 Gyr, and it seems difficult to accumulate those huge masses in such a short time span. In more detail, if accretion onto these early LSs is Eddington-limited, their mass grows as Mbh ∝ et/τ , with the characteristic timescale τ given by τ=

4.5 × 107 η tEdd ≈ yr . (1 − η) λ λ

(27)

Here, λ ≡ L/LEdd is the Eddington ratio between the black hole’s bolometric luminosity and the Eddington limit LEdd ≈ 1.4 × 1038 Mbh /M erg s−1 ; tEdd = Mbh c2 /LEdd ≈ 0.4 Gyr is the Eddington characteristic timescale; and η ≡ L/(M˙ bh c2 ) is the radiative efficiency of the accretion flow, which we have set to

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η = 0.1 (as approximately suitable for describing thin disks around slowly spinning black holes). It is therefore clear that even for λ ∼ 1 (Eddington accretion), a seed of a few ×102 M would grow to  109 M only at z ∼ 7, which is marginally sufficient to explain the observations of [6, 51, 97, 137–139]. Taking into account that accretion is also expected to proceed intermittently (in bouts possibly associated with major mergers and/or starbursts), it seems that a LS formation scenario should require moderately super-Eddington accretion [87]. This is quite possible since at relatively high mass accretion rates M˙ bh  0.5LEdd /c2 , accretion flows are expected to “puff up” and transition from a radiatively efficient, geometrically thin configuration to radiatively inefficient, geometrically slim disks, which can sustain moderately super-Eddington accretion [87]. Another possibility is that MBH seeds may form already with relatively large masses ∼104 –105 M (see [88] for a recent review). Among these “heavy seed” (HS) scenarios are, e.g., the rapid formation of seeds by direct collapse of gas and dust clouds in protogalaxies, induced by mergers, bar instabilities in gaseous disks, or cold-gas inflows along filaments [12, 41, 42, 89, 90, 142], and runaway collisions (favored by mass segregation) of massive stars, e.g., in metal-poor nuclear stellar clusters [39, 81, 106] or in the high-z, strongly star-forming galaxies with dense gas environments that are progenitors to local early-type galaxies [18, 75].

Delays Between Galaxy and Black-Hole Mergers When two DM halos coalesce according to the merger-tree formalism described earlier in this chapter, the smaller one initially retains its identity as a sub-halo (or “satellite”) of the newly formed system. That sub-halo then slowly spirals in (on typical timescales of a few Gyr) as a result of dynamical friction [25]. During this phase, tidal effects (stripping and evaporation) remove mass from both the DM and baryonic components, which in turn affects the evolution of the system (making dynamical friction less and less efficient) [129]. When the sub-halo finally reaches the center of the system, the baryonic components (the “galaxies”) and the contained MBHs do not coalesce immediately, but keep evolving under the same processes (dynamical friction and tidal stripping/evaporation) [9, 45, 132]. This phase, during which the two MBHs go from ∼ kpc to ∼ pc separation, can also last for several Gyr, especially when the merging galaxies have unequal stellar masses [132]. Moreover, tidal effects progressively disrupt the smaller galaxy during its evolution, eventually leaving the MBH naked or at most surrounded by a core of stars [45]. As a result, a significant number of “stalled” MBHs may be left wandering at separations of hundreds of pc [45, 132]. For the MBHs that reach separations ∼ pc and form bound binaries, dynamical friction eventually becomes inefficient compared to other processes. Among the latter, a prominent role is played by stellar hardening [108, 119], i.e., threebody interactions between the MBH binary and individual stars. Stars on low angular momentum orbits (i.e., in the “loss cone”) interact strongly with the binary, removing energy from it via the slingshot effect. Repeated interactions of this sort

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cause the MBH binary to shrink, while stars in the loss cone are progressively removed from the system (being ejected, e.g., as hypervelocity stars [120]). As a result, the loss cone needs to be replenished (by diffusion of stars on the stellar relaxation timescale) if stellar hardening is to efficiently drive the binary’s evolution down to separations ∼10−2 –10−3 pc (where GW emission alone can lead the system to coalescence in less than a Hubble time). Mechanisms that could help enhance stellar diffusion and thus replenish the loss cone (ensuring hardening timescales of a few Gyr) include, e.g., triaxiality of the galaxy potential (resulting, e.g., from a recent galaxy merger) [70, 134–136, 145] or galaxy rotation [64]. It should also be noted that in gas-rich galactic nuclei, planetary-like migration in a gaseous nuclear disk may harden the MBH binary on timescales even shorter (∼107 –108 yr) than those of stellar interactions [34, 47, 83, 85, 98, 101, 113]. Moreover, even in gas-poor nuclei, and even if stellar hardening becomes inefficient due to insufficient loss-cone replenishment, a third MBH will eventually be added to the system by a later galaxy merger (as a consequence of the hierarchical nature of structure formation, described in the first part of this chapter) [9,22,23]. Triple MBH interactions can trigger the merger of the inner binary via Kozai-Lidov oscillations [74, 82], which tend to decrease the inclination of outer binary while increasing the eccentricity of the inner one, or via chaotic three-body interactions [20, 21]. Both processes can drive the inner binary to the GW-dominated regime (i.e., separations of ∼10−2 –10−3 pc) in a sizeable fraction of systems [9, 22, 23]. Remarkably, this triple-MBH merger channel leaves a characteristic imprint on the GW signal observable by LISA, since MBH binaries originating from it are expected to carry a significant residual eccentricity (0.99 when they enter the LISA band and ∼0.1 at coalescence) [23].

Predictions for LISA and PTAs The predictions for how many MBH mergers will be observed by LISA, as well as for the parameters (masses, redshifts, etc.) of those sources, are very sensitive to the physics outlined above, and in particular to the sub-grid modeling of the seeding mechanism, the delays between galaxy and MBH mergers, and the impact of SN feedback on MBH growth. To illustrate this fact, we will review here the results of the semi-analytic galaxy-formation model of [9] (based in turn on the model of [7], with the incremental improvements described in [1, 22, 23, 121] and in [9] itself). We will focus in particular on two competing scenarios for the black-hole seeds, namely, a population-III LS scenario [86] and a representative HS model [142] where seeds form (with masses ∼104 –∼105 M ) by direct collapse, as result of bar instabilities in high-redshift protogalaxies. Moreover, besides considering versions of both the LS and HS models where SN feedback on the nuclear gas and delays are included (SN-Delays models), we also review results obtained when either (or both) of these physical processes are switched off or modified. In particular, in the noSN models, we neglect the effect of SN explosions on the nuclear gas. In the shortDelays

870 Table 1 Predictions of the models of [9] for the total number of MBH mergers and detections in 4 years of observation with LISA. (Adapted from [9])

E. Barausse and A. Lapi

Model SN feedback SN-Delays SN-shortDelays No SN feedback noSN-Delays noSN-shortDelays

LS Total

Detected

HS Total

Detected

48 178

16 36

25 1269

25 1269

192 1159

146 307

10 1288

10 1288

models, we switch off the delays occurring as the MBH pair moves from kpc to pc separations (i.e., we neglect dynamical friction on the satellite galaxy and/or its MBH) while keeping the delays due to dynamical friction of the DM halos, as well as those due to stellar hardening, gas-driven migration, and triple MBH interactions. (Note that the shortDelays models correspond to the nodelays models of [9], since in that work, the “delays” are only meant as those due to dynamical friction on the satellite galaxy and/or its MBH.) The (average) number of MBH mergers detectable by LISA in 4 years of observation is reported in Table 1, alongside the total number of events (i.e., the number of mergers that could be detected in the same observation time if the detector had infinite sensitivity). We assume a signal-to-noise ratio (ρ) detection threshold ρ > 8. As can be seen, LISA will detect essentially all MBH coalescences in our past light cone in the HS models. That is the result of their large masses, which produce larger signal-to-noise ratios (cf. also [71]). Conversely, in the LS models, the fraction of detected MBH mergers is always less than one because of the lower binary masses, which translate into lower signal-to-noise ratios and GW frequencies at the high-end of the LISA sensitivity curve. The detection fraction is typically around 20–30%, but can reach ∼75% in the noSN-Delays (where seed growth is unhampered and the delays allow for longer MBH growth before binaries merge, yielding higher masses and signal-to-noise ratios). Note also that the inclusion of realistic delays tends to decrease both the intrinsic number of MBH mergers and the number of detections. This effect is more spectacular in the case of HSs. The distribution of the signal-to-noise ratios of the detected mergers is shown in Fig. 8. As expected, mergers from HS models are typically louder than those from LS models, as a result of their larger masses. One can also see that the differences between the HS models are mainly due to the dynamics, with realistic delay timescales resulting in louder sources (because longer delays allow the MBHs to grow to larger masses, and shift mergers to lower redshifts, where signal-to-noise ratios are naturally higher). In the LS models, the spread in the predictions is instead due mainly to SN feedback, which lowers the typical mass of MBH binaries by hampering their early growth. Note that the relatively low signal-to-noise ratios of the LS models highlight the importance of the LISA sensitivity level at high frequencies, where these light sources are detected.

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Fig. 8 Distribution of signal-to-noise ratio (normalized to the total number of detections) for the HS and LS models (respectively upper and lower panels). (Adapted from [9])

The distribution of the detected mergers in redshift and in total mass of the MBH binary is shown in Figs. 9 and 10. Again, the effect of the delays, which decrease the number of observed systems and shift the event distribution to lower redshift, is particularly evident. As for SN feedback, it mainly affects LS models. Since it tends to prevent the MBHs from growing when they are located in shallow potential wells at high redshift, the redshift distribution peaks later in the LS models with SN feedback. Those LS models also present a noticeable suppression in the number of mergers with masses below ∼107 M , which is responsible for the lower number of detections in Table 1. Also note that the low-mass peak in the distribution of the HS binaries in the models with short delays occurs because the MBH seeds do not have time to grow significantly before merging. Particularly important for modeling GW signals is the mass ratio of MBH binaries, whose distribution is shown in Fig. 11. As can be seen, most systems feature mass ratios (defined as q = Mbh,2 /Mbh,1 ≤ 1) above 0.01−−0.1, according

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Fig. 9 Detected MBH mergers per unit redshift in 4 years of LISA observations, for the HS models (upper panel) and the LS models (lower panel). (Adapted from [9])

to the specific model. Interestingly, the predicted distribution is very robust in HS models, while in LS ones, SN feedback tends to favor comparable mass systems. This can be understood by noting that LS models with no feedback allow seeds to grow more quickly away from their initial masses, producing a larger range of possible mass ratios. When the seed growth is impeded by SN feedback, however, a significant number of mergers take place between black holes with masses close to their initial values, which are similar as they are produced by the same physical mechanism (explosions of population-III stars). Finally, let us comment that while the number of sources detected by LISA, as well as their parameters, is to a large extent model dependent, the predictions for the level of the stochastic background in the frequency band of PTAs are instead quite robust to the assumptions made about the seeding mechanism, the delays between galaxy and MBH mergers, and the effect of SN feedback on nuclear gas. This is evident from Fig. 12, which shows the predictions for the characteristic

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Fig. 10 Detected MBH mergers as a function of total MBH binary mass for the HS models (upper panel) and the LS models (lower panel). (Adapted from [9])

strain hc of the stochastic background from MBH mergers in the ∼ nHz frequency band, for the eight models presented above, compared with the upper bounds from EPTA, PPTA, and NANOGrav [3, 38, 124]. Note that the slope of the predictions (hc ∝ f −2/3 ) simply follows from the quadrupole formula of general relativity, while the normalization (which depends on the MBH population) only shows a minor scatter according to the model. This robust dependence on the astrophysical model is expected (cf., for instance, also [22]), because the PTA signal comes from comparable mass MBH binaries with masses  108 M at z  2. The predictions for such systems depend very weakly on the initial seed mass function (since memory of the latter is lost due to accretion) and on the modeling of the delays between galaxy and MBH mergers (which for these systems are typically short). One physical ingredient that may affect the normalization of the PTA signal, however, is the overall normalization of the black-hole-galaxy scaling relations [122]. The semianalytic models presented in this chapter were calibrated to the observed scaling

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Fig. 11 Fraction of detected MBH mergers with mass ratio below q, for the HS models (upper panel) and the LS models (lower panel). (Adapted from [9])

relations, accounting for the selection bias affecting the latter [8]. In particular, as shown by [123], the black-hole sphere of influence must be resolved for the MBH mass to be reliably estimated, which may bias the inferred normalization of the black-hole-galaxy scaling relations.

Future Prospects While this chapter was being finalized, NANOGrav reported evidence for a common-spectrum low-frequency stochastic process that affects pulsar-timing residuals in their 12.5-year dataset [4]. The origin of this common “red noise” may be a stochastic GW background from MBH mergers. NANOgrav has so far reported no convincing evidence of quadrupolar angular correlations among the residuals of their pulsars (i.e., the Hellings-Downs correlation that should be present for a GW background [63]). However, if one interprets this common “red

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Fig. 12 Characteristic strain of the stochastic background from MBH binaries in the PTA band, for the eight models considered in this chapter (with the envelope of the LS/HS model predictions shown in purple/light blue, respectively). Also shown are the sensitivities of ongoing PTA experiments [3, 38, 124]. (Adapted from [9])

noise” as tentatively due to GWs, that would correspond to hc = A(f yr)−2/3 , with A = 1.92 × 10−15 (median) and 5 − 95% quantiles A = 1.37 − 2.67 × 10−15 . This would be above the previously published upper bounds shown in Fig. 12, although NANOgrav attributes this discrepancy to the (optimistic) use of log-flat priors in earlier analyses (including theirs). While it remains to be ascertained whether NANOgrav has really observed the stochastic background from MBH binaries, it is possible that the predictions of this chapter for LISA may need to be recalibrated (perhaps reconsidering the effect of selection effects on the scaling relations) and thus possibly revised to higher event rates. In any case, even the models shown in Fig. 12 would be testable by PTAs with ≈15 − −20 year of residual collection (assuming about 50 ms pulsars monitored at 100 ns precision). A much earlier detection would be possible with the Square Kilometer Array (SKA) telescope [48].

Cross-References  Introduction to Gravitational Wave Astronomy  Principles of Gravitational-Wave Data Analysis  Pulsar Timing Array Experiments  Terrestrial Laser Interferometers

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Acknowledgments E.B. acknowledges financial support provided under the European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic Scales” grant agreement no. GRAMS-815673. E.B. thanks I. Dvorkin, M. Bonetti, M. Tremmel, and M. Volonteri for numerous insightful conversations on the astrophysics of galaxies and black holes and for agreeing to adapting the figures of [9]. A.L. acknowledges financial support from the EU H2020-MSCAITN-2019 Project 860744 “BiD4BEST: Big Data applications for Black hole Evolution STudies” and from the PRIN MIUR 2017 prot. 20173ML3WW 002, “Opening the ALMA window on the cosmic evolution of gas, stars and massive black holes.” We thank J. Gonzalez for proofreading this manuscript.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISA Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Population Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source Classes/Observed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detached Binary White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interacting Binary White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron Star-White Dwarf Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black Hole-Neutron Star Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolved Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confusion-Limited Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Within the Galaxy, there are tens of millions of close white dwarf binaries that are emitting gravitational waves within the low-frequency band accessible to the space-based gravitational wave detector LISA. A few tens of thousands of these systems will be individually resolvable in frequency space, while the remainder will contribute to a confusion-limited foreground at frequencies below about 1 mHz. In addition to the close white dwarf binaries, a much smaller population

M. Benacquista () University of Texas Rio Grande Valley, Brownsville, TX, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_19

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of binaries containing neutron stars and black holes is also expected to lie within the LISA sensitivity band. Here, we will describe the basic design of the LISA mission with a focus on those features that allow for the detection and parameter estimation for compact binaries in the Galaxy. We will describe the standard population synthesis methods and computational techniques for dealing with dynamical formation of binaries. Following this, we will then briefly discuss the types of binaries that will comprise the Galactic population. The chapter will conclude with a discussion of data analysis, the generation of catalogs of Galactic binaries, and the expected science that can be done with these detections. Keywords

Gravitational waves · White dwarfs · Neutron stars · Black holes · Binaries · Milky Way

Introduction The Galactic population of stellar-mass compact binaries will provide the greatest number of signals within the LISA data stream. The majority of these will be close white dwarf binaries, but other binary systems containing neutron stars and black holes are possible additional sources. This chapter will start with the minimum description of the LISA mission necessary to understand the detectability of compact binaries in the Milky Way and the capability for determining extrinsic properties (sky location, distance, angle of inclination) and intrinsic properties (masses, spins, orbital separation) of these binaries. Following the description of LISA, we will describe population modeling techniques as they relate to both isolated evolution in the field and dynamical formation in clusters. We will then describe the basic source classes that will be under consideration as well as the observed systems that can be used to put constraints on the total populations. The chapter will conclude with a discussion of the techniques for detection of these systems. There will be some overlap with material covered in other chapters in this book, which can be referred to for more detailed discussion than what is presented here. In particular,  Chaps. 2 “Terrestrial Laser Interferometers” and  3 “Space-Based Gravitational Wave Observatories” describe interferometric detectors in general and LISA in particular. The Galactic population of double neutron stars and double black holes is a subset of the populations described in  Chaps. 12 “Binary Neutron Stars”,  15 “Black Hole-Neutron Star Mergers”, and  16 “Dynamical Formation of Merging Stellar-Mass Binary Black Holes”.

LISA Summary A detailed description of the LISA can be found in another chapter of this book, so we will restrict ourselves to those features of LISA that are relevant for understanding the Galactic population of compact binaries. In particular, we will

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describe the orbital configuration, the orientation of the detector, and the sensitivity curve. With this information, we will be better able to understand how modification to LISA will influence the detectability of Galactic binaries and the science that can be done with these detections. The LISA constellation will consist of three independent spacecraft on heliocentric orbits. The eccentricity and phase of the orbits are carefully chosen so that the constellation appears to be a nearly equilateral triangle that rotates about an axis that makes an angle of 60◦ to the ecliptic. This axis precesses so that it always points toward the sun. The rotation of the triangle and the precession of the axis both have periods of one year. The orbit of the center of mass of the constellation around the sun, the rotation of the constellation around that center of mass, and the precession of the plane of the constellation all induce periodic variations in the frequency, phase, and polarization of the response to a monochromatic gravitational wave. The orbital configuration of LISA is shown in Fig. 1. These variations can be used to infer the sky location of the source. If the wave is comparable to, or shorter than, the arm length of the constellation, the sensitivity of the interferometer will be decreased in a way that depends on both the wavelength and the position of the binary with respect to the plane of the LISA constellation. This introduces

Fig. 1 The orbital configuration of LISA. The top figure shows the relative position of the constellation relative to the earth. The bottom figure shows the tumbling motion of the constellation as the individual spacecraft orbits the sun. This figure is taken from the LISA L3 mission concepts proposal [1]

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an additional periodic variation in the amplitude of the detected signal that gives further position information about the source. The frequency band of sensitivity for LISA is set by force noise at the low end and by position noise convolved with the detector response at the high end. At frequencies below about 1 mHz, the dominant noise source is the force, or acceleration, noise that arises from spurious forces acting on the proof mass. The main causes of force noise are electromagnetic, mechanical, and gravitational coupling between the proof mass and the spacecraft. Generally, we assume that this noise is Gaussian with a flat power spectrum in acceleration. This noise translates through two integrations into a strain amplitude noise that has a f −2 amplitude spectrum (or f −4 power spectrum). This steep rise at low frequencies eventually swamps any known sources. Above about 1 mHz, the dominant noise source will be position noise arising from uncertainty in measuring the position of the proof masses. The primary source of position noise is the influence of shot noise on the measurement of the interference fringes. The strength of the shot noise depends on the mirror size, internal optics, and laser power. Since the uncertainty in position of the proof masses translates directly into strain amplitude noise, this noise has an effectively flat amplitude spectrum. At higher frequencies, the response of the detector is reduced once the wavelengths are comparable or shorter than the arm length; thus, the response of the detector to a signal is eventually too weak to rise above the position noise at high frequencies. The sensitivity curve is one of the more useful ways of describing the capability of LISA to detect individual sources by comparing the characteristic strain of a given source with the convolution of the instrument noise with the sky-averaged response of the detector. The characteristic strain for an effectively monochromatic binary with frequency f at distance d is 5/3

hc ∝ 2Tobs

G5/3 Mc c4 d

(πf )2/3

(1)

where the chirp mass is Mc = μ3/5 M 2/5 , with total mass M and reduced mass μ = M1 M2 /M. A standard LISA sensitivity curve is shown in Fig. 2. Note that the units of the sensitivity curve are Hz−1/2 . This is because the noise curve is computed as the strain spectral density by taking the square root of the power spectral density. Since the power spectral density is dimensionless strain squared divided by the resolvable frequency bin, the strain spectral density is dimensionless strain divided by the square root of the resolvable frequency bin. The overall strength of the noise and the power of the signal response both depend on the duration of the observation time, Tobs . Thus, care must be taken when determining the expected detectability of a given source. In frequency space, the minimum resolvable frequency difference 1/2 −1 is Tobs , and the noise level scales as Tobs . The strength of monochromatic signals scales as Tobs . There are many options for incorporating this into a sensitivity curve, so it is important to know what Tobs was used to calculate the sensitivity curve and how the levels of the noise have been set. In this chapter, we will assume an

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Fig. 2 Sensitivity curve from the LISA mission webpage https://www.elisascience.org/ multimedia/image/lisa-sensitivity. The details behind the sensitivity curve can be found in the LISA L3 mission concepts proposal [1]

observation time of Tobs = 4 yr. For a more detailed look at sensitivity curves and characteristic strains over many different detectors, see [2].

Population Modeling Galactic binaries can form in the field and evolve as binaries throughout their lives, but they can also form dynamically in dense stellar clusters and globular clusters. Thus, modeling the population of Galactic compact binaries requires modeling of both binary and single stellar evolution in addition to the overall dynamical evolution of clusters. Since there are millions of galactic binaries, individually modeling each system is not feasible, and population synthesis methods have been developed to speed up the modeling process. Although direct modeling of the dynamical evolution of many clusters is now possible, several approximate approaches are used to speed up the modeling of dynamical systems to extract the binaries that are formed within. Most of these approaches rely on pre-computing the evolutionary state of a matrix of binary and single stars and then using interpolation or analytical formulae to determine the state of the systems at any point in their evolution. In this section, we will describe the basic features of the dominant modeling codes in use today.

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Field Evolution Modeling of the field population of Galactic compact binaries is done using the methods of population synthesis. Most codes start with an input of basic properties of the binary – zero age main sequence (ZAMS) mass and metallicity of the components and the initial orbital separation and eccentricity. They then use prescriptions to compute the evolution of the binary through various stages (such as evolution of the main sequence, mass loss through winds, Roche lobe overflow, common envelope, supernovae). Although there are differences in the outcomes of the different codes, a comparison of several codes with identical initial conditions has shown that these are subtle [3]. There are four main population synthesis codes that are in common use in modeling the population of binaries in the Galactic field. Based on the Hurley binary stellar evolution work [4], the binary_c code includes nucleosynthesis so that the chemical evolution of binary systems can be followed. The basic binary and single stellar evolution codes can be found on Hurley’s website http://astronomy. swin.edu.au/~jhurley/. The Brussels code uses full binary evolution calculations instead of the analytical formulae common to the other population synthesis codes. The Brussels code is fully reviewed in [5]. Originally developed as part of the Starlab software package, SeBa [6] is one of the rapid population synthesis codes that use analytical formulae to determine the state of the binary at runtime and can therefore easily incorporate changes in the models. Recently, SeBa was incorporated into the Astrophysical Multipurpose Software Environment (AMUSE) [7] and can be found at the AMUSE website amusecode.org. The fourth population synthesis code, StarTrack [8], is a Monte Carlo code that evolves binary and single stars through analytical fitting formulae applied to the evolutionary tracks of detailed stellar models. The code can be updated by adding new physics to the evolutionary models. The least understood, yet most influential, aspect of the binary evolution models that go into population synthesis is the common envelope phase. As the more massive component of a close binary evolves off the main sequence, it can swell to fill its Roche lobe and begin to transfer mass over to its companion. However, if the envelope of the more massive star grows too fast, then the mass transferring through to the companion can exceed its ability to accrete the material. In this case, the envelope will engulf both stars, leading to a common envelope. In this phase of binary evolution, the orbital motion of both stars serves to heat the common envelope, leading to its eventual unbinding from the binary. Since the energy supplied to the envelope comes from the orbital motion, the binary emerges from the common envelope phase with a tighter orbit and shorter orbital period. Prescriptions for the outcome of the common envelope phase have been in existence for decades (see, e.g., [9, 10]), although they remain phenomenological. In its most common form, the final separation of the binary, af , for a binary that starts at an initial separation ai with components of mass m1 and m2 is found from

20 LISA and the Galactic Population of Compact Binaries

  m1 m1,env Gm1,c m2 Gm1 m2 = αCE − + . λR1 2ai 2af

891

(2)

In this equation, the donor star is m1 , its envelope mass is m1,env , and m1,c is the mass of its core. The two phenomenological parameters, λ and αCE , are introduced to describe the structure of the donor star and the efficiency of transferring orbital energy into the envelope, respectively. A thorough review of the current state of the common envelope formalism can be found in [11]. Figure 3 shows several of the outcomes of a common envelope phase that can result in compact binaries. In order for many systems to be brought close enough to enter the LISA

Fig. 3 Typical outcomes of common envelope evolution. Figure from [11]. Their description indicates that the leftmost column describes pathways that result in type Ia supernovae, while the center and rightmost columns indicate pathways to form either low- or high-mass X-ray binaries. The abbreviations in the figure are ZAMS, zero age main sequence; RLO, Roche lobe overflow; CE, common envelope; CO WD, carbon-oxygen white dwarf; He, He star; HMXB, high-mass X-ray binary; LMXB, low-mass X-ray binary; MSP, millisecond pulsar; NS, neutron star; SN, supernova. For example, the center column describes the initial binary as a massive blue star and an average yellow star. The blue star evolves off the main sequence first and transfers mass to the yellow star through Roche lobe overflow, which eventually becomes a common envelope phase that drives the two stars closer together. The remnant of the massive star explodes as a supernova leaving a neutron star that accretes matter from the yellow star once it leaves the main sequence

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frequency band within a Hubble time, they must pass through two common envelope phases. Population synthesis codes are good at predicting the relative numbers of different types of compact binaries. However, in order to determine the absolute numbers of the different binary types that can be expected in the Galaxy, it is necessary to normalize the results to observed objects. The two most common anchors that are used to normalize the results are the supernova type Ia (SNe Ia) rates and the Galactic population of interacting white dwarf binaries (generally AM CVn stars) [12]. Since SNe Ia are rare events, the rates are either determined through extragalactic events or through Galactic remnant observations. There remains a debate on whether SNe Ia arise from the merger of two white dwarfs (the double degenerate scenario) or the accretion of matter from a nondegenerate companion onto a white dwarf (the single degenerate scenario). Thus, there are a number of uncertainties that go into normalizing the population with SNe Ia rates. Normalization with interacting white dwarf binaries is confined to the Galactic population of such objects, but there are two major uncertainties with this method. First, because these systems are dim, the observed population is mostly within a kpc of the earth, and the local density of such objects may not be representative of the Galaxy as a whole. Second, these systems are mostly thought to arise from detached systems that eventually spiral in to the point that mass transfer begins. The systems that persist are those that have stable mass transfer. The conditions that lead to stable mass transfer are not well understood (see Fig. 4), and so, the connection between number of observed masstransferring systems and the underlying population of detached binaries is also not well understood.

Dynamical Evolution Dynamical codes model the evolution of dense stellar systems and need to include interactions between the stars and binaries within the cluster. Since these interactions allow for the formation of binary systems from stars that had not previously interacted, dynamical codes need to consider both single and binary stellar evolution. Naturally, in order to describe the close interactions of the stars in the cluster, it is also necessary to model the global gravitational interactions between the stars in the system. There are two basic approaches to modeling this problem: N-body codes and distribution function codes (commonly called Monte Carlo codes). N-body codes follow the evolution of every star in the cluster and compute the gravitational interaction between all stars in the cluster. This method is computationally expensive and has limited then size of clusters that can be modeled in this way (although advances in computational capabilities have recently allowed for modeling of 106 star clusters). In most cases, smaller clusters are modeled, and then, scaling algorithms are used to infer properties of larger clusters. Monte Carlo codes avoid the computational cost of modeling all gravitational interactions by treating

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Fig. 4 Expected criteria for stable mass transfer [13]. Note that there is a substantial range of mass ratios for which it is not clear whether the mass transfer is always stable or unstable

the cluster as statistical ensemble with multiple distribution functions describing different classes of systems. A detailed (but somewhat outdated) description of these two techniques and their relative advantages and disadvantages can be found in [14]. Here, we give a brief discussion of the most recent incarnations of these two approaches. The N-body approach incorporates direct computation of the gravitational forces between all objects in the cluster. For a recent summary of techniques, see [15]. This approach is very computationally expensive but provides the most realistic representation of the dynamical interactions of stars in the cluster. The evolution of stars within the cluster is usually modeled using standard single and binary stellar evolution codes. A number of unique interactions within a cluster require some care in using the stellar evolution codes. For example, exchange interactions can replace a star within a binary by another star at a different stage of its evolution. Collisions between stars are also possible, and these can interrupt or alter the evolutionary state of a given star. Unless they are run on extremely large and fast computer clusters, most simulations of clusters have been done using smaller cluster sizes which are then extrapolated up to typical globular cluster sizes using scaling relations. Recently, million-body clusters have been evolved [16] using NBODY6++GPU [17]. These calculations required over half a year to complete and included only 5% binaries. Further advances by this group have developed the PeTar N-body code, which is poised to tackle the ten-million-body problem, and can handle 100% binaries in the million-body problem [18].

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Because the N -body approach is so computationally expensive, several approximate methods have been developed to allow for rapid modeling of the evolution of dense stellar systems so that parameter space can be more efficiently covered. These codes are based on Fokker-Planck models which treat the clusters in a statistical fashion with distribution functions for different types of stellar systems. These consider a distribution function fm (x, v, t) for stars of mass m that describes the number of stars in a phase-space volume d3 xd3 v as dN = fm d3 xd3 v. The evolution of the distribution function over time depends upon a treatment of the smoothed gravitational potential, φ, of the cluster and a collisional term, Γ [f ], which describes the effect of random close encounters between stars. The equation governing this evolution is ∂fm ∂fm + v · ∇fm − ∇φ · = Γ [f ] ∂t ∂v

(3)

and is known as the Fokker-Planck equation. The problem with the direct Fokker-Planck approach is that the distribution functions represent a large number of stars of similar mass, and so, it is difficult to include stellar and binary evolution within the codes. The response to this (and other problems) is to treat the cluster as an ensemble of individual stars whose interactions and evolution can be tracked but use a statistical treatment of the small perturbation in energy and angular momentum of each star due to the weak interactions with distant stars. Strong interactions which may result in collisions or binary exchanges are determined by a random likelihood of such an encounter. The details of a strong interaction can then be computed using direct N-body methods but only involving the small number of stars that are participating in the strong interaction. Since each individual star is accounted for, stellar evolution is easy to incorporate into the models. It is the Monte Carlo treatment of the strong interactions that give these codes their name. Two of the more commonly used codes are MOCCA [19] and CMC [20]. Both of these codes have been used recently to model the production of the stellar-mass black holes whose mergers have been observed by LIGO/Virgo. Since the production of other compact binaries is automatically included in the modeling of cluster evolution, estimates of the number and type of potential LISA sources within the Galactic globular cluster system can be made by choosing those globular cluster models that most closely represent Galactic globular clusters.

Source Classes/Observed Systems The frequency range of LISA – from a little below 10−4 Hz to above 10−1 Hz – will be sensitive to gravitational waves from short-period binaries consisting of white dwarfs, neutron stars, and black holes. The bulk of the detectable systems within this frequency range will be binary white dwarfs (BWDs). The majority of these will be detached systems that are slowly inspiraling together through the emission of gravitational waves. Above frequencies of a few mHz, the BWDs will begin to reach

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Roche lobe overflow and will start transferring mass. For some of these systems, the mass transfer will be unstable, and they will coalesce, forming more massive white dwarfs, accretion-induced collapse neutron stars, or SNe Ia. For others, the mass transfer will be stable, and they will spiral outward as the lower-mass white dwarf loses mass to its more massive companion. Most estimates of the number of BWDs (both detached and mass transferring) show around 30 million systems within the LISA sensitivity band. A substantially smaller number of systems within this band will consist of some combination of neutron stars, white dwarfs, and black holes. In this section, we will discuss the expected number of each kind of system and also the accepted formation scenarios. We will identify areas where there is uncertainty in the modeling and where LISA observations can help shed light on the formation of these systems. Hils, Bender, and Webbink [21] produced the first comprehensive discussion of Galactic binaries as gravitational wave sources for LISA in 1990. Nelemans et al. [22] used a refined population synthesis to produce an estimate of the population in 2001. A series of papers using StarTrack [23–25] gave an additional estimate of the population about a decade later.

Detached Binary White Dwarfs Detached white dwarf binaries within the LISA band have orbital periods that are short enough to require that they have gone through at least one common envelope phase as they evolved to become binary white dwarfs. As we have discussed in the previous section, this is the area of their evolution that is the least well understood. Once the cores of a close binary have emerged from their common envelope phases, they will be sufficiently separated that their orbital evolution will then be dominated by the emission of gravitational radiation. As the components grow ever closer, additional influences on the orbit can arise from internal heating of the white dwarfs through coupling of the orbital motion with the internal modes of the star or through tidal stresses [26] and tidal locking [27]. Most white dwarf binaries are expected to emerge from the common envelope phase in nearly circular orbits due to the nature of the interaction between the cores and the envelope. At the low end of the LISA frequency band, nearly all of the detached close white dwarf binaries will be spiraling in toward each other with a frequency evolution given by [28] f˙ = k0 f 11/3 ,

(4)

where ko =

96 (GMc )5/3 . (2π )8/3 5 c5

(5)

If we assume that the birth rate of binary white dwarfs is relatively constant, then we expect the number of binaries passing through each frequency bin to be the same. Thus, the number of detached white dwarf binaries per frequency bin, df , follows

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a power law – dn ∝ f −11/3 df . As the white dwarfs spiral in toward each other, the amplitude of the gravitational waves grows as f 2/3 , and so, the strength of the signal from each binary grows. At high frequency, where the number of binaries per frequency bin is low, individual systems can stand out and be detected. At lower frequencies, the number of binaries per frequency bin grows fast enough that we can expect tens of thousands of weak signals that will add incoherently, producing a stochastic foreground. There are a few known detached white dwarf binaries that can be used to help normalize the results of population synthesis. These systems have been found in the extremely low-mass (ELM) survey [29]. This survey focuses on extremely lowmass helium white dwarfs that cannot have evolved in isolation in a Hubble time. Thus, it is selectively targeting binary white dwarf systems. At the conclusion of the survey, 98 double white dwarf systems had been found [30]. Of these, two are likely to be in the LISA band [31]. Additional normalizations using SNe Ia and AM CVn systems lead to the estimated 30 million systems.

Interacting Binary White Dwarfs As we look at higher frequencies, the power law number density will change once the white dwarf binaries approach contact (or Roche lobe overflow). Once the binaries begin to interact and transfer matter, the outcome of further evolution depends on the stability of the mass transfer. Since the radius of a white dwarf depends on m−1/3 , the lower-mass white dwarf will be the first to fill its Roche lobe. As the mass flows through the Lagrange point, it can be accreted by the more massive companion. As this occurs, two changes happen in the binary: (1) the moment of inertia of the binary decreases, causing the binary to separate and the Roche lobe to expand, and (2) the donor white dwarf loses mass causing its radius to increase. If the radius of the donor white dwarf increases faster than the Roche lobe, the mass transfer becomes unstable, and the two stars coalesce. A general discussion of stable vs. unstable mass transfer can be found in [32]. If the radius of the donor white dwarf does not increase as fast as the Roche lobe, then the binary will bounce apart and become detached again, until gravitational radiation drives the two stars back together. Eventually, the system can settle into a steady state where the change in orbital separation (governed by the change in the moment of inertia and the emission of gravitational radiation) is such that the Roche lobe grows at a rate that exactly matches the increase in size of the donor white dwarf, so that the mass transfer is stable. These systems are the interacting binary white dwarfs in the LISA data stream. The period evolution through stable mass transfer can be found by equating the frequency change due to the loss of angular momentum through gravitational radiation with the frequency change needed to keep the donor star radius equal to the Roche lobe. The loss of angular momentum due to gravitational radiation is given by [33]

20 LISA and the Galactic Population of Compact Binaries

32 G4 μ2 M 5/2 . J˙gr = − 5 a 7/2

897

(6)

Assuming conservative mass transfer (so that M = constant), then the frequency change is related to the mass transfer rate, m ˙ 1 , by −

  m1 1 32 G4 μM 5/2 1 − + ω˙ , = m ˙ 1 5 m2 3 a 9/2

(7)

where ω is the orbital angular frequency and M˙ 1 < 0. This relationship can be further reduced by equating the change in the Roche lobe radius with the change in the donor radius, leaving ω˙ entirely determined by the masses and orbital separation. Interacting white dwarf binaries make up the bulk of the known Galactic binary sources for LISA simply because they are brighter than their detached counterparts. The list of ten known systems are found in Kupfer et al. [31]. However, most of these systems are also observed early in their evolution away from first contact and so are at relatively high frequencies. As the systems evolve, the orbital frequency and the chirp mass both decrease; thus, although these systems also pile up at lower frequencies, their contribution to gravitational wave signal from the Galaxy becomes negligible at the low end of the LISA sensitivity band. As with the common envelope phase of detached systems, there is considerable uncertainty in the criteria necessary for stable mass transfer. This is shown in Fig. 4. As a result, attempts to determine the number of detached binary white dwarfs via a census of interacting binary white dwarfs are fraught with uncertainties.

Neutron Star-White Dwarf Binaries Because the progenitors of binary white dwarfs are two low-mass main sequence stars, their numbers dominate the Galactic population of compact binaries. As we look at higher-mass systems containing neutron stars or black holes, the number of such systems is substantially smaller simply because the number of progenitors is also much smaller due to the initial mass function of stars which is an inverse power law at high masses. Massive stars evolve more quickly than low-mass stars and have higher mass loss rates due to stellar winds. Thus, when a binary system has the potential to result in a neutron star-white dwarf binary, the system will tend to separate due to the steady loss of mass from the more massive star. Once the massive star explodes as a supernova, there are two ways in which the binary can become completely disrupted before the lower-mass star can evolve to become a white dwarf. First, if the explosion causes the system to lose over half its total mass, then the binary will become unbound. Second, most supernovae impart a kick velocity to the newly born neutron star that can exceed the escape velocity of the system, and so, it becomes disrupted. Thus, very few NS-WD systems are known

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in the Galactic field. Most of these are found through searches for pulsars in binary systems. Another formation pathway for NS-WD systems is through accretion-induced collapse of a massive white dwarf in a progenitor double white dwarf system. If the more massive white dwarf is an oxygen-neon white dwarf that is accreting matter from a companion (either a low-mass star or another white dwarf) and its mass grows beyond the Chandrasekhar mass, then the star can collapse directly to a neutron star without a large explosion that would result in a kick or a substantial mass loss to the system. If the more massive white dwarf is a carbon-oxygen white dwarf, then the outcome is more likely to be the complete disruption of the white dwarf in a type Ia supernova. A final formation pathway for these systems is through dynamical interactions that occur in dense stellar systems such as nuclear star clusters or globular clusters. In this case, the neutron star and the white dwarf form separate from each other but are brought together through dynamical interactions. The main bottleneck for this formation scenario in globular clusters is that the kick velocity imparted to the neutron star will likely eject it from the cluster. Furthermore, dynamical interactions between binaries and other stars (or binaries) within a cluster tend to result in an exchange interaction in which the lowest mass star is ejected from the interaction, leaving the two most massive stars in the binary. Thus, it is likely that the white dwarf would be ejected from subsequent interactions. Although nuclear star clusters have a higher escape velocity, so that kick velocities are less likely to eject the neutron star, the dispersion velocity within the cluster is substantially higher as well, so that binaries are more likely to be disrupted unless they are very tight. Nonetheless, formation through dynamical interactions is more likely on a per-star basis. However, the number of stars in globular clusters and the Galactic nuclear star cluster is about four orders of magnitude smaller than the number of field stars, while the likelihood of dynamical formation is not four orders of magnitude greater than field formation. Thus, these systems will still be small in number. The predictions for total numbers of these systems in the Galaxy are on the order of 106 , but most of them will be well out of the LISA sensitivity band. Only a few tens of systems are expected to lie in the LISA band [22]. Searches for these systems in X-rays have found 12 with known or suggested orbital periods [34]. Of these, six are found in globular clusters, and three of these contain an accreting millisecond pulsar.

Binary Neutron Stars Binary neutron stars have been found in the Galaxy through pulsar searches. Most notable is the Hulse-Taylor pulsar which provided the first indirect evidence for gravitational radiation. These systems have been used to estimate the number of binary neutron stars in the Galaxy, but the results are uncertain since they are only found when one of the neutron stars is observable as a pulsar. There are currently 17

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known BNSs in the Galactic field and two more found in globular clusters [35]. Of these 17, only seven have well-measured masses and are expected to merge within a Hubble time [36]. All of the known Galactic systems are at too low a frequency to be observed by LISA. Population synthesis models indicate that there are about 105 -106 of these systems in the Galaxy, but only a small number (on the order of ten) will be observable by LISA [22, 25]. For both of these models, the ability to detect eccentric systems is crucial since many of the systems lie at orbital frequencies below the LISA band, but higher harmonics of the bursts at periapsis can enter the LISA band. The recent detection of a double neutron star merger by LIGO/Virgo and the follow-up observations of the associated kilonova are the best evidence that such systems exist and that they should consequently also be passing through the LISA band, but the number density is low enough that an observation by LISA is uncertain. In addition to the known systems, the Galactic enrichment of r-process elements can be used to infer the ejecta from past mergers of BNSs in the Galaxy [37]. Indirectly, this can be used to help normalize the total number of BNSs predicted from population synthesis. Because they have a higher chirp mass than binaries containing a white dwarf, these systems could be observable in local group satellite galaxies. However, because they have a higher chirp mass, these systems also evolve through the LISA band more rapidly than their white dwarf counterparts. Nonetheless, the detection of a single system would greatly improve our modeling of the formation of double neutron star systems. A good review of the current state of modeling for the formation of BNS can be found in Tauris et al. [38].

Black Hole-Neutron Star Binaries Black hole-neutron star systems are thought to be even more rare than double neutron stars or double black holes. Population synthesis models produce a small number of these systems, but the formation mechanisms are uncertain. Nelemans et al. [22] predicted a few hundred thousand such systems in the field but found that effectively none would be resolvable by LISA. Belczynski et al. [25] find nearly an order of magnitude fewer systems at a few tens of thousands. The number of systems in this model is also effectively zero. Dynamical formation in globular clusters is a more likely formation scenario for BH-NS systems [39], but even then, the overall numbers are expected to be very small, and the orbital periods are likely to be too large for them to be detectable by LISA. The recent LIGO/Virgo detection of a high-mass-ratio inspiral points to the possible existence of a BH-NS binary merger [40]. The low-mass component of the merger is 2.6 M and could either be a very high-mass neutron star of an extremely low-mass black hole. Either interpretation challenges the models for the formation of such an object. The large mass ratio challenges the likelihood of formation dynamically, and so perhaps, field formation is more likely for this object. The rarity of such an event indicates that these systems are very rare in the Galaxy.

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Binary Black Holes Isolated binary black holes are very difficult to identify with electromagnetic observations. Consequently, all known BBHs have been discovered through gravitational waves emitted during their mergers [41]. Needless to say, these have all been extragalactic systems. For the most part, these systems are much more massive than the mass ranges originally predicted for LISA (and LIGO/Virgo) sources. The canonical black hole mass was assumed to be around 10−20 M , but the component masses detected by LIGO/Virgo are in the range 20 − 80 M [42]. The high mass of these black holes implies that they were born in low-metallicity regions so that the stellar winds are reduced and the mass loss is not so great, allowing for the remnant black hole to retain more mass. At the higher end of the mass spectrum of these LIGO/Virgo sources are black holes that cannot be produced by the standard model of stellar evolution. Stars that would be sufficiently massive to form black holes with masses above ≈ 60 M are expected to undergo a pair production instability explosion which completely disrupts the star, leaving no remnant black hole. The existence of these massive black holes points to the production of intermediatemass black holes through repeated mergers of stellar-mass black holes. It is assumed that these systems must be born in dense stellar clusters where black holes can be brought together through dynamical interactions. When merging, black holes are thought to experience a gravitational radiation recoil kick due to asymmetrical emission of gravitational waves at the moment of merger. The recoil kick will give the merger remnant a velocity that is above the escape velocity for globular clusters, so the expectation is that these events occur in nuclear star clusters where the escape velocity is sufficiently high. Simple mergers of stellar-mass black holes (such as the bulk of the LIGO/Virgo sources) can occur both in the field and in dense stellar clusters (including globular clusters). All of the observed BBH systems from LIGO/Virgo point to the existence of a population in the Galaxy. More importantly, the relative numbers of observed BHNS, BNS, and BBH can be used to test the results of population synthesis. Nelemans et al. [22] have about 3.5 more BBH than BNS in the disk of the Galaxy. Belczynski et al. [25] have about twice as many BBH as BNS in their simulations. Given the fact that BBH mergers are observable at much greater distances than BNS mergers, the relative numbers of BBH and BNS are not significantly off, given the uncertainties involved. Thus, models predict on the order of 106 BBH in the Milky Way and its globular cluster system. Most of these systems are at very low orbital frequencies and so are outside of the sensitivity band of LISA. Most of the Galactic BBHs that are predicted to be observed are eccentric systems that emit bursts of gravitational radiation during periapsis passage. The bulk of the power emitted in gravitational waves is pushed to higher harmonics of the orbital frequency in these bursts. The relative power for the harmonics was derived by Peters and Mathews and is [28]  n4 g(n, e)= [Jn−2 (ne)−2eJn−1 (ne)+f rac2nJn (ne)+2eJn+1 (ne)−Jn+2 (ne)]2 32    4 + 1 − e2 [Jn−2 (ne) − 2Jn (ne) + Jn+2 (ne)]2 + 2 [Jn (ne)] (8) 3n

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Fig. 5 Relative power as a function of harmonic number for different eccentricities. (Figure taken from Peters and Mathews [28])

The relative power as a function of harmonic frequency is shown for different eccentricities in Fig. 5.

Detection With the exception of the very rare massive binaries (black holes and neutron stars), most of the Galactic compact binaries will be effectively monochromatic during the lifetime of the LISA mission. The gravitational wave signal from a binary that is circular and not precessing is given by the two polarizations, h+ and hx . Consider a coordinate system that is centered on the binary with the z-axis pointing in the direction of the orbital angular momentum vector, L. For a gravitational wave with frequency F propagating in the eˆ r direction, the two polarizations are defined in terms of the angular unit vectors eˆ θ and eˆ φ and contribute to the strain amplitude according to 5/3  5/3 c h+ = 2 G c4M (πf )2/3 1 + cos2 θ cos (2πf t) d

hx

5/3 5/3 c = −4 G c4M (πf )2/3 cos θ sin (2πf t) . d

(9) (10)

Note that the two polarizations are out of phase by π/2. Recalling that an interferometric gravitational wave detector is effectively a polarizing filter that only responds to the polarization aligned with its arms, we see that the tumbling motion of LISA about the center of mass of the constellation picks up different parts of the polarization of the incoming monochromatic wave, and so,

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Fig. 6 Spreading of a monochromatic signal due to the motion of LISA. The plot on the left is the LISA response to the monochromatic signal on the right

there is an annual phase shift as the detector responds to the incoming plus and cross polarizations. The precession of the plane of the LISA constellation means that the incoming gravitational wave arrives at different points in the sensitivity pattern of the interferometer. This introduces an annual variation in the amplitude of the response to the gravitational wave (which is also varied due to the different amplitudes of the polarizations). Finally, the orbital motion of LISA about the sun introduces a Doppler shift in the observed frequency of the gravitational wave that varies annually and depends upon the degree to which the binary lies out of the plane of the ecliptic. These three variations can be used to determine the sky location of the binary as well as its angle of inclination. The response of LISA to a monochromatic signal around 4 mHz is shown in Fig. 6.

Resolved Systems With an expected observation time of 4 years, LISA will be able to observe and detect tens of thousands of Galactic compact binaries. The vast majority of these will be detached close white dwarf binaries. However, many of the known systems that are considered to be “verification binaries” are mass-transferring systems within a few hundred parsecs of earth. These verification binaries have had updated distances based upon Gaia data [31]. The positions of the known verification binaries are shown in Fig. 7, and their characteristic strain is shown against a LISA sensitivity curve in Fig. 8. These systems and possibly several hundred additional unknown systems will have very large signal to noise ratios and can likely be found using standard matched filtering techniques for isolated signals. These systems are likely to have very high frequencies and may be either detached systems that are close to the onset of Roche lobe overflow or systems that have just recently begun mass transfer. Other easily resolved systems will include high-mass double white dwarfs that may be type Ia supernova progenitors. At frequencies above about 1 mHz, these

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Fig. 7 The sky location of the known verification binaries. The blue line indicates the Galactic plane, and the Galactic center is shown by the blue cross. (Figure taken from [31])

Fig. 8 Characteristic strain of the verification binaries with expected signal to noise ratios above five after a 4-year observation. Black circles are mass-transferring AM CVn systems, red triangles are detached white dwarfs, and the blue square is a hot subdwarf binary. (Figure taken from [31])

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systems should be observable throughout the Galaxy. Thus, LISA should provide a complete census of short-period systems in the Galaxy. Once the loudest signals have been identified, more sophisticated data analysis techniques will be required to perform a global fit on all resolvable systems. This is because the motion of LISA spreads out the signals so that even monochromatic ones will overlap in frequency space. Errors that are introduced in identifying the parameters describing loud signals will thus interfere with weaker signals, causing these to be misinterpreted. In addition, the weaker signals are not Gaussian noise, so that they can contribute to the errors introduced into the louder signals. Thus, the approach needed to fully extract all the resolvable systems will be a global fit [43]. In this approach, a matched filter that includes all the resolvable binaries at once is eventually arrived at. An additional complication that arises when identifying resolved systems is the existence of all the other classes of systems within the LISA band. This includes supermassive black hole binary inspirals and mergers out to very high redshifts, extreme mass ratio inspirals of compact objects into supermassive black holes, and eccentric stellar-mass black hole binaries in the local universe [44, 45]. The catalog of resolved binaries will provide constraints on several aspects of binary stellar evolution. A subset of the resolved binaries may be identified with an electromagnetic source (see, e.g., [46]). These joint EM and GW observations will be able to provide refined measurements of the masses, orbital separation, radii, and angle of inclination. Since many of the resolved binaries will also be at higher frequencies and therefore in close proximity to the onset of mass transfer, an accurate census of the properties of the binaries both before and post mass transfer will provide strong constraints on the conditions for stable mass transfer. This will allow for better constraints on normalizing results from population synthesis codes. Although there will be a selection effect to more massive systems, the mass ratios and orbital frequency distributions can provide insight into the outcomes of some common envelopes. The improved population synthesis estimates and the complete census of potential type Ia supernova progenitors can provide a better understanding of these systems and thus give more accurate information concerning these standard candles. This can affect our understanding of the nature of dark energy and cosmic acceleration. It is possible for precise observations in EM and GW to put constraints on potential violations to general relativity. Finally, deviations from the expected orbital period change due to the emission of gravitational radiation can inform us of the internal structure of white dwarfs [26, 27].

Confusion-Limited Signal Because of the very large number of binaries expected to be within the LISA frequency band, the vast majority of the Galactic binaries will not be individually resolvable. Instead, many thousands of binaries will occupy a single frequency band leading to an incoherent signal that will be above the expected LISA noise level at frequencies between about 0.1 and 1 mHz. This is known as the “confusion signal”

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Fig. 9 A one-year time-domain simulation of the Galactic white dwarf binary signal, showing the semiannual modulation. (Figure taken from [47])

and should be roughly Gaussian, but not stationary. Because the sensitivity pattern of LISA peaks in a direction normal to the plane of the spacecraft constellation, as the sensitivity peak sweeps past the Galactic center, the strength of the confusion signal reaches a maximum. This occurs twice per year. A time-domain simulation of the LISA data stream is shown in Fig. 9. This annual variation of the Galactic foreground noise was studied by Seto [48] to determine if minima in the noise could be used to enhance the signal from extragalactic sources. If we assume that the distribution of white dwarf binaries throughout the Galaxy is related to the star formation history of the Galaxy, then we can use the confusion signal to tease out the shape of the white dwarf binaries and thus recover (or at least constrain) the star formation history of the Galaxy. Initial estimates of detecting the disk scale height were done by looking at the break at which the confusion-limited signal began to dominate the LISA noise curve [49]. More recent work has focused on using spherical harmonic decomposition to determine scale height [50].

Conclusion The contribution of Galactic compact binaries to the LISA signal is of inherent interest to the structure and star formation history of the Milky Way. The Galactic globular cluster system can also contribute to this population, and so, these systems can also provide insight into the evolutionary history of the globular clusters. Because many of the population modeling techniques that are applied to the Galactic population are also applicable to extragalactic systems, there is an overlap between LIGO/Virgo source populations and LISA source population. It is likely that continued LIGO/Virgo observations will help clarify some of the questions

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surrounding the LISA populations, but it is equally likely that LISA observations will help answer some of the questions that are raised by LIGO/Virgo observations. The next few decades will see gravitational wave astronomy come of age.

Cross-References  Binary Neutron Stars  Black Hole-Neutron Star Mergers  Dynamical Formation of Merging Stellar-Mass Binary Black Holes  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range  Introduction to Gravitational Wave Astronomy  Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravita-

tional Waves  Space-Based Gravitational Wave Observatories

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Gravitational Waves from Core-Collapse Supernovae

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Ernazar Abdikamalov, Giulia Pagliaroli, and David Radice

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Overall Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Road to Core Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Core Collapse and Road to Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-rotating and Slowly Rotating Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Quasi-periodic Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PNS Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutrino-Driven Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SASI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Protoneutron Star Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explosion Phase Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapidly Rotating Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounce and Ring-Down Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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E. Abdikamalov () Department of Physics, School of Sciences and Humanities, Nazarbayev University, Nur-Sultan, Kazakhstan Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan, Kazakhstan e-mail: [email protected] G. Pagliaroli Gran Sasso Science Institute, L’Aquila, Italy e-mail: [email protected] D. Radice Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA, USA Department of Physics, The Pennsylvania State University, University Park, PA, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_21

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Non-axisymmetric Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collapse to Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisotropic Neutrino Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quark Deconfinement Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-messenger Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutrino Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

We summarize our current understanding of gravitational wave emission from core-collapse supernovae. We review the established results from multidimensional simulations and, wherever possible, provide back-of-the-envelope calculations to highlight the underlying physical principles. The gravitational waves are predominantly emitted by protoneutron star oscillations. In slowly rotating cases, which represent the most common type of the supernovae, the oscillations are excited by multi-dimensional hydrodynamic instabilities, while in rare rapidly rotating cases, the protoneutron star is born with an oblate deformation due to the centrifugal force. The gravitational wave signal may be marginally visible with current detectors for a source within our galaxy, while future third-generation instruments will enable more robust and detailed observations. The rapidly rotating models that develop non-axisymmetric instabilities may be visible up to a megaparsec distance with the third-generation detectors. Finally, we discuss strategies for multi-messenger observations of supernovae. Keywords

Gravitational waves · Core-collapse supernovae · Convection · Shock · SASI · Protoneutron star · Oscillations · Multi-messenger · Neutrinos

Introduction Core-collapse supernovae (CCSNe), the spectacular explosions of massive stars, play an important role in the evolution of the universe. As the explosive burst sweeps through the interstellar medium, it disseminates newly synthesized elements, profoundly influencing star formation and evolution. They give birth to neutron stars and, in some cases, black holes (BHs). Despite their importance, and after decades of significant research progress, there are still many things that are not understood about CCSNe.

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In CCSNe, all the four fundamental forces of nature cooperate in a particular way to produce an explosion. The collapse of the core to a protoneutron star (PNS) releases ∼1053 erg gravitational binding energy. Most of it (∼99%) escape as neutrinos, while ∼1% goes into the kinetic energy of the explosion. Within 1 s after core collapse, powerful aspherical flows develop in the central few hundred kilometers, a region often referred to as the supernova central engine. Besides playing a crucial role in powering the explosion, these flows generate strong gravitational waves (GWs) with energies up to ∼1046 −1047 erg. Hours later, the explosion propagates to the stellar surface, instigating a burst of photons across the electromagnetic (EM) spectrum with a total energy of around ∼1049 erg. The absolute majority of the supernovae have so far been observed only through their EM signature. While this provided a wealth of information about the supernovae, there is a limit to what photons alone can tell. This is because the EM burst forms at the outer edge of the star, so it contains limited information about the inner regions, where the CCSN central engine is located. Since gravitational waves are generated by the aspherical motion in the inner regions, they contain information about the dynamics of the central engine. In addition, neutrinos carry information about the thermodynamic conditions at the surface of the PNS. Therefore, observations of GWs and neutrinos will enable us to probe the central engine in a completely new way. In particular, this may allow us to extract information about the mechanism that produces the explosion. Moreover, such an observation may reveal the rotation and the structure of the innermost regions of the star. In this chapter, we summarize our current understanding of GW emission from CCSNe with a focus on established results. As we discuss below, most of what we know about CCSN GWs comes from sophisticated multi-dimensional simulations. Wherever possible, we will complement these results with simple back-of-theenvelope calculations to highlight the underlying physical principles. We also briefly discuss GW detection strategies focusing on the potential provided by a possible multi-messenger observation. Finally, we outline the main gaps and limitations in our current understanding of the subject.

Basic Overall Picture In this section, we present a brief and basic overview of CCSNe and the associated GW emission mechanisms.

The Road to Core Collapse Stars spend most of their lifetime in the main sequence phase, slowly burning their hydrogen fuel into helium. For a star with initial mass M, this phase lasts

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∼1010 (M/M )−2.5 years, where M is the solar mass [1]. After exhausting hydrogen, the subsequent stages of nuclear burning proceed at much faster pace, producing heavier and more bound nuclei. Due to their lower temperatures, the least massive stars evolve into white dwarfs (WDs), as electron degeneracy pressure prevents the star from achieving sufficient densities and temperatures to ignite their carbon oxygen cores. The stars with initial mass 8−9M evolve into carbon oxygen white dwarfs, while the stars with 8M  M  9M evolve into oxygenneon-magnesium (ONeMg) WDs. The exact mass limits that separate different evolutionary paths are not known. They depend on a number of free and poorly constrained parameters such as metallicity, mass loss, convection, and interaction with a binary companion [2]. Slightly more massive stars (9M  M  10M ) may also develop ONeMg cores, but instead of evolving into ONeMg WDs, they may produce supernovae. The electrons in the cores become degenerate and get captured by Ne and Mg nuclei. This depletes pressure and triggers collapse. Unless thermonuclear burning of O can stop the infall, the core collapses to a PNS, while the shock launched at bounce expels the stellar envelope and produces an electron-capture supernova (ECSNe). Alternatively, if the thermonuclear burning is strong enough to overpower the collapse, the star may undergo a thermonuclear explosion and leave behind an iron-rich white dwarf [2]. For massive stars with initial masses 9M  M  100M , nuclear burning proceeds all the way beyond silicon burning, forming an iron core. At T ∼ 1010 K, as a result of collision with high-energy photons, iron nuclei dissociate into alpha particles and free nucleons. This absorbs thermal energy, leading to further contraction of the core. In addition, nuclei and free protons capture electrons, further reducing pressure. This triggers dynamical collapse of the core. The stars with mass  100M follow a different path. After central carbon burning, the temperature becomes high enough that photons start producing e+ and e− pairs, converting thermal energy to rest mass. This reduces adiabatic index below 4/3, which triggers gravitational collapse. This collapse is expected to proceed until BH formation for stars with mass M  260M . In stars in the intermediate mass range (100M − 200M ), the thermonuclear reactions may be strong enough to overcome gravitational collapse and power pair-instability supernovae, which might explode with energies as high as ∼1053 erg [2].

Core Collapse and Road to Explosion Once triggered, the dynamical collapse of the core proceeds with an accelerating pace on a free-fall timescale ∼0.3 s. During collapse the core splits into two parts. The outer parts plunge supersonically, while the inner core descends with subsonic speed. Upon reaching supranuclear densities, nuclear matter stiffens, abruptly halting the collapse. The inner core bounces, launching a shock wave into the still-infalling outer core. As it progresses outward, the shock loses its energy to dissociation of iron nuclei, turning into a stalled accretion shock at ∼150 km within

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∼10 ms after formation. In order to produce an explosion and leave behind a stable neutron star, the shock has to “revive” within a few hundred milliseconds and expel the infalling outer shells. Otherwise, the stellar matter keeps piling up on top the PNS, which eventually pushes the PNS beyond its stability limit, leading to BH formation. How exactly the shock revives is a topic of an active research. According to our current understanding (recently reviewed by [3, 4]), the explosion mechanism depends on a number of factors, with rotation perhaps being the most important differentiating factor. As the hot PNS cools and contracts, it emits ∼1053 erg of its binding energy as neutrinos with ∼1052 erg/s luminosity for ∼10 s. Some of these neutrinos are absorbed behind the shock. In the so-called gain region, the neutrino heating exceeds cooling. This drives neutrino-driven hot-bubble convection. In addition, in some cases, the shock may become subject to the standing-accretion shock instability (SASI), which drives large-scale non-radial oscillations of the shock [5]. When developed, these instabilities enhance neutrino heating, which energizes the shock to expel the stellar envelope and produce a supernova explosion with energies up to ∼1051 erg energy. In rapidly rotating models, magnetic fields transfer the rotational kinetic energy of the PNS to the shock. For a millisecond period PNS, the rotational energy is ∼1052 erg, which may empower the supernovae with these energies (aka hypernovae) and may even lead to long gamma-ray bursts (e.g., [3, 4] for recent reviews). While modern simulations of CCSNe confirm this baseline picture, there are still many open questions. The number of 3D simulations is rapidly growing, but a comprehensive understanding of the dependence of the explosion properties on model parameters (including progenitor mass and rotation as well as physical and numerical model assumptions) is still in its infancy. Although it is clear that convection and SASI help the explosion, we do not know how strongly these instabilities develop across different models. For example, there are models that produce explosions but do not exhibit any SASI (e.g., [6]). As a result, despite all the impressive progress, it is not entirely clear which models explode (and if so, how strongly) and which models do not. Due to the close connection between the aspherical motion behind the shock, which is the main source of the GW emission, and the explosion dynamics, there is a potential for probing explosion mechanism of CCSNe using the GWs. A more detailed discussion of the explosion mechanism of CCSNe is beyond the scope of this chapter (see [4] for a recent review and references therein).

Generation of Gravitational Waves The GW strain h emitted by a source at a distance D can be approximately estimated using the quadrupole formula [1] h=

2G d 2 Q , Dc4 dt 2

(1)

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where G is the gravitational constant, c is the speed of light, and Q is the mass quadrupole moment of the system. To an order of magnitude, for an object of mass M undergoing quadrupole motion with speed υ,

hD ∼ εrSch

υ2 , c2

(2)

where ε is a measure of non-sphericity of the source (= 0 for spherical source) and rSch = GM/c2 is the Schwarzschild radius associated with the mass M. The GW luminosity can be estimated as [1]: c5  rSh 2  υ 6 dEGW ∼ ε2 , dt G R c

(3)

where R is a characteristic size of the source. Thus, in order to produce strong GWs, we need a compact object with fast aspherical motion. For CCSNe, this means that the PNS should produce most of the GWs. This intuition is supported by modern CCSN simulations: while the asymmetric flow outside the PNS does produce some GWs, the PNS dynamics is responsible for most of the emitted GWs. Assuming ε = 0.1, υ/c ∼ 0.1, and total emission time of ∼1 s, we obtain crude upper limits for the GW strain hD ∼ 300 cm and the total emitted energy EGW ∼ 1047 erg (∼ 10−7 M c2 ) [7]. The dynamics of PNS in rapidly rotating models is vastly different than that in slowly or non-rotating models. In rapidly rotating models, the centrifugal force leads to slower bounce along equatorial plane than along the rotation axis. As a result, the PNS is born with oblate perturbation. This triggers PNS oscillations that last for ∼10−20 ms [8]. In addition, the PNS may be subject to rotational non-axisymmetric instabilities, which deform the PNS into a rotating nonaxisymmetric shape. Rotation of non-axisymmetric object produces long-lasting GW emission [9]. In slowly or non-rotating models, the centrifugal force has little dynamical impact. Instead, convection and SASI perturb the PNS and excite its oscillations [10]. In the following, we discuss these two cases separately. Furthermore, there are sub-dominant (and/or somewhat exotic) mechanisms for generating GWs: besides perturbing the PNS, convection and SASI can directly emit GWs. The variations in neutrino luminosity in the different regions produce anisotropic flux of neutrinos. The dense matter inside PNS may undergo phase transition, which may lead to a “mini” second core-collapse of the PNS. Finally, if the PNS accumulates more mass than it can support, it collapses to a BH. Each of these processes can emit GWs and we will discuss them in more detail later.

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Non-rotating and Slowly Rotating Case Most stars are expected to rotate slowly (e.g., [12]). As mentioned above, such stars remain quasi-spherical during bounce and early (10 ms) post-bounce phase without emitting any GWs. Instead, the multi-dimensional hydrodynamic flows that develop later lead to GW emission. The shock wave, launched at bounce, leaves behind a negative gradient of entropy while traveling outward within the first few milliseconds. This triggers prompt convection in the post-shock region ∼10 ms after bounce. Prompt convection lasts for 50 ms and is followed by a quiescent period of 100 ms. Afterward, the neutrino heating in the gain region drives the neutrino-driven convection behind the shock, while in some models the shock may develop SASI oscillations. The neutrino emission from the PNS creates negative gradient of lepton number in the region 10  r  25 km, driving vigorous PNS convection for few seconds after formation. These instabilities, schematically depicted in Fig. 1, perturb the PNS and excite

accreting flow

· M

ν

ν

ν ρ∼

neutrino-driven convection

km

10 8 g/

cm 3

on

∼

ν

ga

in

R

ν

oc

sh

re gi

0

15

e av kw

∼

ρ∼

m

PNS convection

10 1

10 9 g/

cm 3

0

PN S

R

k 40

ρ∼

g/

cm 3

ρc ≳ 3 × 1014 g/cm3

Fig. 1 The schematic depiction of the CCSN central engine for slowly rotating case. The neutrinos emitted by the proto-neutron star (PNS) drive neutrino-driven convection in the gain region. The diffusion of neutrinos out of the PNS leads to negative radial gradient of the lepton number, driving PNS convection. The SASI drives large-scale oscillations of the shock with low  number

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its oscillations, which then produce GWs. Once the system starts to transition to explosion, asymmetric shock expansion generates a non-oscillatory shift in the GW strain, while any material that may fall back onto PNS generates additional highfrequency GWs. Since convection and SASI develop from stochastic perturbations, the resulting dynamics and GW signal are expected to have a stochastic time variability. The impact of these instabilities varies from model to model, leading to distinct GW signatures. For this reason, in order to cover a wide range of scenarios, below we discuss eight different models with (initial) progenitor masses ranging from 9M to 60M that were recently simulated by [6, 11] with sophisticated 3D neutrino-hydrodynamics and microphysics (Here, we focus on results only from 3D simulations of CCSNe. In 2D axisymmetric simulations, energy in large-scale motion is artificially large due to inverse energy cascade of 2D turbulence (e.g., [4] for a recent review). Also, SASI is restricted to axisymmetric modes in 2D. These effects lead to inaccurate predictions of GW signal.). Figure 2 shows the radial density profiles of these progenitors. This reveals that the progenitors have vastly different structures, ranging from models with little mass above iron core (e.g. 9M model) to models with significant mass above the core (e.g. 25M model). This leads to a rich variety of outcomes. As we can see in the evolution of the average shock radius versus time, shown in Fig. 3, all of the models, except the 13M model, transition to explosion within 200−500 ms after bounce. All models exhibit strong neutrino-driven convection, but only 13M and 25M models develop SASI. Figure 4 shows the energy of emitted GWs as a function of time. Comparing this to the shock radius evolution, we can see that there is no strong correlation between the shock radius and the GW

1011 13M 19M 25M 60M

1010

ρ [g cm−3 ]

109 108 107 106 105 104 0.0

9M 10M 11M 12M 0.5

1.0

1.5

2.0

2.5

Enclosed mass [M ] Fig. 2 The radial profile of density of the progenitor models studied by Radice et al. [11]

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Fig. 3 The shock radius as a function of time for models. (Reprinted from Radice et al. [11]. c AAS. Reproduced with permission) 

c AAS. Reproduced Fig. 4 GW energy as a function of time. (Reprinted from Radice et al. [11].  with permission)

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energy. For example, the 9M model, which undergoes fast transition to explosion, radiates the smallest amount of energy in GWs (∼4×10−11 M c2  7.2×1043 erg) among these models. This is because in this model, as well as in other lowmass CCSN (and ECSN) progenitors, little mass is available outside of the iron core (cf. Fig. 2). The shock and post-shock flow remains quasi-spherical in the post-bounce phase. As a result, little GW emission takes place, especially beyond ∼300 ms after bounce, as we can see in the plot of GW strain versus time, shown on left panels of Fig. 5. On the other hand, the 13M model, which does not exhibit explosion, experiences strong convection and SASI. The plot of GW strain versus time (cf. Fig. 5) reveals strong high-frequency GW emission throughout postbounce evolution within the time span indicated in this figure. This leads to strong GW emission with  10−9 M c2  2 × 1045 erg energy. Assuming a distance of 10 kpc to the source and a perfect knowledge of waveform, the spectral energy density of GWs is shown on the right panels of Fig. 5 together with the sensitivity curves of the Advanced LIGO and the proposed Einstein telescope in the D configuration. The corresponding optimal SNR for Advanced LIGO ranges from ∼1 for the 9M model to ∼10 for the 19M model. For the third-generation detector, the corresponding SNRs range from ∼10 to ∼100. This suggests that third-generation detectors are necessary for a more reliable detection of GWs from CCSNe across our galaxy. Below, we discuss each of the GW emission mechanisms in more detail and identify the physical mechanisms responsible for them.

Early Quasi-periodic Signal All models experience the early quasi-periodic signal that develops between ∼10 ms and ∼50 ms after bounce, as can be seen in the plot of GW strain as a function of time shown in Fig. 5. It is also visible as a track with ∼100 Hz frequency in the GW spectrogram shown in Fig. 6 for the 25M model. The origin of this signal has been ascribed to a number of processes, including the prompt post-shock convection. The contributions of the shock oscillations [13] and acoustic waves in the postshock region have also been emphasized [14]. Depending on model, the GW strain of the early quasi-periodic signal can reach up to hD ∼ 2 − 5 cm, but due to their short duration, their contribution to the overall signal is somewhat modest (cf. Fig. 4).

PNS Convection The PNS convection has been invoked as a mechanism for exciting PNS oscillations and generating GWs [15]. However, the 9M model, despite exhibiting vigorous convection throughout its evolution [16], emits little GWs in the wake of runaway

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Fig. 5 Left panels: GW strain as a function of time for eight models of [11]. Right panels: The spectral energy distribution of the GW signal together with the sensitivity curves of Advanced LIGO and ET-D detectors for sources at 10 kpc distance. (Reprinted from Radice et al. [11]. c AAS. Reproduced with permission) 

shock expansion at ∼300 ms after bounce (cf. Fig. 5). This means that the PNS convection does not have significant impact on the GW emission, at least for this model. In the rest of the models, it is hard to make a definite statement because the neutrino-driven convection and SASI overlap in time with the PNS convection, at least within the time span covered by the simulations. That said, evidence of stronger contribution of the PNS convection to the GW signal was suggested in [15,17]. This discrepancy points to potential model dependency of the PNS convection, which call for more detailed studies [18].

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Neutrino-Driven Convection As mentioned above, neutrino heating drives neutrino-driven convection in the postshock region (cf. Fig. 1). Since this occurs in an accreting flow, in order to develop, the convection has to grow faster than the advection timescale. Otherwise, it will be accreted out from the gain region before it can develop. This condition can be formulated in terms of the parameter  χ=

Rsh

Rg

Im N dr, |υr |

(4)

where Im N is the imaginary part of the Brunt-Väisälä frequency N. Convection was found to develop for χ  3 [19]. The size of the largest eddies is determined by the width of the gain region: a convective region with height H = Rsh − Rg can fit  pairs of circular vortices of size H along the average circumference of the region. This results in ∼

π Rsh + Rg . 2 Rsh − Rg

(5)

This shows that  is smaller for larger shock radii. When shock is small, eddies with  in range 4−8 have been observed. When shock is large and expanding, large-scale modes with  = 1 and  = 2 become dominant [4]. The contribution of the GWs emitted directly by neutrino-driven convection to the overall signal from CCSNe is somewhat minor. However, as mentioned earlier, convection creates funnels of accretion onto PNS, which excites PNS oscillations that generate stronger GWs.

SASI SASI represents non-radial oscillations of the shock. It develops due to the so-called advective-acoustic cycle. Any perturbation of the shock generates vorticity waves in the post-shock region, which are then advected by the flow toward the PNS. As it approaches the PNS, it encounters steep gradient of density. The vorticity wave distorts the isodensity surfaces of the PNS, producing pressure perturbations and generating acoustic waves. These acoustic waves travel outward and reach the shock. This perturbs the shock more, generating new vorticity and further amplifying the oscillations of the shock [20]. (A purely acoustic mechanism has also been considered [17], but latest detailed analyses favor the advective-acoustic mechanism for SASI. See, e.g., [4] for a recent review.) Due to the advective-acoustic cycle, one SASI period is a sum of advection and sound-crossing time between the shock and PNS. Since post-shock flow is subsonic, the advection timescale is much larger than the sound-crossing time. Using this, we can obtain an approximate estimate for the SASI period

21 Gravitational Waves from Core-Collapse Supernovae



Rsh

TSASI ∼

RPNS

921

dr . |υr |

(6)

The advection velocity scales as υr ∼ υPS r/RPS , where υPS ∼ the post-shock velocity [4]. This yields

TSASI

√

GMPS /RS /7 is

   3/2   Rsh Rsh Rsh Rsh ∼ 20 ms . ∼ ln ln υPS RPNS 100 km RPNS

(7)

This estimate is consistent with the frequency we observe in the spectrogram of the GW signal shown in Fig. 6. It also shows that the frequency of the GWs from SASI should decrease with increasing shock radius. This decrease can be identified in Fig. 6 as the shock starts to expand at around ∼400 ms after bounce. Further expansion of the shock leads to diminishing of the SASI activity as the advectiveacoustic cycle responsible for SASI cannot keep up with rapidly expanding shock. Using Eq. (2), and assuming gain region mass of 10−2 M , υ/c ∼ 0.1, and ε = 0.1, we obtain hD ∼ 1 cm, which is consistent with the results of numerical simulations [17]. In addition, similarly to neutrino-driven convection, SASI modulates accretion onto PNS, which excites PNS oscillations and generates more GWs.

25 M

1000

−15

800

−20

s

on

600

ti illa

−25

c

s SO

400

PN

0

−30

SASI

200

0.1

0.2

0.3

0.4

0.5

0.6

ˆ eff (f ) (D = 10 kpc)] 10 log10 [1022 h

f [Hz]

1200

−10

−35

t − tbounce [s] Fig. 6 Spectrogram of the GW signal for the 25M model. The white dots represent the eigenfrequencies of the quadrupolar f - and n = 1, 2, g-modes of the PNS obtained from the linear c AAS. Reproduced with permission) perturbation analysis. (Reprinted from Radice et al. [11]. 

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Protoneutron Star Pulsations Despite the continued accretion and the presence of hydrodynamics instabilities, the inner region of a CCSN, encompassing the stalled shock and the PNS, can be considered as stationary with good precision, e.g., [21]. In particular, the density stratification is close to hydrostatic behind the shock wave during the shock stagnation phase and at all times in the PNS. For this reason, the PNS oscillations can be described using linear perturbation theory with good accuracy. In the linear theory, the Lagrangian displacement field ξ (t, r ), describing the deformation of the PNS, obeys the equation of motion ∂ 2ξ + Cξ = F , ∂t 2

(8)

where F is the perturbing force and C is a differential operator. To illustrate the basic ideas, we consider the non-rotating, spherical, Newtonian case. We also restrict ourselves to adiabatic perturbations, which means that we neglect heat and compositional changes in the fluid elements as they are displaced. In this case, C is a self-adjoint operator. Its exact expression can be found in, e.g., [22], which also discusses the more general case of rotating stars. For our purposes it is only important to emphasize that C depends only on the unperturbed density, pressure, and composition of the PNS. See Ref. [23] for a review of the general-relativistic formalism and Ref. [24] for a discussion on the impact of non-adiabatic effects. In the case we are considering, it is possible to show that the eigenfunctions associated with the time-independent eigenvalue problem C ξ α (rr ) = ωα2 ξ (rr ),

(9)

with appropriate boundary conditions (more below), form a complete set. They can be normalized as  (10) ξ ∗α  (rr ) · ξ α (rr )ρ(rr )d3r = MR 2 δα  α , where M and R are the total mass and radius of the background solution. This means that any pulsation pattern of the star can be decomposed as ξ (t, r ) =



Aα (t)ξξ α (rr ).

(11)

α

Substituting this expansion into (8), we find the evolution equation for the amplitudes d2 Aα (t) 1 + ωα2 Aα (t) = dt 2 MR 2



ξ ∗α (rr ) · F (t, r )ρ(rr )d3r .

(12)

21 Gravitational Waves from Core-Collapse Supernovae

923

In other words, linear perturbation theory shows that the PNS oscillations can be written as superposition of normal modes, each endowed with its characteristic frequency, and that the normal modes behave like a system of uncoupled forced harmonic oscillators. An important consequence is that the GW spectrum from CCSNe is predicted to contain discrete frequencies, related to the eigenvalues of the operator C , that depend only on the structure of the PNS and not on the nature of the forcing. This suggests that GWs from CCSNe could be used to probe the interior structure of the PNS as it cools and contracts. It is important to remark that the decomposition in normal modes is possible even in the presence of rotation. However, this decomposition is not strictly valid in the GR formalism, in which the modes lose energy due to GW emission (quasi-normal modes) and do not form a complete basis [23]. Some of the first studies of the normal modes of PNSs with realistic temperature and compositional profiles were presented by Refs. [25–27]. With the availability of long-term 3D CCSN simulations with realistic microphysics, it is now also possible to validate perturbation theory with nonlinear simulations [11, 28, 29]. The typical starting point for the perturbation theory is to decompose the Lagrangian displacement in vector spherical harmonics:    ∂ 1 ∂ ξ = ξ r eˆ r + ξ ⊥ eˆ θ + eˆ ϕ Ylm (θ, ϕ). ∂θ sin θ ∂ϕ

(13)

Substituting this expression into Eq. (9) yields a set of decoupled nonlinear onedimensional eigenvalue problems, one for each spherical harmonic l, m. The formulas are particularly simple in the Cowling approximation, that is, when perturbations on the gravitational potential are neglected. This is a good approximation especially for high-order modes, and it is useful to illustrate the key ideas. With this approximation Eq. (9) reduces to [30]

 L2l 1 d(r 2 ξ r ) δp g r − 2ξ + 1 − 2 = 0, r 2 dr cs ω ρcs2

(14)

1 dδp g + 2 δp + (N 2 − ω2 )ξ r = 0, ρ dr ρcs where g, cs , p, ρ are the local gravitational acceleration, sound speed, pressure, and density of the background star, δp is the Eulerian pressure perturbations δp = −γp∇ · ξ − ξ · ∇p,

(15)

γ = (∂ ln p/∂ ln ρ)S,Ye being the adiabatic index, and Ll and N are the Lamb and Brunt-Väisälä frequencies defined as L2l =

l(l + 1)cs2 , r2

 N2 = g

1 d ln p d log ρ − γ dr dr

 .

(16)

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These equations are typically closed with regularity boundary conditions at the center (ξ r = 0) and vanishing pressure perturbation at the surface of the PNS (δp = 0); see [31] for a discussion of possible boundary conditions. These boundary conditions ensure that the operator C is self-adjoint and that the resulting modes form a complete orthogonal basis. We can gain some physical intuition on these equations by solving them in the short wavelength limit (WKB approximation) in which ξ r , δp ∼ exp(ikr r).

(17)

Substituting this expression into Eq. (14), we obtain the (approximate) dispersion relation kr2 

(ω2 − L2l )(ω2 − N 2 ) . ω2 cs2

(18)

Oscillations can propagate in the radial direction if kr is real; otherwise the waves are exponentially damped (evanescent). There are two possibilities: 1. ω2 > L2l , N 2 : these are called p-modes, because their restoring force is pressure. 2. ω2 < L2l , N 2 : these are called g-modes, because their restoring force is gravity. For large frequencies ω  1, kr ∼ cωs which shows that higher-order p-modes have higher oscillation frequencies than lower order p-modes. Conversely, for small frequencies ω 1, kr ∼ l(l + 1)N 2 /(ω2 r 2 ), which shows that higher-order g-modes, instead, have lower oscillation frequencies than lower-order g-modes. The frequencies of the p-modes are typically larger than that of the g-modes. The two families of modes are separated by the so-called f -mode, which is a mode with no radial nodes (ξ r , δp = 0 for 0 < r < R). If rotation is present, modes whose restoring force is the Coriolis force are also present. These are the so-called r-modes. Finally, in GR a new family of modes, the so-called w-modes, is also present. These are modes with characteristic frequencies of several kHz that originate from oscillations in the spacetime metric that have no classical counterpart [23]. It is important to emphasize that, in the case of radial oscillations (l = 0), only the p-modes are present, since the Lamb frequency is zero. Figure 7 shows the characteristic evolution of N and L2 for a PNS taken from [32]. These calculations are for a model simulated in 1D (assuming spherical symmetry) for which the shock stalls and explosion is not successful. However, the structure of the PNS depends only weakly on the outcome of the supernova explosion on the timescales relevant for the GW emission, so this model is representative. It is possible to see that there are two regions where g- and p-modes can propagate: one close to the surface and one in the inner core of the PNS. They are separated by a region that is unstable to convection, corresponding to the PNS convection region [32]. Because of the presence of this convective layer, the g-modes of the inner core are trapped and cannot be excited by perturbations acting

p-waves

10

10 349 ms

evanescence

1

1

N L2 [103 rad s−1 ] N,

925

225 ms

N, N L2 [103 rad s−1 ]

21 Gravitational Waves from Core-Collapse Supernovae

10 599 ms

1

1 10

100 Radius [km]

10

N L2 [103 rad s−1 ] N,

10 475 ms

N, N L2 [103 rad s−1 ]

g-waves

100 Radius [km]

Fig. 7 Propagation diagram for p- and g-modes in a contracting PNS. The solid regions represent regions where the modes can propagate. The vertical black line denotes the position of the shock. There are two regions where modes propagate separated by a convective region where N 2 < 0 (|N | c Gossan et al. Reproduced is shown as a green line there). (Adapted from Gossan et al. [32].  with permission)

on the surface of the star. It is important to remark that the propagation diagram offers only a partial description of the full spectrum of oscillations, since the WKB approximation is only valid for large mode wavenumbers. A full calculation is needed to determine the full spectrum of modes from PNSs. Ab initio CCSN simulations are in good agreement with predictions from linear theory. Simulations show that most of the GW energy is radiated at a frequency coincident with that of a quadrupolar low-order g- or the l = 2 f -mode of the PNS [10, 11, 14, 15, 33]. Additionally, accretion modulated by the SASI, if present, is found to also excite quadrupolar oscillation modes in the PNS and contribute to a low-frequency component of the signal [11, 15]. Figure 6 shows the GW signal of the 25 M of [11]. The signal is characterized by a dominant feature that evolves with time and as the PNS contracts. It evolves from ∼400 Hz at ∼150 ms to ∼1, 100 Hz at ∼600 ms after bounce. This corresponds to the eigenfrequency of the f -mode of the PNS. Other models considered here exhibit a similar behavior.

Explosion Phase Signal Aspherical shock expansion during explosion represents anisotropic flow of matter. This produces non-oscillatory, slowly-varying changes of the quadrupole moment. The resulting GW signal, also known as the GW memory effect, is visible as an

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offset in the GW waveform during the explosion phase, as we can see for, e.g., 12M model. Depending on the model, the offset can be as high as hD ∼ 3 cm. In contrast to this model, the shock in the 9M model remains quasi-spherical during the early explosion phase. As a result, this model does not exhibit any offset in the GW strain. In addition to the offset, the rest of the models exhibit high-frequency variations even after transitioning to explosion. This behavior is caused by the aspherical expansion of the shock, which produce funnels of accretion onto the PNS, triggering its oscillations and producing GWs.

Rapidly Rotating Case While most stars are expected to rotate slowly, in order to explain the energetics of the most powerful ∼1052 erg explosions, some stars must be rapid rotators. Below, we discuss the GW signature of such stars.

Bounce and Ring-Down Signal Due to centrifugal support, the core collapse is slower along the equatorial plane than along the rotation axis. As a result, the PNS is born with axisymmetric oblate  = 2 deformation at bounce [35]. The ensuing PNS oscillations last for ∼10−20 ms. Due to the axisymmetric geometry of the initial perturbation, the PNS pulsations remain axisymmetric in this phase (hence, 2D axisymmetric simulations can accurately predict the signal in this phase). In contrast to the perturbations in slowly or non-rotating models, the centrifugal deformation is not stochastic. As a result, the bounce GW signal in rapidly rotating case is “deterministic” for a given set of model parameters (and for a given model of nuclear and neutrino physics). The degree of centrifugal deformation, and thus the strength of the GW signal, grows with rotation. However, at extremely rapid rotation, the centrifugal support slows the collapse and prevents PNS from reaching higher densities. This limits the GW amplitudes. We can see this trend in Fig. 8, which shows the GW strain along equatorial plane (left panel) as a function of time for three different models with precollapse central angular velocity of 2.0, 5.5, and 9.0 rad/s for a 12M progenitor [34]. The peak GW frequencies of these models are about 800, 750, and 372 Hz [34]. In these models, the rotation respectively has little, strong, and dominant impact of the dynamics of the system. We refer to these models as slowly, rapidly, and extremely rapidly rotating models, respectively. As expected, the slowest rotating model produces little GW signal at bounce. However, it develops prompt convection within ∼10 ms of bounce, which can be seen in the GW signature in Fig. 8. The rapidly rotating models do not exhibit prompt convection because these models have strong positive gradient of specific angular momentum in the post-shock region: a fluid element from an inner region cannot easily rise due to their smaller centrifugal support. This hinders convection. In the model with extreme rotation, we can see that the bounce spike is wider, implying slower dynamics and lower frequencies.

21 Gravitational Waves from Core-Collapse Supernovae

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Fig. 8 GW strain (left panel) and central density (right panel) as a function of time during late collapse, bounce, and early post-bounce phase for three models with slow, rapid, and extremely rapid rotation. (The data is produced by [34])

Due to the axisymmetric geometry of the bounce and ring-down pulsations, the GW signals are only visible for observers close to the equatorial plane, and no signal is emitted along the rotation axis [36]. The evolution of the central density, shown on the right panel of Fig. 8, reveals that the central density is lower in models with faster rotation. This is expected as stronger centrifugal support makes the PNS less compact. We can also see that the slowly rotating models do not exhibit strong oscillations in ρc in the post-bounce phase. That is because, due to small centrifugal support, this model is born with little oblate deformation. Hence, the PNS settles to a quasi-equilibrium configuration within ∼1 ms after bounce. In rapidly rotating models, ρc exhibits strong oscillations as the PNS is born with large centrifugally induced oblate deformations. We can get a more detailed understanding of the impact of rotation if we look at a large sequence of models with varying degrees of rotation and its distribution. This has been attempted by [34], who studied a set of 98 rotational configurations of the 12M model using the two-parameter law for angular velocity:    2 −1 Ω( ) = Ωc 1 + A

(19)

where Ωc is the pre-collapse central angular velocity,  is the distance from the rotation axis, while A is a characteristic distance from the rotation axis over which the angular velocity decreases by a factor of 2. They considered five different values of A ranging from 300 km, which represents extremely strong differential rotation, to 10, 000 km, which represents weak differential rotation. Figure 9 shows the maximum GW strain produced at bounce as a function of the ratio T /|W | of the rotational kinetic energy T and potential binding energy |W | of the inner core at bounce. Because this parameter is a ratio of the two energies, it characterizes the dynamical importance of rotation. As we can see, the GW strain

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Fig. 9 The maximum GW strain produced by core bounce as a function of the rotation parameter T /|W | of the inner core at bounce for models with different degrees of rotation. Here, A is the degree of differential rotation (cf. Eq. 19). The GW waveforms are produced by [34]

h increases linearly with T /|W | for T /|W |  0.06−0.09, but the growth slows for larger T /|W |, and then h starts even decreasing with T /|W | for T /|W |  0.17. Also, for T /|W |  0.06−0.09, the GW signal is not sensitive to the degree of differential rotation A, while for higher T /|W |, the GW signal depends on A. The linear increase of h with T /|W | for T /|W |  0.06−0.09 is easy to understand. Since |W | ∼ GM 2 /R and Ekin ∼ T , the estimate (2) leads to hD ∼

GMΩ 2 R 2 (GM)2 T , ∼ 4 c c4 R |W |

(20)

Thus, the GW signature of the bounce and ring-down oscillations is mostly determined by the mass, radius, and the rotation parameter of the inner core. The fact that, for T /|W |  0.06−0.09, the GW strain is so well approximated by simple relation (20) suggests that the dynamics is governed by the fundamental quadrupole mode oscillations of the PNS. This is supported by detailed oscillation mode analysis [8,37]. The mode frequency is comparable to the dynamic frequency of the PNS, fpeak ∼

1 Gρc , 2π

(21)

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where ρc is the PNS central density [8]. For a typical PNS, fpeak ∼ 750 Hz. For more rapid rotation, T /|W |  0.06−0.09, the h vs. T /|W | curve in Fig. 9 deviates from a straight line. The √ peak frequency of the signal fpeak becomes larger than the dynamical frequency Gρc /2π . More specifically, fpeak now scales as fpeak

1 ∼ 2π

√

Gρc + Ωmax , 2

(22)

for 0.06−0.09  T /|W |  0.17, where Ωmax is the highest angular velocity achieved outside of radius of 5 km [8]. The transition happens when Ωmax √ becomes larger than Gρc , which is a regime where Coriolis force becomes dynamically important. These observations suggest that the dynamics of the PNS is not dominated by the fundamental quadrupole mode anymore. Instead, the PNS oscillations may be transiting to a mode (or modes) supported by the Coriolis force. At extremely rapid rotation of T /|W |  0.17, the dynamics is dominated by the centrifugal force. This leads to complex PNS behavior involving multiple modes with comparable amplitudes. The exact nature of the PNS oscillation modes in this regime is yet to be established. Despite its short ∼20 ms duration, the GW signal from rotational bounce and post-bounce ring-down oscillations is detectable up to a distance of ∼50 kpc with current detectors [38]. Once detected, for rapidly rotating models with optimum orientation, it is possible to extract parameters of the CCSN central engine, such as rotation [34]. A possibility of extracting the parameters of high-density nuclear matter has also been explored [8].

Non-axisymmetric Instabilities Sufficiently rapidly rotating stars may be subject to non-axisymmetric instabilities. These instabilities may last for many dynamical timescales, leading to long-lasting GW emission. There are different types of non-axisymmetric instabilities. (We focus only on the instabilities that may occur in the context of CCSNe. Outside this context, many other types of instabilities have been discussed, especially in the context of accretion disks.) Based on their growth rate, instabilities can be divided into two sub-groups: secular and dynamical. The secular instability is driven by dissipation (viscosity or GW emission) and develops on a dissipation timescale [1], which is ∼1 s or longer for CCSNe. The viscosity drives a non-axisymmetric instability for the  = −m modes if their frequencies pass through zero in the corotating frame [39]. For modes that are counter-rotating in the frame of the star, but appear corotating to a faraway inertial observer, the GWs drive an instability via the Chandrasekhar-FriedmanSchutz mechanism. GWs extract the stellar angular momentum by making the mode angular momentum increasingly negative. The f , r, and w modes have been discussed in this context (see, e.g., [39] for a recent review). Within ∼1 s after formation, PNSs undergo changes that are faster than the secular timescale

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(e.g., accretion of mass and angular momentum), which may interfere with the development of the instability. Also, current CCSN simulations do not cover timescales beyond ∼1 s. For this reason, such instabilities have yet not been observed. Nevertheless, one cannot exclude that these instabilities may develop in a later phase, when the PNS evolves at a slower rate. The dynamical instability develops on the dynamical timescale (Gρ)−1/2 ∼ 1 ms. Two sub-types have been discussed in the context of CCSNe. These are often called high- and low-T /|W | instabilities. The former, also known as dynamical bar mode instability, develops in stars when T /|W | becomes larger than ∼0.24−0.27 depending on the compactness of the star [40]. The low-T /|W | instability develops at lower T /|W |, but requires strong differential rotation [41]. A recent comprehensive review of the instabilities can be found in [39]. Using the Newtonian quadrupole formula (2), we can obtain an estimate of the GW strain [36]: −21

h ∼ 6 × 10

 ε   f 2  D −1  M   R 2 , 0.1 500 Hz 10 kpc M 12 km

(23)

where ε is the ellipticity of the bar. The luminosity of GWs is 32 G 2 2 6 dEGW  I ε Ω , dt 5 c5

(24)

where I is the moment of inertia of the star with respect to the rotation axis [1]. If the deformation persists for N rotation periods, then the SNR of the signal increases by a factor of N 1/2 , significantly improving the detectability [7, 42]. The highest T /|W | observed in CCSN simulations barely exceeds 0.22 [43]. During collapse, due to conservation of angular momentum, the angular velocity increases as r −2 . This means that centrifugal force is ∝ r −3 . On the other hand, the gravity is ∝ r −2 . Therefore, for sufficiently rapid rotation, the centrifugal support becomes stronger than gravity, limiting collapse. This barrier limits the value of T /|W | to  0.22, which is below the threshold for the dynamical high-T /|W | instability. As a result, the high-T /|W | dynamical instability is unlikely to develop in CCSNe [9]. The low-T /|W | instability develops in the presence of a corotation radius, a radius where rotation matches the speed of the mode [44]. This is possible only in stars with differential rotation. The low-T /|W | instability has been observed in a number of simulations (e.g., [45] and references therein). The m = 1 or m = 2 modes are often dominant, but higher-m modes were also observed, albeit with smaller amplitudes [9]. As an example, below we discuss a model with initial mass of 70M that has been simulated by [45] until 270 ms after bounce. Before collapse, this model has a central angular velocity of 2 rad/s and T /|W | of ∼3 × 3−3 . In the post-bounce, T /|W | becomes ∼0.05. The top and bottom panels of Fig. 10 show the GW strain and spectrograms observed along equatorial plane as a function of time. The initial

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Fig. 10 GW strain amplitude (top panel) and GW strain spectrogram (bottom panel) along the equatorial plane as a function of time after bounce. On the left panel, we can easily see m = 1 deformation, while on the right panel, we can identify m = 2 deformation. (Adapted from c Shibagaki et al. Reproduced with permission) Shibagaki et al. [45]. 

Fig. 11 Deviations of density from axisymmetric average at 70 ms (left panel) and 140 ms (right panel) after bounce. The emission at  60 − 70 ms after bounce is caused by the m = 1 mode, while the signal after  120 ms is caused by the m = 2 mode. (Adapted from Shibagaki et al. [45]. c Shibagaki et al. Reproduced with permission) 

rotational bounce and ring-down signal within ∼20 ms after bounce is followed by a quiescent period that lasts until 60 ms after bounce. Between 60 and 80 ms, we see GW emission with frequency ∼400 Hz. The density deviations from axisymmetry, shown in Fig. 11 at tpb = 70 ms at the equatorial plane, reveals m = 1 deformation.

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Another quiescent period is observed between 80 and 140 ms, after which another mode develops and persists until the end of simulation at tpb = 240 ms. The density deviations at tpb = 140 ms reveal m = 2 deformation. In this phase, the GW frequency increases from ∼400 until ∼800 Hz by the end of the simulation. This is caused by cooling of the star, which leads to contraction and increase of the pattern speed of the mode. This resulting signal is easily detectable anywhere within our galaxy with current detectors. The future third generation may be able to detect up to ∼1 Mpc distances [45]. The event rate is not precisely known, but we can make rough estimate. In the local group, the CCSN rate is ∼20−80 events per 100 years [46]. Assuming that ∼1% of all CCSNe rotate rapidly enough to produce low-T /|W | instability, we arrive at ∼0.2−0.8 events per 100 years. Despite significant progress in our understanding of non-axisymmetric instabilities, there are aspects that could benefit from further exploration. In particular, more realistic simulations for a large set of progenitors will shed light on the precise conditions for the development and growth of the instability as well as on the saturation amplitude and the persistence of the non-axisymmetric modes. Here, it is crucial to accurately capture the accretion of angular momentum onto PNS as well as the transfer of angular momentum from the PNS to outer regions by neutrinos and magnetic fields. This is a subject of ongoing research.

Collapse to Black Hole When a PNS collapses to a BH, the collapsing star emits a GW spike at its dynamical frequency of ∼1 kHz, followed by quasi-normal modes of the newly formed BH at a few kHz [47]. Due to the short duration and the high frequency of the signal, the BH formation signal will be challenging to detect.

Anisotropic Neutrino Emission Anisotropic emission of neutrinos produces a flux of outgoing energy with non-uniform angular distribution, which represents a time-changing quadrupole moment. This produces a slowly varying non-oscillatory contribution to the GW strain, somewhat similarly to the way aspherical shock expansion produced an offset in the GW strain. If Eν is the total energy of emitted neutrinos and ΔEani ∼ αν Eν is energy variation due to anisotropy, the resulting signal GW strain can be estimated to an order of magnitude using (2): hD ∼

2G αν Eν c4

In the context of CCSNe, this estimate can be rewritten as [48]

(25)

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   α  Δt Lν ν hD ∼ 1.6 × 10 cm, 1s 10−2 1053 erg/s 2

(26)

Using similar argument, we can obtain an estimate for GW energy EGW,ν ∼ 10−8 αν2



Lν 53 10 erg/s

2 

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 M c 2 .

(27)

where Lν is the total neutrino luminosity. The anisotropy in neutrino emission can be produced by deformations induced by rotation and/or multi-dimensional hydrodynamic instabilities. For Lν ∼ 1053 erg/s, which is typical during the first second after bounce, and for αν ∼ 10−2 , we find hD ∼ 102 cm, which is in line with more detailed calculations [48]. This is comparable to the GW strain produced by asymmetric matter motions in CCSNe. However, the quadrupole moment of a system of anisotropically propagating neutrinos changes slowly with time. As a result, the energy of GWs is ∼10−12 M c2  1.8 × 1042 erg, which is ∼4 orders of magnitude smaller than that from the matter motion. For this reason, and due to the non-oscillatory nature of the signal, the GW signal from the anisotropic neutrino emission is unlikely to be detectable with current detectors. However, detection could be possible with the future third-generation detectors. In order to make a more definitive statement, the CCSN simulations need to cover timescales longer than ∼1 s, which will reveal more precise time evolution of the GW signal from the anisotropic neutrino emission.

Quark Deconfinement Phase Transition When the central density of the PNS is sufficiently large, the nuclear matter may undergo quark-deconfinement phase transition. The exact details of whether and how this happens are not well established. If the phase transition leads to rapid and strong pressure reduction, it may cause a “mini-collapse” of a PNS to a more compact configuration. In rapidly rotating models, the mini collapse may excite the fundamental quasi-radial and and quadrupole oscillation modes of the PNS, which leads to periodic signal with a ∼1 kHz frequency [49]. In slowly rotating models, the radial mini-collapse may interact with the asphericities of the flow outside of PNS and lead to strong GW emission [50]. For a sufficiently strong mini-collapse, such a signal could be detectable for a source within our galaxy.

Multi-messenger Aspects CCSNe are represent a perfect target for a multi-messenger study. Indeed, the enormous energy developed by the stellar explosion, ∼1053 erg, is released through different channels. About 99% of the energy is converted in low-energy (MeV)

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Fig. 12 Time evolution of neutrino, GW (thin blue line), and EM (thick black line) signal luminosities for a non-rotating 17M progenitor of [46]. Thin and thick red lines represent νe and ν¯ e , while magenta line is for νx , which represents νμ , ντ , ν¯ μ , or ν¯ τ . The solid lines are from simulations, while dashed lines are based on approximate estimates. The left and right panels represent time before and after core bounce, respectively. (Reproduction of Fig.1 from Nakamura c Nakamura et al. Reproduced with permission) et al. [46] 

neutrinos, the leftover ∼1% is mainly kinetic energy of the shock wave, while approximately the 0.01% of this energy powers a multi-wavelength electromagnetic (EM) emission and only a few times 1046 erg is released through gravitational waves as discussed in previous sections. The time evolution of the multi-messenger signals expected from a CCSN explosion is reported in Fig. 12. These signals are obtained from the numerical simulation of a neutrino-driven explosion of a nonrotating 17M progenitor [46]. The EM burst of a supernova starts at shock breakout (SBO), when the shock emerges from the stellar surface hours after core collapse (cf. Fig. 12). Characterized by a flash of UV and X-rays, it could provide important information about the CCSN progenitors, such as radius [51]. However, SBO detection is challenging since it is a short-lived phenomenon that can last only hours to days, depending on the density at shock emergence [52]. After the SBO, the EM emission enters the plateau phase lasting about 100 days. During this phase, the luminosity and duration of the plateau can also provide interesting constraints on the progenitor, like its radius and the ejecta mass [46]. As a last remark, it is worth noting that EM emission could be absent (or too faint to be detected). In a small fraction of CCSNe, known as “failed” supernovae,

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the massive star experiences core collapse and bounce, nevertheless is unable to generate a successful explosion and ends up with a BH formation. If EM signal is too faint, GWs and/or neutrino detection represents the only possibility to directly investigate these astrophysical events. As stated above, neutrinos and antineutrinos of all flavors are copiously produced and released during a CCSN event. They are characterized by energies in the few to tens of MeV range. The total neutrino burst duration is a few tens of seconds. Neutrino luminosities are drawn with red lines in Fig. 12, and the reference time is the time of the bounce. The solid lines represent results from numerical simulation, whereas the dashed lines are approximate estimates [46]. The left-hand panel of Fig. 12 shows the pre-SN emission of ν¯ e due to the late silicon burning phase that can release ∼5 × 1052 erg in neutrinos during the few hours just before the star collapse. The detection of this neutrino signal could be exploited as a very useful early warning alert [53]. In the right-hand panel of Fig. 12, neutrino emission can be described as composed of three different temporal phases: neutronization, accretion, and cooling [3]. In the shocked material, electron capture by protons produces a huge amount of electron neutrinos via reaction e− + p → n + νe . Furthermore, the concentration of the positrons, produced through γ → e+ + e− , progressively increases. These positrons interact with electrons, producing neutrinos of all flavors through the reaction e+ e− → νi ν¯i with i = e; μ; τ . At high material density, neutrinos are trapped. During the early shock expansion phase, the density at the shock decreases approaching the value ρ  1012 g/cm3 at which point neutrinos are free to escape. This prompt neutrino emission, called neutronization burst, is characterized by a sharp peak in the electron neutrino luminosity lasting a few milliseconds and releasing an energy of the order 1051 erg. As described in the previous sections, the shock wave stagnates and transforms into an accretion shock. In the nearly transparent shocked region, around the high-density core of the PNS, ν¯e and νe are produced by the symmetric processes e− + p → n + νe and e+ + n → p + ν¯e . This emission generates a prominent hump in the neutrino and antineutrino electronic flavor luminosities lasting a fraction of a second before the final star explosion. Finally the PNS evolves to a hot neutron star that cools down by emitting neutrinos and antineutrinos of all species produced by neutral current processes. During this Kelvin-Helmholtz cooling phase, neutrinos carry away 90% of the total energy emitted in the CCSN with a characteristic diffusion time of a few seconds. Neutrinos and antineutrinos, produced inside the CCSN by all the interaction processes described above, undergo flavor conversion while propagating outward through the star and interstellar medium. The oscillation probability depends on the neutrino mass hierarchy and can be affected by complex non-linear terms due to neutrino-neutrino interactions. In a simple scenario, the Mikheyev-SmirnovWolfestein matter effect will mix flavors during propagation in such a way that the electronic antineutrino flux arriving to the Earth is a mixture of the electronic antineutrinos and heavy lepton flavor fluxes at the production, i.e., Φν¯e = P¯ Φν0¯e + (1 − P¯ )Φν0¯x , where P¯ is the survival probability of ν¯e . A similar expression holds for electronic neutrinos, i.e., Φνe = P Φν0e + (1 − P )Φν0x , where P is the survival probability of νe . Approximated numerical values for the survival probabilities are

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P¯  0.7( 0) and P  0( 0.3) for the normal (inverted) mass hierarchy. This implies that the flux of electronic neutrinos will suffer a strong suppression during propagation. In particular, emission phases of pure νe (e.g. the neutronization burst) can mainly be investigated by the detection of the νx flux.

GW Searches The GW signals expected from CCSNe have been deeply discussed in the previous sections of this chapter. They can originate from a variety of mechanisms and can be characterized by very different frequency ranges that may or may not fall within the sensitive frequency bands of GW interferometers. Moreover, GWs from CCSNe have the common feature of being “short-burst like,” i.e., impulsive signals lasting less than a second. This means that they can be easily confused with the nonstationary part of the noise of GW interferometers, known as glitches. In order to maximize the ability to detect this kind of GWs, the Advanced LIGO and Advanced VIRGO detectors [54, 55] perform all-sky searches of short-duration signals with no prior assumption on the GW signal time of arrival or sky direction [56]. One of the three pipelines used to search short-burst is the Coherent WaveBurst (cWB) pipeline [57], specifically designed to perform searches for unmodelled bursts with a duration of up to a few seconds in the frequency range 32−4096 Hz. In this pipeline, coincident events are ranked according to their coherent network signalto-noise ratio (SNR). This coherent SNR should be higher for cross-correlated signals and lower for uncorrelated glitches, thus favoring real GW signals. The background distribution of triggers is calculated by time-shifting the data of one detector with respect to the other detectors by an amount of time that breaks any correlation between detectors for a real signal. Real temporal coincidences among interferometers, called 0-lag coincidences, are then compared with the accidental ones, and this comparison provides their false alarm rates. The detection efficiency of all-sky searches, performed by Advanced LIGO and Advanced VIRGO, is estimated by injecting simulated GW signals into real detector data and quantifying the percentage of signals recovered by the pipeline with a false alarm rate lower than the threshold of 1/100 years. Such a sensitivity study is performed by considering a set of ad hoc waveforms, for example, Gaussian or Sine-Gaussian signals, that are not derived from specific astrophysical simulations and that can be described by few characteristic parameters, like the central frequency or duration. In the most recent published results [56], for a source located at D = 10 kpc and for a false alarm rate of 1/100 years, 50% detection efficiency √ is obtained for root-mean-square strain amplitude in the range (1−10) × 10−22 H z depending on the central signal frequency. This range of values for the root-mean-square strain amplitude can be converted into an equivalent range for the minimum amount of energy emitted by a GW CCSN to be detected. At the present, Advanced LIGO and Advanced VIRGO, in the frequency band (100 − 200) Hz, can reach 50% of detection efficiency for a GW emitted energy  10−9 M c2 and a source emitting at 10 kpc.

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Simulated GW waveforms from CCSN, as deeply discussed in the previous sections of this chapter, could be very complex and different depending on the emission mechanisms and the progenitor mass. For non-rotating stars that represent the majority of the cases, the GW emission due to prompt convection and neutrinodriven convection, as the one reported in Fig. 12, could be very difficult to identify. Even for a CCSN located in the Galactic Center, i.e., D = 8.5 kpc, the SNR could be too small to claim a detection [46]. The capability to separate a real signal from the noise improves when GW search can profit from additional information/constraint provided by the contemporary detection of other probes from the same sources; see, e.g., Fig. 14.

Neutrino Searches CCSNe are the only sources of extra-solar low-energy neutrinos detected so far. Indeed, in 1987 the neutrino detectors Kamiokande, IMB, and Baksan observed a burst of low-energy neutrinos related to a CCSN explosion in the Large Magellanic Cloud [58]. This event, called SN1987A, represents a milestone for the supernova study. It was not only the first CCSN detected by neutrino experiments but also the first SN visible to the naked eye after the Kepler SN in 1604. The progenitor of SN1987A was found to be a star with mass ∼15M , located on the outskirts of the Tarantula Nebula in the Large Magellanic Cloud, at a distance of about 50 kpc. The data collected by the neutrino detectors, despite the small statistics (∼29 events), confirmed the baseline theory of CCSNe [3]. Several of the present neutrino detectors are sensitive to a neutrino burst from a galactic supernova [59]. The interaction process with the highest cross section in Cherenkov and liquid scintillator detectors is the inverse beta decay (IBD), ν¯e + p → e+ + n. The expected rate of events could be estimated as RI BD (t) = Np Ethr dEν σ (Eν )Φν¯e (Eν , t) where Np is the number of proton targets inside the detector, σ is the differential cross section of the process, and Ethr is the energy threshold of the detector. It is important to note that the effects of source distance, Φν¯e ∝ D −2 ; average energy of neutrino spectrum, Eν¯e ; and neutrino mass hierarchy are degenerate for this rate. Let us consider a neutrino detector with Ethr = 1 MeV and hence sensitive to all of the IBD channel spectrum. The total

number of events expected for a CCSN located at a distance D is Nev = RI BD (t)dt if the average energy of the ν¯e at the source is Eν¯e  = 12 MeV and the neutrino mass hierarchy is the inverted one. However, the same number of events Nev is expected also if the source distance is reduced to 0.9D and the neutrino mass hierarchy is the normal one. To break this degeneracy more, detectors and/or additional information are needed, such as the “a priori” knowledge of the mass hierarchy or of the CCSN distance, e.g., thanks to the detection of the EM counterpart. Finally, several other interaction channels are expected to be observed with smaller statistics and carry important information about the neutrino energy spectrum.

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Super-K, a 50-kton water Cherenkov detector in Japan, could observe some ∼8000 events for a core collapse at the center of the Milky Way, ∼8.5 kpc away. The LVD and Borexino scintillation detectors at Gran Sasso in Italy, and KamLAND in Japan, would observe hundreds of interactions from the same event. The IceCube detector at the South Pole, although nominally a multi-GeV neutrino detector, would observe a coincident increase in count rate in its phototubes due to a diffuse burst of Cherenkov photons in the ice and has sensitivity to a galactic supernova; for a review see Ref. [59] and references therein. The duty factor of neutrino detectors in observing mode is typically 90% or better, and their horizon extends now up to the Large Magellanic Cloud with a very low misidentification probability, i.e., the probability that an events cluster due to background survives selection criteria as a signal [60]. Figure 13 reports the detection efficiencies of the Super-K (left panel) and the Kamland-LVD network (right panel). Super-K can reach a horizon of ∼200 kpc with a detection efficiency of 100% and a misidentification probability of ∼3%. The Kamland and LVD detectors working together can reach the Large Magellanic Cloud with 100% of detection efficiency and 0.2% of misidentification probability. The Super-K, LVD, IceCube, and Borexino detectors are also operating as a part of the SNEWS (SuperNova Early Warning System) network [61], whose goal is to provide prompt alerts to astronomers in the case of a coincident supernova neutrino burst. Recently, the possibility of providing pre-SN alerts has also been discussed based on the expectations for the pre-CCSN neutrino signals. In the case of an extremely nearby CCSN (D ∼ 600 pc), the Kamland detector can send an alert to prepare other detectors for observing upcoming signals [62]. Next-generation neutrino detectors will increase the expected statistics of the supernova signal by scaling up the fiducial mass and increasing the horizon for the detection. For example, Hyper-Kamiokande (HK) [59] could observe tens of events for a CCSN in Andromeda, which is ∼750 kpc away. The Jiangmen Underground Neutrino Observatory (JUNO) [63], a 20 kt underground liquid scintillator detector, will collect ∼2000 all-flavor neutrino-proton elastic scattering events from a typical

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CCSN at a distance of 10 kpc. The Deep Underground Neutrino Experiment (DUNE), 40 kt LArTPC detector to be constructed underground in South Dakota, is expected to detect ∼3000 events due to νe interaction from a 10 kpc supernova, providing a νe sensitivity that complements the ν¯ e interaction channel, dominant for most of other detectors [64].

Combined Searches The multi-messenger nature of CCSNe motivates combined searches, i.e., searches performed by considering the information from different kinds of detectors and signals. In the simplest case, a CCSN detection (a 5σ evidence) with a type of signal, e.g., an EM signal, is exploited for a correlated search for another type of signal expected from the same source, e.g., GWs. In the case of GW searches correlated with EM counterpart (optically targeted search), the EM detection imposes the sky location of the source, the source distance, and a broad time window for the arrival time of the GW signal (generally within a few days) [65]. The detection efficiency can be estimated by injecting simulated signals in the CCSN optical position and looking for these signals by using the on-source window associated to the optical transient. In Fig. 14 as a leading example, the adopted source position is that of SN

Fig. 14 Detection efficiency as a function of the CCSN distance for a source located at the position and time of SN 2017eaw, i.e., with the perfect knowledge of the sky location and an on-source temporal window of about 1 day before the EM detection. (Reproduced from Abbott et al. [65]. c Abbott et al. Reproduced with permission) 

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2017eaw [65], and the observation window is about 1 day. The injected waveforms are the results of different simulations, which considered the case of neutrino-driven explosions of non- or slowly rotating progenitor stars. In this scenario, GWs are emitted in the initial stage after the bounce in the frequency range 100-300 Hz due to prompt convection, while, at later times, the frequency increases up to 2 kHz. Despite the fact that additional information provided by the optical counterpart increase the GW SNR, the detection horizon is less than 5 kpc. The GW SNR enhancement can be stronger when a better suppression of background spikes is implemented by limiting the search to a small temporal window. This is in principle possible by exploiting the neutrino signals, which is characterized by a timescale much smaller than that of the EM signals. Indeed, the neutrino luminosity rises just after the core bounce and has been demonstrated that, by analyzing the ν¯e signal (IBD channel) from a galactic CCSN with the current generation of neutrino detectors, one can identify CCSN core “bounce” time within a window of ∼10 ms or less [66]. This temporal window can be used to optimize a triggered GW search, i.e., to open an ad hoc temporal window (customized to fit the GW emission process around the time of the bounce) to look for GW bursts. The factor of improvement that can be achieved has been investigated in Ref. [46] for a CCSN located at the Galactic Center that emits the signals reported in Fig. 12. The CCSN coordinated observation with Super-K enables to restrict the time window to [0, 60] ms from the estimated time of the bounce. This increases the reconstructed SNR from ∼3.5 to ∼7.5, expanding the detection horizon in comparison to the one reported in Fig. 14 by approximately a factor of two. In addition, using neutrino-electron forward scattering events, a Galactic corecollapse event can be pinpointed to within an error circle of some 5◦ with the current Super-K detector [59]. This uncertainty can be reduced by tagging the inverse beta decay events to ∼3◦ with a tagging efficiency of 95%, and, for large-scale detector as HyperK, this accuracy can be as good as ∼0.6◦ [59]. The DUNE detector is also expected to provide a comparable sensitivity [64]. The pointing capability critically impacts the prospects for the detection of the SBO [46]. For extra-galactic CCSNe, both the GW signal amplitude and the neutrino statistics are low. In these cases, the identification of the astrophysical bursts embedded into the detectors’ noise could be challenging when using GWs or neutrinos alone. However, by exploiting the temporal correlation among the neutrino and GW emissions, it is possible to perform a joint search of distant supernovae. A joint GW-neutrino search would enable improvements to searches by lowering the detection thresholds, increasing the distance ranges, and increasing the significance of candidate detections [67]. Furthermore, neutrino experiments can benefit from a contemporary GW detection by relaxing the detection criteria. For example, SuperK’s search of distantI˙ CCSNe requires two neutrino events (with energy threshold 17 MeV) within 20 s in order to have the accidental rate less than one per year. With these requirements the probability of detecting a supernova in Andromeda is approximately 8%. By requiring the coincidence of a single neutrino event with a gravitational wave signal, the accidental rate could still be less than one per year, but the probability of detecting a core-collapse event in Andromeda would increase to 35%.

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As last remark, it is important to stress that correlated signatures of GWs and neutrino emission can go beyond the pure temporal coincidence, as reported by numerical simulations [68]. In the case of successful explosions, it has been demonstrated that both the neutrino counts and GW evolution are correlated with the core compactness, the PNS mass at 1 sec. post-bounce, and the explosion energy [69]. However, for failed supernova events, both the neutrino and GW observable are correlated with the core compactness and the remnant BH mass [70]. Deep studies of these astrophysical correlations can be essential to maximize the science return from such a rare event.

Conclusion and Prospects In this chapter, we summarized our current understanding of the gravitational wave (GW) emission from core-collapse supernovae (CCSNe). As discussed above, the GWs are predominantly produced by the oscillations of the protoneutron star (PNS). In the most common case of slowly rotating CCSNe, the multi-dimensional hydrodynamic flows that develop in the post-shock region perturb the PNS and excite its oscillations. The resulting signal is marginally observable with current detectors for a source within our galaxy, but future third-generation detectors will enable more detailed observation. In rare rapidly rotating progenitors, the PNS is born with a centrifugal deformation, which excites its “ring-down” oscillations that last for ∼10 − 20 ms. Some rapidly rotating models may develop non-axisymmetric instabilities leading to long-lasting emission of GWs that could be detectable up to megaparsec distances. Despite the significant progress achieved so far, there is much more to do in the near future. Our ability to detect and interpret the GW signal from the next galactic supernova rests not only on the availability of ground-based laser interferometers but also on the development of sophisticated data analysis techniques and reliable theoretical predictions. Progress on all of these crucial components is essential to turn a once in a lifetime event into a ground-breaking discovery. On the data analysis side, techniques for the search of unmodelled signals could be complemented with methods that incorporate theoretical predictions based on PNS perturbation theory. Coincidence analysis of GW and neutrino burst signals should also be developed as a way not only to increase the sensitivity of the searches and decrease their false alarm rate but also to test specific scenarios, such as the presence of correlated modulations of both signals due to SASI, that are predicted by state-of-the-art simulations. Theoretical models need to be improved on several fronts. Most obviously, GW signals should be computed using simulations that consistently treat the gravity sector using general relativity, instead of using ad hoc prescription, such as the use of effective GR potentials in Newtonian calculations, as is done even in some of the most sophisticated simulations to date. Perhaps more importantly, even the longest simulations stop shortly after the onset of explosions, typically well before the end of the GW signal. Long-term simulations are needed to quantify the GW

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signal due to the continued accretion and the PNS inner convection. The impact of neutrino oscillations remains largely unexplored. While the direct impact of neutrino oscillations on the GW signal is expected to be small, neutrino oscillations need to be taken into account in the joint analysis of GW and neutrino data and might even impact the dynamics of the explosion, thus affecting the overall signal. Finally, simulations of larger set of progenitors, including the ones with varying degree of rotation, will reveal the dependence on initial parameters. CCSNe are occur once or twice per century in a galaxy, but the next galactic supernova has already happened. The EM, neutrino, and GW bursts from the event are already on their way toward the Earth, poised for the first coincident observation of neutrinos, photons, and gravitons by the international network of observatories. Their detection will bring a once in a lifetime scientific revolution in our understanding of the end of life of massive stars. With modern neutrino detectors, we will be able to directly observe and measure, among others, the core bounce and the associated breakout burst, the time variability of accretion, and the binding energy of the newborn NS. In the same way in which helioseismology revolutionized our understanding of the sun, GWs from the PNS will allow us to learn about the interior structures of NSs and constrain the turbulent nature of the accretion flows in the core of exploding stars. Finally, EM observations will allow us to learn about the pre-SN structure of massive stars and the degree of asymmetry in the explosion. In conclusion, CCSNe are the ultimate multimessenger source.

Cross-References  Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range  Introduction to Gravitational Wave Astronomy  Isolated Neutron Stars  Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravita-

tional Waves  Terrestrial Laser Interferometers  Third-Generation Gravitational-Wave Observatories Acknowledgments We thank Adam Burrows and Kei Kotake for helpful comments. The work of G.P. was partially supported by the research grant number 2017W4HA7S “NAT-NET: Neutrino and Astroparticle Theory Network” under the program PRIN 2017 funded by the Italian Ministero dell’Istruzione, dell’Universita’ e della Ricerca (MIUR).

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Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range

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Marica Branchesi, Antonio Stamerra, Om Sharan Salafia, Silvia Piranomonte, and Barbara Patricelli

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Astrophysical Sources of Gravitational Waves Detectable by Ground-Based Detectors . . . . Binary Systems of Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Core Collapse of Massive Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expected Electromagnetic Counterparts of Gravitational Wave Signals . . . . . . . . . . . . . . . . . Gamma-Ray Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kilonova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Electromagnetic Follow-Up of Transient Gravitational Wave Sources . . . . . . . . . . . . . . Low-Latency Search for Gravitational Wave Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Branchesi () INFN, Laboratori Nazionali del Gran Sasso, Gran Sasso Science Institute, L’Aquila, Assergi, Italy e-mail: [email protected] A. Stamerra INAF, Osservatorio Astronomico di Roma, Monte Porzio Catone (Roma), Italy SNS, Scuola Normale Superiore di Pisa, Pisa, Italy SSDC, Space Science Data Center, Roma, Italy e-mail: [email protected]; [email protected] O. S. Salafia INAF – Osservatorio Astronomico di Brera, Merate, Italy e-mail: [email protected] S. Piranomonte INAF – Osservatorio Astronomico di Roma, Monte Porzio Catone, Italy e-mail: [email protected] B. Patricelli European Gravitational Observatory, Cascina, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_22

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The Gravitational Wave Alert System: Distribution and Alert Contents . . . . . . . . . . . . . . . Identification and Localization of the Counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counterpart Classification and Follow-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW170817 and Its Electromagnetic Counterparts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

On the 17th of August 2017, the first detection of gravitational waves from the coalescence of a neutron star binary and the observation of its broadband electromagnetic emission first demonstrated the huge scientific potential of multi-messenger astronomy with gravitational waves. Joint gravitational wave and electromagnetic observations are indeed unique tools to unveil the nature of neutron stars and black holes, together with probing the rich physics of energetic transient phenomena in the sky, such as gamma-ray bursts, kilonovae, and supernovae. This chapter aims at giving an overview of the diverse electromagnetic counterparts to gravitational wave sources detectable by current ground-based detectors, providing basic information on their properties and on the physics that governs their emission. In addition, it addresses observational and data analysis strategies to optimize their detection and characterization. Keywords

Gravitational wave astronomy · Multi-messenger astrophysics · Black holes · Neutron stars · Gamma-ray bursts · Kilonovae · Relativistic astrophysics · Nucleosynthesis · Nuclear physics

Introduction The first detection of a binary neutron star (BNS) merger, GW170817, by the Advanced Laser Interferometer gravitational wave Observatory (LIGO) detectors [1] and Advanced Virgo [2] marked the birth of multi-messenger astronomy including gravitational waves. It sparked the most extensive observational campaign in human history which enabled a spectacular detection of electromagnetic emission in essentially all electromagnetic bands. This multi-messenger detection revolutionized our understanding of binary neutron star merger, unveiling the physics governing short gamma-ray bursts and kilonova emissions. The detection of electromagnetic signatures from the gamma-rays to the optical and radio bands enhanced our knowledge of relativistic jets, the nucleosynthesis of heavy elements in the Universe, and the nuclear physics of neutron stars. This event clearly showed the power of detecting the electromagnetic counterpart of a gravitational wave signal. In this context, this chapter focuses on the astrophysical sources emitting high-frequency (hertz to kilohertz) gravitational waves and electromagnetic radiation. It gives an overview of the coalescence of binary systems of compact objects, the core-collapse of massive stars, and isolated neutron stars. It describes

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the basic properties of the physics and observational properties of the prompt and afterglow emissions of gamma-ray bursts and the kilonova emissions. It outlines the observation strategies and data analysis starting from the low-latency gravitational wave event detection to all the steps of the electromagnetic follow-up to detect the gravitational wave counterparts, pointing out challenges and perspectives to make joint detections. Throughout the chapter, the main observations and results from the broadband detection of GW170817 are reported.

Astrophysical Sources of Gravitational Waves Detectable by Ground-Based Detectors The main astrophysical sources of gravitational waves (GWs) in the frequency range (10 Hz–10 kHz) of ground-based interferometers such as Advanced LIGO [1], Advanced Virgo [2], and KAGRA [3,4] are (i) coalescing binary systems of compact objects (CBCs) composed by neutron stars (NSs) and/or stellar-mass black holes (BHs); (ii) core-collapsing massive stars; and (iii) isolated neutron stars. These sources are also expected to emit electromagnetic (EM) radiation, and they will be described in the following sections.

Binary Systems of Compact Objects The frequencies accessible by ground-based GW detectors enable us to observe three classes of CBC systems: stellar-mass binary black holes (BBHs), neutron star black hole (NS-BH) binaries, and binary neutron stars (BNSs). These systems form either through the evolution of isolated stellar binaries in galactic fields (see, e.g., [5, 6]) or through dynamical interactions in dense stellar systems such as globular clusters (see, e.g., [7, 8]), young star clusters (see, e.g., [9, 10]), and nuclear star clusters (see, e.g., [11]). A possible way to distinguish between the two formation paths is through their mass, spin, and eccentricity distributions, which are expected to be different in the two cases (see, e.g., [12, 13] and references therein). CBC systems emit GWs detectable by current GW detectors in the last moments of their life, namely, during the last several orbits of the inspiral, during the merger, and potentially during a brief transient phase in the post-merger. Their GW radiation can be accurately modeled by means of post-Newtonian approximation methods (up to the last few orbits before merger), numerical relativity simulations (for the merger phase and the post-merger), and general-relativistic perturbation methods (for the ringdown in the case of a black hole remnant). The possibility to construct accurate GW waveform models enables matched filter modelled searches (see, e.g., [14, 15]), which represent the optimal strategy for the identification of a known signal in a noise-dominated scenario. As of today, Advanced LIGO and Advanced Virgo detected several BBH mergers [16, 17] and two BNS mergers [18, 19]. No firm detections of NS-BH mergers have been reported by the LIGO Scientific Collaborations and the Virgo collaboration (Collaborations focused on the direct

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detection of gravitational waves, working on the development, commissioning, and exploitation of gravitational wave detectors, and the development of modeling and techniques for gravitational wave detections and their interpretation) so far, except for GW190814 [20], whose nature is uncertain. For BNS and NS-BH systems, during the last few to several orbits before merger, tidal effects become important, causing the deformation and potentially the tidal disruption of one or both the neutron stars, leading to mass ejection. During the merger itself, shocks caused by the collision of the two neutron stars, spiral waves, oscillations of the merger remnant, and angular momentum transport can eject additional amounts of mass (see section “Mass Ejection from Compact Binary Mergers: A Variety of Mechanisms, Compositions, and Morphologies”). A fraction of the ejected mass (0.0001–0.1 M ) is expected to be unbound, forming the so-called “dynamical ejecta,” while another part (0.001–0.1 M ) can form an accretion disk around the merger remnant. Both of these components can power detectable electromagnetic (EM) emission. The observed EM emission depends on the progenitor intrinsic properties, such as component masses, spins, equation of state (EoS) of the NS, and extrinsic properties such as location and geometry (distance, inclination, and orientation with respect to the Earth). All these properties (with some degeneracies) can be inferred from GW measurements. The properties of the EM emissions are also influenced by the nature of the merger remnant. Among the above-described classes of CBC systems, only the EM counterparts of the BNS mergers have been observed so far. BNS mergers have been proven to be the progenitors of short gamma-ray bursts and to give rise to kilonovae (see section “Expected Electromagnetic Counterparts of Gravitational Wave Signals”). For detailed reviews on BNS and NSBH merger ejecta and the corresponding electromagnetic signals, see [21–24]. Stellar-mass binary BH mergers are not expected to produce bright EM signal due to the absence of baryonic matter left outside the merger remnant. However, there are some rare scenarios which predict an unusual presence of matter around the BBH. The matter can come from the remnants of the stellar progenitors [25–27], the tidal disruption of a star in triple system with two black holes [28,29], and when the binaries reside in active galactic nuclei (AGN) [30]. Even if firm observations confirming these scenarios are still missing, it is noteworthy that after the first detection of BBH merger, GW150914, a weak signal was detected by Fermi/GBM above 50 keV 0.4 s after the merger [31]. However, the detection was not confirmed by any other satellites [32, 33]. And more recently, the Zwicky Transient Facility (ZTF) observed a possible optical counterpart of the BBH merger S190521g consistent with the BBH merger occurring in the accretion disc of an AGN [34].

Core Collapse of Massive Stars Core-collapsing massive stars (stars with initial mass M  8 M at their final evolutionary stage) are expected to emit GWs in the presence of explosion asymmetries: this includes non-axisymmetric bulk mass motions due to convection, asymmetric

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emission of neutrinos, and other asymmetries associated with the effects of strong magnetic fields (see [35–37] for an overview). Due to the large uncertainties that affect our knowledge of the collapsing phase, the amount of GW-released energy and the expected GW waveforms are highly uncertain. Detailed multidimensional core-collapse simulations predict the GW energy to be between 10−11 M c2 and 10−7 M c2 . Higher values from 10−4 M c2 to 10−2 M c2 are suggested by phenomenological models exploring “extreme” GW emission scenarios, such as long-lived bar-mode instabilities (see [38] and references therein). As a consequence of this uncertainty on the GW energy emitted, also the maximum distance up to which these sources could be detected is uncertain. For realistic models, SNe will be detectable by ground-based GW detectors within our Galaxy, for the extreme scenarios up to hundreds of Mpc [38]. Due to the uncertain modeling of the signal, GWs from massive star core-collapse are typically searched with the socalled unmodelled searches, with minimal assumptions about the GW waveform (see section “Low-Latency Search for Gravitational Wave Signals”). No GW signal from a massive star core collapse has been observed so far (see, e.g., [38, 39]). The core collapse of a massive star is accompanied by supernova (SN) emission at optical and radio frequencies, generally observed starting from days to weeks after the collapse and lasting for weeks (in the Optical) up to years (in the radio). The SN optical and radio emission can be preceded by the so-called shock breakout (SBO) emission: a bright UV/X-ray flash of radiation created as the shock wave generated by the collapsing core reaches the surface of the star (see, e.g., [40, 41]). Finally, evidence have been found for the association between core collapse of massive stars and long gamma-ray bursts (LGRBs, see section “Jet Launch by the Merger Remnant”), which are GRBs lasting longer than about 2 s and with an average duration of several tens of seconds. In particular, the direct association of some LGRBs with Type Ic SNe [42, 43] implies that these LGRBs are connected to the death of Wolf-Rayet stars. This makes long gamma-ray bursts (most likely observed off-axis) potential electromagnetic counterparts to gravitational wave events from massive star core collapse.

Isolated Neutron Stars Isolated NSs can emit GWs through different mechanisms (see [44] for an overview). Neutron stars are born from the core collapse of massive stars or possibly as remnants of the merger of two neutron stars. When the magnetic field in the neutron star interior is very intense (B  1016 G, expected for newborn neutron stars), a significant deformation of the star is expected, leading to the conversion of a fraction of rotational energy into gravitational waves [45–49]. This transient gravitational wave spin-down signal is potentially detectable up to tens of megaparsecs with third-generation ground-based detectors such as the planned Einstein Telescope [50] and Cosmic Explorer [51]. The spin energy of a newborn NS rotating with a millisecond period can also provide power input to EM emission associated with these sources, potentially powering super-luminous

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SNe [SLSNe – e.g., 52,53] and the shallow decay phase observed in the early X-ray afterglows of a fraction of long and short GRBs (e.g., [54]). Theoretical studies have shown that rotating NSs with very intense magnetic fields (of the order of 1015 G) can undergo starquakes (see, e.g., [55]): when this happens, asymmetric strains can temporally alterate the geometry of the star, and GWs are expected to be emitted (see, e.g., [56]). The expected amount of GW energy emitted (in the range 10−16 M c2 –10−6 M c2 ) and the GW waveform are highly uncertain; therefore, the corresponding GW signals are searched with the unmodelled searches (see section “Low-Latency Search for Gravitational Wave Signals”). EM phenomena possibly associated with magnetar starquakes include soft gamma repeaters and anomalous X-ray pulsars which are sporadic emission of short bursts of gamma-rays and X-rays (see [57]). Neither transient GW emission from newborn NSs nor from isolated NSs have been observed so far. Isolated NSs are also expected to emit GWs continuously if they are asymmetric with respect to their rotation axis. The frequency of this continuous GW emission, fGW , depends on the mechanism responsible for the asymmetry in the mass distribution. Different mechanisms have been proposed in literature, including elastic stress or magnetic fields which induce a deformation in the NS shape (fGW =2frot , with frot the rotation frequency of the NS), free precession around the rotation axis (fGW =k (frot + fprec ) with k=1,2 and fprec the precession frequency), and excitation of long-lasting oscillations (e.g., r-modes, fGW ∼ 4/3frot ). Different methods are used to search for continuous GW emission from isolated NSs: if the source position and rotational parameters are known from EM observations, these parameters are used in the GW search (the so-called “targeted” search) using matched filtering (see, e.g., [58]); otherwise “all-sky” searches are performed (see, e.g., [59]). No continuous GW emission from isolated NSs has been observed so far.

Expected Electromagnetic Counterparts of Gravitational Wave Signals Gamma-Ray Bursts Gamma-ray bursts (GRBs; see [60] for a review) are intense and highly variable flashes of γ -rays (the “prompt” emission), followed by a long-lasting, multiwavelength emission (the “afterglow” emission), typically observed in X-rays, optical, radio, and sometimes gamma-rays. They are believed to be powered by ultra-relativistic jets produced by rapid accretion onto a central compact object, and they are the most luminous objects in the sky, with a peak isotropic-equivalent emitted luminosity in the range ∼1048 –1055 erg/s. The prompt emission light curves are characterized by a great variety of temporal profiles: they range from single-peaked pulses to highly structured multi-pulsed light curves, with a time variability down to 1 ms [61]. The typical prompt emission spectrum has a non-thermal shape and is well described by the phenomenological “Band function”: two power laws smoothly joined by an exponential at a break

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photon energy. Several scenarios have been proposed to interpret the GRB prompt emission, such as synchrotron emission from relativistic electrons accelerated in the internal shocks in the relativistic outflow [62], but no general consensus has been reached yet. The afterglow light curves are typically well represented by a power law decay. However, the advent of the Neil Gehrels Swift Observatory (Swift) shows that the early (first minutes to few hours) X-ray afterglow light curves include one or more of the following parts: an initial very steep decay, followed by a very shallow decay and finally a steeper decay, possibly showing superimposed X-ray flares. In a large fraction of events, the afterglow spectrum is in good agreement with the synchrotron external shock scenario, in which a relativistic population of electrons is accelerated in the shocks produced by the interaction of the relativistic outflow with the ambient medium (see section “Jet Launch by the Merger Remnant”). GRBs are divided into two main classes according to the duration of the prompt emission that is typically expressed in terms of T90 (The T90 is the time interval during which 5% to 95% of the GRB fluence is accumulated.): short GRBs (T90 < 2 s) and long GRBs (T90 > 2 s). Long GRBs are typically observed to take place in star-forming galaxies and reside on the brightest regions of these hosts: this suggests that they are born from very massive stars. This is also supported by the identification of SN signatures in a number of sufficiently close long GRBs (see, e.g., [63], for a recent review of GRB-SNe). Short GRBs occur in both early- and late-type galaxies with low star formation rate and are associated with an old stellar population; furthermore, no underlying SN has been observed in the short GRBs light curves, down to stringent limits in the closest events (e.g., [64]): this suggests a different progenitor with respect to long GRBs. This is also supported by the fact that short GRBs are typically observed with some offset with respect to the center of their host galaxy, with a median value of 5 kpc [64]: the observed offset distribution is in agreement with the population synthesis predictions for compact object mergers, suggesting that these systems are the progenitors of short GRBs. Binary neutron star and black hole-neutron star mergers have long been proposed [65] as the events that produce short GRBs. Such link has been recently confirmed in a spectacular way by the association of the binary neutron star merger gravitational wave signal GW170817 [18, 66, 67] to the short gamma-ray burst GRB 170817A [68, 69] (see section “GW170817 and Its Electromagnetic Counterparts”). In the following section, we describe the current theoretical understanding of the connection between GRBs and these GW sources.

Jet Launch by the Merger Remnant The remnant of a binary neutron star or a black hole-neutron star merger is necessarily a rotating (by angular momentum conservation) compact object, often surrounded by an accretion disk. Kelvin-Helmholtz instabilities during the merger (e.g., [70]) and shear instabilities in the subsequently formed accretion disk [71, 72] have been demonstrated to be able to efficiently and rapidly amplify the seed magnetic field (present in the neutron stars before the merger), leading to favorable conditions for the launch of a relativistic jet by the merger remnant.

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If the remnant is a black hole, and if accretion lasts long enough [73] for the magnetic field to attain a large-scale, predominantly poloidal configuration that threads the black hole horizon, the Blandford-Znajek [74] mechanism can efficiently extract rotational energy from the black hole and convert it into an outflow whose dominant form of energy is magnetic, which leaves the system along the rotational axis: this constitutes the base of a relativistic jet. Several authors (e.g., [75–78]) suggested that a strongly magnetized, highly rotating neutron star merger remnant – a magnetar – could equally act as a jet-launching system, by means of a relativistic, electron-positron pair-loaded wind powered by the rotational energy of the magnetar. Recent high-resolution, long-term general-relativistic magnetohydrodynamical simulations of binary neutron star mergers [79], though, disfavor such a scenario, revealing that baryon-loaded winds from the magnetar remnant would increase too much the baryon density in the region surrounding the remnant for an incipient jet to successfully form and leave the system. An alternative jet-launching mechanism, which has been proposed to work both in the black hole and in the neutron star remnant cases, is that of energy deposition by neutrino-antineutrino annihilations in the funnel above and below the remnant system [65, 80, 81]. The required, extremely high neutrino luminosity would be provided by the inner, hot part of the accretion disk. Global binary neutron star merger simulations [82,83], though, provide evidence against such mechanism, showing that a neutrino-powered jet would not be able to successfully propagate through the rather dense environment surrounding the merger and to break out of it, mainly due to the short duration of the highneutrino-luminosity phase in the accretion flow. Better prospects remain for the black hole-neutron star case [83], where the density of the post-merger environment in the polar region is expected to be much lower. Summing up, the most likely mechanism by which a relativistic jet can be produced following a binary neutron star merger is the Blandford-Znajek mechanism [74], provided that the remnant is a black hole surrounded by an accretion disk. Neutrino-antineutrino annihilation in the polar region might still contribute some energy or even dominate in the case of black hole-neutron star mergers [83], while a jet powered by magnetar spin-down seems disfavored [79].

The Dominant Form of Energy in the Jet Even though the most favored jet-launching mechanism predicts a Poynting-flux dominated (i.e., highly magnetized) jet, during its subsequent propagation through the dense environment surrounding the remnant, the jet can undergo substantial transformations. In particular, magnetic (“kink”) instabilities associated with the jet self-collimation have been suggested [84, 85] to lead to magnetic reconnection events that convert roughly half of the magnetic energy into internal energy (i.e., heat). As typical binary neutron star post-merger environments are thought to be dense enough as to efficiently collimate the jet early during its propagation [86, 87], this means that the dominant form of energy in the jet at the time of breakout could actually be kinetic (as the internal energy eventually contributes to the acceleration of the jet material). The nature of the dominant form of energy in

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the jet is tightly connected to the mechanism that dissipates such energy in order for it to be radiated at the prompt emission stage, which we address later in this chapter.

Jet Interaction with the Progenitor Material The problem of the propagation of a relativistic jet through a dense ambient medium has been first addressed as early as the 1970s (e.g., [88]), as it is relevant to understand the radio emission of some resolvable active galactic nuclei. While for these sources the jet propagation takes place on extremely long timescales compared to a human life, in the case of a GRB, the jet propagation from the central engine up to breakout from the progenitor material (the progenitor star envelope in the case of long gamma-ray bursts or the cloud of expanding merger ejecta in the case of short gamma-ray bursts) happens on a timescale of seconds. Since a relativistic jet is by definition supersonic, its effect on the surrounding matter is naturally that of producing a shock. As long as the jet is active, the resulting structure is that of a forward shock propagating into the progenitor material and a reverse shock propagating backward (as seen in the shocked fluid rest frame) into the jet. The region comprised between the two shocks is usually referred to as the jet “head” (e.g., [89, 90]). Since the jet typically only occupies a small solid angle, the shockcompressed material in the head is left unconfined in the direction perpendicular to the shock propagation: if the head remains causally connected (i.e., if its transverse size can be crossed by sound waves in a time shorter than the propagation dynamical time), the shocked material can flow laterally out of the head and form an overpressured cocoon that surrounds the jet as it propagates [91]. The inward, transverse force exerted by the cocoon pressure onto the jet tends to collimate the latter [92,93], reducing the head transverse size (the “working surface area”) therefore enhancing its propagation speed. This in turn diminishes the energy flow from the head to the cocoon, reducing the pressure in the latter: the result is a self-regulating feedback mechanism which determines the propagation timescale and the properties of the jet and cocoon at breakout [93, 94]. In the case of compact binary mergers, since the jet propagates into the expanding merger ejecta cloud, an additional feedback mechanism arises: the slower the jet head propagates (e.g., due to a low jet kinetic luminosity), the more time the ejecta have to expand and become less dense and vice versa; the successful breakout of the jet therefore does not depend on its luminosity, but only on the ratio of its isotropic-equivalent energy to that of the ejecta [95]. As soon as the head reaches the outer edge of the progenitor material, where the density of the latter drops, the shock-compressed material in the head is free to expand (e.g., [90]) and to accelerate to relativistic speeds, followed by unshocked jet material (if any) that did not cross the reverse shock yet. The over-pressured cocoon also blows out, surrounding the jet. The result is an outflow with an angular distribution of energy and terminal Lorentz factor [93, 96] which depends on the jet properties at launch, on its magnetization, and on the properties of the progenitor material [85, 97]. Such outflow is often referred to as a “structured jet” [98], and its detailed angular distribution of energy and Lorentz factor can have a major impact on the properties of both the gamma-ray burst prompt and afterglow emission, especially if the jet axis is not aligned exactly with the line of sight (which is anyway

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highly unlikely). The final jet structure is customarily described [98] by specifying the energy content per unit solid angle ε = dE/dΩ and the (average) bulk Lorentz factor Γ of the structured jet material as functions of the angular distance θ from the jet axis (Such a description neglects the radial distribution of energy and velocity, and the nature of the energy carried by the jet, both of which can play relevant roles in determining the properties of the emission (e.g., [99]). On the other hand, the afterglow properties are independent of these details once self-similar deceleration has set in [100], and moreover, the processes behind the prompt emission are still too uncertain to allow for a clear link between the detailed radial structure and the observations. For these reasons, the angular structure functions are currently the most useful representation of the jet structure.) (assuming axial symmetry). A considerable effort has been put during the last couple of decades into investigating the dependence of the jet structure on the properties of the central engine and of the progenitor material, with an increased interest (e.g., [93, 94, 96, 97]) after the discovery of clear signatures of a structured jet in GRB 170817A ( [68, 101, 102], see section “GW170817 and Its Electromagnetic Counterparts”).

Prompt Emission After breaking out of the progenitor material (i.e., the merger ejecta in the case of short gamma-ray bursts), the jet (hereafter we use the word “jet” to indicate the entire outflow – the “structured jet” – that forms after breakout, as described in the preceding section) is free to expand into the interstellar medium. Given the typical breakout radii and the jet baryon content, the jet is initially optically thick to Thomson scattering: photons are trapped within the outflow, and they contribute to its acceleration rather than being radiated away. In order to produce high-energy, non-thermal radiation such as that observed in gamma-ray bursts, the jet must therefore expand up to, and beyond, the photospheric radius. Assuming Thomson scattering on the electrons associated with the jet baryonic content to be the dominant form of opacity at the relevant wavelengths, the photospheric radius can be computed as  Rph (θ ) ∼

Ye σT ε(θ ) mp c2 (1 + σ (θ )) Γ (θ )

(1)

where Ye is the electron fraction in the jet material, σT is the Thomson cross section, mp is the proton rest mass, c is the speed of light, and σ is the jet magnetization (i.e., the ratio of the magnetic to the rest-mass energy, in the co-moving frame, which can be a function of the angle θ from the jet axis). Depending on σ , the dominant form of energy beyond the photospheric radius can be associated either with the magnetic field or instead the jet bulk motion kinetic energy. In either case, some process must dissipate such energy and transfer it to particles (most likely electrons, but see [103]) for them to radiate it away. Strikingly, despite over 50 years of gamma-ray burst research, neither the dissipation process nor the radiative process that produces the gamma-ray burst prompt emission has been

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unambiguously identified. Given the non-thermal appearance of gamma-ray burst spectra, the simplest picture places the dissipation above the photosphere, so that the dissipated energy can be promptly radiated away before particles have the chance to thermalize. Depending on the assumed degree of magnetization, the most widely considered dissipation processes are internal shocks ( [104], if the jet energy is mainly kinetic) or magnetic reconnection (e.g., [105,106], in the highly magnetized jet scenario). Independently of the nature of the dissipation and emission mechanisms, these will eventually convert jet energy into radiation with a generally angle-dependent efficiency η(θ ). The isotropic-equivalent energy measured by an observer whose line of sight forms an angle θview with the jet axis can then be computed, accounting for relativistic beaming of radiation [107], as 



2π

Eiso (θview ) =

1

d cos θ

dφ 0

0

dE δ 3 (θ, φ, θview ) η(θ ) (θ ) Γ (θ ) dΩ

(2)

where δ = Γ (θ )−1 [1 − β(θ ) cos α(θ, φ, θview )]−1 is the Doppler factor pertaining to the jet element at coordinates (θ, φ) in a spherical coordinate system whose 1/2  , and cos α = z-axis coincides with the jet symmetry axis, β(θ ) = 1 − Γ (θ )−2 cos θ cos θview + sin θ sin φ sin θview is the cosine of the angle between the line of sight and the velocity (assumed radial) of the jet element at (θ, φ). As long as all parts of the jet, except that within an angle Γ −1 of the line of sight, have Lorentz factors much larger than 1/ sin α, Equation 2 reduces [98] to Eiso = 4π ε(θv ). On the other hand, if the Lorentz factor decreases sufficiently fast with the angle so that the emission from the jet “wings” is not too relativistically de-beamed, then their contribution to the integral in Eq. 2 becomes non-negligible, especially at large viewing angles (see the examples in Fig. 1). This has important consequences on the appearance of structured jets seen off-axis and for their detectability (e.g., [107, 108]), as it can greatly enhance the detectability of not-too-far off-axis jets. Quite generally, one can assume the jet prompt emission to be produced by a series of “episodes” (which comprise jet energy dissipation and the subsequent radiation) similar in nature, but with a stochastic variability in their properties, all taking place around a typical emission radius Rem ≥ Rph . When the energy in Eq. 2 is dominated by jet elements close to the line of sight, the duration of the observed emission from each emission episode is Doppler contracted as seen by the observer, resulting in a rapid variability that tracks that of the conditions at the jet base (i.e., at the central engine). If, on the other hand, the observed emission is dominated by “off-axis” jet elements (That is, jet elements whose velocities form an angle α > arcsin(Γ −1 ) with the line of sight), then the individual episode duration can be much longer than the on-axis case [109]. This can smear out the variability, resulting in a light curve where the emission is composed of several overlapping pulses that appear as a single, broad, and smooth pulse: this signature can help in distinguishing jets whose prompt emission is observed off-axis.

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Eiso(θv) / Eiso(0)

100

θΓ/θc = 1.0 θΓ/θc = 1.4 θΓ/θc = 2.0 Uniform

10-2 10-4 10-6 10-8 1

10 θv [deg]

100

Fig. 1 Example dependencies of the isotropic-equivalent energy Eiso radiated by relativistic jets with different structures toward observers at different viewing angles θv . The blue line shows the ratio Eiso (θv )/Eiso (0) for a jet with uniform bulk Lorentz factor Γ = 400 and energy per unit solid angle (shown by the black dashed line) within a 3◦ half-opening angle. The remaining colored lines show the same quantity for three different jets with Gaussian distributions of energy per unit solid angle ε ∝ exp(−θ 2 /θc2 ) (again shown by the black dashed line) and Lorentz factor Γ = 1+(Γc −1) exp(−θ 2 /θ2 ), for three values of the ratio θ /θc , and with θc = 3◦ and Γc = 400. (Reproduced from Figure 5 of [107])

After the last emission episode has peaked (as seen by the observer), a tail produced by the late arrival of photons emitted by jet elements at high latitude can become visible, if not outshined by other emission components such as the afterglow [110–113]. If the jet structure is uniform and independent of θ , then the flux density from such tail is expected to decay as dF /dν ∝ t −2−β , where β is the spectral index of the co-moving spectrum. If the jet is structured, and the Lorentz factor decreases sufficiently fast with the angle, then the tail decay can become much shallower or even present a rebrightening: this has been suggested as an explanation for the X-ray plateaus observed up to several 104 s after the prompt emission in some GRBs [111, 113] (see Fig. 2 for an example of a gamma-ray burst X-ray and optical light curve explained by a combination of afterglow and the prompt emission tail discussed here). When the jet is observed off-axis [112], such emission component could constitute a promising X-ray counterpart of jet-producing compact binary mergers.

Afterglow Dynamics and Emission After producing the prompt emission, the jet material keeps expanding at relativistic speed, producing a shock that sweeps an increasing amount of interstellar medium mass. As long as the motion is relativistic, parts of the jet at different latitudes θ are out of causal contact, so we can treat them independently. Let us indicate

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Fig. 2 X-ray (cyan) and optical (colored) light curves of gamma-ray burst GRB 100906A, together with the predicted emission (solid lines) from the combined jet afterglow (dashed lines) and “high-latitude” prompt emission tail (dotted lines) from a Gaussian structured jet model. (Reproduced from Figure 7 of [111])

with R the radius of the shock and with m(R) the (isotropic-equivalent) rest mass swept by the shock caused by the portion of the jet whose kinetic energy per unit solid angle is ε (note that in this section, we use ε = dE/dΩ to refer to the kinetic energy that remains after the prompt emission phase, as opposed to the total jet energy as before). As long as m is much less than 4π ε/Γ 2 c2 , the jet does not decelerate significantly. In this phase, a reverse shock forms [114, 115], which propagates backward (as seen in the shocked material rest frame) into the jet. When the reverse shock crosses the jet (at a “deceleration” radius Rdec such that Γ 2 m(Rdec )c2 ∼ 4π ε, unless the central engine activity is very long T > Rdec /Γ 2 c, in which case the reverse shock crosses the jet after Rdec ), the radial structure of the jet is washed out, all its material being compressed into a thin shell behind the contact discontinuity that separates it from the shocked interstellar medium material. From this point on, the shock in the interstellar medium starts to decelerate in a self-similar manner [100], with its Lorentz factor evolving as Γ ∝ R −3/2 in the case of a constant-density interstellar medium.At some point of the shock deceleration, the transverse sound-crossing time in the shocked material becomes shorter than the dynamical time, breaking our assumption of absence of causal contact between parts of the jet at different latitudes. From such point on, the shock starts to spread sideways, sweeping larger amounts of interstellar matter, which

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causes the deceleration to speed up. Eventually, the shock becomes non-relativistic, and (typically at a somewhat later stage) it turns spherical, approaching a spherical Sedov-Taylor blastwave, similar to a supernova remnant (but with a significantly larger radius due to the initial relativistic expansion). Throughout its expansion, the shock compresses and heats up the interstellar medium, accelerating particles through the Fermi process. Weibel-type turbulence in the shock downstream is believed to produce local, small-scale amplification of the interstellar medium magnetic field, causing the accelerated particles to radiate through the synchrotron process. Since both electron acceleration and magnetic field amplification in the shock upstream are poorly understood, it is customary to parametrize the energy density share of non-thermal electrons in the upstream as a fraction εe of the total co-moving energy density e = (Γ − 1)(γˆ Γ + 1)mp c2 /(γˆ −1) (where γˆ is the adiabatic index in the upstream and Γ is the shocked material Lorentz factor as measured in the interstellar medium rest frame) and similarly to assign a fraction εB of the energy density to the magnetic field. These assumptions, together with the shock dynamics Γ (R), allow for the computation of the synchrotron emission in the standard gamma-ray burst afterglow model. The generalization to the structured jet case is straightforward (e.g., [116, 117]), as it essentially amounts to applying the model independently to each angle θ , even though the dynamics can get more complicated due to lateral energy transfer during the sideways expansion phase. The actual appearance of the emission as seen by a (generally off-axis) observer is then obtained by integrating the shock emission over the proper equal-arrival-time surfaces, accounting for relativistic beaming. An example of the resulting light curve, for an off-axis structured jet whose properties are derived by a relativistic hydrodynamic simulation, is shown in Fig. 3.

The Kilonova As discussed in the preceding section, the production of a relativistic jet by the remnant of a BNS or NS-BH merger requires the formation of an accretion disk. Additional electromagnetic emission can arise from gravitationally unbound matter expelled during and after the merger by a variety of mechanisms (described in section “Mass Ejection from Compact Binary Mergers: A Variety of Mechanisms, Compositions, and Morphologies”). Such decompressed NS matter is typically highly neutron rich, and it constitutes an ideal setting for r-process nucleosynthesis [119] to occur. The synthesized nuclei reside far from the valley of nuclear stability: their drift toward stability goes mainly through beta decays but include gamma-ray emission due to de-excitation and emission of α particles and fission fragments. The decay products partly thermalize in the ejecta, powering a supernova-like thermal transient usually called “kilonova” [23, 120, 121]. In addition to being the main source of heating, the heavy elements produced by the r-process also dominate the opacity of the expanding ejecta, especially if lanthanides and actinides are synthesized, due to their extremely complex line

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Fig. 3 Jet structure and afterglow emission of a simulated short gamma-ray burst structured jet. The left panel shows the co-moving rest-mass density in a snapshot of a relativistic hydrodynamic simulation of a gamma-ray burst jet that breaks out of kilonova ejecta. The lower right panel shows the angular isotropic equivalent kinetic energy distribution in the jet (blue line) and the bulk Lorentz factor (orange line). The upper right panel shows the 3 GHz light curves from the afterglow of such a jet decelerating in an interstellar medium with number density 4.2 × 10−3 cm−3 , seen under a 33◦ viewing angle, at a 40 Mpc distance. Different curves show the contribution to the emission from the angular portions of the jet indicated in the left panel. The purple points show the light curve of GRB 170817, which is well explained by the model. (Reproduced with permission from Figure 6 of [118])

structure (e.g., [122, 123]). The presence, or absence, of such elements depends on the electron fraction Ye (the neutron-to-proton ratio) in the ejected material, which in turn is set by the thermodynamical conditions at the time of ejection and by weak interactions [123–125]. Colder ejected material retains a low electron fraction Ye < 0.2, which ensures the synthesis of lanthanides and actinides, resulting in a higher opacity and thus in a redder (optical-infrared) transient emission with a slower evolution (with a peak after days to weeks). On the other hand, the material heated by shocks during the merger, or subject to intense neutrino irradiation by the merger remnant or accretion disk, typically has Ye > 0.2 and produces a smaller amount of lanthanides and actinides, resulting in a lower opacity and thus in a bluer (ultraviolet-optical) emission with a faster (peak time 0.2 M in the most favorable conditions (i.e., a low-mass, highly spinning BH and a stiff neutron star matter equation of state), with typical velocities of 0.1c–0.3 c. Tidal ejecta are expected to power a redinfrared kilonova, while shock-heated ejecta give rise to a bluer component; 2. The wind ejecta are produced tens of milliseconds after the merger by the inner, hot part of the accretion disk during the initial neutrino-cooled phase [133] through neutrino-matter interactions and magnetic pressure (e.g., [134, 135]). The amount of mass in this wind ejecta component, which is typically small (of the order 10−4 M ), likely depends on the BH remnant spin, as this sets the size of the innermost stable circular orbit and thus the temperature (and neutrino luminosity) in the inner part of the disk. If the merger remnant goes through a meta-stable neutron star phase (a hyper-massive neutron star supported by differential rotation or a supra-massive neutron star supported by uniform rotation), the neutrino luminosity is expected to be stronger (due to the contribution from the meta-stable neutron star surface) further enhancing the wind, with the resulting wind total mass being positively correlated to the metastable star lifetime. The typical velocity of this ejecta component is around 0.1c [133], while the electron fraction is expected to be Ye  0.25 following from the enhanced inverse beta decay due to neutrino irradiation. This results in a “blue” kilonova emission; 3. The viscous ejecta are produced by the disk during the advection-dominated phase due to enhanced angular momentum transport driven by magnetically induced viscosity [133]. The timescale of this ejection is similar to the accretion

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duration, which in turn is set by angular momentum transport, viscous heating, and nuclear recombination (e.g., [128, 136]). These winds can unbind 20%–40% of the accretion disk mass (i.e., their mass can be few to several ×10−2 M ), representing therefore the dominant kilonova ejecta component in terms of mass. The velocity of these ejecta is low (v ∼ 0.05c), and the Ye distribution is expected to be broad, resulting in an intermediate opacity. It is worth noticing that different kinds of ejecta feature different geometries, with tidal ejecta being expelled preferentially close to the progenitor binary orbital plane, while shock and wind ejecta move along the polar axis (see [137] Fig. 5). This produces viewing angle dependencies (e.g., [108, 138, 139]), with typically a bluer emission when the system is observed perpendicularly to the orbital plane and a redder emission when the source is observed edges-on, along which the more opaque tidal ejecta can substantially shield the emission from the bluer disk winds. The merger remnant could also provide a further energy input that adds up to the radioactive decay power. This could follow from fallback accretion onto the compact remnant or from electromagnetic spin-down of a millisecond magnetar that may be formed after the merger [23]. The last scenario could also generate a fast (timescale of hours) X-ray and ultraviolet “spindown-powered” transient [140, 141].

R-Process Nucleosynthesis and Ejecta Heating by Nuclear Decay R-process nucleosynthesis in kilonova ejecta produces neutron-rich, unstable nuclei on timescales of a few seconds that decay to stability by a combination of β and α decays, γ -ray emission by de-excitation, and fission. This is the main source of heating of the kilonova ejecta and can power observable electromagnetic emission. Since a diversity of nuclei is expected to be produced, with an abundance pattern that is mostly insensitive to the details of the expansion (the only dominant parameter being the initial electron fraction), the resulting radioactive decay energy rate per unit ejecta mass ε˙ nuc is expected to follow a simple, universal power law ε˙ nuc ∝ t −1.3 [125]. The actual ejecta heating rate depends on the thermalization of decay products (electrons, alpha particles, γ -rays, and fission fragments; e.g., [142]), which depends on time (due to the density and thermodynamic evolution implied by the ejecta expansion). Generally, one can express the ejecta heating rate per unit mass ε˙ heat (t) as the product of the nuclear decay energy rate ε˙ nuc (t) times a thermalization efficiency εth (t). The latter depends on the ejecta composition, magnetization, geometry, and distribution of decay products [142, 143] but is typically close to εth ∼ 1 (full thermalization) in the early stages of expansion (t  1d), after which it decreases to εth ∼ 0.1–0.3 over a week timescale. The late time (10–20 d) heating rate is dominated by β decays which, accounting for the decreasing thermalization efficiency due to the expansion, are expected to provide an effective heating that evolves as ε˙ heat ∝ t −2.8 [143].

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Kilonova Emission Features A simple estimate of the main kilonova emission features (peak luminosity, time of the peak and spectral range of the emission) can be obtained by modelling the kilonova ejecta as an expanding spherical shell whose emission peaks when the expansion timescale equals the diffusion timescale. The relevant parameters in this context are the total mass of the ejecta Mej , the ejecta average expansion velocity  vej , and a represenative grey opacity κ. Setting the diffusion timescale td ∼ κMej /4π vej c equal to the bolometric light curve peak time, one obtains  tpeak ∼ 1.5

Mej 10−2 M

1/2 

κ cm2 g−1

1/2

vej −1/2 d 10−1 c

(3)

The luminosity at peak can then be estimated as Lpeak ∼ ε˙ heat (tpeak )Mej (i.e., assuming that all the heat generated within the ejecta diffuses to the photosphere), which gives Lpeak ∼ 8 × 1040

ε  th

0.5

Mej 10−2 M

0.35 

κ cm2 g−1

−0.65

vej 0.65 erg s−1 10−1 c (4)

where we used ε˙ nuc = ε˙ 0 (t/t0 )−1.3 with t0 = 1 s and ε˙ 0 = 3.7 × 1016 erg g−1 s−1 from [125]. The emission is thermal with a complex (and poorly understood) superimposed structure of Doppler-broadened spectral features. The peak effective 4 2 t 2 , where σ temperature can be estimated as Tpeak ∼ Lpeak /4π σSB vej SB is the peak Stephan-Boltzmann constant, which gives Tpeak ∼ 5200

ε 1/4  th

0.5

Mej −2 10 M

−0.16 

κ 2 cm g−1

−0.41

vej −0.088 K 10−1 c (5)

While early attempts at estimating the peak kilonova luminosity predicted values in the range 1042 –1044 erg/s within a day from the merger, with an effective temperature in the optical and UV spectral range, more realistic models that take into account the high opacity of lanthanides find significantly fainter peak bolometric luminosities of 1040 –1042 erg/s and a peak time around a few days [144] The opacity of lanthanides and actinides is not constrained experimentally, but computations based on their (highly complex) electron shell structure indicate it could be up to two orders of magnitude higher than the opacity of iron groups elements. This translates into relatively long diffusion times, leading to low peak temperatures with a spectral peak in the near infrared (NIR) [144]. On the other hand, the small involved ejecta masses make their evolution rather fast when compared to supernovae. This partly explains why kilonovae have not been found in supernova surveys before and why the only spectroscopically confirmed one has been unambiguously identified only

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thanks to an external trigger (AT2017gfo, [69]). A few previous candidate kilonova signatures have been found in association with short GRBs, the most promising one being an infrared emission observed by the Hubble Space Telescope (HST) in excess of the GRB 130603B afterglow [145, 146], which has been interpreted as a kilonova with a rather large ejecta mass of 0.1M .

Interaction with the Interstellar Medium: The Kilonova Radio Remnant While nuclear heating in kilonovae is expected to become negligible after a few weeks or months, the ejecta still contain a considerable amount of kinetic energy EK =

1 2 2 Mej,−2 erg v Mej ∼ 9 × 1049 vej,−1 2 ej

(6)

where vej,−1 = vej /10−1 c and Mej,−2 = vej /10−2 M . Such energy is expected to power an additional emission component, with a similar mechanism as gamma-ray burst afterglows and supernova remnants: the fast, expanding ejecta produce a shock that propagates into the interstellar medium, where particles can be accelerated to relativistic speeds and can radiate away part of their energy, mainly by synchrotron emission. The most obvious difference with respect to GRB afterglows resides in the lower ejecta velocity, which translates into a non-relativistic shock whose evolution is much slower and whose typical emission frequencies fall in the radio band. From the dynamical point of view, the kilonova ejecta remain in essentially free expansion until they sweep an interstellar medium mass comparable to their own, at a “deceleration” radius [147]  Rdec =

3Mej 8π n mp

1/3

1/3

−1/3

∼ 2.4 × 1017 Mej,−2 n0

cm

(7)

where n0 = n/1 cm−3 and n is the interstellar medium number density. This corresponds to a deceleration time, for an observer on Earth, of tdec = (1 + z)

Rdec 1/3 −1/3 −1/3 ∼ 940 (1 + z)Mej,−2 vej,−1 n0 d vej

(8)

if the source is located at a redshift z. In the simplest case of a negligible ejecta velocity dispersion, the light curve (flux density) produced by the shock steadily increases as t α with α = 1.5, 1.6 or 3, depending on the spectral regime, all the way to the deceleration time, and then decays as t −β with β = −(15p −21)/10, where p is the accelerated electron energy power law index [147]. In the presence of an ejecta velocity profile dEK /dvej (a distribution of kinetic energy in velocity space) with a fast tail, which is predicted by numerical relativity simulations [131], the emission can be significantly enhanced at early times and feature a shallower increase that depends on the steepness of the velocity profile. The peak flux depends (similarly to the gamma-ray burst afterglow) on the microphysical parameters εe (the fraction of shock upstream internal energy that goes into accelerated electrons), εB (the fraction

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that goes into random magnetic field amplified by small-scale turbulence), and p and is given (assuming both the synchrotron self-absorption break and the spectral peak frequency are below the observing frequency νobs ) by [147] 5p−4 p+1 p+1 dFpeak p−1 2 4 ∼ 6.7 Mej,−2 vej,−1 n0 4 εe,−1 εB,−2 dνobs



dL 100 Mpc

−2 

νobs (1 + z) 1.4 GHz

− p−1 2

μJy (9)

where εe,−1 = εe /10−1 and εe,−2 = εB /10−2 , dL is the luminosity distance at redshift z, and the numerical value is strictly correct for p = 2.5. As a side note, the assumption of a constant density interstellar medium could turn out to be inadequate in the presence of a relativistic jet launched by the merger remnant: in such a case, under certain circumstances, the relativistic shock produced by the jet may evacuate the region surrounding the merger before the passage of the kilonova ejecta [148]. In such a case, no emission would be produced until the kilonova ejecta reach the interior boundary of the jet shock (which has decelerated to non-relativistic velocities in the meantime). At such time, the impact of the kilonova ejecta against the jet-shocked shell would produce a relatively bright flare [148], following which the evolution would settle back to the previous case. In such a case, depending on the properties of the jet, kilonova ejecta, and interstellar medium, the impact takes place between few years and several decades.

The Electromagnetic Follow-Up of Transient Gravitational Wave Sources The search for the transient electromagnetic counterparts of gravitational wave signals from compact binary mergers follows different steps, described in the following. The fading nature of the electromagnetic signal requires a rapid detection of candidates, through the observation of the GW sky localization area, using wideand small field-of-view telescopes. The firm identification of the counterpart is then secured through photometric and especially spectroscopic characterization. Some reference papers describing the first searches for EM counterparts during the initial LIGO and Virgo observations and the optimization for the advanced GW detectors are [149–154]. This brought extensive broadband EM observational campaigns during the advanced detector era [155, 156].

Low-Latency Search for Gravitational Wave Signals In order to rapidly detect electromagnetic or neutrino counterparts of gravitational wave signals and to maximize the science return of each gravitational wave detection, the LIGO and Virgo collaborations have developed search pipelines running in low latency (online mode) which are able to detect candidate gravitational

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wave events within a few seconds and infrastructures to promptly send alerts to the astronomical community [157, 158]. Two types of searches are performed, modelled and unmodelled. Modelled searches are based on matched filtering techniques, which use banks of waveform templates. For compact binary mergers, the waveforms cover the parameter space of neutron stars and black holes in binary systems (see, e.g., [159–161]), typically reproducing the inspiral, merger, and ringdown phases as predicted by general relativity. For BNS, only the inspiral phase is used for the low-latency detection, since the merger and post-merger phases occur at high frequencies where the current sensitivity does not permit observations of these signals. Unmodelled searches, on the other hand, make few assumptions on the signal morphology, using time-frequency decompositions to identify statistically significant excesspower transients in the data [162]. The astrophysical targets for unmodelled searches are sources for which the expected waveforms are uncertain, such as core-collapse supernovae, magnetar star-quakes, cosmic string cusps, but also binary neutron star post-merger remnants and as-yet-unknown systems. Unmodelled searches are also complementary to the modelled search for binary black-hole coalescences when dealing with non-standard BBHs, possible non-GR events, BBHs with large eccentricity, or intermediate mass black-hole binaries. In order to reduce the noise for the unmodelled search, coherence of the signal in multiple detectors is required. The modelled and unmodelled searches are carried out in almost real time. Low-latency analyses are also performed to search for coincidences between electromagnetic events or neutrino event candidates and GWs. These GW searches are triggered by EM events, such as GRBs from Fermi/GBM, the Neil Gehrels Swift Observatory, INTEGRAL, and AGILE, or by the high-energy neutrino candidates from IceCube, and by galactic supernova neutrino candidates from the SNEWS collaboration. The time window to perform the externally triggered search is based on the expected time delay between the GW signal and the neutrino or EM signal. For example, the unmodelled GW searches targeting long GRBs adopt a time window from 600 s before to 60 s after the GRB trigger. This time window takes into account the possible delayed emission of the relativistic jet retained by the dense material from the core-collapse of the massive star and the γ -ray precursor observed in long GRB up to several hundred seconds before the main γ -ray emission peak (possibly marking the initial event). For the modelled GW searches targeting short GRBs, the time window is chosen to be 5 s before the GRB trigger to 1 s after the GRB trigger considering the onset of the γ -ray emission delayed only up to a few seconds from the GW emission. This time window is considered large enough to take into account any uncertainty in the definition of the GRB trigger time. As soon as the search pipelines report a candidate event, full parameter estimation analyses begin. These analyses use Bayesian inference to calculate the posterior probability distribution over the parameters (sky location, distance, and/or intrinsic properties of the source). Also for the unmodelled burst signals, parameter estimation codes based on stochastic sampling and Markov Chain Monte Carlo algorithms (able to model both signals and noise glitches) are run over the data.

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These pipelines, exploring a larger parameter space, are computationally more expensive than the search pipelines, and the results are available with a higher latency of hours to days. Over a longer timescale of weeks and months, the offline analyses take the advantage of improved calibration of the data and additional information regarding data quality. The selection of the GW candidate events to be publicly sent is based on their significance with respect to the noise, given by the False Alarm Rate (FAR). The FAR is the rate of accidental events (due to detector noise or transient noise artifacts known as glitches) which are as loud as or louder than the GW candidate event (in the absence of any astrophysical sources). Automated preliminary alerts are sent for all candidate events with a FAR below a fixed threshold and passing data quality automated checks. One of the main challenges for the low-latency search pipeline is the rapid characterization of the transient noise and the identification of noise sources. Information about the detector status are provided to the low-latency analysis pipelines indicating when the data are suitable for the analysis. Glitches that mimic true GW signals but are uncorrelated in the GW detectors can be present in the data stream. Sensors which measure the behavior of the instruments and their environment work to identify them and mitigate their effect. When such glitches are identified, automatic data quality vetoes are generated. More details on the lowlatency searches, the distribution of the alerts, and the alert contents (which will be described in the following sections) can be found in [157, 158] and the LIGO/Virgo Public Alerts User Guide (https://emfollow.docs.ligo.org/userguide/).

The Gravitational Wave Alert System: Distribution and Alert Contents Alerts are distributed using the Gamma-ray Coordinates Network (GCN) system [163], widely used by the astronomical community for rapid communications about the discovery and observations of GRBs. Two types of alerts are used: GCN Notices, which are machine-readable alerts, and GCN Circulars, which are human-readable astronomical bulletins. A Preliminary GCN Notice is issued automatically within minutes after a GW candidate is detected. In order to send the alert, the candidate must have passed automated data quality checks. Later, the candidate may be retracted on the basis of the evaluation of a rapid response team (composed of experts from the detector sites, the analysis teams, the detector characterization team, and the low-latency follow-up team) looking carefully at the detector characterization and data quality. Within a few hours, an Initial GCN Notice is issued when the signal passes human vetting or a Retraction GCN Notice when the signal is rejected. The initial alert is also accompanied by a descriptive GCN Circular. Update GCN Notices and Circulars are issued whenever further analysis leads to improved estimates of the sky localization, significance, or source classification. These updates are sent with different latencies ranging from hours, days, to several weeks after the candidate event detection.

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Exceptionally loud and nearby BNS mergers are detectable up to tens of seconds before merger during the inspiral phase, when the signal sweeps frequencies ranging from tens to thousands of hertz that are within the GW detector sensitive band [164]. For these systems early-warning pre-merger alerts are possible. The GCN Notices (except for the Retraction ones) contain the candidate unique identifier name, the event time, the estimated FAR, and the sky localization. For compact binary merger candidates, they also contain the source distance, the volume localization (3D sky map), the source classification, and properties. Information about candidate GW events are collected in the gravitational wave Candidate Event Database (GraceDB, https://gracedb.ligo.org/).

The Gravitational Wave Sky Localization The sky localization of the GW signal is given in the form of a posterior probability distribution of the source position. In the simplest analysis, the sky position of a GW transient source is obtained by measuring the differences in signal arrival times at the different network detector sites [165, 166]. This yields, per each detector pair, a localization annulus in the sky whose width depends on the accuracy of the arrival time determination. Using a Fischer matrix method, the effective single-site timing accuracy is approximately σt =

1 , 2πρσf

(10)

where ρ is the signal-to-noise-ratio (SNR) in the given detector and σf is the effective bandwidth of the signal in the detector, typically of the order 100 Hz. Thus, a typical timing accuracy is on the order of 10−4 s (about 1/100 of the typical light travel time between sites, which is of the order 10 ms). This simple formula shows that the linear size of the sky localization region scales inversely with the SNR and the frequency bandwidth of the signal in the detector. As stated above, for typical CBC signals, the effective bandwidth in the advanced detectors is ∼100 Hz, but higher mass CBC systems merge at lower frequencies, resulting in a smaller effective bandwidth and thus a worse sky localization. For burst signals, the bandwidth depends on the specific signal: for example, core-collapse supernovae are expected to have relatively large bandwidths, between 150 and 500 Hz. The actual determination of the signal sky-localization takes also into account consistency of the signal amplitudes and phases across the detector network. While the source localization using only the arrival time delay yields an annulus on the sky for a two-site network, requiring consistent amplitudes and phase excludes parts of the annulus, typically resulting in two separate, elongated regions, which typically span hundreds to thousands of square degrees. For three detectors, the time delays restrict the sky location to two regions which are mirror images with respect to the plane passing through the three sites. The amplitude and phase requirements typically eliminate one of the two regions. For a three-detector network, the sky localization covers regions of several tens to hundreds of square degrees. With four or more detectors, the timing information alone is enough to localize into a single

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region, smaller than ten square degrees for some signals (see [158], and references therein). Providing prompt localizations for GW signals is crucial to increase the probability to identify an EM counterpart and to detect the early EM emission. Sky localizations are produced and distributed at several different latencies. After the initial localization, updates resulting from increasingly computationally expensive algorithms, which refine the source parameter estimation, are released as soon as they become available. The first sky localization is generated by BAYESTAR [167], a rapid CBC sky localization algorithm based which computes the posterior probability distribution over the sky location and distance of the source by coherently modeling the response of the gravitational wave detector network and using output from the detection pipeline. This algorithm takes typically tens of seconds to produce the sky map. At higher latency, for CBC candidates, a number of Bayesian inference algorithms implemented in the LIGO/Virgo analysis software suite LALinference [14] explore the full parameter space (sky location, distance, masses, and spins) of the source by using Markov Chain Monte Carlo (MCMC) and nested sampling and produce a refined sky localization. An automated LALInference analysis, which uses the least expensive CBC waveform models and produce a sky map within hours, is run online. More time-consuming analyses with more sophisticated waveform models require a longer latency to be completed, days to weeks. For burst candidates, the unmodelled search pipeline, coherent Wave Burst (cWB), performs a rapid sky localization based on a coherent reconstruction of the GW signal using a wavelet basis and the response of the GW detector network [168]. Refined sky localizations for unmodelled bursts are provided by algorithms using the same MCMC and nested sampling methodology as LALInference [169, 170]. For details on the sky localization properties and perspectives for different detector network configurations in the upcoming years, see [158]. The GW localization is given as posterior probability distribution of the source’s sky position and is encoded as HEALPIX all-sky map in FITS file format. For CBC event candidates, the sky localization is a three-dimensional (3D) sky map: for each line of sight, the map also contains the sky-location-conditional distance posterior probability approximated as a Gaussian multiplied by the distance squared [171]. The 3D sky maps are extremely important to set up galaxy targeting observational strategies (see section “Electromagnetic Counterpart Search Strategies”) and can also be used as an input for constructing more refined (but model dependent) search strategies aimed at maximizing the probability of observing the counterpart at a time when it is above the search sensitivity threshold (e.g. [172]).

Source Classification and Properties The classification of the CBC sources is defined in term of probability that the source belongs to five mutually exclusive categories based on the component masses (see Fig. 4): BNS merger (both component masses 5m ),

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Fig. 4 The four astrophysical categories in terms (BNS, NSBH, BBH, and MassGap) of component masses m1 and m2 . By convention, the component masses are defined such that m1  m2 . (Figure is taken from the LIGO/Virgo Public Alerts User Guide [174])

and Terrestrial (i.e., Noise). The formalism used to compute this probability is given in [173]. Assuming that the GW signal is astrophysical (i.e., non-terrestrial), the probability that one or both components have a mass consistent with a neutron star (mass 100

time off-axis observer flux

neutrino-driven winds ~ 0.1 c

time

edge-on observer

flux

dynamic ejecta ~ 0.1 c accretion disk time

remnant

Fig. 5 Sketch of the different emission components and of how they are observed at different viewing angles. (Figure adapted from [200, 201])

velocity. Then, the brightening of the near-infrared emission is expected from the lanthanide-rich ejecta. As the ejecta expand, the opacity decreases, and the atomic species imprint broad absorption-like lines on the spectral continuum. The identification of the heavy elements produced during the merger is made complicated by the blending of the forest of lines associated with heavy elements broaden by the large velocity of the ejecta. Analogously to the SN case, kilonova polarization can be produced depending on the degree of asymmetry of the ejecta, the optical depth for scattering due to free electrons, and the viewing angle of the system [70, 202]. AT 2017gfo associated with GW170817 showed the broad range of signatures described above and enables to gain a deeper insights emission mechanism of a kilonova (for details see section “GW170817 and Its Electromagnetic Counterparts”). Its evolution from blue to red in ten days was indicative of multi-component ejecta and the production of atoms with a complex electron structure, whose many transitions absorbed optical light. This is exactly what is expected for systems forming the heaviest elements through the r-process. While the selection of the most promising candidates can be done using 1–3-meter telescopes (such as the Palomar 200 inch Hale telescope, the Liverpool Telescope, the Palomar 200 inch Hale telescope, the Telescopio Nazionale Galileo,

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the Nordic Optical Telescope), larger telescopes of 4–10 meters are required to capture detailed features of the kilonova spectrum and its evolution. Some examples of these instruments are the X-shooter spectrograph on the ESO Very Large Telescope, the EFOSC2 instrument in spectroscopic mode at the ESO New Technology Telescope, the Goodman Spectrograph on the 4 m SOAR telescope, and the FLAMINGOS2 near-infrared spectrograph at Gemini-South. The Hubble Space Telescope is another key instrument which avoids challenging atmospheric absorption in the case of infrared spectroscopy.

Radio Observations BNS and NSBH mergers are expected to produce emission in the radio band through a variety of mechanisms [153]. These include potential brief radio bursts from the interaction of the neutron star magnetospheres shortly preceding the merger [203] and much longer-lasting emission from shocks in the interstellar medium caused by the gamma-ray burst jet (section “Jet Launch by the Merger Remnant”) and by the kilonova ejecta (section “The Kilonova”). An observation of the former type of emission would require a wide-field radio facility (such as Aperitif, [204]) to point (either serendipitously or thanks to an early warning alert [164]) the source position before the merger, and it constitutes a challenge to be addressed by future facilities. Observations of the latter types of emission are instead in the reach of current facilities, and here we focus on these. The brightness and time evolution of radio emission from gamma-ray burst jet afterglows depend strongly on the jet viewing angle (see Fig. 5), in addition to being sensitive to the jet properties (energy and structure), those of the interstellar medium (density) and the shock microphysics (encoded in the parameters εe , εB and p, see section “Jet Launch by the Merger Remnant”). As a general trend, the larger the jet angle with respect to the line of sight, the later and fainter is the peak of the radio emission. The light curve peak time for off-axis jets is independent of the observing frequency (being due to the jet core Lorentz factor decreasing to Γ ∼ θv−1 ) and is approximately given by [198] (assuming an electron power law index p = 2.2, and using the same notation as in section “Interaction with the Interstellar Medium: The Kilonova Radio Remnant”) 

θv − θc tp ∼ 520 20◦

8/3

−1/3

1/3

n−3 E52 days

(11)

where θc is the half-opening angle of the jet core and E52 is its isotropic-equivalent kinetic energy in units of 1052 erg. The corresponding peak flux density is [198] dFpk −0.6 1.2 0.8 ∼ 0.2 νobs,GHz n0.8 −3 E52 εe,−1 εB,−2 dν



θv − θc 20◦

2.4 

dL 200 Mpc

−2 mJy

(12) This is in reach of current facilities in the most favorable cases, making the radio detection of short gamma-ray burst afterglows in association with future

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gravitational wave events from binary neutron star mergers a potentially highly rewarding target for future electromagnetic follow-up. The emission preceding the peak time is dominated by off-axis parts of the jet, and observations at these times are therefore precious as tools to constrain the jet structure [102, 205]. After the peak, the light curve is instead dominated by the jet core, and it resembles that of an on-axis jet at a similar observer time. One important caveat is that radio emission (especially at low frequency) from the afterglow of a jet seen on-axis can be suppressed by synchrotron self-absorption at early times: the self-absorption frequency for an on-axis observer (in the relevant regime) is given by [206] (again assuming an electron power law index p = 2.2) −1 νsa ∼ 144 εe,−1 εB,−2 n−3 E52 MHz 1/5

3/5

1/5

(13)

and it remains constant up to the jet lateral expansion (section “Jet Launch by the Merger Remnant”), after which it gradually moves to lower frequencies. This shows that early low-frequency radio observations of a short gamma-ray burst afterglow, for example, with LOFAR [207], are unlikely to be successful if the jet expands in a relatively high-density environment with n  10−2 cm−3 , regardless of the viewing angle (if they are, on the other hand, they can be used to constrain the interstellar medium density). As discussed in section “Interaction with the Interstellar Medium: The Kilonova Radio Remnant”, the emission from the kilonova radio remnant evolves over even longer timescales of several years. At such times, the jet emission is expected to be fainter (as the actual kinetic energy content in the jet is generally smaller than that in the kilonova ejecta, due to collimation), and the kilonova remnant is therefore the dominant radio component (but see the caveat at the end of section “Interaction with the Interstellar Medium: The Kilonova Radio Remnant”). Both electromagnetic counterparts produce potentially detectable radio emission in the reach of the most sensitive single-dish radio facilities (such as the Effelsberg radio telescope, [208]). Potential contamination from the host galaxy (either due to star formation or to the presence of an active galactic nucleus) could be resolved only with the higher spatial resolution of large aperture synthesis interferometers such as the Karl Jansky Very Large Array [209] or even Very Large Baseline Interferometry. The extremely high resolution of the latter observing technique can also be used (in sufficiently bright cases) to put constraints on the source image size [102] or on its proper motion [101], providing a complementary and extremely powerful tool to investigate the properties of the source (see section “GW170817 and Its Electromagnetic Counterparts” for the remarkabke case of GW170817).

High-Energy Follow-Up Observations In order to follow-up the evolution of the non-thermal X-ray emission from the jet of the short GRBs possibly associated with the BNS and NS-BH mergers over timescales of weeks to months, deep sensitivity instruments, such as Chandra or XMM-Netwon, are required. Going to larger distance and mainly for off-axis

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observers, the detection of the X-ray emission will require the new generation of X-ray instruments, such as the Athena space X-ray Observatory. Emission from BNS and NS-BH mergers is also expected up to the MeV energy range, as consequence of the nuclear decays of the kilonova (see section “The Kilonova”, [23, 210]). The MeV component of the kilonova is initially absorbed in the optical thick medium of the ejecta. It will emerge a few days after the merger, and its luminosity is expected to be lower than ∼1041 erg/s [211], thus hardly detectable with present and next future generation of instruments, unless very close to Earth [212]. Other than the kilonova, gamma-rays up to GeV energies are emitted from the short GRB possibly associated with the merger. In short GRB, the non-thermal, gamma-ray emission produced in the on-axis jet is amplified by the beaming effects (see section “Jet Launch by the Merger Remnant”). However, in the majority of the GW events forming a GRB, the collimated jet is seen off-axis, with the line of sight beyond the opening angle of the jet, typically a few degrees. The Doppler factor from off-axis GRB is small, and the corresponding gamma-ray emission, which scales as ∼δ 4 (see Eq. 2), is thus strongly suppressed [213]. The deceleration of the jet during its propagation in the ambient medium, and the possibility of structured jets, as described in section “Jet Launch by the Merger Remnant”, will increase the beaming cone of the relativistic outflow, eventually including the line of sight. The corresponding emission will become mildly Doppler boosted and will bring to observable effects. In these cases, the gamma-ray emission is expected to be delayed by several hours to days/weeks, similarly to the non-thermal emission expected in radio and X-ray [200]. The detection of the gamma-ray prompt emission of the GW transient requires an all-sky gamma-ray monitor, as provided by observatories on satellites or groundbased instruments with large field of view. Targeted searches can be performed by smaller field instruments. In the energy range from MeV to tens of Gev observations are provided by observatories on satellites and in the energy range from tens of GeV up to TeV (teraelectronvolts, 1012 eV) by ground-based telescopes, such as the Cherenkov telescopes. A description of these instruments, and how they contributed to the present stage of the studies on the GeV and TeV emission from the GW counterparts, is provided in the companion  Chap. 23, “Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravitational Waves.”

GW170817 and Its Electromagnetic Counterparts The observations of the GW170817 electromagnetic counterparts [69] and in particular the detection of the associated kilonova AT2017gfo [126] provided the astronomical community with the first practical example of how to identify interesting candidate optical counterparts. In particular, this unique example showed that, at least in this case, the UV to NIR emission is dominated by the kilonova on timescales of 1–2 weeks after the merger. In particular, photometric observations revealed a blue transient fading within 48 hours and an optical to NIR redward

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evolution over about 10 days. One of the key properties, which first pointed to AT2017gfo being an unusual transient, was its rapid luminosity decline in the bluer optical bands within the first few hours. However, the first spectra, showing a blue and featureless continuum, were similar to what is observed for young core-collapse SNe [156]. Only the spectra taken on the second day from the merger were decisive to firmly identify the kilonova and exclude a young SN [126]. They showed an exceptionally fast spectral evolution and the absence of features identifiable with absorption lines common in SN-like transients. X-shooter (the UV-IR spectrograph mounted on the 8.2 m Very Large Telescope at ESO) spectra (see Fig. 6) obtained during about ten nights are by far the best spectral sequence for the GW170817 event, as can be seen in Fig. 6 [126, 214]. They represent the first spectral identification of the kilonova emission. At 2.5 days after merger, the spectrum starts deviating from a black body and revealing the signatures of r-process nucleosynthesis elements as absorption features imprinted on the spectral continuum. These spectra have been used to identify individual r-process elements such as cesium and tellurium [214] and Strontium [215]. Despite the fact that the basic kilonova models are successful in explaining the major features of the X-shooter spectral sequence, the interpretation of the spectra details and evolution are still far from being conclusive. Models assume spherical symmetry, local thermodynamic equilibrium, and uniform abundances. None applies to the actual case, and moreover, the current atomic parameters for the relevant elements, in particular lanthanide, are barely known. High-quality, spectral

Fig. 6 VLT X-shooter and FORS2 spectra. The flux normalization is arbitrary. (Adapted from [126])

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evolutionary sequence are needed as a benchmark for any model development. More kilonova events will allow us to probe the models in different conditions [216] and to disentangle the micro-physics from the viewing effects (line of sight) and the energetics (masses of the neutron stars). The variance of events will allow us to assess whether the neutron star mergers can entirely explain the abundance of heavy nuclei in the Universe. The broadband monitoring of the source position during the first week following the kilonova discovery did not produce any detection in radio and X-rays. In particular, relatively stringent X-ray upper limits from Swift/XRT and from the Nuclear Spectroscopic Telescope Array (NuSTAR) clearly excluded afterglow emission from a relativistic jet pointing toward the Earth [217], despite the gammaray burst detection by Fermi/GBM [218] and INTEGRAL/SPI-ACS [219]. Nine days after GW170817, though, the Chandra satellite first obtained a detection of a faint X-ray counterpart, followed by a second detection 6 days later [200, 220]. On the 16th day after GW170817, the Karl Jansky Very Large Array (JVLA) also detected a faint radio source at 6 GHz, at the same position [221], and announced an additional detection at around 11 days post-GW170817 obtained by stacking four observations which did not separately yield a detection. These four detections altogether provided evidence of an emerging broadband, non-thermal slowly rising counterpart. No further X-ray nor optical monitoring of the source could be carried out for a long time, though, as its projected position stayed too close to the Sun during the next 3 months and a half. The position of the Sun did not constitute a problem for radio observations, though, and the JVLA continued its monitoring silently during the next months, until the results of the first 100 days of monitoring were published [222]. The light curve showed an unexpected steady, shallow increase of the flux density rising as t 0.8 . Soon after, the Sun ceased to constitute a constraint to X-ray and optical observations, and the source was detected in X-rays by XMM-Newton [223] and again by Chandra [224] and also in the optical by the Hubble Space Telescope [225] and by the Large Binocular Telescope [102]. These broadband detections showed [224] that the spectrum of the source consistently remained a power law with dF /dν ∝ ν −0.6 across all observed bands, providing striking support toward the interpretation of the emission as synchrotron from shock-accelerated electrons whose energy distribution was a power law. Observations around that time also first suggested [223] that the source evolution was about to reach the peak (the actual peak was later reached around 160 days after the merger – see Fig. 7 for an overview of the light curves of this emission component at different wavelengths). The unexpected light curve evolution sparked a debate within the community, which led to the emergence of two main interpretations, both of which required the emission to come from a (at least mildly) relativistic shock propagating into the interstellar medium. Either the shock was produced by an off-axis structured jet (e.g., [118, 223]) (see section “Jet Launch by the Merger Remnant” for a description of the underlying mechanism) which was launched by the merger remnant and successfully punched through the kilonova ejecta or it was caused by a quasi-spherical explosion which produced an outflow with a steep velocity

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Fig. 7 Multi-wavelength light curves of the long-lasting, non-thermal electromagnetic counterpart of GW170817, as observed in the Radio, Optical, and X-ray bands up to 300 days after the merger. The flux densities have been re-scaled by the factors shown in parentheses in the legend for presentation purposes. The solid and dashed lines represent the predicted emission from the bestfitting structured jet and quasi-spherical outflow models from [102], respectively. (Reproduced from Figure 3 of [102])

profile (with more energy in the slower ejecta). The latter outflow could have been the result either of the jet being unable to successfully punch through the kilonova ejecta (a “choked jet” which would have deposited all its energy into a hot cocoon, which would then expand and give rise to the outflow [222, 226]) or of a process linked to magnetic amplification during the merger (e.g., [227, 228]). The debate was shown to be unsolvable [99] solely by monitoring the light curve evolution before the peak. The killing observations were instead provided by highresolution Very Large Baseline Interferometry [101, 102] approximately 70, 200, and 230 days after the merger, which revealed an apparently superluminal motion of the radio source centroid and a very small projected size of the source image [102], in clear support of the off-axis structured jet scenario. Later light curve monitoring after the peak of the emission confirmed this interpretation [229], and the detailed study and modelling of the well-sampled, multi-wavelength emission and of the radio centroid motion permitted a relatively detailed reconstruction of the jet properties [102, 230], showing that the jet, if seen on-axis, would have produced an afterglow that resembles that of previously known short gamma-ray bursts [117]. This provided clear, solid evidence in support of binary neutron star mergers being the progenitor of at least a fraction of the known, cosmological short gamma-ray bursts. Knowledge of jet properties also allowed to mitigate the distance-inclination

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degeneracy inherent in gravitational wave emission, improving [230] the constraint on the Hubble constant [231] obtained using GW170817 as a “standard siren.” GW170817 has been a clear example of the importance of using a network of multi-wavelength observatories able to cover huge region of the sky, the importance of using larger telescopes able to characterize the transients, and the importance to repeat pointed multi-wavelength observations over different timescales (up to years after the event).

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Multi-messenger Astrophysics with the Highest Energy Counterparts of Gravitational Waves

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Antonio Stamerra, Barbara Patricelli, Imre Bartos, and Marica Branchesi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Observations in a Multi-messenger Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Physical Parameters Derived from GW Observations . . . . . . . . . . . . . . . . . . . . . . . . . Complementarity of Electromagnetic, Gravitational Waves and Particle Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Energy Neutrino Emission from Gravitational Wave Sources . . . . . . . . . . . . . . . . . UHECR Emission from Gravitational Wave Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The GeV-TeV Astronomy with the Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts . . . . . . . . . . . . GeV-TeV Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Energy Gamma Rays from GRBs and Their Implications on Gravitational Wave Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observation Strategies with Neutrino and Gamma-Ray Instruments . . . . . . . . . . . . . . . . . . . Neutrino Follow-Up of Gravitational Wave Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Stamerra () INAF, Osservatorio Astronomico di Roma, Monte Porzio Catone (Roma), Italy SNS, Scuola Normale Superiore di Pisa, Pisa, Italy SSDC, Space Science Data Center, Roma, Italy e-mail: [email protected] B. Patricelli European Gravitational Observatory, Cascina, Italy e-mail: [email protected] I. Bartos University of Florida, Gainesville, USA e-mail: [email protected] M. Branchesi INFN, Laboratori Nazionali del Gran Sasso, Gran Sasso Science Institute, L’Aquila, Assergi, Italy e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_23

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GeV-TeV Follow-Up of GW Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary on Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The first observation of a binary neutron star merger through gravitational waves (GWs) and the detection of its electromagnetic (EM) counterpart in different energy bands marked the beginning of multi-messenger astronomy with GWs and demonstrated the huge informative power of multi-messenger observations. In this chapter, we review the highest energy counterparts of GW sources detectable by current-generation GW interferometers: this include high-energy photons, neutrinos, and cosmic rays. They are expected from the disruption of the GW progenitor, produced in the accretion of matter and its relativistic ejection during the final coalescence phase. The detection of high-energy neutrinos and cosmic rays requires large-volume detectors, while high-energy gamma rays are observed by means of satellites and ground-based Cherenkov telescopes. We briefly introduce them along with the recent advances in the identification of the neutrino and gamma-ray counterparts of GW. We detail the complementary information we can get observing astrophysical sources with these messengers; we describe the strategies currently used to follow up GWs, the observational status, and the prospects for the next years. Keywords

Multi-messenger astrophysics · Black holes · Neutron stars · Gravitational waves · Neutrinos · Cosmic rays · Gamma rays

Introduction The detection of the first gravitational wave in 2015 [1] and the identification of the electromagnetic counterpart of the neutron star merger GW 170817 (see [2] and references therein) did reveal the richness of information carried by the new astrophysical messenger. The two probes did complement each other in providing the observational clues needed to identify the counterpart and in yielding a deeper knowledge and details of the merger of the two neutron stars. The multi-messenger observations revealed the connection of the merger of binary system of neutron starts with the gamma-ray burst (GRB) phenomenon and with the kilonova emission, as extensively described in the accompanying  Chap. 22, “Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range.” Electromagnetic emission was observed from radio wavelengths up to hundreds of kiloelectronvolts (keV), with a beautiful interplay between the different source components and emission processes. We know that energetic astrophysical sources, powered by compact gravitationally powered objects, do emit up to the highest gamma-ray energies, to

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giga- and tera-electronvolts (109 − 1012 eV ∼ 1023 − 1026 Hz). At these extreme energies, and beyond, other two non-electromagnetic messengers, high-energy neutrinos (E∼10 TeV) and ultrahigh-energy cosmic rays (UHECR, E>∼1018 eV), have been detected. The number of emitted photons at such high energies is relatively small, reaching us with fluxes typically between 10−6 and 10−11 photons/cm2 /s (or ≈ 10−10 Jy, referred to the Crab Nebula at 1 giga-electronvolts) or below, to be compared with the fluxes of the order of 1 Jy in the radio/optical wavelengths. However, the power output associated with their emission can be comparable to that at lower energies, as a consequence of their high energy, with typical values of 10−8 − 10−12 erg/cm2 /s. As such, the high-energy, gamma rays, with possibly the correlated neutrino and cosmic ray emission (see next section), provide a non-negligible contribution to the total energy budget of these sources. The high-energy neutrino and the UHECR are produced in the interaction of relativistic charged particles, protons, and nuclei, with the ambient. These production processes are called hadronic processes. The surrounding ambient can be formed by the interstellar medium, molecular clouds, the ejected material during an explosive event, or the intense photon field (photo-meson production). The details of their acceleration and interaction (and further propagation, for the charged UHECR) and the sources they originate from are still open questions. The high-energy gamma rays are expected to be interwoven with the highenergy neutrinos and cosmic rays, being products of the same emission processes. Hence, observations of high-energy gamma rays, cosmic rays, and neutrinos are complementary. Neutrinos with energies higher than tens of tera-electronvolt (TeV) are still elusive, but their detection is easier due to the larger cross section increasing with energy (see section “High-Energy Neutrino Emission from Gravitational Wave Sources”). Still, detectors with large volumes are needed, such as IceCube. (https:// icecube.wisc.edu) IceCube did prove the astrophysical origin of a fraction of the detected neutrino (see [3] for a review). A strong hint for a direct association of a single energetic neutrino with a gamma ray emitting active galactic nucleus (AGN) has been found suggesting the association with this class of sources [4]. But the association of high-energy neutrinos with gravitational wave (GW) sources, like GRBs and supernovae, is not proven yet. The interactions of relativistic protons and nuclei yielding high-energy neutrinos produce also giga-electronvolt (GeV) and TeV gamma rays. If protons are accelerated in the sources of GW, an emission component in the gamma rays is expected. Such component has been discovered in long GRB and possibly in short ones (see section “The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts”), but its interpretation is still open. In fact, the observed gamma rays are likely originated by the so-called leptonic processes, proceeding through relativistic electrons. Starting from this observational and phenomenological background, this chapter inspects the possible connection of the neutrino and UHECR messengers and of the energetic gamma rays with the gravitational wave events.

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GW Observations in a Multi-messenger Context The Physical Parameters Derived from GW Observations Gravitational waves are emitted by accelerating matter. Given that gravity is extremely weak compared to the other forces, a detectable gravitational wave signal typically requires the acceleration of matter on the solar-mass scale. Therefore, gravitational waves carry information about the bulk dynamics of matter. For the most significant sources of gravitational waves, which are the binary merger of black holes and neutron stars, the gravitational waveform carries information about the merging objects’ masses and spins and the binary’s direction, distance, inclination, and orbital eccentricity. For neutron star mergers, information is also included on tidal deformation prior to merger. We can learn about these parameters by comparing the observed data to computed gravitational waveforms. These computations are often done using post-Newtonian approximations of the Einstein field equations and in some cases, numerical simulations of the binary evolution that fully takes into account the Einstein field equations. The reconstruction of binary parameters are not equally possible, however. Some of them affect the gravitational waveform much more than other; therefore, they can be reconstructed with higher precision. The parameter that can in most cases be measured most precisely is a combination of the two masses within the binary, the so-called chirp mass: Mc =

(m1 m2 )3/5 , (m1 + m2 )1/5

(1)

where m1 and m2 are the masses of the two compact objects (m1 ≥ m2 ). This parameter determines the early inspiral waveform of the binary. It is informative, however, determining the individual component masses is often critical in learning about the origin of the binary, or whether it is expected to produce an electromagnetic or neutrino counterpart. For example, a neutron star merging with a heavy black hole can have the same chirp mass as the merger of two lighter black holes. Other parameters of the compact objects will only affect the waveform closer to merger and are typically more difficult to measure. For the spins of the compact objects, individual spins are typically difficult to reconstruct, and we can most precisely estimate the total spin of the two objects that is parallel to the orbital axis, the so-called effective spin: χeff ≡

c GM



S1 S2 + m1 m2

 ·

L |L|

(2)

where M = m1 + m2 is the total mass of the binary, S1,2 are the spin angular momentum vectors of the two components of the binary, and L is the orbital angular momentum vector. Measuring spin is possible for black holes but is virtually impossible for neutron stars since (i) the neutron star spin will only affect the

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gravitational waveform at very high frequencies, if at all, that is difficult to measure with present observatories; (ii) neutron stars cannot spin too fast as they would get tidally disrupted; (iii) neutron stars in actual mergers are expected to spin much less than they could, based on observations of galactic binary neutron stars BNS) systems (see, e.g., [2] and references therein). A black hole’s spin component that is perpendicular to the orbital axis is typically difficult to reconstruct; however, it is sometimes possible. This parameter is very important in determining the origin of binary mergers (see, e.g., 5). Such spin component is not expected in systems originating from massive stellar binaries, where the stars synchronize their spin to the binary orbit before they collapse and form black holes. If perpendicular spin is present, it will result in the precession of the binary orbit prior to merger, which can be detected at favorable inclination angles and at high signal-to-noise ratio detections. The tidal deformation of neutron stars, while very important in determining the properties of matter supranuclear densities, is difficult to measure from gravitational waves. It will only affect the waveform very late in the binary, i.e., only for a short time, and at very high frequencies (>1 kHz) where current detectors have limited sensitivity. The core collapse of massive stars also produces gravitational waves [6]. Due to the large uncertainties that affect our knowledge of the collapsing phase, the amount of GW released energy and the expected GW waveforms are highly uncertain. Here the reorganization of matter during the formation of a single neutron star is the main driver of gravitational wave emission. Black holes can also form as the end result of stellar core collapse, but this transition is not an efficient gravitational wave source. The core collapse signal is detectable for supernovae occurring in the Milky Way. However, according to the most extreme models, the signal from supernovae can be detected possibly up to tens of hundreds of megaparsecs [7]. These higher distances can be reached if neutron stars born during core collapse are rapidly rotating and develop instabilities that deform them [8]. Such deformations can lead to a rotating, bar-like shape that is an efficient gravitational-wave emitter [9]. The interior magnetic field, if it is in the magnetar range (∼1016 G), might also lead to a significant deformation and to efficient GW emission [10]. If a significant fraction of the rotational energy of a rapidly rotating neutron star can be converted to gravitational waves, this signal could be detected potentially to tens of megaparsecs, within which multiple supernovae occur every year. Fast rotating stars could also fragment during the collapse that could lead to an effective binary merger of the fragments with significant gravitational wave production [11].

Complementarity of Electromagnetic, Gravitational Waves and Particle Observations When multi-messenger information is available about a source, even more information is extractable from gravitational waves. This is a very rich and actively studied area. Here are a few illustrative examples.

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• Gravitational wave emission gives precise information on the time of merger. This can be compared to the time of the onset of other emission, for example, in the case of gamma-ray bursts. For instance, in the case of the first neutron star merger observed through gravitational waves, GW170817, the time difference between the merger and the start of the gamma-ray emission was measured to be 1.7 s [12]. This gives a time frame for the formation of a black hole from the merger remnant, accretion of the matter onto this black hole that was shed by the neutron stars during the merger due to tidal effects, and particle acceleration in the high-velocity outflow that is driven by this accreting black hole and that ultimately leads to the production of high-energy photons, neutrinos, and cosmic rays [13]. • Neutrinos and gravitational waves are both interesting because they can escape from stellar interiors that are opaque to photons. Detecting both gravitational waves and high-energy neutrinos will be informative of how high-velocity jets are launched and then quenched in the aftermath of stellar core collapse. If the jets cannot escape the stellar envelope, only these messengers will be able to escape, enabling us to learn about these choked jets [14]. • Black holes may also produce multi-messenger emission if they are in a sufficiently gas-rich environment [15]. Black hole mergers in gas-rich environments form somewhat differently than in other cases. • For high-energy gamma-ray emission, one of the important properties of the binary is its inclination [16], which determines whether the jet produced in the aftermath of the merger will point toward the Earth or elsewhere. Gravitational waves alone do not precisely tell us the inclination of the binary; instead, only a combination of inclination and the merger’s luminosity distance is measured. If, however, we can measure the merger’s luminosity distance through multi-messenger observations, e.g., by identifying the host galaxy from the precise multi-messenger localization for which distance is readily available, the inclination of the binary can be more accurately reconstructed [17]. • In some cases, inclination of the binary may also be constrained using properties of its environment. For example, if a binary merger occurs within the accretion disk of an Active Galactic Nucleus (AGN), the binary plain can be aligned with the plain of the AGN disk. As the latter can be measured with reasonable accuracy in some cases, by requiring that the host galaxy of the binary merger has the same inclination as the AGN disk, in some cases, we can significantly improve the localization of the binary [18].

High-Energy Neutrino Emission from Gravitational Wave Sources While gravitational waves are produced by the bulk acceleration of matter, highenergy neutrino emission requires hadronic interactions. Internal shocks in relativistic outflows can accelerate protons to high energies. These ultra-relativistic protons then interact with either photons (photohadronic interaction), as first suggested by

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[19], or slower protons or neutrons (hadronuclear interaction). These interactions in turn produce high-energy gamma rays and high-energy neutrinos. Multiple instruments focus on detecting astrophysical high-energy neutrinos. They operate by detecting Cherenkov radiation from neutrinos that interacted in ice, water, or air. They include the IceCube Neutrino Observatory [20], ANTARES [21], and the Baikal neutrino detector [22], which are sensitive to neutrinos in the 102 − 108 GeV energy range, the KM3NeT detector [23] currently under construction focusing on a similar energy band, and the Pierre Auger Observatory (https://www.auger.org), which is sensitive to neutrinos in the 106 − 1012 GeV energy band. Some astrophysical events give rise to both gravitational wave and high-energy neutrino emission. There are two main astrophysical event types relevant for the joint emission of gravitational waves and high-energy neutrinos: (1) the mergers of neutron stars and black holes and (2) core collapse supernovae [8]. The former group can be further divided to BNS mergers, neutron star-black hole mergers, and binary black hole mergers, with different prospects of neutrino emission. We discuss these cases separately below. Binary neutron star mergers – Neutron star mergers are known to produce relativistic outflows, as seen in the aftermath of GW170817/GRB 170817A [12]. This merger was identified as a BNS through gravitational waves, while its associated detection through gamma rays showed that neutron star mergers are the origin of at least some short gamma-ray bursts. During the merger, the neutron stars shed some of their matter through tidal disruption. Some of this matter will fall back onto the newly formed compact object, and the ensuing accretion drives a relativistic jet that produces high-energy emission, including high-energy neutrinos [24]. Some of the tidally disrupted matter from the neutron star will be gravitationally unbound and will produce a slow outflow at 0.1c–0.3c, where c is the speed of light. In addition, some of the accretion disk formed around the central object will also be ejected through winds. As this nonrelativistic outflow is launched before the jet, therefore the jet needs to burrow its way through the slower matter. This interaction could enhance high-energy neutrino emission through boosting inelastic collisions between fast-moving and slower protons. In addition it could reduce the detected gamma-ray flux, making neutron star mergers neutrino luminosity higher than would be expected from gamma rays [25]. The outflow properties likely depend on the properties of the neutron stars in the binary, in particular the neutron star’s equation of state that describes its inner density, as well as the masses of the neutron stars. A binary with different neutron star masses is expected to produce more outflows. Neutron star-black hole mergers – This scenario can lead to two distinct outcomes. If the black hole’s mass is sufficiently small, the neutron star can be tidally disrupted upon getting sufficiently close to the black hole, and the outcome will be similar to the BNS case. This occurs for black holes with masses only up to a few times the solar mass; therefore, it may be the case possibly for a minority

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of such mergers. Black hole masses can be somewhat, but not much, higher for the case of high-spin black holes where the spin reduces the radius of the innermost circular orbit. For higher-mass black holes, the innermost stable circular orbit is reached by the neutron star before it could be tidally disrupted. Therefore, the neutron star will “plunge” into the black hole without shedding any matter, and no multi-messenger counterpart is produced to the gravitational wave signal. Binary black hole mergers – Black holes cannot shed matter, and, therefore, a binary black hole merger in empty space will only emit gravitational waves. Some black hole mergers, however, may take place in a gas-rich environment where they can accrete matter and radiate. Such environments can arise, for example, if a weak supernova leaves some of the stellar envelope near the black holes [26], or due to clumped matter in a collapsing massive star [11]. Nearby gas can also be shocked due to the recoil of the remnant black hole after the merger [27]. Another possible option is the gas-rich environment in the disks of active galactic nuclei (AGNs) [15]. Galactic nuclei harbor a large population of black holes, which can interact with the inflow of gas in AGNs that forms a disk around a central supermassive black hole. The interaction of the disk will align some of the black holes’ orbits with the disk plane. Once in the disk, the black holes will migrate inward. Those getting sufficiently close to each other will form a binary and merge over a short time, aided by dynamical friction within the gas. These black hole mergers will be surrounded by gas, which they can accrete. This accretion can drive relativistic outflows that also produce neutrinos, accompanying gravitational wave emission during the merger. No neutrinos have yet been detected in coincidence with black hole mergers. This scenario is nevertheless plausible given an optical AGN flare found and possibly associated with the black hole merger GW190521 [28], which could have been produced by the black hole merger [29]. This association is not yet clear and is currently under investigation [30]. Core-collapse supernovae – Core-collapse supernovae are known to drive relativistic outflows, e.g. the Type Ib/c supernovae associated with the long GRBs. The core collapse of massive stars can result in the formation of a black hole or neutron star in the center of the collapsing star. Accretion by this black hole from the rest of the collapsing star can produce relativistic outflows that in turn produces highenergy emission, including neutrinos.

UHECR Emission from Gravitational Wave Sources Cosmic rays represent the fourth messenger in multi-messenger astrophysics, and they provide important information about cosmic processes. Due to their charge, however, they are deflected by galactic and extragalactic magnetic fields on their

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way to the observer. This means that the direction in which cosmic rays are observed deviates from the direction of their origin. In addition, due to this deflection, the travel time of cosmic rays is typically much more than that of the other messengers from the source. The candidate astrophysical sources of UHECR are newborn neutron star (remnant of BNS merger or core-collapse of massive stars), gamma-ray bursts, and active galaxies [31]. Some of these astrophysical sources are also the target of gravitational wave and electromagnetic and neutrino observations which are expected to unveil properties and rates of these sources and thus indirectly give information about UHECR.

The GeV-TeV Astronomy with the Gravitational Waves The GeV-TeV astronomy represents a new frontier for the study of the counterparts of gravitational waves, still to be explored. The sources of gravitational waves, either BNS mergers or core-collapse supernovae (section “GW Observations in a Multi-messenger Context”), are expected to emit very-high-energy gamma rays, and evidence has recently increased in this direction (see section “GRB190114C and GRB160821B at the TeV Energies” and also the accompanying  Chap. 22, “Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range.” The single, at the time of writing, GW event with a firm electromagnetic counterpart, GW 170817, did not reveal GeV-TeV gamma-ray emission [12]. The gamma-ray instruments in operation were the Fermi/LAT and the AGILE/GRID covering in the range 100 MeV–1 GeV, the ground-based Cherenkov telescopes observing the very-high-energy (VHE) band with energies greater than 100 GeV, and the High-Altitude Water Cherenkov (HAWC; [32]) observatory, observing energies between 100 GeV and 100 TeV (see section “GeV-TeV Follow-Up of GW Candidates”). These instruments and their basic working principles will be briefly described in section “GeV-TeV Instruments.” The simultaneous detection of GW 170817 and GRB 170817 did prove the association of some BNS mergers with short GRB. There is also evidence that some long GRBs are originated by the core-collapse of massive stars, possible sources of gravitational waves (see section “The Physical Parameters Derived from GW Observations”). Thus, short and long GRBs, viewed either on-axis or off-axis, are the expected counterpart of some GW events. While the gamma-ray emission higher than few MeV is detected in a significant fraction of GRBs, the high-energy (E>100 MeV) gamma-ray emission has been observed for a few GRBs. Recent observations (see section “The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts”) point at the possibility that some GRBs emit up to TeV energies. These observations show that GW sources produce GeV-TeV emission. Here, we provide some physical insight from GeV-TeV detections and a basic description of the instruments able to detect signals in this energy band.

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The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts One of the open questions is which process, or processes, yields the high-energy gamma-ray emission in GRB. A brief overview is provided, and the reader can refer to [33] for a comprehensive review. The spectrum during the prompt phase is peaked, with a maximum in the hard-X to MeV region. Phenomenologically, it is well described by the so-called Band function, built by smoothly connecting two power law functions; the Band function was introduced when the first GRB was observed [34], until the most recent catalogue of Fermi/GBM bursts [35]. The high-energy power law, having a slope lower than -2, is broadly consistent with the synchrotron emission from relativistic electrons, accelerated in the first instants, possibly in internal shocks formed by the ejecta. However, it diverges when trying to have a coherent description of the wide band emission, including the hard power law component behind the peak. A quasi-thermal component from a relativistic fireball has been invoked to explain the observed variety of observed spectra, and both processes might contribute to shape the observed spectra (see [36] and references therein). While the physical description of the prompt phase is still an open question, the observations during the afterglow phase agree with the synchrotron external shock scenario; for its historical development, the reader can refer to the review by Peter Mézsáros [37] and to the references therein. In this scenario, as first suggested by Bohdan Paczy´nski [38], the relativistic population of electron is accelerated in the collimated relativistic outflow (the jet or fireball) launched at the moment of the burst. The jet stems from the core-collapsed star or the merged neutron stars, and it is powered by the gravitational accretion surrounding the collapsed core or the merger remnant. Once the jet sweeps out a sizeable amount of the external interstellar medium, shocks are formed, and the particles are then accelerated through the Fermi process, eventually emitting synchrotron radiation in the surrounding magnetic field. The process was introduced by Roger Blandford and Christopher F. McKee [39] in the context of the collimated outflows in active galactic nuclei and then first applied to GRB by [40] and [41]; for a recent comprehensive review, the reader can refer to [42]. The detailed description of the observed emission is fundamental to get a quantitative estimate of the parameters, such as the magnetic field strength and orientation, the density and distribution of the external medium, the type of accelerated particles (electrons, protons), and their density. In this respect, the energetic gamma rays set a limit on the viability of the synchrotron process at those energies, the burnoff limit [43]. In fact, the synchrotron radiation at the GeV energies trades off the acceleration of the electrons at that energies and their faster cooling. The limit is set around 50 MeV in the co-moving frame and to 50MeV/(1 + z)Γ (t) in the observer frame, where z is the redshift of the source and Γ (t) is the bulk Lorentz factor, changing with time due to the deceleration of the jet interacting with the interstellar medium. With typical values of the Lorentz factor Γ of the expanding fireball of few hundreds up to thousand, gamma rays with energies higher than 100 GeV would be difficult to be reconciled with the standard synchrotron scenario. Photons with

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such energy would require a contribution from other processes, such as the inverse Compton emission from the same population of electrons on their synchrotron photons (the self-Synchtrotron model) or from different seed photons (external Compton), or hadronic processes (see section “High-Energy Neutrino Emission from Gravitational Wave Sources”). These processes would manifest themselves in a new emission component at the highest gamma-ray energies. Hence, gammaray observations can be used to investigate the viability of hadronic processes in GRBs, and thus in GW sources, and are complementary to the quest for high-energy neutrinos (see section “High-Energy Neutrino Emission from Gravitational Wave Sources”). We will see that recent observations did prove the existence of this second component at least in long GRB and provided hints of its presence in short GRB.

GeV-TeV Instruments The main interaction process of high-energy gamma rays with energies above 100 MeV is pair production. The typical interaction length of gamma ray with this energy is ∼37 g/cm2 (radiation length). The telescopes built to observe gamma rays in this energy range detect the electron-position pair produced by the gamma-ray interaction in the detector and the particles and photons produced in the subsequent cascade. These telescopes can be space-based or ground-based. In general terms, such a detector comprises a tracker, where first gamma interaction and pair production happen and their tracks are recorded, and a calorimeter. The pair formed in the tracker interacts with the material of the calorimeter forming a cascade of high-energy particles that deposit their energy through successive interactions. The tracker is used to determine the direction of the primary gamma ray; the calorimeter its energy. Other than the gamma rays, charged nuclei, constituting the energetic cosmic rays component impinging on Earth, hit uniformily and continuously the detector. They represent the main background source, with a flux thousands of times higher than gamma-ray flux from bright sources, which needs to be rejected to let the gamma-ray signal to be significantly detected. Cosmic rays are selected and rejected by means of anti-coincidence shields in space-based detectors, or through the reconstruction and identification of the cosmic-ray signal with respect to the gamma-like signal.

Space Gamma-Ray Observatories The Large Area Telescope (LAT) onboard the Fermi observatory, launched in 2008 and still in operation at the moment of this writing, will be used as an example of the space-based gamma-ray observatory, devoted to observations in the ∼100 Mev to hundreds of GeV energy range. Refer to [44] for a comprehensive review. Some key results obtained by the Fermi observatory in the study of GRBs and for the follow-up of GW events are described in section “The GeV-TeV Astronomy with the Gravitational Waves”. The Fermi observatory comprises two instruments: the gamma-ray burst monitor (GBM) covering the hard X-ray to gamma-ray energies

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8 keV–40 MeV and the gamma-ray large area telescope (LAT) observing at energies greater than 30 MeV. Fermi/LAT operates in survey mode, and its field of view of 2.4 sr covers the whole sky approximately in every 3 hours. The tracker has 36 layers of silicon strips detector alternated with 16 layers of tungsten foils to facilitate the pair conversion of the gamma ray, whose arrival direction is reconstructed from the e+ to e− tracks. The angular resolution depends on the primary energy and goes from a few degrees down to 0.1 deg at the highest energies. The calorimeter has a depth of 8.6 radiation lengths and measures the energy released by the particles formed by the cascade initiated by the pair with an energy resolution around 15%. A segmented anti-coincidence system removes the background due to the isotropic flux of cosmic rays. A similar instrument concept is used in the satellite Astro-rivelatore Gamma a Immagini Leggero (AGILE), which is a project of the Italian Space Agency (ASI) launched in April 2007 [45]. Typically, space detectors are limited in their dimension, so that their collection area is of the order of 1 m2 or less (the collection are of Fermi/LAT is ∼6500cm2 ). This poses a limitation to the maximum energy that can be explored. Typically, the flux of gamma-ray photons in astrophysical sources decreases as a power law with energy, and the rate of photons impinging on the detector decreases at the highest energies, where very long integration times are needed. To reach a significant number of gamma rays at higher energies, larger collection areas are needed, as the one endowed by the ground-based telescopes.

Ground-Based Gamma-Ray Telescopes: Cherenkov Telescopes and Particle Detectors The atmosphere has a total thickness of 1050 g/cm2 , equivalent to ∼28 radiation lengths, and it is not transparent to high-energy gamma rays. Despite the impossibility for the primary gamma ray to impinge on a tracker on ground, the atmosphere itself can be used as a calorimeter and becomes part of the detector. In fact, the interaction of the gamma ray with the atoms of the atmosphere produces a particle cascade. The cascade develops in the atmosphere through successive pair production and bremsstrahlung interactions with the air atoms (see Fig. 1,left). Eventually, a so-called electromagnetic atmospheric shower is formed. This atmospheric shower can be observed through the final particles reaching the ground and by means of the Cherenkov light produced by all the relativistic particles of the shower and integrated throughout the cascade. It is noteworthy that other than the electromagnetic atmospheric showers initiated by gamma rays, the majority of atmospheric showers are generated by the highenergy cosmic rays. In this case, the interactions with the atoms involve different processes (hadronic interactions, nuclear fragmentation) and yield to the so-called hadronic shower. The gamma-ray signal from the electromagnetic showers needs to be discriminated and selected from the overwhelming hadronic background. Differently than space-based telescopes, it is not possible to build an anti-coincidence shield to reject the cosmic rays. Instead, the selection is made through the signal

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Fig. 1 Schematic description of the imaging air Cherenkov technique. Left: schematic reproduction of the interactions of the primary gamma ray with the atmosphere. The shower of particles develops through successive interactions, pair production, and bremsstrahlung, with the air atoms. The shower fades when the energy of the secondary particle drops below the critical energy for the processes ∼80 MeV. Right: a representation of the Cherenkov light cone and its pool on ground, sampled by the large reflective surface comprising the Cherenkov telescopes. The image of the shower development is reconstructed (center) by the camera at the focus of the reflector. The image shape, its orientation, and the light content are used to identify the primary particle, its direction, and energy. (Credits for the central image: MAGIC collaboration. The image is a real gamma-like image recorded by the MAGIC-I telescope)

reconstructed by the detector at ground, carrying the imprint of the primary particle nature and of the different interaction processes in atmosphere. In the framework of the time domain astronomy at very-high-energy gamma rays, the big advantage of the Cherenkov telescopes is their large collection area. In fact, the Cherenkov light-pool of an atmospheric shower (whether electromagnetic or hadronic) has a diameter typically of 200 meters, and the telescope samples a small fraction of this area. Consequently, the collection area is of the order of 104 −105 m2 . In fact, wherever the atmospheric shower develops within such an area around the telescope, its Cherenkov light will be detected. In fact, thanks to the Cherenkov light detected by the telescope, the main parameters of the primary energetic particle, such as its nature (gamma ray or cosmic ray), its direction, and its energy, can be measured. This is achieved through the reconstruction and parameterization of the image produced by the Cherenkov light. The reconstruction technique is referred to as imaging air Cherenkov technique, and for this reason, the Cherenkov telescopes are usually named with the acronym IACT (imaging air Cherenkov telescopes). The image has an ellipse-like shape which reflects the longitudinal development of the shower and its particle content. While the longitudinal development is related

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to the main axis of the shower, that is, to the direction of the primary particle, the light content of the image is related to the total energy of the shower and thus to the energy of the primary particle. The different processes responsible for the electromagnetic and hadronic showers yield to different images. This enables us to discriminate among electromagnetic and hadronic showers with a rejection of hadronic showers higher than 99.9% and an acceptance of gamma-ray images around 50%. The proper reconstruction of the image and the comparison with the expected parameters derived by dedicated simulations yield to the selection of the gamma-ray signal over the background of cosmic rays and to the computation of the arrival direction of the primary particle and its energy. The typical performance of the Cherenkov telescopes is given by sensitivities around 10−11 − 10−13 erg/cm2 /s in the 100 GeV–10 TeV range, angular resolution of 0.1 degree, and energy resolution of 15% [46]. The energy threshold of a Cherenkov telescope depends on the number of Cherenkov photons that can collect. The larger the reflective area and the more sensible the light-detector is, then the greater the number of Cherenkov photons and correspondingly the smaller the threshold in the detectable energy. The IACT in operation, such as H.E.S.S., (https://www.mpi-hd.mpg.de/hfm/HESS/) MAGIC (https://magic.mpp.mpg.de), and VERITAS, (https://veritas.sao.arizona.edu) have energy thresholds from 50 to 100 GeV. With the future of Cherenkov telescope array (CTA), this threshold will be reduced remarkably up to ≈ 20 GeV [47]. The highest reachable energy is a few tens of TeV, mainly limited by the low flux at the highest energies and by the collection area. The maximum energy can be stretched to more than 100 TeV with dedicated techniques [48], or extending the area covered by the telescopes, that is, increasing their number, as planned by the CTA observatory [47]. The Cherenkov telescopes collect the light by means of large reflective area (above 10 meters in diameter) and focus it on a camera made of several – few hundreds – fast photomultipliers arranged pixel-wise. The camera has a field of view of few degrees and can reconstruct the Cherenkov image, which lasts few nanoseconds and subtends ∼1 deg. A more detailed description of these instruments can be found in the references provided. The impact of Cherenkov telescopes in the observation of GRB and GW counterparts is briefly outlined in Sections “GRB190114C and GRB160821B at the TeV Energies” and “GeV-TeV Follow-Up of GW Candidates.” The second method to detect gamma-ray showers at ground is based on the detection of the particles of the atmospheric shower reaching the ground. The particle cascade develops for few kilometers in the atmosphere starting at height around 20 km. Hence, the higher the altitude of the detection, the greater the number of particles that can be detected and the lower the energy threshold. This threshold is in any case higher than few hundreds of GeV, since cascades with lower energy do develop completely in the atmosphere before reaching the detector. The particles can be detected with different techniques and detection methods. The HAWC observatory, (https://www.hawc-observatory.org) placed at 4100 m of altitude in Mexico and previously mentioned, is one of such detector. It comprises a set of 300 water tanks tightly distributed in an area of ∼22000 m2 . The particles of the cascade that survive up to that altitude produce Cherenkov light within the

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water tank. The Cherenkov light is collected by four photomultipliers. An external outrigger array made by 345 tanks, sparsely distributed around the main compact detector, increases the area and the sensitivity, so that currently the collection area of HAWC is ∼104 −105 m2 , which is comparable to that of IACTs. The reconstruction technique has a lower discrimination power of the hadronic background with respect to the imaging technique implemented by the IACTs up to energy around few TeV. Hence, HAWC can only detect the fast fading signal of the brightest transients [49].

High-Energy Gamma Rays from GRBs and Their Implications on Gravitational Wave Observations The electromagnetic emission associated with gravitational wave signals from binary system of neutron stars or neutron star black hole is strictly connected to the gamma-ray burst phenomenon. The firm detection and characterization of the very energetic gamma-ray emission in the GeV domain by Fermi/LAT represents one of the recent major advances in the understanding of the physics governing their emission [50]. Recently, a further step forward was achieved with the significant detection of TeV emission from a few long GRBs (GRB 190114C, GRB 180720B, and GRB 190825A). This detection enabled the identification of a second component and a different emission process at this energy range (see section “GRB190114C and GRB160821B at the TeV Energies”). Hints of the presence of an extra component were previously found in a few GRB (e.g. the short GRB 090510 and the long GRB 090926A [50] and GRB 090417a [51]). These detections show that the search for the GeV-TeV from GW counterparts represents a new window to be explored. In the first 10 years of operation, the Fermi/LAT did detect 169 bursts emitting photons with energy higher than >∼100 MeV, over a total of 2357 GRB detected in the lower energy band by the Fermi/GBM instrument [35]. The comparison with the GRB detected at lower keV-MeV energies, e.g., by the Swift/BAT and the Fermi/GBM instruments, shows that the GRBs detected in the GeV range by Fermi/LAT are the brightest in the distribution of luminosities and the most energetic [35]. This result might have an implication on the nature of the central engine, namely, a magnetar or a black hole. This is one of the still open key questions on the fate of the BNS merger, and the presence of the high-energy gamma-ray emission can provide a clue. Of the whole sample seen by the Fermi/LAT, 16 (5% of the total) are short GRBs, as such possibly associated with BNS mergers. Short and long GRBs show a similar behavior in the high-energy gamma bands, thus suggesting a common emission mechanism, as described by the external shocks in the ISM. For this reason, it is important to indicate the parameters driving the presence or the lack of the GeVTeV emission in GRB. The lack of GeV emission in the prompt phase can be ascribed either to the low luminosity or to the presence of a cutoff in the extrapolated Band function (see section “The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts”).

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Indication of a cutoff was found in few bursts not detected in gamma rays, but for the majority of them, no conclusions can be obtained [52]. Similar constraints can be derived from the lack of GeV emission in the afterglow phase. In this case the peak luminosity depends also on the bulk Lorentz factor of the fireball decelerating in the ISM, so that the lack of GeV emission can be used to set a limit on the Lorentz factor [52]. Gamma photons of energy higher than ∼TeV (in the observer frame, so depending on the Lorentz factor) can be strongly affected by internal absorption. TeV photons are also heavily absorbed by the interaction with the intergalactic background light (see section “GRB190114C and GRB160821B at the TeV Energies”). These constraints can be derived with the assumption that the high-energy gamma-ray emission belongs to the same component and process observed at lower energies, so that the straight extrapolation is meaningful. In fact, a few GRBs detected by Fermi/LAT show a harder component, e.g., the short GRB 090510, the long GRB 090926A, and GRB 090902B [50]. The arrival of E > 10 GeV photons tens of seconds after the burst challenges the leptonic acceleration models used to explain the emission in the X-rays and lower frequencies. During the deceleration of the fireball in the ISM, the bulk Lorentz factors decrease, and the maximum observed energy of gamma rays corresponds to leptons with cooling times shorter than the time needed to accelerate them. Despite these different indications, until the discovery of TeV emission from the long GRB 190114C, there was not a clear evidence of an extra component at higher energies.

GRB190114C and GRB160821B at the TeV Energies The short GRB 170817, the single electromagnetic counterpart of a GW event discovered during the first three scientific runs by LIGO-Virgo, did not reveal any emission beyond 1 MeV, as it was outlined in section “GeV-TeV Follow-Up of GW Candidates”; more details can be found in the accompanying  Chap. 22, “Electromagnetic Counterparts of Gravitational Waves in the Hz-kHz Range.” The late detection of X-ray and radio emission, presumably originated from synchrotron nonthermal processes in the jet seen off-axis, makes it possible that inverse Compton emission is produced. The search of GeV-TeV emission did not produce any detection [53] but provided upper limits on its flux. The feasibility of an inverse Compton emission component in GRB, which should peak in the GeV-TeV range, was finally proved with the long GRB 190114C. This bright and nearby (z = 0.34) GRB was promptly observed by the MAGIC Cherenkov telescopes, starting at 58 s from the burst. A strong fading emission was detected for ∼30 min, at energies higher than 100 GeV, up to more than 1 TeV [54]. The arrival time of the energetic photons rules out the synchrotron emission as origin of these gamma rays, and the study of the simultaneous multifrequency spectra (see Fig. 2) clearly shows the presence of a second component at the highest energies. The emitted power is similar to the one at lower (optical X-ray) energies. The simultaneous modeling of the two components brings to tighter constraints on the parameters and on the environment. This detection opens a new window in the study of GRB and has implications on the detectability of the GW counterparts in the GeV-TeV energy range.

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Fig. 2 The second high-energy GeV-TeV component in the long GRB 190114C. The plot shows the spectral energy distribution of the high-energy component, from X-ray to TeV gamma rays, of the long GRB 190114C. Simultaneous spectral points and fits in two successive time intervals T0 + 68 − T0 + 110 s (cyan butterflies) and T0 + 110 − T0 + 180 s (yellow butterflies) are shown. The spectra measured by MAGIC at later times are also reported. (Figure from [55] Comment: Permission not needed since one of coauthors (see NATURE editor email )

The detection of the TeV emission on a GRB has been corroborated by the successive indication of VHE emission by H.E.S.S. in the late afterglow of GRB 180720B [56] and in the long GRB 190829A [57] 4 hours after the burst. Even more interesting for the prospects of gamma-ray detection from the GW counterparts is the strong hint of TeV emission from the short GRB 160821A, as seen by the MAGIC IACT [58, 59]. GRB 160821A is a short burst, one of the nearest known, with measured redshift z ∼ 0.16. A kilonova has been identified as an optical-infrared excess superimposed to a nonthermal afterglow, thus supporting the association with a BNS merger event [60, 61]. MAGIC found an indication of emission at energies higher than 500 GeV, at a significance level of 3 sigma, 4 h after the burst [58]. Assuming the derived flux as real, [58] tries to model the TeV emission as the afterglow component from the external forward shock. The broadband emission, from X-ray to TeV energies, is modeled as synchrotron and self-synchrotron Compton (SSC) emission, similarly to what was invoked in GRB 190114C (see section “The GeV-TeV Emission from GRBs as Gravitational Wave Counterparts”). The observed TeV emission, if real, can hardly be reconciled with the simplest one-zone SSC model within the constraints set by the broadband spectral energy distribution. To this aim, further observations by

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IACTs of short GRB and follow-ups of GW associated events are needed. The case of GRB160821B shows the importance of TeV observations in removing the degeneracies of the models based on the synchrotron component alone and in shedding light in the nature of short GRB and their connection with the BNS mergers. A strong limitation to the detection of GRB by the IACTs derives from the interaction of the GeV-TeV photons with the optical-infrared extragalactic background light (EBL, see [62] for a review). The EBL is the result of the ultraviolet and optical light emitted by the stars during their evolution in the galaxies, from their early formation all along their evolution, and then reprocessed by the dust to the infrared. As a consequence, the EBL spectrum extends from 0.1 to 1000 μm, and it shows two main components, peaking in the optical and in the far-infrared bands. The γEBL − γTeV interaction produces an electron-positron pair, when their energies allow the production of the pair in the center of mass, and according to the γ − γ cross-section. For example, a 1 TeV photon interacts with the EBL photons with wavelength in the range λ ≈ (0.1 − 2.4)μm Eγ [TeV]×(1 + z)2 , where z is the distance of the source. The resulting opacity of the gamma rays to the EBL depends on the energy of the gamma ray, on the density of the EBL photons and on the column of EBL that is met on the way, that is, on the distance of the source. As a result, gamma photons with an energy higher than ∼10 GeV are affected by this interaction, and sources beyond a given distance – where the opacity become greater than one (which defines the gamma-ray horizon) – are completely absorbed. With the present generation of IACT, with the energy threshold around 100 GeV, this horizon is at z ∼ 1, thus limiting the observation of distant GRB. This limit will be reduced by the smaller energy threshold of the future Cherenkov Telescope Array (CTA) (see section “GeV-TeV Follow-Up of GW Candidates”). On the other hand, the GW sources from the present generation of interferometers are limited to smaller distances and, as such, well within the gamma-ray horizon.

Observation Strategies with Neutrino and Gamma-Ray Instruments Neutrino Follow-Up of Gravitational Wave Candidates Neutrinos are detected by Cherenkov detectors in water or ice. When neutrinos interact, they produce charge particles that in turn produce photons through Cherenkov radiation. Due to the transparent medium, this radiation can be detected by photodetectors. Neutrino detectors are essentially sensitive to neutrinos arriving from any direction, monitoring the whole sky at all time. The neutrino follow-up of gravitational wave candidates has two main advantages that can help their joint use in multi-messenger follow-up: (i) both gravitational wave and neutrino detectors monitor effectively the whole sky at all times, and (ii) both types of emissions are expected within a short time frame following violent cosmic events. Consequently, it is possible to search for coincident neutrinos

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for all gravitational wave detections without needing dedicated telescope time or early warning, and a potential candidate can be used to motivate further multimessenger follow-up. In particular, since neutrino observations have a much better reconstructed direction than gravitational waves, it is substantially easier to follow up the better neutrino pointing than to cover the gravitational wave skymap that can be as large as a 1000 deg2 or more [63]. Joint searches for gravitational waves and neutrinos have been carried out for over a decade, from before the participating detectors were completed (e.g., [64]). Participating observatories included IceCube, ANTARES, and the Pierre Auger Observatory on the high-energy side and Super-Kamiokande, Borexino, and Baikal sensitive to thermal, MeV energies. No common gravitational-wave-neutrino source has been discovered so far. The most astrophysically interesting constraints came from the non-detection of neutrinos from the neutron star merger GW170817. Give the close distance of this source, neutrinos by themselves could constrain the viewing angle of the relativistic outflow produced by the merger [65]. In addition, the non-detection of neutrinos also probed the interaction of relativistic outflows from the merger with the slower dynamical ejecta that produced the observed kilonova emission [25].

GeV-TeV Follow-Up of GW Candidates The joint observation of GW170817 and GRB170817A provided the first direct evidence that at least a fraction of BNS mergers are progenitors of short GRBs [12]. As already said (see section “The GeV-TeV Astronomy with the Gravitational Waves”), GRBs are also expected to emit high-energy (HE, > 100 MeV) and veryhigh-energy (VHE, > 100 GeV) EM radiation. HE and VHE detectors followed up GW170817, but no EM counterpart has been found. Fermi-LAT was not collecting data at the time of the GW trigger (T0 ) due to a passage through the South Atlantic Anomaly; therefore, it was not possible to place constraints on the existence of HE emission associated with the moment of the BNS coalescence. A search has been performed on longer time scales, but no candidate EM counterpart was detected on timescales of minutes, hours, or days after the GW detection [66]. The AGILE satellite was occulted by the Earth at T0 ; the first exposure started about 935 s after T0 . The GRID observed the field before and after T0, but no gamma-ray source was detected in the energy range 30 MeV– 30 GeV [67]. VHE EM follow-up of GW 170817 has been performed with the H.E.S.S. telescopes (see section “GeV-TeV Instruments”). Specifically, H.E.S.S. observations started ∼5 h after GW170817 and three observation runs of 28 min each were initially performed, looking at different regions of the GW skymap (see Fig. 3); the fields to be observed have been scheduled with the so-called “galaxy targeting” approach that combines the three-dimensional information of the location of the GW event provided by the GW skymap with the location of galaxies within that volume. No EM source has been detected during these first observations.

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Fig. 3 Left: Pointing directions of the first night of H.E.S.S. follow-up observations starting August 17, 2017, at 17:59 UTC. The circles illustrate a FoV with radius of 1.5◦ , and the shown times are the starting times of each observation with respect to GW170817. The LALInference map of GW170817 is shown as colored background; the red lines denote the uncertainty contours of GRB170817A. Right: map of significances of the gamma-ray emission in the region around SSS17a obtained during the first observation of GW170817. The white circle has a diameter of 0.1◦ , corresponding to the H.E.S.S. point spread function and also used for the oversampling of the map. (Figure from [69], reproduced by permission of the AAS)

After the detection of the optical counterpart to GW170817, SSS17a, first reported by [68], the monitoring campaign was extended over several days, with focus on the sky region containing the source, but no significant gamma-ray emission has been found. However, the derived upper limits on the VHE gamma-ray flux allowed to constrain nonthermal, high-energy emission following the merger of a BNS system [69]; furthermore, the joint analysis of X-ray and radio data, together with the H.E.S.S. upper limits, allowed to characterize the Self-Synchrotron Compton (SSC) emission and, specifically, to put constraints on the magnetic field strength [53]. A search for a possible EM counterparts to GW170817 at VHE has been performed also with the High Altitude Water Cherenkov detector (HAWC, [32]). The LIGO-Virgo sky localization first became visible for HAWC on August 17 between 19:57 and 23:25 UTC; SS17a was observed for 2.03 hr starting at 20:53 UTC. Also in this case, no significant gamma-ray emission has been found, and an upper limit was derived [70]. So far, GW170817 represents the only GW event with clear EM counterparts, although without an associated GeV-TeV emission. Observational campaigns at GeV-TeV energies have been performed also to follow up other GW candidates/events (see, e.g., [71]), but no EM counterpart has been found. In the next years, Advanced LIGO and Advanced Virgo will take data with increased sensitivity, possibly jointly with KAGRA and LIGO-India [63]; at the same time, the Cherenkov Telescope Array (CTA, see [72]; see also

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section “GeV-TeV Instruments”) will become operative. CTA represents an ideal instrument to perform the EM follow-up of GWs at VHE, given its unprecedented sensitivity, its rapid slewing capabilities, and its large field-of-view (FOV). CTA will have a large collection area and will explore a wide energy range (20 GeV– 300 TeV) thanks to three types of telescopes, large-, medium and small-size telescopes (LST, MST, SST), the latter distributed over a large area. Despite the large FOV of CTA, the poor sky localization of the GW events (from tens to hundreds of square degrees [63]) requires the definition of an observational strategy for the search of the VHE EM counterpart, through an optimized scan of the GW skymap. The capability of CTA to perform the EM follow-up of GWs has been investigated by different authors. For instance, [73] considered CTA operating in “survey mode” and assumed the same observing time for each consecutive CTA observation needed to tile the GW skymap (as typically done during the EM followup campaigns also in other energy bands); this observing time has been estimated considering two possible values for the GW sky localization area (200 deg2 and 1000 deg2 ) and taking into account the total duration of the GRB emission (assumed to be 1000 s). They found that short GRBs with emission extending up to 100 GeV can be detected by CTA if observations are delayed no more than 100 s with respect to the GRB onset, while short GRBs with emission at lower energies can only be detected with lower delay times (0.1 mHz) 3 × 10−16 (>1 mHz) 3 × 10−15 (>0.1 mHz) 3 × 10−15 (>0.1 mHz)

Fig. 1 Sensitivity curves of mission designs with different choice of baseline parameters, with ALIA, LISA, and eLISA included for the purpose of comparison

levels are converted from estimations by Hils and Bender [10] and Farmer and Phinney [11]. For black hole binaries with a mass ratio 1 : 4, typical of what one would expect from hierarchical black hole growth at high redshift, the all-angle averaged detection range is plotted in Fig. 2. Apart from galactic confusion noise, upper level (dashed curve) and lower level (dotted dashed curve) of confusion noise generated by extragalactic compact binaries as those estimated by [11] are also taken into account. In calculating the averaged SNR, we have used hybrid waveforms in the frequency domain with black hole spin not taken into account [68, 69]. For 1 year of observation before merger, the contributions in SNR due to large spin are indeed negligible according to our calculations. Spin is relevant only in the parameter estimation stage, which will not be discussed in the present work. As may be seen from Fig. 2, for a given redshift, the proposed mission concept is capable of detecting lighter black hole binaries in comparison with eLISA/LISA

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Fig. 2 All-angle averaged detection range of a single Michelson channel with threshold SNR of 7 for 1 : 4 mass ratio intermediate black hole binaries, 1 year observation prior to merger. For each mission option, both upper and lower confusion noise levels (represented by the dashed curve and dotted dashed curve, respectively) due to extragalactic compact binaries are considered

and thereby provides better understanding of the hierarchical assembling process in early Universe (Fig. 3). Apart from intermediate-mass black hole binaries at high redshift, the designed sensitivity at around 0.01 Hz measurement band means that the instrument is also capable of detecting IMRIs harbored at globular clusters or dense young star clusters at low redshift (z < 0.6). See [57] for a further discussion of the capture dynamics of an IMRI in dense star clusters. Displayed in Fig. 4 are the detection ranges of IMRIs with different mass ratios 1 year prior to merger. The stellar black hole is fixed to be 10 M , while the mass of an intermediate-mass black hole is subject to variation in order to generate different mass ratios in the figure (Fig. 5).

Simulations of Cosmic Growth and Merger of Black Holes and Event Rate Estimates To understand the detection capability in high redshift Universe of the mission options given in Table 2, we carry out a Monte Carlo simulation of black hole merger histories based on the EPS formalism and semi-analytical dynamics. Pop III remnant black holes of 150 M are placed in 3.5σ biased halos at z = 20 with initial spins of the seeds generated randomly. By prescribing VHMtype dynamics [25, 26], we trace downward the black hole merging history. The halo mass ratio criteria for major merger is set to be greater than 0.1. Both the prolonged accretion and the chaotic accretion scenario are considered. Black hole

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Fig. 3 The signal-to-noise ratio contours for detecting IMRIs at different redshifts by a single Michelson channel with 3 × 106 km arm length and 1 year observation before merger. (a) 5pm · H z−1/2 , z = 2, (b) 5pm · H z−1/2 , z = 6, (c) 5pm · H z−1/2 , z = 10, (d) 8pm · H z−1/2 , z = 2, (e) 8pm · H z−1/2 , z = 6, and (f) 8pm · H z−1/2 , z = 10. Confusion noise considered is at an intermediate level between the upper and lower limit

spins coherently evolve through both mergers and accretions processes, and their magnitudes influence strongly the mass-to-energy conversion efficiency. We assume efficient gaseous alignment of the black holes so that the hardening time is short and only moderate gravitational radiation recoils take place. Numerical simulations [71] suggest that the hardening and merger time scales remain short even in gas-free environment. In calculations relevant to GW observations, we assume a threshold SNR of 7 for detection in the sense of single Michelson interferometer and 1 year observation prior to merger. The results are schematically given in Figs. 6 and 7. We assess our simulations by fitting the black hole mass functions and luminosity functions at six almost equally divided successive redshift intervals ranging from z = 0.4 to 2.1 (see Fig. 8). In the prolonged accretion scenario, the results deviate from the observational constrains given by Soltan-type argument when going up to redshift z > 1.5. It may therefore underestimate the black hole growth rate and perhaps also the coalescence rate. Observationally the existence of very high redshift (z > 6) AGNs implies that feedback mechanisms may be very different at early epoch so that fast growth of the seed black hole could be possible. In terms of coalescence rate, our result displayed in Fig. 4 is in overall agreement with the results given by Sesana et al. [32, 34] and Arun et al. [70], though the coalescence counts given by their simulations are about two or three times higher. It is likely due to various numerical discrepancies in the simulations. Overall, our black hole mass growth is slower, particularly in the prolonged accretion scenario.

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Fig. 4 All-angle averaged detection range under a single Michelson threshold SNR of 7 for IMRIs with reduced masses of 10 M , 1 year observation prior to merger. For each mission option, both upper and lower confusion noise levels (represented by the dashed curve and dotted dashed curve, respectively) due to extragalactic compact binaries are considered

Fig. 5 The signal-to-noise ratio contours for detecting IMRIs with the reduced mass of 10 M ) by a single Michelson channel with arm length of 3 × 106 km, 1 year of observation before merger. (a) 5pm · H z−1/2 , and (b) 8pm · H z−1/2 . The confusion noise considered is at an intermediate level between the upper and lower limits

At z = 15, the total mass of the black hole binaries typically is still less than 600 M in the prolonged model, and this may lead to smaller counts in detectable sources. Our results are expected to give a very conservative (pessimistic) estimate of black hole binaries merger event rate.

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Fig. 6 Coalescence rate predicted by the Monte Carlo simulations

The astrophysics encapsulated in our simulation represents the state of the art understanding of structural formation after the dark age. Due to our poor understanding of the evolution of the Universe at this epoch, it is likely that the simulation overlooks many details of the physical processes involved. The event rate count should be looked upon in a cautious way. Instead of reading into the precise numbers, it serves as an indication what spaceborne gravitational wave detector is capable of and in our case, the advantage of setting the most sensitive regime of the measurement band from a few mHz to 0.01 Hz.

Event Rate Estimate for the Detection of IMRIs in Dense Star Clusters Consider the following scenario [53–55]: (1) In the accessible range of the Universe for the mission design, the spatial number density of star clusters is a constant in respect to the volume calculated by the luminosity distance. It has the same value as that of the local Universe. For globular and young clusters, nGC ≈ 8h3 · Mpc−3 , nY C ≈ 3h3 · Mpc−3 , h = H0 /(100 km·s −1 ·Mpc−1 ), and we take h = 0.73. In the event rate estimate in what follows, we consider only the globular clusters and assume that the total

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Fig. 7 Event rates estimates for the mission designs

number density of dense star clusters is given by nC ≈ 8h3 · Mpc−3 , which will give a conservative estimation of the event rate. (2) The probability of a small compact celestial body being captured by an intermediate-mass black hole at the center of a cluster is given by ν(M, μ, z) ≈ 10−10

M −1 a , μ

i.e., it is directly proportional to the mass of the intermediate-mass black hole and inversely proportional to the reduced mass. This assumption is motivated by the dynamical analysis on globular clusters [51–54]. In the event rate calculations, for wave sources with non-negligible redshift effect, a scaling factor 1 + z is required in principle, but the coalescence rate itself is just an of order of magnitude estimate. Further, the accessible range of redshifts considered by the current mission options is quite small. This scaling factor will be neglected here. (3) In dense star clusters where the intermediate-mass black holes are located, the mass distribution function of the intermediate-mass black holes is assumed to be f (M) =

ftot max ln M Mmin

1 . M

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Fig. 8 The degree of coincidence of the luminosity function of quasars given by the Monte Carlo simulation with observed data

Mmin and Mmax are, respectively, the lower and upper limits of the massdistribution range of intermediate-mass black holes, they are taken respectively as 102 M and 104 M , and beyond this range f (M) = 0. ftot is the fraction of dense star clusters containing intermediate-mass black holes. Its value is highly uncertain and we shall take a conservative estimate that ftot = 0.1 [53, 55]. Take the observational time of 1 year before merger, the reduced mass of an IMRI system is taken to be 10 M , and the threshold value of signal- to-noise ratio of a single Michelson detection is taken as 7. The event rate may then be calculated using the following formula [53, 56]: R=

4π 3



Mmax

[DL (M, μ)]3 ν(M, μ, z)nc f (M)dM.

Mmin

and the result is given in Table 3. The above event rate estimate is subject to many uncertainties, and perhaps we should not attach too much importance to the precise numbers. Instead, the calculations serve as an indication of the detection potential of the mission concept as far as IMRIs at low redshifts are concerned. Further, as event rate goes up as the cubic of the improvement in sensitivity, it also brings out the advantage of shifting

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Table 3 Prospective detection rate for IMRIs in globular clusters Mission option ALIA 5pm (D = 0.6 m) 8pm (D = 0.45 m) LISA (5 × 106 km in armlength)

Upper level of confusion zc = 5 : ∼8000 zc = 3 : ∼6000 ∼90 ∼26

Lower level of confusion ∼12,000 ∼7000 ∼130 ∼32 ∼3

slightly the most sensitive region of the measurement band to a few hundredth H z, as far as detection of IMRIs is concerned. It should also be remarked that collision of dense star clusters [57] constitutes possible intermediate-mass black hole binaries gravitational wave sources, while the inspiral of massive black holes (∼103 M to ∼104 M ) into the supermassive black hole at the center of a galaxy is also a promising IMRI source [56, 72]. However, the corresponding event rates would be difficult to estimate.

Concluding Remarks Gravitational wave detection in space promises to open a new window in the quest for understanding of our Universe, in particular as a new way to probe the formation of galactic structure at early Universe discussed here. With the recent revised definition of the LISA mission in which the arm length of the laser interferometer is shortened to 2.5 million kilometers, modulo some minor variations in the baseline parameters, there is basically no longer any difference between LISA and TAIJI in terms of the mission definition. A global effort to realize such a mission seems to be the next natural step forward, though it does not seem possible in view of the current political climate. Still it is an option we should keep in mind, in the hope that when the occasion is right this becomes a realistic step to be taken. Acknowledgments We were very grateful to the director of the Morningside Center of Mathematics, Prof. Shing-Tung Yau, for very strong financial backing that enabled us to conduct the study in a very comprehensive and thereby laid the foundation for current development. The feasibility study was conducted with some very generous technical assistance from the Albert Einstein Institute, Hannover and Peter Bender. Our thanks also go to Profs. Wenrui Hu and Shuangnan Zhang for encouragement and support.

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Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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Chiara Caprini and Daniel G. Figueroa

Contents Introduction: Gravitational Waves in Friedmann-Lemaître-Robertson-Walker Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties . . . Characterization of a Stochastic Gravitational Wave Background . . . . . . . . . . . . . . . . . . . . . Evolution of a Stochastic Gravitational Wave Background in the Expanding Universe . . . . Signal from a Generic Stochastic Source of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . Cosmological Gravitational Wave Backgrounds: Relevant Examples . . . . . . . . . . . . . . . . . . First-Order Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmic Defect Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

We discuss gravitational wave (GW) signals of cosmological origin, generated in the early universe. We argue that early universe GW backgrounds are always of stochastic nature and describe their general properties and their evolution during cosmological expansion. We present examples of relevant early universe GW generation mechanisms, as well as the properties of the GW backgrounds they produce. In particular, we discuss (1) GWs from first-order phase transitions and (2) GWs from topological defects, with a particular emphasis on cosmic strings. The phenomenology of early universe GW sources is extremely rich, possibly leading to GW backgrounds within the reach of near-future detectors.

C. Caprini () Université de Paris, CNRS, Astroparticule et Cosmologie, Paris, France e-mail: [email protected] D. G. Figueroa Instituto de Física Corpuscular (IFIC), CSIC-Universitat de Valencia, Valencia, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_25

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The detection of any of these stochastic signals would be a milestone in physics, providing crucial information on the high-energy physics characterizing the early universe, probing energy scales well beyond the reach of present and planned particle accelerators. Keywords

Gravitational waves · Stochastic backgrounds · Cosmological backgrounds · Phase transitions · Cosmic defects · Cosmic strings

Introduction: Gravitational Waves in Friedmann-Lemaître-Robertson-Walker Backgrounds Gravitational waves are naturally described as small, propagating perturbations over a Minkowski background metric. In appropriate sets of coordinate systems, linearization over Minkowski allows to easily distinguish the flat, background spacetime from the weak, space-time-varying gravitational field, which constitutes the waves (see, e.g., Chapter 1 of [158]). The small metric perturbation manifestly obeys a wave equation in the Lorentz gauge, which can be picked without loss of generality, exploiting the invariance of the linearized theory under slowly varying infinitesimal coordinate transformations [158]. If one further restricts to globally vacuum space-times, the wave equation becomes invariant under Lorentz preserving infinitesimal coordinate transformations. In this case, performing the right coordinate choice allows one to reduce completely the gauge freedom, defining the transverse-traceless (TT) gauge, in which only the two physical radiative degrees of freedom of the metric perturbation remain. The TT gauge clearly exhibits the intrinsic nature of the gravitational interaction, mediated by the graviton, a spintwo massless field that has only two independent helicity states (see, e.g., Chapter 2 of [158]). However, if one wants to explore more in general GW propagation and generation processes operating in the universe, in many cases, it becomes necessary to go beyond linearized theory over Minkowski and define GWs over the FriedmannLemaître-Robertson-Walker (FLRW) space-time background. In general, characterizing GWs over a curved background implies to be able to distinguish the background from the fluctuation, which becomes difficult when the background metric components are also time- and space-dependent. One usually resorts to possible separations of scales/frequencies: if the typical length or time scale of variation of the background is much longer than the typical wavelength or inverse frequency of the GWs, the latter can be identified as perturbations on a smooth background (from “their” point of view), or rapidly varying fluctuations over a slowly varying background. In practice, this is implemented by averaging physical quantities over intermediate length and/or time scales [40]. In the case of a GW source operating in the early universe, one could compare, for example, the Hubble scale at the time of action of the source, redshifted to

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today (the typical wavelength of the GW signal), with the Hubble scale today (the typical “variation” of the background), and conclude that GWs are indeed well defined in the cosmological context, where a clear separation of scales is manifest. However, there are cosmologically relevant phenomena for which the situation is less clear, notably the case of inflation (treated elsewhere in this book): when metric perturbations cross the Hubble scale during inflation, the dynamics of the GWs cannot be distinguished from the one of the background, and separation of scales does not hold. However, the definition of GWs over the FLRW background can still formally be performed, thanks to the FLRW space-time symmetries. Since the hyper-surfaces of constant time are homogeneous and isotropic, the splitting of the metric into a curved background component plus linear time- and space-dependent perturbations can be defined uniquely. Furthermore, the two-index tensors representing the perturbations can be irreducibly decomposed on constant-time hyper-surfaces under spatial translations (harmonic analysis) and rotations (scalar, vector, and tensor modes). Cosmological gauge-invariant perturbation theory analyses the characteristics and dynamics of the physical degrees of freedom of the so-defined perturbations (see, e.g., [21, 82, 135, 201], and [53, 95] for the flat space-time limit). In this setting, GWs correspond to the tensor mode. The GW tensor is by definition zero in the background, so GWs are automatically gauge invariant, i.e., invariant under arbitrary infinitesimal coordinate transformations [82, 194]. This fact simplifies considerably their treatment with respect to scalar and vector perturbations. Gauge-invariant cosmological perturbation theory directly brings out the TT gauge, highlighting the two independent components of the metric perturbation representing the physical radiative degrees of freedom. Given that scalar, vector, and tensor modes are decoupled from each other at linear order in perturbation theory [82], we neglect the first two in what follows. Ultimately, GWs in the FLRW universe are represented by the tensor space-time perturbation hij (x, t) (i, j = 1, 2, 3): ds 2 = −dt 2 + a 2 (t) (δij + hij ) dx i dx j .

(1)

Since hij is symmetric, the transverse and traceless conditions ∂i hij = hii = 0 leave only two independent degrees of freedom, which correspond to the two GW polarizations. The linearization of Einstein equations over a FLRW background and to first order in hij leads to the GW equation of motion: ∇2 h¨ ij (x, t) + 3 H h˙ ij (x, t) − 2 hij (x, t) = 16π G ΠijT T (x, t) , a

(2)

where a dot denotes derivative with respect to t, H = a/a ˙ is the Hubble rate, ∇ 2 = ∂i ∂i is the Laplacian associated to the co-moving coordinates x i in (1), and ΠijT T is the transverse and traceless part of the anisotropic stress. In general, the anisotropic stress is

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a 2 Πij = Tij − p a 2 (δij + hij ) ,

(3)

where Tij denotes the spatial components of the energy-momentum tensor of the GW source and p is the pressure of the background fluid. In the RHS of Eq. (3), the term a 2 p δij is a pure trace that does not contribute to ΠijT T , while the term in a 2 p hij cancels out with an identical term of opposite sign that emerges in the derivation of Eq. (2). In order to extract the transverse and traceless part of the anisotropic stress, it is easier to go to Fourier space:  Πij (x, t) =

d 3k Πij (k, t) e−ik·x . (2π )3

(4)

ˆ = δij − kˆi kˆj the projector on the subspace orthogonal to k, We define Pij (k) satisfying Pij ki = 0 and Pij Pj l = Pil . The transverse and traceless anisotropic ˆ Πlm (k), where the TT projector Oij,lm is stress becomes then ΠijTT (k) = Oij,lm (k) given by (see, e.g., Ref. [162]) ˆ = Pil (k) ˆ Pj m (k) ˆ − Oij,lm (k)

1 ˆ Plm (k) ˆ . Pij (k) 2

(5)

Note that ki ΠijTT = ΠiiTT = 0. It is also convenient to express the transverse and traceless perturbation hij in terms of its two polarization states r = +, ×, like hij (x, t) =

  r=+,×

d 3k r ˆ hr (k, t) e−ik·x eij (k) (2π )3

(6)

+ ˆ where the two polarization tensors can be written as eij (k) = m ˆi m ˆ j − nˆ i nˆ j and × ˆ ˆ and nˆ the two unit vectors orthogonal to kˆ and to eij (k) = m ˆ i nˆ j + nˆ i m ˆ j , with m each other. We now turn to the derivation of the homogeneous solutions of Eq. (2), describing the free propagation of GWs through the FLRW space-time. In particular, there are two regimes of interest in the cosmological case: GW wavelengths smaller and larger than the Hubble radius. We adopt conformal time dη = dt/a(t), in terms of which the metric (1) reads

  ds 2 = a 2 (η) −dη2 + (δij + hij ) dx i dx j .

(7)

In conformal time and Fourier space, Eq. (2) becomes hij (k, η) + 2 H hij (k, η) + k 2 hij (k, η) = 16π G a 2 ΠijT T (k, η) ,

(8)

where primes denote derivatives with respect to η, k = |k| is the co-moving wavenumber, and H = aH denotes the conformal Hubble factor.

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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We consider the case without source, ΠijT T (x, η) = 0 (for the solution in the presence of a generic stochastic source, see section “Signal from a Generic Stochastic Source of Gravitational Waves”), and we are interested in solving for the time dependence of the Fourier amplitudes hr (k, η) in Eq. (6). We first define the variable Hij (k, η) = a hij (k, η), decompose it as done in Eq. (6), and derive the equation of motion for the amplitudes Hr (k, η) = a hr (k, η) from (8), without the source Hr (k, η) +

  a  2 Hr (k, η) = 0 . k − a

(9)

The general solution for a generic scale factor with power law behaviour a(η) = an ηn reads hr (k, η) =

Ar (k) Br (k) η jn−1 (kη) + η yn−1 (kη) , a(η) a(η)

(10)

where jn (x), yn (x) are the spherical Bessel functions and Ar (k) and Br (k) are dimensional constants, to be determined by the initial conditions. This covers the cases of radiation (n = 1) and matter (n = 2) domination, as well as De Sitter inflation (n = −1) (see, e.g., [82]). Note that, for a power law scale factor, a  /a ∝ H 2 in Eq. (9). It is therefore straightforward to study the limits of super-Hubble (k  H ) and sub-Hubble (k  H ) scales, either taking the limits kη  1 and kη  1 in Eq. (10) or directly from Eq. (9), as we now proceed to do. Considering sub-Hubble scales, the term a  /a is negligible compared to k 2 , and the solution of Eq. (9) becomes hr (k, η) =

Ar (k) ikη Br (k) −ikη e + e , a(η) a(η)

for k  H

(11)

(note that the dimension of the constants Ar (k) and Br (k) is different than in Eq. (10)). From the reality condition on hij (x, η), one obtains Ar (−k) = Br∗ (k) and Br (−k) = A∗r (k). The free solution for the tensor metric perturbation on subHubble scales becomes naturally a superposition of plane waves with wave-vectors k and amplitude decaying as 1/a(η) because of the expansion of the universe: hij (x, η) =

 1  d 3k r ˆ e (k)[Ar (k)eikη−ik·x + c.c.] = a(η) r=+,× (2π )3 ij

(12)

=

 1  d 3k r ˆ e (k)[Br (k)e−ikη−ik·x + c.c.] , a(η) r=+,× (2π )3 ij

(13)

where c.c. stands for complex conjugate.

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In the case of super-Hubble scales, the term a  /a dominates over k 2 (except in the radiation era, when it is zero – this case has to be treated separately; see, e.g., [51]), and the solution reads  hr (k, η) = Ar (k) + Br (k)

η

dη a 2 (η )

,

for k  H .

(14)

This solution is relevant in particular in the case of quantum fluctuations of the tensor metric perturbation generated during inflation, treated in detail elsewhere in this book. Due to the quasi-exponential expansion of the background during inflation, the second term in the RHS of (14) becomes quickly subdominant with respect to the first one, so that the super-Hubble solution is constant in time. Ar (k) represents therefore the amplitude with which the tensor perturbations re-enter the Hubble scale after the inflationary phase. After Hubble re-entry, the tensor modes start to oscillate and become thereby standard GWs, behaving as in Eq. (11). Figure 1 shows the properly normalized solution for hr (keq , η), derived from Eq. (9) with scale factor performing the transition across the radiation to matter eras, for the wave-number entering the Hubble scale at the time of matter-radiation equality. Note that, in terms of the GW definition based on the separation of characteristic scales between the background and the metric perturbation (discussed at the beginning of this section), GWs correspond to the sub-Hubble solution. The super-Hubble tensor mode, on the other hand, cannot, strictly speaking, be referred to as GWs. After having defined GWs in a cosmological setting, given by the FLRW metric, in the following we focus on GW signals produced in the early universe. The next section presents the general properties of these signals as they would appear today,

Fig. 1 Evolution of the tensor perturbation hr (keq , η), normalized to the super-Hubble amplitude Ar (keq ), for the wave-mode entering the Hubble scale at matter-radiation equality, as a function of conformal time

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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discussing in particular why it is appropriate to describe GWs from early universe sources as stochastic GW backgrounds (SGWB), statistically homogeneous and isotropic, Gaussian, and unpolarized.

Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties We move now into discussing general aspects of cosmological GW backgrounds. We argue first that GW backgrounds generated by cosmological sources are always expected to be stochastic. We then explain the properties expected for any cosmological SGWB, generated in the early universe. We postpone the discussion on the properties of the spectrum of any such SGWB for section “Signal from a Generic Stochastic Source of Gravitational Waves”. We speak of SGWBs whenever the amplitude hij (x, η) of the tensor perturbation representing the background is a random variable that can only be characterized by its statistical properties, e.g., by means of ensemble averages. To perform an ensemble average, many copies of the system should be available. This is however not possible in our case, as there is only one observable universe. In cosmology, we can invoke nonetheless the ergodic hypothesis, which amounts to equate ensemble averages with either spatial or temporal averages. In other words, by observing a large enough region of the universe (alternatively, a given region for a long time), we can access many realizations of the background. For this to hold, two conditions are necessary: first, the initial condition of the source generating the GWs must be the same at every point in space (even if only in a statistical sense), and second, the GW source must respect causality, operating within regions in the universe of size smaller than the causal horizon at the time of GW generation. If these conditions are met by a GW source in the early universe, the expected GW signal takes the form of a stochastic background, and we can study its properties by means of the ergodic theorem. Let us characterize this in more detail. Due to causality, a GW generation mechanism acting at a given time in the early universe cannot produce a signal correlated at length/time scales larger than the causal horizon Hp−1 at the time of GW production. The physical correlation scale p of the emitted GW background must satisfy p ≤ Hp−1 , and the signal can be correlated at best on a time scale Δtp ≤ Hp−1 (here a subscript p denotes the time of production). Note that we use the inverse Hubble factor Hp−1 as the causal horizon, as this becomes a very good approximation during any stage of the evolution of the universe with the scale factor being a power law in cosmic time (like in radiation and matter domination). At the present time, we can access much larger length/time scales than today’s redshifted scale associated to Hp−1 : therefore, a cosmological GW background today is composed by the superposition of many individual signals, uncorrelated in time and space. The number of such independent signal contributions can be calculated knowing the correlation length of the GW source at the time of production and the evolution of the universe from tp till today.

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To obtain a better understanding of the above statements, let us compare the size of the horizon today H0−1 , versus the correlation length redshifted to today, 0p = p (a0 /ap ). We can write the ratio 0p H0−1

=

Hp−1 a0 p a0 a0 /ap ≤ = , −1 a −1 a Ωmat (zp ) + Ωrad (zp ) + ΩΛ H0 H0 p p

(15)

where in the last equality √ z denotes the redshift, and we have inserted the Friedmann equation H (z) = H0 Ωmat (z) + Ωrad (z) + ΩΛ , with H0 = 100 h km s−1 Mpc−1 the Hubble rate today, Ω∗ (z) = ρ∗ (z)/ρc0 , ρc0 = 3H02 /(8π G) the critical energy density today, and ρ∗ (z) denoting the energy densities of the radiation fluid ρrad (z), nonrelativistic matter ρmat (z), and a cosmological constant ρΛ ≡ Λ/(8π G) mimicking dark energy. We are only interested in sources creating GWs deep inside the radiation era, because only these produce observable GW backgrounds in the appropriate frequency range for GW detectors. Therefore the term proportional to Ωrad (z) above is dominating in the square root. Furthermore, the expansion of the universe is adiabatic and hence driven by the conservation of entropy per co-moving volume, gS (T ) T 3 a 3 (t) = const., where T is the photon temperature at time t and gS is the effective number of entropic degrees of freedom at the same time [136]. As the universe cools down, gS decreases when some species become nonrelativistic. When this occurs, they release their entropy to the species that are still in thermal −1/3 equilibrium, causing the temperature to drop as T ∝ a −1 gS , slower than the usual decay T ∝ a −1 . With all of this in mind, we can quantify the amount of expansion of the universe between GW production at a temperature Tp and, today, by the ratio a0 = ap



gS (Tp ) gS (T0 )

1/3 

Tp T0





1.25 × 10

13

gS (Tp ) 100

1/3 

Tp GeV

 ,

(16)

where we have used T0 2.35 × 10−13 GeV for the photon temperature today and gS (T0 ) 3.91 for the Standard Model degrees of freedom with three light neutrino species [136]. It is noteworthy to observe that gS (T0 ) must be evaluated as if the neutrinos were still relativistic today, as they decouple from the thermal plasma when they are relativistic (at T ∼ MeV, while mν < 2 eV), but do not release their entropy to the photons later on when they become nonrelativistic. In the Standard Model, the last drop of gS occurs when electrons and positrons become nonrelativistic (at a temperature T ∼ me 0.5 MeV), so that the photon temperature evolves from then on, simply as T ∝ a −1 . In the radiation era, the 2 total energy density is given by ρrad = π30 g∗ (T ) T 4 , where g∗ (T ) is the effective number of relativistic degrees of freedom at temperature T . The radiation fraction can be then written like

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

 0 Ωrad (T ) = Ωrad

gS (T0 ) gS (T )

4/3 

 g∗ (T ) a0 4 , g∗ (T0 ) a

1049

(17)

0 = 2.47 × 10−5 is the radiation energy density today, g (T ) = 2, and where h2 Ωrad ∗ 0 we note that g∗ (T ) gS (T ) for T  0.1 MeV. Considering the largest possible correlation scale of the signal, p ≡ Hp−1 , and keeping only the dominant term during the radiation era, Eq. (15) leads to

0p H0−1

   gS (Tp ) 1/3 g∗ (T0 ) T0 1

 =√ g∗ (Tp ) Tp Ωrad gS (T0 ) Ωrad (Tp )     100 1/6 GeV −11 ,

1.3 × 10 g∗ (Tp ) Tp a0 /ap

(18)

clearly exhibiting that the correlation scale today of a GW background generated in the early universe is tiny comparable to the present Hubble scale. Furthermore, we can find the number of uncorrelated regions from which we are receiving today independent contributions to the stochastic GW background, by calculating the angle Θp in the sky subtending the size p at zp Θp =

p 1 , dA (zp ) = dA (zp ) H0 (1 + zp )

 0

zp

dz , (19) √ Ωmat (z ) + Ωrad (z ) + ΩΛ

where dA (zp ) is the angular diameter distance. At the present time, we have access to a total of ∼ dA (zp )2 /2p = Θp−2 uncorrelated regions (projected areas in the sky). Let us consider, for example, the GW background from the electroweak (EW) phase transition at TEW ∼ O(102 ) GeV, embedded in some extension of the Standar Model of particle physics (SM), so that the transition is a first-order one (c.f. section “First-Order Phase Transitions”). Taking gS (TEW ) ∼ 100 for simplicity, today’s redshifted scale corresponding to the horizon scale at the epoch of the EW −1 phase transition is (a0 /aEW )HEW

2.7 × 10−4 pc. Using that h2 Ωmat 0.12 and ΩΛ = 1 − Ωmat , Eq. (19) leads to ΘEW 2 × 10−12 deg. This implies that the GW background due to a causal process occurring at the EW scale (i.e., ∼10−11 s −2 after the beginning of the universe) is composed by the superposition ∼ΘEW ∼ 1024 independent signals emitted by an equal number of uncorrelated regions (even more, −1 if the inequality EW ≤ HEW is not saturated). This implies that the GW signal must be described at a statistical level, and cannot be resolved beyond its stochastic nature: in order to resolve individual realizations of a GW background from the first second of the universe, a GW detector should have an angular resolution as good as ∼Θp , which, as illustrated by the example of the GW signal produced at the EW transition, corresponds always to a tiny resolution unreachable by any currently conceivable technology. In light of the above discussion, GW emissions from active sources in the early universe always lead to GW signals perceived today as stochastic backgrounds.

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Furthermore, since any cosmological background is composed of the superposition of signals emitted by the same physical process taking place in many causally disconnected regions, this immediately justifies the use of the ergodic theorem, allowing us to interpret spatial averages as ensemble averages. Actually, the above considerations apply even to much lower energy scales than the EW one. For the sake of the exercise, let us (optimistically) assume that a stochastic background with high enough amplitude was produced in a late stage of the evolution of the universe (after CMB decoupling), so that future GW detectors could detect it, say with an angular resolution of about ∼10 deg. According to Eq. (19), the redshift at which the background should have been generated should be zp 17. Consequently, the stochastic nature of any cosmological GW background persists until well inside the matter-dominated era. It is worth mentioning that the above arguments remain valid even for causal GW sources that are not localized in time, but rather operate continuously during several Hubble times. This is the case of topological defects, which we will introduce in section “Cosmic Defect Networks”. In the paradigmatic example of a network of cosmic strings, GWs are continuously emitted all the way since the epoch of the phase transition that generated the strings, until today. The GW signal in this case is the sum of two components. One is given by GWs generated by the anisotropic stress of the network at around the horizon scale at each time. The second contribution is the superposition of the GWs emitted from subhorizon cosmic string loops (chopped off from the main network). The GW signals from these two components form also a stochastic background, as it is contributed by the superposition of many signals coming from many independent sources emitting from regions of current redshifted size much smaller than today’s horizon scale. Hence, again, this background cannot be resolved beyond its stochastic nature, for the same reasons discussed above. The main difference between this continuously sourced signal and the one arising from sources localized in time, is that the former extends over many frequencies, precisely due to the fact that the source emits GWs for many Hubble times. Finally, note that during inflation, the causal horizon and the above arguments do not apply (as they were implicitly based on assuming the causal horizon being of the order of the Hubble radius, which grows linearly in time whenever the scale factor is a power law a ∝ t p ). The expected background of GWs from inflation is, however, also stochastic. In this case, the reason for its stochastic nature is due to the intrinsic quantum nature of the generating process, as discussed elsewhere in this book. In general, stochastic GW backgrounds from early universe sources are expected to be statistically homogeneous and isotropic, unpolarized, and Gaussian. The reasons for all this can be easily understood as follows: • Statistical homogeneity and isotropy: This property is inherited from the same symmetries of the FLRW background metric, i.e., it is based on the homogeneity and isotropy of the universe. It implies that the two-point spatial correlation must satisfy hij (x, t1 ) hlm (y, t2 ) = Fij lm (|x − y|, t1 , t2 ), where . . . represents the ensemble average (that turns into a volume/time average through the ergodic hypothesis), hij (x, t) are the tensor perturbations representing the GW

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

1051

cosmological background, and Fij lm is a function that depends only on the modulus of the separation between the two tensor amplitudes. In the cases discussed above, of a source operating during the radiation era, where the GW background amplitude is given by the superposition of signals emitted from many uncorrelated regions, the (statistical) homogeneity and isotropy of the universe imply that such regions must share the same statistical properties, say the same temperature and averaged particle number densities. Therefore, if a GW generating event, say a phase transition, happens everywhere in the universe at the same time, the produced GW background will be statistically homogeneous and isotropic. • Chiral symmetry: Cosmological GW backgrounds are typically assumed to be unpolarized, because most sources in the universe are parity preserving. If the source of the GWs is based on interactions that are symmetric under parity, the two polarizations + and × of the resulting GW background will be uncorrelated. This means in particular that the correlator h+ (k, t)h× (k, t) = 0 vanishes. The connection with the parity symmetry of the source can√be made more ˆ = (m ˆ ± i n) ˆ i / 2, where m, ˆ nˆ explicit introducing the helicity basis i± (k) ˆ We can are the unit vectors transverse to the GW’s direction of propagation k. construct a basis for the transverse-traceless tensor space representing the two ±2 + × = (eij ± i eij )/2. This basis transforms independent helicity states ±2, as eij ±2 ±2 ±2iθ ˆ The tensor field as eij = e eij under rotation by an angle θ around k. representing the GW can be expressed as a linear combination in this basis, +2 −2 hij = h+2 eij + h−2 eij . Using the definitions just introduced, it then follows that h+2 (k, t)h+2 (k, t) − h−2 (k, t)h−2 (k, t) = h+ (k, t)h× (k, t) = 0, where the last equality holds if the background is unpolarized. The absence of a net polarization in a GW cosmological background is thus equivalent to the condition that the two independent helicity modes are produced (on the average) with identical (quadratic) expectation values. If this is not the case, the GW background would be chiral and must have been produced from some parityviolating source. • Gaussianity: This property follows from the fact that a stochastic GW background from the early universe is formed by the emission of many uncorrelated sources. Since the signal is composed by a large number of independent GW emissions, the central limit theorem implies that the signal resulting by the superposition of many independent signals must follow a Gaussian distribution [8]. Note, however, that Gaussianity also applies to the case of the inflationary GW background, originated by vacuum fluctuation, due to the quantum origin of such background. Finally, let us point out that there are exceptions to the properties just listed above (statistical homogeneity and isotropy, Gaussianity, and absence of net polarization), even though they are satisfied to a good approximation by most cosmological GW backgrounds. For example, a certain level of large-scale anisotropy in the universe is allowed by present CMB constraints [5]. An example of a cosmological background with large anisotropies is the GW signal generated during

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post-inflationary preheating, via parametric resonance, by a preheat field with a coupling to the inflation chosen so that it behaves as a spectator field deep inside inflation and as a massive field toward the end of inflation (see Refs. [26, 27]). Another example of GW source from the early universe, leading to a statistically anisotropic GW background, is the excitation of a gauge field during inflation (although this has been studied mainly for the scalar mode; see, e.g., [24, 25, 167]). The GW background generated by gauge field dynamics during inflation is also non-Gaussian, since the GW source is quadratic in the fields [58], and it can be polarized if the interaction between the gauge field and the inflation is paritybreaking [12, 192]. As a matter of fact, it has been pointed out recently that any SGWB from the early Universe exhibits today a certain degree of inhomogeneity and anisotropy, simply due to the GW propagation through the large-scale structure of the universe [22, 23].

Characterization of a Stochastic Gravitational Wave Background In this section, we introduce various quantities that define and characterize the power spectrum of a SGWB. By construction, the stochastic variable is hr (k, t), a Fourier mode of the SGWB with momentum k and polarization r. For a SGWB enjoying the properties described at the end of section “Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties” (statistical homogeneity and isotropy, lack of net polarization and Gaussian distribution), the power spectrum of the tensor modes can be defined as hr (k, t) h∗p (q, t) =

8π 5 (3) δ (k − q) δrp h2c (k, t) , k3

(20)

with hc a dimensionless real function, depending only on the time t and the modulus of the wave-number k = |k|. The delta function acting over the momenta k, q, is due to statistical homogeneity, whereas the fact that hc does not depend on the direction kˆ is due to statistical isotropy. The delta function over the polarization indices r, p, is due to the absence of a net polarization. Finally, the fact that the distribution is Gaussian implies that the above expectation value contains all the relevant information on the statistical distribution of the stochastic variable hr (k, t). In other words, once we know the power spectrum h2c (k, t), we can build any other correlator as a function of it, as higher-order even-point correlators can be written in powers of h2c (k, t) (thanks to the Wick theorem), whereas odd-point correlators simply vanish. The factor 8π 5 in Eq. (20) is made explicit simply for convenience, so that the variance of the stochastic tensor field hij (x, t) representing the GW background in real space can be written as 

+∞

hij (x, t) hij (x, t) = 2 0

dk 2 h (k, t) , k c

(21)

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

1053

where the factor 2 is a convention to reflect the fact that h2c only characterizes the typical amplitude of one polarization, while the LHS in Eq. (21) involves contributions from two independent polarizations. It is clear from Eq. (21) that h2c (k, t) represents the characteristic (quadratic) amplitude of the GWs, per logarithmic wave-number and polarization state, at any time t. Let us recall that, as argued in section “Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties”, the ensemble average in the LHS of Eq. (20) can be interpreted both as a volume average over sufficiently large regions compared to the GW wavelengths and as a time average over several periods of oscillation. The time behavior of the Fourier amplitudes for freely propagating (subHubble) modes is given by Eq. (11). Inserting this solution into Eq. (20), we see that, except for the terms quadratic in the oscillatory functions, cross terms average out over oscillations, so one obtains hr (k, t) h∗p (q, t) =

1 [ Ar (k) A∗p (q) + Br (k) Bp∗ (q) ] . a 2 (t)

(22)

For free waves, the amplitude of a GW mode behaves therefore as hc (k, t) ∝ 1/a(t) (once the mode is well inside the Hubble radius). Besides the amplitude of the tensor modes hc , a relevant quantity of interest to characterize the SGWB is its energy density spectrum per logarithmic unit of wavenumber, dρGW /dlogk. The GWs energy density is given by the 00-component of the GW energy-momentum tensor [158] T GW μν =

∇μ hαβ ∇ν hαβ , 32π G

(23)

which in the FLRW universe becomes ρGW =

 +∞ hij (x, η) hij (x, η) h˙ ij (x, t) h˙ ij (x, t) dk dρGW = , (24) = 32π G k dlogk 32π G a 2 (η) 0

where in the second equality we switched from coordinate time t to conformal time η ≡ dt/a(t), while the third equality simply defines dρGW /dlogk. It is worth noticing that the energy-momentum tensor of GWs cannot be localized inside a volume smaller than the typical wavelengths of the GW modes. The energymomentum tensor (and hence the energy density carried by GWs) only becomes a meaningful concept when defined by performing a volume or time average (or an average over several wavelengths and/or frequencies, for its Fourier components). We refer the reader to [51, 158, 162] for discussions on this subtle aspect. For a cosmological stochastic GW background, the volume/time average performed in Eq. (23), and inherited in Eq. (24), can be interpreted, via the ergodic hypothesis, as the usual ensemble average of Eq. (20). An expression for the GW energy density power spectrum dρGW /dlogk, valid for freely propagating waves inside the Hubble radius, can be derived from Eq. (24). Let

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us first consider the power spectrum of the conformal time derivatives of the Fourier modes hr (k, η), written in a similar structure as Eq. (20): ∗

hr (k, η) hp (q, η) =

8π 5 (3) 2 δ (k − q) δrp hc (k, η) . k3

(25)

Here the characteristic amplitude hc 2 (k, η) is defined in analogy to h2c (k, η). Now, because of the delta function, k = q, and hence the cross oscillatory terms are averaged out [as done before in Eq. (22)]. Besides, one can neglect the (a  /a)2 term arising due to the time derivative of Eq. (11) against the term k 2 , since in the case under analysis, k  (a  /a). We then find a simple relation among amplitudes as hc (k, η) k 2 h2c (k, η) ,

(26)

k 2 h2c (k, η) dρGW . = dlogk 16π G a 2 (η)

(27)

2

which leads to

Since hc (k, η) ∝ 1/a(η) for sub-Hubble modes, as previously shown, we find that the GW energy density is diluted like radiation with the expansion of the universe, i.e. ρGW ∝ a −4 , as expected for any propagating mass-less degree of freedom. In order to provide detection forecasts for SGWB signals, the SGWB spectrum today must be expressed in terms of physical frequencies f = k/(2π a0 ), which correspond to the co-moving wave-number k, redshifted to today (hence the 1/a0 factor, where we remind the reader that a subscript “0” indicates evaluation today). We define then hc (f ) = hc (k, η0 ), which matches the definition given, e.g., in Ref. [159]. One can also define the SGWB spectral density Sh (f ) =

h2c (f ) , 2f

(28)

a quantity with dimension Hz−1 , which can be directly compared to the noise spectral density Sn (f ) of a GW detector. The GW energy density spectrum is typically conveniently normalized as ΩGW (f ) =

1 dρGW , ρc dlogf

(29)

with ρc = 3H 2 /(8π G) the critical energy density of the universe at time t. In (0) cosmology, one rather uses the quantity h2 ΩGW , which is independent of the uncertainty in the measured value of H0 . Eqs. (27) and (28), together with f = k/(2π a0 ), lead to the following useful expression

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

(0)

ΩGW (f ) =

1055

4π 2 3 f Sh (f ) . 3H02

(30)

Inserting the value of H0 , we can write the following relations: Sh (f ) = 7.98 × 10−37



hc (f ) = 1.26 × 10−18

Hz f



3

Hz f

1 h2 ΩGW (f ) Hz ,



(31)

h2 ΩGW (f ) .

(32)

It is important to remark that the expansion of the universe is negligible on the time scales of interest of GW direct detection experiments. This allows us to connect the definitions given in section “Introduction: Gravitational Waves in Friedmann-Lemaître-Robertson-Walker Backgrounds”, and in particular expansion (6), with another commonly used expansion of the present-day GW tensor; see, e.g., [158, 159, 183]: hij (x, t) =

 



+∞

df

r=+,× −∞

ˆ

ˆ ei 2π f (t−k·x) er (k) ˆ , d 2 kˆ h¯ r (f, k) ij

(33)

where the integration over negative frequencies comes from the definition ˆ ≡ h¯ ∗r (f, k), ˆ required for hij (x, t) to be real. To connect the above h¯ r (−f, k) expansion with Eq. (6), one starts from (12), fixes the scale factor to its value today, and performs a time Fourier Transform allowed by the fact that the expansion of the ˆ and extending the universe is negligible. Rewriting then d 3 k = (2π a0 )3 f 2 df d k, ˆ ≡ A∗r (f, k), ˆ integration domain to negative frequencies by identifying Ar (−f, k) one gets hij (x, t) = a02

 

+∞

r=+,× −∞

 f 2 df

ˆ

ˆ ei 2π f (t−k·x) er (k) ˆ . d 2 kˆ Ar (f, k) ij

(34)

ˆ = a 2 f 2 Ar (f, k) ˆ (the same can be done with Br (f, k)), ˆ Identifying h¯ r (f, k) 0 ˆ usual expressions for the power spectrum of the Fourier amplitudes h¯ r (f, k), given, e.g., in Refs. [158, 159, 183] and for the power spectral density, can be recovered. In order to do this, we just need to observe that hr (k, t) h∗p (q, t) = 2 Ar (k) A∗p (q) . This leads therefore to a 2 (t) ˆ h¯ ∗p (g, q) ˆ = a04 f 2 g 2 Ar (k) A∗p (q) = h¯ r (f, k) =

1 ˆ δrp Sh (f ) . δ(f − g) δ (2) (kˆ − q) 8π

(35)

From this equation, we can appreciate that the definition of the spectral density given in [158, 159, 183] is equivalent to the one we introduced in Eq. (20).

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Evolution of a Stochastic Gravitational Wave Background in the Expanding Universe The evolution of GWs after generation by a cosmological source can be very well approximated as free propagation through the FLRW space-time. Effectively, one can assume a weak interaction rate for the GWs, with Γ (T ) = n σ v, where n is the number density of particles n ∼ T 3 , σ ∼ G2 T 2 is the cross-section (with Newton constant G = 1/MP2 l , MP l the Planck mass), and v ∼ 1. Comparing this rate with the Hubble rate in the radiation-dominated era, H (T ) ∼ T 2 /MP l , gives [159] Γ (T ) G2 T 5 = ∼ 2 H (T ) T /MP l



T MP l

3 .

(36)

Therefore, at any temperature in the universe T < MP l for which our present knowledge about gravitation holds, the GW interaction rate is smaller than the Hubble parameter. One can therefore safely assume that GWs propagate through the universe essentially without self-interactions, nor interacting with the surrounding matter; the occasional interaction events do not perturb the essential nature of the signal (e.g., they occasionally interact with our detectors nowadays!). This implies that a GW signal from the early universe can have imprinted unique information about the process that produced it and therefore about the state of the universe at epochs and energy scales unreachable by any other means. It is important to remark that the energy scales that GWs can probe extend far beyond the reach of presently available observational probes of the universe, mostly based on electromagnetic emission. On a practical level, this means that the solution given in Eq. (11) properly describes the GW signal as soon as the generating process has ended (or as soon as the tensor mode has entered the Hubble scale, in the case of the SGWB produced during inflation). This leads to the natural result that the GW energy density scales as a radiation field, ρGW ∝ a −4 (c.f. Eqs. (27) and (11)), while the physical wave momentum redshifts as k/a. The SGWB spectrum therefore retains its initial shape while redshifting with the expansion of the universe. We can rewrite the GW energy density spectrum today as (0)

h2 ΩGW (k) =

h2 ρc



ap a0

4

 ρp

1 dρGW ρ dlogk

 for k  H ,

(37)

p

where ρp denotes the total energy density in the universe at the time of GW production. The GW frequency of the signal today, accounting for redshifting, is given by f =

1 k xk ap = Hp , 2π a0 2π a0

(38)

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where we have defined the parameter xk given by the ratio of the physical wavenumber at the time of GW production to the Hubble rate at that time Hp xk =

k/ap . Hp

(39)

The significance of the parameter xk becomes clear in the case of GW sourcing processes that take place at some time tp in the early universe and last only for a brief amount of time, comparable to the Hubble time at tp . In this case, we expect the GW signal to be uncorrelated on length/time scales larger than Hp−1 . The characteristic length/time scale of the sourcing mechanism imprints in the GW energy density spectrum. Consequently, one typically expects this latter to be peaked (or to show a feature) at a characteristic wave-number k∗ /ap ≥ Hp . The parameter xk∗ , evaluated at the characteristic wave-number of the SWGB signal, must therefore satisfy xk∗ ≥ 1. Its precise value contains an indication of the typical time/length scales involved in the GW sourcing process, with respect to the Hubble scale. A classical example of fast SGWB production mechanisms is a first-order phase transition, discussed in section “First-Order Phase Transitions”. There are however examples of early universe GW sources that produce GWs continually during a long period of time, i.e., for a large range of values of ap Hp in Eq. (38). These sources lead to SGWB spectra that show similar amplitude on broad frequency ranges, instead of clearly identified peaks. Typical examples are cosmic strings (presented in section “Cosmic Defect Networks”), and inflation, for which ap Hp varies almost exponentially (discussed elsewhere in this book). Often, the SGWB signal produced by this kind of long-lasting sources is approximately flat over a wide range of frequencies, generally covering the sensitivity range of direct GW detectors as well as pulsar timing arrays (see, e.g., [51]). This means that the source converts into GWs an amount of energy that remains a constant fraction of the total energy density ρ during the radiation era. The resulting spectrum (37) is then approximately flat because ap4 ρp is approximatively constant during the radiation era (up to variations of the number of relativistic species; c.f., e.g., [120]). In the case of GW signals generated during the radiation era, it is possible to rewrite Eqs. (37) and (38) in a more instructive way. Using (16) and (17), the present-day GW amplitude and GW frequency can be directly related to the temperature of the universe Tp at the time of GW production:

h

2

(0) ΩGW

−5

= 1.6 × 10



100 g∗ (Tp )

1/3 

1 dρGW ρ dlogk

 (40) p

and f = 2.6 × 10−8 Hz xk



g∗ (Tp ) 100

1/6

Tp GeV

(41)

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where we have used H 2 = 8π Gρ/3 and again g∗ (Tp ) = gS (Tp ) for Tp  MeV [136]. Using the above equations, it is easy to predict whether a GW signal coming from a source acting in the early universe falls in the right frequency/intensity range of sensitivity of specific GW detectors. Furthermore, Eq. (38) and its analogue in the radiation era Eq. (41) illustrate the potential of GW detectors to probe the early universe through SGWB signals. One can identify the frequency in Eq. (38) with the characteristic frequency of the SGWB (e.g., the frequency at which it peaks). In this case, the precise value of xk∗ can only be determined within a specific GW generation process; however, setting xk∗ = 1, Eq. (41) represents the lowest possible characteristic frequency, today, of a SGWB emitted by a process operating at a given time in the universe, parametrized by Tp . It is then possible to establish a correspondence between the frequency sensitivity range of a given GW experiment and a range of temperatures in the early universe to which it would grant access through possible SGWB detection. Figure 2 shows this correspondence taking as example several GW detectors: ground-based interferometers with 1 Hz  f  103 Hz, LISA with 10−5 Hz  f  0.1 Hz, pulsar timing arrays with 3 × 10−9 Hz  f  10−6 Hz, and CMB (in this case the observable frequency window corresponds, respectively, to the Horizon today and at the epoch of photon

Fig. 2 Black line: GW frequency today as given in Eq. (38) (which reduces to Eq. (41) in the radiation era), for xk∗ = 1, as a function of temperature in the universe (and corresponding redshift in the upper x-axis). Shaded regions: the frequency ranges of detectability of several GW experiments, from right to left, respectively, 1 Hz  f  103 Hz for ground-based interferometers, 10−5 Hz  f  0.1 Hz for LISA, 3 × 10−9 Hz  f  10−6 Hz for pulsar timing arrays, and 3.4 × 10−19 Hz  f  7 × 10−18 Hz for the CMB. (Figure taken from [51])

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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decoupling: H0 /(2π ) ≤ f ≤ Hdec (a0 /adec )/(2π ), with Tdec 0.26 eV. Note that Eq. (41) does not hold in the case of the CMB, which extends beyond the radiation era, while Eq. (38) is generically valid.) with 3.4 × 10−19 Hz  f  7 × 10−18 Hz. One can appreciate that the potential of GW experiments to probe the early universe extends beyond the one of other cosmological probes, e.g., Large Scale Structures, the CMB, and Big Bang Nucleosynthesis (BBN). In particular, it covers the energy scale interval between inflation (probed by the CMB) and BBN (probed by light elements abundance) which is not directly accessible by other means, since the universe was opaque to photons at that epoch. Detection is far from being guaranteed, as very little is understood about the status of the universe at these high energy scales, and only speculations are possible concerning the physical processes occurring at this epoch. Conversely, this warrants GW detectors a great discovery potential.

Signal from a Generic Stochastic Source of Gravitational Waves We start by characterizing the source of the GWs, the transverse-traceless (or simply “tensor”) part of the anisotropic stress given in Eq. (3), upon application of the projector (5). Analogously to hij , ΠijT T can be decomposed in two polarization states (here and in the following, we omit the superscript TT for brevity):

Πij (x, t) =

  r=+,×

d 3k r ˆ Πr (k, t) e−ik·x eij (k) . (2π )3

(42)

For the reasons discussed in section “Cosmological (Ergo Stochastic) Gravitational Wave Backgrounds: General Properties”, we consider the source as a stochastic variable, assuming again that the properties of statistical homogeneity and isotropy, Gaussianity, and absence of net polarization apply equally, as previously assumed for the GWs (indeed, these latter enjoy such properties precisely because the source enjoys them in the first place). We then write the unequal-time correlator of the Fourier modes of the tensor anisotropic stress as Πr (k, η) Πp∗ (q, ζ ) =

(2π )3 (3) δ (k − q) δrp Π (k, η, ζ ) . 4

(43)

Note that the normalization is different than in Eq. (20) and that we have written the correlator at different (conformal) times, for future convenience. When the times are the same in the above equation, we speak then about the power spectrum of the tensor anisotropic stress. In terms of conformally transformed Fourier amplitudes Hij = a hij , the evolution equation for the GW modes is

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C. Caprini and D. G. Figueroa

Hr (k, η) +

  a  2 Hr (k, η) = 16π G a 3 Πr (k, η) . k − a

(44)

Let us first consider the case of a GW source operating in the radiation dominated era, with a(η) = a∗ η. The above equation in terms of the dimensionless variable x = kη becomes then d 2 Hrrad (k, x) 16π Ga∗3 3 + Hrrad (k, x) = x Πr (k, x) . 2 dx k5

(45)

The Green’s function associated with the differential operator in the LHS of this equation is G (x, y) = sin(x − y), while {sin x, cos x} are the homogeneous solutions. Let us consider now that the source is turned on at a time xin = kηin and stops producing GWs at xfin = kηfin , with ηfin still deep inside the radiationdominated era. Assuming vanishing initial conditions for the GWs, the solution to the above equation during the time when the source is active, xin ≤ x ≤ xfin , is given by the particular solution Hrrad (k, xin < x < xfin ) =

16π Ga∗3 k5



x

dy y 3 sin(x − y) Πr (k, y) .

(46)

xin

As we are interested in the GW spectrum today, we need to find the solution at x0 = kη0  xfin . This requires to match Eq. (46) with the homogeneous solution (corresponding to a freely propagating GW – c.f. Eq. (11)) rad Hrrad (k, x > xfin ) = Arad r (k) cos x + Br (k) sin x ,

(47)

and with its first derivative. This procedure leads to finding the matching coefficients rad as Arad r , Br . Arad r (k)

16π Ga∗3 = k5

Brrad (k) =

16π Ga∗3 k5



xfin

dy y 3 sin(−y) Πr (k, y),

(48)

dy y 3 cos(y) Πr (k, y).

(49)

xin



xfin

xin

By applying now Eq. (22) together with Eq. (20) and Eq. (43), we can express the power spectrum amplitude today, due to a GW source acting in the radiation era, as h2c (k, η0 )|rad

G2 a 6 = 64 2 ∗7 a0 k





xfin

dy y xin

3

xfin

dz z3 cos(y − z) Π (k, y, z) . (50)

xin

Equipped with this expression, it is straightforward to obtain the GW energy density power spectrum from Eq. (27):

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

dρGW (k, η0 )

dlogk rad  ηfin  4 G 3 ηfin 3 = k dη a (η) dζ a 3 (ζ ) cos[k(η − ζ )] Π (k, η, ζ ) , π a04 ηin ηin

1061

(51)

where the integrals are written in terms of the conformal time. Equation (51) represents the energy density power spectrum of a generic stochastic source, operating during a finite interval (ηin , ηfin ) well inside the radiation-dominated era, given implicitly in terms of the unequal time correlator of the anisotropic stress sourcing the GWs. Equivalently, if there is a GW source operating during the matter-dominated era, the scale factor behaves as a(η) = a∗ η2 , and the GW equation of motion is given by   d 2 Hrmat (k, x) 2 16π Ga∗3 6 mat H + 1 − (k, x) = x Πr (k, x) . r dx 2 x2 k8

(52)

After the source has ceased producing GWs, the freely propagating GW solution becomes Hrmat (k, x > xfin ) = Amat r (k)

cos x x

 

sin x + sin x + Brmat (k) − cos x ,(53) x

with matching coefficients Amat r (k) = Brmat (k)

16π Ga∗3 k8

16π Ga∗3 = k8



xfin

dy y 5 [y cos y − sin(y)] Πr (k, y),

(54)

dy y 5 [cos y + y sin y] Πr (k, y).

(55)

xin



xfin

xin

As we are interested in wave-numbers satisfying x0  1 today, the terms proportional to x0−1 in Eq. (53) are subdominant and can be dropped, so that the solution takes again the same form as the sub-Hubble one in Eq. (11). Proceeding as done before in the case of the radiation-dominated era, the GW energy density power spectrum for a source active during the matter era is, in terms of the source unequal time correlator:

 ηfin  ηfin

dρGW 4 G 5/2

(k, η0 ) = k a∗ dη a (η) dζ a 5/2 (ζ ) Π (k, η, ζ ) dlogk π a04 ηin ηin mat × [(1 + k 2 ηζ ) cos(k(η − ζ )) + (kη − kζ ) sin(k(η − ζ ))] .

(56)

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We now proceed to analyze two examples of SGWB sources related to possible phase transitions operating in the early universe: bubble percolation during a firstorder phase transition and topological defects.

Cosmological Gravitational Wave Backgrounds: Relevant Examples First-Order Phase Transitions Phase transitions (PTs) are a generic prediction in quantum field theories. It is reasonable to expect that some PTs occurred in the early universe, for instance, during the thermal radiation-dominated era, as the temperature drops due to the expansion of the universe. The Standard Model (SM) of particle physics predicts two PTs: the quark-gluon confinement (QCDPT) around ∼150 MeV and the electroweak (EW) PT around ∼100 GeV. The Large Hadron Collider (LHC) has enlarged the present knowledge of high-energy particle physics up to the TeV scale. However, the fundamental theory describing the primordial universe at scales well above the EW scale is still unknown. In Beyond the Standard Model (BSM) scenarios, other PTs can be envisaged to take place at higher energy scales. Establishing the occurrence of a primordial PT in the early universe would provide direct information on the underlying high-energy theory. It is therefore important to investigate possible observational signatures of primordial PTs, as this would constitute a valuable test of new physics. Gravitational waves are a plausible observational remnant of primordial PTs [115, 131, 141, 142, 202]. According to Eq. (41), the typical frequencies of a GW signal emitted at the energy scale of the EWPT lie within the LISA frequency detection range, while those corresponding to the QCD energy scale fall within the PTA range. However, if the system evolves through several stages essentially without quitting thermal equilibrium, as predicted for both the QCD [14, 193] and the EW PTs [15, 104, 127, 128, 151] within the SM, the expected GW signal is observationally negligible [97]. On the other hand, if a PT is of the first order, i.e., it proceeds through the nucleation of bubbles of the new phase, it can lead to a significant GW background: for a recent review, see [111] (GWs can also be generated by the topological defects left over after a PT, as discussed in section “Cosmic Defect Networks”.). In the following, we analyze this scenario and present the conditions under which such a signal could be measured by future GW detectors. We concentrate, in particular, on the compelling case of the EWPT at LISA frequencies [10, 49] (for a recent review, see [50]). During a cosmological first-order PT, the order parameter (generally a scalar field in the cosmological case) changes from a metastable vacuum configuration to a more energetically favored vacuum state, via thermal fluctuations or quantum tunnelling over a potential barrier [114, 154, 196]. Bubbles of the new phase are nucleated and expand, ultimately percolate, and complete the transition. If the transition occurs in a thermal environment as during the thermal era in the early universe, the latent heat released by the PT is converted into thermal energy of the

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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background, kinetic and gradient energy of the scalar field, and kinetic energy of the surrounding fluid [85, 121, 148]. It is in fact natural to expect that the field performing the transition is coupled to the particles of the surrounding fluid (e.g., in the EWPT, the particles get a mass from the SM Higgs field), so the fluid is set into motion by the expanding bubbles [121,160,163]. The field-fluid coupled system can be described via near-equilibrium hydrodynamics, and the microscopic coupling is usually modelled by an effective macroscopic friction term [86, 121]. GW production occurs if shear stresses are generated, either in the scalar field configuration as the bubble collide toward the end of the PT [65, 66, 118, 141, 142] or by the bulk fluid motion linked to the bubble expansion and collision [131]. The fluid shear stresses are initially in the form of pressure waves around the bubbles [67,106–109], which get partially converted into kinetic and magnetohydrodynamic turbulence if the average velocity is sufficiently high [38, 47, 78, 101, 125, 140, 170, 184]. As we will see, the SGWB depends on four main parameters: the PT temperature T∗ , the PT strength α, the PT inverse duration with respect to the Hubble time β/H∗ , and the bubble wall speed vw [31, 86]. It is in principle possible to calculate these parameters for a given PT scenario, though the use of perturbative techniques might lead to substantial errors on their values, and more refined techniques require model-by-model evaluation [50, 61, 102]. The transition from the stage of colliding pressure waves to the turbulent phase and the spectral shape of the SGWB from turbulence remain yet to be fully understood. The prediction of the SGWB signal from a first order PT is therefore still affected by large theoretical and computational uncertainties. However, it is important to remark that the SGWB features only depend on a few parameters, related to the thermodynamics of the system and to its hydrodynamical behaviour. The extraction of these parameters from the GW signal detection can be achieved in favorable circumstances. Moreover, these parameters can (at least in principle) be calculated from the quantum field theory leading to the PT. Extracting information on fundamental high-energy theories leading to first-order PTs through GW detection is therefore feasible, and the great payoff in case of a signal detection justifies the large effort and growing interest in developing this demanding field.

Examples of Cosmological First-Order Phase Transitions The discovery of the Higgs boson at the LHC, and the determination of its mass, confirms that the EW symmetry breaking corresponds to a cross-over if embedded only in the SM [1,15,104,127,128,151]. The LHC provides so far no solid indication of new physics slightly above the EW energy scale. However, many compelling and well-motivated BSM scenarios predict first-order PTs at the EW scale and beyond. Some of these scenarios are particularly interesting, as they not only give rise to a potentially observable GW background but provide as well dark matter candidates and baryogenesis mechanisms or help to alleviate the hierarchy problem (see, e.g., [2, 74, 79, 90, 117, 150, 173]). One of the simplest possibility is to extend the SM with a scalar singlet (see, e.g., [55, 57, 60, 64, 87, 88, 165]). These models can be difficult to probe experimentally at colliders but yield a strong first-order EWPT with good GW

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detection prospects. The EWPT can also proceed in two steps, being preceded by a first-order transition in the singlet direction [87, 176]. One can also introduce EW charged scalar fields, like in the two-Higgs-doublet-model scenario [96, 129] or triplet SM extensions [56, 173]. These models are generally more suitable to be probed at colliders, in some cases providing both colliders signatures and detectable GW signals at LISA [50]. The SM can also be embedded in supersymmetric theories; in particular, nonminimal supersymmetric scenarios are less constrained and most promising for leading to a first-order EW PT (see, e.g., [16, 73, 116, 144, 145, 175]). If the new physics affecting the EWPT is sufficiently heavy, of the order of the TeV, one can model it via effective operators in the Higgs potential, of dimension-6 and higher [75, 103, 119]. Reference [50] finds that LISA will provide the first actual test of these models, because they cannot account at the same time for signatures at the LHC and a strong first-order EWPT (see also [54, 75, 119]). It is important to point out that current predictions of the GW signal from the above-mentioned EWPT scenarios rely in most cases on perturbative treatments of the PT dynamics. The PT parameters evaluated via the perturbative approach can be significantly inaccurate, thereby affecting the final detection forecasts (see, e.g., [61, 102, 169]). Some models, featuring small effects from higher-dimensional operators and new fields (sufficiently heavy not to play a dynamical role at the PT), can be mapped onto the 3D effective theory of the standard model [102]. Some analyses also suggest that perturbative approaches can give an estimate of the PT parameters which is correct within one order of magnitude [126, 169]. For the time being, this is just one of the several uncertainties affecting the evaluation of the SGWB signal from first-order PTs, as we will see in the following. However, as our understanding advances and the SGWB predictions become more accurate, running lattice simulations to evaluate the action on a model-by-model basis will become a necessary task in order to obtain sufficiently reliable forecasts of the GW signal [61]. There are also high-energy scenarios that do not rely on weakly coupled new physics nor polynomial scalar potentials and can therefore generically predict a very significant supercooling. Five-dimensional warped models [179] and composite Higgs models [41, 172] provide a natural and well-motivated framework for a very strong first-order PT, a compelling target for LISA [139, 161, 166]. Other wellmotivated scenarios are those in which the dark matter is a stable bound state of a confining dark sector, decoupled from the (beyond-)SM visible sector, except gravitationally [3, 4, 13, 19, 20, 39, 55, 62, 63, 89, 122, 157, 188, 195]. Finally, the QCDPT is also predicted to be a cross-over by lattice simulations run at zero baryon and charge chemical potentials (in the absence of lepton and baryon asymmetries) [14, 193]. However, the lepton asymmetry is poorly constrained in the early universe, since it could be hidden in the neutrino sector. Large lepton asymmetry, still compatible with present constraints, might affect the dynamics of the cosmological QCDPT [189, 203]. A viable test of this hypothesis would be the detection of the SGWB thereby emitted, which would fall within the PTA frequency sensitivity range (see, e.g., [11,48,130]). Note that the PTA collaboration NanoGrav has just announced the detection of an extra noise component [17], and

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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alternative interpretations to the background from supermassive black hole binaries have been explored (see e.g. [168], and c.f. discussion in section “Gravitational Wave Background from the Decay of Cosmic String Loops”).

Elements of the Dynamics of the Expanding Bubbles and the Surrounding Fluid A cosmological first-order PT is expected to occur in the early universe when an order parameter, usually a scalar field such as the Higgs, drives the system through a change of phase. For a comprehensive, recent review, see [111] and references therein. In the thermal case, the field potential at finite temperature develops a second minimum, at a nonzero field value, as the temperature decreases. At temperatures lower than the critical temperature, the new minimum becomes the global one, representing the stable ground state, and the PT can start. During the transition, the two phases coexist and are separated by phase boundaries: bubbles of the energetically favorable phase are nucleated via quantum or thermal fluctuations and then expand into the old phase, if their interior pressure is larger than the exterior one. The rate of nucleation of the bubbles per unit volume can be evaluated from the free energy of the system and reads P(t) = A(t)e−Sc (t) , where A is a prefactor with unit of energy to the fourth power and Sc is the critical bubble action (see, e.g., [152, 178]). The onset of the PT is characterized by the nucleation of one bubble per Hubble volume per Hubble time, which defines the nucleation time tn [196]. At this time, the universe is still almost entirely in the metastable phase. The fractional volume in the metastable phase then changes very rapidly around the percolation time t∗ and settles to zero on a time scale given by the transition rate parameter [111]

d ln P(t)

. β= dt t∗

(57)

Relevant for the GW production is H∗ , the Hubble rate at percolation [49, 50]. For PTs without significant supercooling and reheating, this is approximately equivalent to the one at nucleation temperature, H∗ ≈ Hn . Note that in scenarios predicting strongly supercooled transitions, the vacuum energy difference between the phases can come to dominate the energy density of the Universe: one must therefore ensure that percolation indeed occurs and the PT completes [43,77,137,139,161,166,179]. The transition rate parameter (57) also determines the mean bubble center separation, which can be evaluated from the bubble number density at late times and reads [111] R∗,b = (8π )1/3

vw , β

(58)

where vw denotes the wall speed, defined as the speed of the phase boundary after nucleation in the rest frame of the plasma, far from the wall. Although vw is a

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C. Caprini and D. G. Figueroa

fundamental parameter for the PT dynamics and the GW production, this is a rather difficult parameter to evaluate. For a PT in a thermal environment, such as the EWPT, one has to analyze the out of equilibrium dynamics of the coupled system of the scalar field and the surrounding fluid [113,138,143,163,164]. The masses of the fluid particles depend on the value of the Higgs field undergoing the PT and become space-time dependent. In this setting, the field and the fluid energy momentum tensors are not conserved separately but coupled via terms representing the fact that gradients of the scalar field drive the fluid out of equilibrium by changing particle masses [121,148,149]. Such coupling term subsequently enters in the Klein-Gordon equation governing the scalar field evolution and can be represented as an effective friction depending on the field value and the temperature, proportional to the fluid 4-velocity, the scalar field gradient, and the rate of change of the particle masses [86, 121]. The bubble wall speed must be derived from the Klein Gordon equation, and it is evident that it has to be evaluated on a model-by-model basis, as it depends on the specific particle content and interactions of the theory (see however [80]). In general, the bubble wall settles to a steady state with constant velocity vw , due to the friction exerted by the coupled fluid particles [37] (notable exceptions can arise in scenarios with very large supercooling; c.f. [43, 77, 137, 139, 161, 166, 179] and the discussion in [50]). After the wall has reached the terminal velocity, the latent heat released by the PT is converted for the most part into bulk motion of the surrounding fluid, as opposed to kinetic energy of the scalar field (c.f. the discussion after Eq. (61)). In the context of GW production, it is therefore also very relevant to characterize the fluid bulk motion, as well as the bubble dynamics. The fluid flow around the bubbles can be studied by modelling the phase boundary as a combustion front in a hydrodynamical setting [86, 100, 106, 147]. By enforcing energy and momentum conservation across the boundary, one obtains the boundary conditions at the wall, represented by the fluid velocity ahead and behind the wall (in its reference frame). The fluid profile surrounding the bubble is then derived, integrating the continuity equations imposing spherical symmetry, for a given fluid equation of state and sound speed. Three solutions can be obtained, which satisfy appropriate boundary conditions at the bubble center (fluid at rest), at the wall as described above, and at a distance r > t from the bubble center (fluid at rest by causality). These solutions are: (1) Subsonic deflagrations, characterized by a compressional flow outside the bubble in the symmetric phase, terminating with a shock front, while the fluid is at rest inside the bubble, and the wall moves at subsonic speeds; (2) Detonations, characterized by a rarefaction wave behind the bubble wall in the broken phase, while the fluid outside the bubble is at rest, and the wall moves at supersonic speeds; (3) Hybrids, for which the wall speed is supersonic and the fluid is moving both behind and ahead of the wall. When studying GW production from the fluid bulk motion, the characteristic scale of the latter is a parmeter of fundamental importance, as it sets also the scale of the shear stresses generating the GW signal [46, 47]. In the case of shear stresses by pure bubble collision, or by collision of the detonation fronts, the relevant scale is simply R∗ . In the case of deflagrations, though, this has to be corrected as the shell of moving fluid resides outside the broken phase bubble and the wall expands

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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at subsonic speed. The characteristic scale to be used in the GW evaluation process must then be generalized as [108] R∗ =

(8π )1/3 min(vw , cs ) . β

(59)

During a thermal first-order PT in the early universe, the released latent heat gets converted into thermal energy, kinetic/gradient energy of the bubble walls, and kinetic energy of the surrounding fluid due to the field-fluid coupling. The shear stresses associated with the last two components source in turn the GW signal. One can parametrize the shear stresses amplitudes as first proposed in Ref. [107], starting from the space-space parts of the respective energy momentum tensors (the only ones entering the transverse-traceless projection relevant for the GW sourcing), f φ i.e., Tij = ∂i φ∂j φ and Tij = wγ 2 vi vj , with w = e + p the enthalpy density. The root-mean-square bulk fluid velocity can be written as (a bar denoting averaged quantities) 1 U¯ f2 = V w¯

 d 3 xwγ 2 v 2 ,

(60)

V

and the equivalent for the scalar field becomes 1 U¯ φ2 = V w¯

 d 3 x(∂i φ)2 .

(61)

V

These quantities enter directly in the pre-factor of the SGWB signal, as we will see. For a PT occurring in vacuum, or with very large supercooling, U¯ f 0 and all the released energy gets converted into kinetic and gradient energy of the bubble walls, which accelerate up to the speed of light. However, U¯ φ2 is proportional to the total area of the phase boundary, while U¯ f2 is proportional to the volume, so the former quickly becomes subdominant in the presence of a fluid coupled to the scalar field (as demonstrated by simulations [107–109]). The scalar field contribution being negligible for a thermal PT, Eq. (60) can be related to the kinetic energy fraction of a bubble Kb , identified with the spherical fluid configuration set into motion by the phase boundary [131] Kb =

3 3 es vw

 dξ ξ 2 wγ 2 v 2 =

w¯ ¯ 2 U , es f

(62)

with es denoting the average energy density in the symmetric phase and ξ = r/t. The kinetic energy fraction of a single bubble is a reasonable estimate of the kinetic energy fraction of the entire flow, K = (w/ ¯ e) ¯ U¯ f2 [109]. The latter is usually represented by the efficiency factor κ, defined, e.g., in [86]. This work analyses the hydrodynamics of the bubble growth and of the coupled fluid using the bag equation of state: es = as T 4 + , ps = (as /3)T 4 − , eb = ab T 4 , and pb = eb /3, where

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C. Caprini and D. G. Figueroa

subscripts s, b denote the symmetric and broken phases. The efficiency factor κ is defined there as the ratio of the kinetic energy of the flow and the vacuum energy  (c.f. also [131]). Within the bag equation of state, the relative importance of the bag constant to the thermal energy can be used to parameterize the strength of the PT, with a strength parameter α defined as α=

 . as T 4

(63)

One then finds, to summarize κ=

3 3  vw

 dξ ξ 2 wγ 2 v 2 ,

K=

w¯ ¯ 2 κα . U = e¯ f 1+α

(64)

Reference [86] provides several fits relating κ to α and vw , for the different bubble propagation modes (deflagrations, detonations, hybrids). We provide here the one valid for high wall speed, which is widely used in the literature to evaluate the GW signal from first-order PTs:  −1 √ κ α 0.73 + 0.083 α + α for vw ∼ 1 .

(65)

Note that Ref. [109] proposes another, more general, parametrization of the PT strength, which is valid beyond the bag equation of state and makes use of the trace anomaly of the fluid energy momentum tensor θ = (e − 3p)/4: αθ =

4 θs − θb . 3 ws

(66)

Parametrization of the GW Signal The GW production by first-order PTs has very rich phenomenology. The final result one aims at is the SGWB spectral shape and amplitude today, as a function of the system parameters. As previously mentioned, SGWB depends only on a few PT parameters: ΩGW = ΩGW (T∗ , α, β/H∗ , vw ). The latter are related to the thermodynamics of the PT and can be evaluated in the context of a specific theory. However, predicting the SGWB spectral shape is far from trivial, as it depends both on the scalar field dynamics and on the fluid response to the expanding bubbles. To accomplish this task, one must resort to numerical simulations of the coupled field-fluid system, complemented with analytical interpretation of their outcome [67, 106–109]. In particular, it is necessary to precisely evaluate the space-time behavior of the transverse-traceless component of the source tensor anisotropic stress. Only numerical simulations solving at the same time the scalar field KleinGordon equation coupled to the fluid continuity equations allow to gauge the efficiency with which the gradient energy of the scalar field and the kinetic energy of the fluid are converted into shear stresses and ultimately into GW.

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It is believed that the GW sourcing process proceeds through three subsequent stages: • Bubble collision: toward the end of the PT, the bubble walls collide, breaking spherical symmetry and generating a nonzero tensor anisotropic stress. The characteristic duration of this phase is fixed by β/H∗ and is typically short compared to the following stages, dominated by the fluid motion, unless the PT lasts close to one Hubble time. As discussed around Eqs. (60) and (61), one also expects the total energy involved in this phase (and consequently the shear stresses) to be subdominant with respect to the one characterizing the fluid motion, unless the PT is strongly supercooled so that it effectively occurs in vacuum. For this reason, we will not develop further about GW production from this stage, and we refer to the literature for the spectral shape of the GW signal [65, 66, 118, 141, 142]. • Sound waves: as discussed in the previous section, spherical shells of nonzero fluid velocity are produced as the bubbles sweep through the fluid, by the fieldfluid coupling. These are in the form of compression and rarefaction waves and overlap toward the end of the PT leading to shear stresses and thereby sourcing GWs [67, 106–109]. As the kinetic viscosity of the primordial universe’s fluid is typically very small [47], this so-called “acoustic” phase is expected to subsist well after bubble percolation, and its contribution to the GW spectrum is expected to dominate the signal. We elaborate further on this particular source in the following. • Turbulence: if the typical velocity of the bulk fluid motion is high enough, nonlinearities in the fluid evolution become important. These latter might lead to vorticity in the flow, evolving into a fully developed turbulent stage. Tensor anisotropic stresses are then present because of the chaotic distribution of the velocity field. Though the development of the turbulent stage is very plausible for strong enough PTs, numerical simulations started only recently to explore the nonlinear regime where vorticity and turbulence generation are expected to occur [67]. Analytical evaluations of the GW signal exist [47, 78, 101, 140, 170], as well as numerical simulations of the GW production in the turbulent regime [38, 184], which however do not link this latter to the scalar field and PT dynamics. Therefore, more insight is needed to (1) properly characterize the typical turbulent kinetic energy involved and link it to the actual PT evolution; (2) work out the details of the SGWB spectral shape from this stage; and (3) establish the role of the possible presence of magnetic fields and consequent magnetohydrodynamic turbulence. Note that, if confirmed, the contribution to the GW signal from the turbulent phase is expected to be comparable to the one of the acoustic phase for strong PTs [49, 50]. In order to evaluate the GW spectrum, one needs to model the unequal time correlator of the source anisotropic stress (c.f. section “Signal from a Generic

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Stochastic Source of Gravitational Waves”). Here we concentrate on the case of GW production from bulk fluid motion, i.e., sound waves and kinetic turbulence, as they provide the dominant contribution to the signal in the case of a thermal PT (we follow Ref. [50]). Concisely denoting with ΩGW the SGWB spectrum at the time

GW (k, η) ), from Eq. (51), we get of action of the source (i.e., dρ

dlogk rad

 ΩGW ∼ K 2 (kR∗ )3

dη η



dζ ˜ cos(k(η − ζ )) Π(k, η, ζ ) , ζ

(67)

where K is the kinetic energy fraction of the fluid defined in the previous section (c.f. Eq. (64)) and the mean bubble separation R∗ sets the flow length scale, at which the kinetic energy of the fluid motion is maximal. Note that we have normalized the spatial part of the fluid energy momentum tensor Tij (x, t) = wγ 2 vi vj as T˜ij = 3/2 ˜ ij (k, t) = Oij k T˜k (k, t). ¯ Tij /(eKR ∗ ), leading to the normalized shear stress Π Some fairly general considerations dictate the scaling of ΩGW . One expects the source to be active for a given time interval τv , which sets the integration range in (67). Furthermore, the source is expected to decorrelate over an autocorrelation time τc < τv , which represents the time interval that it takes for the anisotropic stress to decorrelate. If the duration of the source is less than one Hubble time τv H∗ < 1, the factor (ηζ )−1 in (67) becomes (H∗ a∗ )2 , where a∗ denotes the scale factor at the bubble collision time. Usually one then finds, for k < τc−1 ΩGW ∼ K 2 (kR∗ )3 (H∗ τv )(H∗ τc ) P˜gw (k)

(τv H∗ < 1 , k < τc−1 )

(68)

where P˜gw (k) parametrizes the dependence on wave-number inherited directly from the source. If the duration of the source is more than one Hubble time, τv H∗ > 1, one expects decorrelation after a time τc < H∗−1 , yielding instead ΩGW ∼ K 2 (kR∗ )3 (H∗ τc ) P˜gw (k)

(τv H∗ > 1 , k < τc−1 )

(69)

It appears that the effective lifetime for GW production is of the order τv H∗−1 for a source persisting for longer than a Hubble time [46, 47, 108]. From the scaling in Eqs. (68) and (69), one can estimate the expected GW energy density from bulk fluid motion. We concentrate first on the acoustic phase. In this case, the decorrelation time is τc ∼ R∗ /cs [105], so that for long-lasting acoustic production (c.f. (69)) ac ΩGW ∝ K 2 (H∗ R∗ /cs ) .

(70)

The acoustic phase is expected to last until shocks are generated by the nonlinearities in the fluid flow [153, 174]. This happens on a time scale τsh ∼ R∗ /U¯  , where U¯  is the RMS longitudinal velocity associated with the sound waves (c.f. Eq. (60)). For high enough kinetic energy in the system, so that U¯   H∗ R∗ , the lifetime

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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of the sound waves is less than the Hubble time, since shocks develop within one Hubble time. The GW energy density parameter becomes (c.f. (68)) ac ∝ K 3/2 (H∗ R∗ )2 /cs . ΩGW

(71)

If the shock formation time is shorter than one Hubble time, one expects turbulence to form [153,174]. Part of the fluid kinetic energy K will be transferred to the vortical component of the velocity field, U¯ ⊥ . The relative importance of acoustic and turbulent GW production depends on how efficient this conversion is. This is a subject of ongoing study, and no reliable estimate can yet be provided on the turbulent kinetic energy fraction K⊥ = (w/ ¯ e) ¯ U¯ ⊥2 . Nevertheless, we derive scaling tu predictions for ΩGW . The turbulent cascade is expected to set in after one eddy turnover time at the flow length scale τtu ∼ R∗ /U¯ ⊥ , followed by decorrelation of the flow on small scales, and overall free decay of the turbulent kinetic energy [72]. Given the low kinetic viscosity of the early universe fluid, dissipation is inefficient, and turbulence is expected to last longer than one Hubble time [47]. Setting the decorrelation time to the eddy turnover time, Eq. (69) yields in this case 3/2

tu ΩGW ∝ K⊥ (H∗ R∗ ).

(72)

However, the decorrelation on the small scales time might effectively reduce the turbulence lifetime as a source of GW. In this case, one would estimate τv ∼ τc ∼ τtu [47], leading to tu ∝ K⊥ (H∗ R∗ )2 . ΩGW

(73)

Only thorough numerical simulations of the GW generation by turbulence would confirm whether the scaling given in (72) or the one given in (73) is the appropriate one and provide a reliable estimate of K⊥ . Given the large uncertainties still remaining concerning GW production from turbulence, in the following, we focus exclusively on the case of sound waves, for which one can rely on the results from numerical simulations [98, 107–109] and their analytical interpretation [105, 106]. The insight provided on the SGWB spectrum by the simulations goes far beyond the general scalings of Eqs. (70) and (71), since it allows to predict the efficiency with which shear stresses are produced (i.e., the pre-factor in the abovementioned equations) and the spectral shape of the signal P˜gw (k) (c.f. Eqs. (68) and (69)). However, for now, only the case of a source lasting τv ∼ 1/H∗ has been simulated, i.e., small RMS fluid velocity, U¯   H∗ R∗ . Equations (70) and (71) guide us in heuristically correcting the SGWB spectrum resulting from simulations when the velocities are large, so that the shock formation time scale is less than a Hubble time, i.e., H∗ τsh = H∗ R∗ /U¯  < 1. Let us start with the case in which sound waves are long-lived in the fluid, i.e., τv ∼ 1/H∗ : a good fit of the SGWB spectrum today from the simulations result is [50]

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C. Caprini and D. G. Figueroa

h

2

(0) ΩGW (f )

= 0.687 F0 K

2 H∗ R∗

cs

Ω˜ gw



f f∗

3 

7 4 + 3(f/f∗ )2

7 2

,

for τv ∼ 1/H∗ ,

(74)

where the numerical factor ensures that the total GW power is K 2 (H∗ R∗ /cs )Ω˜ gw ; F0 = 1.6 × 10−5 is the pre-factor in Equation (40); Ω˜ gw ∼ 10−2 is obtained from the numerical simulations; and the peak frequency today is  f∗ 26

1 H∗ R∗



  zp

T∗ g∗ 16 μHz. 10 100 GeV 100

(75)

The value of the parameter zp is determined from simulations; note that it can deviate from 10 if the wall speed approaches the Chapman-Jouguet speed, invalidating the fit in Eq. (74) [105, 106, 109]. For PT scenarios providing a shock formation time scale shorter than one Hubble time, the scaling should be changed to the one given in (71), leading to [50]  h

2

(0) ΩGW (f )

= 0.687 F0 K

3/2

H∗ R∗ √ cs

for τv ∼ τsh < 1/H∗ .

2

Ω˜ gw



f f∗

3 

7 4 + 3(f/f∗ )2

7 2

, (76)

It is worth noticing that, in the case of the EWPT, most models of interest for LISA predict H∗ τsh < 1; therefore, Eq. (76) is often the relevant result to be used for the SGWB signal estimation [84]. In this case, the overall GW signal might receive additional non-negligible contributions from turbulence. We end this section by showing an example of the SGWB signal from a specific, first-order realization of the EWPT, which can be detected by the LISA interferometer. Reference [50], a publication of the LISA Cosmology Working Group, presents a selection of PT scenarios, both related and not related to the EW symmetry breaking, that can produce SGWBs peaking in the frequency range of LISA. For each scenario, it is possible to evaluate benchmark values for the PT parameters T∗ , α, β/H∗ , vw , which can be realized within the model. This allows to give predictions for realistic GW signals, plausible in well-identified particle physics scenarios. Reference [50] also introduces a web tool, PTplot (https:// www.ptplot.org/ptplot/), which helps in plotting the SGWB power spectrum from a first-order PT against the LISA sensitivity curve, taken from [10]. The SGWB spectrum is calculated from Eqs. (74) and (76), setting the duration of the source to the Hubble time or the shock formation time, whichever is shorter, as the most conservative estimate possible. PTPlot carries a database of benchmark points for all the theories considered in Ref. [50] and also provides contour plots of the signalto-noise ratio in the space of parameters (α, β/H∗ ) or (U¯ f , H∗ R∗ ) together with

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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the benchmark points of a given PT scenario. Figure 3 has been produced with the PTPlot website: to give a concrete example, we have chosen a model extending the SM with a scalar singlet with Z2 symmetry (c.f. [50]).

Cosmic Defect Networks As said before, a phase transition (PT) in the early universe is driven by the dynamics of some scalar field(s), which acquire a nonzero vacuum expectation value within a vacuum manifold M . Cosmic defects may be produced as an aftermath product of a phase transition [112, 200], if the vacuum manifold is topologically nontrivial, i.e., it has a nontrivial homotopy group πn (M ) = I . If that is the case, topological field configurations are then created, in the form of strings (n = 1), monopoles (n = 2), or textures (n = 3) [134]. For larger n, non-topological field configurations (field gradients) arise. Depending on whether the symmetry broken during the PT is global or gauge, the defects generated are of the “global” or ”local” type, respectively. Independently of their type, local or global, topological or nontopological, we will refer to all of them as cosmic defects.

Irreducible Gravitational Wave Emission Due to Scaling Local strings and global defects of any type share in common that they are cosmic defects that exhibit a scaling regime after the completion of the PT [81, 112, 197, 200]. This regime is characterized by a self-similar distribution of the number density of the defects. Namely, the number of defects within a causal volume at every moment of cosmic history is always the same. In the following, we describe first the irreducible GW background emitted by any network of cosmic defects, simply due to the fact of being in scaling. In section “Gravitational Wave Background from the Decay of Cosmic String Loops”, we specialize further our analysis on the more specific production of GWs from loops chopped off from a string network. As a network of cosmic defects evolves maintaining the scaling regime, its energy-momentum tensor adapts itself continuously to each new configuration. This leads to a relativistic evolution of the defect’s network energy-momentum tensor, the transverse-traceless part of which sources GWs. This was first appreciated in Ref. [146], where it was argued (using the quadrupolar approximation) that in a global phase transition, the field dynamics after the transition is completed should emit gravitational radiation with an approximately scale-invariant spectrum, as the fields self-order themselves in order to maintain scaling. Later on, in [91,124], a full treatment of the tensor metric perturbation was used, demonstrating that in the large N limit of a global phase transition [197], the self-ordering process of the resulting global defects generates during radiation domination, in fact, an exactly scaleinvariant GW background. Furthermore, Ref. [99] studied lattice simulations after a second-order phase transition, concluding that the resulting global defects generate a GW background consistent with scale invariance, even though the numerical spectra

Fig. 3 GW signal from the EWPT in the Z2 symmetric scalar singlet model, from PTplot (https://www.ptplot.org/ptplot/) [55, 60, 176, 177]. Left panel: SGWB spectrum (black) and LISA sensitivity (light blue region) for the benchmark values T∗ = 50 GeV , α = 0.16 , β/H∗ = 100 , vw = 1. Right panel: LISA signal-to-noise ratio contours in the parameter space (α, β/H∗ ) (colored lines), together with the possible benchmark points for Z2 symmetric scalar singlet model which have been identified in [50]

1074 C. Caprini and D. G. Figueroa

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

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exhibited some tilt and oscillatory fluctuations. The origin of the scale invariance of this background was finally clarified in [94], where it was shown that any scaling source during the radiation domination period necessarily radiates GWs with an exact scale-invariant energy density GW spectrum. It is then important to remark that the previous result is not related to the particular topology of the cosmic defects, neither to the order of the phase transition that created them nor the global/local nature of the underlying symmetry-breaking process. The scale invariance of the emitted GWs emerges only as a consequence of maintaining the scaling regime during RD. Recently, Ref. [93] has reinforced numerically the above conclusion with new larger lattice simulations, where different techniques for the extraction of the GW signal have been confronted, leading to the same result. Furthermore, Ref. [93] has extended the calculation to all cosmological history, considering the GW emission during both the radiation- and matter-dominated eras. This results in new scale-dependent features in the GW spectrum, which does not remain scaleinvariant anymore within the entire frequency range. The abovementioned GW background represents an irreducible GW emission from any type of cosmic string network (let it be global, Abelian, non-Abelian, or semi-local strings), as well as from any type of global defect networks (let it be domain walls, monopoles, or textures). Let us review now the derivation of the emission of GWs by a defect network in scaling. We will also present afterward the latest numerical results (for global defects) from [93]. The starting point is to determine the unequal-time-correlator (UTC) of the defect network, Π 2 (k, η1 , η2 ) (see Eq. (43)). The spectrum of GWs emitted by the network can be obtained by simply plugging Π 2 (k, η1 , η2 ) into Eq. (51). We can actually predict the form of the GW spectrum using Eqs. (43) and (51), by simply exploiting the concept of scaling. In a scaling regime, the UTC can only depend on co-moving scales k through the dimensionless variables x1 = kη1 and x2 = kη2 . A simple dimensional analysis implies therefore that the UTC must be of the form (here v is the vacuum expectation value of the symmetry breaking field) Π 2 (k, η1 , η2 ) = √

v4 U (kη1 , kη2 ) . η1 η2

(77)

Using this, we can express now the normalized GW energy density spectrum, as [93, 94] 1 dρGW (k, η) ρc d log k    v 4 k2 a1 a2 16 cos(x1 −x2 )U (x1 , x2 ), (78) dx1 dx2 √ = 3 MPl H 2 a(η)4 x1 x2

ΩGW (k, η) ≡

with a1 ≡ a(x1 /k) and a2 ≡ a(x2 /k).

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C. Caprini and D. G. Figueroa

Before the late time dark energy dominance in the universe, but after the electronpositron annihilation, the scale factor can be written as  2 2 √

(0) H η (0) a(η)=aeq [( 2−1)(η/ηeq )+1]2 −1 =a03 Ωmat 0 +a02 Ωrad H0 η , 4

(79)

with aeq the scale factor at the time of matter-radiation equality, ηeq . The change in the number of relativistic degrees of freedom (dof ) during RD can actually be taken into account by correcting the solution deep in RD as aRD (η) =

 a02

(0) Ωrad H0







Rη dη ,

Rη ≡

gs,0 gs,η

4/3 

gth,η gth,0

 ,

(80)

with gs,η and gth,η the entropic and thermal energy density number of relativistic dof at time η. For most of cosmic history, gs,η gth,η , so to a good approximation, we can treat R as a piecewise constant function. In particular, we can take Rη

RQCD 0.39 before the quark-gluon QCD phase transition η < ηQCD , Rη

Re− e+ 0.81 between QCD and electron-position annihilation ηQCD < η < ηe− e+ , and Rη R0 = 1 after electron-position annihilation η > ηe− e+ . The scale factor during RD can then be approximated as aRD (η)



(0) Ωrad R∗ a02 H0 η ,

(81)

with R∗ RQCD , Re− e+ or R0 , depending on the time η. Plugging this into Eq. (78) leads to a GW energy density spectrum, evaluated at sub-horizon scales x ≡ kη  1 during RD, as [93, 94]  ΩGW (x, η) = Ωrad (η) [U ] (x) ≡ FRD

64 3





x

dx1

x

dx2

v MPl

4

R∗ [U ] F (x) , Rη RD

√ x1 x2 cos(x1 − x2 ) U (x1 , x2 ) ,

(82) (83)

where Ωrad (η) = 1 while η  ηeq , and Ωrad (η) < 1 for η > ηeq . At sub-horizon scales, U (x1 , x2 ) is peaked near x1 = x2 ≡ x, decaying along the diagonal as a power law ∝ x −p , with p a positive real number [81]. The convergence of the integration is then guaranteed as long as U (x1 , x2 ) decays fast enough, i.e. p>2. [U ] (x) becomes gradually more and more insensitive to the In such a case, FRD integration upper bound, asymptotically approaching a constant value for x  1. (∞) [U ] [U ] Namely, FRD (x  1) approaches the constant value FRD ≡ FRD (x → ∞). In other words, the emitted GW background has a scale-invariant energy density spectrum at sub-horizon wavelengths. For every type of defect, there is a characteristic function U (x1 , x2 ), leading to (∞) a value FRD , which ultimately determines the amplitude of the GW spectrum. The

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin

1077

spectral amplitude today is  (0) h20 ΩGW (k)

=

(0) h20 Ωrad

v MPl

4

(∞)

R∗ FRD .

(84)

As just demonstrated, the GW background produced by the evolution of any network of defects in scaling during RD is scale-invariant, modulo small step-wise jumps in R∗ due to changes in the number of relativistic species in thermal equilibrium in the (0) medium. Today’s amplitude of the GW spectrum is suppressed by Ωrad ∼ O(10−5 ) and by the 4th power of the VEV as (v/MPl )4 . It also depends on the type of defect through the shape of the UTC, which ultimately determines the amplitude through (∞) FRD . We can now obtain the GW spectrum emitted by a scaling network during MD, analogously as how we did for RD. When the radiation component is completely subdominant at times η  ηeq , the scale factor Eq. (79) can be written (0) 2 2 H0 η . As soon as the defect network during approximately as a(t) 14 a03 Ωmat MD is back to scaling, the UTC can be written again as in Eq. (77). Assuming for simplicity that scaling is maintained for η ≥ ηeq , the spectrum of GW at sub-horizon scales x ≡ kη  1 during MD can then be written as ΩGW (x, η) = Ωrad (η)



v MPl

4

2 keq k2

[U ] FMD (x) ,

(85)

√ x [U ] 2 x 3/2 cos(x − x ) U (x , x ) , (86) (x) ≡ 64 FMD 1 2 (x1 x2 ) 1 2 1 2 xdx 3 ( 2 − 1) xdx eq eq with xeq ≡ kηeq and keq ≡ 1/2ηeq . By construction, we identify keq with the mode with wavelength λeq = 2π/keq , equal to twice the horizon 1/ηeq at the moment of matter-radiation equality. Red-shifting the spectrum today leads to (0) h20 ΩGW (k)

1 ≡ ρc



dρGW d log k



 =

(0) h20 Ωrad

v MPl

4 

keq k

2

(∞)

FMD .

(87)

Finally, let us note that the spectrum can also be obtained at super-horizon scales x = kη < 1. In such a case, the integration in both Eqs. (82) and (85) becomes sensitive to the upper bound x. For super-horizon scales, the signal actually scales as ∝ x 3 , independently of whether we are in RD or MD. It is clear therefore that today’s spectrum does not retain a scale-invariant shape over all frequencies. The overall spectrum rather raises as ∝ f 3 for small frequencies f  f0 , reaches, a maximum at the frequency f0 corresponding to the present horizon scale today, and then decays as ∝ 1/f 2 for larger frequencies f > f0 , eventually settling down at frequencies f  feq > f0 to the characteristic RD scale-invariant plateau discussed before, where feq is the frequency today corresponding to the horizon scale at the matter-radiation equality. The form of the GW spectrum spanning over

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C. Caprini and D. G. Figueroa

a wide range of frequencies, including both sub- and super-horizon, is shown for various examples in Fig. 4. In the particular case of non-topological defects created after the spontaneous breaking of a global O(N) symmetry into O(N − 1), Π 2 (k, η1 , η2 ) can be estimated analytically when N  1 [91,92,124]. The latest update on an analytical expression for the GW background today, corresponding to the modes that entered during RD, is obtained as [92] h

2

(0) ΩGW (f )

≡

1 (0)

ρc



dρGW d log f

 0

650 (0) Ωrad

N



v MPl

4 .

(88)

This result exhibits expected features, for instance, the fact that the signal is reduced the larger the number of field components N we consider. This is expected because spatial gradients among the field components decrease with N , and hence the larger the N , the smaller the source of the GWs, Π∗∗ ∼ ∂∗ φ∂∗ φ (see [91] for a detailed discussion on this aspect). For a general defect network, Π 2 (k, η1 , η2 ) can only be obtained from actual field theory simulations, like those used for CMB studies [156]. Using lattice simulations as an input, [93, 94] have computed numerically the GW amplitude from a system of global O(N) defects. The lattice computations demonstrate actually that the numerical results converge well to the large-N analytical result Eq. (88), as we go increasing N progressively. The case of global strings (N = 2), however, cannot be possibly interpreted as a large-N limit, so the numerical result deviates significantly from the analytical prediction Eq. (88), when evaluated at N = 2. The numerical amplitude of the GW energy density spectrum for global strings is actually a factor ∼O(100) larger than the analytical prediction [93, 94]. One can estimate the maximum amplitude of the GW RD-plateau expected from a defect network with energy scale compatible with CMB constraints [156]. Reference [93] obtains h20 ΩGW < 9.7 · 10−15 , N = 2 , (0) h20 ΩGW < 6.4 · 10−15 , N = 3 . (0)

(89)

We note that these amplitudes are large, for instance, larger than the maximum amplitude expected (as bounded by current CMB constraints [6, 7]) for the quasi(inf) scale invariant GW background from slow-roll inflation [51], h20 ΩGW  10−16 . The amplitudes in Eq. (89) are however not large enough to be observed by any near future direct GW detector. For instance, based on the expected capabilities of LISA to detect a stochastic GW background [52], the GW plateau from global strings cannot be detected by LISA with any significant signal-to-noise ratio, as this would (0) require at least h20 ΩGW  10−13 at frequencies around fp ∼ 10−3 Hz. Some of the signals shown in Fig. 4 are therefore forbidden by CMB observations and were shown only for pictorial reasons, to show the scaling with v and N of the signal. The futuristic Big Bang Observer (BBO) mission [59], as well as the Deci-hertz

(0)

Fig. 4 Red-shifted gravitational wave spectra today h2 ΩGW (f ) as a function of frequency, using R∗ = 1. The spectra are obtained using as input the UTC’s from lattice simulations of global defect networks, where the symmetry breaking field has N ≥ 2 components. In the left panel, we fix the vev to v = 1016 GeV and vary N from 2 to 20. In the right panel, we consider N = 2 (solid lines, corresponding to global strings) and N = 4 (dashed lines, corresponding to textures) and vary the vev from 1017 to 1015 GeV. We note that the range of energy scales in v is chosen only for illustrative purposes (see discussion around (0) Eq. (89), about the physically allowed upper bound on the plateau amplitude of h2 ΩGW ). (Figures taken from Ref. [51])

25 Stochastic Gravitational Wave Backgrounds of Cosmological Origin 1079

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Interferometer Gravitational wave Observatory, DECIGO [132, 133, 187, 190, 204], (0) both with similar expected sensitivities h20 ΩGW  10−17 , could possibly detect the signal coming from global strings.

Gravitational Wave Background from the Decay of Cosmic String Loops Whereas in section “Irreducible Gravitational Wave Emission Due to Scaling” we have discussed the irreducible emission of GWs by any network of cosmic defects in scaling, we are going to focus now in the case when the defects are specifically strings. Cosmic strings are particularly relevant because (i) they represent the best motivated case for defects from a high-energy particle physics point of view, and (ii) they emit a significant extra amount of GWs via a mechanism only proper to them, overtaking in amplitude the previously discussed GW emission due to the scaling regime. Cosmic strings are one-dimensional topological defects, formed whenever the fundamental homotopy group π1 of the corresponding vacuum manifold M in a phase transition (PT) is nontrivial, i.e., π1 (M ) = I . Cosmic strings are present in well-motivated inflationary models, e.g., local strings are always expected to be formed at the end of inflation in supersymmetric GUT models of hybrid inflation [123], for quite a variety of generic assumptions about the underlying GUT physics. Furthermore, cosmic strings can also be fundamental superstrings (as opposed to field theory configurations), arising naturally in scenarios like brane inflation [186]. For either type of cosmic strings, there is a fundamental quantity that characterizes them: their linear energy density μ. In the Nambu-Goto picture, which assumes that strings are infinitely thin, μ is actually the string tension. For convenience, μ is often measured in units of Planck mass squared, through the dimensionless quantity Gμ, where G is the Newton constant. For field theory strings, Gμ actually indicates the scale of the vacuum expectation value v of the ordering field(s) responsible for the phase transition, typically related through Gμ = π(v/MP l )2 (though the prefactor π depends on conventions). What makes special a network of cosmic strings (from the point of view of GW emission) is that contrary to any other type of defect, the string network is composed by two populations: at any time, there are “small” string loops, with a diameter smaller than the causal horizon, and “large” loops, with a diameter larger than the horizon. As it is customary in the literature, we will denote from now on the large loops as “infinite” (or long) strings, labelled with the subscript >, whereas by “loops” we will refer only to those of sub-horizon size, labeled with the subscript tb is then simply obtained from l(t, tb ) = αH −1 (t) − Γ Gμ(t − tb ). In flat space-time, the instantaneous period of oscillation is l/2, so the frequencies of the gravitational radiation are harmonics of the inverse of this period, i.e., fn ≡ 2n/ l. (n) The power radiated in GWs by each harmonic mode is then expressed as E˙ GW =   2 ˙ ˙ (n) Pn Γ Gμ2 , with ∞ n=1 Pn = 1, so that EGW ≡ n EGW = Γ Gμ . It can be shown that the fraction of energy emitted per harmonic can be expressed as a power law, in terms of a spectral index q (characteristic of the type of loop), as Pn ≡ Dq /nq+1 (see, e.g., [18]). If the loop contains cusps, kinks, and kink-kink collisions, then for large n, we expect Pn scaling as n−4/3 , n−5/3 , and n−2 , respectively [30, 198].  From the condition n Pn = 1, we obtain Dq = 1/ζ (q + 1), with ζ (p) the Riemann Zeta function. We can then approximate the discrete   mode emission of GWs into a continuous emission. We take n Pn = Dq 2l n (Δfn ) × 1/(lfn /2)q+1

∞ l 2/ l df P(f ), with P(f ) ≡ Cq /(f l)q+1 . The normalization constant is fixed as ∞ Cq = 2q q, in order to guarantee the normalization condition l 2/ l P(f ) = 1 in the continuum. The “spectral power” emitted by a loop of size l, i.e., the total energy emitted in GWs (per unit of time) within the frequency range [f, f + df ), can then be written as dPGW (f ) = Γ Gμ2 lP(f )df .

(90)

We note that the effect of “radiation back-reaction” in the loop is however ignored in the previous calculation, so in order to somehow include this effect, Ref. [45] proposed to impose a cutoff n∗ , so that Pn = 0 for n > n∗ . This is not a rigorous procedure, but at least it represents a quantification of the gross effect due to back-reaction. Using this approach, the radiation per mode is modified into (br) the fraction Pn ≡ Aq n−(q+1) , with the normalization constant now changed to   ∗ (br) ∗ n−(1+q) , so that the new normalization condition nn=1 Pn = 1 is Aq ≡ nn=1 preserved. Even though the value of n∗ is not known a priori, we can speculate that it should not be larger than n∗ ∼ R/dc , i.e., the ratio between the string curvature R and the diameter of the string core dc . We can consider it in any case as a phenomenological parameter within the range n∗ ∼ 103 –105 (as further contributions from higher harmonics just become negligible). Translating this into the modelling of the GW emission in the continuum, the spectral power emitted by a loop of size l is now written as dPGW (f ) = Γ Gμ2 lP(f )df , where in q (br) (br) this occasion P(f ) ≡ Cq /(lf )q+1 , with Cq ≡ 2q q/(1 − 1/n∗ ) a constant 2n∗ preserving the required normalization condition 2 dxP(x) = 1. Using all of the above, we write formally the spectrum of the stochastic background of GWs, as formed by the superposition of the GW harmonics emitted by the distribution of loops at different moments of time in cosmic evolution. Denoting as dρGW (t) the energy density in GWs emitted by loops of size within

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the length interval [l, l + dl), during the time interval [t, t + dt), and within the frequency range [fe , fe + dfe ), it follows that dρGW (t) ≡ dPGW (f )dt n(l, te ) dl. As the energy density of GWs redshifts as that of relativistic species ∝ 1/a 4 , and the corresponding frequencies today are simply the redshifted values of those at the time of emission f = fe ae /a0 , we can write  (0) dρGW ≡

a(t) a0

4

 dρ(t)=

a(t) a0

3 Gμ2 Γ l P((a0 /a(t))f l) df dt n(l, t) dl . (91)

The final spectrum today of the GWs produced by the loops of a string network, emitted all through cosmic history from an initial time t∗ until today t0 , is then (0)

dρGW ≡ Γ Gμ2 df





t0

dt t∗

a(t) a0

3 

α/H (t)

dl l n(l, t) P((a0 /a(t))f l) .

(92)

0

Something worth mentioning is that the GW signal from cosmic string loops includes not only the stationary and nearly Gaussian background that we have just described, but also strong infrequent bursts that could be detected individually [68, 69]. Such bursts are produced by configurations in the loops known as cusps, which correspond to a highly boosted piece of the loop where the string folds up. The bursts are also produced by kinks, which correspond to shape discontinuities that propagate along the strings at the speed of light. Here we rather focus only on the stationary stochastic background that we have described so far, due to the continuous emission of GWs from the loops decay through all cosmic history (The individual bursts emitted by nearby strings create a “popcorn” discontinuous noise [180], which actually could be potentially detected at an individual event level.). It is therefore necessary that the strong infrequent GW bursts are not included in the computation of the stationary background, in order not to overestimate its amplitude. In practice, for sufficiently large initial loop sizes compared to the horizon, removing the rare burst has a negligible effect on the present-day GW spectrum [31,171,191]. At least this is true whenever the number of cusps and kinks per loop oscillation period is of O(1). The latest simulations [32–34] clearly favor the birth of large initial loop sizes, of the order of α ∼ O(0.1). It is therefore safe not to be concerned any further with the rare emissions from cusps and kinks. To evaluate the amplitude of the stationary background Eq. (92), it is in any case very important to assess whether there are cusps and/or kinks in the loops, as the presence of either of these determines essentially the spectral index q. For instance, q = 1/3 is obtained analytically for loops with cusps [44, 198] (and supported by numerical simulations in [9]), whereas q = 2/3 is obtained for the contribution from kinks. It is true however that as n increases, the contribution from kinks decays faster than the one from cusps, so that generally speaking, the total power at large n is expected to be dominated by the cusp contribution. It is therefore common to use q = 1/3 for loops with cusps, independently of the presence of kinks. Such case is referred to as

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“cuspy loops.” On the contrary, for loops that have kinks but no cusps, referred to “kinky loops,” one rather uses q = 2/3. The above spectral mode emission Pn ∝ 1/nq+1 , or equivalently P(f l) ∝ 1/(f l)q+1 , is clearly based on the asymptotic behavior expected at large n. It is implicitly assumed therefore that the asymptotic behavior of the power is valid down to n = 1, something that in reality does not need to be true. The presented modeling may be therefore inaccurate for the lower harmonics, especially for n = 1. Since Eq. (92) was however derived independently of the functional form of P(f l), it might be interesting to compute the GW emission assuming that only the fundamental mode n = 1 of the GW harmonics is emitted. This could be modeled by simply considering a Dirac Delta distribution as P(f l) = δ (1) (f l − 2). The dependence of the final GW spectrum on the different cosmic string parameters has been studied repeatedly (see, e.g., [31, 76, 171, 185, 191] or the most recent LISA collaboration paper [18]). The constraints on the different parameters characterizing the string network {Gμ, α, q, n∗ , p}, and in particular the constraint in the tension Gμ (under various assumptions on the other parameters), have been re-assessed by successive analysis (see, e.g., [32, 34, 35, 182, 185]). Following, for example, Ref. [185], one can obtain a constraint on the cosmic string tension in the most conservative manner. We will later compare such conservative constraint against other more recent results. The strategy of Ref. [185] is to consider all parameter combinations leading to a GW spectrum saturating the upper limit on stochastic backgrounds from the EPTA data available in that moment. They proceeded by first obtaining, for each set of fixed n∗ , q and p values, the amplitude h2 ΩGW (Gμ, α). They introduce a spectral index nΩ (Gμ, α), assuming that the background can be written as a power law h2 ΩGW (f ) ∝ (f/f1 )nΩ around the relevant frequencies. This then characterizes completely the SGWB as a function of Gμ and α, around a reference frequency f1 = (1 yr)−1 . One can then obtain a constraint curve for each set of n∗ , q and p values in the Gμ − α parameter space, requiring that h2 ΩGW (Gμ, α; nΩ ) = h2 ΩGW,EPTA (f1 , nΩ ), where h2 ΩGW,EPTA (nΩ ; f1 ) represents the EPTA 95% exclusion limit. Setting p = 1, they obtain constraints for various values of q and n∗ which satisfy the EPTA limit. The most conservative constraint on the cosmic string tension can be set by the parameter values where the constraint curve Gμ vs α presents an absolute maximum (see Fig. 13 from [185]). This corresponds to cosmic string networks with n∗ = 1 and α ≈ 10−5 , leading to [185] Gμ < 5.3 × 10−7 (conservative bound) ,

(93)

which represents a 95% upper bound. The constraints from [185] are based on an analytical approximation to the loop number density n(l, t) function. Simulations of Nambu-Goto string networks [33, 34, 181, 182] are however capable to extract numerically, in a more precise manner, the actual form of n(l, t). Unfortunately, possibly due to the different approaches for obtaining the loop number density, the shape of n(l, t) differs significantly between the different groups performing the simulations. For instance,

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Refs. [33, 34] favor large loops at birth only, in particular of a size of α ∼ 0.1. The constraints obtained from the EPTA bound by Ref. [185] just discussed above become then much stronger, of the order of Gμ  10−10 . Actually, a recent analysis presented in Ref. [32] yields in fact a 95% confidence level constraint as Gμ < 1.5 × 10−11 . Reference [182], on the other hand, has also presented recently a new analysis, yielding 95% confidence level constraints as Gμ < 7.2 × 10−11 and Gμ < 1.0 × 10−11 , depending on various network modelling assumptions. In general, each of these works makes different assumptions about cusps, kinks, gravitational back-reaction, and the procedure to obtain the numerical ingredients, so it is very difficult to compare the results among them. It seems clear in any case that all of the most recent simulation analyses yield a strong constraint on the strings tension, of the order of Gμ  10−10 − 10−11 , depending on the details. This is actually a stringent upper bound, and it is worth noting that unless future PTA observations observe soon a stochastic GW background, the limits on Gμ will only improve marginally: a string network with tension much smaller than Gμ ∼ 10−11 only sustains an amplitude for the GW background at frequencies higher than the observational frequency range accessible to PTA. If that was the case, a detection of the GW background from cosmic string networks becomes then only possible on observatories like LISA or futuristic missions like DECIGO and BBO. As a matter of fact, the ability of LISA to detect and characterize a SGWB produced by a network of cosmic strings is just spectacular. LISA will be able to probe cosmic string with tensions down to Gμ  O(10−17 ), what presents an improvement of ∼6 orders of magnitude over current constraints from PTA’s and potentially ∼3 orders of magnitude over estimated future constraints from next generation of PTA experiments. As a final remark, we note that the NANOGrav Collaboration has recently reported strong evidence for a stochastic process [17], which could be interpreted as a SGWB. The collaboration warns however clearly about the fact that no statistically significant evidence is found for quadrupolar spatial correlations in the signal, something considered necessary to claim a SGWB detection consistent with general relativity. The process has neither monopolar nor dipolar correlations, which may arise from, e.g., reference clock or solar system ephemeris systematics, respectively. So at the time of writing (September 30, 2020), there is no clear identification of the detected stochastic process with a SGWB. Still, if we insist in the interpretation of the signal as a SGWB, in the case of a string network with α ∼ 0.1 (as supported by the latest NG simulations), this would correspond to a string tension Gμ ∼ 10−11 − 10−10 [18, 36, 42, 83]. The SGWB produced by cosmic strings with such tension values is beyond the reach of LIGO but could be measured at much higher frequencies by planned and proposed GW detectors. Acknowledgments We are very grateful to our collaborators from the LISA Cosmology Working Group for the work developed within the group, some of which we review in this chapter. DGF (ORCID 0000-0002-4005-8915) is supported by a Ramón y Cajal contract by Spanish Ministry MINECO, with Ref. RYC-2017-23493, and by the “SOM: Sabor y Origen de la Materia” grant from the Spanish Ministry of Science and Innovation, under no. FPA2017-85985-P. This work was supported by CNES, in the framework of the LISA mission.

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Primordial Gravitational Waves

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Gianmassimo Tasinato

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cosmological Inflation as Source of Primordial Gravitational Waves . . . . . . . . . . . . . . . . . . Motivations for Cosmological Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Basic Mechanism of Gravitational Wave Production by Inflation . . . . . . . . . . . . . . . Realizations of Inflation with a Single Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond the Standard Mechanism of Inflationary Gravitational Wave Production . . . . . . Propagation of Inflationary Gravitational Waves Through Cosmic History . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Cosmological inflation represents a successful paradigm for early universe cosmology, which makes well-definite predictions in excellent agreement with current observations. One of these predictions, the existence of a stochastic gravitational wave background from the early universe, has not been tested yet and represents one of the major challenges and opportunities for future research. In this chapter, we provide a brief theoretical overview of the mechanism underlying gravitational wave production during inflation. We explain how the detection of inflationary gravitational waves can provide invaluable information on the physics driving inflation, as well as on aspects of high-energy physics and gravity at the highest energy scales. In particular, we discuss how different realizations of inflation – from single-field inflation to more sophisticated scenarios motivated from high-energy physics – lead to distinctive properties for the gravitational wave background they produce. Given the future prospects

G. Tasinato () Physics Department, Swansea University, Swansea, UK © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_26

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on the observational and experimental side of gravitational wave physics, a detailed theoretical analysis of these properties will be essential for a correct interpretation of data. The theoretical ideas we discuss here aim to provide a general orientation on well-motivated physical effects from the early universe, to be searched for with experiments aimed to detect a stochastic background of gravitational waves. Keywords

Primordial gravitational waves · Cosmological inflation · Early universe cosmology

Introduction Can we explore the very first moments of our universe life using gravitational waves? This chapter aims to theoretically investigate this exciting possibility. There are two main motivations for considering this question. The first reason is that gravitational waves can be produced in the early universe by cosmological inflation, a short phase of accelerated cosmological expansion that is believed to have occurred within the first seconds of our universe history. This scenario was initially proposed in [1–4] for solving basic problems of standard big bang cosmology. After the original proposal, it was soon realized that inflation is accompanied by a mechanism of particle production, an inevitable consequence of quantum mechanics applied to an expanding space-time (see, e.g., [5]). Such mechanism is responsible both for generating primordial scalar anisotropies that source the evolution of cosmic structures at large scales and for producing a stochastic background of primordial gravitational waves [6, 7], whose amplitude and properties depend on the details of inflation. A measurement of the inflationary stochastic gravitational wave background – which has not been detected yet – would provide important information on the physics driving cosmological inflation in the early universe. Moreover, it would be the first experimental demonstration of the quantization of gravitational interactions, given that the production of gravitational waves is based on quantum mechanical concepts applied to cosmology. The second reason for being interested on primordial gravitational waves is that, after production, they travel through large cosmological distances, only experiencing very feeble gravitational interactions with matter density fluctuations. Their characteristics bring us pristine information on the properties of our universe in the first epochs of its expansion history, which cannot be obtained by any other means. Given these motivations, a theoretical characterization of the properties of primordial gravitational waves is essential, both for fully understanding and exploiting their physics content and for devising new ways for detecting their presence in the present-day universe. This chapter aims to be a brief introduction to the subject. We start in section “Cosmological Inflation as Source of Primordial Gravitational Waves” with some motivations for cosmological inflation, to then explain in detail the mechanism

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of cosmological production of gravitational waves in the early universe and their basic features in the simplest scenarios of inflation. We then briefly discuss theoretical issues with embedding inflation in modern theories of high energy and quantum gravity and how attempts to address these issues lead to scenarios with distinctive and testable consequences for primordial gravitational waves. section “Propagation of Inflationary Gravitational Waves Through Cosmic History” then discusses the propagation of gravitational waves within the observed universe and how they can affect cosmological observables. Section “Conclusions” presents our conclusions. The literature on these topics is very large – we limit our references to some of the original literature and to reviews or books that have been useful for preparing this chapter, as, for example, [8–13]. Throughout this chapter, we set c = h¯ = 1, call 2 , and work with the mostly plus convention for the metric tensor. 8π GN = 1/MPl

Cosmological Inflation as Source of Primordial Gravitational Waves Motivations for Cosmological Inflation Cosmological inflation is a period of quasi-exponential expansion of our universe that we believe occurred within the first second of our universe life. We briefly discuss here some of its motivations, referring to cosmology textbooks (see, e.g., [8]) for more details. Our universe expands with time and is described at large cosmological scales by a homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry, whose line element reads 

 dr 2 2 2 2 2 ds = −dt + a (t) + r (dθ + sin θ dφ ) , 1 − Kr 2 2

2

2

(1)

where K = 0, ±1 denotes the spatial curvature, and a(t) is the scale factor whose time dependence parameterizes the cosmological expansion. The geometry of the universe is governed by Einstein gravity and is controlled by an energy momentum tensor that at large scales acquires a homogeneous, perfect fluid form μ T ν = diag [ρ(t), −p(t), −p(t), −p(t)]. Pressure and energy density are related by an equation of state, p = ωρ, and the evolution equations for the scale factor and energy density read H2 =

ρ K − 2, 2 a 3 MPl

ρ˙ + 3H (ρ + p) = 0 ,

(2)

with H (t) = a(t)/a(t) ˙ being the Hubble parameter. This quantity is important in cosmology since its inverse defines the scale of horizon, H −1 ; its value today is parameterized as H0 = h0 100 km/(s Mpc), with h0 a dimensionless number

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approximately 0.7 (we denote with an index 0 quantities evaluated today and fix the present-day scale factor a0 = 1). It is sometimes convenient to work with conformal coordinates, defining the conformal time dτ = dt/a(t) and placing the scale factor a(τ ) as an overall coefficient in the line element (1). Cosmological observations convincingly indicate the validity of the hot big bang model for cosmology. Our universe was very hot and dense in the first period of its expansion. It was dominated by radiation with an equation of state parameter ω = 1/3 and characterized by a scale factor increasing as a(t) ∝ t 1/2 ∝ τ . This radiation epoch was followed by a period of matter domination starting around 10, 000 years after the big bang, characterized by an equation of state ω = 0 and a scale factor scaling as a(t) ∝ t 2/3 ∝ τ 2 . Finally, in relatively recent cosmological times, we are entering a period of dark energy domination, with ω  −1, that will eventually lead to an exponentially growing scale factor if it continues forever. The universe cosmological history outlined here leads to an elegant scenario for present-day cosmology within the framework of the cold dark matter model ( CDM, with denoting a cosmological constant), whose predictions are in very good agreement with current data. Yet, it leaves various questions unanswered about the initial conditions of our universe. For example, why the universe appears so homogeneous and isotropic at large scales? In standard cosmology, the horizon size grows faster than the scale factor, implying that distinct patches in the present sky have never been in causal contact in the past, even if today they have very similar properties. And why our universe appears to be spatially flat with vanishing curvature parameter K? This choice corresponds to an unstable fixed point for the dynamical evolution in standard cosmology and requires extremely fine-tuned initial conditions. These and other questions are solved by cosmological inflation [1–4], a short period of quasi-exponential expansion that occurred before the onset of standard big bang cosmology, during which the Hubble parameter was nearly constant and cosmological expansion exponentially growing: a(t)  eH t = −1/(H τ ). The accelerating expansion rate ensures that during inflation the scale factor grows faster than the horizon size, causally connecting distinct regions of the universe. Moreover, it makes the condition of small spatial curvature an attractor for cosmological evolution, explaining why the universe is spatially flat today. In order to have successful inflation, we need to impose small values for the so-called slow-roll parameters and η:

= −

H˙  1, H2

η =

˙  1, H

(3)

where a dot indicates derivatives along time t. The first condition is imposed to maintain a nearly constant Hubble parameter, while the second condition ensures that keeps small for a sufficiently long time. In fact, in order to solve the aforementioned puzzles of standard big bang cosmology, we need to impose that the number of e-folds of inflation is at least of order 60:

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 Ninf =

tend

H dt ≥ 60

(4)

tin

where tin, end indicates the times of beginning and end of inflation. During inflation, then, our universe expands at least of a factor aend /ain  eNinf  e60 in an extremely short time scale (expressed in seconds, it is around 10−34 s), to then quickly connect with a standard, power-law expansion after inflation ends. Remarkably, cosmological inflation not only solves the initial condition problems mentioned above but also provides a mechanism for amplifying initially small quantum fluctuations, which source the subsequent evolution of large-scale cosmological structures. Such mechanism has been successfully tested by measuring the distribution of anisotropies in the cosmological microwave background and large-scale structures, strongly supporting the paradigm of cosmological inflation (although there exist alternatives, not discussed here, that aim to address the same problems; see, e.g., the review [14]). In addition, inflation also predicts the existence of a stochastic background of primordial gravitational waves, which has not been detected yet, but that would provide us with crucial information on the physics of the early universe and on the behavior of gravity at high energy scales. In what follows, we discuss this mechanism of primordial gravitational wave production and its consequences.

The Basic Mechanism of Gravitational Wave Production by Inflation Cosmological inflation offers a mechanism for converting the drastically changing space-time geometry in the early universe into the production of primordial gravitational waves. This mechanism is based on the application of quantum mechanical concepts to an expanding space-time. More complete discussions can be found in the excellent reviews [5, 9–11], on which this section is based. We consider spin-2, transverse-traceless tensor fluctuations hij (τ, x) around a conformally flat FLRW metric, described by the line element     ds 2 = a 2 (τ ) −dτ 2 + δij + hij (τ, x) dx i dx j .

(5)

The small tensor fluctuations hij behave as spin-2 objects under spatial rotations and are characterized by two independent polarization states that we denote with (+) and (×). These quantities are at the origin of the primordial gravitational waves we are interested in. Expanding the Einstein-Hilbert action up to second order in fluctuations, we find ST =

2 MPl 8



  dτ d 3 x a 2 (τ ) hij2 − (∇hij )2 ,

(6)

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where primes from now on denote derivatives along conformal time τ . We expand the field hij in Fourier space, summing over polarizations 

hij (τ, x) =

λ = +,×

d 3 k (λ) i k·x

(k) h(λ) , k (τ ) e (2π )3 ij

(λ)

(7)

(λ)

and the polarization tensors ij satisfy the transverse-traceless conditions ii 0 =

(λ) k i ij ,

with normalization

Fourier mode

h(λ) k (τ )

(λ )

(λ)

ij ij

= 2δ

λλ

=

. We canonically normalize the

by the redefinition (λ)

ψk (τ ) ≡

MPl a(τ ) (λ) hk (τ ) . 2

(8)

Substituting these expressions in Eq. (6), we find the effective action in Fourier space for the canonically normalized fluctuations ST

1 = 2

 λ = +,×





 a  (λ) 2 (λ) 2 2 . ψk − k − dτ d x ∂τ ψk a 3

(9)

This action describes two independent harmonic oscillators with time-dependent mass, controlled by the scale factor a(τ ). The corresponding equations of motion for each polarization state read ψk

a  2 ψk = 0 , + k − a

(10)

where we understand the polarization index (+, ×), since our arguments apply equally to both cases. We consider a cosmological space-time where a period of quasi-de Sitter expansion associated with inflation is followed by cosmological evolution within the framework of standard big bang cosmology: for definiteness, we focus on a radiation-dominated epoch. Each of these phases is characterized by its own solution for the evolution equation (10), given the different scale factors: a(τ ) = −1/(H τ ) in the de Sitter limit of inflation, and a(τ ) ∝ τ during radiation domination. The resulting system can be quantized using the standard rules of quantum field theory. Promoting ψk to an operator, we can expand it in terms of a basis of creation and annihilation operators. During inflation, we write † ∗ (τ ) aˆ −k , ψˆ kinf (τ ) = fk (τ ) aˆ k + f−k

(11)

where (fk (τ ), fk∗ (τ )) denotes the complete set of mode function solving Eq. (10) and the operators aˆ k and aˆ k† satisfy the standard commutation relations [aˆ k , aˆ k† ] =

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(2π )3 δ(k − k ). The inflationary vacuum state |0 inf is by definition the element in the Hilbert space annihilated by the operators aˆ k : aˆ k |0 inf = 0 . We can proceed analogously for quantizing the field within the post-inflationary, radiation-dominated epoch. In general, we need to expand in a different basis † ∗ ψˆ krad (τ ) = gk (τ ) bˆk + g−k (τ ) bˆ−k ,

(12)

where (gk (τ ), gk∗ (τ )) denotes the corresponding complete set of mode function solving Eq. (10) in radiation domination. The annihilation operators relative to this phase bˆk act on the corresponding vacuum state |0 rad as bˆk |0 rad = 0 . Notice that since the scale factor in the space-time metric depends on time, time translation is not a symmetry of the system, and the two eras of inflation and radiation domination can be characterized by different vacuum states. Since (fk (τ ), fk∗ (τ )) and (gk (τ ), gk∗ (τ )) form sets of complete basis in each of these epochs, we can relate each of them in terms of a Bogoliubov transformation: gk (τ ) = αk fk (τ ) + βk fk∗ (τ ) ,

(13)

where, thanks to the background spatial isotropy, there is no mixing between modes of different momentum k. Using this relation in the previous expressions, we find that creation and annihilation operators in the two cosmological phases are related by aˆ k = αk bˆk + βk∗ bˆk† ,

aˆ k† = αk∗ bˆk† + βk bˆk ,

(14)

and the commutation relations impose the condition |αk |2 − |βk |2 = 1. We now consider the evolution of these quantum fields from inflation to radiation domination. If the transition between these two phases is sudden and occurs on a short time scale with respect to the typical period of oscillation of the fluctuations, an inflationary state ψˆ kinf (τ ) has no time to evolve and remains the same in the postinflationary era. It is interesting to compute the occupation number of the quantum states under consideration, i.e., to evaluate at what extent the quantum state is filled with particles of the quantum mechanical system. If during inflation the quantum state is characterized by an occupation number nk associated with the operator aˆ k† ak , after the transition, its new occupation number Nk can be computed in terms of the operator bˆk† bk , relative to radiation domination. Using the relations (14), one finds that the occupation number in the new phase is 

Nk = 1 + 2|βk |2 nk + |βk |2 .

(15)

Equation (15) implies that a rapidly changing space-time geometry induces spontaneous creation of tensor modes in a cosmological setting. In fact, if |βk | is very large, the occupation number Nk is much amplified after the transition, and the resulting

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modes can be described in terms of a stochastic distribution of classical transversetraceless spin-2 fields – these are gravitational waves produced in the early universe. We can proceed with an explicit computation of Nk in this setup, to physically understand what are the momentum scales involved in the amplification mechanism. The mode functions ψk (τ ) that solve the evolution equation (10) during inflation (in the pure de Sitter limit) and during radiation domination read, respectively

i 1− , kτ

(16)

ψkrad (τ ) = Ak e−ikτ + Bk eikτ .

(17)

e−ikτ ψkinf (τ ) = √ 2k

These solutions have interesting physical implications. During inflation, the Hubble parameter H is nearly constant: tensor fluctuations with large physical momenta k/a  H (corresponding to small wavelengths) are deep in a sub-horizon regime where they are expected to oscillate as free fields in flat space. In fact, a  /a = 2a 2 H 2 , and its contribution can be neglected with respect to k 2 in Eq. (10). However, as time passes, the physical wavelength of each mode grows, and when k/a  H , we are in a super-horizon regime, where fluctuations get frozen: in fact, the corresponding solution of Eq. (10) is characterized by a time-independent profile (λ) for the tensor mode function h(λ) k ∝ ψk /a. The exact solution (16) interpolates between these two regimes; we impose the Bunch-Davies vacuum condition [5] and consider only positive frequency fluctuations. These considerations do not apply to the radiation-dominated regime after inflation, when the universe is not accelerating: the solution in Eq. (17) only consists of oscillatory modes. The free parameters Ak and Bk characterizing the combination in (17) are determined by imposing continuity of the functions and their first derivatives at transition time τtr . The Bogoliubov coefficients αk and βk appearing in Eq. (13) can be readily computed, to give αk = 1 −

i 1 − , kτtr 2 k 2 τtr2

βk =

1 . 2k 2 τtr2

(18)

Substituting this result in Eq. (15), we find

atr4 H 4 atr4 H 4 n Nk = 1 + + , k 2 k4 4 k4

(19)

where atr is the value of the scale factor at the transition time τtr , while H is the Hubble parameter during inflation. Then, we learn that only the occupation number of super-horizon modes with k  atr H , whose physical wavelength is well larger than the Hubble scale during inflation, is amplified by the inflationary process. The physical scale controlling the transition epoch is the Hubble scale, the only physical scale available. Super-horizon modes that vary slowly with respect to this scale do

26 Primordial Gravitational Waves

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not have time to adapt to the transition and experience the amplification mechanism described before. We now have the tools for better characterizing the spectrum of tensor fluctuations during cosmological inflation at super-horizon scales. For simplicity, we model inflation in terms of pure de Sitter geometry. A physically important quantity to compute during inflation is the two-point correlator of tensor fluctuations in Fourier space that controls the amplitude of fluctuations. Using the formulas above, one finds during inflation (we reintroduce here the polarization indexes)  2

 2H (λ) (λ ) 2 2 0|hˆ k (τ ) hˆ k (τ )|0 = (2π )3 δ(k + k ) δ λλ 1 + k . τ 2 k3 MPl 

≡ (2π )3 δ(k + k ) δ λλ Ph(λ)

(20) (21)

Super-horizon modes correspond to momenta satisfying the inequality k/(aH ) = |kτ |  1: in this limit, the two-point correlator is time-independent, and the quantity (λ) Ph is controlled by the value of the Hubble parameter H during inflation. It is convenient to rescale this quantity to form the dimensionless tensor power spectrum at super-horizon scales for each polarization (λ), as well as a total spectrum which is the sum of the two: =

k 3 (λ) 1 4H2 P = 2 2π 2 h (2π )2 MPl



2H2 2 π 2 MPl λ=+,× (22) The quantity Pt is important since it provides us information about the value of the 2 ), we learn that Hubble parameter H during inflation. Since H 2 = ρinf /(3 MPl (λ)

Ph

;

1/4

Pt

=

2 3 π2

Pt =

1 4

(λ)

Ph

=

1/4

ρinf , MPl

(23)

Hence the value of the yet-to-be-measured quantity Pt would inform us on the energy density scale during inflation. If Pt is not excessively small, this relation implies that inflation occurs at very high energy scales and can provide an observational window on the high-energy behavior of gravitational interactions. It is interesting to notice that, so far, for deriving all our arguments, we made no assumptions about the inflationary dynamics, but we only used the properties of Einstein gravity and of the space-time geometry during inflation. In fact, although the computations leading to Eq. (22) have been made under the hypothesis of exact de Sitter space-time, the results remain true also in the quasi-de Sitter limit in which slow-roll parameter = −H˙ /H 2 is positive and small (see, e.g., [9]). This implies that we can readily compute the tilt nt of the tensor power spectrum, associated with the change in its amplitude as tensor fluctuations of momentum k leave the horizon during inflation. Using the relation d ln k = H dt for the scale of horizon crossing, valid at leading order in a slow-roll expansion, we have

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G. Tasinato

nt ≡

d ln H 2 2 dH d ln Pt  = 2 = −2 . d ln k H dt H dt

(24)

where we used the definition in Eq. (3), and we assume that is small and positive during inflation. This fact implies that in the regime of quasi-de Sitter inflationary expansion, the tensor spectrum has a weak scale dependence, related with the time dependence of the Hubble parameter during inflation. Since nt is negative, the tensor power spectrum slowly decreases as k increases, that is, from large to small scales. While in this section we only discussed two-point correlation functions, higherorder point functions involving tensor modes can be computed as well, and their amplitude is generally small in the simplest realization of inflation. See [15] for a detailed discussion on the techniques for computing these quantities.

Realizations of Inflation with a Single Scalar Field So far, the analysis of primordial tensor modes from inflation has been very general, with no need to specify the physical mechanisms for concretely realizing inflation. To make further progress, we need to examine what physics can drive the inflationary expansion. Inflation consists of a brief phase of quasi-exponential amplification of the scale factor, connected with the standard power-law epochs of cosmological evolution. We can describe this process by means of a single scalar field φ, whose time-dependent profile φ(t) controls the different phases of evolution of the scale factor during the inflationary era. Single-field models of inflation are standard textbook material – see, for example, [8, 9] for comprehensive reviews. We briefly discuss this topic in its simplest single-field realization, to discuss its connections with the properties of primordial gravitational waves. The action we consider consists of an Einstein-Hilbert part and a scalar Lagrangian  S =

4

d x

√

 −g

2 MPl R + Lφ 2

 ,

(25)

where Lφ consists of a kinetic and a potential term: 1 Lφ = − ∂μ φ∂ μ φ − V (φ) . 2

(26)

Inflation occurs when the kinetic energy of the scalar field is much suppressed in comparison with its potential energy: in this case, the scalar potential V (φ) provides a nearly constant contribution to the energy budget, driving an exponential expansion of the scale factor. In terms of the small slow-roll parameter , the inflationary dynamics requires [9]

26 Primordial Gravitational Waves

=

1105

2 MPl φ˙ 2 = 2 H2 2 2 MPl



V,φ V

2  1,

(27)

where H is the Hubble parameter during inflation, while φ˙ is the time derivative of the homogeneous inflaton profile φ(t), and V,φ is the derivative of the potential with respect to φ. The scalar field φ is dubbed the inflaton. As the simplest concrete example, we consider chaotic inflation with quadratic potential V (φ) = m2 φ 2 /2: this is one of the most straightforward cases to 2 /φ 2 . theoretically analyze. Using Eq. (27), one finds in this model that = 2 MPl Choosing a large φin initially ensures that the parameter starts very small: then the scalar slowly rolls from large to small values, and inflation ends when  1 where the value of φ is of order of the Planck scale. A simple computation of the number 2 /(4M 2 ) = 1/(2 ): to obtain around 60 of e-folds of inflation gives Ninf = φin Pl e-folds of inflation, at the onset of inflation should be of the order of 10−2 . Single-field inflation make well-definite predictions for the evolution of scalar fluctuations, besides the tensor fluctuations analyzed above. The scalar components of the metric fluctuations can be combined with the fluctuations of the field φ to form a convenient gauge-invariant variable that we denote with ζ (τ, x ). The quadratic action for ζ has a structure very similar to the one of the quadratic action for the tensor fluctuations described in the previous section, and the system can be quantized following an identical procedure. See [15] for a particularly lucid presentation and the lectures [9] for a pedagogical treatment of these topics. Scalar fluctuations are also characterized by a nearly scale-invariant power spectrum Pζ at super-horizon scales, whose expression reads Pζ =

H2 H2 H2 , = 2

(2π )2 φ˙ 2 8π 2 MPl

(28)

where in the last equality, we used the relation (27). Contrary to tensor fluctuations, the amplitude of scalar fluctuations ζ depends on the inflationary mechanism, namely, the dynamics of the inflaton field φ. An important parameter to consider is the tensor-to-scalar ratio between the amplitude of tensor and scalar power spectra. Using Eqs. (22) and (28), it reads r =

Pt = 16 . Pζ

(29)

Interestingly, the tensor-to-scalar ratio can be related with the tensor spectral tilt of Eq. (24) to form a consistency relation r = −8 nt ,

(30)

that must be true in any single-field model of inflation described by the action (25).

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The quantity r has not been measured yet, and major observational efforts are planned for the upcoming future. In fact, the inflationary mechanism of particle production that amplifies scalar and tensor fluctuations can leave imprints on cosmological observables: the cosmic microwave background (CMB) radiation, the distribution of large-scale cosmological structures, and (for tensor modes) the stochastic gravitational wave background that can be probed by gravitational wave experiments. In particular, both types of scalar and tensor fluctuations affect the properties of CMB at the largest cosmological scales. In fact, inflationary predictions for the scalar sector have been accurately confirmed by studying the distribution of CMB temperature fluctuations, determining both the amplitude (Pζ  2 × 10−9 at pivot scale k = 2 × 10−3 Mpc−1 ) and tilt of the scalar spectral index: in fact, these measurements strongly constrain several models of inflation [16]. So far data [17] only provide the upper bound r < 0.06 on the tensor-to-scalar ratio r at the momentum scale k = 5 × 10−2 Mpc−1 . Future CMB polarization experiments are expected to be able to probe the tensor-toscalar ratio r with statistical errors of the order Δr  10−3 , probing the range predicted by the Starobinsky model of inflation [2] that is one among the most favored by present-day data [16]. For more details, see the brief discussion at the end of section “Propagation of Inflationary Gravitational Waves Through Cosmic History,” as well as the comprehensive contribution by Benjamin Saliwanchik to this encyclopedia. Notice that choosing r in the range 10−3 − −10−2 , and using Pζ = 2 × 10−9 , a straightforward computation leads to the value H  10−5 MPl . The value of the Hubble parameter during inflation is quite large, although well below the Planck scale. The dynamics of single-field inflation has a further connection with high-energy physics, besides what we have discussed so far. It is based on the simple yet deep observation by Lyth [18] that there is a relation between the quantity r and the field excursion of the inflaton field φ during inflation. Using Eq. (27), Eq. (29) can be expressed as r = 16 = 8

1

dφ MPl H dt

2



dφ = 8 MPl dN

2 .

(31)

Hence, integrating over the number of e-folds from a pivot value (at large CMB scales) to the end of inflation, the total field excursion Δφ from beginning to end of inflation can be expressed as Δφ = MPl



 dN

r 1/2 N r inf  6.7 . 8 0.1 60

(32)

As mentioned above, inflation requires at least around 60 e-folds of expansion for solving the basic problems of standard big bang cosmology. Hence, a tensor-toscalar ratio in the range 10−3 ≤ r ≤ 10−2 , still compatible with present-day limits and in the range that can be probed by future CMB experiments, implies that the inflaton field excursions must be of the order of the Planck scale. This fact can

26 Primordial Gravitational Waves

1107

indicate that the inflaton dynamics might be affected by high-energy effects, possibly related with quantum gravity, that become important when the value of φ becomes sufficiently large. Recent approaches to inflation from the perspectives of quantum field theory and string theory use this point as a motivation for considering more sophisticated inflationary scenarios, with interesting consequences for gravitational waves, as we discuss in the next section.

Beyond the Standard Mechanism of Inflationary Gravitational Wave Production The paradigm of cosmological inflation finds many realizations in models based on ideas from particle physics, field theory, and quantum gravity (see, e.g., [19, 20] for reviews). In fact, over the past decades, early universe cosmology has been a fertile ground for interactions between different fields of physics and an opportunity to test new theoretical ideas with cosmological observations. The goal of detecting and characterizing primordial gravitational waves from inflation represents an important chance to further test the mechanism of inflation and, more in general, aspects of physics and cosmology at very high energy scales. While single-field realizations of inflation are in excellent agreement with current data [16], there are compelling theoretical motivations for considering more general setup. In order to ensure that slow-roll parameters remain small during inflation, the inflationary potential must be very flat as a function of φ: Eq. (27) requires MPl ∂φ ln V  1. This might be an issue since corrections associated with quantum loops might contribute to the inflaton potential and spoil its flatness. This is called the η-problem of inflation: see, for example, [20] for a review. Such an issue is particularly worrisome in scenarios with large field excursions (see the discussion around Eq. (32)). In that case, Planck-suppressed quantum corrections to the potential are not necessarily small, being generically weighted by coefficients scaling as powers of (φ/MPl ), hence making the setup sensitive to high-energy effects. In fact, modern advances in quantum gravity related with the swampland program and with weak gravity conjectures in string theory make these considerations more precise, also suggesting that in this case additional fields can become light and influence the inflaton dynamics (see, e.g., [21] for a review). Since many high-energy scenarios realizing inflation contain additional fields besides the inflaton, it is worth exploring whether they can help in controlling the stability of the inflationary process. We discuss representative examples, with the aim of demonstrating the sophistication of current approaches to inflationary model building and their relations with high-energy physics and for stressing their distinctive consequences for the physics of primordial gravitational waves. The production of primordial tensor modes can be different with respect to the basic mechanism discussed in section “The Basic Mechanism of Gravitational Wave Production by Inflation” and can amplify the corresponding spectrum in frequency ranges that cannot be tested by the CMB data mentioned in the previous section. In fact, while CMB can test frequency ranges

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G. Tasinato

in the interval 10−19 Hz ≤ f ≤ 10−17 Hz, other gravitational wave experiments can test frequency intervals: 10−9 Hz ≤ f ≤ 10−6 Hz for pulsar timing arrays; 10−5 Hz ≤ f ≤ 10−1 Hz for space-based interferometers like LISA; and 1 Hz ≤ f ≤ 103 Hz for ground-based interferometers. These experiments can provide complementary tests of gravitational waves from inflation. The first scenario we examine for addressing these questions is axion inflation. As we mentioned above, perturbative loop effects normally lead to corrections to the inflationary scalar potential, whose details depend on the high-energy physics controlling the field dynamics. An elegant way out is to impose a shift symmetry φ → φ + c, with c a constant, protecting the flatness of the inflationary potential from perturbative corrections. This possibility is naturally realized in natural inflation models [22] where the inflaton is a pseudo-Nambu-Goldstone boson – as, for example, one of the axion fields that appear among the constituents of several particle physics and string theory models. The shift symmetry is usually broken by non-perturbative effects to a discrete subgroup φ → φ + 2π n f with n an integer and f the so-called axion decay constant:   φ 4 ¯ V (φ) = 1 − cos , f

(33)

¯ is an energy scale (see, e.g., [20]). In order to ensure a sufficiently flat where potential, the axion decay constant should be larger than the Planck mass, f > MPl . Single-field natural inflation is not favored by the CMB constraints [23] and in theoretical tension with general requirements from quantum gravity (see, e.g., [21]). But the potential (33) is not the only ingredient characterizing axion models. In general, axions are coupled with gauge fields through interactions that preserve the shift symmetry Lint = −

g φ Fμν F˜ μν 4f

(34)

where g is the gauge coupling, Fμν = ∂μ Aν − ∂ν Aμ is the field strength associated with a gauge field Aμ , and F˜ μν = μνρσ Fρσ is the dual field strength, with

μνρσ a tensor totally antisymmetric in its indexes and 0123 = 1. Lagrangian (34) should be included to the inflaton Lagrangian (26), together with a standard kinetic term for the gauge fields. Couplings as (34) can slow down the inflaton velocity even for steep inflaton potentials, since during inflation, the inflaton potential energy can be converted into gauge vector energy instead of inflaton kinetic energy. This possibility was first proposed in [24] for Abelian gauge fields, by studying dissipative effects associated with the production of gauge quanta. Couplings between the inflaton and non-Abelian fields were first explored in [25, 26], pointing out that in this case, gauge fields can acquire a vacuum expectation value and the energy conversion from inflaton to gauge fields can occur in a classical way. In general, interactions as (34) have important implications for the production of primordial gravitational waves: we discuss here the Abelian case for simplicity.

26 Primordial Gravitational Waves

1109

Selecting a Coulomb gauge, the gauge vector can be decomposed in Fourier space in terms of helicity operators as

A(τ, x) =

 λ=±

 d 3 k  (λ) (λ) i k×x A (τ, k) e (k) e + h.c. (2π )3

(35)

where the helicity vectors are chosen to satisfy the relations k × e(λ) = 0, k × e(±) = ∓i k e(±) , and we consider a standard kinetic term for vector fields. During inflation, the evolution equation for the gauge modes reads   ∂ 2 (±) ξ 2 A(±) (τ, k) = 0 , A (τ, k) + k ± 2 k τ ∂ τ2

g φ˙ . 2f H (36) Choosing for definiteness a positive sign for ξ , the positive helicity of the vector field is characterized by a tachyonic instability at large scales, for k/(aH ) ≤ 2 ξ . In the limit of de Sitter space a(τ ) = −1/(H τ ) and assuming a constant inflaton velocity, an exact solution for Eq. (36) is found in [27] in terms of Whittaker functions. The resulting gauge spectrum is chiral: when ξ ≥ O(1), the positive helicity mode gets amplified by a factor of order exp (π ξ ), while the negative helicity mode is not amplified. The exponentially amplified vector mode generates an effective anisotropic stress at second order in perturbation theory, which sources chiral gravitational waves during inflation, as well as scalar perturbations. The relevant evolution equation for tensor modes in position space reads hij + 2

with

a  2 hij − ∇ 2 hij = T EM , 2 ij a MPl

ξ =

(37)

where TijEM denotes the transverse-traceless projection of the electromagnetic stress tensor; see [27]. The resulting primordial tensor spectrum can be computed with Green function methods and is a sum of uncorrelated intrinsic and induced contributions [27]. To make manifest the consequences for the tensor spectrum of the asymmetric amplification of one vector mode with respect to the other, it is convenient to express tensor spectra in a left- and right-handed (L, R) basis, (L,R) (+) (×) √ connected with the (+, ×) basis by the linear relation eij = (eij ∓ eij )/ 2 between polarization tensors. Using this basis, one finds

PhL, R

⎧ ⎪ ⎪ 1 + 8.6 × 10−7 ⎪ ⎪ ⎨

H2 = × 2 ⎪ π MPl ⎪ ⎪ ⎪ ⎩ 1 + 1.8 × 10−9

H 2 e4π ξ 2 ξ6 MPl H 2 e4π ξ 2 ξ6 MPl

(L) polarization ,

(R) polarization .

(38) Hence, the tachyonic instability affecting the positive helicity vector mode enhances one of the tensor polarizations with respect to the other, leading to a chiral spectrum.

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G. Tasinato

In the Abelian vector case discussed above, this amplification occurs at second order in perturbations: in the non-Abelian SU (2) case, the decomposition of vector perturbations has a tensor component that allows one to transmit at linear order the tachyonic vector instability into the primordial tensor mode sector [28]. The inflationary mechanism we discussed has distinct observational consequences for the properties of primordial tensor modes. Although, for Abelian gauge field scenarios, large values of ξ source non-Gaussian scalar fluctuations that are severely constrained by CMB measurements [29, 30], the parameter ξ is generally time-dependent, being controlled by the Hubble parameter and the inflaton velocity during inflation (see Eq. (36)). This implies that scenarios can be designed where the value of this quantity keeps small at times associated with modes leaving the horizon early during the inflationary era, which control the large scales affecting CMB measurements. Then ξ increases with time as inflation proceeds, amplifying the amplitude of the tensor spectrum toward smaller scales (see Eq. (38)). As ξ increases even further, the gauge field production backreacts on the inflaton dynamics, reducing the inflaton velocity, possibly avoiding trans-Planckian field excursions. This scenario has the opportunity of enhancing the size of primordial gravitational wave modes that can be detected in the frequency range of gravitational wave interferometers: see, e.g., [31] for a detailed study of capabilities of LISA to test this possibility. Moreover, both scenarios based on Abelian and non-Abelian vector fields are characterized by a chiral primordial tensor spectrum (see Eq. (38)) and have sizeable tensor non-Gaussianity, since nonlinear tensor interactions are enhanced by the vector instabilities. These and other features can be in principle probed by CMB observations and/or interferometer detections (see, e.g., [32, 33] for recent detailed studies), making models of inflation based on gauge fields very interesting candidates for inflationary scenarios that address some of the theoretical issues of single-field inflation and that can be specifically tested by studying the properties of primordial gravitational waves. Notice that the gauge field inflationary modes discussed here are not the only examples where the anisotropic stress plays an important role in enhancing gravitational waves from inflation. Other possibilities are scenarios generating primordial black holes, where enhancements in the scalar sector amplify the spectrum of primordial gravitational waves once they re-enter the horizon after inflation ends. See the contribution by Juan Garcia-Bellido in this volume for a discussion of these scenarios. Slowing down inflation through couplings with extra fields is one among various possibilities for achieving a prolonged period of inflation without relying on a flat inflationary potential. For example, models where inflation is driven by scalars with large derivative couplings can offer technically natural scenarios where the inflationary dynamics is protected by shift symmetries. As non-Abelian gauge field inflation relies on vector fields acquiring space-dependent vacuum expectation values, scenarios of solid [34] and supersolid inflation [35] do so with derivative couplings involving scalar fields only, which break space diffeomorphisms during inflation. These models are characterized by distinctive properties for the primordial gravitational wave spectrum that can be amplified at the large frequency scales that

26 Primordial Gravitational Waves

1111

can be probed by interferometers. In fact, once space diffeomorphisms are broken, a mass term for tensor modes is allowed during inflation. Massive tensor modes are characterized by a positive tensor spectral tilt, not a negative one as in single-field inflation (recall the discussion around Eq. (24)). Neglecting inflationary slow-roll corrections, and denoting with mh the tensor mode mass, one finds nT =

2 m2h > 0. 3H2

(39)

As a consequence, the tensor power spectrum Pt grows toward large momenta (equivalently small scales), contrary to the single-field results discussed earlier, rendering primordial tensor modes potentially detectable with interferometers: see, e.g., [31] for an analysis of this possibility. Large tensor non-Gaussianities, that can be realized in this class of scenarios, can in principle be probed by a quadrupolar modulation of the tensor spectrum [36]. After inflation terminates, the spatial diffeomorphism symmetry is restored, and the mass term for tensor modes vanishes. The scenarios we discussed in this section aim to demonstrate that mechanisms based on cosmological inflation are able to produce primordial tensor modes with a rich variety of distinctive features that can be in principle tested with a variety of cosmological observables or gravitational wave measurements. Primordial tensor modes can be chiral and non-Gaussian, and their amplitudes enhanced at scales directly detectable with gravitational wave experiments. In the next section, we discuss the dynamics of primordial gravitational waves once they propagate inside our cosmological horizon after inflation ends, within the standard framework of big bang cosmology.

Propagation of Inflationary Gravitational Waves Through Cosmic History Cosmological inflation provides a mechanism for amplifying tensor fluctuations at super-horizon scales. As we have seen, once they cross the horizon during inflation, tensor modes rapidly become time-independent and acquire a nearly scaleinvariant almost Gaussian distribution associated with the inflationary mechanism of gravitational wave production – see the discussion in section “The Basic Mechanism of Gravitational Wave Production by Inflation.” When inflation ends, the energy density stored in the inflaton field gets transmitted into Standard Model particles, and the standard framework of big bang cosmology starts [8]. An initial period of radiation domination in the early universe is followed by matter domination, until recent cosmological epochs when dark energy becomes the dominant content of the energy density budget. During the epochs of radiation and matter domination, the cosmological horizon size increases, and the primordial tensor fluctuations – whose wavelengths were exponentially stretched during inflation – slowly re-enter the horizon, since as time passes their physical wavelength becomes smaller than the inverse of the Hubble parameter. At this stage, the gradients of tensor fluctuations

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are no more negligible with respect to the rate of expansion, and we expect these modes to oscillate, with initial conditions dictated by their value at horizon crossing: these modes become primordial gravitational waves that can then travel all the way from the early universe to today. Moreover, once they re-enter the horizon, inflationary gravitational waves propagate almost undisturbed and interact only feebly with the universe content, implying that their properties are mostly sensitive to the background geometry of the universe. In good approximation, an analysis of their linear evolution is sufficient for characterizing their present-day features. We consider the linearized evolution for the transverse-traceless tensor components hij (τ, x) of the metric fluctuations around a conformally flat FLRW universe in matter and radiation domination, described by the line element of Eq. (1), which we rewrite here as     (40) ds 2 = a 2 (τ ) −dτ 2 + δij + hij (τ, x) dx i dx j . We identify the fluctuation hij (τ, x) with the primordial gravitational wave originating from inflation. The evolution equation reads hij + 2H hij − ∇hij = 0 ,

(41)

with prime indicating derivative along conformal time and H = aH = a  /a the conformal Hubble parameter. We make the simplifying assumption of neglecting possible sources associated with anisotropic stress (see the reviews [12, 13] for a discussion of its effects and for references to the relevant literature). Expanding the tensor modes in Fourier space as in Eq. (7), and understanding the polarization index (λ), we rewrite Eq. (41) in terms of hk (τ ) as hk + 2H hk + k 2 hk = 0 .

(42)

Depending on the value of the comoving momentum k, a Fourier mode of tensor fluctuations re-enters the horizon at a time given by the relation H(τ ) = k, which can be during radiation or matter domination eras. Given that dark energy started to play a relevant role only at a relatively recent cosmological epoch, for simplicity, we do not consider its effects in our discussion. If a tensor mode re-enters the horizon and propagates during radiation domination, the conformal scale factor scales as a(τ ) ∝ τ . As we already mentioned around Eq. (17), the two independent solutions of Eq. (42) read h(τ ) = (sin kτ )/(kτ ) and hk (τ ) = (cos kτ )/(kτ ). Denoting with τin the early time when radiation domination starts, and assuming k τin → 0, we impose the initial conditions hk (τin ) = hinf k

,

hk (τin ) = 0 ,

(43)

in order to match with the inflationary, constant solution hinf (λ) (k) at super-horizon scales. We then find that during radiation domination, the tensor mode function reads

26 Primordial Gravitational Waves

1113

hRD k (τ )

=

sin (kτ ) kτ

hinf k .

(44)

An analogous procedure applies for tensor modes re-entering the horizon well within the era of matter domination, giving the solution  hMD k (τ ) =

3 (kτ )2



 sin (kτ ) − cos(kτ ) hinf k . kτ

(45)

The time of equality that in first approximation separates the radiation and the matter domination epochs defines a characteristic scale keq given by the relation keq = HRD (τeq ). Converting keq in present-day gravitational wave frequency using the relation f = k/(2π ) (valid in our conventions where the present-day scale factor is a0 = 1), one finds [12] −17

feq  1.7 × 10



h0 0.7

2 Hz ,

(46)

for the present-day frequency of primordial tensor modes that re-entered the horizon at radiation-matter equality. Modes with larger frequencies re-entered the horizon during radiation domination, while modes with smaller frequencies re-enter during the epoch of matter domination. In the previous discussion, we made various simplifying hypothesis in order to derive simple analytical expressions for the postinflationary behavior of tensor modes. For example, we assumed the scale factor to scale as a(τ ) = const × τ during the entire epoch of radiation domination. This description is slightly oversimplified, since as temperature decreases, some of the species of the Standard Model particle content leave thermal equilibrium or become non-relativistic, changing the effective number of degrees of freedom and the dependence on time of the scale factor. See [37] or the comprehensive reviews [12, 13] for a detailed discussion of these effects. In any case, in each epoch, we can generally parameterize the sub-horizon evolution of primordial tensor modes in terms of a transfer function T (τ, k) starting from the initial inflationary mode solutions as hk (τ ) = T (τ, k) hinf k ,

(47)

where the quantity T (τ, k) can be evaluated numerically; for modes well within the horizon, the transfer function is typically described in terms of oscillatory functions and powers of the combination kτ . After discussing how a single inflationary tensor mode evolves after re-entering the horizon during inflation, it is natural to investigate the global properties of the post-inflationary spectrum of primordial gravitational waves. Once they reenter the horizon after inflation ends, inflationary tensor modes form a stochastic gravitational wave background. As we have seen, the mechanism of gravitational wave production during inflation, based on quantum mechanics, greatly amplifies

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their occupation number, so that the distribution of inflationary gravitational wave can be characterized in terms of classical, stochastic random variables. The resulting gravitational wave spectrum is stationary, statistically homogeneous, and isotropic, reflecting the spatial isotropy and homogeneity of the background space-time during inflation. These are indeed the defining characteristic of a stochastic gravitational wave background. In the simplest inflationary scenarios, the tensor spectrum is also nearly scale-invariant, almost Gaussian, and not polarized – although, as we have discussed in section “Beyond the Standard Mechanism of Inflationary Gravitational Wave Production,” there are exceptions to these latter predictions in well-motivated scenarios of inflation. A convenient way for characterizing the properties of the stochastic gravitational wave background from inflation is in terms of its energy density ρGW and of the dimensionless energy density parameter ΩGW (τ, f ). This latter quantity is defined in terms of the logarithmic derivative of the energy density along frequency, evaluated at conformal time τ after inflation ends: ΩGW (τ, f ) =

1 d ln ρGW , ρc d ln f

(48)

2 is the critical value of the energy density which sustains where ρc = 3H02 /MPl the present-day Hubble parameter in a spatially flat FLRW universe. The energy momentum tensor associated with gravitational waves at time τ is [38]

1 ρGW (τ ) = h h = 32π GN a 2 ij ij

 0

∞

dk dρGW , k d ln k

(49)

where . . . denotes averaging over several wavelengths. Making use of expression (47), we find that the density parameter ΩGW at present time τ0 is related with the primordial inflationary tensor spectrum Pt as (we substitute k = 2πf ) ΩGW (τ0 , f ) =

1 |T  (τ0 , f )|2 Pt (f ) . 12 H02

(50)

The frequency dependence of ΩGW is related to the frequency dependence of the transfer function, in particular on whether the associated primordial tensor modes re-enter the horizon during the radiation or the matter domination eras. The characteristics, limits, and future detection prospects for stochastic gravitational wave backgrounds with gravitational wave experiments and the cosmological microwave background (CMB) are discussed at length in other contributions to this encyclopedia; see in particular  Chaps. 6 “CMB Experiments and Gravitational Waves” and  25 “Stochastic Gravitational Wave Backgrounds of Cosmological Origin”. In the last part of this section, we make some general considerations for what respects a stochastic cosmological background from inflation. The small frequency limit f  feq is associated with large-scale tensor modes that re-enter the horizon during matter domination. In this case, the best tests and constraints on primordial tensor modes come from the study of CMB, which probes

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frequency ranges approximately in the interval 10−19 Hz ≤ f ≤ 10−17 Hz. Primordial gravitational waves affect CMB physics in various ways, and the seminal works [39–41] on this topic have been followed by very intense theoretical and observational activity. Gravitational waves contribute to the CMB temperature fluctuations through the Sachs-Wolfe effect, since the stochastic background of primordial tensor modes deforms the photon geodesics, and induce a quadrupolar pattern on temperature anisotropies. Moreover, gravitational waves induce a distinctive B-mode polarization pattern in the CMB, sourced by Thompson scattering of electrons with the anisotropic distribution of photons at last scattering. As mentioned after Eq. (30), the lack of detection of these effects so far sets the upper bound r ≤ 0.06 at CMB scales, with the prospects of setting in the future an upper bound of r ≤ 10−3 in absence of detection. We refer to the contribution of Benjamin Saliwanchik in this encyclopedia and to the technical review [42] for a detailed theoretical analysis of how to measure primordial gravitational waves through the CMB. Interestingly, the application of CMB physics to gravitational waves can test not only the standard single-field inflationary models analyzed in section “Realizations of Inflation with a Single Scalar Field” but also more sophisticated scenarios as the ones discussed in section “Beyond the Standard Mechanism of Inflationary Gravitational Wave Production.” In fact, forecasts exist for testing the value of nT [43], hence probing the validity (or the violation) of the single-field consistency relation of Eq. (30). Also a chiral spectrum and tensor non-Gaussianity – motivated, for example, by the axion inflation model discussed in section “Beyond the Standard Mechanism of Inflationary Gravitational Wave Production” – are observables that can in principle be probed in case of a detection of primordial tensor modes: see, respectively, [44] and [33]. This makes CMB a valuable observable tool for distinguishing different scenarios for inflation through the properties of the gravitational wave spectrum. For the case of larger frequencies, we have the possibility of testing stochastic gravitational wave backgrounds with direct gravitational wave experiments. As mentioned in section “Beyond the Standard Mechanism of Inflationary Gravitational Wave Production,” different experiments probe different frequency ranges: 10−9 Hz ≤ f ≤ 10−6 Hz for pulsar timing arrays; 10−5 Hz ≤ f ≤ 10−1 Hz for space-based interferometers like LISA; and 1 Hz ≤ f ≤ 103 Hz for groundbased interferometers. Tensor modes with frequency f > feq , with feq given in Eq. (46), re-enter the horizon during radiation domination. Modes with f ≥ 10−4 Hz re-enter the horizon at a stage where the universe temperature is sufficiently high that all Standard Model particles are in thermal equilibrium, and one can obtain an expression for the transfer function T that accurately takes into account their effects without the oversimplifications mentioned after Eq. (46). The book [12] presents a simple expression for the value of ΩGW (f ) today, as a function of frequency and of various inflationary quantities:

h20 ΩGW

−17

= 1.36 × 10



H 10−5 MPl

2

f f

nt

for f ≥ 10−4 Hz . (51)

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In the previous expression, H is the value of the Hubble parameter during inflation, f is a pivot frequency f = 7.7×10−17 Hz (corresponding to primordial momenta of k = 0.05 Mpc−1 so as to compare with CMB scales), and nt is the primordial tilt of the tensor spectrum. When focusing in single-field scenarios of inflation, where nt is small and negative (see Eq. (24)), Eq. (51) suggests that the sensitivity required to gravitational wave experiments for measuring ΩGW from inflation is h20 ΩGW  10−18 − 10−17 . This is orders of magnitudes below the sensitivity that can reach with current experiments or with detectors approved for construction. For example, the spaceborn interferometer LISA [45] will be able to impose upper bounds at the level of h20 ΩGW ≤ 10−13 , few orders of magnitude below ground-based interferometers. On the other hand, future planned projects as DECIGO [46] aim to reach sensitivities sufficient to probe the amplitude of the gravitational wave spectrum predicted by single-field inflation. The goal of measuring a primordial ΩGW from inflation might be easier to achieve when considering scenarios beyond single-field inflation, as discussed in section “Beyond the Standard Mechanism of Inflationary Gravitational Wave Production.” Then the primordial tensor spectrum can grow toward large frequencies, amplifying its amplitude in ranges tested by interferometers: see, e.g., [31] for a study of the opportunities offered by LISA. In this case, there is the possibility to probe more complex frequency shapes of the primordial power spectrum, beyond a single power law (constant nt ) in Eq. (51) – see, for example, the analysis in [47, 48]. Finally, also the chirality of primordial tensor spectra is a property that can be probed with interferometers, under appropriate conditions: see, for example, [49–51]. All these features are particularly interesting because, in case of detection, they can help in distinguishing an inflationary source from astrophysical or other cosmological sources to the stochastic gravitational wave background – see the contributions of Chiara Caprini to this encyclopedia.

Conclusions Cosmological inflation represents a successful paradigm for early universe cosmology, which makes well-definite predictions in excellent agreement with current observations. One of these predictions, the existence of a stochastic gravitational wave background from the early universe, has not been tested yet and represents one of the major challenges and opportunities for future research. In this chapter, we provided a brief theoretical overview of the mechanism underlying gravitational wave production during inflation. We explained how the detection of inflationary gravitational waves provides invaluable information on the physics driving inflation, as well as on aspects of high-energy physics and gravity at the highest energy scales. In particular, we discussed how different realizations of inflation – from single-field inflation to more sophisticated scenarios motivated from high-energy physics – give different predictions that can be distinguished by measuring the stochastic gravitational wave background they produce. Given the future prospects on the observational and experimental side of gravitational wave physics, a detailed

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theoretical analysis of the properties of primordial gravitational waves is then essential for a correct interpretation of data. We might hope that future detections of a stochastic gravitational wave background will teach us much about the physics of the early universe.

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Primordial Black Holes

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Juan García-Bellido

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primordial Black Holes as Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter I give a personal overview of the new scenario of broad mass and clustered primordial black holes that could constitute all of the dark matter, be responsible for the binary black hole events detected with gravitational waves by LIGO/Virgo interferometers, and also for the microlensing events in our galaxy detected by OGLE and GAIA. I describe their plausible origin as quantum fluctuations during inflation, their clustering and evolution since their formation at the quark-hadron transition in the early universe, as well as their signatures, both cosmological and astrophysical. I make emphasis on the fact that this scenario does not require new physics beyond the Standard Model of particle physics. I then give an overview of the extremely rich phenomenology that this scenario opens for exploration in the next few decades, where a multiprobe, multi-scale, and multi-epoch approach is required to connect the different phenomena.

J. García-Bellido () Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_27

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Keywords

Gravitational waves · Dark matter · Primordial black holes · Inflation · Particle physics · Higgs field

Introduction Primordial Black Holes (PBH) have gone through a renaissance period since the first detection by LIGO of pairs of massive black holes [1] that don’t seem to match the expectations from astrophysical black holes, while having characteristics that resemble those of primordial origin [2, 3]. In this review I will describe briefly how PBH can arise naturally in cosmology without requiring the introduction of new physics. I will apply a naive version of Occam’s razor to Cosmology by assuming that no ingredients beyond the Standard Model of particle physics are needed, in the context of the basic laws of Quantum Mechanics, General Relativity, and Thermodynamics. The only assumption is that we can extrapolate to the Early Universe the known phenomena, as explored in high energy particle physics experiments up to the electroweak (EW) scale. The first ingredient needed to describe the origin of structure in our universe is a concrete realization of cosmological inflation, where quantum fluctuations seed structure. We propose that the role of the scalar field responsible for inflation could be played by the Higgs field of the Standard Model (SM) [4]. The only new ingredient is a non-minimal coupling to gravity through a term ξ H † H R with dimensionless coupling constant ξ . This new SM parameter cannot be measured in particle physics experiments at the energies accessible by our present colliders, nor in solar system gravitational experiments due to the tiny ratio of the EW to the Planck scale. Nevertheless, such a coupling is inevitable in the context of Quantum Filed Theory (QFT) in curved space which is the proper realm in the Early Universe. Moreover, taking into account the QFT running of fundamental couplings with scale, both the Higgs quartic self-coupling λ and the non-minimal coupling ξ are expected to change their values w.r.t. those measured at the EW scale, in such a way that the Higgs alone is enough to provide a sufficiently long period of inflation to explain the observed flatness and large-scale homogeneity of the universe and, at the same time, give rise to a spectrum of curvature fluctuations with an amplitude and tilt like those observed in the CMB temperature anisotropies by the Planck satellite [5]. These fluctuations generate matter concentrations that eventually, under gravitational collapse, will give rise to the observed large-scale structures (LSS) of galaxies, clusters, and superclusters that constitute the cosmic web [6]. It also predicts a small amplitude of primordial gravitational waves which could still be detected in near future experiments searching for the curl (so-called B-mode) in the CMB photons’ polarization pattern. Moreover, the Higgs coupling to SM particles is well measured and allows us to compute the reheating temperature of the universe immediately after inflation. It turns out that reheating is very efficient and the universe acquires a thermal state

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just below Grand Unification scales, thus initiating the Hot Big Bang scenario that so many successful predictions have been harvested. We are then left with three mysteries: the nature of dark matter, the origin of the matter-antimatter asymmetry, and the recent cosmic acceleration. I will concentrate here only on the first of these mysteries and leave for the future a review on the resolution of the other two in the context of PBH.

Primordial Black Holes as Dark Matter Dark matter (DM) is a component of the universe that acts like a pressureless fluid and whose presence ensures the gravitational collapse and formation of structures like galaxies and clusters after recombination, when baryons decouple from photons and fall in the potential wells (halos) of DM, initiating thermonuclear reactions and filling space with light, reionizing the universe. The amount of ordinary matter (baryons and electrons) in our universe is surprisingly close to that of DM. There is a factor ∼10 more DM than baryons in galaxies to support their rotation curves, and a fraction ∼6 in the whole universe. The only property required for DM to explain all the observed phenomena is their gravitational interaction, so how did it originate? For the last four decades, it has been assumed it was made of fundamental particles of theories beyond the SM of particle physics, and that it was bound to be discovered in the near future. No such new physics has been found at the high energy frontier at the LHC or in complementary astrophysical searches. An alternative which had relative impact in the community was that DM was made of PBH. The idea that black holes may have formed in the early Universe comes back to the late 1960s with the precursor work of Zel’dovich and Novikov [7] and to the 1970s with the works of Hawking and Carr [2, 3] and of Chapline [8]. Already in [3, 8] it was mentioned that such PBHs could contribute to the suspected DM in the Universe or to the seeds of supermassive black holes. The first formation scenarios in the context of inflation were proposed in the 1990s [9, 10], but these usually led to (evaporating) PBHs of very small mass. In a seminal paper [11], it was proposed that PBH from peaks in the matter power spectrum could have solar masses and explain all of the dark matter. By the late 1990s, stellar-mass PBHs had already been seriously considered as a cold DM candidate, following the possible detection (e.g., in the MACHO survey) of several microlensing events toward the Magellanic clouds [12, 13]. However, the EROS [14] and OGLE [15–18] surveys later set more stringent limits on the PBH abundance, and at the same time, very stringent constraints from cosmic microwave background (CMB) observations were claimed in [19]. Since 2016, the real game-changer that has renewed the idea that PBHs may exist and constitute the totality of the DM [20–22] has been the first gravitationalwave (GW) detection from a black hole merger by Advanced LIGO/Virgo [1]. Nowadays, the importance of the different PBH binary formation channels, the possible abundance of PBHs, their viable mass function, etc. are subject to an

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intense activity and are debated (for recent reviews, see, e.g., [23]). Furthermore, since PBHs are formed by the collapse of large density perturbations, there is also a stochastic GW background (SGWB) sourced by these perturbations at secondorder accompanying PBHs. It has been calculated by [24, 25] that if BHs detected by LIGO have primordial origin, there is an inevitable accompanying SGWB, (This SGWB is inevitable in the sense that it does not require any further assumption other than general relativity and large density perturbations. It is a standard SGWB formed by anisotropic stresses which are second order in scalar perturbations.) peaking around pulsar-timing-array frequencies. In September 2020 NANOGrav has claimed the possible detection of a SGWB at nanohertz frequency using pulsar timing arrays [26], which could have been sourced by the density perturbations at the origin of stellar-mass PBH formation [27, 28]. In this context, LISA will search for the GW signatures of PBHs [25, 29–31] and will be complementary to groundbased GW detectors [32, 33] and electromagnetic probes [34], in order to prove or exclude the existence of PBHs, to evaluate the possible contribution to the DM, to the seeds of SMBHs at high redshift, and to distinguish primordial from stellar origins of black holes, on a wide range of mass scales. Any firm detection would open a new window on the physics at play in the very early Universe and a possible way to solve various long-standing astrophysical and cosmological puzzles [35,36]. We will now provide a rapid overview of the principal mechanisms that can lead to large curvature fluctuations and PBH formation, which can be related to some particular models discussed in other parts of this chapter (inflation, phase transitions, cosmic strings, etc.). There exists a broad variety of scenarios leading to large curvature fluctuations and PBH collapse, which can be classified as follows. Single-field inflation: Slow-roll single-field inflation models can produce a power spectrum amplitude of curvature fluctuations on smaller scales larger than the ones probed by CMB and large-scale structures observations. The simplest example is to have two subsequent inflationary phases, the second one lasting less than 50 efolds (roughly). Potentials with inflection points [6], like critical Higgs inflation [5], can also lead to a transient enhancement of the primordial power spectrum. More generically, in the slow-roll approximation, any potential leading to a transient reduction of the speed of the inflaton will lead to large curvature fluctuations, eventually leading to PBH formation. Another possibility is to invoke a variation of the sound speed during inflation; see, e.g., [37]. Multi-field inflation: Large curvature fluctuations may also arise in multi-field models, e.g., during the waterfall phase of hybrid inflation [38] or due to turning trajectories in the inflationary landscape [39]. Eventually, the power spectrum will not only exhibit a broad or sharp peak on scales that are relevant for PBHs, but for a sufficiently sharp turn, this is accompanied by oscillatory features that may lead to specific signatures in the PBH population and to oscillatory patterns in the scalar-induced SGWB [40–42]. It is also possible that curvature fluctuations are generated by the tunneling of the inflaton toward a local minimum of the field space [43–45] and the subsequent bubble collapse. Another intriguing possibility has been proposed by employing axions and their interaction with the gauge fields to enhance primordial density perturbations and producing PBHs [25,46–50]. In Natural Inflation [51], the inflaton

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is a pseudo-scalar particle protected from quantum corrections for super-Planckian excursions via the shift symmetry. Also, theories with UV completion predict a large number of pseudo-scalar particles that could be the inflaton or spectator fields. In this scenario, curvature fluctuations are sourced by enhanced vector modes and can have a non-Gaussian distribution. Quantum diffusion: The backreaction of quantum fluctuations during inflation makes the dynamics of the fields stochastic and allows them to explore wider regions of the inflationary potential. This makes the tails of the distribution functions of primordial density fluctuations much heavier [52–54], since they decay exponentially instead of in a Gaussian way, which boosts the production of PBHs afterward. Curvaton and stochastic spectator fields: PBHs could have formed due to the existence of a curvaton field, e.g., with a simple modification of the original curvaton scenario [55]. The primordial curvature perturbations on CMB scales are produced by the inflaton, which acts very similarly to the standard single-field scenario, while the curvaton field becomes responsible for perturbations on smaller scales, at the origin of PBH formation. In another scenario [56], a stochastic spectator field experiences quantum fluctuations during inflation making it exploring a wide range of the potential, including its slowroll part but without having any impact on the inflationary dynamics. In regions where the field acquires a value allowing slow-roll, after inflation but when these are still super-horizon, additional expansion is produced locally, which generates curvature fluctuations that later collapse into PBHs. This model has the advantage that the primordial power spectrum remains at the CMB level almost everywhere, except in PBH-forming regions. Preheating: If inflation is followed by a preheating, when the inflaton oscillates coherently at its ground state and decays to other degrees of freedom, resonant amplification of the quantum field fluctuations takes place [57–59]. These are accompanied with a resonant amplification of curvature fluctuations [60–63], which may collapse and form PBHs [64, 65]. The PBHs that form are typically very light, and only if reheating completes at very low energy (below the electroweak scale), the formed PBHs would have a relevant mass for the LISA frequency range (for GW radiation from PBH binaries). It is however also possible that these curvature fluctuations source a SGWB at second order [66], and in this case LISA is sensitive to much lower masses (down to 10−14 M ). Phase transitions: The formation of PBHs may have been facilitated in first-order phase transitions [67], in non-equilibrium second order phase transitions [68], and in specific realizations of the QCD transition [69]. Early matter era: The curvature threshold for the PBH collapse depends on the equation of state of the Universe. It goes to zero in a matter-dominated era, and so it is possible that PBH have formed from standard O(10−5 ) inflationary curvature fluctuation if the early Universe has undergone a transient matter-dominated era. Several mechanisms have been proposed to be at the origin of such early matter era; see, e.g., [70, 71]. Cosmic strings, domain walls: Topological defects may have led to the production of PBHs. For instance, cosmic string cusps can collapse gravitationally into PBHs with a mass function that could extend up to stellar-masses [72]. The collapse of domain walls, e.g., produced by tunneling during inflation [73, 74] or in QCD axion models [75, 76], is another class of PBH formation scenarios. Primordial

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magnetic fields: It is suspected that the required seeds for extra-galactic magnetic fields have an origin in the early Universe [77]. These primordial magnetic fields induce an anisotropic stress that can act as a source of large super-Hubble curvature fluctuations, leading to PBH formation with a broad range of masses [78]. A very conservative alternative proposed 25 years ago is that DM arises from the same mechanism as the other matter fluctuations we observe in the CMB and LSS, but on vastly smaller scales [11]. It turns out that quantum diffusion during inflation is a universal phenomenon of QFT in curved space and responsible for a significant deviation from Gaussian of the statistics of curvature fluctuations, giving rise to large exponential tails to otherwise purely Gaussian (i.e., free field) fluctuations [54]. These highly non-Gaussian (NG) tails generate, upon reentry during the radiation era, localized regions where curvature gradients are unusually large. In such regions, radiation pressure from the plasma cannot prevent gravitational collapse, and the whole causal horizon becomes a black hole [38]. Those are PBHs, formed well before primordial nucleosynthesis (BBN), with masses given by that within the horizon at reentry. They are widely separated at formation, but their highly non-Gaussian statistics induce aggregations or clusters of PBH when they become causally connected, well before recombination [21]. The plasma in the early universe is made of very energetic photons, gluons, intermediate gauge bosons, and relativistic particles like quarks and leptons. When quarks and gluons confine to form baryons (protons and neutrons) and mesons (pions) at the quantum chromodynamics (QCD) scale, a significant fraction of the radiation pressure exerted by these particles disappears. Since radiation pressure opposes gravitational collapse, its sudden absence makes it extremely more probable for PBH to form precisely at that time, out of the high tails of the curvature spectrum of fluctuations. Perhaps not surprisingly, the mass within the causal horizon at the QCD epoch coincides with the Chandrasekhar mass, MCh ∼ MP3 /m2p ∼ M , and thus to the masses of stars in our present universe [56]. Therefore, a prediction of this simple scenario [36], based solely on known physics, is that PBH should have masses similar to those of stars. Moreover, radiation pressure drops not only at QCD (although this gives the largest change) but whenever known particles become non-relativistic (NR), when the expansion of the universe cools below their mass-energy and photons can no longer repopulate them. The heaviest particles we know are the top quark and the Higgs, then the W and Z bosons, all with similar masses around the EW scale. As they become NR, radiation pressure drops and a small fraction of PBH form, with masses around those of planets like Neptune. Later on in the evolution of the universe, when the temperature falls below the electron mass, positrons annihilate with electrons and disappear. The mass within the horizon at that scale corresponds to supermassive black holes (SMBH) of masses M ∼ 106 M , most probably responsible for the seeds of galaxies after recombination. Both of these populations constitute not more than 0.1% of all DM, but it is enough to explain astronomical observations [79]. A fascinating prospect would be to use mass spectroscopy of PBH to deduce the way in which the universe cooled down, learning about the nature of the cosmological QCD transition, a cross-over in numerical lattice simulations, but so

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far untested in experiments. One could furthermore explore the possible existence of new particles beyond the SM, above the EW scale, since they would appear as broad peaks in the PBH mass spectrum. For example, if Beyond Standard Model (BSM) particles above 100 TeV exist in nature, we may search for their imprint in the PBH spectrum at masses M ∼ 10−10 M , a range still consistent with being an important component of all DM [79]. Note that in order to explain PBH as DM, we only require known physics: quantum mechanics for the origin of curvature fluctuations, general relativity (GR) for the gravitational collapse of BH, and thermodynamics of the known particle content of the SM extrapolated to the Early Universe, for their mass spectrum. A similar extrapolation of known physics was done by George Gamow in the 1940s to explain the observed He abundance in terms of the then measured nuclear reaction rates in the context of an expanding universe at temperatures comparable to those of nuclear bound states and times of order a few seconds to a minute after the Big Bang [80]. This new extrapolation pushes our understanding of higher energy particle physics to the cosmological quark-hadron transition at times of order 50 μs and energies of order the proton mass. The confirmation of PBH at planetary scales would open the possibility to explore the cosmological nature of the EW scale, still uncharted territory in particle physics experiments. Black holes in nature come with three “hairs” or properties: mass, charge, and spin. We have discussed the PBH mass origin, but what about their charge and spin? In a charge-symmetric universe like ours, black holes are neutral. The same amount of protons and electrons fall onto a stellar BH created in a supernova (SN) explosion, while the charge within the horizon in the early universe was zero to great accuracy, and thus PBH are also electrically neutral. But what about spin? Supernova explosions of massive stars leave behind, after shedding the outer layers, either a neutron star (NS) or a black hole. Conservation of angular momentum implies that the remnant of massive low-metallicity stars should be highly spinning, and indeed spinning BH in X-ray binaries have been detected. (See the  Chap. 17, “Formation Channels of Single and Binary Stellar-Mass Black Holes” by Michela Mapelli in this same volume.) In the PBH case, however, the fact that the whole volume within the (spherical) horizon collapses to form a PBH of essentially the same size (within a factor two at most) implies that they should be essentially spin-less [81, 82]. They may acquire some spin afterward, from gas accretion or collisions with other BH, but the spin distribution should be highly peaked at zero. This is a robust prediction that seems to be in agreement with what LIGO/Virgo have measured [83]. What is still uncertain is the total amount of PBH in the universe, distributed in the aforementioned, characteristic and fundamental-physics-derived mass spectrum. Do they constitute all of the DM, or only a small fraction? Their actual abundance depends on the quantum dynamics during inflation, which itself depends, in this scenario, on the microphysics of the Higgs field on scales much beyond the EW scale. If there are indeed no new particles beyond those found up to date in particle physics experiments, then one can extrapolate, using the renormalization group equations (RGE), the running of the fundamental masses and couplings all the

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way to the inflationary scales, taking into account gravity at the classical level. As mentioned above, with a very reasonable (order one) non-minimal coupling to gravity at the EW scale, the Higgs itself may give rise to the observed amplitude of CMB anisotropies. The surprise when running the Higgs self-coupling λ is that, due to the unusually large Yukawa coupling of the top quark, λ drops at inflationary scales to values very near zero and then grows again. The point at which the Higgs self-coupling has a minimum corresponds to a plateau in the potential energy density of the Higgs, giving rise to an enhanced amplitude of curvature fluctuations [5], together with a growth of quantum diffusion that generates important non-Gaussian tails, increasing exponentially the probability of collapse to PBH upon reentry after inflation [54]. Thus known particle physics, together with QFT, could have given rise to the PBH that constitute most if not all of the DM. The quantum nature of curvature fluctuations responsible for CMB anisotropies and matter perturbations responsible for the observed LSS has been a subject of debate for years since the fundamental equivalence of quantum free fields in their ground state and stochastic Gaussian random fields leaving the QFT of inflation untested. However, the fundamental nature of quantum diffusion gives rise to inevitable NG tails and ultimately to gravitational collapse and PBH. Therefore, the confirmation of the quantum origin of PBH will be also a confirmation of the paradigm of inflation. At present we have only partial evidence of PBH, but it is becoming a field of research in itself as improved observations keep coming in, specially thanks to the new window opened into the universe opened by GW detections with interferometric antennas. The mass distribution inferred from present LIGO/Virgo sensitivities (1 − 200 M in the last run O3a [84]) indicates that there are black holes with properties that are common to a single population in the whole mass range, without discontinuity [85]. On the other hand, astrophysicists acknowledge that there are two mass gaps in the BH mass distribution arising from stellar evolution. The lower mass gap appears between two and five solar masses. Stellar simulation models have not been able to generate BH in that mass range. Only masses above 5 M appear as the result of SN explosions, even in the case of low metallicity stars (those with few electrons per nucleon, composed of Hydrogen and Helium alone). The upper mass gap appears in the range 50 − 130 M and has a clear physical origin: As very massive stars above 200 solar masses start burning heavy elements like Carbon and Oxygen in their inner layers, there is not enough thermal pressure to maintain equilibrium and the star contracts, heating up to temperatures above pair creation of electrons and positrons, which further reduce thermal pressure, inducing further contraction and making the star catastrophically unstable, exploding as a supernova that completely obliterates the star, leaving no remnant behind. The actual boundaries of both mass gaps are somewhat uncertain, and new simulations keep improving the models. However, if BH appear well within the boundaries of those mass gaps, there is consensus that they cannot have stellar origin, and most probably they are primordial [86]. In fact, just before GW antennas started to detect BH mergers and accurately measure their masses, the only known stellar black holes were those associated with

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X-ray binaries, where a BH captures mass from a stellar companion and forms an accretion disk that emits in the X-rays. From the orbital motion of the companion a mass distribution has been determined, peaked at 8 M and sharply falling beyond the ends at 5 M and 10 M . This expectation was responsible for the surprise when LIGO/Virgo GW detections in runs O1-O2 showed a mass distribution around 30 M , clearly above the observed X-ray BH mass range [87]. In run O3, the improved sensitivity of the detectors allowed in principle to detect BH with masses in the range 1 − 200 M . In fact, from the four published run O3 events, one finds that there are BH in both the lower and upper mass gaps, thus creating a challenge for stellar models, and giving the opportunity to assigning a PBH origin. In particular, some of the assumed binary neutron star (BNS) events like GW190425 are most probably BBH in the lower mass gap, due to their masses being well above the measured NS distributions, the lack of optical or radio counterparts, as well as total absence of tidal deformations in the final orbits before merger. On the other hand, the extraordinarily heavy BBH of GW190521, with 66 and 85 M components of the binary clearly inside the upper mass gap, suggests these BH are not of stellar origin but most probably primordial [86]. Furthermore, a very important argument in favor of the primordial origin of LIGO/Virgo BH is their absence of spin. All BBH measured so far have, within errors, zero effective spin (a mass-weighted average of the spin projection onto the orbital angular momentum), χeff = 0, and final spin af = 0.686 (as predicted by GR from the conversion of angular momentum into spin of the merger of two spinless BH). This is a strong hint for the absence of spin of the original members of the binary. Moreover, event GW190814 presents the largest mass ratio so far, 1:10, between the components of the binary, again an unexpected ratio for stellarorigin BBH due to mass transfer between members before BH formation. But more importantly, the large mass ratio allows one to measure very accurately the heaviest BH spin, which turns out to be less than 0.07 at 95% c.l., a property that simply cannot be explained by stellar evolution due to angular momentum conservation. Even if a single BH may have ended up having zero spin through a complicated dynamical process, one cannot explain that essentially all LIGO/Virgo BH are spinless. Therefore, one comes to the conclusion that we have encountered a new population of BH with a broad mass distribution and without spin, which very strongly suggest a primordial origin [83]. There is further support of this PBH scenario by computing the expected merger rates of black holes in the case of clustered PBH with the mass distribution of QCD (i.e., with a proton peak at ∼2 M and a pion plateau at ∼70 M ), which exactly reproduce the observed events seen by LIGO/Virgo, with the presence of three distinct bumps in the plane (m1 , m2 ), at (2, 2), (2, 70), and (70, 70) in solar masses [36]. With the future run O4 sensitivity, we should begin to measure the whole mass distribution and merger-rates plane and determine whether or not we have convincing evidence for the PBH scenario outlined above. In summary, LIGO/Virgo have detected a new population of BH, but are they really primordial? And if so, do they contribute to the totality of the DM? How can we further test this hypothesis? Are there other observations that support this

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scenario? Indeed, if PBH of these masses constitute all of DM, they should populate the halos of all galaxies, in order to explain their rotation curves. We have seen that they necessarily cluster in pockets of about a thousand PBH. However, three body interactions within the cluster will heat them up and make them “evaporate,” ejecting isolated PBH as well as tightly bound binaries out of the cluster [88]. Therefore, we should see those wandering PBH in our halo, moving at large speeds. Moreover, close hyperbolic encounters of PBH [89, 90] inside clusters in our galactic halo should eventually be detectable with LIGO/Virgo or in the future with the Einstein Telescope [32]. Another way to test this scenario is by looking at microlensing of stars in our galaxy. The amplification of light of a distant star due to microlensing by a massive compact halo object (MACHO) used to be the best way to constrain the abundance of PBH in the solar mass range, but since the broad-mass-distributed and spatially clustered scenario we proposed predicts very low optical depths toward the Large Magellanic Cloud (LMC), all those historical bounds have disappeared. In fact, what is ruled out is the monochromatic and uniformly distributed PBH scenario, not the PBH idea as the MACHO component of our Milky Way halo. Instead, we can search for them by looking at stars nearby, like OGLE collaboration did. They found more than 58 microlensing events toward the galactic bulge, of which they could trace 18 compact lenses, thanks to GAIA astrometry, to break the degeneracy between mass and distance in the Einstein radius, and thus measure the masses of the dark lenses. To their surprise, they found black holes with masses between 0.1 and 20 M , including several BH in the lower mass gap (note that at a distance of a kpc, a few-solar-mass star emits more light than the GAIA limit of 18th magnitude, and these were darker objects) [91]. But perhaps much more interesting is that they found BH with masses below the Chandrasekhar limit of 1.4 M , where they should not exist unless they are primordial. Acknowledgely, these measurements have significant uncertainties and BH masses, but soon LIGO/Virgo/Kagra run O4 (expected to start in the summer of 2022) will be able to measure routinely events with less than one solar mass companions, if they exist. This will create a revolution in our understanding of BH formation. So let us assume for the moment that LIGO/Virgo and OGLE have detected PBH. What is their present abundance? How are they distributed in space? Inevitable non-Gaussian tails from quantum diffusion during inflation predict they should be clustered today, with a fraction of their original concentration diluted by evaporation. This induces a segregation by mass of the original PBH population, with the heaviest BH at the centers of PBH clusters, while the lightest ones ejected and distributed more or less uniformly in the halos of galaxies. The order of a few thousand solar mass PBH cluster, of a few parsecs in size, constitute the building blocks of DM halos today [88]. They give rise to the small-scale structure of DM, a scale too small to be accessible in cosmological N-body simulations, whose present mass resolution reaches at best a billion solar mass “particle,” and scales of order a kpc. These simulations cannot distinguish particle DM from PBH DM; they are in the fluid regime of the halo mass distribution. However, we expect the granularity of DM to be observed on dwarf galaxy scales, where stars and BH coexist on the same footing. Remember that ultra-faint

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dwarf galaxies (UFDG) are faint because they have fewer stars and essentially no gas, reaching mass-to-light ratios of a thousand. Perhaps these galaxies originally started with more stars and lost them via close encounters with the PBH in their halos, exchanging energy and momentum with them and thrown out of their shallow potential wells via the gravitational slingshot effect. This could explain the lack of observations of substructures around our galaxy, until recently, where, thanks to increased exposure, tens of dwarf galaxies have been detected by SDSS and DES surveys, thus resolving the so-called missing satellites or substructure problem [35]. This new scenario suggests a change of paradigm with respect to particle DM, similar to that of the Rutherford versus Thomson model of the atom. In the PBH DM scenario, rather than moving essentially unimpeded in the dilute and uniform medium of DM particles, where stars only feel the gravitational field of the other stars, here a few stars can have close encounters with the hard cores of PBH in the halo, and strongly scatter off them, just like alpha particles on atomic nuclei in Rutherford’s experiment, and be ejected from the galaxy. This could be the reason why UFDG are almost devoid of stars. But if they have been ejected out of their halos, we should see them elsewhere. Indeed, we may observe some of those stars as hypervelocity stars, recently detected by GAIA astrometry, moving out of dwarf galaxies like Sagittarius in our own galaxy [92]. Another way to measure the microstructure of DM in the form of PBH clusters is via their perturbations on tidal streams. These are dwarf galaxies that have crossed the galactic disk and have been stretched by tidal forces. They are elongated stellar structures very sensitive to their gravitational interaction with DM clumps. If the clumps are large and massive, they are less numerous, and the chance of interaction with the tidal stream is small, but smaller DM clumps, as small as 103 − 105 M , are ubiquitous in the halo and leave multiple hits on tidal streams, inducing gaps in them, just like it has been observed in GD1 and Palomar-5 stellar streams. A proper statistical analysis has recently been proposed [93]. Moreover, PBH clusters of thousands of solar masses may encounter and be swallowed whole by dense molecular clouds, driving a rapid accretion of gas onto them, which ends into a single intermediate mass black hole (IMBH) at the center that injects energy into the molecular cloud and makes it shine in X-rays and radio, as has been observed in several molecular clouds near the galactic center. There are many ways to search for black holes in the universe, mostly by the light that accelerated matter emits when falling onto them. What may be difficult is to distinguish between stellar BH and primordial ones, unless you measure their spin like we do with LIGO/Virgo, or their spatial distribution, since stellar BH are born out of stars and typically populate the inner parts of galaxies, while PBH clusters inhabit the outer halos of galaxies. In the future, we may detect with an array of GW antennas the location of BBH mergers and correlate with the spatial location of galaxies and clusters of galaxies. The correlation with the halos of DM should give us information about their nature and origin. Moreover, massive black holes tend to move by dynamical friction toward the centers of their potential wells in dwarf galaxies, which should then act as stabilizers of stellar orbits, just like the SMBH at the center of our own galaxy.

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Therefore, in the PBH scenario of DM, the mass distribution in galaxies is characterized by a central massive black hole, surrounded by stars and other lighter black holes, which leave a relatively empty inner core due to hard three-body interactions, building up in density as we move away until it reaches a maximum at around a kiloparsec from the central black hole, in the case of galaxies of our size, and correspondingly smaller in dwarf galaxies. The density then falls off as in the usual DM profile found in N-body simulations [88]. Thus the so-called corecusp problem of DM is resolved with neither of them being correct. The inner cores of galaxies are devoid of DM except for the central SMBH, which could have originally been seeded by a massive million-solar-mass PBH and later acquired its present mass, up to billions of solar masses, due to gas accretion immediately after formation and through the first few million years, driving the cores of active galactic nuclei and quasars up to high redshifts [38]. Although SMBH may have played some role in the reionization of the universe, most of it was probably due to the solar mass PBH, which constitute the bulk of the DM. Gas falling onto them after recombination and baryon decoupling injected gamma and UV photons into the universe, slowly reionizing it around dense pockets, rather than explosively as some other scenarios propose. In the case of PBH-induced reionization, the optical depth to the CMB will gradually grow toward one as a function of redshift, instead of changing abruptly at z ∼ 6, as the explosive stellar model of patchy reionization suggests. At the moment we only measure, with the Planck satellite, the integrated optical depth, but future measurements of the 21cm line of Hydrogen with full-sky surveys like the Square Kilometer Array will determine both the redshift and spatial dependence of reionization up to z ∼ 20, distinguishing both scenarios. It is interesting to confirm that a strong level of spatial correlation has been observed between fluctuations in the source-subtracted Cosmic Infrared Background (CIB) and the diffuse soft X-ray background. These correlations could hint at a common origin between the redshifted hard X-ray and UV backgrounds, arising from energy injections at redshift z ∼ 15 − 20, as gas falls onto solar-mass black holes of the type discovered by LIGO/Virgo, if their abundance is as large as that of DM at that epoch. Another way to search for PBH is by looking at spectral distortions in the CMB photons [94]. If PBH existed well before recombination, the accretion of gas directly falling onto the PBH, or via accretion disks, will reinject energy into the plasma. If this happens before redshift z ∼ 106 , the plasma reequilibrates itself via Compton scattering, and there is no trace of it left. However, if there remains a significant amount of PBH up to recombination, as is expected in the case they are all of the DM, the energy injected into the plasma induces spectral distortions in the high frequency tails of the black body spectrum [95]. So far, COBE-FIRAS measurements exclude distortions as large as one part in 105 . In the near future we may launch a satellite that reaches a thousand times better resolution and detect the injections due to PBH. Apart from the energetic interactions with its surroundings, the cleanest way to detect a black hole is through its gravitational distortion of space, which affects

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the path of distant light sources via the famous gravitational lensing effect. We have seen that microlensing has traditionally been one of the best methods used to put constraints on the MACHO content of the halo. However, their clustering in dense compact clumps makes them evade the bounds [96, 97]. Nevertheless, if we look at the light of really distant objects, like quasars or supernovae, we should be able to determine the mean amplification induced by the overall permeating population of PBH along the line of sight to those sources, inside galaxies and in the intergalactic medium [98]. For the moment, the supernovae catalogs are still systematics dominated and not precise enough to determine the full distribution of lensing amplifications at a given redshift, to distinguish PBH clusters and individual isolated black holes from larger substructures [99]. There is already evidence from QSO microlensing that there is a population of solar-mass black holes that could constitute a significant fraction of the DM in the universe [100]. Some studies argue that we are seeing the stellar distribution but PBH is a far more likely interpretation, given their abundance. Alternatively, one could look into the different images of a distant quasar which is strongly lensed by a galaxy or a cluster of galaxies along the line of sight. The stochastic quasar variability can be used to determine the difference in path lengths to the various images, and thus the actual rate of expansion of the universe. But even more interesting is to look for microlensing amplifications along one of the paths and not the others, which should be apparent in the residuals after time-shift matching the detailed variability of the quasar. There are indications in a few strong lensed QSO of long duration microlensing events over periods of decades, which would indicate massive BH in the LIGO/Virgo range [101]. This would be an independent confirmation of such a population of BH that cannot be measured in any other way. In the case of the strong lensed system, one can determine the path taken by light around the galaxy that acts as lense, and confirm that it is an object in the outer halo, most probably a PBH if they are the DM. Furthermore, smaller black holes of the size of planets like Neptune are difficult to detect since their gas accretion is too feeble to be seen at great distances. There is the tantalizing possibility that planet 9, recently discovered in the outskirts of the solar system from the gravitational clustering it induces on extreme trans-Neptunian objects orbiting the sun at 250 AU, could be a PBH [102]. Such an object can only be detected gravitationally, unless another large object like an asteroid falls on it and emits a flash of X-rays that reaches Earth’s satellites, but that’s rather implausible. It is important to clarify that PBH may lurk in our solar neighborhood but should not be considered as dangerous since they orbit around the galaxy just like stars, and although they are dark and could approach us unseen, they distort space via gravitational lensing and we may detect them via their gravitational effects on other bodies. Some of these PBH may be moving at large speeds after their ejection from a nearby cluster, but the vast size of our galaxy makes it very improbable that one such BH should happen to cross the Sun’s path. A fascinating possibility, which may turn out to be the key to the discovery of PBH, is that of finding the SGWB generated at the time of their formation during the radiation era. Generic stochastic backgrounds arise when one cannot

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resolve the individual sources and their emission is more or less uniform across the sky, and in particular coming from BBH coalescence over the entire history of the universe [103]. In the case of PBH formation, the size of the horizon at that time is so ridiculously small projected in the sky today (of order microarcseconds) that the corresponding GW emission leaves a perfectly smooth SGWB today. There are several sources of gravitational waves at PBH formation. Since the process of gravitational collapse is very violent, one would expect that it would emit a substantial SGWB; however, the causal horizon is essentially spherical by construction, and, although large curvature gradients are needed to overcome radiation pressure, the highly non-Gaussian tails suggest that the process is a very rare one, and it can be shown that rare peaks in a random density field are essentially spherical. However, general relativity predicts that perfectly spherical collapse will not emit GW, and thus we don’t expect a large amplitude of the SGWB from the slightly aspherical collapse at PBH formation. On the other hand, since scalar curvature perturbations need to have a large amplitude in order for gravitational collapse to occur against radiation pressure, second order anisotropic stresses at horizon crossing can source the production of a SGWB [104]. Such a background should have a typical frequency of order a few nano Hertz, corresponding to the redshifted size of the QCD horizon today. This is the realm of Pulsar Timing Arrays (PTA), where the precise time of arrival of pulses from a collection (of order 30) pulsars acts as a sensitive detector that can recognize the passage of a GW between us and the pulsar. The near quadrupolar pattern of correlations in different directions in the sky would be a clear signature of such a SGWB. There has been a recent claim by NANOgrav [26] of marginal detection (less than 3 sigma) of a SGWB with a peak at 5 nHz and a strain amplitude of a few 10−15 , which is compatible with the signal expected from PBH formation at the quark-hadron transition [28]. However, it is still too early to claim detection, and it is expected soon a similar analysis by the other PTA collaborations in Europe and Australia. Moreover, unresolved BBH mergers similar to those detected by LIGO/Virgo that happened in the past, through the whole history of the universe, add up to create a SGWB with sufficient amplitude to be detected by the next generation of detectors like A+, Voyager, Einstein Telescope, Cosmic Explorer, and LISA. The characteristic feature of such a SGWB is its spectral dependence (ΩGW h2 ∝ f 2/3 ), which should be easily recognizable [103]. The amplitude will depend on the abundance and mass distribution of BH, both astrophysical and primordial, as well as on the redshift dependence of their merger rate. It is clear that astrophysical black holes could not have started their formation before stars; they are later captured to form binaries and finally merge via the emission of GW millions to billions of years later, depending on their mass. On the other hand, PBH in dense clusters form binaries early on, but three-body interactions prevent a SGWB to build up too quickly. Only a fraction of the binaries are ejected from the clusters and finally merge to be observed by GW antennas [88]. Those PBH binaries give rise a SGWB, but the redshift dependence is still uncertain, depending on the evolution of PBH clusters throughout the history of the universe.

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The next generation of GW antennas will be sensitive to both the low and high mass part of the PBH spectrum, and therefore should be able to detect massive PBH mergers at redshift z ∼ 20, where no other type of BH could exist. Moreover, their sensitivity is such that they should be able to measure the whole mass distribution above and below the Chandrasekhar mass limit, with better than a few % accuracy, and determine whether they could constitute all of the DM, according to their merger rate as a function of redshift, which is a very sensitive probe of their mass function and total abundance.

Conclusions To conclude, more than 20 years ago we predicted [11] that massive PBHs would form via the gravitational collapse of radiation and matter associated with high peaks in the spectrum of curvature fluctuations and that they could constitute all of the DM today. In 2015, we predicted [38] the clustering and broad mass distribution of PBH, which peaks at several M , and whose high-mass tails could be responsible for the seeds of all galaxies. Since then, LIGO/Virgo interferometers have detected gravitational waves from at least 50 merger events of very massive and spin-less black hole binaries, and we propose that they are all PBH. We have recently understood [36] that a universal mechanism associated with rapid changes in the number of relativistic species in the early universe could have been responsible for the formation of PBH at specific scales and thus have a very concrete prediction for the mass spectrum of PBH as DM, with broad peaks at 10−5 , 2, 70, and 106 M . In particular, the QCD quark-hadron transition could be responsible for the efficient production of baryons over antibaryons at PBH collapse, thus explaining the presence of baryons today and the relative abundance of DM. We predict [86] that within a few years a less than one solar mass PBH will be detected by AdvLIGO/Virgo and that an array of GW detectors will allow us to determine in the near future the mass and spin distribution of PBH DM with 10% accuracy [32]. Thus, gravitational wave astronomy could be responsible for a new paradigm shift in our understanding of the nature of DM and galaxy formation. Acknowledgments The author thanks his colleagues for endless discussions on the ideas that have helped to shape the present scenario of broad-mass and clustered PBHs as DM and acknowledges funding from the Research Project PGC2018-094773-B-C32 (MINECO-FEDER) and the Centro de Excelencia Severo Ochoa Program SEV-2016-0597.

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28

Testing the Nature of Dark Compact Objects with Gravitational Waves Elisa Maggio, Paolo Pani, and Guilherme Raposo

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models of Exotic Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECO Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multipole Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal Deformability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ringdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ergoregion Instability of ECOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational-Wave Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Open Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Within Einstein’s theory of gravity, any compact object heavier than a few solar masses must be a black hole. Any observation showing otherwise would imply either new physics beyond general relativity or new exotic matter fields beyond the standard model and might provide a portal to understand some puzzling properties of a black hole. We give a short overview on tests of the nature of dark compact objects with present and future gravitational-wave observations, including inspiral tests of the multipolar structure of compact objects and of their tidal deformability, ringdown tests, and searches for near-horizon structures with gravitational-wave echoes.

E. Maggio · P. Pani () · G. Raposo Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_29

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Keywords

Gravitational waves · Black holes · Tests of gravity

Introduction Originally considered just as bizarre solutions to Einstein’s theory of general relativity (GR), black holes (BHs) have now acquired a prominent role in astrophysics, theoretical physics, and high-energy physics at large, to a level in which “BH physics” is living a new golden age. Gravitational-wave (GW) observations have provided the strongest and most direct evidence to date for the existence of BHs with masses ranging from few to hundred solar masses. Meanwhile, progress in modeling the electromagnetic emission from accreting BHs and new X-ray and VLBI facilities has provided new probes of the region near dark compact objects, both in the mass range explored by LIGO and Virgo and in the supermassive range, i.e., million to billion solar masses. All observations so far are beautifully compatible with the predictions of GR and with the so-called Kerr hypothesis, namely, that any compact object with mass larger than a few solar masses is described by the Kerr BH metric, as predicted by some remarkable uniqueness and “no-hair” theorems established within GR. Given this state of affairs and the robustness (both at astrophysical and observational level) of the BH picture, it is natural to question the motivation for further tests of the nature of compact objects. In fact, in synergy with GW-based tests of gravity discussed in other chapters, tests of the nature of compact objects are emerging as one of the cornerstones of strong-gravity research, for several theoretical and phenomenological reasons: • One of the most urgent open problems in theoretical physics is the so-called information-loss paradox [136], which is related to loss of unitary at the end of the BH evaporation due to Hawking’s radiation. This problem should be hopefully resolved within a consistent quantum gravity theory. Several attempts to address this issue (most notably the fuzzball scenario in string theory [123, 134– 136, 139] but also nonlocal effects [79–83], firewalls [13], etc.) predict drastic changes at the horizon relative to the classical BH picture, regardless of the curvature of the object. • Although within classical GR nothing special or dramatic happens in the vicinity of the BH horizon, the BH interior is pathological. It harbors curvature singularities (where Einstein’s theory breaks down) and may contain closedtimelike curves (which violate causality) and Cauchy horizons. Some of the attempts to regularize the BH interior also affect the near-horizon structure and would leave some imprint in the exterior. • From a more phenomenological standpoint, BHs and neutron stars might be just two “species” of a larger zoo of compact objects. New species might have very different properties that can be used to devise precision searches with current and future experiments. In this context, it is interesting that current LIGO/Virgo

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observations (especially the recent GW190814 [5] and GW190521 [4, 6], respectively, in the lower-mass and upper-mass gap forbidden for standard stellarorigin BHs) do not exclude the possibility that some of the GW mergers involve exotic objects [43]. • Under general conditions, Penrose’s theorem [161] implies that an apparent horizon always hides a curvature singularity. Thus, BHs are not only a unique prediction of Einstein’s theory, but they are in fact crucial for the self-consistency of the latter. On general grounds, the evidence supporting the existence of horizons should be properly quantified in the most accurate way. This implies, on the one hand, devising model-agnostic tests of the “BH-ness” of compact astrophysical sources and, on the other hand, confronting the BH scenarios with more exotic ones, for example, using Bayesian model selection. This chapter is devoted to an overview of tests of dark compact objects using GW probes, a topic that has acquired significant attention in the last few years. We refer to other chapters of the book for electromagnetic tests, which are complementary to GW ones in several ways.

Models of Exotic Compact Objects Deviations from the BH hypothesis require either corrections to GR or beyondstandard model fields coupled to gravity (see [44, 47] for some reviews). Hypothetical dark compact objects without a classical BH horizon that, nonetheless, can mimic the phenomenology of BHs at the classical level are generically called “BH mimickers” [48, 118] or exotic compact objects (ECOs) [85]. ECOs may be classified [47] in terms of their (see Fig. 1) • compactness, i.e., the inverse of their – possibly effective – radius r0 in units of the total mass M. It is customary to define a “closeness” parameter  such as r0 = r+ (1 + ), where r+ is the location (For spherical objects, the definition of the closeness parameter  is coordinate-independent (2π r0 is the proper equatorial circumference of the object). More in general, one can directly relate  to gauge-independent quantities.) of the horizon of the corresponding Kerr BH with the same mass and spin. Therefore, the BH limit corresponds to  → 0; • reflectivity R at their (possibly effective) surface. This quantity is generically complex and frequency-dependent and can also depend on other object’s parameters such as the spin. Note that R → 0 in the limit of perfect absorption by a classical BH horizon. Another mention-worthy property used to classify ECOs is their so-called softness, associated with the spacetime curvature at its surface. When the ECO’s underlying theory is associated to a new length scale L  M, the curvature (e.g., the Kretschmann scalar K ) at the surface can be much larger than the corresponding horizon curvature K  1/M 4 . On the other hand, ECOs not motivated by new length scales other than M (or parametrically close to it) cannot sustain larger

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Fig. 1 Schematic representation of the parameter space of ECOs. The closeness parameter  is related to the (effective) radius of the object by r0 = r+ (1+), where r+ is the horizon location for a Kerr BH with same mass and spin. Models with quantum corrections at the horizon scale imply r0 − r+ ≈ Planck ; hence,  = O (10−50 − 10−40 ) for stellar to supermassive objects, depending on their mass. The reflectivity is related to the object’s interior and is generically complex and frequency-dependent. For some reviews on different ECO models, see [44, 47]. For the distinction between tight and diffuse fuzzballs, see [89]

curvatures at their surfaces. To the former, we call “hard” ECOs while the latter are denoted by “soft ECOs.” A useful compass to navigate the ECO atlas is provided by the Buchdahl’s theorem [41], which states that, under certain assumptions, the maximum compactness of a self-gravitating object is M/r0 = 4/9 (i.e.,  ≥ 1/8). This result prevents the existence of ECOs with compactness arbitrarily close to that of a BH. Relaxing some of these assumptions [192] provides a way to circumvent the theorem and suggests a route to classify ECOs [47]. In addition to some technical assumptions, Buchdahl’s theorem assumes GR, spherical symmetry, and especially the fact that the matter sector is described by a single perfect fluid which is at most mildly anisotropic (tangential pressure smaller than the radial one). Besides the assumption of GR (which is therefore violated in any modifiedgravity theory), a quite common property of ECOs is the presence of an anisotropic pressure. Strong tangential stresses are necessary in many models to support very compact self-gravitating configurations. This is the case, for instance, of

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boson stars [106, 173], gravastars [55, 138, 139, 144], ultracompact anisotropic stars [38, 168], and wormholes [119, 142, 194]. Although other classifications are possible, from a phenomenological perspective, it is useful to divide ECOs into two classes: (i) “ab initio” models that are solutions to consistent field theories coupled to gravity and (ii) phenomenological models that are studied to test possible generic implications of the absence of a horizon in dark compact objects, without a complete embedding in a concrete model. Examples of the first category are boson stars [106, 120, 173] and fuzzballs [123, 134–137, 139], whereas the second category comprises horizonless, possibly regular, phenomenological metrics that do not arise from a given theory or that require ad hoc matter fields in order for these metrics to solve Einstein’s equations, as in the case of certain wormhole metrics. The zoo of proposed ECO models is large and ever growing. It is not our scope to describe each model, for this we refer the interested reader to recent reviews [44, 47]. However, some ECO models stand out for being particularly interesting or representative of a generic class. In the following, we shall discuss two examples of “ab initio” models. The most studied example of ECOs are boson stars, self-gravitating solutions formed by massive complex (In the case of real bosonic fields, similar selfgravitating solutions called oscillations exist in the same theories [176]. They have a weak time dependence and slowly decay but can be very long lived, and their phenomenology is similar to that of boson stars, whose metric is instead stationary.) bosonic fields, minimally coupled to GR [106, 173]. Their properties depend strongly on the bosonic self-potential and various models with different classes of self-interactions [61, 87, 88, 107, 141, 175], and different field contents [12, 40, 97] have been considered. Boson stars are the most robust model of ECOs, since they do not require modified gravity and their formation, stability, binary coalescence, etc. can be studied from first principles [30, 120, 152, 153, 174]. These ECOs are not meant to replace all BHs in the universe, for various reasons: indeed, just like ordinary neutron stars, their compactness is lower than the BH one, and equilibrium solutions have a maximum mass above which they are unstable against the gravitational collapse and classically form an ordinary BH. Finally, their mass scale is set by the mass of the bosonic field, which implies that a single bosonic field could give rise to boson stars only in a certain mass range. A family of boson stars ranging from (say) the stellar-mass to the supermassive range would require several different bosons with masses across several orders of magnitude. More ambitious models of ECOs aim instead at replacing classical BHs completely, providing at the same time a quantum description of the horizon. A representative example is the fuzzball proposal of string theory [122, 123, 134, 135], wherein a classical BH is interpreted as an ensemble of regular, horizonless geometries that describes its quantum microstates [18, 23, 24, 134, 145]. These geometries are solutions to (the low-energy truncations of) string theory – hence circumvent Buchdahl’s theorem – and have the same mass and charge of the corresponding BH. For special classes of extremal and charged BHs, one can precisely count the microstates that account for the BH entropy [99, 129, 179], whereas in other cases,

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the entropy counting is still an open problem. In the fuzzball paradigm, all properties of the BH geometry emerge as an average over an ensemble of a large number of microstates or as a collective behavior of fuzzballs [25, 26, 32, 33, 35], regardless of the curvature of the object [137]. In the following, we shall focus on the ECO phenomenology in a model-agnostic way, only occasionally referring to specific models.

ECO Phenomenology GW astronomy allows us for unprecedented tests of the nature of BHs and to search for new exotic species of compact objects. In this section, we overview the main classes of GW-based tests, including inspiral tests of the multipolar structure and of the tidal deformability, ringdown tests, and searches for near-horizon structures with GW echoes.

Multipole Moments Multipole moments were first introduced in the context of Newtonian mechanics (resp. electromagnetism) as a set of scalar quantities that appear on a multipolar expansion used to describe the gravitational (resp. electrostatic) potential Φ(x) resulting from a distribution of masses (resp. charges),  ∞   Mm Φ(x) = r +1 =0 m=−



4π Ym (θ, ϕ) , 2 + 1

(1)

where the expansion coefficients Mm are the mass multipole moments of the body (for the gravitational case, on which we shall focus in the following) and Ym are the usual spherical harmonics. In expansion (1), the multipole moments can in general be real or complex numbers. Nonetheless, if the potential Φ is real, the multipole moments must satisfy, ∗ M−m = (−1)m Mm .

(2)

From the perspective of the interior of the object, these multipoles are related to the nonspherical distributions of matter within the body and they can be written as  Mm =

4π 2 + 1



∗ 3 d x. ρ(x)r  Ym

(3)

where ρ is the mass density. (The normalization of the multipole moments was chosen to coincide with the relativistic multipole moments (discussed below) when

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taking the Newtonian limit of the latter. Other work may use different normalization factors for the multipole moments, e.g., [164].) This simple Newtonian definition of multipole moments breaks down in GR due to the nonlinearity of Einstein’s equations. Nonetheless, two independent formulations to define the relativistic multipole moments were developed, firstly by the works of Geroch and Hansen [77, 92] and later by Thorne [184]. While the former presents an elegant mathematical definition of multipole moments, it is not suitable for computations for most astrophysical scenarios. Thus, henceforward we shall focus mostly on Thorne’s approach. Remarkably, it was shown that the two definitions are equivalent [91]. In Thorne’s formulation, the multipole moments of stationary and asymptotically flat spacetimes can be read-off directly from the metric as an extension to the procedure used to read the mass and the angular momentum from the asymptotic metric. Thorne’s procedure requires the spacetime metric to be written in a specific class of coordinate systems called “asymptotically Cartesian and mass centered” (ACMC) where the metric approaches the asymptotically flat Minkowski space sufficiently fast and the mass dipole term vanishes. This last condition is equivalent to set the origin of the coordinates at the center of mass of our system. In this ACMC form the metric reads, ds 2 = dt 2 (−1 + c00 ) + c0i dt dxi + (1 + c00 ) dxi2 ,

(4)

with c00 and c0i admitting a spherical-harmonic decomposition, c00 = 2

 ∞   =0 m=−

c0i = 2

 ∞   =1 m=−

1



r 1+  1 r 1+

 4π  Mm Ym +  <  , 2 + 1

(5)

4π( + 1)  B Sm Yi,m +  <  , (2 + 1)

√ B where Yi;m = ij k nj r∂k Ym / ( + 1) is the magnetic vector spherical harmonic and Mm and Sm are the relativistic mass and current multipole moments, respectively. In the Newtonian limit, the former reduces to well-known Newtonian mass multipole, whereas the latter has no Newtonian analog. Remarkably, as long as the metric is written in ACMC coordinates, the multipole moments are coordinate-independent and all coordinate dependence is pushed to the subleading terms in the metric. In Eqs. (5), we already adopted the correct normalization factors to coincide with Geroch-Hansen multipole moments which are most commonly adopted throughout the literature [45]. When the source of the gravitational field can be covered by so-called “de Donder” coordinates, one can properly define the mass and current multipole moments using some “effective” mass and momentum densities, respectively [184]. Multipole moments of compact stars and ECOs can be computed using this method;

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however, multipole moments of BHs can only be defined in terms of the external spacetime geometry.

Testing the Nature of Compact Objects with Multipole Moments A set of uniqueness and no-hair theorems predicts that, within GR, the outcome of the full gravitational collapse must be a Kerr BH [54, 95, 172]. In spite of having an infinite multipolar structure, all the multipole moments of a BH can be related uniquely to only two parameters, its mass M and the angular momentum J [77]. In the absence of rotation, BHs are spherically symmetric and the geometry reduces to the well-known Schwarzschild metric, where the only nonvanishing multipole moment is the mass. The multipolar structure of a Kerr BH can be elegantly written as [92] MBH + iSBH = M +1 (iχ ) ,

(6)

where χ := J /M 2 is the dimensionless spin and M = M00 ,

J = S10 ,

M = M0 ,

S = S0 .

(7)

Since the Kerr metric is axisymmetric, only multipole moments with m = 0 appear in Eq. (6). In addition, Kerr BHs have vanishing mass multipole moments when  is odd and vanishing current multipole moments when  is even. This is a consequence of the Kerr solution being axially and equatorially symmetric. This elegant simplicity does not hold for other types of compact objects. There are no physical reasons for compact objects to have this same multipolar structure, and nothing prevents them from being deformed even when nonrotating (e.g., there is no analog to Birkhoff’s theorem beyond spherical symmetry). There is also no argument for the multipolar structure to be such as to satisfy the axial and the equatorial symmetry. In the most general scenario, it can be argued that no symmetries are expected for ECOs and their multipolar structure will be rich and nontrivial. In general, one can summarize this statement by parametrizing the multipole moments of an ECO as ECO Mm = MBH + δMm ,

ECO Sm = SBH + δSm ,

(8)

where δMm and δSm are some model-dependent corrections to the mass and current multipole moments, whose value can be found by matching with the interior solution of the object or by other microphysical arguments. Generically, the most dominant smoking-gun signals of this “non-Kerrness” property are the current quadrupole moment S2 (which breaks the equatorial symmetry of the Kerr solution) and a complete mass quadrupole tensor with three independent components M2m for m = 0, 1, 2 (which breaks the axisymmetry of the Kerr solution).

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Some particular ECO solutions provide specific examples of this complex multipolar structure (see Fig. 2), as in the case of multipolar boson stars [96] and fuzzball microstate geometries [21, 22, 34, 36]. “Soft” ECOs motivated by some new physics effect whose new scale L is comparable to the mass (and for which the curvature at the surface is comparable to the corresponding horizon curvature) cannot have arbitrarily large deviations from the BH multipole moments. The “softness” property of these ECOs requires that the multipole moment deviations vanish in the BH limit sufficiently fast [169]. The characteristic vanishing behavior depends on the nature of the multipole moments. For axisymmetric spacetimes, spin-induced moments must vanish logarithmically (or faster), while nonspin-induced moments vanish linearly (or faster) δM χ → a + b  . . .  log  M +1

(9)

and equivalently for the current multipole moments. In the expression above, a and b are numbers of order unity and the ellipsis stands for subleading terms. This particular behavior suggests that any multipole moment detection will be dominated by the spin-induced quadrupole instead of any generic intrinsic quadrupole moment, unless the angular momentum is sufficiently low:

Fig. 2 Representation of the multipolar structure of different compact objects in isolation. The shape describes the embedding plot of constant t and r surfaces of the metric, while the colors were weighted according to gtφ to represent the leading current multipole moments. BHs (on the left) are described by the Kerr metric and have a unique multipolar structure. Nonrotating BHs are spherical, while rotating BHs have an oblate shape. In addition, BHs have the remarkable feature that their moments are equatorially and axially symmetric. On the other hand, ECOs can have a richer multipolar structure. Fuzzballs (on the right), a string-theory motivated horizonless compact object which can be interpreted as a multicenter (represented by the red dots) extension of the BH model, can have a nontrivial multipolar structure, and in general scenarios, their multipole moments can break both the equatorial and axial symmetry [34, 36, 167]

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χ



| log | .

(10)

The multipole moments can be measured through GW observations of binary coalescences. The multipole moments of compact objects in a binary system affect the point-particle phase of the emitted GW signal. The dominant term for this effect is the quadrupole moment M2 which enters at 2PN order as [112]

ψ=2

 (1) (2) 75 m2 M2 + m1 M2 1 , = 64 v (m1 m2 )2

(11)

(i)

with mi the mass of the i-th body, M2 its (m = 0) quadrupole moment, and v the orbital velocity. Putting constrains on the quadrupole provides a way to test the nature of compact objects already at 2PN level. Constraints on parametrized PN deviations using GW events [2, 3, 7] can be mapped into a constraint on δM20 , in particular using light binaries which perform many cycles in band before merger. However, such tests are challenging due to the fact that the 2PN term in the GW phase depends also on the binary component spins, which have not been measured accurately so far. This introduces correlations between the spin and the quadrupole moment. While current GW constraints will become (slightly) more stringent in the next years as the sensitivity of the ground-based detectors improves [104, 105, 111, 113], much tighter bounds will come from extreme mass-ratio inspirals (EMRIs), one of the main targets of the future space mission LISA [16]. Although EMRI data analysis is challenging [17,59,60,64], a detection of these systems can be potentially used to measure the (m = 0, mass) quadrupole moment δM2 of the central supermassive object with an accuracy of one part in 104 [17, 19] and of a large set of high-order multipole moments [105], offering unprecedented tests of the nature of supermassive objects [71, 86, 169].

Tidal Heating If the binary’s components are dissipative systems, energy and angular momentum will be dissipated in their interior in addition to the GW emission to infinity. For BHs, energy and angular momentum absorption by the horizon is responsible for tidal heating [94,100]. This effect is particularly significant for highly spinning BHs and in the latest stages of the inspiral, since it enters the GW phase at 2.5PN × log v order (4PN × log v order) for spinning (nonspinning) binaries. In the absence of the log v term, such corrections would be completely degenerate with the time and phase of coalescence in the waveform and therefore unmeasurable. They anyway remain mildly correlated with other parameters and therefore hard to measure. For LISA binaries, constraints of the amount of dissipation would be stronger for highly spinning objects and for binaries with large mass ratios [70, 132].

28 Testing the Nature of Dark Compact Objects with Gravitational Waves

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On the other hand, tidal heating can contribute to thousands of radians of accumulated orbital phase for EMRIs in the LISA band [27, 69, 70, 93, 100, 180]. This would allow to distinguish between binary BHs and binary involving other compact objects (for which tidal dissipation is often negligible). For EMRIs in the LISA band, this effect could be used to put a very stringent upper bound on the reflectivity of ECOs, at the level of 0.01% [70].

Tidal Deformability As the two compact objects in an inspiraling binary approach each other, tidal effects become increasingly relevant. The gravitational field of each object acts as a tidal field on its companion, deforming its shape and inducing some multipolar deformation in the spacetime. This effect can be quantified in terms of “tidalinduced multipole moments.” A weak tidal field can be decomposed into the electric (or polar) tidal field moments Em and the magnetic (or axial) tidal field moments Bm . In the nonrotating case, the ratio between the multipole moments and the tidal field moments that induces them defines the tidal deformability of the body: ()

λE =

Mm , Em

()

λB =

Sm . Bm

(12)

These quantities are independent of m and of the external tidal field (if the latter is sufficiently weak). It is convenient to introduce the dimensionless tidal Love numbers (TLNs) kE and kB : ()

kE = const

λE , M 2+1

()

kB = const

λB , M 2+1

(13)

where different (dimensionless) constant prefactors have been used in the literature, depending on the adopted conventions. When the object is rotating, the angular momentum couples with the electric (resp. magnetic) tidal field moments to induce current (resp. mass) multipole moments on the body according to a set of selection rules [114, 159, 160, 163]. This effect allows us to define some new classes of “rotational TLNs” for rotating objects, which have no Newtonian analog. A remarkable result in GR is that the TLNs of BHs are precisely zero. This was first demonstrated for nonrotating BHs [37, 67, 90] and then extended for slowly rotating BHs [114, 160, 163], but more recently it has been extended to Kerr BHs without any approximations [57] (see also Refs. [115, 116]). This characteristic behavior is associated to the regularity conditions for tidal perturbations at the horizon. Thus, due to the absence of a horizon, the TLNs of ECOs will in general be nonzero and in principle can provide a smoking-gun test of the nature of dark ultracompact objects [52].

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The TLNs were explicitly computed for different models of ECOs such as boson stars [52, 140, 177], gravastars [52, 154, 190], anisotropic stars [168], and other simple ECOs with stiff EOS at the surface [52]. As expected, it was found that the TLNs are generically nonzero and vanishing in the BH limit. For the particular class of “stiff” ECOs, i.e., with Robin-type boundary conditions at the surface [133] (e.g., due to a perfectly reflective surface), it was found that the TLNs exhibit a logarithmically vanishing behavior in the BH limit [52]: kECO →

a , 1 + b log()

→0

(14)

where a and b are some model-dependent ∼O(0.01 − 1) dimensionless constants. This remarkable logarithmic dependence suggests that, although small, the TLNs are not infinitesimally vanishing even for ultracompact objects. For spherical objects with surfaces located at Planckian distances from the corresponding horizon location, this magnifying effect yields k2 ∼ O(10−3 ). In Fig. 3, we show the leading (electric and magnetic) TLNs for these models of ECOs. In addition to this behavior, we see that the TLNs of this class of ECOs exhibit an interesting isospectrality in the BH limit, i.e., polar and axial Love numbers coincide for ultracompact configurations ( → 0). In contrast, other ECOs such as anisotropic stars with a “smoother” transition between the vacuum external region and the interior fluid matter show some approximately polynomial vanishing behavior in the BH limit [168]: kECO → a

  n , M

(15)

where a and n are some model-dependent parameters.

Fig. 3 TLNs of “stiff” ECOs (left panel) [52] and (scalar) TLNs of an anisotropic star model dependent on the anisotropic scale (right panel) [168]. The latter depends on a parameter C¯ that controls the anisotropy scale of the model (the isotropic limit is obtained by taking C¯ → 0). The behavior of the TLNs depends on the particular boundary conditions at the surface. ECOs with Robin-type boundary conditions at the surface (left) show a logarithmic vanishing behavior in the BH limit, while the TLNs of anisotropic stars vanish with an approximately polynomial relation in this regime (right). Note that in the former case, the axial and polar TLNs coincide as  → 0

28 Testing the Nature of Dark Compact Objects with Gravitational Waves

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Similarly to the effect of multipole moments discussed above, the effect of tidal deformability alters the GW signal of a compact object inspiral by adding a 5PN correction to the GW phase: ψTD = −ψN

624Λ 10 v , m5

(16)

where Λ is the weighted tidal deformability that contains the contribution of the  = 2 polar tidal deformability of both compact objects in the binary. Other TLNs enter the waveform at even higher order. Current (and especially future) GW interferometers could measure the TLNs of BH mimicker in order to distinguish it from a BH [52,132,177]. In the comparablemass case, a precise measurement requires highly spinning supermassive ECO binaries up to 10 Gpc detectable by LISA. LISA may also be able to perform model selection between different families of BH mimickers [133], although this will in general require detection of golden binaries, with very large signal-to-noise ratio [11]. Finally, EMRI observations can set even more stringent constraints, since the measurement errors on the Love number scale as q 1/2 , where q  1 is the mass ratio of the binary, potentially constraining the Love number of the central object within one part in 105 [156]. Typically the effects discussed above are included independently in inspiral waveform templates. However, for concrete models like boson stars, it is possible to consistently include several corrections (multipolar structure, tidal heating, TLNs) in the inspiral signal, improving the accuracy on the measurement of the (fewer) free parameters of the template [151].

Ringdown The ringdown is the final stage of a compact binary coalescence when the remnant relaxes to an equilibrium solution. When the remnant is a BH, the ringdown is dominated by its complex characteristic frequencies, the so-called quasi-normal modes (QNMs), which describe the response of the compact object to a perturbation [28, 56, 109, 117, 165, 182, 183]. In the linear regime, the spacetime metric can (0) (0) be written as gμν = gμν + hμν , where gμν is the background metric of the compact object and hμν  1 is the perturbation. The BH ringdown signal can be modeled as a linear superposition of exponentially damped sinusoids h=



Amn (r)e−t/τmn sin(ωmn t + φmn ) −2 Ym (θ, ϕ) ,

(17)

mn

where ωmn are the characteristic frequencies of the remnant, τmn are the damping times, Amn (r) ∝ 1/r is the amplitude of the signal at a distance r, φmn is the phase, and s Ym (θ, ϕ) ∝ eimϕ are the spin-weighted spheroidal harmonics which depend on the location of the observer with respect to the source. Each mode is

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described by three integers, namely, the angular number ( ≥ 0), the azimuthal number m (such that |m| ≤ ), and the overtone number (n ≥ 0). From the detection of the ringdown, it is possible to infer the QNMs of the remnant and understand the nature of compact objects. Due to the GR uniqueness theorem, the QNM spectrum of Kerr BHs depends uniquely on two parameters, i.e., their mass and angular momentum [54, 172]. As a consequence, a test of the Kerr hypothesis would require the identification of at least two QNMs, whereas a detection of several modes would allow for multiple independent tests of the null hypothesis. Up to date, the least-damped QNM ( = m = 2, n = 0) has been observed in the ringdown of several GW events and is compatible with a Kerr BH remnant with the mass and the spin predicted by the inspiral [1, 2, 7, 78]. Few loud GW signals show evidence for the first overtone ( = m = 2, n = 1) whose frequency measurements could set constraints on GR [84,101] (but see [31,103,150] for a related discussion), whereas the damping times are less constrained [7, 78]. Third-generation groundbased detectors and the future LISA mission will allow for unprecedented tests of the BH paradigm given the expected large signal-to-noise ratio in the ringdown [16, 29, 128, 171]. ECOs can be distinguished from BHs from their different linear response. For simplicity, let us analyze a static ECO whose exterior spacetime, assuming GR as a reliable approximation, is described by the Schwarzschild metric (see Ref. [125] for the extension to the spinning case): 1 dr 2 + r 2 (dθ 2 + sin2 θ dϕ 2 ) , f (r)

ds 2 = −f (r)dt 2 +

(18)

where f (r) = 1 − 2M/r and M are the total mass of the compact object. In order to derive the QNM spectrum of the ECO, let us perturb it with a spin-s perturbation where s = 0, ±1, ±2 for scalar, electromagnetic, and gravitational perturbations, respectively. The perturbation can be decomposed as Ψs (t, r, θ, ϕ) =



s Ym (θ, ϕ) s ψm (r)e

−iωt

,

(19)

m

where in the following we will omit the s, , m subscripts for brevity. The radial component of the perturbation is governed by a Schrödinger-like equation [170, 199, 200]: d 2 ψ(r)  2 + ω − V (r) ψ(r) = 0 , dr∗2

(20)

where r∗ is the tortoise coordinate r∗ = r + 2M log

 r −1 , 2M

(21)

28 Testing the Nature of Dark Compact Objects with Gravitational Waves

1153

and the effective potential is  ( + 1) 2 2M , + (1 − s ) r2 r3 

2 q (q + 1)r 3 + 3q 2 Mr 2 + 9M 2 (qr + M) , = 2f r 3 (qr + 3M)2

Vaxial = f

(22)

Vpolar

(23)

where q = ( − 1)( + 2)/2. The potential in Eq. (22) describes scalar, electromagnetic, and axial gravitational perturbations, whereas the potential in Eq. (23) describes polar gravitational perturbations. The effective potential as a function of the tortoise coordinate is shown in Fig. 4 for axial gravitational perturbations. It has a maximum approximately at the photon sphere r ≈ 3M, which is the unstable circular orbit of photons around the compact object. In the ECO case, the absence of the event horizon at r∗ → −∞ implies the existence of a cavity between the ECO radius and the photon sphere. The cavity can support long-lived trapped modes which are responsible for a completely different QNM spectrum with respect to the BH case. In order to derive the QNMs of ECOs, let us impose boundary conditions at infinity and at the radius of the ECO. Indeed, Eq. (20) with the addition of two boundary conditions defines an eigenvalue problem whose complex eigenvalues are the QNMs of the object, ω = ωR + iωI . According to the convention in Eq. (19), a stable mode has ωI < 0, whereas an unstable mode has ωI > 0, with damping (instability) timescale τdamping (inst) ≡ 1/|ωI | for the former (latter) case. At infinity we impose that the perturbation is a purely outgoing wave:

M 2V

0.15

Blackhole

0.10 0.05

M 2V

0.00 0.15

ECO

0.10 0.05 0.00 40

20

0

20

40

r M Fig. 4 Effective potential for axial gravitational perturbations of a Schwarzschild BH (top panel) and an ECO with  = 10−6 (bottom panel). The effective potentials have a maximum approximately at the photon sphere, r ≈ 3M. In the ECO case, the effective potential features a cavity between the radius of the ECO and the photon sphere [47, 50, 51]

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ψ ∼ eiωr∗ ,

as r∗ → +∞ .

(24)

In the BH case, the event horizon would require that the perturbation is a purely ingoing wave as r∗ → −∞ (r → r+ = 2M). In the ECO case, the solution is generally a superposition of ingoing and outgoing waves: ψ ∼ Cin e−iωr∗ + Cout eiωr∗ ,

as r∗ → r∗0 ,

(25)

where r∗0 ≡ r∗ (r0 ) and r0 = 2M(1 + ). The surface reflectivity of the ECO is then defined as [124] R=

Cout 2iωr∗0 e . Cin

(26)

A perfectly reflecting ECO has |R|2 = 1, whereas a totally absorbing compact object has |R|2 = 0 as in the BH case. The boundary conditions that describe a perfectly reflecting ECO are [124, 125] ψ(r0 ) = 0 Dirichlet on axial , dψ(r0 )/dr∗ = 0 Neumann on polar ,

(27) (28)

where for the Dirichlet (Neumann) boundary condition, the waves are totally reflected with inverted phase R = −1 (in phase R = 1). Figure 5 shows the QNM spectrum of a perfectly reflecting ECO compared to the fundamental  = 2 gravitational QNM of a Schwarzschild BH, where  ∈ (10−2 , 10−10 ) from the right to left of the plot. As shown in Fig. 5, an important feature of ECOs is the breaking of isospectrality between axial and polar modes unlike BHs in GR [28, 53, 127]. Furthermore, as  → 0, the deviations from the BH QNM are arbitrarily large and the QNMs are low frequencies (MωR  1) and long-lived (τdamping  1) [50]. For   1, the QNMs can be derived analytically in the low-frequency regime [47, 125, 193]: 

s(s + 1) Mπ , MωR  − 0 q + 2 2|r∗ | MωI  −βs

M (2MωR )2+2 , |r∗0 |

(29) (30)

where q is a positive odd (even) integer for polar (axial) modes and βs = 2  (−s)!(+s)! [178]. Low-frequency modes are trapped in the cavity of the effective (2)!(2+1)!! potential between the ECO radius and the photon-sphere barrier. The real part of the QNMs scales with the width of the cavity as ωR ∼ | log |−1 , whereas the imaginary part of the QNMs depends on the tunneling probability through the potential and scales as ωI ∼ −| log |−(2l+3) .

28 Testing the Nature of Dark Compact Objects with Gravitational Waves

1155

Fig. 5 QNM spectrum of a perfectly reflecting ECO with  ∈ (10−2 , 10−10 ) compared to the fundamental  = 2 gravitational QNM of a Schwarzschild BH. Axial and polar modes are not isospectral at variance with the BH case. As  → 0, the ECO QNMs are low frequencies and long-lived [50]

We note that Eq. (25) is valid only in a region where the effective potential vanishes, so it requires   1. In order to define the ECO boundary condition more generally, one can make use of the BH membrane paradigm and generalize it to the case of horizonless compact objects. According to the BH membrane paradigm, a static observer outside the horizon can replace the interior of a perturbed BH by a fictitious membrane located at the horizon [66, 166, 185]. The properties of the membrane are fixed by the Israel-Darmois junction conditions [68, 102]: [[Kab − Khab ]] = −8π Tab ,

[[hab ]] = 0 ,

(31)

where hab is the induced metric on the membrane, Kab is the extrinsic curvature, K = Kab hab , Tab is the stress-energy tensor of the membrane, and [[...]] is the jump of a quantity at the membrane. The fictitious membrane is such that the extrinsic curvature of the interior spacetime vanishes. Consequently, the junction conditions impose that the fictitious membrane is a viscous fluid with the stress-energy tensor [185]: Tab = ρua ub + (p − ζ Θ)γab − 2ησab ,

(32)

where η and ζ are the shear and the bulk viscosities of the fluid; ρ, p, and ua are the density, the pressure, and the three-velocity of the fluid; Θ = ua;a is the expansion;   σab = 12 ua;c γbc + ub;c γac − Θγab is the shear tensor; γab = hab + ua ub is the projector tensor; and the semicolon is the covariant derivative compatible with the induced metric, respectively.

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The BH membrane paradigm allows to describe the interior of a perturbed BH in terms of the shear and bulk viscosities of a fictitious fluid located at the event horizon. The generalization of the BH membrane paradigm to horizonless compact objects allows to analyze several models of ECOs with different internal structures in terms of the properties of a fictitious membrane located at the ECO radius [10, 127]. The shear and the bulk viscosities are generically complex and frequency-dependent and are related to the reflective properties of the ECO. In particular, for each model of ECO, the shear and the bulk viscosities are uniquely determined. In the following, we shall focus on models of ECOs which are described by a Schwarzschild exterior. The junction conditions in Eq. (31) with the stress-energy tensor in Eq. (32) allow to derive generally the boundary conditions at the ECO radius (for details on the derivation, see Ref. [127]): r 2 Vaxial (r0 ) dψ(r0 )/dr∗ iω =− − 0 , ψ(r0 ) 16π η 2(r0 − 3M)

axial ,

(33)

dψ(r0 )/dr∗ = −16π iηω + F (r0 , ω, η, ζ ) , ψ(r0 )

polar ,

(34)

where F (r0 , ω, η, ζ ) is a cumbersome function given in Ref. [127]. Let us notice that in the BH limit, the boundary conditions in Eqs. (33), (34) reduce to the BH case. Indeed, according to the BH membrane paradigm, ηBH =

1 . 16π

(35)

For η → ηBH and r0 → 2M ( → 0), both the axial and polar boundary conditions in Eqs. (33) and (34) describe a purely ingoing wave as in the BH case. Moreover for η = 0 and   1, the boundary conditions in Eqs. (33), (34) reduce to Dirichlet and Neumann boundary conditions on axial and polar modes, respectively, as in Eqs. (27) and (28), thus describing a perfectly reflecting ECO. The case of a partially absorbing surface is analyzed by considering η ∈ (0, ηBH ). Interestingly as the ECO radius approaches the photon sphere, r0 → 3M, the axial boundary condition reduces to ψ(r0 ) = 0 for any η ∈ C. As a consequence, an ECO with r0 = 3M is a perfect reflector of axial modes regardless of its interior structure. The same universality does not occur in the polar sector. Figure 6 shows the ratio between the QNM frequencies of an ECO with the same reflective properties of a BH (η = ηBH , ζ = ζBH ≡ −1/(16π )) and the fundamental  = 2 QNM of a Schwarzschild BH as a function of the ECO compactness. As  increases, the ECO QNMs start deviating from the BH QNM. The highlighted regions in Fig. 6 correspond to the maximum allowed deviation (with 90% credibility) for the least-damped QNM in the event GW150914 [1] with respect to the Kerr BH case and correspond to ∼16% and ∼33% for the real and the imaginary part of the QNM, respectively [78]. Horizonless compact objects with

28 Testing the Nature of Dark Compact Objects with Gravitational Waves 2.5

1.1

Axial Polar BH

BH BH,

BH

2.0

I

0.9

I

R

R

BH

1.0

0.8

Axial Polar

0.7 0.6 10

1157

4

0.001

1.5

BH BH,

BH

0.010

1.0 0.100

1

10

4

0.001

0.010

0.100

1

Fig. 6 Real (left panel) and imaginary (right panel) part of the QNMs of an ECO with the same reflective properties of a BH (described by a fictitious fluid with shear viscosity η = ηBH and bulk viscosity ζ = ζBH ) compared to the fundamental l = 2 gravitational QNM of a Schwarzschild BH, as a function of the ECO radius, where r0 = 2M(1 + ) [127]. The highlighted region corresponds to the maximum deviation (with 90% credibility) for the least-damped QNM in the event GW150914 [1] with respect to the Kerr BH case [78]

  0.1 are compatible with current measurements. Future ringdown detections will allow us to set more stringent constraints on the nature of compact objects.

Ergoregion Instability of ECOs In a stationary spacetime, the ergoregion is the region where the Killing vector field ∂t = (1, 0, 0, 0) becomes spacelike. The outer boundary of the ergoregion (sometimes called the ergosphere) coincides with the horizon for a Schwarzschild BH, whereas for a Kerr BH, the ergoregion extends outside the horizon. In the ergoregion, no static observers can exist, and negative-energy states are allowed. This is the key mechanism of the Penrose’s process which allows to extract energy and angular momentum from a Kerr BH [162]. Let us consider a particle with energy at infinity E0 decaying in two particles inside the ergoregion. One of the two particles can have a negative energy E1 < 0, and therefore the other one must have energy larger than the initial value, E2 > E0 . The event horizon forces the negative-energy particle to fall into the BH, while the positive-energy particle can escape at infinity and extract energy from the BH [39]. Spinning compact objects with an ergoregion but without an event horizon is prone to the so-called ergoregion instability. Assuming the horizonless object is non-dissipative, the negative-energy particle remains in orbital motion inside the ergoregion, since it cannot be absorbed. It is therefore energetically favorable to cascade toward more negative-energy states leading to a runaway instability. This infinite cascade can be prevented only if the compact object can efficiently absorb the negative-energy states. The ergoregion instability was proven by Friedman in ultracompact stars under scalar and electromagnetic perturbations [76] and analyzed in uniform-density stars [39, 62, 110, 143, 198], gravastars [58], boson stars [48], superspinars [157], and

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ultracompact Kerr-like ECOs [49,124,125]. The origin of the ergoregion instability in horizonless ultracompact objects is due to the existence of long-lived modes. As shown in Fig. 5, the imaginary part of the QNM frequencies of a static ECO tends to zero in the limit of large compactness ( → 0). In the rotating case, these modes can turn unstable due to the Zeeman splitting of the frequencies as function of the azimuthal number. In the small-spin limit, the QNM frequencies can be written as [158] (0)

(1)

ωR,I = ωR,I + mχ ωR,I + O(χ 2 ) ,

(36)

(0) are the real and the imaginary parts of the QNM frequencies in the static where ωR,I (1) case and ωR,I are the first-order corrections to the QNM frequencies in the spin. For (0)

an ultracompact ECO with   1, ωI ∼ 0 and the first-order correction in Eq. (36) can turn the sign of the imaginary part of the frequency to be positive for a certain value of the azimuthal number. The symmetries m → −m, ω → −ω∗ guarantee that the ergoregion instability generically affects ECOs above a critical value of the spin. Figure 7 shows the fundamental gravitational ( = m = 2) QNM frequencies of a perfectly reflecting Kerr-like ECO

as function of the spin with a given radius r0 = r+ (1 + ), where r+ = M(1 + 1 − χ 2 ) and  = 10−10 [125]. The real part of the QNM frequency has a zero crossing at some critical value of the spin which depends on the axial or polar nature of the perturbation. Most importantly, the imaginary part of the QNM frequency changes sign above the same critical values of the spin, turning the ECO from stable into unstable. For perfectly reflecting Kerr-like ECOs with   1, the critical value of the spin can be computed analytically from the generalization of Eqs. (29), (30) to the spinning case which are accurate when ωR  ωI  0 (for details, see Refs. [47, 125]). The ergoregion instability occurs for [47, 125]

Fig. 7 Real (left panel) and imaginary (right panel) part of the fundamental gravitational ( = m = 2) QNM of a Kerr-like ECO as a function of the spin. The surface of the ECO is located at r0 = r+ (1 + ), where  = 10−10 , and is perfectly reflecting, |R |2 = 1. The ECO is affected by an ergoregion instability above a critical value of the spin that differs for axial and polar perturbations [125]

28 Testing the Nature of Dark Compact Objects with Gravitational Waves

χ > χcrit

  π s(s + 1) ∼ q+ , m| log | 2

1159

(37)

where q is a positive even (odd) integer for axial (polar) modes. For example, for an ECO with Planckian corrections at the horizon scale ( = 10−40 ), χcrit  0.03, 0.05 for gravitational  = m = 2 axial and polar perturbations, respectively. We conclude that even slowly spinning ECOs are unstable due to the ergoregion instability. The timescale of the instability is defined as τinst ≡ 1/|ωI |. From Fig. 7, for an ultracompact ECO with  = 10−10 and spin χ = 0.7, the instability timescale of the  = m = 2 mode is 

M s, (38) τinst ∈ (5, 7) 10M where the lower (upper) bound is for polar (axial) perturbations. Let us notice that the low-frequency approximation of ωI is not accurate for χ = 0.7, since MωR ∼ 0.3. The ergoregion instability acts on a timescale which is short compared to the accretion timescale of astrophysical BHs, i.e., τSalpeter ∼ 4 × 107 yr. However, the instability timescale is longer than the decay time of the ringdown of BHs, i.e., τringdown ∼ 0.5 ms for a 10M object, and in general it is parametrically longer than the light-crossing time of the object. If the remnant of a compact binary coalescence was an ECO, the ergoregion instability could spin down the remnant over a timescale τinst until the condition for the stability, χ = χcrit , is satisfied [39]. This process would lead to a stochastic GW background due to spin loss [73,74]. The absence of such background in the first observing run of Advanced LIGO already puts strong constraints on perfectly reflecting ECOs which can be a small percentage of the astrophysical population [20]. One way of quenching the ergoregion instability is by assuming that the surface of the ECO is partially absorbing [124]. This model is more realistic than a perfectly reflecting surface since the compact object can absorb part of the radiation through viscosity, dissipation, fluid mode excitation, etc. The minimum absorption rate to have a stable ECO is related to the maximum amplification factor of BHs [125]. In order to derive this result, let us analyze a spin-s perturbation in the background of a spinning ECO. It is convenient to introduce the Detweiler’s function which is governed by the master equation [72]: d 2 Ψ (r) − V (r, ω)Ψ (r) = 0 , dr∗2

(39)

where the effective potential reads V (r, ω) = with

(r 2

UΔ dG + G2 + , 2 2 dr∗ +a )

(40)

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G=

s(r − M) rΔ + 2 , r 2 + a2 (r + a 2 )2

U = VS + VS = −

(41)

2α + (β Δs+1 ) , βΔs

(42)

1  2 K − isΔ K + Δ(2isK − λs ) Δ

(43)

where Δ = r 2 −2Mr+a 2 , K = (r 2 +a 2 )ω−am, the prime denotes a derivative with respect to r, α and β are chosen such that the potential in Eq. (40) is real [72, 125] and a = χ M. The two independent solutions of Eq. (39) have asymptotic behavior: 

e+iωr∗ Ψ˜ + (ω, r∗ ) ∼ Bout (ω)e+ikr∗ + Bin (ω)e−ikr∗ Ψ˜ − (ω, r∗ ) ∼

 Aout (ω)e+iωr∗ + Ain (ω)e−iωr∗ e

−ikr∗

as r∗ → +∞ as r∗ → −∞ as r∗ → +∞ as r∗ → −∞

,

(44)

,

(45)

˜ ˜ where the Wronskian of the solutions is conserved WBH = ddrΨ∗+ Ψ˜ − − Ψ˜ + ddrΨ∗− = 2ikBout , k = ω − mΩ, and Ω = χ /(2r+ ) is the angular velocity of a Kerr BH at the event horizon. Let us define the reflection and transmission coefficients of a wave coming from the left of the photon-sphere barrier with unitary amplitude as

RBH =

Bin , Bout

TBH =

1 . Bout

(46)

As shown in Fig. 8, after each bounce in the cavity between the ECO surface and the photon sphere, the perturbation acquires a factor RRBH , where R is the ECO surface reflectivity and RBH is defined in Eq. (46). Due to the conservation of the Wronskian, |RBH | = |Aout /Ain | where Ain and Aout are the coefficients of the incident and reflected wave, respectively, at the photon sphere for a left-moving wave originating at infinity. The latter coefficients are related to the amplification factor of BHs for a wave of spin s by Zslm

   Aout 2  − 1.  = Ain 

(47)

The condition for the energy in the cavity to grow indefinitely is |RRBH |2 > 1 which implies that the object is unstable due to the ergoregion instability if |R|2 >

1 . 1 + Zslm

(48)

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Fig. 8 Schematic diagram for the wave propagation in the spacetime of an ECO [9, 47, 193]

Since the surface reflectivity is defined to be |R|2 ≤ 1, Eq. (48) implies that the ergoregion instability occurs when the real part of the QNM frequency is in the superradiant regime, i.e., Zslm > 0. In order to quench the ergoregion instability at any frequencies, the surface absorption, 1 − |R|2 , needs to be larger than the maximum amplification factor of superradiance, namely, 1 − |R|2  Zmax ,

(49)

where Zmax  1. Figure 9 shows the amplification factor of a BH as a function of the frequency under scalar, electromagnetic, and gravitational perturbations and for several values of the BH spin. In order to have a stable spinning ECO under any type of perturbation, the surface absorption needs to be at least 0.3% (6%) for an ECO with χ = 0.7 (χ = 0.9). Let us notice that the maximum amplification factor of an extremal BH is ≈138% for  = m = 2 gravitational perturbations [39, 183]; therefore, an absorption rate of ≈60% would allow for stable ECOs with any spin [125]. Some models of quantum BHs have a frequency-dependent reflectivity R = e−|k|/TH , where TH is the Hawking temperature, which allows for stable spinning solutions against the ergoregion instability [149].

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Fig. 9 Superradiant amplification factor of a BH as a function of the frequency for ( = m = 1) scalar, electromagnetic, and ( = m = 2) gravitational perturbations. The minimum absorption rate to have a stable ECO for any type of perturbation is 0.3% (6%) for χ = 0.7 (χ = 0.9) [125]

Gravitational-Wave Echoes GW echoes are an additional signal that would be emitted in the postmerger phase of a compact binary coalescence when the remnant is a horizonless ultracompact object [51, 75, 108, 109] (see also [14, 15, 186, 187] for related studies). Possible sources of GW echoes are near-horizon quantum structures [50, 51, 196], ultracompact neutron stars [75, 155], and BHs in modified theories of gravity in which the graviton reflects effectively on a hard wall [148,201]. The key features of the sources of GW echoes are the existence of a photon sphere in the exterior spacetime and the absence of an event horizon [46, 47]. As shown in Fig. 4, the effective potential of a perturbed ultracompact object displays a maximum near the photon sphere. If the radius of the object is smaller than the photon sphere, the effective potential features a cavity. If sufficiently compact, the latter can support quasi-trapped modes that leak out of the potential barrier through tunneling effects and are responsible for the GW echoes [50]. In order to describe the dynamical emission of GW echoes, let us analyze the scattering of a Gaussian pulse by the compact object. When the pulse crosses the

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photon sphere and perturbs it, a prompt ringdown signal is emitted at infinity as shown in Fig. 8 [9, 47, 193]. The prompt ringdown emitted by an ultracompact horizonless object is almost indistinguishable from the BH ringdown because the photon sphere is approximately at the same location and has a similar shape [50]. Afterward, the perturbation travels inside the photon-sphere barrier and is reflected by the surface of the compact object. A fraction of the radiation is absorbed by the compact object depending on its reflective properties [42, 125, 149]. After each interaction with the photon sphere, a GW echo is emitted at infinity with a progressively smaller amplitude. The photon-sphere barrier acts as a frequencydependent high-pass filter. In particular, the characteristic frequencies governing the prompt ringdown are very similar to the BH QNM frequencies, even if the latter are not part of the spectrum of horizonless ultracompact objects. The frequencies governing each subsequent GW echo become progressively smaller, and at late times the GW signal is dominated by the low-frequency QNMs of the ECO [126, 130, 195]. The morphology of the GW echoes gives us information about the properties of the ECO, in particular its compactness and its reflectivity. The delay time between subsequent GW echoes is associated with the light-crossing time and depends logarithmically on the compactness of the ECO [50, 51]. For a nonspinning object (see Ref. [9] for the spinning case), τecho = 2M (1 − 2 − 2 log ) ;

(50)

therefore, the more the object is compact (  1), the (logarithmically) longer is the time delay between GW echoes. In principle, the logarithmic dependence on  would allow to detect even Planckian corrections ( ∼ Planck /M) at the horizon scale few ms after the merger for a remnant with M ∼ 10M . However, a crucial parameter that regulates the morphology of the echo signal (in particular the damping factor between subsequent echoes) is the reflectivity, as shown in the left panel of Fig. 10 [127]. For a perfectly reflecting ECO, the relative amplitude of the GW echoes is maximum, whereas it decreases for a partially absorbing ECO, and it goes to zero in the limit of a perfectly absorbing object as in the BH case. When the remnant of a merger is an ECO with   0.01, the delay time of the GW echoes is comparable with the decay time of the prompt ringdown where the latter is associated to the decay time of the fundamental QNM of a BH, τdamping ≈ 10M. As a consequence, the prompt ringdown emitted by the direct excitation of the photon sphere interferes with the first GW echo in an involved pattern as shown in the right panel of Fig. 10 [127]. In particular when the two pulses sum in phase, the interference produces high peaks in the GW signal. Furthermore, subsequent echoes are suppressed because the cavity between the photon sphere and the radius of the compact object is small and does not trap the modes efficiently. Let us derive the gravitational waveform that would be emitted by an ultracompact horizonless object in the ringdown. We require the GW signal emitted at infinity ˜ to be a purely outgoing wave, ψ(ω, r∗ → ∞) = Z˜ + (ω)eiωr∗ . In frequency domain,

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Fig. 10 Left panel: GW echoes emitted by a static ultracompact horizonless object with a given compactness (  5 × 10−16 ) and several values of the surface reflectivity (R = 1, 0.5). Right panel: ringdown of an ECO with radius r0 = 2M(1 + ) and the same reflective properties of a BH [126, 127]

the GW signal emitted by an ECO can be written in terms of the GW signal that would be emitted by a BH and reprocessed by a transfer function. In particular [130], + − (ω) + K (ω)Z˜ BH (ω) , Z˜ + (ω) = Z˜ BH

(51)

± where Z˜ BH (ω) are the responses of a BH at infinity and near the horizon, for the plus and minus sign, respectively, to a source S˜ : ± (ω) = Z˜ BH

1 WBH



+∞

−∞

dr∗ S˜ Ψ∓ ,

(52)

where Ψ± are the independent solutions of the homogeneous equation (39) with asymptotics in Eqs. (44), (45). The transfer function is defined as [47, 130] TBH R(ω)e−2ikr∗

0

K (ω) =

1 − RBH R(ω)e−2ikr∗

0

(53)

.

According to Eq. (51), the GW signal emitted by an ECO is the same as the one emitted by a BH with an extra GW emission that depends on the reflectivity of the ECO. In order to get an insight of the additional GW emission, let us expand the transfer function in Eq. (53) as a geometric series [65, 130]: K (ω) = TBH R(ω)e−2ikr∗

0

∞ 

[RBH R(ω)]j −1 e−2i(j −1)kr∗ . 0

(54)

j =1

In view of Eq. (54), the GW signal takes the form of a series of pulses where the index j represents the signal emitted by the j -th echo. The phase factor 2ikx0 corresponds to the time delay between each pulse due to the round-trip time between the photon sphere and the surface of the ECO. Each echo can have a phase inversion

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relative to the previous one when the factor RBH R(ω) has a negative sign [126,181]. Eq. (51) allows to construct a template for the GW signal emitted by an ECO that depends only on the black-hole ringdown parameters and the parameters of the ECO, i.e., its compactness and reflectivity. In time domain, the gravitational waveform is obtained by an inverse Fourier transform: 1 h(t) = √ 2π



+∞

−∞

dωZ˜ + (ω)e−iωt .

(55)

Several phenomenological templates have been developed in order to perform matched-filter searches of GW echoes [10, 47]. In time domain, some templates are based on standard GR ringdown templates with extra parameters that are related to the morphology of the GW echoes, i.e., the delay time and the damping factor [9, 146]. Several time-domain templates approximate the GW echoes by complex Gaussians [195] and by a superposition of sine-Gaussians with free parameters [131]. In frequency domain, some analytical templates depend explicitly on the physical parameters ECOs [130] and are obtained with analytical approximations of the transfer function in Eq. (53) [126, 181]. Some searches for GW echoes have been performed [10]. A tentative evidence for GW echoes has been claimed in the postmerger phase of compact binary coalescences detected by Advanced LIGO and Advanced Virgo in the first two observing runs [8, 9, 63]. However, the statistical significance of GW echoes has been claimed to be low and consistent with noise [147, 197]. Some independent searches confirmed to find no evidence for GW echoes based on morphologyindependent searches with a decomposition of the signal in terms of generalized wavelets [188,189] and template-based searches [121,191]. Moreover, no evidence for GW echoes has been found in the binary BH events from the GWTC-2 catalog [7] confirming previous results. Third-generation detectors like the Einstein Telescope [98,128], the Cosmic Explorer [171], and the Laser Interferometer Space Antenna [16] will allow to detect GW echoes even for objects with small reflectivity or to put strong constraints on ECO models, given the large signal-to-noise ratio in the ringdown of O(100) [126, 181].

Conclusions and Open Issues We conclude with a list of some of the most outstanding open problems in this area: • The formation channel of ECOs is mostly unmodeled, except for boson stars [120,176]. For ultracompact ECOs, it relies on the idea that some quantum effects can prevent the formation of the event horizon; however, there are no available simulations. • Beside the case of boson stars, very little is known about the dynamics of isolated and, especially, ECO binaries. Numerical simulations would be crucial to study

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the stability of ECOs as well as ECO mergers and to develop consistent inspiralmerger-ringdown waveform templates. • Many studies so far have assumed the ECO reflectivity to be constant, but in realistic models, R is a complex function of the frequency and the spin. Modeling this property is important for both the viability and the phenomenology of realistic models [42, 149]. • It would be interesting to extend the membrane paradigm for ECOs [127] to spinning objects and to other configurations. • Phenomenological studies of fuzzballs are in their infancy and should be extended in various directions; e.g., the ringdown, tidal effects, and the impact of the multipole moments on the inspiral waveform are uncharted territories. Acknowledgments We acknowledge financial support provided under the European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480. We also acknowledge support under the MIUR PRIN and FARE programs (GW-NEXT, CUP: B84I20000100001) and networking support by the COST Action CA16104 and support from the Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001).

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Contents Introduction: Why Quantum Gravity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic GW Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Dispersion Relation and Propagation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luminosity Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

We review the present status of quantum-gravity phenomenology in relation to gravitational waves (GWs). The topic can be approached from two directions, a model-dependent one and a model-independent one. In the first case, we introduce some among the most prominent cosmological models embedded in theories of quantum gravity, while in the second case we point out certain common features one finds in quantum gravity. Three cosmological GW observables can be affected by perturbative as well as non-perturbative quantum-gravity effects: the stochastic GW background, the propagation speed of GWs, and

G. Calcagni () Instituto de Estructura de la Materia – CSIC, Madrid, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_30

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the luminosity distance of GW sources. While many quantum-gravity models do not give rise to any observable signal, some predict a blue-tilted stochastic background or a modified luminosity distance, both detectable by future GW interferometers. We conclude that it is difficult, but still possible, to test quantum gravity with GW observations. Keywords

Quantum gravity · Theories beyond Einstein gravity · Phenomenology · Modified dispersion relations · Inflation · Stochastic GW background · GW propagation speed · Luminosity distance

Introduction: Why Quantum Gravity? Gravitational-wave (GW) astronomy is the new frontier of knowledge on astrophysics, the fundamental gravitational interaction and the properties of spacetime. In the recent history of gravitational physics, seldom have experiment and theory converged in such a synergy. So far, observations of mergers by the LIGO-Virgo network have confirmed all predictions of general relativity (GR) about GWs emitted by binary systems of black holes and other compact objects [1, 2], but there is still much to be explored about the nature of gravity. However, while GWs are acknowledged as a promising test of modifications of GR, their role in exploring such an extreme physics as quantum gravity is unclear [3–20]. This issue can have lasting repercussions, since the discovery of a signature of quantum gravity would change forever our way of understanding the fundamental interactions of Nature. The general expectation is that the low curvature, low energies, and large distances characterizing the production and propagation of GWs make it unlikely, or at least difficult, to probe Planck-size effects, unless they are cumulatively amplified by some cosmological mechanism. As we will see, in some isolated cases, such a mechanism is in action. Before entering into the topic of this chapter, let us briefly recall what quantum gravity is and why we are interested in it. Quantum gravity is a generic label denoting any theory unifying the gravitational force and quantum mechanics in a consistent way. Although there is no experimental evidence that gravity should be a quantum interaction, theoretically, one would expect all forces of Nature to follow about the same rules, while at the moment gravity is described with quite different tools than those employed in the Standard Model of electroweak and strong interactions. Sometimes, Gedankenexperimente are invoked [21, 22] to explain local observations [23] and show that it is not possible to have a purely classical gravitational force interact with quantum matter fields as in the semi-classical Einstein equations Gμν = κ 2 Tμν , where κ 2 = 2π G is Newton’s constant, Gμν is the Einstein tensor, and Tμν  is the expectation value of the energy-momentum tensor. However, one can change the quantum setting in such a way as to make classical gravity compatible with present observations [24]. Other circumstantial arguments against classical gravity are the presence of singularities (black-hole

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singularities and the big bang) where the laws of GR break down and the inability of the latter to explain satisfactorily the cosmological constant. Therefore, we have no compelling proof that quantum gravity is needed. Quantum gravity is not necessary to get a rich GW phenomenology either. Cosmological models beyond Einstein gravity such as f (R), Horndeski, DHOST, or infrared nonlocal models are not embedded in a fundamental gravitational quantum theory and, yet, deserve interest for their consequences in GW astronomy [9, 12, 19]. It is also worth mentioning that other types of deviations from the standard cosmological model, for instance, in the matter sector while leaving GR untouched, can also trigger a signal, for instance, as a large high-frequency amplitude of the stochastic GW background [25, 26]. Still, for many, it is more rewarding to extract phenomenology and predictions from a robust, consistent theoretical setting rather than from ad hoc scenarios which are either unfalsifiable or, if falsifiable or validated by observations, difficult to place inside a bigger picture. This research trend is further strengthened by the fact that quantum gravity is no longer a mirage and some concrete proposals exist [27–29], such as string theory, asymptotic safety, loop quantum gravity, group field theory, and nonlocal quantum gravity, just to name some. Due to lack of space, we cannot review all theories of quantum gravity, but we present a reduced list of scenarios where GW observables have been calculated. • Stelle gravity [30–33], a non-unitary, renormalizable quantum field theory of gravity on a continuous spacetime where the fundamental action contains secondorder curvature invariants (R 2 , Rμν R μν , Rμνσ τ R μνσ τ ). • String theory in its low-energy limit [34–36] and the corresponding cosmological models [37]. Here gravity is not quantized directly, but it emerges in the spectrum of quantized stringy fundamental blocks and is unified with the other Standard Model interactions. • Asymptotic safety [38–44], a non-perturbative quantization of gravity via the functional renormalization group approach. • Quantum gravities with discrete pre-geometries (LQG/SF/GFT), where spacetime emerges from a structure (“pre-geometry”) characterized by discrete labels (spin group representations). The two main theories under this paradigm are loop quantum gravity (LQG) [45, 46] and spin foams (SF) [47–49], possibly different manifestations of a more general setting known as group field theory (GFT) [27, 28, 50–53]. • Causal dynamical triangulations [54–61], a non-perturbative quantization of gravity via a path integral on discrete triangulated geometries. • Nonlocal quantum gravity [62–67], a perturbative quantum field theory of gravity on a continuous spacetime with scale-dependent nonlocal operators in the fundamental action. • Hoˇrava–Lifshitz gravity [68–70], a perturbative quantum field theory of gravity on a continuous spacetime where the time dimension scales anomalously. Stelle gravity and Hoˇrava–Lifshitz gravity are known to have issues, but we include them nonetheless because some of their features are common to the other

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approaches and can be calculated easily. To this list, we add other approaches that do not reach the same level of completeness but that have given valuable insights into the problem of quantum gravity. • Canonical quantum cosmology (reviewed in, e.g., [29]), models of the early universe built on the Hamiltonian formalism. They include the original approach based on the Wheeler–DeWitt equation as well as loop quantum cosmology, based on the LQG quantization of gravity in Ashtekar–Barbero variables. • String-gas cosmology [71, 72], a model producing primordial spectra via a thermal mechanism alternative to inflation and involving strings. • New ekpyrotic scenario [73, 74]. At the density of the string scale, new degrees of freedom govern the effective four-dimensional cosmological dynamics. At the time when such density is reached, the dynamics is dominated by an S-brane, a space-like hypersurface with zero energy density and negative pressure that induces a transition between a cosmological contracting phase (ekpyrosis) and an expanding one. • Brandenberger–Ho non-commutative inflation [75,76], where time and space coordinates do not commute, and as a consequence, the inflaton scalar field driving the early phase of acceleration obeys a modified dynamics. • Non-commutative κ-Minkowski spacetime [77–80], a spacetime with noncommuting coordinates whereupon one can construct a field theory of gravity and matter. • Padmanabhan’s nonlocal field theory [81–83], an effective field theory with nonlocal operators assumed to be valid near the horizon of black holes. • Multi-fractional spacetimes [7], a class of models with scale-dependent spacetime geometries. Integrals and derivatives acquire an anomalous multiscaling, the action gets a new discrete symmetry at short scales, and standard cosmology is modified accordingly The background upon which we will construct cosmological observables is the flat homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) line element ds 2 = −dt 2 + a 2 (t) dxi dx i = a 2 (τ ) (−dτ 2 + dxi dx i ) ,

(1)

 where t is proper time and τ := dt/a is conformal time. We set the speed of light to c = 1. For each observable, we will recall the GR expression, its generalization to quantum gravity, and the constraints placed to date in the literature.

Stochastic GW Background Basics Let Pt (f ) be the primordial tensor spectrum of tensor perturbations as a function of the frequency f = k/(2π ), where k = |k| is the comoving wave number. These

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fluctuations arise in the early universe during an era of inflation or from alternative mechanisms. The amplitude of the tensor spectrum is small compared to that of scalar fluctuations Ps (f ), and in fact, the bound on the tensor-to-scalar ratio r := Pt /Ps by the PLANCK Legacy release [84] from the PLANCK+TT+TE+EE+lowE+ lensing+BK15+BAO data set at the pivot scale f0 = 7.7 × 10−17 Hz (comoving wave number k0 = 0.05 Mpc−1 ) is r < 0.068 at the 95% confidence level (CL). Other observables of interest are the tensor spectral index nt := d ln Pt /d ln f and its running αt := dnt /d ln f . The primordial tensor spectrum is thus parametrized as  Pt (f ) = Pt (f0 )

f f0

nt (f0 )+ 1 αt (f0 ) ln 2

f f0

.

(2)

In standard cosmology, both nt and αt are negative, and the spectrum is said to be red-tilted. The amplitude of the stochastic GW background as observed today is characterized by the dimensionless density parameter ΩGW (f ) :=

π 2f 2 1 dρGW = Pt (f ) T 2 (f ) . ρcrit d ln f 3H02

(3)

2 H 2 is the critical energy density and ρ :=[M 2 /(8a 2 )] where ρcrit :=3MPl GW 0 Pl 2 (∂τ hij ) + (∇hij )2  is the energy density of GWs (spatial average of the kinetic energy of the transverse-traceless perturbation hij ), and the shape of the transfer function T , which can be found in [85], depends on the expansion history of the universe. Substituting the values of cosmological parameters provided by PLANCK [84] in the transfer function and making standard assumptions on the number of relativistic degrees of freedom in the early universe, the GW amplitude of the mode which enters the horizon during the radiation-dominated era can be written as

A ΩGW (f ) = 2 r h



f f0

nt + αt ln 2

f f0

,

A ≈ 1.4 × 10−15 .

(4)

A blue-tilted spectrum can produce a stochastic GW background with increasing amplitude that can reach the sensitivity thresholds of GW interferometers at high frequencies.

Results in Quantum Gravity For any given model, one can compare the theoretical prediction of the GW spectrum with the sensitivity curve of LIGO-Virgo-KAGRA [86–88], LISA [25,89], Einstein Telescope (ET) [90], and DECIGO [91–93], taking into account the cosmic microwave background (CMB) bound corresponding to r < 0.068 and current and future sensitivity curves of pulsar timing experiments, NANOGrav [94] and SKA [95].

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Obtaining a primordial blue-tilted tensor spectrum in quantum gravity is difficult. Despite the abundance of viable cosmological inflationary models in quantum gravity, a close scrutiny reveals that most of them predict a red tilt and those that have a blue tilt often lead to unobservable effects because nt or r, or both, are too close to zero. • The large class of flux-compactification models in string cosmology is uniformly characterized by nt < 0 [29,37]. A case apart is the old ekpyrotic scenario, which predicts a strongly blue-tilted tensor index nt = 2 [96]. While early versions of the model are ruled out because they have also a blue-tilted scalar spectrum, in a recent single-field version, perturbations are generated before the ekpyrotic phase, and the scalar spectrum is safely red-tilted [97, 98]. However, in all the realizations of the model, the tensor-to-scalar ratio is exceptionally small, and the resulting stochastic GW background is well below the detection threshold of any present or future interferometer [99]. • In Wheeler–DeWitt canonical quantum cosmology [100–102], the semi-classical limit of the Wheeler–DeWitt equation for the wave function of the Universe admits two solutions such that the tensor spectrum is  Pt (k) 

(0) Pt (k)

 1 ± c (Pl H )

2

k0 k

3  ,

(5)

where Pt(0) (k) ∝ H 2 is the standard spectrum at horizon crossing (k = aH ), c > 0 is a known numerical constant, Pl ≈ 10−35 m = 5 × 10−58 Mpc is the Planck length, and k0 is the pivot scale of the experiment: for the CMB, typically k0 = 0.05 Mpc−1 , or k0 = 0.002 Mpc−1 . The − (+) sign corresponds to a blue (respectively, red) tilt. The strong suppression of the (Pl H )2  1 term is further increased at late times by the (k0 /k)3 factor, since at the frequencies of LISA and DECIGO k0 /k ∼ 10−15 − 10−13 . Therefore, the quantum correction is unobservable. • In loop quantum cosmology, there are three main approaches to cosmological perturbations. – In the dressed-metric approach, the tensor spectrum is red-tilted [103, 104]. – In the effective-constraints or anomaly-cancellation approach, one can consider quantum corrections coming from inverse-volume operators, from holonomies or, more realistically, from both. If one considers only inversevolume corrections, the inflationary spectra are compatible with observations, but the tensor spectrum is red-tilted [105, 106]. On the other hand, the case with only holonomy corrections predicts a blue-tilted tensor spectrum, but it is ruled out observationally [107]. To the best of our knowledge, the case with both types of corrections has not been explored yet. – In the hybrid-quantization approach, the value and sign of the spectral index nt (k) depend on the background effective solution and, even more importantly, on the vacuum on which to perturb such background. The k-dependence of the

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tensor index can be found via a numerical analysis, and it turns out that for some choices of vacuum nt < 0, while for others, the spectrum oscillates rapidly, and a blue tilt can be generated at certain frequencies [108, 109]. However, what matters for the formation of a stochastic GW background is the average trend of the spectrum, and in all these cases, it decreases in k when k is sufficiently large. Therefore, it is unlikely that this model could generate a detectable stochastic background, for any vacuum choice. A detailed numerical study, which we will not pursue here, could give a more precise answer. • Nonlocal quantum gravity offers a natural embedding of Starobinsky gravity into a fundamental theory. The early universe is described by a period of inflation driven by a nonlocal quadratic gravitational action [110–114]. In the Jordan frame, the perturbed action on a quasi de Sitter background is S=

2 MPl 2

 d4 x



  R ˜ eH2 () hij , |g| hij  − 6

(6)

˜ 2 (z) := H2 (z − 4z∗ ), H2 is where R is the Ricci scalar on the background, H a function such that the form factor exp H2 is entire, and z∗ = R/(6M∗2 ), M∗ being the fundamental mass scale of the theory. The background accelerating solution is the same as in local Starobinsky inflation, so that on this quasi de Sitter background, the slow-roll approximation holds. The approximate primordial tensor spectrum as a function of the comoving wave number is



2 ˜ 2 2H (k) −H 3 M∗2 e , 2 ln(ke /k)  1   1 ke 2 ke − 2 1 H (k) ∼ ln ln + , k 12 k

Pt (k) ∼ 1 −

(7)

(8)

up to proportionality factors, where ke is the wave number of the last perturbation exiting the horizon at the end of inflation. Equation (8) tells us that the Hubble parameter is approximately proportional to a positive power of ln(ke /k) > 0 during inflation. Thus, both H and z∗ ∝ H 2 decrease when k increases. Since ˜ 2 increases with z∗ , then a characteristic of nonlocal quantum gravity is that H ˜ H2 decreases when k increases and vanishes asymptotically. Therefore, at high frequencies (high k), the nonlocal term in the tensor spectrum (7) tends to unity, ˜ 2 (z∗ )] = 1. Therefore, nonlocal Starobinsky gravity is out of limk→∞ exp[−H reach of any present or future GW interferometer because at high frequencies, it reduces to standard Starobinsky inflation, which has a red-tilted spectrum [114]. Other models of, or related to, quantum gravity manage to generate a bluetilted tensor spectrum and a detectable stochastic GW background: new ekpyrotic scenario, Brandenberger–Ho non-commutative inflation, and multi-fractional

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spacetimes. We will discuss them separately, contrary to the model-independent approach we will follow for the luminosity distance.

String-Gas Cosmology In a compact space, the excitation modes of a thermal ensemble of strings are momentum modes and winding modes. The energy of winding modes decreases with the size of the available space, and their number increases with the energy, so that, for an adiabatic process, winding modes dominate the thermal bath in a small space. The temperature of this bath cannot rise indefinitely and reaches a maximal temperature TH called Hagedorn temperature. The universe starts with an almost constant scale factor and a temperature slightly lower than TH . Both scalar and tensor spectra are generated thermally. In particular, tensor modes are generated by anisotropic pressure terms in the energy-momentum tensor, but near the Hagedorn temperature, the thermal bath is dominated by winding modes, and the pressure decreases. Thus, there is a decrease of power at low k and a slight blue tilt. Eventually, winding modes decay, and three spatial directions open up, while the others stay compact, leading to a radiation-dominated era. In this scenario, inflation is replaced by a quasi-static era where thermal fluctuations generate an almost scale-invariant primordial scalar and tensor spectrum, the latter being    ˆ (k) 1 − T 2 Pt (k)  Tˆ (k) 1 − Tˆ (k) ln , 4(MPl lst )4 lst2 k 2 1

(9)

where lst is the string length scale, Tˆ (k) := T (k)/TH , and the temperature T (k) is evaluated at the time when the mode with comoving wave-number k exits the horizon. The form of T (k) is unknown except for its behavior during the Hagedorn phase (T ≈ const  TH ) and in the following radiation-domination era (T ∼ 1/a). In the scalar sector, an increase of power is observed for small k (ns − 1 < 0), while the tensor index and its running read nt  (1 − ns ) −

4 > 0, ˆ ln[(1 − T )(lst k)−2 ]

αt  −αs .

(10) (11)

A blue-tilted tensor spectrum is one of the characteristic predictions of string-gas cosmology that could be tested if a primordial gravitational signal was discovered. However, in order to achieve detection, the small blue tilt (10) nt ≈ 0.035 [84] should be accompanied with as high as possible a tensor-to-scalar ratio   2 ˆ 1 − T 25 2 r= 1 − Tˆ ln . 9 (lst k)2

(12)

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Fig. 1 Stochastic GW background of string-gas cosmology compared with the sensitivity curves of LIGO-Virgo-KAGRA (LVK), SKA, LISA, ET, and DECIGO. The black and blue solid curves −1 −1 correspond to lst = 103 MPl and 5 × 103 MPl , respectively. We used the central values of the PLANCK constraints ns = 0.9658 and αs = −0.0066. Note that the approximation (2) breaks down for high-frequency GWs, since the corresponding curves go beyond the upper bound on the GW amplitude (dashed lines) obtained from the full expression (9). (Credit: [114])

The stochastic GW background predicted by string-gas cosmology is shown in Fig. 1. In the most optimistic case of a tensor-to-scalar ratio saturating the CMB bound, the model reaches DECIGO sensitivity.

New Ekpyrotic Scenario In the ekpyrotic universe, two flat 3-branes constitute the boundary of a fivedimensional spacetime and interact with an attractive potential V (ϕ) along a compact fifth dimension parametrized by the radion ϕ. As the branes get closer, the gravitational energy in the bulk is converted into brane kinetic energy, and the branes collide and oscillate back and forth their center of mass along the extra direction. During the collision, part of the brane kinetic energy is converted into matter and radiation. An observer on one of the branes experiences the brane collision as a big bang after a period of contraction called ekpyrosis. During the slow contraction phase, a pattern of inhomogeneities is developed. In the new scenario of [73, 74], the tensor index is small and positive nt = 1 − ns > 0 ,

αt = −αs .

(13)

The tensor-to-scalar ratio at the pivot scale is r(k0 ) 

2ns 25   (k0 τB )2(1−ns ) (1 − ns )2 , Γ 2 1 − n2s

(14)

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Fig. 2 Stochastic GW background of the new ekpyrotic scenario compared with the sensitivity obs curves of LIGO-Virgo-KAGRA, SKA, LISA, ET, and DECIGO. Denoting as nobs s ±δns and αs ± δαs the PLANCK values, we plot the worse case minimizing the tensor blue tilt and maximizing obs + the negative tensor running at the 2σ -level (nt = 1 − (nobs s + 2δns ) ≈ 0.026, αt = −(αs 2δαs ) ≈ −0.007, red solid curve), the intermediate case taking the central values of the parameters (nt = 1 − nobs ≈ 0.034, αt = −αsobs ≈ 0.007, black solid curve), and the best case maximizing s the tensor blue tilt and the positive tensor running at the 2σ -level (nt = 1 − (nobs s − 2δns ) ≈ 0.042, αt = −(αsobs − 2δαs ) ≈ 0.021, blue solid curve). The dotted curves correspond to the above cases with no running. (Credit: [114])

where Γ is Euler’s function and τB is the conformal time at which the string density is reached and the cosmological bounce takes place. Taking the grand-unification scale τB = (1016 GeV)−1 ≈ 5 × 10−55 Mpc and the observed value of the scalar index ns ≈ 0.9658, one has r(k0 ) ∼ (10−4 − 10−2 ) (1 − ns )2 . Figure 2 shows the stochastic GW background of this model. The signal reaches the DECIGO curve only in the most optimistic case of a strong blue tilt and a strong positive tensor running.

Brandenberger–Ho Non-commutative Inflation The graviton action of this model is decorated with *-products. This structure alters the scalar and tensor primordial spectra in a way that depends on whether the perturbation modes have a wavelength smaller (ultraviolet limit, UV) or larger (infrared limit, IR) than the fundamental length scale nc appearing in the commutation algebra. In the particular case of natural inflation, the IR limit of this model has a blue-tilted tensor spectrum and is compatible with PLANCK data [76]. Here the mechanism is directly related to the time–momentum uncertainty, which induces a k-dependence in the effective mass term of the GW propagation equation. This dependence is inherited by the tensor spectrum, and, for a certain choice of

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parameters, it leads to an enhancement at small scales. In the IR limit (0)

Pt = Pt

Σ 2 (nc H ) ,

(15)

(0)

where Pt = Pt (Σ = 1) is the amplitude in the commutative limit (Einstein gravity) and Σ is a function encoding the non-commutative effects. The factor Σ multiplies both the tensor and scalar amplitudes, so that their ratio r is unchanged. When the FLRW 2-sphere is factored out of the total measure, the tensor spectral index is positive nt 

r  4, 4

αt  8( − η) ,

(16)

¨ ˙ are the first two slow-roll parameters. The where  := −H˙ /H 2 and η := −φ/(H φ) scalar spectrum is red-tilted for some choices of inflation potential, such as natural inflaton [76]. The dependence of the observables (16) from the number of e-foldings N and the parameter A = (φ∗ /MPl )2 /3, where φ∗ is a characteristic scale of the scalar-field potential, can be found in [76, 114]. In particular, CMB data constrain 5.8 < A < 11 at the 95% CL for 50 < N < 60. As expected by the fact that non-commutativity changes the sign and value of the coefficients in the slow-roll expressions of the observables but not their order of magnitude, the spectrum is enhanced up to the DECIGO sensitivity curve, but barely so, and it does not reach ET [114].

Multi-fractional Spacetimes Multi-fractional spacetimes are spacetimes where the clocks and rulers used by the observer register different scaling laws (for instance, linear size versus volume) in copies of the same object with different sizes (for instance, a human-size hyperball compared with a microscopic one) [7]. This ever-changing geometry is typical of spacetimes arising in quantum gravity [115, 116]. Multi-fractional theories implement this dimensional flow via a modification of the integration measure in the field action and of the kinetic operators acting on the fields, including gravity. In the particular case of the so-called theory with q-derivatives, the tensor spectrum reads 



 αt (k0 ) p(k) 2 p(k) + ln Pt (k) = r(k0 ) Ps (k0 ) exp nt (k0 ) ln , p(k0 ) 2 p(k0 )

(17)

where nt < 0, αt < 0, and r = −8nt as in standard inflationary models and 

1 p(k)  k 1 + |α|



k k∗

1−α −1 ,

(18)

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where k∗ is a fundamental comoving scale and the parameter α is related to the Hausdorff dimension of space (the way volumes scale with the linear size; see space below) by dH = 3α in three topological dimensions. If 0 < α < 1 (respectively, α > 1), the tensor spectrum is red-tilted but less (respectively, more) than in Einstein gravity. If α < 0, for k  k∗ (large scales, small frequencies), one recovers GR, while an exotic regime is reached for k k∗ (small scales, large frequencies) where 1 2 p(k)  k α and Pt (k) ∼ k αnt + 2 αt α ln(k/k0 ) . The effective spectral index αnt is positive definite, and the spectrum increases at large k, while the effective tensor running at large frequencies stays negative, α 2 αt < 0. The geometric interpretation is that in the IR, the Hausdorff dimension of spacetime is smaller than 4, and there is an increase in the number of modes given the same density. As one can see in Fig. 3, for the typical signs and values of r, nt , and αt of standard inflationary scalar-field models, the theory can reach the DECIGO sensitivity curve if the tensor-to-scalar ratio r is close to the CMB bound (in the figure, r = 0.06). The running decreases the spectral amplitude at the typical frequencies of interferometers. Results with zero running are insensitive of the choice of scale k∗ (position of the bending point of the spectrum) because αnt is very small, while there is a visible effect in the presence of running due to the amplification by α 2 .

Fig. 3 Stochastic GW background of multi-fractional inflation with no running (αt = 0) and α = −1/2, −3, −5 (respectively, red, black, and blue solid curve), compared with the sensitivity curves of LIGO-Virgo-KAGRA, SKA, LISA, ET, and DECIGO. Here r = 0.06 and nt ≈ −0.0075 is given by the consistency relation r = −8nt . The αt = 0 plots are unaffected by different choices of k∗ , which we took in the range k∗ = 10−20 –1010 Mpc−1 . The dotted curves correspond to the above cases with nonzero running αt = −0.0001 (typical order of magnitude of inflationary models) and k∗ = 10−3 Mpc−1 . (Credit: [114])

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Modified Dispersion Relation and Propagation Speed Basics A feature common to all theories of quantum gravity is dimensional flow, the change of spacetime dimension with the probed scale. Quantization of spacetime geometry or its emergence from fundamental physics introduces, directly or indirectly, two types of change relevant for the propagation of GWs: an anomalous spacetime measure d(x) (how volumes scales) and a kinetic operator K (∂) (modified dispersion relations), so that the perturbed action for a small perturbation hμν over (0) a background gμν is S=

1 22Γ ∗



  d |g (0) | hμν K hμν + O(h2μν ) ,

(19)

up to a source term, where ∗ is a (or the) characteristic length scale of the geometry and Γ is a constant. Splitting the perturbation into the usual polarization + + h e× , the modes h Γ decomposition hμν = h+ eμν × μν +,× /∗ are dimensionally and dynamically equivalent to a scalar field. The measure defines a geometric observable, the Hausdorff dimension dH () := d ln ()/d ln , describing how volumes scale with their linear size . In a classical spacetime, dH = D. Also, spacetime is dual to a well-defined momentum space characterized by a measure (k) ˜ with Hausdorff dimension dHk , in general different from dH . The kinetic term is related to dHk and to another observable, the spectral  dimension dS () := −d ln P ()/d ln , where P () ∝ (k) ˜ exp[−2 K˜ (−k 2 )]. In any plateau of dimensional flow, where all dimensions are approximately constant, k (k) ˜ ∼ dk k dH −1 and 2−2β K˜ (−k 2 ) = −∗ k 2 + k 2β ,

(20)

where β := [K ]/2 is a constant given by half the energy scaling of K . Here momentum–space coordinates have the usual dimensional units [k μ ] = 1. Then, k β−1 one finds that P ∝ (∗ )−dH /β , implying dS = 2dHk /[K ] and β=

dHk . dS

(21)

In such a plateau region, since [S] = 0, from (19), we have Γ 

dk dH − H, 2 dS

(22)

and Γ ≈ const. We assume that dS = 0 at all scales. The case of nonlocal quantum gravity, where dS = 0 at short scales, must be treated separately. In the GR limit in

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D topological dimensions, dH = dHk = dS = D and Γ = D/2 − 1, the usual scaling of a scalar field. In D = 4, Γ = 1. The value of the dimensions dH , dHk and dS and of the parameter Γ for different quantum gravities can be found in [18]; in the UV, −3  ΓUV  2. The modified dispersion relation (20) arises in higher-order non-unitary theories (β = 2, 3, . . . ) [30, 32, 33], the propagation of low-energy particles in non-critical string theory (β = 3/2 at mesoscopic scales and β = 2 in the deep UV) [117], asymptotic safety (β = 2) [118], Hoˇrava–Lifshitz gravity (β = 3 only in the spatial directions) [69], multi-fractional spacetimes with fractional and q-derivatives (β > 0; Lorentz invariance is broken here) [7], causal sets (β = 2) [119], and as an effective dispersion relation in loop quantum gravity [120–123]. Last, ∗ is a length scale around which quantum-gravity effects become relevant. If there is only one fundamental scale ∗ , then it is very small (of order of, or not too far from, the Planck scale Pl ), and the UV regime corresponds to very short wavelengths. If, however, (20) is an effective dispersion relation, ∗ is a mesoscopic scale that could be relatively far from the actual UV and closer to the IR regime.

Results in Quantum Gravity The dispersion relation (20) of the spin-2 graviton field has been used to impose constraints on quantum-gravity theories exhibiting dimensional flow using the LIGO-Virgo merging events [6,8,9]. Given the dispersion relation (20), the velocity of propagation of a wave front is given by the group velocity cGW :=

dω . d|k|

(23)

The signal of GW150914 is peaked at a frequency f = ω/(2π ) = 100 Hz, corresponding to ω ≈ 630 Hz ≈ 4.1 × 10−13 eV. At that frequency, the difference Δc := cGW − 1 between the propagation speed of the signal and the speed of light is bounded from above as [1] |Δc| < 4.2 × 10−20 .

(24)

For the usual Lorentz-invariant massive dispersion relation ω2 = |k|2 + m2 , one gets cGW  1 − m2 /(2ω2 ) if the mass is small, so that Δc  −m2 /(2ω2 ) and m < 1.2 × 10−22 eV. Consider now the more general dispersion relation ω2 = |k|2 + O(1)n∗ |k|n+2 . The case n = 1 stems from generic quantum-gravity arguments [117], while the case n = 2 can be obtained either as the low-momentum expansion of Padmanabhan’s nonlocal model or from an argument matching the logarithmic leading-order LQG correction to the entropy-area law for black holes [122]. Since the GW frequency is much lower than the Planck frequency, one gets very weak bounds on ∗ when n = 1, 2 [6, 8]. However, setting instead

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−1 ∗ > 10 TeV (quantum-gravity scale larger than the LHC scale), one finds the bound 0 < n < 0.76 [8]. We can connect these results with the kinetic operator (20) if we allow for a breaking of Lorentz invariance. Expanding the dispersion relation K˜ = 0 for small ω/|k|, one finds  ω = |k| 2

2

2β−2 − ∗ |k|2β

ω2 1− 2 |k|

β 2β−2

 |k|2 − ∗

|k|2β ,

(25)

where n = 2β − 2. As said, too weak bounds on ∗ are obtained when n = 1, 2 (β = 3/2, 2), while when −1 ∗ > 10 TeV, we have β = dH /2 − Γ < 1.38 when dH ≈ 4. This happens at intermediate scales where the corrections to GR are small but non-negligible. In this mesoscopic regime, the above bound implies Γmeso > 0.62 ,

(26)

weaker than the constraint (33) discussed below. At any rate, one should apply this bound with care to Lorentz-invariant theories, as it entailed an assumption (ω/|k|  1) that may strongly depend on the model and on the symmetry-breaking mechanism. Multi-fractional spacetimes can produce ad hoc models [7, 9] saturating the bound (26) which, in turn, would constrain the two-dimensional parameter space (∗ , α) of the simplest version of the theory to a region that should be checked against other types of cosmological observations and mechanisms (inflation, dark energy, and so on). While this procedure can select specific viable models, it is purely phenomenological. The modified dispersion relation of the graviton in nonlocal quantum gravity requires a separate but rather quick analysis. The linearized perturbation equation is [64] h˜ = 0 ,

h˜ := eH2 () h ,

(27)

where H2 is a nonlocal form factor such that exp H2 is an entire function of the background Laplace–Beltrami operator . The dispersion relation is k 2 exp H2 (−k 2 ) = 0, which leads to the usual pole k 2 = −ω2 + |k|2 = 0 in the propagator. Since ˜ the propagation speed of GWs the perturbation equation is the standard one for h, equals the speed of light, and the theory avoids the bound (24). Early-universe scenarios such as Brandenberger–Ho non-commutative inflation and the new ekpyrotic scenario have no say about the propagation of late-time GWs. Therefore, they are not constrained by interferometric observations on the propagation speed or the luminosity distance of individual sources. We conclude that, in general, modified dispersion relations in quantum gravity are not an efficient way to constrain the theory because the correction is too small and the graviton speed is very close, or equal to, the speed of light, thus evading the bound (24).

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Luminosity Distance Basics The luminosity distance dLEM of a source of electromagnetic radiation is defined by the relation between the flux F of light reaching an observer and the power L per unit area emitted by the source: F =:

L . 4π(dLEM )2

(28)

In standard GR, the luminosity distance as a function of the redshift 1 + z = a0 /a (with a0 = 1) is  dLEM (z) = (1 + z)

t0

t (z)

dt = (1 + z) a



z 0

dz , HGR

(29)

where a(z) = (1 + z)−1 . The Hubble parameter HGR (z) is determined by the 2 = (κ 2 /3)ρ and includes all energy contributions, first Friedmann equation HGR including dark energy. At small z, dLEM  z/H0 , where H0 is the Hubble parameter today. Sources of GWs admit another definition of luminosity distance. Let hμν be a metric perturbation around the Minkowski background ημν = diag(−, +, · · · , +), and call h one of the graviton polarization modes. The scalar h is the amplitude of a gravitational wave emitted by a source such as a black-hole or a neutron-star binary system. In D topological dimensions, a direct calculation [124] or a scaling argument [20] yields the asymptotic dependence of h on the distance r = |x − x | in the local wave zone of the source, i.e., a region of space much larger than the wavelength of the metric perturbation but smaller than cosmological scales: h

κFh (t − r) r

D=4

D−2 2

∝

1 , r

where Fh is a function of retarded time. If the observer is at a cosmological distance, the cosmic expansion must be taken into account. Since r is nothing but the comoving distance from the source, after rescaling with the scale factor a, one gets (see [125, section 4.1.4] for the derivation in four dimensions) h∼

1 EM

(dL )

D=4 D−2 2

=

1 . dLEM

(30)

For sources of GWs and light, called standard sirens, both sides of this equation can be measured: the left-hand side is the strain measured in an interferometer, while the right-hand side is determined by observations in the optical spectrum.

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In theories beyond GR, the relation between h and dLEM can be different. Defining the GW luminosity distance as h =: 1/dLGW (up to a retarded time-dependent function), in these theories, the ratio dLGW (z)/dLEM (z) deviates from 1. This is a cosmological observable that can be determined from standard-sirens data and is parametrized in several forms [19].

Results in Quantum Gravity To extract the luminosity distance in quantum gravity in a model-independent way, we appeal again to dimensional flow. In a multiscale spacetime such as those arising in quantum gravity, the measurement of a generic distance r in the absence of curvature in a non-relativistic regime (Newtonian approximation) deviates from the one in ordinary space by a power-law correction, so that [7, 11, 126, 127] 

 α−1  r r → r˜ = r 1 +  , l∗

(31)

where the parameter  accounts for two types of corrections (deterministic if  = ±1, stochastic if  is a random variable averaging to zero [11]). According to the flow-equation theorem [7], when measuring r with physical rods, one can identify α with the UV Hausdorff dimension of spacetime divided by D [11]. Generic quantum-gravity arguments select the values α = 1/3, 1/2 as especially appealing [11, 126, 127], although they are not really preferred in most of the concrete theories listed here. It turns out that the luminosity distance follows a similar multiscale power law. It is not difficult to show that (22) is the scaling of the GW amplitude h (subscripts +, × omitted) with respect to the radial distance r in the local wave zone, h(t, r) ∼ fh (t, r) (∗ /r)Γ . On cosmological distances, it is sufficient to replace r → ar. Assuming that quantum-gravity corrections to dLEM are negligible at large scales and absorbing redshift factors and all the details of the source (chirp mass, spin, and so on) into the dimensionless function fh (z), one has h(z) ∼ fh (z)

∗ EM dL (z)

Γ .

The final step is to generalize this relation, valid only for a plateau in dimensional flow, to all scales. An exact calculation is extremely difficult except in special cases, but a model-independent approximate generalization is possible because the system is multiscale (it has at least an IR and a UV limit, respectively Γ → 1 and Γ → ΓUV ). In fact, multiscale systems such as those in multi-fractal geometry, chaos theory, transport theory, financial mathematics, biology, and machine learning are characterized by at least two critical exponents Γ1 and Γ2 combined together as a sum of two terms r Γ1 + A r Γ2 + . . . , where A and each subsequent coefficients

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G. Calcagni

contain a scale (hence the term multiscale). In quantum gravity, lengths have exactly this behavior, which has been proven to be universal [7,11,126,127] in the flat-space limit. In particular, it must hold also for the luminosity distance because one should recover such a feature in the sub-cosmological limit dLEM → r. Thus [20]  EM γ −1 dLGW dL = 1 ± |γ − 1| , dLEM ∗

(32)

with γ = 0. In the presence of only one fundamental length scale ∗ = O(Pl ), (32) is exact, and γ = ΓUV takes values different from 1, in the interval −3  ΓUV  2 for the theories considered in this chapter. Conversely, if ∗ is a mesoscopic scale much larger than the Planck scale, then (32) is valid only near the IR, close to the end of the flow, where γ = Γmeso ≈ 1. This equation resembles the GW luminosity–distance relation expected in some models with large extra dimensions, where gravitons leak into a higher-dimensional space [128–131]. Observations can place bounds on the two parameters ∗ and γ in a modelindependent way, by constraining the ratio (32) as a function of the redshift of the source. An analysis based on two standard sirens (the binary neutron-star merger GW170817 observed by LIGO-Virgo and the Fermi telescope [132] and a simulated z = 2 super-massive black hole merging event that could be observed by LISA) shows that no constraint can be placed on the deep UV limit of any quantum gravity unless (32) were valid at all scales and 0 < γ − 1 = ΓUV − 1 = O(1), in which case κ-Minkowski spacetime and Padmanabhan’s nonlocal effective model would be ruled out. The other alternative is that (32) was valid in a near-IR regime and γ = Γmeso was very close to 1 from above, in which case one finds the bound [20] 0 < Γmeso − 1 < 0.02

(∗ = Pl ) .

(33)

Examining (22), one concludes that this case is realized only for geometries with a spectral dimension reaching dS → 4 from above. The only theories in our list that do so are those where ΓUV > Γmeso > 1 (κ-Minkowski spacetime with ordinary measure and bicross-product or relative-locality Laplacians and Padmanabhan’s model) or Γmeso > 1 > ΓUV (LQG/SF/GFT). One can exclude observability of the models with ΓUV > Γmeso > 1, since they predict Γmeso − 1 ∼ (Pl /dLEM )2 < 10−116 . Thus, only LQG/SF/GFT could generate a signal detectable with standard sirens, unless some yet unknown theoretical constraints limited the size of the effect. Here dS runs from small values in the UV, but before reaching the limit dSIR = 4, it overshoots the asymptote and decreases again [133]: hence Γmeso > 1 > ΓUV . A complementary solar-system constraint on the spin-2 sector can arise from modifications of Newton’s potential and can be much stronger than (33), but it heavily relies on model-dependent assumptions on the scalar perturbation sector which are under poor control [18]. On the other hand, the bounds obtained from dL are stronger than the ones found from the modified dispersion relation (20). This is

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one reason behind the recent surge of interest in the luminosity distance to probe theories beyond GR. Let us now discuss the case of nonlocal quantum gravity [67]. Using the same ˜ for the entire full calculation or the scaling argument as in GR with h replaced by h, form factors, we have h˜ =

1 dLGW

⇒

h = e−H2

1 , dLGW

(34)

where h˜ is defined in (27). We can estimate the nonlocal correction in the righthand side for the minimal form factor H2 () = −2∗  = 2∗ (∂t2 + 3H ∂t ) in the homogeneous approximation and, crudely, an approximately constant Hubble 2 parameter H  H0 , so that at large redshift h  H0 e−10(∗ H0 ) e2H0 (t−t0 ) , while at 2 small redshift h  H0 e−3(∗ H0 ) e2H0 (t−t0 ) . Overall e−c(∗ H0 ) , h dLGW 2

c = O(1) − O(10) .

(35)

Assuming, to maximize the effect, that light is not affected by nonlocality, we have dLGW  1 + c(∗ H0 )2 , dLEM

(36)

and, for ∗ = Pl , the right-hand side is of the order of 1 + 10−120 , an effect completely unobservable compared with the estimated error ΔdL /dL ∼ 0.001 − 0.1 of present and future interferometers [134–137]. For a power-law expansion a = (t/t0 )p , dL ∝ (t0 /t)2p (t0 − t), and one can show that, again, the correction in the ratio (36) is of the order of (∗ /t0 )2 ∼ 10−120 . Increasing ∗ to particlephysics scales does not magnify this correction enough, since it is governed by the cosmological scale H0−1 ∼ t0 ∼ 1017 s. Therefore, no nonlocal effect is observable in the luminosity distance for this theory. Regarding multi-fractional spacetimes, it is possible to construct models with large deviations from the standard luminosity distance [18], but just as in the case of the modified dispersion relation, one enters the realm of ad hoc phenomenology.

Strain Noise The above constraints can be complemented by a bound on the Hausdorff dimension of spacetime in the UV (dHUV ) coming from the strain noise of interferometers [3, 5, 11]. The correction in (31) can also be regarded as a threshold on the minimal uncertainty in physical measurements (spacetime fuzziness) of distances. Quantum gravity may manifest itself as an intrinsic noise with variance

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G. Calcagni

 2 σQG = 2∗

L ∗

2α (37)

.

If L is the typical length of a GW interferometer (e.g., the linear size of its arms), we can compare this quantum-gravity noise with the instrumental or strain noise  2 = σexp df S 2 (f ) of a GW interferometer, where S is the spectral noise. For a signal dominated by the characteristic frequency 1/L (in c = 1 units), a rough 2  f S 2 (f )| estimate is σexp f =1/L . The strain noise is dimensionless, and (37) has the dimensionality of (length)2 , so that a signal of spacetime fuzziness would be detectable if 

∗ L

2(1−α) =

2 σQG

L2

 2 ∼ σexp  f S 2 (f )f = 1 , L

leading to  S (f ) =

1 ∗

α−1 f

1 2 −α

⇒

α=

ln ln

S √ ∗ f

1 ∗ f



 .

(38)

Results in Quantum Gravity In the worst-case scenario where ∗ is of Planckian size, one might believe it impossible to probe with an instrument of a macroscopic size L of order of the kilometer (or millions of kilometers, in the case of LISA). However, L does not appear in (38), and if α is small enough, the detector may even catch the stochastic background from spacetime fuzziness. Setting ∗ = Pl in (38), we get an upper bound on α for all the main interferometers in operation, under construction, or proposed for the near future, ranging from α < 0.47 for Ligo-Virgo-KAGRA and DECIGO to α < 0.54 for LISA [18]. This translates into a bound on the small-scale Hausdorff dimension of spacetime dHUV < 1.9 ,

(39)

very close to the value found in certain kinematical states of LQG/SF/GFT. Note that an infinitely sensitive instrument not detecting quantum-gravity noise would push the bound to α ∼ 0, which would mean that there is no dimensional flow (dH = 4), i.e., the condition of application of the flow-equation theorem does not hold. Therefore, one could not interpret this result as having a spacetime with zero UV dimension. In particular, the Hausdorff dimension in nonlocal quantum gravity does not change, and we do not expect any imprint in the strain noise. Any constraint from (38) would be physically helpful only in the case where a strain noise of quantumgravity origin were actually detected.

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Conclusions Table 1 summarizes the cosmological models of quantum gravity that we have explored here in relation with GW physics. As one can appreciate, their great majority does not give rise to any observable signal, as expected on the grounds that Planck scale corrections do not have an impact on the production and propagation of gravitational waves. Still, in some cases, there is margin for moderate hope to see something, or at least to place meaningful constraints, in the near future. All these results would deserve further critical scrutiny because they are as good as the assumptions made to obtain them. For example, it is not yet clear whether the bump in the spectral dimension found in kinematical quantum states in LQG/SF/GFT and giving rise to a potential detectable deviation from the luminosity distance [18] is an artifact or a physical feature of the model [133]. The number of ticks in the table is not related to the quality of science one can do with each theory. Some models have many ticks because they can be tuned more easily than others, while some have no tick at all because their predictions are rigorous but consistently below detection threshold. Also, the table is by no means exhaustive. While we have spent some time on the propagation of GWs, we have not discussed the constraints that can come from the production of GWs at inspiral and merger phases [9, 12] or from the horizon structure of merging black holes [16], topics that would require another review and more advances from the theoretical side than those currently available for the theories in our list, list that, as we already stressed, is not comprehensive.

Table 1 Observability in GW data of various theories of quantum gravity using the stochastic GW background (ΩGW ), the propagation speed of gravitons (cGW ), the luminosity distance (dL ), and the strain noise. A tick indicates that the theory might give a detectable signal, empty cells correspond to theories which cannot produce such a signal, and question marks are placed where no prediction has been calculated so far ΩGW Stelle gravity String theory (low-energy limit) Asymptotic safety LQG/SF/GFT loop quantum cosmology Causal dynamical triangulations (phase C) Nonlocal quantum gravity Hoˇrava–Lifshitz gravity κ-Minkowski spacetime Brandenberger–Ho non-commutative inflation New ekpyrotic scenario Padmanabhan’s nonlocal model Multi-fractional spacetimes

cGW

dL

Strain noise

✓

? ✓

?

? ✓

? ✓ ✓ ✓

?

✓

✓

✓

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G. Calcagni

The main message to take on board from this review is that in certain models of quantum gravity, short-scale modifications can leave an imprint on GWs when they are accumulated on cosmic distances or amplified by the cosmic expansion via mechanisms that go beyond effective field theory. By the first half of this century, GW interferometers should be able to give a deeper insight into the physics of quantum gravity. Acknowledgments The author is supported by the I+D grant FIS2017-86497-C2-2-P of the Spanish Ministry of Science and Innovation and acknowledges networking support by the COST Action CA18108.

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Patrick Brady, Giovanni Losurdo, and Hisaaki Shinkai

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIRGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KAGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Gravitational Wave Network (IGWN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observational Science Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Run 1 (O1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Run 2 (O2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Run 3: The First 6 Months (O3a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The first detection of gravitational waves was made by the two LIGO detectors in the United States one hundred years after general relativity was first described

P. Brady () Department of Physics, Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin-Milwaukee, Milwaukee, WI, USA e-mail: [email protected] G. Losurdo Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy e-mail: [email protected] H. Shinkai Osaka Institute of Technology, Hirakata City, Osaka, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_51

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by Einstein. Two years later, Virgo joined LIGO in the second advanced gravitational-wave detector observing run. As of May 2021, 50 gravitationalwave events from mergers of binary black-holes or neutron stars have been published by the LIGO-Virgo Collaboration. KAGRA in Japan is part of this international gravitational wave network since April 2020, and joint observations are anticipated in the next observing run. We briefly introduce the LIGO, Virgo and KAGRA detectors and the remarkable results of gravitational-wave observations up to now. The other articles in this handbook provide a comprehensive overview of the subject at this time. Keywords

Gravitational wave · LIGO · Virgo · KAGRA · Detector · Observation · Black hole · Neutron star

Introduction The general theory of relativity was derived by Einstein in 1915 and revolutionized our understanding of gravity. Based on general relativity, Einstein explained the perihelion advance of Mercury, he predicted the bending of light by massive objects and the redshift of light emerging from a gravitational potential, and he showed that gravitational waves should exist and propagate at the speed of light. As physicists grappled with the mathematics of general relativity, some argued that gravitational waves might be a mathematical artifact of how the theory is formulated and not physical phenomena. In 1956, after Einstein’s death, Pirani provided a mathematical formulation that demonstrated that the gravitational waves predicted by general relativity carry energy and are thus physical entities. Bergman [1] raised the question whether a binary star system gives rise to gravitational waves that carry energy proportional to the square of the amplitude in his summary of the 1957 Chapel Hill conference without giving any answer. Peters and Matthews subsequently calculated the impact of gravitational-wave emission on a binary system under the assumption that the waves do carry energy [2]. With the discovery of the binary pulsar in 1974, the existence of gravitational waves was indirectly confirmed by demonstrating that the orbit of PSR B1913+16 is shrinking as predicted by Peters and Matthews thus answering Bergman’s question in the affirmative. Since then, relativity researchers have made significant progress to understand the theory, to invent experimental measuring devices, to implement simulation techniques, and to construct data analysis methods for the direct detection of gravitational waves. In the United States, construction of the National Science Foundation’s Laser Interferometer Gravitational-wave Observatory (LIGO) was approved in 1990 with the first year of funding in 1991. Civil construction on LIGO facilities at Hanford, WA, and Livingston, LA, started in 1994. Each site houses detectors in a pair of 4 km long, orthogonal, arms. The first coincident science operations took place in 2002 using the Initial LIGO detectors and the British-German GEO600 detector

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in Germany [3]. LIGO was envisioned as a facility to house multiple generations of detectors. After a period of scientific data taking interspersed by incremental improvements to the initial LIGO detectors [4], the upgrade to the Advanced LIGO (aLIGO) [5] detectors began in 2010. Advanced LIGO commenced observations in 2015 immediately delivering the first direct detection of gravitational waves [6]. France and Italy realized the laser interferometer Virgo with 3 km arms in the suburbs of Pisa, Italy (construction started in 1996 and was completed in 2003). In 2007, LIGO and Virgo signed an agreement to collaborate on the search for gravitational waves with full data exchange and a joint publication policy. This marked the beginning of operations as LIGO-Virgo network. The Virgo detector was upgraded to a second-generation detector through the Advanced Virgo [7] program. The upgrade started at the end of 2011 and Advanced Virgo started joint observation with Advanced LIGO in August 2017. During the second observing run (O2), the LIGO-Virgo Collaboration reported eight events including the first detection of a binary neutron star merger [8]. In Japan, the TAMA interferometer with 300 m arms was constructed on the premises of the National Astronomical Observatory of Japan in Mitaka City, Tokyo. TAMA started observations in 1998. Then, following the low-temperature mirror demonstration interferometer CLIO, construction of the KAGRA interferometer with 3 km arms started in 2012 in Kamioka, Gifu, Japan [9]. The implementation of the device was completed in 2019, and KAGRA began joint observations with LIGO and VIRGO in 2020. As of 2021, the LIGO-Virgo-KAGRA (LVK) Collaboration is working to establish the International Gravitational Wave Network (IGWN) which is conceived as an organization to facilitate coordination among ground-based, gravitationalwave-detector projects across the globe. This chapter briefly summarizes gravitational-wave observations carried out up to December 2020.

Gravitational Wave Detectors Basic Concepts In the 1970s, Rainer Weiss began thinking about building gravitational-wave (GW) detectors using laser interferometers. At the same time, Kip Thorne was studying the theoretical properties of gravitational waves and the astrophysical sources that generate them. Gravitational waves induce a strain in spacetime which is characterized by the change in length per unit length as the waves pass. The strain sensitivity of a laser interferometer gravitational-wave detector is therefore inversely proportional to its arm length. Weiss completed an analysis of the fundamental noise sources impacting these detectors in 1972. Given the weakness of gravitational waves, the instruments would need to have kilometer-scale arms and use a number of tricks to enhance their sensitivity to the waves. Over the next two decades, new ideas

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led to improved sensitivity of these detectors and the mergers of binary neutron stars and black holes were identified as likely sources. In 1989, the LIGO detectors were proposed to the US National Science Foundation. In Europe, Adalberto Giazotto and Alain Brillet pursued studies on low frequency vibration isolation, lasers and optics which led them to propose the project for a detector in Europe: Virgo. And a group led by James Hough proposed the British-German detector project. At fixed arm length, the sensitivity of these detectors is limited by noise generated in the measuring instruments. There are many different noise sources, but the sensitive band of the detectors is determined primarily by three types. The first is noise from the light source. Laser interferometric detectors monitor changes in the interference pattern induced by passing gravitational waves. Since the rate at which photons interfere follows a Poisson process, the light fluctuates accordingly. This is photon shot noise. The number of photons collected by the detectors is approximately determined by the power multiplied by the period of the gravitational waves. Thus photon-shot noise limits the sensitivity at high frequencies. It can be reduced by increasing the input power of the laser and by using power recycling to recover the light that would otherwise leave the interferometer through the symmetric output. Even if the entire optical path is evacuated to suppress fluctuations due to refraction, noise will be generated depending on the frequency, intensity, and output beam of the light source. Moreover, as the lightpower increases, radiation pressure on the mirrors begins to limit the sensitivity at low frequencies. The second noise comes from the vibration of the instrumentation arising from heat and gives rise to thermal noise which limits the sensitivity in mid frequencies (∼100 Hz). Up to now, LIGO and Virgo have relied on reducing the internal losses by judicious choices of materials to reduce thermal noise. KAGRA includes a system to cool its mirrors to ∼20 K; successful cryogenic operations will further control the thermal noise. The third noise is seismic vibration. The ground is constantly vibrating due to the unique vibration of the earth itself, and it also vibrates with a great influence on the surrounding environment. Sophisticated vibration isolation systems have been conceived, capable of suppressing the seismic noise by many orders of magnitude, in order to extend the detector bandwidth down to ∼10 Hz and, at the same time, allowing to control the mirrors at the required level. There are many more sources of noise that are addressed by the development of new instrumental techniques and by careful design and engineering. Research into better ways to control all of these noise sources is continuing. Even in stable operations, an interferometer is susceptible to myriad disturbances that can mimic a gravitational wave in a single detector. Moreover, multiple interferometers are required to determine the direction, orientation and distance to a gravitational-wave source, to measure the polarization of gravitational waves, and to maximize the scientific information that can be extracted. Today, LIGO, Virgo, and KAGRA operate four detectors in unison and analyze the data together. Table 1, Figs. 1, and 2 show the locations and their arm orientations of these detectors.

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Table 1 Geometry of LIGO, Virgo & KAGRA detectors Detector LIGO Hanford (LHO) LIGO Livingston (LLO) Virgo KAGRA

Arm length 4 km 4 km 3 km 3 km

Latitude 46◦ 27 19 30◦ 33 46 43◦ 37 53 36◦ 24 36

Longitude N 119◦ 24 28 W N 90◦ 46 27 W N 10◦ 30 16 E N 137◦ 18 36 E

X-arm N 36◦ W N 18◦ S N 19◦ E E 28.3◦ N

Y-arm W 36◦ S S 18◦ E W 19◦ N N 28.3◦ W

Fig. 1 Location of GW detectors; LHO (LIGO Hanford Observatory), LLO (LIGO Livingston Observatory), Virgo and KAGRA. The size of the arms is drawn exaggeratedly

LIGO The LIGO observatories at Hanford, WA, and Livingston, LA, were constructed in the 1990s. Each observatory includes two orthogonal, evacuated arms of 4 km in length which house the detectors. The first coincident science operations took place in 2002 using the Initial LIGO detectors and the GEO600 detector in Germany [3]. This was followed from 2002 to 2010 by a sequence of science runs (S2-S6) that were interleaved between periods of detector improvements. LIGO and Virgo operated their detectors as a network starting with S5. No gravitational-wave signals were identified in the data from those runs. LIGO was envisioned as a facility to house multiple generations of detectors. The upgrade to the Advanced LIGO (aLIGO) detectors began in 2010 and the aLIGO instruments began operating in 2015 with the first observing run (O1) of the new era starting in September of that year. The LIGO detectors are Michelson laser interferometers with suspended test mass mirrors. The interferometer configuration is modified to enhance their sensitivity to gravitational-wave strain. Resonant cavities in each arm increase the effect of gravitational waves on the phase of the light by ∼300 times; a power recycling mirror increases the input light power from

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Fig. 2 Orientation of GW detectors on the Earth surface; LHO (LIGO Hanford Observatory), LLO (LIGO Livingston Observatory), Virgo, and KAGRA

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20 to 700 W on the beam splitter leading to about 100 kW of circulating power in each arm cavity; the bandwidth of the coupling to the differential mode is broadened using signal recycling; and a Nd:YAG laser with amplitude, frequency, and beam geometry stabilization provides light at 1064 nm. Together these modifications reduce the impact of photon shot noise on the detection of gravitational waves. The test-mass mirrors are suspended by a quadruple pendulum system in which the final stage supports the mirrors using fused silica fibers. Along with active seismic isolation, seismic noise is suppressed by 10 orders of magnitude above 10 Hz. Finally the test masses are 40 kg fused silica substrates with low-loss dielectric coatings providing low-thermal noise. The resulting instruments are ∼3–5 times more sensitive than the initial LIGO instruments at frequencies 100–300 Hz and 10 times more sensitive around 60 Hz. On 14 September 2015, the detectors at LIGO Hanford and LIGO Livingston registered gravitational waves from the merger of a pair of black holes [6] for the first time in history. This first direct observation of gravitational waves took place at the very beginning of O1. The presence of the same signal in both detectors modulo a time and phase shift consistent with their separation and orientation provided compelling evidence that the signal was astrophysical in origin. Careful examination of data from arrays of seismometers, accelerometers, microphones, magnetometers, and other environmental sensors was used to rule out an environmental origin. Auxiliary information about the instruments’ operational state indicated that they were both in stable operating mode for several hours at the time of the event. Improvements to the LIGO detectors continue to be interleaved between observing runs. In addition to fixing a number of technical problems, the following significant changes were made between O1 and O3: the injection of squeezed

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vacuum at 2–3 dB was achieved; a signal recycling mirror with lower transmission was installed; the input power was increased to 40 W using an amplifier and tuned mass dampers applied to the test masses to address parametric instabilities at high power; end mirrors were replaced with versions that achieve lower optical losses; and many light baffles were installed to reduce scattered light. The result is an angle averaged binary neutron star range of over 100 and 125 Mpc for LIGO Hanford and Livingston, respectively, during O3. Detector improvements have been mapped out by the LIGO Laboratory through the middle of the 2020s. For O4, work is under way to improve stray light control, to introduce a new high power laser amplifier, and to reduce the impact of point absorbers on the test masses. Additional improvements are expected from early implementation of certain A+ Project [10] upgrades, including adaptive mode-matching and frequency-dependent squeezing. Together these changes should support a BNS range of up to 190 Mpc for O4. Plans for O5 include installation of large aperture beam splitters and suspension improvements, implementation of a balanced homodyne readout system, and reduction of coating thermal noise via improved test-mass coatings. These changes could bring the detection rate of binary black hole mergers above one per day with commensurate increases in the detection ratess of other transients.

VIRGO Virgo [11] is an interferometric detector of GW located in Cascina, near Pisa, Italy, with an arm length of 3 km. It was funded by CNRS (France) and INFN (Italy). Virgo was designed with a particular attention to the low frequency range: its vibration isolation system, the Superattenuator [12], conceived by A. Giazotto, allows to push the “wall” of the residual seismic noise down to a few Hz. The construction of Virgo was completed in 2003 and a long activity of commissioning then started to progressively improve the detector sensitivity. Virgo performed its first science run in 2006. Since 2000 the detector site is managed by the European Gravitational Observatory, a CNRS-INFN consortium, recently joined by NIKHEF (The Netherlands). In the years 2011–2017 Virgo underwent a major upgrade through the Advanced Virgo [7] project, approved in December 2009, which allowed to significantly improve its sensitivity. The main features of the upgrade realized before the O2 run were: a larger beam spot on the test masses, heavier test masses (42 kg instead of 21 kg), thermal compensation system to cope with the aberrations on the test masses, improved stray light control, cryopumps at the end of the 3 km tubes. Advanced Virgo has started taking data on August 1, 2017, joining the two LIGO interferometers in the last part of the O2 run, with a sensitivity corresponding to a BNS inspiral range of ∼30 Mpc. Two weeks later Virgo detected its first GW event (GW170814 [13]). The event, also detected by the two LIGO interferometer, was the first triple detection. Three

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days later the three interferometers detected GW170817 [8], the coalescence of a two neutron stars, which marked the start of the multi-messenger astronomy. Between the end of O2 and the start of O3 Advanced Virgo underwent a phase of further upgrading, with the installation of fused silica suspensions on the test masses and a squeezer [14], and an extensive noise hunting campaign. The BNS inspiral range eventually achieved in the last part of O3 was ∼60 Mpc. The optical scheme of Advanced Virgo in the O3 configuration is shown in Fig. 3. Currently, Virgo is pursuing a two-phase upgrade named Advanced Virgo+. The main novelties of Phase 1 (before O4) are: a new fiber laser, the signal recycling, the implementation of frequency-dependent squeezing, with a target sensitivity of

Fig. 3 Optical scheme of the Advanced Virgo detector during the O3 run. NI,WI,NE,WE are the test masses, BS the beam splitter, PRM/SRM the power/signal recycling mirrors, CP the compensation plates, POP the pick-off plate, SIB1/2 the suspended injection benches, SDB1/2 the suspended detection benches, SPRB the suspended power recycling bench, SNEB/SWEB the suspended end benches, whereas Bn are the photodiodes. (Courtesy: Virgo Collaboration)

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90–115 Mpc BNS inspiral range. Between O4 and O5 new mirrors with improved coatings will be installed. The end test masses will have a diameter of 55 cm (instead of 35), allowing to enlarge the beam spot and reduce the thermal noise. The target sensitivity of the Phase 2 configuration is 145–260 Mpc.

KAGRA Compared to LIGO and VIRGO, KAGRA is technologically unique in two ways [9, 15]. First, it is located in an underground site in order to reduce seismic noise. In addition, KAGRA’s test masses are sapphire mirrors that are designed to be operated at cryogenic temperatures (∼20 K) in order to reduce thermal noise. KAGRA is designed as a resonant sideband extraction (RSE) interferometer, and quantum non-demolition techniques are planned to be applied to beat the standard quantum limit of displacement measurements. As a result, KAGRA is expected √ to reach an equivalent sensitivity to those of Advanced LIGO/Virgo; 2 × 10−24 / Hz at 100 Hz. The designed sensitivity is 140 Mpc in BNS range. In Japan, plans to construct interferometric gravitational wave detectors started in the 1980s. After the comparisons of a 100-m delay-line Michelson interferometer (TENKO-100) at the Institute of Space and Astronautical Science (ISAS) and a 20m Fabry-Perot Michelson interferometer at the National Astronomical Observatory of Japan (NAOJ), Japanese decided to construct a 300-m Fabry-Perot Michelson interferometer, called TAMA at NAOJ, which was successfully operated for more than 1000 h in 2001. However, since NAOJ is in a suburb of Tokyo, significant seismic noise due to human activities was inevitable below 100 Hz. In order to overcome the large seismic noise, it was decided to put a planned future interferometer underground. Although various experiments showcased the possible scientific achievements of the project and the plausibility of fundamental technologies, the proposal for developing a km-scale cryogenic detector took years to be formally approved in Japan. This was mainly due to the fact that there was no gravitational wave detection reported in 2000s, and thus it was judged to be too expensive and too risky. The project (named Large-scale Cryogenic Gravitational wave Telescope (LCGT)) was finally approved in 2010 for construction, and the excavation of the tunnels in Kamioka began in 2012 and finished in early 2014. During the construction, LCGT was given its nickname, KAGRA, chosen from a public naming contest. The name KAGRA is taken from KAmioka (the location) plus GRAvity; the Japanese word kagura reminds a type of traditional sacred dance accompanied by music dedicated to gods. KAGRA is constructed in Kamioka, Gifu, Japan, which is located 220 km northwest of Tokyo (Fig. 4). Kamioka also hosts the neutrino detectors, SuperKamiokande and KamLAND, which share the area with KAGRA. All the operations of KAGRA are controlled remotely from the office 5 km apart from the site (Fig. 5). The installations of the principal instruments, such as sapphire mirrors, large suspensions (Fig. 6), and cryogenic instruments, were completed in the summer of

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Fig. 4 Concept image of KAGRA: a 3-km cryogenic interferometer inside Ikenoyama mountain in Kamioka, Gifu, Japan. (Courtesy: KAGRA Observatory, ICRR, The University of Tokyo)

2019. Meanwhile two engineering runs were performed in April 2016 (as iKAGRA operation) [16] and April 2018 (bKAGRA phase-1 operation) [17]. In the latter, the performance of the large vibration isolation systems and cryogenic technology (down to 20 K) were demonstrated. From the summer 2019, the commissioning of the detector was initiated. This continued until March 2020, when the sensitivity got over 1 Mpc in BNS range with power-recycling technique, and KAGRA joined the international GW network.

International Gravitational Wave Network (IGWN) It was understood early in the development of interferometric gravitational-wave detectors that establishing confidence that a minute strain originated from a passing gravitational-wave signal would require its observation in more than one detector. Even in stable operations, an interferometer is susceptible to myriad disturbances that can mimic a gravitational wave in a single detector. It is also extremely difficult to measure the background in a single detector since we cannot shield the effect of gravitational waves. Data that is time-shifted between multiple detectors is used to go off-source and estimate the background in a multi-detector analysis. Moreover, multiple interferometers are required to determine the direction, orientation, and

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Fig. 5 Schematic image of the KAGRA interferometer. All the mirrors shown are suspended inside the vacuum tanks with four types of vibration isolation systems (Type-A, B, Bp, and C). IMMT/OMMT: input/output mode-matching telescope, IFI/OFI: input/output Faraday isolator, BS: beam splitter, PRM/SRM: the power/signal recycling mirrors, ITMX/ITMY: initial test masses, ETMX/ETMY end test masses. (Figure from [17])

distance to a gravitational-wave source, to measure the polarization of gravitational waves, and to maximize the scientific information that can be extracted. In short, having more detectors is better. In recognition of these facts and with the goal of establishing a long-term collaborative relationship, LIGO and Virgo first established an agreement in 2007 to work together to detect gravitational waves and use them as physical and astronomical probes. A key demonstration of the importance and success of this collaboration was the identification of GW170817 and its associated optical counterpart. In 2019, the agreement was revised to include KAGRA and the LIGO-VirgoKAGRA Collaboration now carries out its observational science program in unison, coordinating observing runs, and jointly planning and carrying out analyses. The LIGO-India detector, currently under construction in Maharashtra, India, is expected to join the network in the late 2020s. With the transition to routine astronomical operations, LIGO-Virgo-KAGRA are establishing the International Gravitational Wave Network (IGWN) to coordinate multiple facets of the research, development, construction, and operations of

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Fig. 6 Four types of KAGRA mirror suspension system. Type-A system is for the four main mirrors (ITMs and ETMs). Type-B system is for beam splitter and signal recycling mirrors. TypeBp system is for power recycling mirrors. Type-C system is used for other auxiliary mirrors. Type-A has the height of 14 m, of which top is located in the higher layer of the tunnel. (Figure from [17])

earth-based gravitational-wave detectors. As an initial step toward establishing an IGWN organization, the LIGO-Virgo-KAGRA computing and software development activities have been formulating a unified plan to provide the infrastructure and resources needed to maximize the scientific impact of the network. We envision IGWN as providing a structure in which to work together on common problems, to share infrastructure, to synchronize observation runs, and to collaborate on acquiring and analyzing data. IGWN also provides a common interface with other astronomy and physics communities and supports open access to gravitational-wave data and tools through the Gravitational-Wave Open Science Center (GWOSC). Interested readers can find more information using the links provided in Table 2.

Observing Runs The first observation run (O1) of Advanced LIGO started in September 2015 and lasted for 4 months. After a break to further improve the detectors, Advanced LIGO started the second observing run (O2) in November 2016. Advanced Virgo joined O2 in August 2017 and the run concluded after 9 months and many spectacular discoveries. The most recent observing run (O3) started on 1 April 2019 with both LIGO and Virgo operating and was planned for a year. Due to COVID-19, LIGO and Virgo O3 operations were interrupted at the end of March 2020 just as

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Table 2 List of webpages. GraceDB (Gravitational-Wave Candidate Event Database) provides information about candidate GW events. GWOSC (GW Open Science Center) provides data from gravitational-wave observatories, along with access to tutorials and software tools URL www.ligo.org www.virgo-gw.eu gwcenter.icrr.u-tokyo.ac.jp/en/ gracedb.ligo.org www.gw-openscience. org

LIGO Virgo KAGRA GraceDB GWOSC

Table 3 List of Observing Period Obs. Runs O1 O2 O3a O3b O3GK

Advanced LIGO Sep 12, 2015 to Jan 2016 Nov 30, 2016 to Aug 2017 Apr 1, 2019 to Sep 2019 Nov 1, 2019 to Mar 2020 –

Advanced Virgo 19, –

KAGRA –

25, Aug 1, 2017 to Aug 25, – 2017 30, Apr 1, 2019 to Sep 30, – 2019 27, Nov 1, 2019 to Mar 27, 2020 – Apr 7, 2020 to Apr 21, 2020

KAGRA was preparing to join the run. KAGRA started operating in April 2020, joining GEO600 which was operating as “Astrowatch” with the sensitivity ∼1 Mpc. Coincident GEO600 and KAGRA data is being analyzed as a part of O3b. Table 3 and Fig. 7 show the list of observing runs and its period.

Observational Science Highlights Since the beginning of observations with the advanced detectors in 2015, gravitational-wave astronomy has yielded many remarkable results.

Observing Run 1 (O1) On 14 September 2015, the LIGO detectors at Hanford and Livingston registered gravitational waves from the merger of a pair of black holes [18] for the first time in human history. This first direct detection of gravitational waves took place as the LIGO Scientific Collaboration was closing out the engineering run that preceded O1. Detailed analysis of the data around this event revealed the following:

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Fig. 7 Timeline table of Observing Runs up to O3b together with the sensitivities of the detectors in binary NS range, and future plans of Observing Runs. The planned start of O4 is in the middle of 2022 [10]

•

GW150914 [18] was the first detection of gravitational waves from a BBH merger. The masses of the pre-merger components are measured to be 30 M and 36 M (M : mass of the Sun), and the final remnant mass is 62 M . The luminosity distance to the source is approximately 400 Mpc. In addition to being the first direct measurement of gravitational waves, this event was also the first confirmed observation of a stellar-mass black-hole binary, and the first direct measurement of black holes of this mass.

The first detection was announced publicly on February 11, 2016, making headline news around the world. For the scientific community, the observation also brought a wealth of new results. The rate of BBH mergers similar to GW150914 was estimated to be 2–600 Gpc−3 yr−1 providing the first ever lower bound on the merger rate [19]. The observation enabled tests of general relativity via the dynamics of gravitational waves generated in the regime of strong field gravity alone [20]. The discovery also opened new research directions in astrophysics [21]. The first observing run (O1) lasted just over 4 calendar months and resulted in two additional detections [22]. GW151226 was identified in low-latency analysis of the data for BBH signals; it was a lower mass binary resulting in a final black hole of about 20 M . A third BBH signal LVT151012, with lower significance in the initial searches, was also identified.

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Observing Run 2 (O2) The second observing run (O2) started with LIGO’s two detectors in November 2016. Virgo joined O2 from July 2017. In the single month of August 2017, the three detector LIGO-Virgo network detected gravitational waves from 4 BBH mergers and the BNS merger GW170817. The localization of GW170817 using information from all three detectors enabled the discover of an optical counterpart in NGC4993 leading to an unprecedented follow-up effort by astronomers around the globe. Notable detections during O2 include: •

GW170814: The first gravitational-wave signal measured by the three-detector network came from a binary black hole merger. • GW170817 [8, 23]: The first gravitational-wave signal measured from a BNS merger – and also the first event also observed in light, by dozens of telescopes across the entire electromagnetic spectrum. This event is widely considered the beginning of the modern era of multi-messenger astronomy. The first Gravitational-Wave Transient Catalog (GWTC-1) [24] was released on Dec. 3, 2018, and includes 10 BBH mergers and 1 BNS merger.

Observing Run 3: The First 6 Months (O3a) The LIGO-Virgo Collaboration started observing run 3 (O3) on April 1, 2019, with the sensitivity 120–130 Mpc (LIGO) and 50 Mpc (Virgo) in BNS range. They took a 1-month commissioning break in October 2019 designating the first 6 months of observations as O3a. From the start of O3a, event alerts were publicly distributed within minutes of data acquisition. The events were available via GraceDB (Gravitational-Wave Candidate Event Database) including estimates of the false alarm rate for the event, localization of the source, probability that the signal was astrophysical, and the likelihood that source included a neutron star in the case of mergers. By the end of O3, these alerts were automatically distributed within about 10 min with updated information provided within hours or when available. As of May 2021, the second Gravitational-wave Transient Catalog (GWTC-2) [25] has been released including events from O3a. This includes 46 BBHs, 2 BNSs, and 2 other binaries for which may be BBH or NSBH. Figure 8 shows the mass posteriors for events in GWTC-2. Notable events from O3 include: •

GW190412: The first BBH with definitively asymmetric component masses, which also shows evidence for higher harmonics in its waveform. • GW190425: The second GW event consistent with a BNS, following GW170817. • GW190426_152155: A low-mass event consistent with either an NSBH or BBH.

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Fig. 8 Results after O3a (Apr.1/2019–Sep.30/2019). GWTC-2 (released on Oct. 28, 2020) has 46 BBHs, 2 BNS, and 2 unknown companion binary with BH. The graphic shows BHs (blue), NSs (orange), and compact objects of uncertain nature (gray). (Credit: LIGO Virgo Collaboration/Frank Elavsky, Aaron Geller/Northwestern University)

• • •

•

•

GW190514_065416: A BBH with the smallest effective aligned spin of all O3a events. GW190517_055101: A BBH with the largest effective aligned spin of all O3a events. GW190521: A BBH with total mass over 150 M , the first evidence of the intermediate mass BH (over 100 M , and less than the super-massive BH ∼104 M ). GW190814: A highly asymmetric system, corresponding to the merger of a 23 M BH with a 2.6 M compact object, making the latter either the lightest BH or heaviest NS observed in a compact binary. GW190924_021846: The lowest-mass BBH, with both BHs exceeding 3 M .

GWTC-2 was released together with companion papers analyzing the population properties [26] and using the events to test general relativity [27]. As we see in Fig. 9, the number of detected events is monotonically increasing according to the effective BNS spacetime volume V T surveyed; V T is defined by the spatial sensitive volume of the search integrated over time. Intuitively, this is the average spatial sensitive volume of the second-most sensitive detector in the network multiplied by the live-time of the that network configuration. Figure 10 shows 90% credible region of mass M and mass ratio q for all candidate events. Using the events in GWTC-2, we estimate merger rates for BBH to be RBBH = −3 −1 −3 −1 23.9+14.9 yr , and for BNS to be RBNS = 320+490 yr . The −8.6 Gpc −240 Gpc primary mass distribution of BH is not well-described as a simple power-law model, but with additional single peak around 37 M . The minimum mass of black holes

30 LIGO, VIRGO, and KAGRA as the International Gravitational Wave Network Fig. 9 The number of compact binary coalescence detections versus the effective volume-time to which the gravitational wave network is sensitive to BNS coalescences. (Fig. 1 from the GWTC-2 paper [25])

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Fig. 10 Credible region contours for all candidate events in the plane of total mass M and mass ratio q. Each contour represents the 90% credible region for a different event. (Fig. 6 from the GWTC-2 paper [25])

in BBH systems is constrained to be Mmin < 6.6 M . There are a couple of events with masses in the range of 2–5 M for which it has not been possible to determine if they are black holes or neutron stars. There is evidence of spin-induced, general relativistic precession of the orbital plane and examples of component spin that is anti-aligned with respect to the orbital angular momentum of the binary. The latter suggests the dynamical formation of the binary. There is no evidence of violation of general relativity.

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Outlook The direct measurement of gravitational waves in 2015 established gravitationalwave detectors as powerful new tools for exploring the Universe. When combined with electromagnetic telescopes and particle detectors, they extend our view into the nature of gamma-ray bursts, neutron star structure, and cosmology. Over the next 5 years, the sensitivity of the LIGO-Virgo-KAGRA network should improve by up to 2–3 times in amplitude sensitivity over O3 [10]. That will bring the detection rate of compact binary mergers close to 1 per day with signal-tonoise of the most significant candidates in the 50–100 range. The majority of these detections will involve binary black hole mergers providing a rapidly increasing sample with which to study the properties and mechanisms of binary black hole formation. Moreover, the reduced statistical uncertainty on measured parameters of the loudest events will enable more stringent tests of general relativity in the context of binary black hole mergers. While GW170817 is the only binary neutron star merger observed both in gravitational and electromagnetic waves to date, early identification of gravitationalwave transients and distribution of alerts about them is a high priority for the community. By the end of O3, the LVK was publicly releasing alerts within minutes of data acquisition for compact binary mergers of all sorts. Unfortunately, the candidates thought most likely to contain a neutron star were not well localized during O3 making the identification of electromagnetic counterparts extremely challenging, and no high-confidence associations were found. It was gratifying to see improved coordination of follow-up observations across the community: this bodes well for future runs. Moreover, an improved alert system was deployed and shown to be capable of identifying coalescing neutron-star systems and alerting the broader community from seconds to minutes before merger [28, 29]. As the sensitivity of the IGWN detectors improves and the number of useful baselines grows, transient events will be better localized. Improving the sensitivity of all detectors operating in the network is the best way to improve the likelihood of identifying the electromagnetic counterpart of gravitational-wave sources. We look forward to the identification of multi-messenger sources becoming more routine over the course of the decade. While it is appropriate at this time to emphasize the breakthroughs associated with the observations of compact binary mergers over the past years and the opportunities that increased detection rate affords, the gravitational-wave discovery space remains largely untapped beyond compact binaries. Searches for a gravitationalwave stochastic background have constrained the dimensionless energy density ΩGW ≤ 5.8 × 10−9 at the 95% credible level for a flat (frequency-independent) GWB [30]. Searches for gravitational waves from spinning neutron stars in our Galaxy continue to set tighter limits on gravitational-wave emission from known pulsars and from as yet unidentified sources [31–34]. The possibility of detecting gravitational waves hypothesized sources such as cosmic strings [35], dark matter [36], supernovae [37], or as yet unknown astrophysical sources or events is tantalizing.

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As we enter a new decade, physicists and astronomers are considering the future paths for gravitational-wave detectors. The LIGO-Virgo-KAGRA Collaboration are establishing the International Gravitational Wave Network (IGWN) which is conceived as an organization to facilitate coordination among gravitational-wave detector projects across the globe. The initial effort is being invested in establishing a unified computing plan under the IGWN umbrella to introduce efficiencies by sharing common infrastructure and tools. The A+ and AdV+ projects involve significant upgrades to the detectors in the LIGO and Virgo facilities that will be completed for O5 which is anticipated to start around 2025. Both LIGO and Virgo have initiated studies to consider options for upgrades in the existing facilities that would dovetail with the constructions and operations of next generation facilities. The LIGO-India [38] project is also under way and should join the international network toward the end of the decade. Work has continued to ramp up on next generation facilities featuring detectors with arm lengths of 10 to 40 km that could begin operations in the early to mid2030s. Einstein Telescope [39] is being pursued in Europe as an underground facility to house multiple detectors in triangular configuration. Cosmic Explorer [40, 41] is being pursued in the United States as an aboveground facility with interferometers at different locations. Specialized detectors, such as the Australian NEMO [42], are also being proposed to pursue specific science goals. With the successes of the past decade and the possibilities of future observations and facilities, the dream of gravitational-wave astronomy is now reality.

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32. Abbott R et al (LIGO Scientific Collaboration, Virgo Collaboration, KAGRA Collaboration) (2021) Constraints from LIGO O3 data on gravitational-wave emission due to r-modes in the glitching pulsar PSR J0537-6910. arXiv:2104.14417 33. Abbott R et al (LIGO Scientific Collaboration, Virgo Collaboration, KAGRA Collaboration) (2021) Diving below the spin-down limit: constraints on gravitational waves from the energetic young pulsar PSR J0537-6910. Astrophys J Lett 913 34. Abbott R et al (LIGO Scientific Collaboration, Virgo Collaboration) (2021) All-sky search in early O3 LIGO data for continuous gravitational-wave signals from unknown neutron stars in binary systems. Phys Rev D 103(6):064017 35. Abbott R et al (LIGO Scientific Collaboration, Virgo Collaboration, KAGRA Collaboration) (2021) Constraints on cosmic strings using data from the third Advanced LIGO-Virgo observing run. Phys Rev Lett 126:241102 36. Abbott R et al (LIGO Scientific Collaboration, Virgo Collaboration, KAGRA Collaboration) (2021) Constraints on dark photon dark matter using data from LIGO’s and Virgo’s third observing run. arXiv:2105.13085 37. Abbott BP et al (LIGO Scientific Collaboration, Virgo Collaboration) (2020) Optically targeted search for gravitational waves emitted by core-collapse supernovae during the first and second observing runs of Advanced LIGO and Advanced Virgo. Phys Rev D 101(8):084002 38. Saleem M et al (2021) The science case for LIGO-India. arXiv:2105.01716 39. Punturo M et al (2010) The Einstein telescope: a third-generation gravitational wave observatory. Class Quant Grav 27:194002 40. Reitze D et al (2019) Cosmic explorer: the U.S. contribution to gravitational-wave astronomy beyond LIGO. arXiv:1907.04833 41. Abbott BP et al (LIGO Scientific Collaboration) Exploring the sensitivity of next generation gravitational wave detectors. Class Quant Grav 34(4):044001 (2017) 42. Ackley K et al (2020) Neutron star extreme matter observatory: a kilohertz-band gravitationalwave detector in the global network. Publ Astron Soc Aust 37:e047

Part IV Gravitational Wave Modeling

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Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspirals Soichiro Isoyama, Riccardo Sturani, and Hiroyuki Nakano

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Goal and Relation to Other Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Essence: Quadrupole Radiation from a Mass in Circular Orbit . . . . . . . . . . . . . . . . . . . Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stationary Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Post-Newtonian Gravitational Waveforms for Spinning, Nonprecessing Binary Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PN Binding Energy, Energy Flux, and BH Horizon Flux . . . . . . . . . . . . . . . . . . . . . . . . . . Balance Equation for Slowly Evolving Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy of the Post-Newtonian Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time- and Frequency-Domain Inspiral Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Taylor Time-Domain Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Taylor Frequency-Domain Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond Spinning, Nonprecesssing Binary Black Hole Cases . . . . . . . . . . . . . . . . . . . . . . Full Inspiral-Merger-Ringdown Waveform Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective-One-Body (EOB) Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenological (IMRPhenom) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S. Isoyama () School of Mathematical Sciences, University of Southampton, Southampton, UK e-mail: [email protected] R. Sturani International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal, Brazil e-mail: [email protected] H. Nakano Faculty of Law, Ryukoku University, Kyoto, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_31

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Abstract

To enable detection and maximize the physics output of gravitational wave observations from compact binary systems, the availability of accurate waveform models is crucial. The present work aims at giving an overview for nonexperts of the (inspiral) waveforms used in the gravitational wave data analysis for compact binary coalescence. We first provide the essential elements of gravitational radiation physics within a simple Newtonian orbital dynamics and the linearized gravity theory, describing the adiabatic approximation applied to binary systems: the key element to construct the theoretical gravitational waveforms in practice. We next lay out the gravitational waveforms in the postNewtonian approximation to general relativity and highlight the basic input for the inspiral waveform of the slowly evolving, spinning, nonprecessing, and quasicircular binary black holes, namely, post-Newtonian energy, fluxes, and the (absorption-corrected) balance equation. The post-Newtonian inspiral templates are then presented both in the time and frequency domain. Finally, including the merger and subsequent ringdown phase, we briefly survey the two families of the full waveform models of compact binary mergers currently implemented in LSC Algorithm Library Simulation: the effective-one-body approach and the phenomenological frequency-domain model.

Keywords

Binary black holes · Compact object binaries · Gravitational waves · Post-Newtonian approximation · Adiabatic approximation · Stationary phase approximation · Inspiral-merger-ringdown waveforms · LSC Algorithm Library

Introduction The recent detection of gravitational waves (GW) from compact binary coalescences made by the large interferometers LIGO [1] and Virgo [19] opened the era of GW astronomy, triggering scientific interests over all aspects of GW production and detection. This new, exciting branch of scientific activity is even bursting now, with the third GW interferometer: KAGRA online [27]; for a useful summary of these detectors, see  Chap. 2 “Terrestrial Laser Interferometers”. Because the GW signals are in general drowned in a much larger noise, their extraction from data stream and correct interpretation crucially depend on the quality of theoretically predicted template waveforms. Experimental data are processed via matched-filtering techniques; see, e.g., Ref. [28], which are particularly sensitive to the phase of GW signals; hence, a prediction of waveforms with absolute  O(1) precision (in the phase) is very important, especially for a correct parameter estimation and in general for maximizing the physics output of detection (see,

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1 0.8 0.6 0.4 0.2 0 -0.2

NR GW waveform Match

-0.4 0

0.2

0.4

0.6 Time [s]

0.8

1.2

Fig. 1 Inspiral-merger-ringdown (IMR) signal from the numerical-relativity simulation SXS:BBH:0305 [47] which models GW150914. The horizontal axis is the real time. Here, we present only the (scaled) “+” GW mode (blue thin curve). The red thick curve shows the “match,” i.e., the accumulation of the (normalized) signal-to-noise ratio in white noise

e.g.,  Chap. 41 “Principles of Gravitational-Wave Data Analysis” for differences between GW searches and parameter estimations). From the point of view of theoretical modelling, the process of binary coalescence can be divided into three distinct phases: inspiral, merger, and ringdown (Fig. 1). During the inspiral phase, due to large binary separation with respect to their size, the dynamics can be efficiently described within the post-Newtonian (PN) approximation to general relativity (GR), the small expansion parameter being GM v2 ∼ 2  1, 2 c R c

(1)

where M, v, and R denote, respectively, the total mass, characteristic orbital velocity, and characteristic separation of the binary system. A n-PN correction to the gravitational potential is then given by any term of the type (M/R)n−j v 2j with integers 0 ≤ j ≤ n. The GR corrections to the Newtonian dynamics around flat Minkowski spacetime are then incorporated into the equation of motion and Einstein’s field equations (expanded around the Minkowski metric) order by order in that small parameter. An excellent textbook devoted to the PN theory is Gravity by Poisson and Will [157]; more expert-oriented reviews (including more recent developments) are provided by Blanchet [42], Schäfer and Jaranowski [169], and Futamase and Itoh [88]. Recently, the PN approximation to GR has been developed in an effective field theory description [92], dubbed non-relativistic GR (NRGR), which resulted in an alternative, Independent, and equivalent derivation of the PN

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dynamics (see Refs. [84, 91, 124, 158] for reviews, and  Chap. 32 “Effective Field Theory Methods to Model Compact Binaries”). It is a remarkable feature of GR that the two-body dynamics in the inspiral phase is exactly the one of two (structureless) point particles up to 5PN when finite finitesize effects come into play, according to the effacement principle [64]. However, finite size effects that tidally “deform” a black hole (BH) have been shown to vanish for a spherical, non-spinning BHs [41, 69, 94, 115] in the static limit, and there are indications that they vanish for spinning BHs, too [151,155] (see also [59,121,122]). On the other hand, the tidal deformation effects for material bodies like neutron stars (NS) are expected, depending on the equation of state of matter in NSs, and there have been some evidence of their detection in GW170817 [11]. Considering circular orbit, in the nonprecessing case, is usually accurate since angular momentum radiation is more efficient than energy radiation; thus, binary orbits tend to circularize at early stage [152]. However, while unlikely, it is not excluded that some observed GW signals can show a non-negligible eccentricity due to, e.g., dynamical formation of binary BHs (BBHs) presented in  Chap. 16 “Dynamical Formation of Merging Stellar-Mass Binary Black Holes”. When spins are included in the binary dynamics, in general, one has to also take into account orbital precession caused by the orbital angular momentum-spin coupling, as well as the relativistic precession of the spins. As the binary’s orbital separation shrinks by decreasing the binding energy through the emission of GWs (with an increase in relative velocity of the constituents), the system enters the merger phase where two compact objects form one single object. At the merger phase, the PN approximation breaks down, v  1, and full numerical-relativistic simulations have to be used, where the nonlinear radiative dynamics of the binaries is obtained by directly solving the exact Einstein field equations numerically: see, e.g., a recent text by Baumgarte and Shapiro [34] and  Chap. 34 “Numerical Relativity for Gravitational Wave Source Modeling”. Immediately after the merger, a perturbed BH forms which rapidly damps its excitation in a ringdown phase, whose GW emission is described by a superposition of damped sinusoids [39, 114, 146]. Current inspiral-merger-ringdown (IMR) waveform models for data analysis applications combine input from the PN theory and NR simulations. The models implemented in the LSC Algorithm Library (LAL) Simulation [127] are summarized in “Enumeration Type Documentation” in Ref. [61]. The GW signal models used for the analyses of observed GW events so far are described in: GW150914: first BBH detection [8, 9], GW170817: first binary NS (BNS) detection [4, 11], events from the first Gravitational Wave Transient Catalog: first and second observation runs, O1 and O2 henceforth, including the above two GW events, along with GW151012, GW151226, GW170104, GW170608, GW170729, GW170809, GW170814, GW170818, and GW170823 [3], GW190412: O3, unequal mass (mass ratio ∼3 with nonzero spin for the heavier object) BBH [13],

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GW190425: O3, second BNS detection [7], GW190814: O3, highest mass ratio between binary constituents (∼10) and lighter object in the low “mass gap” [15], GW190521: O3, whose remnant can be considered as the first ever evidence of a light intermediate mass (∼102 M ) BH [14, 17], events from the second Gravitational Wave Transient Catalog: the first half of the third observing run (O3a), including 39 GW events [16].

Goal and Relation to Other Chapters Our purpose with this chapter is twofold: (i) to make a compilation of the PN-based GW waveform models for the binary inspirals in literature and (ii) to provide a “catalogue” of IMR waveform families implemented in the LALSimulation. The main target audience is graduate students (as well as researchers) who work in areas outside of the GW source modelling of binaries, but who need a working knowledge of waveform models. We set the stage with the basics of gravitational radiation physics, using the simple quadrupole formalism in the linearized gravity theory. We next cover the GW waveforms in the PN theory, collecting the state of art of PN binding energy and energy flux of slowly evolving, spinning, and nonprecessing BBHs, and present their inspiral PN templates both in the time and frequency domains. Then, we briefly overview the two main families of the full IMR waveforms: the effectiveone-body approach and the phenomenological frequency-domain model, used and implemented in LALSimulation so far. We conclude our chapter with some of remaining issues and future prospects.

Notations We use the metric signature (− + ++) and the standard geometrized unit (c = 1 = G) with the useful conversion factor 1M = 1.477 km = 4.926 × 10−6 s, throughout. Greek indices μ, ν, . . . denote 0, 1, 2, 3 (or, e.g., t, x, y, z), while the Latin letters i, j, . . . run over 1, 2, 3. For a binary with masses ma and spin vector Sa (where a = 1, 2 labels each compact object), we will use the notation defined in Ref. [3]. The total mass of a binary is M ≡ m1 + m 2 ,

(2)

and the mass ratio is q≡

m2 ≤ 1. m1

(3)

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We also use the symmetric mass ratio η≡

m1 m2 , M2

(4)

and the reduced mass μ ≡ η M. The spin vector Sa has a conserved magnitude, as long as absorption effects are neglected, by using an appropriate spin supplementary condition to remove unnecessary degrees of freedom associated with the spin, and we denote its dimensionless magnitude by χa ≡ |χ a | ≡

|Sa | . m2a

(5)

Finally, the O(v 2n ) terms relative to the Newtonian dynamics will be referred to as the n-th PN order.

The Essence: Quadrupole Radiation from a Mass in Circular Orbit The essence of GW generation formalism for a binary inspiral can be understood from the Newtonian orbital dynamics and the Einstein’s quadrupole formalism based on the linearized gravity theory. While a raw approximation, the linearized gravity theory catches the basic concepts behind the GW signal calculation, without the intricacies of the full GR nonlinearity (see, e.g., Chapter 11 of Poisson and Will [157] for rigorous derivation of the quadrupole formula and details about the PN treatment). We here review the main line of reasoning and results, modelled on nice tutorials by Flanagan and Hughes [83] and the LVC collaboration [12]. The material covered in this section is fairly standard, and the details of derivation are given in various introductory GW and GR texts, including Landau and Lifshitz [119] and Maggiore [130]. We start by supposing that the full spacetime metric (i.e., gravitational field) gμν deviates only slightly from the background flat metric ημν ≡ diag(−1, 1, 1, 1): gμν = ημν + hμν ,

||hμν ||  1 ,

(6)

where hμν is a small metric perturbation (i.e., weak gravitational field) of the background Minkowski spacetime and ||hμν || is a typical magnitude of hμν . In the linearized gravity theory, everything is consistently expanded to linear order in hμν , neglecting all higher-order terms, and all indices are raised and lowered with the Minkowski metric ημν . Also, we assume that the decomposition of the metric (6) is always preserved in any coordinate system that we can chose. The general covariance of full GR is then restricted to an infinitesimal coordinate transformation x = xμ + ξ μ , μ

(7)

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where ξ μ (x) is an infinitesimal vector field, and the metric perturbation changes via h μν = hμν − ∂μ ξν − ∂ν ξμ .

(8)

(Note the close analogy to the gauge transformation of the vector potential Aμ in electromagnetic theory, i.e., A μ = Aμ − ∂μ χ with a scalar field χ . This is why Eq. (8) is often referred to as a gauge transformation in the linearized gravity theory). The Einstein’s field equations in the linearized theory are best described under the Lorenz-gauge conditions ∂ μ h¯ μν = 0 ,

(9)

in terms of the trace-reversed perturbation 1 h¯ μν ≡ hμν − ημν hρ ρ , 2

(10)

which can be enforced without loss of generality by making use of the coordinate freedom (8). The Lorenz-gauge conditions reduce the Einstein’s equations to a simple, decoupled wave equation for h¯ μν given by  h¯ μν = −16π Tμν ,

(11)

where  ≡ ∂ μ ∂μ is the d’Alembertian operator and Tμν is the energy-momentum tensor of the matter. Equation (11) can be solved by the method of Green’s function G(x, x  ) (for the operator ), imposing suitable boundary conditions. For the problem of the GW radiation, just like in electromagnetism, the appropriate choice is the retarded Green’s function: G(x, x  ) = −

1 δ(t  − tret ) , 4π |x − x |

(12)

where tret ≡ t − |x − x | is the retarded time, accounting for the propagation delay between the source at x and the observer at x. The associated retarded solution of Eq. (11) is then given by h¯ μν = 4



d 3x

Tμν (t − |x − x |, x ) . |x − x |

(13)

For the GW calculation, again like the radiation in the electromagnetism, we are particularly interested in the behavior of the solution (11) in the (far-away) wave zone, where the length of the position vector r ≡ |x| is larger than the characteristic wavelength of the GW radiation λChar (defined by λChar = tChar in terms of the characteristic time scale of the change in the source tChar ). The characteristic radius rChar ∼ v tChar of the matter distribution is then smaller than λChar , justifying the

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2 /r) with ni ≡ x i /r, the zero-th solution expansion |x − x | = r − ni x  i + O(rChar in the (far-away) wave zone is given by

4 h¯ μν = r



d 3 x  Tμν (t − r, x ).

(14)

Here, not only the terms that fall off as O(1/r 2 ) and higher are neglected, but there is an additional approximation: all elements of the extended source contribute to field at the same retarded time. At this point, we must identify the truly radiative degrees of freedom contained in Eq. (14). Apparently, all six degrees of freedom in hμν (hμν have ten components, and four are constrained by the Lorenz-gauge condition (9)) look radiative. However, they are not; only two are actually radiative. The remaining four are nonradiative degrees of freedom tied to the matter (like the Coulomb piece of the electromagnetic field), and their wave-like behaviours are merely artefact due to the Lorenz-gauge formulation of linealized Einstein equation (11). Indeed, a detailed analysis of the linerlized gravity theory by Flanagan and Hughes [83] reveals that the radiative degrees of freedom are just encoded in the spatial transverse-traceless components hTT ij of hμν , which satisfies the following four conditions (in addition to the Lorenz gauge condition): ∂ i hTT ij = 0,

hiiTT = 0.

(15)

Importantly, hTT ij is coordinate-invariant (gauge-invariant) under the transformation (8) (as long as the full metric (6) remains asymptotically flat), and hence, it is the direct observable of GW detectors, e.g., LIGO, Virgo, and KAGRA. In the (faraway) wave zone, hTT ij can be obtained by projecting hμν onto the plane orthogonal to the direction of propagation n = x/r and subtracting its trace. We introduce the projector (this is sometimes referred to as Lambda tensor [130]) 1 Λij, kl ≡ Pik Pj l − Pij Pkl , 2

(16)

where Pij = δij − ni nj , and we obtain 4 Λij, kl h¯ TT ij = r



d 3 x  T kl (t − r, x ).

(17)

For the final step, we simplify the integral on the right of Eq. (17), making use of the energy-momentum conservation ∂μ T μν = 0 repeatedly to obtain the convenient relation    3  ij 2 3  00  i  j . (18) 2 d x T = ∂t d x T x x

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Defining a (Newtonian) symmetric trace-free quadrupole moment by  Qij (t) ≡

  1   2 , d x ρ(t, x ) x i x j − δij x 3 3 



(19)

where ρ ≡ T 00 is the Newtonian mass density and δij ≡ diag(1, 1, 1) is the Kronecker-delta. Inserting Eq. (19) back into Eq. (17), we arrive at hTT ij (t, x) =

2 d 2 Qkl Λij, kl (n) (t − r). r dt 2

(20)

This is the well-known “quadrupole formula” for the GW signal. The two radiative degrees of freedom contained in hTT ij can be conveniently extracted by introducing two unit polarization vectors p and q, which are orthogonal to the propagation direction n and to each other (satisfying ni nj + pi pj + qi qj = δij ). The tensor hTT ij is then decomposed into two independent “plus” and “cross” polarization modes of GWs: hTT ij = h+ (pi pj − qi qj ) + h× (pi qj + qi pj ).

(21)

For example, if we use a Cartesian coordinate system x = (x, y, z), h+ and h× modes of GWs that propagate in the z-direction (so that n = (0, 0, 1)) are   d 2 Qyy 1 d 2 Qxx − , h+ (t, z) = r dt 2 dt 2 2 d 2 Qxy h× (t, z) = . r dt 2

(22)

We now apply the GW polarization modes (22) to the Newtonian binary system in a fixed circular orbit with the separation R and the orbital frequency Ω (we will momentarily delay to discuss the back-reaction on the motion due to the GW emission from the system). To compute the quadrupole moment Qij , we continue to use the Cartesian coordinate system and assume that the orbit lies in the x–y plane whose center of mass is at the coordinate origin. In this setup, we have   R2 Qij = μ xi xj − δij 3

(23)

with x = R cos(Ωt + π/2), y = R sin(Ωt + π/2), and z = 0 (so that the binary initially at x(0) = 0 and y(0) = R when t = 0). The second derivative of Eq. (23) is

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d 2 Qyy d 2 Qxx 2 2 = 2 μ R Ω cos(2Ω t) = − , dt 2 dt 2 d 2 Qxy = 2 μ R 2 Ω 2 sin(2Ω t). dt 2

(24)

Making substitution in Eq. (22), we obtain 4μ (ΩR)2 cos(2Ωtret ) , r 4μ (ΩR)2 sin(2Ωtret ) , h× (t) = r h+ (t) =

(25)

where the retarded time tret ≡ t − r has been introduced. We notice that the quadrupole radiation is at twice the orbital frequency Ω. From the observational point of view, the GW detector only sees the radiation in the direction that points from the binary to the detector, and it is not always the same as z-direction like Eq. (25) in general. When the GW propagates in a direction of the line of sight to the binary ni ≡ (sin θ sin φ, sin θ cos φ, cos θ ), the GW polarization that an observer measures is conveniently rewritten in the form (see, e.g., Chapter 3 of Maggiore [130]) 4 h+ (t) = M 5/3 Ω 2/3 r h× (t) =



1 + cos2 θ 2

 cos(2Ω tret + 2φ) ,

4 5/3 2/3 M Ω cos θ sin(2Ω tret + 2φ) , r

(26)

where we use the Kepler’s law for the circular orbit frequency Ω 2 = M/R 3 . Here, the combination of body’s mass, the chirp mass:M , is defined by M ≡ η3/5 M.

(27)

We remark that the amplitudes of the GW polarization at fixed Ω depend on the binary masses only through the chirp mass M (within the quadrupole approximation).

Adiabatic Approximation Our discussion has so far assumed that the GW is emitted from a given, fixed, (Newtonian) circular orbit. However, GW can transport energy, momentum, and angular momentum away from the binary, and hence the GW radiation actually drives an evolution of the circular orbit at the same time; the orbital separation R will slowly shrink, causing the gradual radiative inspiral of the binary. In this

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subsection, we improve the GW signals (26), accounting for the inspiral motion of the binary due to the radiative losses. The total radiated power in all directions around the binary (i.e., the total energy flux of GWs) that results from the quadrupole GW signal (20) is in general given by (see, e.g., Chapter 12 of Poisson and Will [157] for a derivation) Newt F∞

3 1  = 5

i,j =1



d 3 Qij d 3 Qij dt 3 dt 3

,

(28)

where . . . stand for time average, which is well-known as the quadrupole (“Newtonian”) formula for the GW energy flux. Inserting here the quadrupole moment of the circular binaries (23), we obtain Newt = F∞

32 2 10 η v , 5

(29)

√ where v ≡ M/R = (MΩ)1/3 is the (relative) orbital velocity. We now assume the so-called adiabatic approximation, in which a circular orbit evolves with a slowly changing orbital velocity v, so that its fractional change over an orbital period is negligibly small. In this approximation, the source of the GW energy fluxes comes from the sum of the kinetic and the (Newtonian) binding energy of the binary Mη 2 v , 2

(30)

dE Newt Newt = −F∞ . dt

(31)

E Newt = − and hence, it obeys the balance equation

The balance equation implies that the typical timescale of the radiation reaction is tRR ≡

E Newt ∼ Mη−1 v −8 , dE Newt /dt

(32)

which is much longer than the orbital time scale tOrb ∼ M v −3 ,

(33)

as far as the orbital velocity is small (or the orbital separation is large), v = √ M/R  1. Therefore, the adiabatic approximation is mostly valid during the inspiral phase of the orbital evolution, but it breaks down as the binary separation approaches to the last stable orbit (whose typical orbital radius is close to that of the

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Innermost Stable Circular Orbit (ISCO) in Schwarzschild geometry with the mass M, rISCO = 6 M), where the transition from the inspiral phase to the merger phase approximately occurs. In the adiabatic approximation, we can use the balance equation (31) to derive the evolution equation for any binary parameters. For instance, we have the evolution equation of the orbital velocity v(t): dv 32 η 9 = v . dt 5 M

(34)

The time to coalescence tc is formally defined by the time it takes the velocity to evolve from an initial value vIni to v → ∞, and it is tc =

5 M 5 = 8 256 η vIni 256



RIni M

4

(1 + q)2 M, q

(35)

−2 where RIni = MvIni denotes the initial orbital velocity and radius, respectively. Similarly, the differential equation

dR = dt



dE dR

−1

64 ηM 2 dE =− , dt 5 R2

(36)

determines the shrinking rate of the orbital separation R(t). Assuming r(tc ) = 0, we have the solution  R(t) =

256 η M3 5

1/4 (tc − t)1/4 .

(37)

In turn, recall the Kepler’s third law Ω 2 = M/R 3 , it leads the increasing orbital frequency  Ω(t) =

1 5 256 tc − t

3/8

M −5/8 .

(38)

Importantly, the frequency evolution depends on the binary masses only through the chirp mass M in the zero-th order approximation leading to Eq. (20). We next wish to describe the impact of these time-dependent binary parameters on the GW signals (26) within the adiabatic approximation. Recall the discussion of the quadrupole GW generation formalism in the preceding subsection; the orbital frequency Ω and the GW phase 2Ω tret have to be promoted to {Ω, 2Ω tret } → {Ω(tret ), φ(tret )} .

(39)

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Here, we introduce a new accumulated phase of GW (associated with the timedependent orbital frequency (38)) by  φ(t) ≡ 2



1 tc − t Ω(t)dt = −2 5 M

5/8 + φc ,

(40)

and φc ≡ φ(t = tc ) is the phase at the time to coalescence tc . Inserting Eq. (39) into Eq. (26), we arrive at

    1 + cos2 θ 1 tc − t 5/8 cos −2 + φc , 2 5 M

     5M 1/4 M 1 tc − t 5/8 h× (t) = cos θ sin −2 + φc . (41) r tc − t 5 M M h+ (t) = r



5M tc − t

1/4 

This is the GW polarization from inspiring quasicircular binaries that an observer measures in a quadrupole approximation. Both the amplitudes and the frequency in Eq. (41) increase as the coalescence time is approached: this is referred to as “chirping” (in fact, the “chirp mass” M is named after it), and we often call Eq. (41) the chirp signal. Also, we note that both the amplitude and the phase still depend on the binary masses only through the M . This explains why the chirp mass is well determined in the GW data analysis, compared with the component masses of the binary, ma .

Stationary Phase Approximation An alternative to the time-domain chirp signal (41) is its frequency-domain (i.e., Fourier domain) representation, which is also commonly used for the GW data analysis applications. The time-domain chirp signal takes the schematic complexexponential form of h(t) = A(t)e−iφ(t) , and its Fourier transform is ˜ )= h(f

 dt A(t) ei(2πf t−φ(t)) ;

(42)

˜ note that h(−f ) = h˜ ∗ (f ), so we can assume f > 0. Since the amplitude of the chirp signal evolves much slower than the phase, i.e., d ln A(t) dφ(t)  , dt dt

(43)

the stationary phase approximation to the integral provides a good approximation of ˜ ): h(f

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˜ )  √ A(t) h(f ei (ΨSPA (t∗ )−π/4) , dF /dt (t∗ )

(44)

ΨSPA (t∗ ) ≡ 2πf t∗ − φ(t∗ ) ,

(45)

where

and t∗ is a function of f defined as dφ(t∗ ) ≡ 2πf , dt∗

(46)

and the time at the stationary point t∗ is determined by the Fourier variable f being equal to the instantaneous frequency dφ(t)/dt at t = t∗ , The explicit calculation of the stationary phase approximation to the time-domain chirp signals (41) is worked out in, e.g., Chapter 4 of Maggiore [130]. The resultant frequency-domain chirp signal is h˜ + (f ) = h˜ × (f ) =

1 π 2/3 1 π 2/3



5 M 5/6 ei Ψ+ (f ) 24 f 7/6 r



1 + cos2 θ 2

 ,

5 M 5/6 ei Ψ× (f ) cos θ , 24 f 7/6 r

(47)

where the phases are given by Ψ+ (f ) ≡ 2πf (tc ) − φc − Ψ× (f ) ≡ Ψ+ (f ) +

3 π + (π M f )−5/3 , 4 128

π . 2

(48)

Post-Newtonian Gravitational Waveforms for Spinning, Nonprecessing Binary Black Holes In our discussion so far, we have restricted our analysis to the Newtonian orbital dynamics in the linearized gravity theory (with the quadrupole formula for the GW fluxes). While this treatment has provided an adequate description of the dynamics of the binary pulsars (e.g., the secular change in the orbital period of the Hulse-Taylor binary: PSR B1913+16; see a review [129]), it is not accurate modelling of the GW signals emitted from inspiralling astrophysical binary system; the chirp signals (41) are just inconsistent with the observed GW data of, e.g., BBHs GW150914 [8] and BNS GW170817 [11]. For the latter case, one must rely on an improved approximation method in GR to have a more refined wave-generation formalism.

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The perfect starting point for that discussion is the “post-Newtonian (PN) theory,” applied to the two-body problem. The PN theory is a systematic approximation method to exact GR, solving the Einstein field equation (and the equation of the motion for a source) in the form of power series in small physical parameters v  1,

M ∼ v2  1 , R

(49)

namely, the two-body system is assumed to move slowly (with a large separation) and to be in the weak gravitational field. Therefore, the PN theory provides the most natural tool to model the early inspiral stage for LIGO-Virgo-KAGRA binary mergers. The technical developments of the general wave-generation formalism in the PN theory is far more involved than the quadrupole moment formalism in the linearized theory, and we shall continue to specialize our discussion to GW signals emitted by a binary system in the (slowly evolving) circular orbit for simplicity. We refer the reader to the text by Poisson and Will [157], Blanchet’s Living review article [42], and  Chap. 32 “Effective Field Theory Methods to Model Compact Binaries” for the effective field theory approach. In the PN theory, the GW polarizations h+, × produced by a circular binary system have the following general structure (p is an integer number) h+, × =

  2 μ v 2  p (p) 1 , v H+, × + O r r2

(50)

p≥0

where the variable v that was a relative orbital velocity in the previous section is conveniently redefined as the frequency-related parameter by v 2 ≡ (MΩ)2/3 =

  1 M . 1+O 2 R c

(51)

The leading-order terms of Eq. (50) explicitly reads (ignore the static non-linear (0) memory contribution to H+ : see, e.g., Chapter 9 of Ref. [42]) (0)

H+ = −(1 + cos2 θ ) cos 2ψ ,

(0)

H× = −2 cos θ sin 2ψ.

(52)

Here, we introduce the “tail-distorted” phase, ψ = φ(t) − 6v 3 ln v ,

(53)

where the binary’s orbital phase φ(t) receives a correction from the scattering of the GW off the static curvature generated by the binary itself, note that since φ(t) ∼ v −5 in the quadrupole approximation, see Eqs. (38) and (40), this is a relative v 8 correction to the leading order.

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It is convenient to decompose the GW polarizations (50) onto the (spin-weighted) spherical harmonic mode (see, e.g., Section 3.1 of Blanchet’s review [42]) when comparing the PN waveform with, e.g., the numerical-relativity waveform. This is often called GW modes hm , which is expressed as h+ (t) − i h× (t) =

 ∞  

hm (t) −2 Ym (θ, φ) ,

(54)

=2 m=−

where −2 Ym is the spin (−2)-weighted spherical harmonics, and we note that h−m = (−1) h¯ m . The dominant quadrupole GW modes (, m) = (2, 2) is (see, e.g., Ref. [24]) h22 =

8μ r



  π 2 −i2ψ 1 + O v3, 2 , v e 5 r

(55)

and the other GW modes starts at O(v 3 ) or higher. For the GW data analysis applications, the phasing of the GW signal is significantly more important than its amplitude due to the matched-filter searching of the GW signal. Thus, the (so named) “restricted” PN waveform [63] is commonly (0) used, in which only leading term H+, × (or h22 ) of the waveform is retained, while all the PN correction to the orbital phase evolution φ(t) in Eq. (53) are included. (we note, however, that the higher-order amplitude terms become more pronounced especially for precessing or unequal mass-ratio binaries, e.g., GW190412 [13]). Because the early inspiral phase of binary is in the adiabatic regime, where the typical radiation-reaction time scale is much longer than the typical orbital time scale tOrb /tRR ∼ O(v 5 )  1 (recall Eqs. (32) and (33)), the orbital phase evolution φ(t) in the restricted PN approximation can be computed efficiently, making use of the adiabatic approximation discussed previously. The orbital phase φ(t) nonetheless has to be calculated up to a very-high-order PN term. The PN phase evolution can be parametrized by the following general structure (discard the constant phase): φ(v) = −

 1 1  2 3 4 5 1 + O(v ) + O(v ) + O(v ) + O(v ) + . . . . 32 η v 5

(56)

Because the leading term in φ(v) scales like O(v −5 ), one needs to compute the PN corrections at least 2.5PN order (or even higher order) to keep the absolute phase error to less than O(1), which is needed for the effectiveness of the matched-filtering search. Throughout the rest of this section, assuming the adiabatic approximation and making a further specialization to the BBHs with aligned spins and without orbital eccentricity, we will review how the orbital phase φ(t) can be computed up to the 3.5PN order [106]; adding the neglected effects for other binary configurations will be discussed in the next section. In this simpler setup, we will just need (i) the

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PN (center-of-mass) binding energy, (ii) the energy flux emitted to the infinity and across the BH horizons, and (iii) the certain balance equation associated with them, as inputs.

PN Binding Energy, Energy Flux, and BH Horizon Flux We schematically express the 3.5PN corrections to the Newtonian (center-of-mass) binding energy (30) as

 Mη 2 v3 v4 v7 8 v ENS + 2 ESO + 4 ESS + 6 ESSS + O(v ) , E≡− 2 M M M

(57)

where “NS,” “SO,” “SS,” and “SSS” denote the non-spinning, spin-orbit (linearin-spin), spin-spin (quadratic-in-spin), and spin-spin-spin (cubic-in-spin) contributions, respectively; the explicit expressions for ENS are given in, e.g., Ref. [42], while those for ESO , ESS and ESSS are given in, e.g., Ref. [44] and [131], respectively (see also Refs. [135] and references therein). In the parenthesis of the right-hand side of the above equation, we have factored out v and M where v shows the leading PN order of each term and M is related to the powers of the spins. We note that the complete binding energy is also available up to 4PN order both in the spinning sector [125] and the non-spinning sector [35, 75, 85, 108], the latter of which has very recently been obtained: 

 3969 123671 9037 2 896 448 4PN + − + π + γE + ln(16v) η ENS = v8 − 128 5760 1536 15 15    498449 3157 2 301 3 77 4 2 η + (58) + − + π η + η . 3456 576 1728 31104 Similarly, the 3.5PN corrections to the Newtonian energy flux (29) emitted to the infinity is written as F∞

 32 2 10 v3 v4 v7 8 FNS + 2 FSO + 4 FSS + 6 FSSS + O(v ) . η v ≡ 5 M M M

(59)

The explicit expressions for FNS and FSO are given in, e.g., Ref. [42] while those for FSS and FSSS given in, e.g., Ref. [44] and [131], respectively (see also Refs. [135] and references therein). Currently, the complete form of GW fluxes beyond 3.5PN order is missing (only relative O(η) piece is available up to 11PN order [86]), and its derivation is a frontier in the PN calculations. When the coalescing binary has at least one BH component, there is a part of the GW flux that goes down to the BH horizon, due to absorption effects of the energy and angular-momentum GW fluxes across the BH horizon. Such horizon flux appear at 2.5PN order for spinning BHs [173] (but it is pushed to 4PN order for non-

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spinning BHs [93, 156]) relative to the leading Newtonian-order energy flux (29), and it is known up to relative 1.5PN order for arbitrary mass ratio [57] (i.e., 4PN order beyond leading order energy flux) as well as 11PN order for the linear in the mass-ratio part [86], beyond the leading-order fluxes. The horizon fluxes (absorbed by the BH labelled by a) may have the following factorized from FHa (t ; ma , χa ) = Ωtidal (ΩH − Ωtidal ) Cva ,

(60)

where Cva

     16 m4a 2 12 2 2 1 + v 1 + O(v ≡− η 1 − χ ) + . . . . a 5 M2

(61)

Here, Ωtidal = O(v 3 ) and ΩH are the angular velocities of the tidal field (caused by a companion BH) and the BH horizon, respectively. The coefficient Cva denotes the remaining factor determining the horizon flux, and it is needed to next-to-leading (relative 1PN) order to achieve 3.5PN order precision.

Balance Equation for Slowly Evolving Black Holes The most important equation in the adiabatic approximation, analog to Eq. (31) in linearized theory, is the energy balance equation. In the case of BBHs, it is written as  dE = −F∞ − FHa , dt a

(62)

where the change rate of the center-of-mass binding energy E (related to conservative dynamics) is equated to the energy fluxes of the GW emission carried out to infinity F∞ and down to the BH horizon FHa (related to dissipative dynamics). A key assumption made in Eq. (62) is that the BH masses ma and BH spins Sa remain constant in the inspiral phase until the adiabatic approximation itself breaks down. However, this assumption is violated by horizon absorption effects. Each BH can be slowly evolving at the same time by changing its mass and spin via horizon absorption, namely dma = FHa , dt

1 dma dSa = . dt Ωtidal dt

(63)

The absorption-corrected energy balance equation up to 3.5PN order is schematically given by Ref. [106] 

∂E ∂t

 = −Feff ≡ −F∞ − m, S

 a

(1 − ΓHa ) FHa ,

(64)

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where the absorption-corrected binding energy E and fluxes F∞, H are defined by corresponding E and F∞, H , promoting the constant BH mass ma and spin Sa to the slowly evolving mass ma (t) and spin Sa (t). The BH’s growth factors ΓHa = O(v 2 ) account for the fact that the derivative on the left hand is now partial one.

Accuracy of the Post-Newtonian Approximants At this point, we must be mindful of the accuracy of the PN approximations. Two possible approaches to this problem have been extensively investigated in the literature. The first approach is to directly compare the PN waveform against the exact waveform from full NR simulations, performed in, e.g., Ref. [172]. The second approach is to compare physical quantities computed, such as the GW energy fluxes and binding energy (as well as linear momentum) that can be computed at higher PN order than the waveforms themselves: the modern development on both approaches is nicely summarized by Le Tiec’s review [120]. This subsection is an additional contribution to the latter, based on the recent comparison of the GW energy fluxes [165]. The general strategy is to rely on Black Hole Perturbation (BHP) theory (see, e.g., Refs. [134, 143, 167] for reviews as well as  Chap. 36 “Black Hole Perturbation Theory and Gravitational Self-Force”) outlined by earlier work in Refs. [183,185] in which very high PN order calculations can be achieved systematically by further expanding the PN series in the mass ratio q assumed to be small; see also “Black Hole Perturbation Toolkit” at http:// bhptoolkit.org/ and “Black Hole Perturbation Club” at https://sites.google.com/ view/bhpc1996/home for further details of the BH perturbation theory. Consider a small body moving along the quasi equatorial-circular orbit in the Kerr spacetime with the mass M (note that M is not the total mass in this subsection) and Kerr spin parameter a/M. Assuming a “small-mass-ratio” limit (q → 0), the 11PN GW energy flux radiated to the infinity at the leading order in q has been computed [86]: F (N ) =

N [k/6]  

F (k,p) {ln(v)}p v k ,

(65)

k=0 p=0

where p and k are integer numbers and N = 22, and the leading order has been to normalized to unity: F (0,0) = 1. The PN coefficients F (k,p) are approximately fitted by a linear function with respect to k in the log-linear plot. As a specific example, the PN coefficient F (k,0) versus k in the cases of the Kerr spin parameters of a/M = 0.99 (for the retrograde orbit), 0, and 0.99 (for the prograde orbit) is displayed in Fig. 2. The inverted (blue) triangles, (black) circles, and (red) triangles denote F (k,0) in the three cases, respectively. The dashed lines are the fitting lines for each case, and we find that F (k,0) (k = 0, . . . , 22) are fitted as (2.00014)k for a/M = 0.99 (retrograde), (1.94120)k for a/M = 0.0, and (1.83690)k for a/M = 0.99 (prograde). These

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1e+007 a/M = 0.99 (retrograde) a/M = 0 a/M = 0.99 (prograde)

1e+006 100000 10000 1000 100 10 1

0

5

10

15

20

k Fig. 2 The inverted (blue) triangles, (black) circles, and (red) triangles denote F (k,0) in Eq. (65) for the Kerr spin parameter a/M = 0.99 (retrograde orbit), 0, and 0.99 (prograde orbit), respectively. The dashed lines are the fitting and show (2.00014)k for a/M = 0.99 (retrograde orbit), (1.94120)k for a/M = 0, and (1.83690)k for a/M = 0.99 (prograde orbit)

fittings suggest that the approximate radius of convergence in terms of the velocity parameter v is expected to be vconv (a/M = 0.99) ∼ 0.499965 (retrograde) ; vconv (a/M = 0.00) ∼ 0.515145 ; vconv (a/M = 0.99) ∼ 0.544395 (prograde).

(66)

These values should be compared with the frequency at the innermost stable circular orbit (ISCO) because the quasicircular inspiral of a small body lasts until the orbital separation shrinks to the ISCO radius rISCO (in the Boyer-Lindquist coordinate). The ISCO radius in the equatorial plane of the Kerr spacetime is given by [33]   rISCO = M 3 + Z2 ∓ {(3 − Z1 )(3 + Z1 + 2Z2 )}1/2 ,

(67)

where Z1 ≡ 1+(1−χ 2 )1/3 {(1+χ )1/3 +(1−χ )1/3 } and Z2 ≡ (3χ 2 +Z12 )1/2 . Here, the upper/lower sign refers to prograde/retrograde orbits. In Table 1, we summarize the ISCO radius (rISCO /M), ISCO frequency (M ΩISCO = {(rISCO /M)3/2 + χ }−1 ), and the frequency parameter at ISCO (vISCO ≡ (MΩISCO )1/3 ) for the representative Kerr spin parameters. Comparing the values of vconv (66) and vISCO in Table 1, we expect that the PN series to the GW energy fluxes would work well up to ISCO for most of the retrograde orbits, but not for the prograde orbit with the high BH spin.

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Table 1 The ISCO radius (rISCO /M), ISCO frequency (M ΩISCO ), and the frequency parameter at ISCO (vISCO ) for the representative Kerr spin parameters (a/M); compare them against the values of the approximate radius of convergence vconv in Eq. (66) a/M 1.0 (retrograde) 0.9 (retrograde) 0.5 (retrograde) 0.0 0.5 (prograde) 0.9 (prograde) 1.0 (prograde)

rISCO /M 9.000000000 8.717352279 7.554584713 6.000000000 4.233002531 2.320883043 1.000000000

MΩISCO 0.03571428571 0.03754018063 0.04702732522 0.06804138173 0.1085883589 0.2254417086 0.5000000000

vISCO 0.3293168780 0.3348359801 0.3609525320 0.4082482904 0.4770835292 0.6086179484 0.7937005260

Time- and Frequency-Domain Inspiral Templates In this section, we shall construct PN GW templates for the early adiabatic inspiral of spinning, nonprecessing BBHs, including the secular evolution of BH mass and spin. Our main goal is to obtain the phase function φ(t) (53) to 3.5PN order, making use of the (BH-absorption corrected) PN binding energy (57), PN energy fluxes (59) and (60), and the generalized balance laws for the slowly evolving BHs (64) introduced in the previous section. The obtained phase in this way is called Taylor PN approximants [54,72,73]. For time-domain templates, we will present TaylorT1, TaylorT2, TaylorT3, TaylorT4, and TaylorT5 approximants. For the frequencydomain templates, we will show TaylorF1 and TaylorF2 approximants. These Taylor approximants are formally equivalent up to the 3.5PN order, but the uncontrolled (higher-order) PN order terms are truncated differently. Throughout the rest of this section, the labels “NS”, “SO”, “SS,” and “SSS” denote the spinning, point-particle’s contributions, namely, without BH absorption of non-spinning, spin-orbit (linear-in-spin), spin-spin (quadratic-in-spin), and spinspin-spin (cubic-in-spin) terms to the phase, while all the BH-absorption corrections are labelled by “Flux, 5” (from LO (2.5PN) horizon flux), “Flux, 7” (from NLO (3.5PN) horizon flux), and “BH, 7” (from slowly evolving, BH mass and spin). The explicit expressions for the point-particle contributions, not including absorption effect, are implemented in LALSimulation [127] (see also Ref. [54]) as “Module LALSimInspiralTaylorXX.c” at https://lscsoft.docs.ligo.org/lalsuite/ lalsimulation/group___l_a_l_sim_inspiral_taylor_x_x__c.html, while all the BHabsorption contributions are listed in Ref. [106]. Before proceeding, we note that our construction in this section is not complete. The frequency evolution due to changes in the BH’s mass and spin is constrained by (adiabatically invariant) “first laws” of compact binary mechanics [43, 87, 123], which will be disregarded here; accounting for the first-law effect will await future work. See the recent work by Hughes [104] for details.

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Taylor Time-Domain Approximants Using the balance equations (64), the evolution equations of the phase φ(t) are [106] dφ v3 =Ω= , dt M(v)

(68)

dv Feff (v) =− ; dt (∂E /∂v)M, S

(69)

note that M(v) is a slowly evolving total BH mass, and E and Feff are the absorption-corrected binding energy and (effective) fluxes. The different Taylor approximants integrate this system of ordinary differential equations differently. In each cases, the time-domain waveforms (in the restricted PN approximation) are obtained by inserting the resulting phase into, e.g., the leading-order GW polarizations h+, × (50), or the dominant (, m) = (2, 2) mode h22 (55). In the spinning, point-particle case (without BH absorption), these Taylor families of time-domain waveforms are implemented as “Module LALSimInspiralTaylorXX.c” at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_ l_sim_inspiral_taylor_x_x__c.html in LALSimulation; the equations for the pointparticle binaries are recovered by simply replacing E and Feff with a “standard” expressions of E (57) and F∞ (59) as well as the constant total mass M [54, 127].

TaylorT1 The TaylorT1 phase, φ T1 (t), is obtained by solving the system of two ordinary differential equations represented by Eqs. (68) and (69) with respect to time t. One may use v T1 (t) to calculate the GW amplitude. TaylorT4 Based on Ref. [51], first Taylor expands the ratio in the right-hand side of Eq. (69) and then truncate at appropriate PN order before integrating: dv T4 dv T4 dv T4 = ∞ + H , dt dt dt

(70)

T4 /dt describes the spinning point-particle contribution, while dv T4 /dt where dv∞ H accounts for the BH-absorption correction. Their formal PN structures are

 T4 dv∞ 32 η 9  T4 T4 T4 T4 = v v˙NS + v 3 v˙SO + v 4 v˙SS + v 7 v˙SSS + O(v 8 ) , dt 5 M

(71)

and T4    dvH 32 η 14  T4 T4 T4 = v v˙Flux,5 + v 2 v˙Flux,7 + O(v 3 ) . + η v˙BH,7 dt 5 M

(72)

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One then integrates two ordinary differential equations (68) and (70) at the same time. The resulting solutions v T4 (t) and φ T4 (t) are TaylorT4 approximants. We note that among Taylor-based approximants, this waveform in the case of non-spinning quasicircular orbits is the one that agrees more with NR waveforms for moderate values of the mass ratio 0.5  q ≤ 1 [46].

TaylorT2 We rewrite Eqs. (68) and (69) as dφ v 3 dt = , dv M(v) dv

dt (∂E /∂v)M, S =− . dv Feff (v)

(73)

One can then analytically integrate this system with respect to v after reexpanding the right-hand sides of these expressions in PN series. The resulting solutions are TaylorT2 phase φ T2 (v) and time t T2 (v). We express them as T2 T2 + φ∞ (v) + φHT2 (v), φ T2 (v) = φref

T2 T2 t T2 (v) = tref + t∞ (v) + tHT2 (v),

(74)

T2 and t T2 denote the reference phase and time as integration constants. The where φref ref spinning point-particle contributions give rise to

 1  T2 3 T2 4 T2 7 T2 8 φ + v φ + v φ + v φ + O(v ) , SO SS SSS 32ηv 5 NS  5M  T2 3 T2 4 T2 7 T2 8 t =− + v t + v t + v t + O(v ) , SO SS SSS 256ηv 8 NS

T2 φ∞ =− T2 t∞

(75)

while the BH-absorption corrections are    1  T2 T2 T2 ln (v) φFlux,5 + O(v 3 ) , + v 2 φFlux,7 + η φBH,7 32η    5M  T2 2 T2 T2 3 t t + O(v =− + v + η t ) , Flux,7 BH,7 256ηv 3 Flux,5

φHT2 = − tHT2

(76)

Here, all the terms are expressed in the PN series (analytically). TaylorT2 approximants are useful to understand the contributions of each term because the total expressions are summed up.

TaylorT3 Inverting the PN series t (v) (like TaylorT2 time t T2 (v)) to obtain v(t) analytically, we can re-express the TaylorT2 phase with respect to t via the relation φ(t) ≡ φ(v(t)). These are the TaylorT3 approximants. With the dimensionless time variable T2 is analogue to the coalescence t introduced in Eq. (35)), (tref c θ≡

 η −1/8 T2 (tref − t) , 5M

(77)

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one obtains T3 T3 + φ∞ (θ ) + φHT3 (θ ), φ T3 (θ ) = φref

T3 F T3 (θ ) = F∞ (θ ) + FHT3 (θ ),

(78)

where F ≡ (2dφ/dt)/(2π ) = v 3 /(π m) is the GW frequency of the dominant (, m) = (2, 2) GW mode, and the meaning of the labels “∞” and “H” are the same as Eq. (74). Their formal PN structures are  1  T3 3 T3 4 T3 7 T3 8 φ + θ φ + θ φ + θ φ + O(θ ) , SO SS SSS ηθ 5 NS  θ 3  T3 T3 T3 T3 FNS + θ 3 FSO = + θ 4 FSS + θ 7 FSSS + O(θ 8 ) , 8π M

T3 φ∞ =− T3 F∞

(79)

and    1 T3 T3 T3 ln (θ ) φFlux,5 + O(θ 3 ) , + θ 2 φFlux,7 + η φBH,7 η    θ 8  T3 T3 T3 FFlux,5 + θ 2 FFlux,7 + O(θ 3 ) . = + η FBH,7 8π M

φHT3 = − FHT3

(80)

The TaylorT3 waveform is useful when plotting inspiral GW waveforms in the time domain because the GW phase (by using φ T3 ) and amplitude (by using F T3 ) are written with respect to t directly.

TaylorT5 A variant of the TaylorT2 construction has been adopted to define the TaylorT5 approximants in Ref. [21]. It consists in Taylor expanding in v the right-hand side of the second equation of Eq. (73), the one for dt/dv, truncating it to the appropriate order, then taking its inverse, and integrating it to obtain v(t). The phasing is then obtained by substituting v(t) inside the analytical expression of φ(v) and direct integration of the Taylor expanded dφ/dv.

Taylor Frequency-Domain Approximants For an adiabatic BBH inspiral, the frequency-domain waveform is conveniently constructed using the stationary phase approximation (SPA), described in the subsection “Stationary phase approximation” (see also, e.g., Ref. [135] and references therein). In the case of the dominant (, m) = (2, 2) GW modes h22 , for instance, we model the Fourier amplitude A(f ) and the Fourier-domain phase Ψ (f ) defined by h˜ 22 (f ) = A(f ) ei(ΨSPA (f )−π/4),

ΨSPA (f ) = 2πf t (f ) − Ψ (f ).

(81)

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The SPA Fourier amplitude with evolving reduced mass μ(v) is 8μ(v) A(f )  r



π 2 v 5



3v 2 dv π M dt

−1/2    

,

(82)

v=vf

where vf ≡ (π Mf )1/3 is a dimensionless Fourier parameter (normalized by the initial values of the total mass M = MI ) and dv/dt is given in Eq. (69). At the same time, the SPA Fourier-domain phase is obtained by solving the set of ordinary differential equations: dΨSPA − 2π t = 0, df

dt π M (∂E /∂v)m, S + 2 = 0. df Feff (v) 3v

(83)

Again, E and Feff are the absorption-corrected binding energy and (effective) fluxes: recall Eq. (64). The different Taylor approximants integrate Eqs. (82) and (83) differently. In the spinning, nonprecessing point-particle case (without BH absorption) TaylorF2 waveforms are implemented as “Module LALSimInspiralTaylorF2.c” at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_inspiral_ taylor_x_x__c.html in LALSimulation; the equations for the point-particle binaries are recovered by replacing E and Feff in Eq. (83) with E (57), F∞ (59), and the constant total mass M [54, 127].

TaylorF1 The direct numerical integration of Eqs. (82) and (83) gives the TaylorF1 amplitude F1 (f ), respectively. One may use the TaylorT4 AF1 (f ) and the TaylorF1 phase ΨSPA T4 dv /dt (70) as input PN expression for dv/dt when solving Eq. (82).

TaylorF2 Drawing an analogy with TaylorT2, we re-expand the right-hand sides of Eqs. (82) and (83) in the PN series and analytically integrate them with respect to f . The TaylorF2 phase is then given by F2 F2 (f ) = 2πf tc − Ψc + Ψ∞ (f ) + ΨHF2 (f ), ΨSPA

(84)

where tc and Φc are constants that we can choose arbitrarily. The spinning pointparticle contribution is F2 Ψ∞ (f ) =

  3 F2 3 F2 4 F2 7 F2 8 Ψ + v Ψ + v Ψ + v Ψ + O(v ) f SS NS f SO f SSS f , 128ηv 5

(85)

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while the BH-absorption contributions are ΨHF2 (f ) =       vf 3 F2 2 F2 F2 3 ΨFlux,5 + vf ΨFlux,7 + η ΨBH,7 + O(vf ) , 1 + 3 ln 128η vreg (86) where the constant vreg can be chosen arbitrary. The TaylorF2 amplitude AF2 (f ) is given by inserting TaylorT4 dv T4 /dt (70) into Eq. (82) and re-expanding it in the PN series and truncating at 3.5PN order. The most commonly used TaylorT2 amplitude is, however, in the restricted PN approximation given by   2 M 5/6 F2 A (f )  f −7/6 1 + O(v 2 ) . (87) 1/3 r 3π It is worth noting that the spin effects at the leading 1.5PN order (i.e., the leading spin-orbit term) in the SPA amplitude and phase are encoded in a single spin parameter, the effective aligned spin [65], χeff ≡ (m1 χ1 + m2 χ2 )/M, or the reduced spin parameter [21] χPN ≡ χeff −

38 η (χ1 + χ2 ), 113

(88)

used for waveform calibration against NR waveform in inspiral-merger-ringdown waveforms which we will introduce in the next section.

Beyond Spinning, Nonprecesssing Binary Black Hole Cases Until this subsection, we have confined our attention to the simplest adiabatic inspiral of a spinning, nonprecessing BBH without orbital eccentricity. The leap from this narrow case to a generic case of binary configurations – in particular, one of the component compact objects is a NS rather than a BH – comes with a number of consequences. Below we briefly review some of the key elements when constructing PN templates for generic inspirals; the inclusion of merger and ringdown phase with completely generic waveform models will be summarized in the next section. Eccentricity — The radiative loss of the orbital energy and angular momentum to GWs circularizes the orbits of inspirals. The averaged rates of change of orbital eccentricity e(≤ 1) due to the radiative losses in the adiabatic approximation are estimated as (see, e.g., Chapter 12 of Poisson and Will [157], and Ref. [152]) de 304 e =− η dt 15 a



M a

3

2 −5/2

(1 − e )

  121 2 1+ e , 304

(89)

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where a is the semi-major axis of the ellipse (not the Kerr spin parameter here), and we assume Newtonian elliptic orbits for simplicity. That is, the eccentricity always decreases when the orbit shrinks as approaching the merger phase. In fact, assume a small eccentricity limit e  1, Eq. (89), can be integrated (with the help of da/dt, etc.) to give e  eini (a/aini )19/12 , where aini and eini are initial values. We clearly see that the orbit is circularized quite fast toward the late inspiral phase. Nevertheless, there is emerging need for GW models for the quasieccentric inspirals because it will be an important source for the next-generation GW detectors both on the ground (e.g., KAGRA+ [133], Voyager (https://dcc.ligo. org/LIGO-G1602258/public), Einstein Telescope [101], Cosmic Explorer [163], etc.) and in the space (e.g., LISA [29], (B-)DECIGO [110], TianQin [132], etc.), which have wider sensitivity band for the early inspiral phase; see also  Chaps. 3 “Space-Based Gravitational Wave Observatories” and  7 “Third– Generation Gravitational-Wave Observatories”. There is an ongoing program of development on the waveforms for the quasieccentric binaries. A theoretical GW waveform for non-spinning eccentric binaries has been developed in Refs. [103, 137, 174, 178, 179] with the quasi-Keplerian formalism (see also Chapter 10 of Ref. [42] and references therein). This waveform is valid for the small initial orbital eccentricity, and its frequency-domain model is implemented in LALSimulation as TaylorF2Ecc at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_ l_sim_inspiral_taylor_f2_ecc__c.html (see also EccentricTD/FD modules). However, a GW waveform that performs well even at large eccentricities requires further development. Recent efforts can be found in, e.g., Refs. [136, 138] and  Chap. 33 “Repeated Bursts”. Spin precession— In the nonprecessing system, the spins of the compact objects are (anti-)parallel to the orbital-angular momentum; the orbital place of binary is fixed. If the spins are not aligned with the orbital angular momentum, the spin-orbit and spin-spin effects lead to precession of the spins and orbital plane, while the total angular momentum is constant, modulo angular momentum loss via radiation. The precession timescale for the spinning binaries is tP =

|Sa | ∼ Mv −5, |dSa /dt|

(90)

which is shorter than the radiation reaction timescale tRR /M ∝ v −8 given in Eq. (32), hence neglecting losses the orbital and spin angular momentum precess around the total angular momentum. Schematically, the (orbital-averaged) spin precession equations that govern the conservative evolution of spin vectors are (see, e.g., Chapter 9 of Poisson and Will [157] and Ref. [52] for the explicit expressions)   dSa SS = Ω SO × Sa, + Ω a a dt

(91)

SS where Ω SO, are spin-orbit and spin-spin pieces of the precessional angular a velocity. The frequency Ω SO a describes the spin-orbit (geodetic) precession of the

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spin vectors, while Ω SS a is responsible for the spin-spin (frame-dragging) precession of those spin vectors. The precession determined by Eq. (91) then modulates the waveform, adding rich periodic structure, as the angle between the normal to the orbit and the line of sight also precesses. As a result, the GW mode (, m) = (2, 2) is no longer guaranteed to be the “dominant” one; all the  = 2 modes become relevant. These complications make the modeling of precessing waveform challenging, but they gain us access to more binary parameters that can be hard to measure in nonprecessing binaries, due to their degeneracies with other parameters that can be disentangled by observing the source’s plane from different angles during the very same inspiral. For more recent effort to develop the precessing binary waveforms, see, e.g., a concise review by Hannam [95] (and references therein, also see [58]). Various spin precessing Taylor waveforms are implemented in LALsimulation and are found at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_ inspiral_spin_taylor__c.html. Matter effects — When one of binary components is not a BH, but a material body as NS, it can acquire a quadrupole moment induced by, e.g., the tidal field generated by its companion. It is particularly interesting since the deformation giving rise to the tidally induced quadrupole bears the imprint of the equation of state of the NS – i.e., the microscopic property of strongly interacted nuclear matter under extreme conditions of pressure and density. The induced quadrupole moment of NSs is mainly characterized by two physical effects. The first one is due to (static) tides, as non-spinning NS of mass ma and radius RaNS  R in a binary is subjected to a tidal field by the companion. In a Newtonian gravity (for simplicity), a tidally induced quadrupole moment is given by Qtidal ∼ λa a

mb , R3

(92)

where λa = ka (RaNS )5 is the (static) tidal deformability defined in terms of the dimensionless (gravitational) Love number k that depends on the NS’s equation of state. A measurement of ka or λa through the use of GW therefore provides unique insight into NS matter; we note that ka = 0 for a non-spinning, Schwarzschild BH (the tidally induced quadrupole moments of Schwarzschild BH vanish in the static case [41, 69, 115] and is proportional to the inducing field time derivatives, being a dissipative effect, giving rise to the horizon fluxes discussed in the previous section [154]). The GW astronomy on the NS tidal deformability is described in detail, e.g., by Chatziioannou [56]; see also  Chaps. 12 “Binary Neutron Stars” and  15 “Black Hole-Neutron Star Mergers”. For non-spinning BNSs, the family of PN templates with tidal interactions is nicely summarized in, e.g., Ref. [145] and Chapter 4 by Dietrich et al. [78]. As an example, we briefly summarize the Kyoto’s phenomenological model by Kawaguchi et al. [109], which combines the PN tidal corrections with phenomenological terms obtained by fits to Kyoto’s high-precision NR waveforms. In this model,

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˜ ) = A(f ) eiΨ (f ) with the tidal contributions may take the split TaylorF2 (strain) h(f form of A(f ) = APP (f ) + ATidal (f ),

Ψ (f ) = ΨPP (f ) + ΨTidal (f ),

(93)

where “PP” and “tidal”are individual contributions from the non-spinning point particle and tidal interactions, respectively. The tidal terms ATidal (f ) and ΨTidal (f ) are then given by (note that x ≡ v 2 )

KyotoTidal ATidal KyotoTidal

ΨTidal

  5π η M 2 27 449 6 ˜ −7/4 × − x 5 − x − bx r , Λx 24 r 16 64

   39 3 − Λ˜ 1 + a Λ˜ 2/3 x p x 5/2 = 128η 2   3115 28024205 2 4283 5/2 3/2 , (94) x − πx + x − πx × 1+ 1248 3302208 1092 =

with a = 12.55, p = 4.240, b = 4251, and r = 7.890. Here, Λ˜ is the combination of the individual (dimensionless) tidal deformability Λa ≡ λa /m5a defined by [82] Λ˜ =

16 (m1 + 12m2 )m41 Λ1 + (m2 + 12m1 )m42 Λ2 . 13 M5

(95)

This parameter characterizes the leading-order (relative 5PN) tidal effects in the waveforms. TaylorF2 with tidal effects was used to analyze the first BNS event, GW170817 [11], and it is implemented in LALSimulation as a part of “Module LALSimInspiralTaylorXX.c” at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_ l_sim_inspiral_taylor_x_x__c.html. For more improved analysis of GW170817 [4], a different model was used as a reference. In GWTC-1 [3], this is also used as a frequency-domain model for GW170817. For example, we will see the SEOBNRv4_ROM_NRTidal and IMRPhenomPv2_NRTidal models in the next section. The second effect that gives rise to the induced quadrupole moment is the rotation. The spinning motion deforms the NS by creating a distortion in its mass distribution, which also depends on the NS’s equation of state [118]. The resultant (relativistic) spin-induced quadrupole moment is given by [153] spin

Qa

 α χa2 m3a,

(96)

where α is the (dimensionless) spin deformability. Although the known NS has low spin in general (χa ∼ 0.2 or less [7, 11]), in principle, the measurement of α for highly spinning BNSs would provide another GW probe to NS matter [100]. We note that α = 1 for Kerr BHs, followed by its well-known “no-hair” property.

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By likewise writing ΨTidal , the correction to the TaylorF2 phase due to the NS’s quadrupole spin-deformation is given by [117, 153]. ΨQM =

 3  −25Q˜ v −1 + O(v), 128η

(97)

˜ is a certain combination of the individual spin-induced quadrupole where Q spin deformation Qa and spins χa , which characterize the leading-order (relative 2PN) effects in the waveform. It is implemented in LALSimulation within “Module LALSimInspiralSpinTaylor.c” (https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/ group___l_a_l_sim_inspiral_spin_taylor__c.html), and the quadrupole spin-deformation contribution to energy, flux, and phasing terms are coded in https://lscsoft. docs.ligo.org/lalsuite/lalsimulation/_l_a_l_sim_inspiral_p_n_coefficients_8c.html.

Full Inspiral-Merger-Ringdown Waveform Models In our discussion so far, we have focused exclusively on the GW waveforms from adiabatic inspirals within the PN approximation. However, this is just a part of GW signals from the coalescence of compact-object binaries that can be observed by LIGO, Virgo, and KAGRA. As the separation of binary shrinks (by “chirping” the frequency v → 1), the binary dynamics moves on to the merger phase, and the PN calculations become more and more inaccurate; recall our discussion on the accuracy of the PN approximants. The modelling in the late inspiral phase should be modified to enhance the accuracy of PN approximation. The transition frequency from inspiral to the merger phase is roughly estimated by the GW frequency at ISCO of Schwarzschild BH with the mass MSch : 

fISCO

MSch ≈ 73.28 60M

−1



MSch Hz ≈ 1570 2.8M

−1 Hz.

(98)

This is well in the sensitive frequency band of LIGO, Virgo, and KAGRA. Furthermore, the ringdown phase followed by the merger phase has the typical GW frequency at [39]  fRing = 538.4

MRem 60M

−1   1.5251 − 1.1568(1 − αRem )0.1292 Hz,

(99)

where MRem and αRem are the mass and non-dimensional spin of the remnant Kerr BH after merger. The above expression gives 198.3 Hz for MRem = 60M and αRem = 0. This ringdown frequency, especially for BBHs, is again well in the sensitivity band of the Advanced LIGO, Virgo, and KAGRA. Therefore, it is

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indispensable to have a “full” waveform, including the merger and ringdown phases in addition to the PN model for the inspiral phase, in order to maximize our ability to GW data analysis. The goal of this section is to briefly survey such a full, inspiral-merger-ringdown (IMR) waveform actually implemented and used for the GW data analysis of the first, second, and third observing runs (O1, O2, and O3) of Advanced LIGO and Virgo; the IMR waveform models used in the up-to-date GWTC-2 [16] are summarized in Table III of that paper. There are two main families of the IMR waveform; the effective-one-body (EOB) approach (in the time-domain) and the “phenomenological” (IMRPhenom) models (in the frequency-domain). All the details of these two methods in LALsimulation can be found at https://lscsoft. docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_i_m_r__h.html. We should note, however, that these methods in LALsimulation ignore any contributions of BH absorption discussed in the previous section.

Effective-One-Body (EOB) Approach An effective-one-body (EOB) approach [49,50] is an analytical framework to cover the full range of the inspiral, merger, and ringdown phases, making use of a variety of analytical approximation methods, such as the PN theory and the Black Hole Perturbation (BHP) theory, and NR data as calibrations. A brief review of the EOB approach is given in Refs. [66, 67]; see also, e.g., Refs. [30, 68] (and reference therein) for latest developments. The starting point of the EOB approach is to precisely describe the orbital dynamics of binaries (as a source of GW waveform). First, one conveniently maps the real two-body PN Hamiltonian (for their relative motion) to an “effective” testparticle Hamiltonian H eff of non-geodesic motion in a fictitious effective spacetime, so as to construct the so-named EOB (or “improved real”) Hamiltonian:  H

EOB

≡ M 1 + 2η



 H eff −1 . μ

(100)

In general, H EOB improves the convergence of PN series of the original PN Hamiltonian. Second, the radiation reaction to the system, which is another piece to describe the radiative dynamics of binaries, is prepared from, e.g., the PN and BHP results of the GW fluxes with a resummation such as a Padé approximation or the factorized resummation [70, 71]. Third, one introduces some adjustable free parameters to the Hamiltonian and fluxes and calibrates them against the results of NR simulations (and that of the BHP and self-force theory [32] in the small-massratio limit, q → 0). In particular, the EOB models calibrated to NR simulations are dubbed as “EOBNR.”

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The next step is to construct GW waveforms from the obtained EOB orbital dynamics. The inspiral-plus-plunge GW waveform hinsplunge is derived from the orbital motion, based on the improved resummation of PN (multipolar) inspiral waveforms [74, 149], including non-quasicircular effects [70, 71]. This waveform is connected smoothly to a ringdown GW waveform hringdown which consists of several quasinormal modes of the remnant BH after merger, around a matching time tmatch (see Ref. [40] and references therein for the ringdown phase). The full GW waveform is then schematically written as [71] hEOB (t) = θ (tmatch − t) hinsplunge (t) + θ (t − tmatch ) hringdown (t),

(101)

where θ (t) is the Heaviside-step function. In Ref. [53], the EOBNRv1 model was proposed for non-spinning BBHs. This was calibrated to NR simulations with mass ratios, m1 /m2 = 1, 3/2, 2, and 4. In Ref. [148], the EOBNRv2 model was proposed for the same nonspinning case. This was also calibrated to NR simulations with mass ratios, m1 /m2 = 1, 2, 3, 4 and 6. The above two GW waveform approximants include not only the dominant (, |m|) = (2, 2) modes, but also some subdominant harmonic modes; see also Section II-a in Ref. [2]. In the classical GW data analysis of the LIGO fifth science run (S5) [2], these two EOBNR models were used as an IMR theoretical template, and the EOBNRv2 model is available at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_i_m_ r_e_o_b_n_rv2__c.html; note, however, that EOBNRv1/v2 models have been superseded by more recent developments (such as SEOBNR family below), and they are no longer used in the modern LAL simulation.

SEOBNR Family SEOBNRv1/v2 — In Ref. [175, 176] (and references therein), EOB models have been presented for spinning, nonprecessing BBHs (the first character, “S” in SEOBNR denotes spin). The SEOBNRv1 and SEOBNRv2 models are available at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_i_m_ r_spin_aligned_e_o_b__c.html. The parameters of the first BBH event, GW150914 [8], was evaluated by the SEOBNRv2 model. This was also used in the detailed study on the properties of GW150914 [9]. SEOBNRv2_ROM_DoubleSpin [81] which speeds up the waveform generation with reduced-order modelling (ROM) [162] was also used. SEOBNRv3 — In Ref. [150], the SEOBNRv3 model has been presented as a fully precessing waveform model for BBH coalescence. The SEOBNRv3 model is available at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/_l_a_l_sim_i_m_r_ spin_prec_e_o_b_8c.html. This SEOBNRv3 model was used in the detailed study on the properties of GW150914 [9]. Also, in GWTC-1 [3], the SEOBNRv3 model was used to analyze generic two-spin precession dynamics. SEOBNRv4/v4HM — For spinning, nonprecessing BBHs, the SEOBNRv4 model [45] is an improvement of the SEOBNRv2 model with calibration to 141 NR waveforms including the spin effects. As a further improved version of the SEOBNRv4 model with higher harmonics, the SEOBNRv4HM model is presented

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(HM standing for “Higher Modes”) [62]. The SEOBNRv4 and SEOBNRv4HM models are available at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___ l_a_l_sim_i_m_r_spin_aligned_e_o_b__c.html where all aligned spin, i.e., nonprecessing models, are summarized. For the analysis of GW170817 in Ref. [4] with the help of the NRTidal model for the tidal effects (a hybrid model mainly based on the PN tidal phase corrections up to the 7.5PN order and the calibration against the high-precision NR data) [76,77,79] SEOBNRv4_ROM_NRTidal = SEOBNRv4_ROM + NRTidal was used as a signal model of BNS mergers. In GWTC-1 [3], this was also used as a frequency-domain model for GW170817. In the analysis of GW190814 [15], there was no measurable tidal signature although the SEOBNRv4_ROM_NRTidalv2_NSBH model with phenomenological tidal effects and the NS’s tidal disruption was applied. The SEOBNRv4T model [102] which gives a time-domain waveform with analytical dynamic tide effects has been also used in GWTC-1 [3] for GW170817. SEOBNRv4P/v4PHM — The EOBNR model with higher multipoles for precessing binaries (see Refs. [62, 147] and references therein) was used for GW190412 [13]. Also, this model has been used for GW190814 [15], GW190521 [14], and GWTC-2 [16] as a BBH waveform model. The precessing model SEOBNRv4PHM also includes its restriction to the dominant GW modes, SEOBNRv4P.

TEOBResumS The TEOBResumS has been the first to implement a tidal description up to merger verified with NR simulations [25, 36, 37, 142], and it is the only model that implements a binary NS post-merger completion [48] so to give a complete description of the signal emitted by binary NSs. Note that especially the method of calibration against NR simulations is different from the above EOBNR family; see Section VI of Ref. [142]. Spin interactions in the BNS waveforms are included at next-next-leading order [139] and include precession effects [26]. The BBH sector implements higher modes [141] and eccentricity [60] and can model hyperbolic mergers [140]. It is a time-domain waveform that has been used in GWTC-1 [3] for GW170817 and GWTC-2 [16]. TEOBResumS is available at https://bitbucket. org/eob_ihes/teobresums/wiki/Home and included in its non-spinning, reducedorder modeling (ROM) version in the LAL’s module https://lscsoft.docs.ligo.org/ lalsuite/lalsimulation/_l_a_l_sim_inspiral_t_e_o_b_resum_r_o_m_8c.html.

Phenomenological (IMRPhenom) Models In the frequency domain, phenomenological IMR (IMRPhenom) models have been presented as another way to construct full GW signals by combining the analytical PN/BHP results with NR simulations.

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The IMRPhenom model basically consists of three parts: the PN inspiral (Ins) part, merger-ringdown (MR) part, and intermediate (Int) part between the former two parts. The amplitude AIMR (f ) and the GW phase ΦIMR (f ) are written in the form of ΦIMR (f ) = φIns (f ) θf−1φ + θf+1φ φInt (f ) θf−2φ + θf+2φ φMR (f ), AIMR (f ) = AIns (f ) θf−1A + θf+1A AInt (f ) θf−2A + θf+2A AMR (f ),

(102)

where the function θf±0 is defined by θf±0 =

1 {1 ± θ (f − f0 )} ; 2

θ (f − f0 ) =

−1, f < f0, 1, f ≥ f0 .

(103)

Here, we basically pattern after the notation in Refs. [105, 160], and we introduced the certainly prepared transition frequencies f1φ , f2φ , f1A , and f2A . Each individual component is parametrized and calibrated against NR simulations. For example, the amplitude and phase models of the inspiral part are based on extensions of those of TaylorF2 models with calibration parameters. Their model functions in the state-of-the-art PhenomX framework take the form [160]

3  2η −7/6 f ρi (πf )(6+i)/3, 3π 1/3 i=1   1 3 4/3 3 5/3 1 2 3 7/3 , (104) σ0 +σ1 f + σ2 f + σ3 f + σ4 f + σ5 f = φTF2 + η 4 5 2 7

AIns = ATF2 + φIns

where ATF2 and φTF2 are (essentially) the same as TaylarF2 models in Eqs. (87) and (85). The free parameters ρi and σj (j = 0, 1, 2, 3, 4 and 5) are phenomenological, pseudo-PN coefficients calibrated against NR data sets. In Ref. [22], the first frequency-domain IMR waveform, IMRPhenomA model, was presented for non-spinning BBHs. The spinning, nonprecessing BBH waveform is called the IMRPhenomB model [23], and it is later improved to IMRPhenomC model [166]; see also Section II-b in Ref. [2] about IMRPhenomA/B models. In the classical GW data analysis of the LIGO fifth science run (S5) [2], the IMRPhenomA and IMRPhenomB models were used as an IMR theoretical template; note, however, that IMRPhenomA/B/C models have been deprecated and they are no longer used in the modern LAL simulation. All models in IMRPhenom family are available at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_ a_l_sim_i_m_r_phenom__c.html

IMRPhenomD Family IMRPhenomD/HM — In Refs. [105,111], the IMRPhenomD model is presented for the dominant (, |m|) = (2, 2) modes of spinning, nonprecessing binaries. For the

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analysis of GW170817 in Ref. [4], using the IMRPhenomD model with the help of NRTidal for the tidal effects approximants [76, 77, 79] IMRPhenomD_NRTidal = IMRPhenomD + NRTidal was used as a signal model of BNSs. In Ref. [128], based on the IMRPhenomD model, the subdominant harmonic modes have been included in the waveform model, giving origin to the IMRPhenomHM model. IMRPhenomP/Pv2/Pv3/Pv3HM — The IMRPhenomP [96] and IMRPhenomPv2 models [105] are for precessing binaries, based on the nonprecessing IMRPhenomC and IMRPhenomD models, respectively, and have the single precession spin (to rotate the nonprecessing signals in the co-precessing frame). The improved IMRPhenomPv3 model [112] has two independent spins in the precession dynamics, making use of the results of the multi-timescale analysis of the (conservative) PN precession dynamics [58] to cover a broader region of the parameter space than that of the IMRPhenomPv2 model; recall the radiation-reaction and precession time scales in Eqs. (32) and (90) that imply tp /tRR ∼ v 3  1. This model is further extended to IMRPhenomPv3HM model [113] to include subdominant GW modes, which is based on the (nonprecessing) IMRPhenomHM model. The IMRPhenomPv2 model was used in the detailed study on the properties of GW150914 [9]. In GWTC-1 [3], the IMRPhenomPv2 model was also used as a model for BBH coalescence. The IMRPhenomPv3HM model was used for GW190521 [14] and GW190814 [15] as well as Ref. [13] and GWTC-2 [16] to analyze potential BBH signals (we note that a fast and accurate NR surrogate model NRSur7dq4 [181] has also been used for GW190521 and GWTC-2). IMRPhenomPv2_NRTidal/NRTidalv2 — Including spin-precessing with the Pv2 style [105] and tidal interactions with the NTRidal [76], we have IMRPhenomPv2_NRTidal = IMRPhenomPv2 + NRTidal. This is called as the IMRPhenomPv2_NRTidal (PhenomPv2NRT) model [79]. In Ref. [77], an improved version of IMRPhenomPv2_NRTidal has been presented as the IMRPhenomPv2_NRTidalv2 model. In the detailed analysis of GW170817 [4], this IMRPhenomPv2_NRTidal model has been used as the reference model. In GWTC-1 [3], this was used as a frequency-domain model for GW170817. Also, to analyze GW190425 [7] with total mass ∼ 3.4M and any sources in GWTC-2 [16] that had evidence for at least one binary component below 3M , this was used as the signal model. IMRPhenomNSBH — For spinning, nonprecessing NSBH binaries (with a nonspinning NS and a spinning BH), the IMRPhenomNSBH model has been developed [177], based on the amplitude of the IMRPhenomC model and the phase of the IMRPhenomD_NRTidalv2 model. The IMRPhenomNSBH model was used for the analysis of GW190814 [15] and potential NSBH sources in GWTC-2 [16].

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IMRPhenomX Family IMRPhenomX family is an entirely new Phenom pipeline, superseding the IMRPhenomD family. The main improvements from PhenomD include, e.g., (i) the larger number of input NR waveforms for calibrations increased from 19 to 652, broadening the coverage of the mass ratio from 1 : 18 to 1 : 1000 with the help of BBH merger simulations in the small mass-ratio limit produced by EOBBHP approach [97–99]; (ii) the higher dimensionality the model parameter space enlarged from 2 to 3, using the symmetric mass ratio and two spin components (orthogonal to the orbital plane); further improvements are summarized in Sec.X of Ref. [160]. These improvements resolve various shortcoming of PhenomD family and drastically improve the accuracy. The baseline models for the dominant (, m) = (2, 2) modes of spinning, nonprecessing binaries is called IMRPhenomXAS [160]. This model is then generalized to IMRPhenomXHM model [89, 90] to include subdominant harmonic modes of nonprecessing binaries and further to the IMRPhenomXPHM model [159] for precessing binaries with “twisting-up” the nonprecessing waveform using the (Pv3-style) double-spin approach developed in Ref. [58]. All the models in IMRPhenomX family are available at https://lscsoft.docs. ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_i_m_r_phenom_x__c.html; see Appendix C of Ref. [89] for the technical details of implementations, leading to significantly faster waveform production without compromising on accuracy. IMRPhenomTP IMRPhenomTP Ref. [80] is a time-domain phenomenological model for the dominant (, m) = (2, 2) modes of spinning precessing BBHs, making use of the “twisting up” approximation [168] (see also, e.g., Section 5.3 of Ref. [95]) to the nonprecessing BBHs, based on TaylorT3 approximants. GIMR for Modified Theory of Gravity Until now we have (implicitly) assumed that the gravity theory is described by Einstein’s GR. However, it is not the only relativistic theory of gravity. Indeed, motivated by the recent observation of accelerating expansion of the universe, there is a growing interest to consider alternative gravity theory other than GR. Yet, each candidate gravity theory has to be experimentally verified, and the GW signals from the coalescence of a compact object binary allow a unique test of gravity theories in the strong curvature regime; this topic is covered in  Chap. 39 “Testing General Relativity with Gravitational Waves” with a lot more details. Although we can consider model-dependent GW waveforms for each modified theory of gravity, it is possible to formulate a model-independent waveform that phenomenologically captures the main features of a wider class of the modified theory of gravity. For instance, the GIMR model is prepared by introducing deformations to the phase of the frequency-domain IMRPhenom waveform model in GR (see also a parametrized post-Einsteinian framework in Ref. [184]). The standard pipeline for model-independent GIMR waveforms is called TIGER (Test Infrastructure for GEneral Relativity) [20].

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In Refs. [10] for GW150914, [5] for GW170817, [6] for GWTC-1, and [18] for GWTC-2, the GIMR model has been also used to test the dipole radiation at −1PN order, which is absent in GR (recall our previous discussion on the quadrupole formalism for GW generation) but is a common prediction of modified theory of gravity due to the existence of additional scalar degrees of freedom mediating long-range interactions. See  Chap. 38 “Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories” for more details.

Conclusion The development of the theoretical GW templates of coalescing compact object binaries was initiated within the PN approach, focusing on the adiabatic inspiral phase. However, after the 2005 breakthroughs in NR simulations of BBHs [31, 55, 161] and the observations of GW events by LIGO and Virgo, learning that almost all the GW signals detected so far have both merger and ringdown in the sensitive frequency band of LIGO, Virgo, and KAGRA, complete models with inspiral, merger, and ring-down phases are under vigorous development. The theoretical construction of GW waveforms for the entire coalescence makes full use of known analytical approximation scheme to other methods than the PN approximation to GR, e.g., the BHP (and self-force) theory as well as cutting-edge NR simulations. This motivation has continuously driven a concerted effort by GW theorists and data analysists to develop accurate and efficient waveforms of compact-object binary mergers [127], and we have provided a broad (but yet small-corner) overview of this active subject. We conclude our chapter by listing some of the open challenges and prospects. • Although we have exclusively discussed quasicircular binaries in this chapter, the observation of nonzero eccentricity will be a smoking gun of the formation scenario of binary systems: evolution of isolated binaries (e.g., circularized by binary interactions and GW radiation; recall Eq. (89)) versus dynamically formed binaries in dense stellar environments (e.g., the Kozai-Lidov mechanism [116, 126]). • The ground-based GW detectors, LIGO, Virgo, and KAGRA, have a peak of sensitivity around 100 Hz. For future plans of ground-based detectors, e.g., KAGRA+ [133], Voyager (https://dcc.ligo.org/LIGO-G1602258/public), Einstein Telescope [101], and Cosmic Explorer [163], significant sensitivity improvements are expected both in low- and high-frequency bands. Enhanced low-frequency part would stretch the visible range of BBHs and NS-BH inspirals to heavier masses. At the same time, improvements in the high-frequency part would enable us to observe the BNS merger phase more accurately, where the (equation-of-state dependent) finite-size effects of NSs become particularly pronounced.

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• The observation of GW190814 showed a binary system with mass ratio around 10:1. The remnant compact object of GW190521 [14] has been considered as an intermediate mass BH. The combination of these two events may suggest the existence of intermediate mass black holes and the possibility to observe in the future binary systems with an individual mass ratio around 1 : 100. In LALSimulation [127], a program for “extreme” mass ratios is present at https://lscsoft.docs.ligo.org/lalsuite/lalsimulation/group___l_a_l_sim_i_m_r_ phenom_x__c.html, calibrated to NR waveforms for mass ratio from q = 1 to 20, and the region between q = 20 and 1000, where no NR waveforms are available, is covered by the waveform of extreme mass-ratio inspirals (EMRIs) (based on the BHP theory with the EOB orbital dynamics in the small mass-ratio limit [97– 99]), with the mass ratio region 200 < q < 1000 covered by the extrapolation based on these waveforms. Further synergy with EMRI waveforms is ongoing: see, e.g., Refs. [38, 164, 180] and references therein. • The planned space-based GW observatory, e.g., LISA [29], (B-)DECIGO [110, 144, 171], and TianQin [132], will observe much longer-length inspiral signals of compact object binaries than LIGO, Virgo, and KAGRA. They will allow us to measure the binary parameters with exquisite precision, particularly in the context of the multiband GW astronomy [107, 170, 182]. At the same time, however, the benefit of such an observation is gained only when one is able to construct much more accurate templates for the inspiral phase than currently available models: their phase coherence has to be maintained over O(106 ) GW cycles (and one should also devise a consistent data analysis technique to process such a long-length GW signals). • Our presentation does not cover the efficiency aspects in the waveform modelling. The matched-filtering search of GW signal needs a (so named) “template bank” by nature (see, e.g.,  Chap. 41 “Principles of Gravitational-Wave Data Analysis”), designed to efficiently cover a parameter space as large as possible. This is a computationally expensive and challenging task, demanding different investigations. A common strategy is to use “effective” parameters such as the chirp mass M (27), the reduced spin χPN (88), and the binary tidal deformability Λ˜ (95), to reduce the numbers of the dimension in the parameter space by focusing on those combinations of astrophysical parameters which affect most prominently the waveform. An alternative strategy is a reduced-order modeling (ROM) [162] and a surrogate model (for NR waveforms) [181].

Cross-References  Binary Neutron Stars  Black Hole Perturbation Theory and Gravitational Self-Force  Black Hole-Neutron Star Mergers

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 Dynamical Formation of Merging Stellar-Mass Binary Black Holes  Effective Field Theory Methods to Model Compact Binaries  Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories  Formation Channels of Single and Binary Stellar-Mass Black Holes  Numerical Relativity for Gravitational Wave Source Modeling  Principles of Gravitational-Wave Data Analysis  Repeated Bursts  Space-Based Gravitational Wave Observatories  Terrestrial Laser Interferometers  Testing General Relativity with Gravitational Waves  Third-Generation Gravitational-Wave Observatories Acknowledgments The authors warmly thank Maria Haney for reading the manuscript and improving it with her suggestions. S. I. acknowledges support from STFC through Grant No. ST/R00045X/1. S. I. also thanks the financial support from the Ministry of Education, MEC, during his stay at IIP-Natal-Brazil and acknowledges networking support by the GWverse COST Action CA16104, “Black holes, gravitational waves and fundamental physics.” The work of R. S. is partially supported by CNPq. H. N. acknowledges support from JSPS KAKENHI Grant Nos. JP16K05347 and JP17H06358.

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Effective Field Theory Methods to Model Compact Binaries Riccardo Sturani

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method of Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational Self-Interactions, Wick Contraction, and Power Counting . . . . . . . . . . . . . The Conservative Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Radiative Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The gravitational two-body problem is currently subject of intense investigations, under compelling phenomenological and theoretical motivations. The gravitational wave detections from compact binary coalescences will demand even more accurate description of the source dynamics as the sensitivity of detectors increases over years. The analytic modeling of classical gravitational dynamics has been enriched over the last decade of powerful methods borrowed from field theory originally developed to describe fundamental particle quantum scatterings. This work aims at presenting a review of a specific effort, initiated by the seminal paper by Goldberger and Rothstein, dubbed nonrelativistic general

R. Sturani () International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal, Brazil e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_32

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relativity, which applies effective field theory methods to describe the twobody dynamics in general relativity. It models the classical interaction between astrophysically massive objects via field theory methods, showing that many features usually associated with quantum field theory, e.g., divergences and counter-terms, renormalization group, loop expansions, and Feynman diagrams, have all to do with field theory, be it quantum or classical. Keywords

Post-Newtonian approximation to general relativity · Effective field theory · Nonrelativistic general relativity · Two-body problem · Inspiral dynamics

Introduction The analytic modeling of the two-body problem has received a lot of attention since the detections of gravitational waves (GWs) [2–6] by the large interferometric detectors LIGO [1] and Virgo [7]. The sources of all detections performed so far are coalescences of compact binary systems, and looking for their signals, the detector’s output is processed via the matched filtering [8] technique, which is particularly sensitive to the phase of the GW signals. As a result, source parameter reconstruction depends crucially on the availability of accurate models of signals, which in turns depend on the quantitative details of general relativity (GR) in its highly nonlinear regime. To construct waveform templates for the LIGO/Virgo data analysis pipeline, different approaches have been adopted: perturbative analytic, like the post-Newtonian (PN) approximation to GR (see [19] for a review); radiation reaction (see [10] for a review); exact numeric (see, e.g., [25] for one of the most recent and complete catalogs of numerical GW forms); and phenomenological approaches collecting the results of all the above (see [72] and [58] for latest waveform developments). The present work is an overview of the PN approximation scheme approached with effective field theory methods, introduced in [49] and known as nonrelativistic general relativity (NRGR), for which reviews are already available; see, e.g., [34, 45, 64, 80]. We treat the two-body problem within the PN approximation scheme, in which the dynamics is expanded around the Newtonian result, with expansion parameter being the relative velocity v, where v 2 ∼ GN M/r for Kepler’s law (with GN the Newton’s constant, M the total mass of the binary system, r the binary constituent mutual distance, using natural units for the speed of light c = 1), and n-PN n−j +1 2j corrections corresponding to terms of the order GN v , with 0 ≤ j ≤ n + 1. Note that this is not the type of expansion one usually deals with in particle physics, where the coupling constant, in this case GN , is the expansion parameter. Expansion in powers of GN in the two-body problem is known as postMinkowskian (PM) expansion [30], which has received a lot of attention after the field theory derivation of the 3PM dynamics [12] and partial results up to 6PM order within the syncretic approach of [28].

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The effective potential at any PN order is instantaneous; the retardation effects necessary for a consistent relativistic theory will manifest themselves via Taylor expansion of binary constituent trajectories, hence contributing to velocitydependent potentials that will be present at any GN order. As it will be shown, interest in the problem goes beyond its applications to GW physics, due to its richness in intriguing theoretical aspects, representing a highly nontrivial test bed for classical field theory. The clear separation of scales of the underlying physical problem allows a paradigmatic application of effective field theory techniques and a systematic expansions of interactions in terms of Feynman diagrams. Divergences are encountered in purely classical computations, which are regularized and renormalized; applications of the renormalization group flow equations can be used to derive non-perturbative results. This review is structured as follows. First, NRGR is introduced, with emphasis on classical path integral and how to split the integration region of Green function momenta into a near and a far zone. Then, a summary of the methods allowing to obtain the conservative dynamics within NRGR is presented, followed by a section dedicated to the radiative sector of the problem. The final section is dedicated to a compilation of the present status of the investigations and prospects for future developments.

Notation We adopt the mostly plus metric diag (−, +, +, +) and natural units for the speed of light c = 1 but keep Newton’s constant GN explicit. Vector are indicated in boldface   dd k character, e.g., x, and for integration over momenta, the notation k ≡ (2π is )d adopted. Given a quantity A(t, x) in direct space, its Fourier transform and inverse Fourier transform are defined as  ˜ A(ω, k) = dtdd x A(t, x)eiωt−ik·x ,  dω ˜ A(t, x) = A(ω, k)e−iωt+ik·x . k 2π

The Setup To describe the gravitational dynamics of a two-body system, we introduce the bulk Lagrangian of gravity, which is given by the gauge-fixed Einstein-Hilbert action SEH

1 = 2Λ2

 d+1

d

  1 μ x R − Γ Γμ 2

(1)

where d is the number of pure space dimension (d = 3 in our world); the μ μ harmonic gauge condition involves Γ μ ≡ Γαβ g αβ , being gαβ the metric and Γαβ the

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Christoffel symbol; Λ is a constant with dimension of (lengthd−2 /mass)1/2 defined √ d−3 by Λ ≡ (32π Gd )−1/2 , being Gd ≡ GN 4π eγE /2 L0 , with L0 an arbitrary length needed to ensure the correct dimension of the Newton’s constant in generic d + 1 dimensions; and the irrational number in its definition is due to an arbitrary but convenient choice that will be justified later. To complete the description of the dynamics, one needs to complement action (1) with the coupling of gravity to world-line sources. The most general world-line action for an object with mass, spin, and multipole moments can be written as 

 1 dτ − dτ S μν Ωμν 2    1 4 ij 1 ij k ij + dτ Q Eij + J Bij + I Eij,k + . . . , 2 3 3

Swl = −m

(2)

where S μν is the spin antisymmetric tensor (which provides a redundant description of the physical spin, which can be defined as Si = 12 ij k S j k ; see [78]); Ω μν are the components of the relativistic generalization of the angular velocity of the object; Qij and J ij are, respectively, the electric and magnetic quadrupole moment; I ij k is the octupole electric moment (with higher multipoles understood); and Eij (Bij ) is μ the electric (magnetic) part of the Riemann curvature tensor R νρσ , defined in the rest frame of the source via the standard relations Eij = R0i0j , 1 Bij = ikl R0j kl , 2

(3)

being ij k the standard three-dimensional Levi-Civita tensor. The multipole tensors can be intrinsic to the object or induced by the external field; in the latter case, it is useful to separate the contribution from induced multipoles by adding them explicitly to (2)  Sind =

  dτ cE0 Eij2 + cB0 Bij2 + cE1 E˙ ij E ij + cB1 B˙ ij Bij + . . . ,

(4)

where cEi,Bi are constant scaling mrs4+i , where rs is the typical source size, and dots stand for terms with higher time derivatives. Series (2) and (4) are expansions in terms, respectively, of rs k and rs ω, being k (ω) the typical inverse space (time) variation scale of the Riemann tensor of appropriate parity, so they are expansions in size of the source vs. background curvature length (in case Riemann curvature is determined by radiation k = ω). Actually, all terms with an odd number of time derivatives in (4) are total derivatives; hence, they do not contribute to equation of motions derived from (4) with time-symmetric boundary conditions. When dealing with an action principle to derive equation of motions, usually, the in-out formalism, with time-symmetric boundary conditions, is understood, which however cannot account for dissipation,

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which is exactly what terms with odd number of time derivatives are accounting for; see [50, 76] for the values of cE1 , cB1 . The correct treatment of dissipative terms consists in adopting the in-in formalism [42,53], whose description goes well beyond the scope of this review. However, it will be shown later in this work how to extract information about time-averaged dissipation from the imaginary part of the time Fourier transformed Lagrangian in the in-out formalism. For the moment, it can be anticipated that dissipative terms contribute to the imaginary part of the Lagrangian as the Fourier transform of the Lagrangian term E˙ ij (t)E ij (t) ∼ iω|E˜ ij (ω)|2 , which would vanish when integrated over dω to give the action. The following step is to map actions (2, 4) into the description of a definite physical system. When applied to individual black holes, which can be described solely by mass and spin (neglecting charge which is astrophysically insignificant), one can drop all terms in the second line of (2) but the quadrupole one, since Kerr black holes have a spin-induced quadrupole (spin)

Qij

=

CES 2 Sik S kj 2m

(5)

with CES 2 = 1 [75] for Kerr black holes. The induced quadrupole term cE0 ,B0 have been shown to vanish for a Schwarzschild black hole [17, 29, 60], and there are indications that they vanish for Kerr black holes too [73, 77]. Note that for spinning black holes, a new class of induced multipole terms appears, coupling multipoles of different order. For matter sources like neutron stars, independently on their spin, individual object can be endowed with permanent multipoles, as well as induced multipoles of any kind, coupling to the companion’s gravitational field. The spin-induced quadrupole multipole (5) can have CES 2 ∼ 4 − 8 depending on the equation of state [61], and none of the cEi , cBi coefficients are expected to vanish in general. When applied to the binary system as a whole, one can discard (4) and consider action (2) as the coupling of the binary system multipoles to emitted radiation. In this case, the multipole expanded action (2) is an expansion over the internal velocity of the source, as k = ω ∼ rs /v, so truncation of the expansion to a finite order correctly represents source with nonrelativistic or mildly relativistic internal velocities.

Classical Path Integral One can use the coupling of the world-line action for either point particles with no permanent multipoles, i.e., first line in Eq. (2), or extended objects, to derive the effective potential ruling the dynamics of a binary system via Gaussian integration. Formally, the effective action is obtained by integrating out the gravitational field mediating the interaction

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 Z[mi , vi , . . .] ≡ eiSeff (mi ,vi ,...) =

Dhμν ei(SEH +Swl ) ,

(6)

where Swl can be taken from (2), possibly considering multipoles, and (4) for finite size objects, and . . . stand for all possible world-line degrees of freedom (like spin, acceleration, eventually higher derivatives of the trajectory, multipoles, tidal coefficients). Note that strictly speaking, a quantity with the dimension of an action should appear at the exponents in (6) dividing Seff to give the correct dimensions, that is, h¯ in quantum application of the path integral. However, we will work here at the classical level and consider only terms homogeneous in 1/h, ¯ thus making irrelevant its value to determine the classical dynamics. Neglecting gravity self interactions, i.e., keeping in SEH only the quadratic, gauge-fixed, kinetic term, the Gaussian integral for the continuously infinite variables h˜ μν (ω, k) can be performed in analogy to the simple one-dimensional case 



1 dx exp − Ax 2 + J x 2



 =

  1 2π −1 exp JA J . A 2

(7)

Then, taking the logarithm and neglecting terms that do not depend on the sources, one gets effective action for matter particles which schematically at leading order in gravitational self-interaction is  1 dτ1 dτ2 [J1 (t1 )G(t1 − t2 , x1 (t1 ) − x(t2 ))J2 (t2 ) 2 +J1 (t2 )G(t2 − t1 , x2 (t2 ) − x1 (t1 ))J2 (t1 )] ,

iSeff (J ) = −

(8)

where Ji stands for source charges, which in our case will be the energy-momentum δSm tensor (As usual the energy momentum tensor is defined by T μν = − √2−g δg . μν Note that in GR, the source can also couple nonlinearly to the gravity field of particle i;) G is gravitational field Green function; ti is understood to be functions of τi ; and Lorentz indices have been suppressed here for simplicity. The effective potential derived in this construction is manifestly symmetric under 1 ↔ 2 as this construction is tailored for determining the conservative dynamics. The natural choice for the Green function boundary conditions is the Feynman one  GF (t, x) ≡ k

dω e−iωt+ikx , 2π ω2 − |k|2 + ia

(9)

where a stands for an arbitrary small positive number, necessary to ensure convergence of the Gaussian integral. (We could not use the character here as it is common in literature, since we prefer to keep it to denote d − 3, being d the number of space dimensions.) Other commonly used boundary conditions define the retarded and advanced Green functions

32 Effective Field Theory Methods to Model Compact Binaries

 GA,R (t, x) ≡ k

e−iωt+ikx dω 1 δ(t ± |x|) . =− 2 2 2π (ω ∓ ia) − |k| 4π |x|

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(10)

Introducing the Wightman functions  Δ± (t, x) ≡

k

e∓ikt+ik·x , 2k

(11)

one finds that [43] (To derive (12) the representation of the Heaviside theta function  ∞ dω eiωt θ (t) = i −∞ 2π ω+ia is used.) GF (t, x) = −i(θ (t)Δ+ (t, x) + θ (−t)Δ− (t, x)) , GR (t, x) = −i(θ (t) [Δ+ (t, x) − Δ− (t, x)]) , GA (t, x) = i(θ (−t) [Δ+ (t, x) − Δ− (t, x)]) ,

(12)

from which one can derive GF (t, x) =

1 −i (GA (t, x) + GR (t, x)) + (Δ+ (t, x) + Δ− (t, x)) . 2 2

(13)

One then finds that Feynman Green function is complex, its real part being given by the time-symmetric, real combination GA + GR , and its imaginary part given by the symmetric combination of Wightman functions known as Hadamard function. Plugging GF into Eq. (8), one sees that in our convention, Re(Gf ) gives the conservative potential via the time-symmetric Green function 1/2(GR + GA ), and the I m(GF ) is related to the imaginary part of the effective action, which is related to the “decay width” (divided by h¯ ) Γ of the process, i.e., it represents the rate of particle emitted, in our case gravitons. We can thus write  I mSeff = h¯

dω Γ˜ (ω) , 2π

(14)

where via a Fourier transform Seff has been conveniently expressed in terms of an integral over Fourier variable ω conjugate to time. Despite Γ˜ (ω) being an intrinsically quantum object, i.e., h¯ −1 i.e. the number density of particle emitted, once multiplied  by ω the integrand in (14), the righthand side of (14) becomes the quantity dω ¯ ωΓ (ω) which is the emitted energy, 2π h hence a classical quantity in the case of macroscopically large number of emitted graviton. Equivalently, one could have determined the emitted flux by first obtaining the classical gravitational perturbation by inverting the linearized Einstein equation, using the TT gauge for convenience  hTijT (t, x) = −16π GN Λij,kl

dt d3 x GR (t − t , x − x )Tkl (t , x ) ,

(15)

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Fig. 1 Optical theorem at work: imaginary part of the process in which a particle emits and reabsorbs a radiative mode that is related to the product of the emission amplitude by the absorption one

and then computing the energy flux F via the standard formula F =

1

h˙ ij h˙ ij  , 32π GN

(16)

see Sec. 3 of [80] for an explicit demonstration. At the hearth of the equivalence between the two methods lies the optical theorem which equates the imaginary part of the process of emission and absorption of the same radiation mode with the modulus square of the emission process, i.e., the product of amplitude for emission by the amplitude of absorption, see Fig. 1. At the end, imaginary part of the effective action depends on the imaginary part of the ˜ F = π δ(ω2 − k2 ). Feynman Green function: I mG Thus, in the in-out formalism, one can get the averaged emitted flux, but for describing the emission waveform (15) and the radiation-reaction force onto the source, one must use retarded Green functions within the in-in formalism. To conclude this section, we find it useful to recall the explicit form of (15), expanded for small-source internal velocities 

hTijT (t, x)

Tkl (t − |x − x |, x ) |x − x | 

(17) 4GN 3 Λij,kl dt d x Tkl (t − r, x ) + T˙kl (t − r, x )nˆ · x + . . . , r

= 4GN Λij,kl

dt d3 x

where |x − x | has been expanded in the argument of the energy-momentum tensor as r − nˆ · x . The expansion parameter of (17) is v, the same as in the multipolar action (2).

Method of Regions The main goal of the two-body problem modeling within PN approximation is determining the dynamics in terms of the two-body trajectories (and derivatives), by integrating out, i.e., substituting with their values fixed by Einstein’s equations, the gravitational field mediating the interaction. This is very efficiently performed via path-integral Gaussian integration. A crucially useful tool in determining the PN dynamics is to separate the contributions of the gravitational modes mediating the gravitational interaction in

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a near and a far zone, thanks to the clear separation of scales between the size of the binary r and the gravitational wavelength 1/T ∼ r/v v, being T the orbital period. ¯ say, with the property such that ω ∼ Let us choose a momentum scale k, ¯ v/r  k  1/r. Physically, in the near zone (distances shorter than k¯ −1 ), the source appears as composed of particles, eventually with finite size; in the far zone (distances larger than k¯ −1 ), the binary system is described as a composite object of negligible size and endowed in general with mass, spin, and all sort of multipoles. The potential between the two particles with trajectories xi (ti ), taking the pointparticle part of the world-line action (2) for the source, can be expressed in terms of the particle energy-momentum tensors (in mixed direct space in time and Fourier space in space representation) uμ (t)uν (t) ik·x1 μν e T˜i (t, k) = m , u0 (t)

(18)

and considering for simplicity only the contribution of the 00 gravitational polarization to the effective action, one has 





1 . 2 − ω2 − ia k k (19) We want to show that the exact O(GN ) potential (hence neglecting gravity selfinteraction, i.e., correction G2N or higher order) can be reproduced by expanding the Green function for ω  k (k ≡ |k|), i.e., by replacing the k integral in (19) with Seff = m1 m1 GN 4π

dt1 dt2

 e

dω −iω(t1 −t2 ) e 2π

ik·(x1 −x2 )

k

eik·(x1 (t1 )−x2 (t2 ))

 m 1  ω2 , k2 k2

(20)

m≥0

with some caveats that are going to be explained. Qualitatively, one expects that the effective potential is mediated by the exchange of off-shell longitudinal modes with k  ω, not by on-shell radiative modes with k = ω. A complementary expansion can be taken in the far zone kr < 1, where one can write  k

 (ik · x)n 1 . n! k2 − ω2

(21)

n≥0

Note that both integrals (20) and (21) are performed over all values of k; hence, none of the two are expected to reproduce quantitatively the exact result (19). As it will be shown, the exact result is given by the sum of (20) and (21), provided dimensional regularization [85] is adopted in computing the integrals, which consists in extending the three-dimensional integrals to arbitrary d = 3 + dimensions and taking the limit → 0 only at the end of the computation.

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The sketch of the proof goes as follows: the interested reader can look at the formal proof in [52]. Let us define 1 , − ω2 m 1  ω2 , N ≡ eik·r 2 k k2 m≥0  (ik · r)n 1 F ≡ , n! k2 − ω2 I ≡ eik·r

k2

(22)

n≥0

and rewrite the original k integral of (19) as the trivial identity 

 I = k

 F+

k

N k

¯ (I − N − F ) + θ (k − k)

+ k



θ (k¯ − k) (I − N − F ) ,

(23)

k

being θ the usual Heaviside step function, and k¯ a scale separating the orbital and radiative scale. It will be shown that the second line in Eq. (23) vanishes. Since for k > k¯ (near zone) one has k > ω and for k < k¯ (far zone) kr < 1, the full integral in each region can be written as 

¯ = θ (k − k)I

k

θ (k¯ − k)I =

 k

k

¯ , θ (k − k)N θ (k¯ − k)F .

(24)

k

To reach our conclusion, we just need to show that 

¯ + θ (k − k)F k



θ (k¯ − k)N = 0 ,

(25)

k

which holds because both integrands of F at large k and of N at small k admit the same parametrization 

¯ = θ (k − k)F k

 k

 m  1  (ikr)n ω2 = θ (k¯ − k)N , n! k2 k2 k

(26)

m,n≥0

which vanishes in dimensional regularization. The procedure of this simplified example carries over to the multi-loop integrals defined in the next section, generated by gravitational self-interaction. The derivation of the validity of the method of region is shown in [52], and it is explicitly demonstrated up to 4PN order in [40]. Note that using in (19)

32 Effective Field Theory Methods to Model Compact Binaries



dω −iω(t1 −t2 ) 2 e ω = ∂t1 ∂t2 δ(t1 − t2 ) , 2π

1289

(27)

factors of ω2 are replaced by ∼ (k · v1 )(k · v2 ) (vi ≡ x˙ i ); hence, the expansion in (ω/k)2 is equivalent to an expansion in v 2 , and the expansion makes the potential effectively instantaneous. In particular, a perturbative, mixed direct space in time and Fourier space in space representation of the Green function are useful  2  ∂ 1 . GF (t, k) = δ(t) 2 1 + O t2 k k

(28)

On the other hand, in the far zone, the expansion parameter is ∼ krs (being rs the size of the source, whether it is a single fat object or a binary system), which is the expansion parameter of the multipole part of the action (2).

Gravitational Self-Interactions, Wick Contraction, and Power Counting When gravitational self-interactions or nonlinear source coupling are considered, the Gaussian integral (6) involves bulk terms hm with m ≥ 3 or J hn terms with n≥2, and it is convenient to treat all interactions, both gravitational self-interactions and the ones with the source perturbatively. E.g., in case of the one-dimensional Gaussian variable in the example (7), one Taylor expands from the exponential everything but the quadratic term Ax 2 to be left with integration of the type 

  1/2      2π d k 1 2 1 −1  JA J  dxx exp − Ax = exp . (29) 2 A dJ 2 J =0 k

Performing this integral is equivalent to substituting pairs of x variables with one Green function for each pair in all the possible ways to form pairs. When dealing with infinite number of fields, like for h˜ μν (ω, k), the path integral integration has the effect of replacing each pair of fields in the integrand out of the exponential with the inverse of what multiplies the quadratic term |h˜ μν |2 , i.e., the Green function: this substitution is called Wick contraction. Examples of representations in terms of a Feynman graph of processes involving the direct linear interaction of two sources, interaction mediated by a cubic selfcoupling and a nonlinear source interaction, are shown in Fig. 2. Another fundamental concept for the derivation of the effective action in terms of perturbative path integral is the definition of a connected diagram: if following Green function lines all the vertices (both gravitational self-interaction and gravitysource ones) can be connected, the diagram is said to be connected; otherwise, it

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Fig. 2 Examples of Feynman diagrams representing near-zone contributions at leading order (first on the left) and at v 2 order with respect to leading (both remaining diagrams)

is disconnected. Only connected diagrams contribute to the effective action. This statement will not be demonstrated here, but a sketch of the proof is the following. Let us collectively indicate with J all the source degrees of freedom in (7), and let us define Z0 [J ] by setting to zero in Z[J ] all interactions. Taking the logarithm of (6) and Taylor expanding Z[J ] around Z0 [J ], one gets Seff [J ] = −i log Z0 [J ] − i

∞  (−1)n+1 n=1

n

Z0−n [J ] (Z[J ] − Z0 [J ])n .

In the expression for Seff , all terms with n > 1 describe disconnected diagrams, and some disconnected diagrams are also present, along with connected ones, in the n = 1 term. However, the n = 1 disconnected contribution is precisely canceled by the n = 2 terms, and an exact cancellation happens at all n. All the examples in Fig. 2 correspond to connected diagrams, two of them being loop diagrams, i.e., involving an integration over internal momenta. However, they all describe classical processes. For a generic graph, the number of loops L is given by L = I − V + 1, being V the number of vertices and I the number of internal lines (including the solid line for massive particles). Let us denote with Ih the number of internal gravitational lines and by Im the number of massive source internal lines, with I = Ih + Im . Each Green function, being the inverse of the quadratic part of (SEG + SGF )/h, ¯ brings a factor h; ¯ each vertex, as it comes from the expansion of SEH /h, ¯ a factor h¯ −1 . Note that the massive bodies are non-dynamical source/sinks of gravitational mode; they are not associated to any Green function; thus, for any graph, we have the quantum scaling h¯ Ih −V = h¯ L−Im −1 ,

(30)

where h¯ −1 is the scaling for classical graphs, and it can be straightforwardly verified that all three diagrams in Fig. 2 are classical. The reader familiar with scattering amplitudes in relativistic quantum field theory may want to check this result with the h¯ counting performed there. The two-body potential V (x) is a classical concept, and one can derive it from the computation of relativistic scattering amplitudes Ai→f for a process taking from

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some two-body initial state i to a two-body final state f via; see, e.g., Chap. 6 of [70]  V (x) = −

Ai→f eiq·x ,

(31)

q

where q is a wave number (i.e., momentum divided by h) ¯ and the nonrelativistic normalization for the quantum states is understood. In the computation of a matrix element between states k and k mediated by the energy-momentum tensor T μν for a particle of mass m, the relativistic and nonrelativistic amplitudes are related by

p|T˜μν |p (N R)

m h¯ 2

= p|T˜μν |p (R) .

Then, for the tree-level graph in Fig. 2, assuming for simplicity, a scalar field φ with action Sφ given by S=−

1 2



  m2 dtdd x ∂μ φ∂ μ φ + 2 φ 2 , h¯

(32)



2 2 and using that p|T 00 |p (R) ∼ p0 /h¯ ∼ (m/h) ¯ for small velocities, one finds the scaling of the first diagram in Fig. 2 h¯ ¯4 (R) (R) h

p |T |p 

p |T |p  1 1 2 2 1 2 m1 m2 h¯ 2     2 2 4 m2 h¯ m1 m1 m2 h¯ , = 2 ∼ m1 m2 h¯ h¯ h¯ h¯

(33)

which correctly reproduces the classical scaling with h¯ (and masses) of the diagram. The second diagram in Fig. 2 is slightly more involved, and it includes a loop integral  k

1 1 1 dω , 2 2 2 2 2π ω − k + ia ω2 − (q − k) + ia (E + ω) − (p + k)2 − m2 /h¯ 2 + ia (34)

where (0, q) is the overall wave number exchanged between the two objects, and the massive body with two gravitational insertions has initial wave number (E, p). In the nonrelativistic limit, E ∼ m/h, → 0 gives [18] ¯ hp/m ¯ h¯ 2m

 k

  1 hq 1 1 dω ¯ 1+O , (35) 2 2 2 2 2π ω − k + ia ω − (q − k) + ia ω + ia m

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which can be integrated by closing the integration contour in the upper half plane to pick the contribution from the residues at the poles ω = −|k| + ia and ω = −|k + q| + ia to get −i

h¯ 4m

 k

  1 1 h¯ q 1 + O . 2 2 m k (k + q)

(36)

This result is proportional to the loop integral in the potential region of momenta, see the Green function representation (28), and it is understood that the region of integration is |k|  m/h. ¯ This exercise shows that the loop integral (34) has both a classical and quantum pieces, the former being isolated by the limit hq/m → 0. Taking such limit before ¯ performing the loop integration over k kills the quantum part of the diagram. Going back to h¯ power counting, one gets h¯ 4 h¯ h¯ 4 m2



m1 h¯

2 

m2 h¯

4

m1 m22 h¯ 4 , ∼ m1 m2 h¯

(37)

hence, it scales as a classical diagram, with appropriate mass powers. As a result, we have shown how classical contributions from loop diagram arise by taking the large mass limit.

The Conservative Sector Besides the scaling with h, ¯ it is instructive to check the scaling of diagrams in NRGR for astrophysical parameters m, v. Using the parametrization (28) and considering that |k| ∼ 1/r, t ∼ r/v, one can associate the scaling laws: • • • •

 source vertex m dt → mr/(hv) ¯ , −1 , interaction vertex dt∂i2 hn → (GN hrv) ¯ δ(t) Green function k2 → GN hrv ¯ , space integration over loop momentum d 3 k → r −3 .

Putting these ingredients together, one has that the three diagrams in Fig. 2 scale as  L mr 2 1 Gm2 ∼ , Fig. 2a ∼ × 3 × h¯ rvGN ∼ vh v h¯ h r  ¯ 3  2 Lv 2 mr Gm2 1 1 3 ∼ , (38) Fig. 2b ∼ ∼ × 3 × (GN h¯ rv) × v h¯ GN h¯ rv v h¯ h r  3  2 mr 1 Gm2 Lv 2 Fig. 2c ∼ × 3 × (GN h¯ rv)2 ∼ ∼ , v h¯ v h¯ h r 

32 Effective Field Theory Methods to Model Compact Binaries

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where in the last passage in each line we used Kepler’s law GN m/r ∼ v 2 and substituted for the angular momentum L ∼ mvr. Clearly, quantum effects are negligible for astrophysical system which have L ∼ 8 × 1077 h¯



M 3M

2 

v −1 , 0.1

(39)

even though the leading quantum correction to the two-body gravitational potential has been computed; see [31]. To efficiently compute the PN expanded two-body potential, it is useful to distinguish the gravitational polarization according to what powers of v they involve. For this reason, the following metric decomposition [59] ⎛

gμν

⎞ Aj −1 ⎜ ⎟ Λ  = e2φ/Λ ⎝ A  Ai Aj ⎠ σij i −cd φ/Λ − δij + , e Λ Λ Λ2

(40)

with cd ≡ 2(d − 1)/(d − 2), inspired by the Kaluza-Klein (KK) decomposition, can be adopted. The corresponding bulk action, truncated to cubic interactions, then reads [32]    −cd φ   1 (∇σ )2 − 2(∇σij )2 − σ˙ 2 − 2(σ˙ ij )2 e Λ ⊃ d x −γ 4  2   Fij cd φ  cd φ c φ 2 2 2 − 2 − ˙ e Λ e dΛ + (∇·A) − A −cd (∇φ) − φ˙ e Λ + 2 

SEH +GF

d+1

√

  2  ˙ ˙ Fij Ai A˙j + A·A(∇·A) − cd φA·∇φ Λ 

σ˙ ij  ij j l ˙ ˙ −δ Al Γˆkk +2cd φ∇·A + 2Ak Γˆijk − 2Ai Γˆkk − A·∇φ + Λ     1 σ ij ij ,k ,l ,k ,k ,l ,k δ −σ σik σj l − σik σj l + σ,i σj k − σik,j σ , − Λ 2 +

(41) where Fij ≡ Aj,i − Ai,j and Γˆjik are the connection of the purely spatial d-dimensional metric γij ≡ δij + σij /Λ, which is also used above to raise and contract spatial indices. All spatial derivatives are understood as simple (not covariant) derivatives, and when ambiguities might arise, gradients are always meant to act on contravariant fields, e.g., ∇·A ≡ γ ij Ai,j and Fij2 ≡ γ ik γ j l Fij Fkl . The world-line action for point particles expressed in terms of the KK variable is

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 Spp−wl = −ma

   va2 ma + . . . + Ai vi (1 + . . .) dt −φ 1 + dτ ⊃ Λ xa 2

(42) 1 φ2 + σij v i v j (1 + . . .) + (1 + . . .) + . . . , 2 2

where ellipses stand for terms with higher powers of v or higher powers of the gravity fields. Besides coupling to point particles with definite v−scaling, the variables φ, Ai , σij have the further advantage of having diagonal Green functions, as their kinetic terms do not mix: ⎫ ⎪ ˜ φ,φ (ω, k) = − 1 ⎪ G ⎪ ⎪ 2cd ⎪ ⎬ δij 1 ˜ × 2 .(43) GAi ,Aj (ω, k) = 2 − ia k − ω 2  ⎪ ⎪ ⎪ 1 2 ⎪ ˜ σij ,σkl (ω, k) = − ⎭ G δik δj l + δil δj k − δij δkl ⎪ 2 d −2 As an example, we give the actual computation of the 1PN potential, which involves the first diagram in Fig. 2 with φ, Ai polarizations mediating interaction and the last with only the φ polarization involved. The middle one turns out to be a 2PN diagram, as it will be shown momentarily. The first diagram via φ exchange boils to   dω e−iω(tA −tB )+ik(xA −xB ) 3 2 v1 + v22 + dtA dtB 1 + 2 k2 − ω2 − i k 2π  ik(xA −xB )   ∂t1 ∂t2 im1 m2 e 3 2 2 v 1 + dt δ(t − t ) + + v dt A B A B 2 2 1 8Λ2 k2 k2 k

  



3 2 1 GN m1 m2 1+ v1 + v22 + (v1 · v2 ) − v1 · nˆ v2 · nˆ ,(44) = i dt r 2 2

F ig. 2aφ =

im1 m2 8Λ2



where the formulae 

1 Γ (d/2 − α)  r 2α−d = , k 2α (4π )d/2 Γ (α) 2 k

   d ki kj Γ (d/2 − α + 1)  r 2α−d−2 δ ij i j ,(45) − − α + 1 n eik·x 2α = n 2 2 2 k (4π )d/2 Γ (α) k eik·x

have been used.

32 Effective Field Theory Methods to Model Compact Binaries

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The same graphs with an Ai exchange gives 

im1 m2 F ig. 2aA = − 2Λ 2 = −4i

dt

j dtA dtB δij v1i v2

GN m1 m2 v1 · v2 , r

 k

dω e−iω(tA −tB )+ik(xA −xB ) 2π k2

(46)

and finally, the last diagram in Fig. 2 gives

F ig. 2cφ 2

  im1 m22 eik·(x1 −x2 ) =− dt 128Λ4 k2 k 2 2 G m m2 = i N 21 . 2r

2

(47)

The sum of the three sub-processes (the last diagram must be supplemented with its image under 1 ↔ 2) returns the 1PN Einstein-Infeld-Hoffman potential VEI H = − +





 GN m1 m2   2 3 v1 + v22 − 7v1 · v2 − v1 · nˆ v2 · nˆ 2r G2N m1 m2 (m1 + m2 ) . 2r 2

(48)

The middle diagram in Fig. 2 is a genuine loop diagram involving three gravitational Green functions, which we take to be φ-polarized for illustration F ig. 2bφ 3 = =

im1 m22 32Λ4 i





eik·(x1 −x2 )

dt k,p

k2 (k − p)2

(2k · v1 p · v2 + p · v2 (p − k) · v2 )

 G2N m1 m22  2v1 · v2 − 4(nˆ · v1 )(nˆ · v1 ) − v22 + (nˆ · v2 )2 . (49) 2 2r

To obtain the result (49), one needs the following standard integral 

1 k

k 2a (p − k)2b

2 d/2−a−b p Γ (d/2 − a)Γ (d/2 − b)Γ (a + b − d/2) . = d/2 Γ (a)Γ (b)Γ (d − a − b) (4π ) (50)

At higher GN order, one hits on higher-order nested loop integrals: when combined with standard integration by parts techniques [26, 84, 87], Eq. (50) is enough to integrate the new two-loop master integrals that appear at O(G3N ) and the three three-loop master integrals appearing at O(G4N ) [35]. At O(G5N ), seven new master integrals appear, one of which vanishing and one of which has been computed for the first time in d = 3 in [37].

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Divergences When dealing with loop integrals, it is not unlikely to encounter divergent quantities, which need regularization to save predictivity of the computations, and the regularization mechanism naturally embedded in effective field theory methods is dimensional regularization. As an introductory example, let us analyze the structure of divergences of (50), considering that the gamma function has simple poles for all nonpositive integer argument. The poles for a = d/2 and b = d/2 correspond to infrared (IR) divergences of  −d/2 the type ∼ k k2 for k → 0, whereas an ultraviolet (UV) divergence appears for a + b = d/2. The integral vanishes identically when either a = 0 or b = 0, but  −a care is needed for the scaleless integral k k2 , which is always vanishing, but it can be considered as the sum of mutually compensating UV and IR divergences for the specific case a = (3 − ) /2 [81]  k

L−2 1 0 = 3− 2 4π 2 2 k



1 U V

−

1



I R

,

(51)

where the arbitrary length scale L0 has been added on the right-hand side to adjust dimensions. Clearly, all dependence on L0 must vanish when → 0, i.e., d → 3, and the notation U V ,I R has been introduced to identify from which integration region the divergences arise, (We could have also introduced a L0I R and L0U V to clarify from which limit the arbitrary L0 scale originates from.) as it is important to keep track of the nature of divergences, which have different origins and very different physical meanings, as we now discuss. Note that the divergences of the scaleless integral (51) do not depend on r, so one may think it could be simply discarded as it does not affect the potential; however, it could still enter as a sub-diagram of a larger, factorizable diagram overall dependent on r or involve derivatives of the particle velocity, like v˙ ∼ v 2 /r, thus having an implicit r dependence. In general, in gauge theories, UV divergences signal the failure to describe the short-distance features of the physical problem being modeled: they point to some short-distance incompleteness of the model. Divergences in dimensional regularization appear as a pole for d → 3 (poles for → 0 can be reabsorbed in dimensional parameter redefinitions); however, since the gravitational coupling has a dependence on the arbitrary length scale L0 for d = 3 of the type Gd ∝ Ld−3 0 , one has 1 1 Gd → (1 + log L0 ) , (d − 3)

(52)

so that the effective potential acquires a spurious dependence on the arbitrary quantity L0 in presence of divergences.

32 Effective Field Theory Methods to Model Compact Binaries

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Since the physical result cannot depend on L0 , there are two main possibilities: 1. The divergences are unphysical as one can get rid of them by a field redefinition, i.e., a mere change of (field) coordinates which does not affect observables. 2. A physical parameter (like the mass or a multipole moment) acquires a logarithmic dependence on the typical source-observer distance scale, say r, at which the physical parameter is measured, i.e., ∝ log(r/L0 ), to compensate for the log L0 dependence from (52). It turns out that the UV divergences of the near-zone description up to 4PN are of type 1. Considering a generic field redefinition like (index i running over the binary constituents) gμν → gμν = gμν + δgμν , μ

μ

μ

μ

xi → xi = xi + δxi ,

(53)

that changes the action by the amount  δS μ δS δgμν + μ δx , δgμν δxi i 2

δS =

(54)

i=1

a necessary condition for UV divergences to be absorbed by such redefinition is that they vanish on the equation of motions, and this is indeed what happens in the conservative sector of the two-body dynamics as explicitly checked up to 4PN order in [40]. Equivalently, using the language of field theory, it is possible to introduce into the action counter-terms to cancel the UV divergences, and since they vanish on the equation of motions, they do not affect any physical observables. The explicit counter-terms respecting the symmetry of the problem are made of curvature invariants and particle trajectory derivatives, and at lowest order, they are Sct =

2  

  μ dτi cia a iμ v˙iμ + civ Rμν ui uνi + ciR R ,

(55)

i=1 μ

being R and Rμν , respectively, the Ricci scalar and tensor and where a term aiμ ai has not been included since it is proportional to the square of the equations of motion (as long as geodesic motion is considered); hence, it can be considered exactly zero. The Ricci scalar and tensor appearing in (55) on the ith particle world line are the ones sourced by the other particle; hence, they vanish on the equation of motions μ (i.e., Einstein equation), and ai = 0 because of the geodesic equation. (Note that for a spinning particle one can indeed build a term linear in the Riemann of the type Qij Eij with a permanent quadrupole given by Eq. (5).)

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In [40], the values of the counter-term coefficients have been computed, resulting in 11 2 3 G m + c¯ia (L0 ) , 3 N i = G2N m3i + c¯iv (L0 ) ,

cia = − civ

(56)

ciR = 0 , the coefficients c¯i corresponding to arbitrary finite term addition of the counterterms which in general must be fixed by matching to an observable whose value is affected by their values. However, as stated earlier, these terms are proportional to equation of motions; hence, the values of the c¯i are completely irrelevant. At second order in the curvature, a possible invariant term is

SctR 2 ∼

2  

dτi ciR 2 Rμνρσ R μνρσ ,

(57)

i=1

which is exactly of the type of the finite size effect for spinless object introduced in Eq. (4). However, as it is well known [17, 29, 60], static tidal effects that would be generated by this term for non-spinning black holes vanish in 3+1 dimensions. The lowest-order diagram in which the Riemann-squared term can enter has the same μ topology as the last one in Fig. 2, and considering that Rνρσ ∼ ∂ 2 h, power counting shows that this diagram with a Riemann-squared insertion would contribute to Seff /h¯ as Lv 10 /h, ¯ i.e., a 5PN term, in agreement with the effacement principle [27]. It is then expected that the conservative two-body dynamics, when available, should not need a counter-term of this kind.

Infrared Divergences IR divergences are of a complete different nature, showing that the model is inconsistent at large distances. Since gravitational interactions are smaller at larger distances, IR divergences clearly point to an inconsistency of the modeling. The source of the inconsistency in our case can be readily identified in the Taylor expansion defining the near-zone integral (20): higher-order expansions have more powers of k2 at the denominator, only partially compensated by the typical term appearing at the numerator, ∂t2n eik·x ∼ k(d 2n−1 x/dt 2n−1 ) + O(k2 ); hence, the very approximation used in the near zone is bound to create infrared divergences. However, the problem has a straightforward fix, named zero-bin subtraction [71], which consists of deleting all IR divergences because they do not belong to the appropriate near region (|k| > v/r); after properly identifying IR and UV divergences with the procedure described earlier, zero-bin subtraction can be

32 Effective Field Theory Methods to Model Compact Binaries

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Fig. 3 Diagrams describing far-zone processes. Sub-diag. (a) is the leading-order nonzero process, which has only an imaginary part. Sub-diag. (b) represents emission/scattering/absorption of radiation (leading-order tail process) with both real and imaginary part. Sub-dig. (c) represents three-radiation modes interaction, or leading-order memory process, which happens to be purely real [36]

adopted, a simple recipe justified by the general description of adding near– and far-zone integrals described earlier. The far-zone diagrams involve a source endowed with multipoles; it describes the composite object made by the binary system, the simplest diagram being the first one in Fig. 3. The lowest-order diagram is given in Fig. 3a, where a gravitational mode is emitted and reabsorbed, its contribution to the effective action being [36]  F ig.3a = −iπ GN

∞

−∞

dω 2π

 k

˜ kl (−ω) ˜ ij (ω)Q Q 2 k − ω2 − i



1 −ω4 δ ik δ j l + 2ω2 δ ik k j k l − k i k j k k k l 2  ∞ GN dω ˜ ij (ω)|2 , |ω|ω4 |Q = 10 −∞ 2π

(58)

where the first lines of Eq. (58) show separately the contribution from the three polarizations σij , Ai , and φ. Diagram in Fig. 3a gives a real contribution to iSeff , which as discussed previously can be related to the emitted energy. Note that had we inserted in the same diagrams mass M or angular momentum L at the source-gravity interaction vertex instead of a quadrupole Qij , the amplitude would have vanished in both the real and imaginary part because both M˜ and L˜ are ˜ ij with higher-order multipoles ∝ δ(ω), being conserved quantities. Substituting Q of the electric (IL ) or magnetic (JL ) type would give their contribution to the emitted flux, (L)

F = GN cI,J

 0

∞

dω 2+2l ˜ ω |IL (ω)|2 , 2π

(59) (L)

where L denotes collectively l space indices, with the numerical coefficients cI,J given by

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(n + 1)(n + 2) , n(n − 1)n!(2n + 1)!! 4n(n + 2) cJn ≡ , (n − 1)(n + 1)!(2n + 1)!! cIn ≡

(60)

as computed within effective field theory methods in [36], matching standard GR predictions [86]. We will see in the next section how to express the multipole moments in terms of binary individual constituent parameters. At the next order in the coupling, one hits on the tail diagram 3b; see [20] for the first historical computation of tail diagrams, which describes the emission, scattering, and absorption of radiation [33, 44] 2G2 M F ig.3b = − N 5

 0

∞

dω 2π



1 U V

 41 ˜ ij (ω)|2 . − iπ + 2 log (L0 ω) ω6 |Q − 30 (61)

This diagram contains both a real and an imaginary part, contributing to the real part of Seff a UV divergent quantity with its related log(L0 ω) and a finite piece. Result (61) has to be combined with the near-zone result as per the discussion of the previous section to obtain the full result for the binary dynamics and give a finite answer by canceling the IR divergence there. To perform such check, one would first need to express the quadrupole in (61) in terms of individual binary components parameters, according to a standard procedure that goes under the name of matching that will be deferred to the next section. However, the leading-order quadrupole expression in terms of binary constituents can be easily guessed on general grounds   1 d3 xT 00 x i x j − δij x2 3   !2 j 1 2 i = n=1 mn xn xn − 3 xn , 

Qij =

(62)

and plugging it into (61), one recovers the correct finite result for the conservative dynamics, with the UV divergence in (61) exactly compensating the IR divergence from the near zone and the L0 dependence of the logarithm being canceled, as explicitly demonstrated in [40]. Note that the coefficients of the divergence, of the related logarithm, and of the relative imaginary term are in simple relationship with one another. This is not an accident of the quadrupole case, but the relationship can be generalized to arbitrarily multipole [36]    ∞ 1 dω 2+2l Stail = 2G2N M − iπ + 2 log (L0 ω) ω6 × ω U V   0 2π (I ) (J ) 2 cl |Iij (ω)| + cl |Jij (ω)|2 .

(63)

32 Effective Field Theory Methods to Model Compact Binaries

1301

This shows that coefficients (60) of the leading-order flux fix the tail correction to the flux and also the logarithmic contribution to the conservative dynamics: the logarithmic term coming from high PN order from the near-zone dynamics is fixed because its L0 dependence must match the L0 dependence in (63). In the PN expansion, far-zone UV and near-zone IR divergences first appear at 3PN order [40]. This shows that the near-zone IR and far-zone UV divergences are two sides of the same coin: the arbitrary replacement of the full theory integral with the sum of two partial representations introduces spurious divergences that cancel out in the full theory. In this case, the near zone acts as a UV completion of the far-zone theory. There is an additional lesson we can take from this: what if we had at our disposal only the far zone, i.e., low-energy theory, without knowing its UV completion? If that was the case, we could have rather imposed L0 independence of the physical result and obtain renormalization group equation evolution for physical parameters like mass, angular momentum, and multipoles. To be able to pursue this idea, one needs to compute processes involving emission of external gravitational modes; that will be the topic of the next section.

The Radiative Sector So far, we have considered processes in which initial and final states are the same: two massive objects going into two massive objects after exchanging some gravitational interaction or one composite object interacting with itself. This section is dedicated to processes involving one external gravitational mode, like the ones represented in Fig. 4. While only the radiative polarization of the gravitational fields can carry away energy from the binary system, it is instructive to consider diagrams where the external field has any polarization, including longitudinal non-radiative modes. E.g., the first graph in Fig. 4 represents the coupling of a 00-polarized gravitational mode to the mass of the composite binary object or the coupling of the 0i-polarized

Fig. 4 Leading-order and next-to-leading-order coupling of external gravitational mode to a composite source indicated by a double line. Black dashed lines indicate longitudinal gravitational modes; green wavy lines represent radiative modes

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Fig. 5 Diagrams describing leading-order radiative process from a spinless binary system. The first one must be supplemented by its mirror image under particle exchange. Green wavy lines represent radiation and black dashed φ-polarized longitudinal modes

gravitational field to its angular momentum (and linear momentum, if any). The graph in Fig. 4b represents the Qij Eij coupling of (2). To express the multipoles in terms of binary constituent parameters, it is necessary to map diagrams of the type of Fig. 5 that represent coupling to radiation as seen in the near zone, into those of Fig. 4. Diagrams in Fig. 5 give the following contributions for the radiative field m1 − iF ig. 5 = 2

  GN m2 r i r j σij i j v1 v1 − + 1 ↔ 2. Λ 2r 3

(64)

Using the Newtonian equation of motion a1 = −GN m2 r/r 3 , one can recast Eq. (64) into the standard 12 Qij Eij of Eq. (2) where Qij at lowest order is given as expected by Eq. (62). The derivation of the equivalence between Eqs. (64) and (62) is a standard textbook exercise which can be obtained in complete generality by repeated μν use of the energy-momentum conservation law T ,ν = 0. See [63] for matching calculation of multipoles up to electric hexadecapole and the magnetic octupole. Diagram in Fig. 4c is the radiative analog of the tail process Fig. 3b, and it gives GN Mω correction to (64). Relative to the leading-order Fig. 4b, it has a finite real part and an IR divergent imaginary part [83]     2 ω 11 π +i . +2 log(ωL0 ) + γE − F ig. 4c=F ig. 4b× 1+GN Mω |ω| I R 6 (65) The imaginary part can actually be resummed to a phase and hence drops out of the modulus squared of the (corrected) quadrupole to determine the emitted flux. Such phase however can in principle leave an observable imprint on the waveform, as its shift is not universal in its finite rational part for all multipoles. This happens because once fixed a reference time at which to start measuring the phase, e.g., the time at which the GW signal enters the detector’s sensitive band, phase differences

32 Effective Field Theory Methods to Model Compact Binaries

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Fig. 6 Coupling of radiative mode to a composite source at G2N M 2 order with respect to leading radiative diagram given in Fig. 4b

will depend logarithmically on the frequency and on rational number differences between different multipoles, but the unphysical divergent (and the irrational) part(s) will be unobservable. At the next order in the calculation of radiation, one has to compute the diagrams in Fig. 6, which are (GN Mω)2 corrections to the leading order. Such diagrams present a UV divergence, which makes manifest the failure of the farzone description to treat the binary as a single object of arbitrary small scales. The modulus squared of the sum of diagrams in Figs. 4c and 6 gives the (GN Mω)2 ∼ v 6 , i.e., 3PN gravitational self-interaction (or tail-squared) corrected quadrupole moment entering the flux formula (16), which is [48]

(v 6 )

|Qij |2 (0)

|Qij |2

= 1 + 2π GN M|ω| + (GN Mω)

2

214 105



1 U V

 − 2 log (ωL0 ) + . . . , (66)

where . . . stands for a finite part. This correction represents the only gravitational correction to the multipoles scaling with an expansion parameter that has an odd power of v: GN Mω ∼ v 3 . Additional corrections to the multipoles come from nearzone nPN corrections to the “matching” diagrams in Fig. 5 and by v 2n corrections in the expansion (17). Differently from the conservative sector, here, the computation in the near zone to resolve this far-zone UV divergence has not been performed; however, we have the possibility to use renormalization group equation to know about logarithmic terms in multipoles. After subtracting the infinite part, one can write the renormalization group flow equation for the renormalized quadrupole (which will still be indicated by Qij ) by imposing μ ≡ L−1 0 independence of the flux, which is a physical observable, to obtain [48] μ with βI ≡

214 105 ,

d ˜ Qij (ω, μ) = −βI (GN Mω)2 Qij (ω, μ) , dμ

which can be solved immediately for constant M to give [48]

(67)

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Fig. 7 Coupling of longitudinal modes to a composite source at G2N Q2ij order with respect to leading radiative diagram given in Fig. 4a [51]

 Q (ω, μ) = ij

μ μ0

−βI (GN Mω)2

Qij (ω, μ0 ) .

(68)

An analog exercise can be done for tail-squared diagrams to determine the G2N corrections to the mass M [51] and angular momentum L [15], from diagrams of the type in Fig. 7 to obtain  2G2  (1) (5) d log M (2) (4) (3) (3) = − N 2Qij Qij − 2Qij Qij + Qij Qij , d log μ 5  8G2N M ij k  dLi (5) (1) (4) (2) (3) μ =− Qj l Qkl − Qj l Qkl + Qj l Qkl . d log μ 5

μ

(69)

We first observe that μ dependence in M starts only at G2N order, so (68) is a consistent solution of (67). Short-circuiting Eqs. (68) and (69), one finds [21] "

# +∞ 

n−1 (n+2) (n+2) M(μ) (2 log μ) ¯ n = 1 − G2 βI G2 M 2

Qij Qij , M n! n=1

+∞

J i (μ) = J i (μ0 )−

n−1 (n+1) (n+2) ¯ n 12G2 M ij k  (2 log μ) ε βI G2 M 2

Qj l Qkl , 5 n! n=1

(70) where the brackets  denote the time average, μ¯ ≡ μ/μ0 , and M and Qij are understood to be evaluated at the scale μ0 . Such result can be made in more interesting form by finding the leading logarithms at all order in v of the energy of

32 Effective Field Theory Methods to Model Compact Binaries

1305

circular orbit, which is a relationship between two physical observables: the energy and the frequency of circular orbits. To do so, we must express the energy and the quadrupoles in Eq. (70) not as functions of the velocity v and radius r of the orbit, which have no invariant meaning being coordinate dependent, but as a function of the invariant orbital frequency ω. To obtain a relationships between r, v, and ω, the last one usually traded for the PN parameter x ≡ (GN Mω)2/3 , we need an extra piece of information, which is usually provided by the binary constituents equation of motion. Here however, it is more convenient to use the information than on circular orbits [41, 62] dE d|J | =ω , dω dω

(71)

also known as “thermodynamic” relation, derived from the first law of binary black hole mechanics for constant individual masses. Eq. (71), together with the two in Eq. (70), permits to determine r(x) and v(x), leading to invariant functions E(x) and J (x) containing all the leading logarithms of the energy and angular momentum functions. Remarkably, these functions admit a resummation, and we report here only the one relative to the energy [21] E(x) = −



 8mη2 x 2  3 1 + 24βI x 3 log x x 4βI x − 1 . 15βI

(72)

This result for the leading logarithmic terms in the energy of circular orbits involves only up to the next to leading order in the symmetric mass ratio η ≡ m1 m2 /M 2 ; hence, it can be checked with extreme mass ratio calculations, from which it is known up to 21PN order [57], showing perfect agreement.

Conclusions The effective field theory modeling of compact binary dynamics is a program currently under vigorous development, whose main phenomenological motivation lies in improving the templates used for gravitational wave detections. At the moment of writing (October 2020), the 5PN spinless conservative sector has been partly computed within the nonrelativistic general relativity approach by two groups independently; see [38, 39] and [22, 23], with agreement in the overlapping 5PN static sector (∼ G6N ); moreover, see [24] for partial results at 6PN, and work is ongoing to complete the 5PN static sector. The calculation of dissipative effects within the NRGR approach has been completed to 2PN level [63], and the tail-squared quadrupole contribution to the flux has been obtained at 3PN level in [48]. Spin degrees of freedom were only briefly mentioned in this review, but they can straightforwardly be included in the NRGR approach, as per the pioneer work of [78]. Recent results in the conservative sector reached next3 -to-leading-order level

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for linear in spin [67] and spin-squared sub-sectors [68] and next to leading order in the spin-cube [66] and spin-to-the-fourth sub-sectors [65], whereas in the flux, the investigations reached next to leading order in linear-in-spin terms ([74,83]) and spin-squared [82] sectors. Another topic only briefly mentioned in this review is the one represented by absorption effects, which have been investigated within NRGR both in the nonspinning [50] and in the spinning sector [79]. As the effective field theory approach leading to NRGR resulted in a large effort to investigate all aspects of binary dynamics (energy and flux functions, radiation reaction forces, all order renormalization group results, spin effects, absorption), still, it is not the only effective field-theory-based approach to tackle the two-body problem. Recent progress in effective field theory techniques, not directly related to NRGR, but promising fruitful developments in the near future, goes into the direction of computing the post-Minkowskian effective action, which reached G3N level [12–14], and solving the conservative dynamics of binary systems by computing post-Minkowskian expanded scattering deflection angle [54–56], as well as application of cutting-edge field theory methods like double copy relationships [11] to bound states [46, 47, 69] and NRGR [9]. We can expect in the near future advancements both in the NRGR approach, reaching for higher perturbative post-Newtonian orders, and in the postMinkowskian computations parallely improving using novel methods. Using yet other methods like the powerful resummation technique represented by the effective one-body formalism and input from the extreme mass ratio computations, as well as PN results, partial results in the conservative dynamics have been obtained at 6PN order [16]. The gravitational two-boy problem is presently being investigated under several “perpendicular” angles, the possibility of realizing nontrivial consistency checks among different methods and the cross-fertilization among different approaches is a powerful engine to the development of new investigation methods and computational tools and to reaching new results, with the expectation that they may shed some light beyond perturbation theory.

Cross-References  Binary Neutron Stars  Introduction to Gravitational Wave Astronomy  Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspi-

rals Acknowledgments This work has been partially supported by CNPq. The author wishes to thank Stefano Foffa for long-lasting collaboration and discussions. The author would like to thank ICTPSAIFR FAPESP grant 2016/01343-7.

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Repeated Bursts Gravitational Waves from Highly Eccentric Binaries

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Nicholas Loutrel

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eccentric Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matched Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power Stacking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of Black Hole Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binaries with Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tidal Interactions and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Post-merger and Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Compact object binaries formed from dynamics interactions will generically have nonzero orbital eccentricity. The gravitational waves from such binaries can change drastically depending on how large the eccentricity is, ranging from emitted in small a subset of orbital harmonics at low eccentricities to being concentrated into intense bursts of radiation from each pericenter passage at large eccentricities. Gravitational waves from such highly eccentric binaries present themselves an intriguing systems for probing fundamental physics but also present interesting challenges in terms of detection. The presence of orbital

N. Loutrel () Department of Physics, Princeton University, Princeton, NJ, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_33

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eccentricity in the gravitational wave signature gives an unequivocal method of determining the origin of the binary, while the highly dynamical nature of pericenter passage often enhances the physics associated with matter and gravity. The generation of faithful models of the repeated burst signals has proven challenging, and often, sub-optimal detection strategies like power stacking are considered for detection. This chapter reviews our current understanding of eccentric binaries, the leading efforts to model their gravitational wave emission, and the physics that can be probed with these detections. Keywords

Gravitational wave bursts · Eccentric binaries · Post-Newtonian approximation · Tests of general relativity · Binary neutron stars · Dynamical tides · F-modes

Introduction The advent of the era of gravitational wave astronomy has seen a remarkable number of important developments in our understanding of compact objects. From the first detection of binary black holes [1], along with constraints on the dynamical strong field regime of gravity [2], to the detection of binary neutron stars [3] that allowed for constraints on the nuclear matter equation of state and astronomical observations revealing the origins of r-process heavy elements within the universe [4], the ground-based detectors of the LIGO [5] and Virgo [6] observatories have opened a new window through which to elucidate our universe. Despite the detection of multiple compact object binaries [7], an open question still remains: what is the origin of such systems detected by ground-based observatories? Arguably, the canonical picture of binary formation is that of binary stellar evolution, described in detail in [8]. Mass transfer and common envelope phases of main sequence stars in binaries can harden the system such that the binary black holes and neutron stars formed as the end state of the binary’s supernovae can merge within the lifetime of the universe. A plausible example of such a binary is the Hulse-Taylor binary [9], composed of a neutron star and a pulsar, observations of which were used to infer the existence of gravitational waves long before the first detection [10]. Despite it’s importance to probing the strong field regime of gravity, there is one particular property of the binary that is worth examining. The binary is actually on a fairly elliptical orbit, with orbital eccentricity of e ∼ 0.7. The binary is approximately three hundred million years from merger, and thus, its gravitational wave emission is outside of the detection band of ground-based observatories. Nevertheless, one can perform a basic calculation to determine the orbital properties of the binary once it enters their detection band using the quadrupole approximation of gravitational radiation. The back reaction of quadrupole radiation on the orbital dynamics was first calculated by Peters and Mathews [11], who realized that

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gravitational radiation generally causes the orbital eccentricity to decrease. A back of the envelope application to the Hulse-Taylor binary reveals that the binary will have negligibly small eccentricity by the time it enters the detection band of groundbased observatories. Indeed, this is the general result of stellar evolution binaries, and all of the current detections are consistent with this prediction. In recent years, studies of dynamical stellar environments, such as globular clusters, have predicted that there is a non-negligible population of compact object binaries that will enter the detection band of ground-based detectors with measurable eccentricity (e > 0.05) [12]. One particular class of binaries of interest in such environments is gravitational wave capture binaries, wherein two compact objects on an unbound orbit can form a bound binary through the emission of gravitational waves during the closest approach. Such binaries can be formed with sufficient compactness that their gravitational wave emission is within the frequency band of ground-based observatories but with a wide range of possible orbital eccentricities, spanning 0.1 < e < 1 [13]. A small subset of these binaries will form with large, close to unity (e ∼ 1) orbital eccentricity. For binary black holes, the merger rate is estimated to be 1–2 per year per cubic gigaparsec [14], while for binary neutron stars or black hole-neutron star binaries, the rate is significantly lower at 0.04 per year per cubic gigaparsec [15]. While gravitational wave captures play an important role in determining the population of eccentric binaries of relevance to ground-based detectors, there are a host of dynamical interactions at play in globular clusters. Binary-single interactions, wherein a binary system is disrupted by a third compact object, are chaotic in nature but can settle into quasi-stable configurations comprising an eccentric binary, whose components need not be those of the original binary, orbiting around a third body [12]. There are two possible outcomes for this scenario. The first is the eccentric binary receives a sufficient kick to its center of mass to eject it from the cluster. In the second, the eccentric binary merges before the third object disrupts it and sends the total system back to a chaotic phase. Both of these scenarios form binaries that will merge within the age of the universe and are formed in the detection band of space-based observatories with non-negligible eccentricity. If the masses of the binary components are small enough, these binaries will merge within the detection band of ground-based observatories but with eccentricities too small to be detected, making it difficult to distinguish them from stellar evolution binaries. This highlights the central challenge in determining the origin of the systems observed by LIGO and Virgo. Another possible channel to form eccentric binaries is galactic nuclei, wherein many of the same dynamical interactions take place as in globular clusters. Binarysingle and single-single interactions within active galactic nuclei (AGN) lead to distributions tilted toward higher masses and higher eccentricities [16, 17]. In the case of gravitational wave capture events, ∼26–50% of binaries will form at peak gravitational wave frequencies >10 Hz with eccentricities >0.95, making them the most eccentric source population for ground-based detectors to date [18]. In addition to these dynamical interactions, the orbit of a binary system may be perturbed by the gravitational interaction of the supermassive black hole at the center of the

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galaxy. Such systems are called hierarchical triple systems [19], due to there being a separation of scales; specifically, the orbital period of the binary is much shorter than the period associated with motion around the third body. In such systems, the effect of the third object causes a trade-off between the mutual inclination angle of the orbits with the orbital eccentricity of the inner binary. Under exceptional circumstances, a resonant state can be achieved via the Kozai-Lidov [20] or postNewtonian effects [21], wherein the eccentricity can be enhanced by orders of magnitude and become close to unity, i.e., the unbound limit. From simulations of dynamical stellar environments, it is known that triple systems are a common possibility, but it is expected that resonant triples are not likely to occur in nature. This chapter reviews our current understanding of the gravitational waves from eccentric systems, with particular emphasis on the high-eccentricity limit. The fundamentals of the orbital dynamics and gravitational wave emission of eccentric binaries, the methods of detecting gravitational waves from eccentric systems, the approximations and methods used to model the waveforms from binary black holes, and the physics that can be probed with eccentric systems with emphasis on tests of general relativity and neutron star physics will be explored. Units where G = c = 1 are used throughout this chapter.

Eccentric Dynamics Before we proceed, it is useful to have a clear picture of the dynamics of eccentric binaries and the gravitational wave signals that they generate. A basic understanding of these can be obtained by studying eccentric binaries at so-called leading postNewtonian [22] order, where the conservative orbital dynamics of the binary are governed by Newton’s law of gravitation and the dissipative radiation reaction is controlled by quadrupole radiation. We begin with the orbital dynamics, where the equations of motion are a = −

M n r2

(1)

where M = m1 + m2 is the total mass of the binary, r is the relative separation of the two bodies, a is the relative acceleration, and n is the unit normal to the radial separation. The motion of the binary is restricted to a plane, which we choose to be the x − y plane, with the z coordinate orthogonal to the orbit. With this choice, r = r n, with n = [cos φ, sin φ, 0], and (r, φ) are functions of time. Then, a = d 2 r/dt 2 , and the equations of motion separate into  2 d 2r dφ M − r =− 2 , dt dt 2 r   dφ d r2 = 0. dt dt

(2) (3)

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There are two constants of motion associated with these equations. The first arises from the latter of these equations, which implies that r 2 dφ/dt = h = constant, which is recognized as the orbital angular momentum. The second, namely, the orbital energy, is found by realizing that the above equations are conserved under a constant shift of the time coordinate, specifically 1 2

=



dr dt

2 +

h2 M − 2. 2 2r r

(4)

The existence of these conserved quantities allows us to directly integrate the equations of motions. Making use of the orbital angular momentum, Eq. (2) can be written purely in terms of r and its time derivatives, specifically h2 M d 2r + =− 2 . dt 2 r3 r

(5)

Direct integration of this equation once with respect to time gives Eq. (4). The solution to this equation is r=

p 1 + e cos φ

(6)

where p is the semi-latus rectum of the orbit and e is the orbital eccentricity. This is supplemented by an evolution equation for the orbital phase φ derived from the definition of the orbital angular momentum, specifically dφ = dt



M p3

1/2 (1 + e cos φ)2 .

(7)

Making use of these solutions, the quantities (p, e) can be related to the orbital energy and angular momentum =−

M (1 − e2 ) , 2p

h = (Mp)1/2 .

(8) (9)

This completes the solution to the equations of motion. The geometry of the orbit is controlled by the value of the eccentricity e. For e = 0, the radial coordinate r becomes a constant, and the orbit is a circle. For 0 < e < 1, the orbit is an ellipse. In an effective one-body frame, a small mass μ = m1 m2 /M orbits around a large mass M centered at the focus of the ellipse. Closest approach of μ to M is referred to as pericenter, and the orbital velocity (called the pericenter velocity at this point) is at its greatest value. For e = 1, the orbit is a parabola, and the pericenter velocity is now the escape velocity. In this

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case, the binary still sweeps out a total phase Δφ = 2π , but the orbit extends out to spatial infinity, and as a result, the orbital period is infinite. Finally, for e > 1, the orbit is a hyperbola and no longer closes. Instead, the orbital sweeps through Δφ = 2 arccos(1/e). While the parameterization of the orbit in terms of φ provides a complete solution to the orbital dynamics, it unfortunately does not admit an explicit closedform solution in terms of time due to the nonlinear nature of Eq. (7). To provide more complete solutions in terms of time, alternative parameterizations are often employed. The starting point is to define the eccentric anomaly u, which is related to the orbital phase φ through tan

    u φ 1 + e 1/2 = . tan 2 1−e 2

(10)

Combining this with Eq. (7), one obtains the evolution equation for u, which can be directly integrated to give Kepler’s equation,  = u − e sin u ,

(11)

where  = (2π/Torb )(t − tp ) is the mean anomaly, with Torb the orbital period of the binary and tp the time of pericenter passage. While Kepler’s equation provides a useful relationship between the eccentric anomaly and time, it is unfortunately transcendental and has no known closed-form solutions. The main usefulness of Kepler’s equation is for the Fourier decomposition of the Newtonian two-body problem, which allows one to write relevant quantities in terms of summations on harmonics of the orbital period. For example, ∞

 2 cos φ = −e + (1 − e2 ) Jk (ke) cos(k) , e

(12)

k=1

sin φ = 2(1 − e2 )1/2

∞ 

Jk (ke) sin(k) ,

(13)

k=1

where Jk (x) is the Bessel function of the first kind and the prime corresponds to differentiation with respect to the argument. For small eccentricities, e  1, Jk (ke) ∼ ek , and the summations can be truncated at a finite number of terms while retaining accuracy compared to a numerical solution to Eq. (11). This completes the discussion of the conservative dynamics of the binary. At leading PN order, the gravitational waves from the binary are controlled by the quadrupole approximation, where the metric perturbation takes the form hij =

2 d 2 Iij , DL dt 2

(14)

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where DL is the luminosity distance to the source and Iij is the quadrupole moment associated with the binary’s orbital motion. The observable gravitational wave is given by the transverse traceless (TT) projection of Eq. (14). Defining N to be the line of sight from the observer to the source, specifically N = [sin ι cos β, sin ι sin β, cos ι]

(15)

where ι is the inclination angle of the source and β is the angle between the x-axis of the orbital frame with the principle axis defining the plus polarization, the TT projection is defined by ij

hTT = P ij mn hmn

(16)

with the projector P ij = δ ij − N i N j , 1 P ij km = P i k P j m − P ij Pkm 2

(17) (18)

The plus and cross polarizations (h+ , h× ) of the waveform can be defined by  The end result for eccentric binaries is constructing the vectors orthogonal to N.   1 2μM  5 1 + cos2 ι cos(2φ − 2β)+ e cos(φ − 2β) + e cos(3φ − 2β) pDL 4 4 

 1 1 , (19) + e2 cos(2β) + sin2 ι e cos φ + e2 2 2  2μM 5 h× = − cos ι 2 sin(2φ − 2β) + e sin(φ − 2β) pDL 2 1 + e sin(3φ − 2β) − e2 sin(2β) . (20) 2 h+ = −

The polarizations can be further specified as explicit functions of time using the Fourier series in Eqs. (12)–(13), specifically h+,× = −

∞ 2μM   C+,× (e, ι, β) cos(k) + S+,× (e, ι, β) sin(k) , pDL

(21)

k=1

where C+,× and S+,× are given in Eqs. (A15)–(A16) in [23]. The gravitational waves carry energy and angular momentum away from the binary, with the energy and angular momentum fluxes governed by [22]

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P=

1 32π

DL2

1 ij k Ji =  32π

ij dhTT ij dhTT

dt

DL2



dt

(22)

dΩ ,

dhTT k hTT jm dt

m

 TT ∂h 1 dhmn TT dΩ , − xj mn 2 dt ∂x k

(23)

respectively. To the leading order, radiation reaction is governed by the balance laws, which directly relate the loss of orbital energy and angular momentum to the fluxes averaged over the wavelength of the gravitational waves. For binary systems, the wavelength of the waves is related to the orbital period, so the average simply becomes the orbit average of the above equations, with the evolution equations for the orbital energy and angular momentum becoming [11] 32 dE = − η2 dt 5



dL 32 = − η2 M dt 5

M p 

 5  3/2  73 37 1 + e2 + e4 , 1 − e2 24 96   7/2  3/2 M 7 1 − e2 1 + e2 . p 8

(24) (25)

The effect of radiation reaction is now fully specified to leading PN order. Compact object binaries with eccentric orbits have a markedly different gravitational wave signature than their quasi-circular counterparts. Quasi-circular (e ∼ 0) binaries emit gravitational waves with frequency twice the orbital frequency and with a characteristic chirping pattern, becoming louder (higher amplitude) and with increasing frequency as the binary inspirals. This behavior is shown in the time domain in the top left panel of Fig. 1. Another common method of analyzing the waveforms from compact binaries is the use of spectrograms, a time-frequency representation of the signal. There are a number of ways of achieving these representations, such as a short-time Fourier transform or a wavelet transform. The top right panel of Fig. 1 displays the spectrogram of the corresponding quasi-circular waveform using a Q-transform, a wavelet transform that uses sine-Gaussians as the basis function. The spectrogram explicitly shows the chirping behavior of the binary, characterized by the increasing frequency of the signal as a function of time. For moderately eccentric binaries, the behavior of the waveform changes to a deformation of the quasi-circular sequence. As the eccentricity increases, the binary begins emitting at multiple harmonics of the orbital frequency. In the time domain, the emission of multiple harmonics creates a beating-like behavior in the waveform as can be seen from the middle left panel of Fig. 1. The explicit emission in multiple harmonics can be seen in the spectrogram in the middle right panel of Fig. 1. Each harmonic resembles the typical chirping behavior of quasi-circular waveforms. At sufficiently large eccentricities (e ∼ 1), the waveform changes drastically. The power from gravitational radiation scales as a high power of the orbital velocity (specifically v 10 ). For eccentric binaries, the orbital velocity changes throughout the orbit, achieving its maximum at pericenter passage. Thus, as the orbital eccentricity

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Fig. 1 (Left) Plot of the time-domain inspiral waveform for a binary with masses m1 = m2 = 30M and initial eccentricities e = 0 (top), e = 0.35 (middle), and e = 0.95 (bottom). The waveforms are normalized to their maximum amplitude. As the eccentricity increases, the waveform becomes increasing peaked at pericenter, until it resembles a repeated burst signal. As the binary inspirals, the bursts become louder due to the decreasing of the pericenter distance in each subsequent passage and occur more rapidly due to the decreasing orbital period. (Right) Q-transform of the inspiral waveforms in the left plots, using wavelets with quality factor Q = 20. The circular binary (top) only emits power in the second orbital harmonic, while the moderately eccentric binary (middle) emits in multiple harmonics. At high eccentricity (bottom), the power no longer splits into recognizable harmonics and instead becomes localized in time

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increases, the gravitational radiation becomes more concentrated near pericenter passage. For large orbital eccentricities, the gravitational wave emission no longer resembles the continuous chirping signal of quasi-circular and low-eccentricity binaries but instead becomes a set of discrete bursts emitted during each pericenter passage, as can be seen from the time-domain waveform in the bottom left panel of Fig. 1. The spectrogram from such a signal, displayed in the bottom right panel of Fig. 1, reveals that while the bursts are localized within a small time window, they are spread over a broad range of frequencies.

Detection Strategies Having a clear picture in mind of the morphology of eccentric signals, it is useful to discuss the plausible methods through which these signals may be detected before discussing the accuracy of models for binary black holes. This section provides a brief overview of the two common methods described to detect eccentric systems: matched filtering and power stacking.

Matched Filtering Matched filtering [24] is considered to be the gold standard for detecting any gravitational wave signal. In this method, one develops a phase-accurate model for the gravitational wave signal desired to be detected and then filters the detector data using this waveform model as a kernel. The basic quantity for performing such a search is the noise-weighted inner product, written in the frequency domain as  (A|B) = 4Re

fhigh

df flow

˜ )B˜ † (f ) A(f , Sn (f )

(26)

˜ ) and B(f ˜ ) are two frequency-domain signals, † corresponds to complex where A(f conjugation, Re corresponds to the real part, and Sn (f ) is the noise spectral density of the detector. The limits of integration are chosen to be the lower and upper frequencies that define the detection band of the relevant observatory. If the waveform model is a perfect representation of the signal being searched for, then the detector’s data stream s can be written as s = h + n, where h is the waveform model and n is the noise of the detector. Using the above inner product, one can define the match-filtered signal-to-noise ratio (SNR) as ρ = (h|h)1/2 . The search is performed by filtering the data s with waveform h and calculating the statistic ρ as a function of time. If the detector noise is stationary and Gaussian, one would claim a detection of a signal when ρ achieves a predetermined threshold. In practice, however, the noise of ground-based detectors is not stationary and Gaussian and typically contains a large number of glitches [25], spurious noise that creates regions of excess power in spectrograms whose origin may or may not

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be known. In some cases, glitches may resemble the types of signals being searched for, which is especially true in the case of bursts from highly eccentric systems. When this happens (which is very often for ground-based detectors), it is necessary to calculate the false alarm rate (FAR) of the signal, i.e., how often would a particular signal be created purely by detector noise. The match-filtered search would then not just require a threshold SNR, but also a FAR above the threshold determined by the background noise. How does one determine if a particular waveform model is sufficiently accurate to be used in a match-filtered search? This is probed using the match statistic, which is also sometimes referred to as the faithfulness or effectualness of the model [26]. The match is defined as M = max(hexact |hmodel (tc , φc )) tc ,φc

(27)

where hexact is a waveform considered to be an exact representation of what one might expect a signal to look like in nature and hmodel is the waveform model being considered, which is dependent on a time tc and phase φc shift. The exact procedure for maximizing over these parameters depends on how complicated the model is. In the simplest case, such as a PN quasi-circular waveform, the model can be written as hmodel = h0 (μa )e2iπf tc +iφc , where μa are the remaining parameters of the model. If the waveform can be written in this manner, then the maximization over tc may be achieved by computing the inverse Fourier transform of the integrand in the inner product, while φc can trivially be maximized over after this procedure. In general, the match is in the range 0 ≤ M ≤ 1. Using the analogy between the dot product of two vectors, if M = 1, then the waveforms are said to be perfectly in phase, while if M = 0, then the waveforms are maximally out of phase. The closer the match is to unity, the more accurate a representation the model is to the exact waveform. The match is also related to the loss in number of detections if one used a particular waveform model for a match-filtered search. The higher the match, the less signals one would miss in a search. From this notion, a threshold is typically chosen for the waveform model to achieve in order to be used in such a search, specifically M ≥ 0.97, which ensures that one misses at most 10% of signals [26]. The match statistic will be important in the discussion of waveform models for binary black holes in the next section “Modeling of Black Hole Binaries”.

Power Stacking One of the primary challenges of a match-filtered search is obtaining models with sufficient phase accuracy compared to the signals that we expect from nature. As we will discuss in the section “Numerical Relativity”, there are few accurate numerical relativity waveforms for highly eccentric binaries that cover more than one pericenter passage. This means that in the high-eccentricity limit, we are currently missing the requisite models to perform a match-filtered search. We must

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then turn our attention to sub-optimal strategies for achieving detections of highly eccentric binaries. A commonly discussed strategy is that of power stacking [27]. In this strategy, one combines the power in multiple signals to boost the SNR in order to achieve the threshold needed for detections. For highly eccentric binaries, these signals are the individual bursts of the inspiral sequence of a single binary. However, power stacking is a general technique that can also be used to combine multiple lowsignificance signals from different systems [28]. Assuming each signal is the same SNR, then the total SNR of the stacked signal scales as N 1/4 , where N is the number of signals. This is sub-optimal in the sense that for match filtering, the enhancement in the SNR scales as N 1/2 through the stacking of amplitude as opposed to power. Power stacking for highly eccentric binaries has been considered in the idealized scenario of stationary Gaussian detector noise [27]. To simulate the waveforms generated by highly eccentric binaries, the authors made use of the effective Kerr spacetime model explained in detail in section “Effective Kerr Spacetime”. The analysis carried out therein makes use of wavelet transforms, with basis functions ψk , to characterize the bursts of the simulated signal. From the  wavelet transform,  2 one can define two relevant statistics. The first is the sum C = N k=1 || ,  where h is the whitened signal h (t) =



˜ ) 2π if t h(f df √ , e Sn (f ) −∞ ∞

(28)

and < | > defines the time-domain inner product  < f |g >=

∞ −∞

dtf (t)g † (t)

(29)

with † corresponding to complex conjugation. Note that the statistic C is the sum of power in each wavelet. The N-burst signature is then the set of wavelets SN = {ψ1 , ..., ψN } that maximizes C . With the N-burst signature, one can then compute statistic E (τ ) =

N  i=1

max | < ϕi (t)|x(t + τ ) > |2

ϕi ∈Bi

(30)

where x is the detector output and ϕ are the wavelets in the set Bi ={ϕ|D(ϕ, ψi )≤ξi }, with D the Euclidean distance between wavelets. In order to achieve detection, one requires E to achieve a minimum value set by the false alarm rate Nfalse = η(T /δt), where T is the total observation time and δt is determined by the sampling rate. For a false event rate of 1yr−1 and step size of 10 ms, one requires − ln η ∼ 22, which sets the threshold for E . Simulated searches were performed using this method through the use of Monte Carlo simulations. The search is generally more efficient at detecting equal mass

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binaries with higher total masses and smaller pericenter distances, as can be seen from Figs. 17–19 in [27]. The strategy is also significantly more robust to modeling errors in the fluxes of gravitational radiation than matched filtering (see Fig. 20 in [27]). The drawback of this search is that signals generally need to be closer than for matched-filtering searches by as much as a factor of three, which is visualized in Figs. 10–16 in [27]. However, the search is more sensitive than single burst searches.

Modeling of Black Hole Binaries The detection strategies described in the previous section require one to have models of the types of signals being searched for. For matched filtering, this requires the construction of phase-accurate waveforms, while in the case of power stacking, timing models can aid in locating bursts in time-frequency space. This section details some of the relevant models for approximating the gravitational waves from eccentric binary black holes.

Numerical Models Generically, the equations governing the conservative and dissipative dynamics of eccentric binaries in general relativity are too complicated to be solved in closed form. Thus, there are two ways one may proceed to obtain the waveforms from these systems. The first is to simply numerically integrate the equation of motion for the orbital dynamics, which in turn determines the gravitational waveform of the binary. Many of the methods and approximations used to achieve this are detailed below. The second method is to apply analytic approximations to solve the necessary equations, a discussion of which can be found in section “Analytic Models”.

Numerical Relativity Currently, the most accurate waveforms compared to signals in nature are achieved through numerical relativity, which seeks to solve the Einstein field equations as an exact system numerically for binary mergers. As a basic concept, the simulations model black holes as punctures on a coordinate grid that defines a slice in time. Initial data constructed from constraint equations is specified on the initial slice and then evolved according to a set of evolution equations. There are challenges related to eccentric binaries in numerical relativity and some limited results for eccentric binary black holes that are worth discussing at this stage. Generic initial data leads to punctures moving on eccentric orbits, which can be seen qualitatively from oscillations in the coordinate radial velocity of the punctures. Historically, the presence of eccentricity has been viewed as a negative aspect of the simulations, with many procedures developed to quantify and reduce the eccentricity [29–31]. Primarily, this is a result of the expectation that most binaries of relevance to ground-based detectors will be quasi-circular. However, in recent

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years, the Simulating Extreme Spacetimes (SXS) [32] program has explored binary black holes with small to moderate eccentricity in detail. The highly eccentric limit, on the other hand, poses a challenge for numerical simulations. The main issue is a result of the multiple timescales associated with the problem. The gravitational wave emission is concentrated during pericenter passage and thus occurs on a small timescale. However, the time between bursts is effectively the orbital period of the binary, which can be orders of magnitude larger than the pericenter passage timescale in the high-eccentricity limit. Thus, one must invest sufficient computation resources to integrate over the long orbital timescale while also retaining sufficient resolution to accurately resolve the dynamics at pericenter, and it is difficult to retain convergence over multiple orbits. The few simulations of the high-eccentricity limit typically only resolve a single pericenter passage, except for dynamical captures with sufficiently small-impact parameter. These simulations have primarily focused on black hole-neutron star binaries and binary neutron stars, so the results will be presented in section “Binaries with Neutron Stars”. There are a few results from numerical relativity concerning black holes in highly eccentric orbits that are worth mentioning here. The first is the work of Healy, Levin, and Shoemaker [33], who investigated the generality of zoom-whirl orbits, wherein the binary “whirls” around the center of mass are in nearly circular motion during closest approach before “zooming” out along a highly eccentric orbit. Such behavior is well known in extreme mass ratio inspirals (EMRIs), but at the time, it was not known how general this behavior was especially for more comparable masses. It was expected that dissipation due to gravitational radiation would cause the binary to circularize before zoom-whirl behavior could be initiated. Healy, Levin, and Shoemaker showed that zoom-whirl behavior can be found in highly eccentric binaries of comparable masses along highly eccentric orbits with sufficiently small-impact parameters. In general, the zoom-whirl behavior was more prevalent in systems with larger mass ratios and larger spins. Second, Damour et al. [34] considered the scattering of black holes on hyperbolic-like orbits using numerical relativity, specifically computing the scattering angle χ as a function of the initial impact parameter. Comparisons therein to predictions of the scattering angle as computed in post-Newtonian theory and the effective one-body framework showed that the post-Newtonian predictions become increasingly inaccurate the smaller the initial impact parameter, while the effective one-body predictions have significantly improved accuracy with errors of at most ∼5%.

Effective Kerr Spacetime One of the most important models of highly eccentric inspirals is the effective Kerr spacetime [35]. The model takes some motivation from the effective one-body formalism [36], where inspiral dynamics of non-spinning binaries are modeled as geodesics of a deformed Schwarzschild black hole. Further, zoom-whirl dynamics in equal mass binaries is typically better approximated by geodesics of Kerr spacetimes rather than Schwarzschild spacetimes [37]. The central idea of the model is that, for small-impact parameters, the spacetime resembles a single throat of an effective

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black hole, with the orbital dynamics governed by geodesics in this spacetime. This approximation is similar to those used in the close-limit approximation [38, 39]. The Kerr spacetime is given by the metric, in Boyer-Lindquist coordinates,   2Mr Σ dt 2 + dr 2 + Σdθ 2 ds 2 = − 1 − Σ Δ   2Ma 2 r sin2 θ 4Mar sin2 θ 2 2 + r +a + sin2 θ dφ 2 − dtdφ , Σ Σ

(31)

Σ = r 2 + a 2 cos2 θ ,

(32)

Δ = r 2 − 2Mr + a 2 ,

(33)

where M and a describe the mass and spin of the black hole. In general, a < M for astrophysical sources, with a = M called the extremal limit. The model is described as an effective Kerr black hole since M is taken to be the total mass of the binary and ˜ the model sets a = aeff , where aeff = μL/M + aBH , where μ is the reduced mass of the binary, L˜ is the reduced angular momentum of the geodesics, and aBH is the total spin of the component black holes of the binary. As a result, for large orbital separations, aeff > M, and the spacetime resembles that of a super-extremal black hole (which is actually a naked singularity). One may worry that this creates possible pathologies within the model. However, this only occurs when L˜ is sufficiently large. Assuming a Newtonian mapping L˜ = (Mp)1/2 where p is the semi-latus 2 ). For rectum of the orbit, aeff = M when p = (M/μ2 )(M 2 − 2aBH M + aBH equal mass μ = (1/4)M and non-spinning aBH = 0 binaries, this corresponds to p = 16M, which corresponds to an orbit that is outside the region where pathologies exist. The inspiral phase of the binary coalescence is described by the geodesics of the effective Kerr spacetime, with the geodesics described by the reduced orbital ˜ Radiation reaction is included in a postenergy E˜ and angular momentum L. Minkowski [40]-style approach, where, under the emission of gravitational waves, these quantities are promoted to functions of time through the quadrupole formula μ ... ...j k d E˜ = − I jkI , dt 5 ˜ ... k 2μ dL = −  zij I¨ik I j dt 5

(34) (35)

where Ij k is the quadrupole moment of the geodesic and ij k is the Levi-Civita symbol. The time-domain waveform is then given by h=

 2  ¨ Ixx + i I¨xy , DL

(36)

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where DL is the distance to the source. The merger and ringdown of the gravitational wave signal are modeled through the implicit rotating source (IRS) model, where the multipolar moments of the gravitational radiation are associated with the mass multipole moments of a rigidly rotating source, specifically the remnant black hole in this case [41, 42]. The phase of the model is provided by a function that asymptotically approaches the dominant quasi-normal mode frequency of the remnant black holes. This phase function, given in Eqs. (6)–(7) in [35], has three free parameters, plus one for the amplitude, that must be fit to numerical relativity simulations. In [35], the model was validated against both post-Newtonian waveforms and numerical relativity flyby waveforms. For large orbital separations, the loss of energy and angular momentum of the geodesic model converges to that of the 2.5and 3.5-PN approximations, but these approximations diverge for close pericenter passages, specifically rp < 10M. On the other hand, the model was compared to numerical relativity flyby waveforms using the match statistic. The free parameters of the geodesic model, namely, the pericenter distance rp and eccentricity e, were fit by finding the highest match. The match was found to be more sensitive to the parameter rp than to the eccentricity of the orbit. The comparison was performed with binary black holes, black hole-neutron star binaries, and binary neutron stars. For the binary black hole case, the match was very high at 0.99, while for the cases with neutron stars, the match was very sensitive to the closest approach. This is due to the fact that the model does not include finite-size effects, which become increasingly important as the pericenter distance decreases. Thus far, the model is designed to describe binary black holes, since it does not include the matter effects necessary to describe binaries with neutron stars. The model has also only been validated with non-spinning binaries and only for single pericenter passages, due to the lack of fully inspiral waveforms from numerical relativity simulations. Finally, the model has not been validated in a simulated data analysis study.

Inspiral-Merger-Ringdown waveforms Generically, one cannot solve for the full inspiral-merger-ringdown structure of the waveform of binary systems without using numerical relativity. However, even numerical relativity has its limits in terms of the number of gravitational wave cycles it can accurately simulate. On the other hand, the early inspiral evolution where the orbital velocity of the binary is small compared to the speed of light can be accurately modeled by the post-Newtonian approximation. A number of approaches have been used to combine these two approaches to obtain complete waveform that accurately model the full coalescence of binary systems. Two of these, IMRPhenom and SEOBNR models, are standard models employed by the LIGO and Virgo collaborations, while a third, the ENIGMA model, has been developed specifically for eccentric systems.

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The IMRPhenom [43, 44] family of models combines post-Newtonian approximation for the inspiral phase of the coalescence with numerical relativity simulations for the merger-ringdown phase. In the post-Newtonian approximation, waveforms can be computed analytically in the Fourier domain by application of the stationary-phase approximation [45], and thus, the matching is typically done in the frequency domain. The two separate zones, early inspiral and merger-ringdown, are interpolated in between in an intermediate matching zone. The resulting hybrid waveform is then fit to a parameterized model with phenomenological coefficients that are related to the masses and spins of the binary. While much recent work has gone into extending the IMRPhenom models to include spin precession, there are currently no eccentric IMRPhenom models. However, the first step toward this, namely, developing a post-Newtonian eccentric inspiral waveform, has been achieved in [46] and is detailed in section “Adiabatic Waveforms”. The effective one-body (EOB) approach [36] takes some inspiration from techniques in quantum field theory. Specifically, the generic two-body problem in general relativity is mapped to an effective one-body system described by a deformed Schwarzschild or Kerr spacetime metric characterized by effective potentials, which are matched to known limits. In particular, much of the free parameters in the potentials are known from the post-Newtonian expansion, while any unknown coefficients are fit by matching to numerical relativity simulations. The potentials also properly reduce to the correct test mass limit of the self-force formalism [47]. The orbital dynamics and gravitational waves are then characterized by geodesics of the effective spacetime. Much of the work on SEOBNR waveforms have focused on quasi-circular and spinning binaries. However, more recently, two versions of these models have been developed for moderately eccentric binaries and shown to provide good agreement with numerical relativity SXS simulations [48–50]. Finally, Nagar et al. [51] have developed an EOB model for dynamical capture binary black holes, which reproduces the scattering angle of numerical relativity hyperbolic encounters to within a few percent. This last model constitutes the only currently available inspiral-merger-ringdown model for highly eccentric binaries. Finally, the ENIGMA model [52,53] makes use of time-domain post-Newtonian approximations of the inspiral dynamics of the binary. However, these approximations are only known up to third post-Newtonian order for generic eccentric systems. The ENIGMA model also incorporates higher post-Newtonian order corrections to the energy flux and binding energy from the self-force formalism up to sixth postNewtonian order. For the merger-ringdown part of the waveform, the model makes use of Gaussian process emulation, wherein a machine learning algorithm is trained to interpolate between numerical relativity waveforms in a dataset. To match the inspiral and merger-ringdown parts together, the model finds the attachment point by maximizing the overlap, given by Eq. (26), between quasi-circular ENIGMA waveforms with SEOBNRv4 waveforms. The model was validated against Einstein Toolkit numerical relativity waveforms up to eccentricity e = 0.2 ten cycles before merger in [53].

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Analytic Models Complementary to the numerical approaches detailed above, much work has gone into the development of analytic waveform models for eccentric binaries. These models usually only cover the inspiral phase of the coalescence but have the benefit of typically being faster to evaluate than numerical models. The radiation reaction equations for eccentric binaries are more difficult to solve analytically than their quasi-circular counterparts, and usually, one has to employ assumptions and approximations to solve them accurately. Most eccentric waveform models use some version of the post-circular expansion [54], an expansion about low eccentricity. An excellent, short review of these current models can be found in Sec. of [55]. The following sections provide an overview of analytic waveform models that are tailored to highly eccentric systems.

Adiabatic Waveforms Most inspiral waveforms are usually constructed in the adiabatic approximation, where the loss of orbital energy and angular momentum is balanced by the orbitaveraged energy and angular momentum fluxes of gravitational radiation. The current state-of-the-art of adiabatic inspiral waveforms for eccentric binaries are the NeF model of [55] and its post-Newtonian extension [46]. The main challenge of developing adiabatic eccentric models is to solve for the relationship between the orbital frequency F , the mean anomaly , and eccentricity e. At leading PN order, these quantities evolve under radiation reaction according to   121 2 e 1 + e 304 de 304 η =− (37) (2π MF )8/3 dt 15 M (1 − e2 )5/2   73 2 37 4 1 + e + e 24 96 96 η dF = (38) (2π MF )11/3 2 2 7/2 dt 10π M (1 − e ) d = 2π F (39) dt Generically, these equations can only be solved by change of variables from t to e to obtain F (e), t (e), and (e), which the latter two depend on hypergeometric functions. There are two issues with this. The holy grail of analytic waveforms is to obtain them in the Fourier domain analytically, which can be achieved by application of the stationary-phase approximation. In order to do this, one needs then t (F ), and the stationary point of the phase gives the relationship between the Fourier frequency f and the orbital frequency. One thus has to invert F (e) to obtain e(F ), which is challenging since this is a transcendental equation. The second issue is a result of t (e) and (e) depending on hypergeometric functions, which are typically slow to evaluate numerically. These problems can be overcome by use of post-circular (low-eccentricity) expansions. Comparisons of the low-eccentricity inversion of F (e) to numerical

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inversions reveal that the approximation is well behaved and controllable depending on the order of the expansion. The residuals can be further improved by use of resummations of the series using Chebyshev and Legendre polynomials. On the other hand, the hypergeometric functions in t (e) and (e) are relatively simple functions of the eccentricity and are well approximated by sufficiently high-order post-circular expansions, which are significantly faster to evaluate. With the analytic solutions to the radiation reaction equations in Eqs. (37)– (38) in hand, the time-domain waveform can be written analytically through a decomposition in terms of harmonics of the orbital period, specifically h+,× (t) ∼



(+,×)

Aj

[e(F ), F ]eij (F )

(40)

j

where h+,× are the waveform polarizations, Aj are amplitude functions written in terms of Bessel functions of the first kind, and F is a function of time through Eq. (38). This decomposition was originally considered in the calculation of adiabatic fluxes by Peters and Mathews [11] and was later extended to the waveforms by Moreno-Garrido et al. [23]. These waveforms can be analytically Fourier transformed through application of the stationary-phase condition, with the stationary point given by j F (t ) = f , where t is the time associated with the stationary point, f is the Fourier frequency, and j is the harmonic number. The NeF model is then, schematically, h˜ +,× (f ) ∼



Aj(+,×) [e(f ), f ]eiψj (f )

(41)

j

where Aj are known amplitude functions, and ψj (f ) = j c − 2πf tc −

15j σ (e0 )−5/2 e(f )30/19 I [e(f )] 304η (2π MF0 )5/3

(42)

with (c , tc ) the phase and time of coalescence, (e0 , F0 ) the initial eccentricity and orbital frequency, and σ (e) given in Eq. (15) in [55]. The function I (e) is given in closed form in terms of hypergeometric functions in Eq. (58) in [55] or in a post-circular expansion in Eq. (59) therein. Each harmonic of these waveforms has qualitatively similar behavior to quasi-circular waveforms, solidifying the fact that eccentric orbits can be thought of as epicycles of circular orbits. This can be seen through the spectrograms in Fig. (4) in [55]. The post-Newtonian extension of the NeF waveform was considered in [46]. Much of the machinery developed for the NeF waveforms also applies at higher PN order. However, there is one critical feature that is present at higher postNewtonian order, namely, the splitting of harmonics. Periastron precession enters the post-Newtonian two-body problem at relative 1PN order, which causes eccentric orbits to be characterized by two frequencies: an azimuthal (orbital) frequency and a

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radial frequency. Physically, this causes a Zeeman splitting-like effect of each orbital harmonic within the waveform, as can be seen in Fig. (4) in [46]. This model has been shown to achieve matches greater than 0.97 against numerical post-Newtonian waveforms up to eccentricities of 0.8 for widely separated binaries. The maximum eccentricity at which it is valid decreases with decreasing separation due to issues related to the convergence of the PN sequence for eccentric systems.

Effective Flyby Approach A recent development in pursuit of analytic waveforms for highly eccentric binaries is the effective flyby (EFB) formalism [56]. The formalism seeks an accurate description of the gravitational wave bursts from individual pericenter passages, which may be strung together to create inspiral waveforms using the timing models described in section “Timing Models”. At leading post-Newtonian order, the relationship between the orbital phase and coordinate time is given by the transcendental Kepler’s equation, which admits no known closed-form analytic solution. One circumvents this issue by using Kepler’s equation to write functions that depend on the orbital phase as functions of time using Fourier series, as is done in Eqs. (12)–(13). At low eccentricities (e  1), these series can be truncated at a small, finite number of terms to obtain an accurate representation of the orbit. However, at high eccentricities (e ∼ 1), this cannot be achieved, and one would have to keep hundreds to thousands of terms to obtain the same accuracy. At it’s core, the formalism relies on a procedure to re-sum series of the structure in Eq. (12). Series that depend on Bessel functions of the form Jk (ke) are known as Kapteyn series. In a select few cases, these series can be re-summed exactly, as was done in the quadrupole radiation calculations of Peters and Mathews [11]. In general, this is not the case, and one has to rely on approximate methods to resum these series. The general procedure that is utilized in the EFB formalism was originally developed in [57] for use in the calculation of nonlinear effects in the gravitational wave fluxes. The procedure follows a few basic steps: 1. Replace the Bessel function Jk (ke) with it’s uniform asymptotic expansion about k → ∞. To the leading order, Jk (ke) ∼ K1/3 [(2/3)ζ 3/2 k] + O(1/k), where Kn (x) is the modified Bessel function of the second kind, and ζ is a known function of the orbital eccentricity e (see Eq. (26) in [56]). For a comparison of the asymptotic expansion to the exact function, see Fig. 2 in [57]. 2. Replace the sum over k with an integral. There is a practical issue of whether or not the integral can be evaluated analytically. In the case of Eq. (12), this can be done once the integral is extended down to k = 0. 3. Expand about high eccentricity, specifically 1 − e2  1. The previous two steps are most accurate in the high-eccentricity limit. The expansion also serves to speed up numerical computations, since the result of the previous step generally produces hypergeometric functions that are slow to evaluate.

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In the case of Eq. (12), this procedure produces cos φ ∼ −1 +

2 cosh[(1/3) sinh−1 ψ]  + O(1 − e2 ) , 1 + ψ2

(43)

where ψ = (3/2)/ζ 3/2 . The time-domain quadrupole waveform can then be written purely in terms of ψ and is given in Eq. (45) in [56]. The above re-summation procedure solves the problem of the conservative dynamics of the binary at leading post-Newtonian order. Using the balance laws in the quadrupole approximation, one can obtain evolution equations for the semi-latus rectum p and orbital eccentricity e in the adiabatic limit (see Eqs. (36)–(37) in [56]). To obtain an approximate solution to these coupled equations near pericenter, the EFB formalism relies on a simple Taylor series expansion of the form  μ = a

μa0

+

dμa d

 (t) + O[(t)2 ]

(44)

0

where μa is either p or e, and the underscore designates the value of the relevant quantity at  = 0. Once radiation reaction is considered, the mapping between  and coordinate time t is no longer linear, and one has to solve the differential equation d/dt = n[p(), e()]. Making use of the Taylor series in Eq. (44), this equation can be solved to obtain (t) given in Eq. (44) in [56]. A comparison of these analytic functions to the numerical integration of the quadrupole radiation reaction equations can be found in Fig. 2 of [56]. The time-domain radiation reaction model and the gravitational wave polarizations combined with the re-summed conservative dynamics constitute the EFB-T model. Validation of the EFB-T model was also considered in [56]. The match statistic was used to quantify the faithfulness of the model against numerical leading PN order quadrupole waveforms, which was found to be above 0.97 for much of the parameter space of relevance to ground-based detectors. While this is useful in quantifying the errors associated with the approximations used to develop the model, an important question is whether or not the model accurately captures a burst from an eccentric binary that we might expect in nature. To do this, the EFB-T model was compared to several numerical relativity flyby waveforms computed in [58]. For pericenter distances of rp > 10M, the match is greater than 0.94 but peaks at different orbital parameters than those used in the numerical relativity simulations, indicating systematic bias as a result of the limited leading post-Newtonian order approximation used to develop the model. For smaller pericenter distance, the match decreases rapidly, owing to the fact that nonlinear effects break the symmetrical structure of the burst in the numerical relativity simulations (see Figs. 6–8 in [56]). There is still much open work that remains to be done in the EFB formalism. So far, studies have focused on the leading post-Newtonian order description of both the conservative and dissipative dynamics of the binary. Beyond this order, the

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conservative dynamics becomes significantly more complicated, and it’s not clear if the same re-summation techniques can be applied. A more tantalizing question is whether this approach can be applied to effective description of the two-body problem, such as the effective Kerr spacetime in section “Effective Kerr Spacetime” or the effective one-body formalism. Further, the modeling currently only suffices for test particles, which may be a good approximation for black holes, but will fail to capture the rich physics that is present if at least one of the binary components is a neutron star.

Timing Models One of the possible detection strategies for gravitational wave bursts from highly eccentric binaries is power stacking, discussed in section “Power Stacking”. However, the issue with this method is distinguishing the bursts from glitches, anomalous noise sources in ground-based detectors, many of which resemble the time and frequency morphology of the bursts. Thus, this detection method requires a means to distinguish the bursts from glitches. The key is to realize that eccentric binaries will emit a burst with each pericenter passage as it inspirals and that the sequence of bursts will follow a predictable track in time-frequency space, whereas glitches are generally random events. The goal is then to use our knowledge of the two-body problem in general relativity to construct a timing model (called a burst model in earlier literature), for the time-frequency track of the inspiral sequence [59, 60]. The starting point to construct a timing model is a visualization of the burst in time-frequency space; see, for example, Fig. 2. The bursts are effectively twodimensional shapes in this space, so a first approximation to this is to characterize the bursts by rectangles in time and frequency. One could choose a more apt shape, such as an ellipse, but this can be achieved by a simple reinterpretation of the widths of rectangles. Each rectangle will then be characterized by a centroid (ti , fi ) and widths (δti , δfi ) (or, in the case of ellipses, the major and minor axes). To construct a timing model of the bursts, three ingredients are required: 1. Orbital evolution: A model that determines the orbital parameters of one orbit from those of the previous orbit under the effect of radiation reaction. 2. Centroid model: A model that determines the centroid of one burst (ti , fi ) from the centroid of the previous burst (ti−1 , fi−1 ). 3. Volume model: A model that determines the extent (δti , δfi ) of each burst in time and frequency. First, consider the orbital evolution. In the time domain, the orbital evolution is described by a steplike behavior near pericenter, due to the rapid emission of gravitational waves during pericenter passage. This can be approximated by treating the orbital parameters as constant throughout the orbit but allowing them to change instantaneously at pericenter. The orbit in between consecutive pericenter passages is described by two parameters, namely, the orbital energy E and orbital angular momentum L. The change in orbital energy and angular momentum due to radiation reaction are dependent on these parameters, as well as the orbital phase; namely,

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Fig. 2 Schematic of the timing model for bursts emitted during the inspiral of a highly eccentric binary. In the simplest model, each burst is described by a box characterized by a centroid with frequency fi and time ti and widths (δti , δfi ). The time between bursts is simply the orbital period of the binary, while the central frequency and widths are related to the properties of the power spectrum of each burst

˙ ˙ E˙ = E(E, L, φ), and L˙ = L(E, L, φ). From these, one must compute the change in orbital energy and angular momentum over the orbit. This can be achieved via the adiabatic approximation, where the average of these rates of change allows one to directly compute the change in these quantities if one knows the orbital period. Thus, the mapping between the orbital periods of one orbit and the previous orbit is ˙ Ei = Ei−1 + E(E i−1 , Li−1 )Pi−1 ,

(45)

˙ Li = Li−1 + L(E i−1 , Li−1 )Pi−1 ,

(46)

˙ and L ˙ are the orbit-averaged energy and angular momentum fluxes where E due to gravitational waves and P is the orbital period. These mappings are true regardless of the approximations that one applies to the orbital dynamics. However, these quantities have so far only been computed within the post-Newtonian formalism, using the quasi-Keplerian description of the orbit. With this description, one can alternatively work in terms of the pericenter velocity vp and time eccentricity

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et , which are known functions of the orbital energy and angular momentum. The mappings vp,i (vp,i−1 , et,i−1 ) and et,i (vp,i−1 , et,i−1 ) are given in a post-Newtonian expansion about vp,i−1  1 in Eqs. (53)–(59), (71)–(74), (88)–(95), and (107)– (112) in [60]. Now, consider the centroid mapping. Similar arguments regarding the steplike behavior of the orbital evolution can be applied to write the time mapping purely in terms of the orbital period, specifically ti = ti−1 + Pi .

(47)

Just like the mappings in Eqs. (45)–(46), this applies to any generic orbital description. Within the quasi-Keplerian formalism, the orbital period is a known function of the orbital energy and angular momentum or, alternatively, the pericenter velocity and time eccentricity. The mapping Pi (vp,i−1 , et,i−1 ) is given in Eqs. (62), (77), (97), and (118) in [60]. The frequency mapping, on the other hand, requires knowledge of the power spectrum of the gravitational wave burst. However, from simple scaling arguments, one might expect the peak frequency of the power spectrum to scale inversely with the pericenter timescale τ , namely 1 , 2π τi

(48)

pericenter distance . pericenter velocity

(49)

fi = where τ=

At leading post-Newtonian order, calculations of the power spectrum of bursts from parabolic orbits have provided evidence of this (see Fig. (7) of [61]), but this argument likely holds for generic orbital descriptions. The mapping fi (vp,i−1 , et,i−1 ) is given in Eqs. (61), (80), (98), and (124) in [60] for the quasi-Keplerian formalism. Finally, consider the volume mapping, which is characterized by the widths δti and δfi . The models developed in [59, 60] choose the widths to be proportional to the pericenter timescale and peak frequency of the burst, specifically δti = ξi τi ,

(50)

δfi = ξf fi .

(51)

The proportionality factors (ξt , ξf ) must be chosen from data analysis considerations. One possible way of fixing them is to choose their values such that a certain percentage of the power in each burst is contained in the box characterized by (δti , δfi ). Alternatively, if one wanted to associate the bursts with sine-Gaussian wavelets, the proportionality factors are related to the Q-factor of the wavelet. The entire sequence of N-bursts is characterized by the set {ti , fi , δti , δfi |i = 1, ..., N}. Naively, one might expect that 4N parameters are needed to describe the

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entire track of the bursts in time-frequency space. However, the conservative and dissipative dynamics of the binary are only described by four parameters (neglecting spins), specifically (m1 , m2 , E0 , L0 ) where m1,2 are the component masses and (E0 , L0 ) are the orbital energy and angular momentum in the first burst in band of the gravitational wave detector. From Eqs. (45), (46), (47), (48), (49), (50), and (51), the entire track of the bursts in time-frequency space is determined by only these four parameters. As a result, these four parameters can be determined by a finite number of bursts in principle and neglecting statistical uncertainties. Since the widths are determined by τ , each burst only actually provides at most two independent parameters. Thus, at the very least, one needs only two bursts to invert the parameters (ti , fi , δti , δfi ) to (m1 , m2 , E0 , L0 ). While the timing model reviewed here applies in any generic orbital description, it has only been computed to third post-Newtonian order in the quasi-Keplerian formalism. Further, the model has only been validated against numerical integration of the post-Newtonian equations of motion. It has not yet been validated in a mock data analysis study. It is also worth noting that these timing models don’t just apply for power stacking searches but can also be used in matched-filtering searches using the EFB waveforms. In [56], it was shown that the EFB waveforms combined with a leading post-Newtonian order timing model achieve a high match against a numerical leading post-Newtonian order quadrupole waveform. However, if a small error is introduced into the timing model, the match drops off rapidly, and one could potentially lose detection of the bursts. One would thus likely need to work beyond the post-Newtonian formalism to obtain the most accurate timing model compared to binaries that we would observe in nature.

Tests of General Relativity One of the most interesting applications of eccentric binaries is as a means of testing general relativity. As the eccentricity increases, the velocity at closest approach also increases up to approximately 41% from the quasi-circular case. Further, the innermost stable orbit, which one may consider the end of the inspiral phase of the coalescence, is generically smaller for eccentric systems. In the Schwarzschild spacetime, the innermost stable circular orbit corresponds to r = 6M, where M is the mass of the black hole. For eccentric orbits, the shortest the pericenter distance can be is 4M for bound orbits, corresponding to a pericenter velocity of vp ∼ 0.7c, where c is the speed of light in vacuum. These considerations are important when considering alternative theories of gravity. Orbital dynamics in many of these theories can be considered as deformations of the dynamics within general relativity, specifically the corrections to orbital quantities scale with the characteristic velocity of the system. As an example, consider the correction to the orbital energy in Einstein-dilaton-Gauss-Bonnet gravity, a theory of gravity that modifies general relativity through the introduction of quadratic curvature invariants in the action. The orbital energy and angular momentum in this theory are [62]

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ζ 1+ 6

E = EGR  L = LGR



1−e 1+e



ζ 3 + e2 1− 12 (1 + e)2



M rp M rp

2  (52) 2  (53)

where ζ is the coupling constant of the theory and M/rp ∼ vp2 by the viral theorem. These expressions also illustrate an important point, namely, that corrections to orbital dynamics in theories of modified gravity are eccentricity dependent, and as a result, constraints on the coupling constants of the theory from parameter estimation will also depend on the eccentricity of the binary. Projected constraints on example modified theories have been considered in both the low- and high-eccentricity limits. In the low-eccentricity limit, waveforms have been computed in Jordan-Brans-Dicke-Fierz theory using the stationary-phase and post-circular approximations [63]. This theory reduces to general relativity in the limit ω → ∞, with ω the coupling constant. With analytic waveforms, the projected constraints on ω were obtained using a Fisher analysis, an analytic parameter estimation technique valid in the high SNR limit and when the detector noise is stationary and Gaussian. Ma and Yunes [63] found that as the orbital eccentricity increases from zero (the quasi-circular limit), the constraint on ω initially deteriorates but recovers for eccentricities above 0.03 (see Fig. 1 therein). Similar nontrivial behavior of the constraint with increasing eccentricity was found by Moore and Yunes in [64], who studied constraints on Brans-Dicke theory and Einstein-dilaton-Gauss-Bonnet gravity. Using Markov chain Monte Carlo methods, Moore and Yunes found that the constraints are strongest for moderately eccentric systems with e ∼ 0.4, with the projected constraints being an order of magnitude better than current constraints with quasi-circular binaries. In the high-eccentricity limit, constraints on modified theories were considered with the timing models considered in the previous section in [59]. At the time, analytic waveforms for the gravitational wave bursts from highly eccentric systems were not available, but one could compute the timing model for generic deformations from general relativity to develop a parameterized post-Einsteinian (ppE) model. To obtain projected constraints, the authors considered variations of time and frequency centroid mappings with respect to the ppE parameters and required them to be at least as large as variations of the sequence with respect to the parameters within general relativity (the masses, orbital eccentricity, etc.), i.e., ∂(Δt, Δf ) a ∂(ΔtGR , ΔfGR ) a δλppE > δλGR . a ∂λppE ∂λaGR

(54)

Constraints on two example theories were considered in [59], namely, Einsteindilaton-Gauss-Bonnet and Brans-Dicke gravity. In both of these theories, the leading-order corrections to the timing model come from the existence of scalar dipole radiation, which are −1PN order effects on the flux of energy and angular

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momentum from gravitational waves. Making use of Eq. (54), the projected constraints on the coupling constants for these theories were found to be comparable, but not better than astrophysical constraints at the time. These upper limits are strictly estimates, since there are currently no waveforms for highly eccentric binaries in modified theories for use in parameter estimation studies.

Binaries with Neutron Stars So far, the discussion in this chapter has focused on black holes, which can be modeled as effective point particles in post-Newtonian theory and beyond. However, several studies have considered the possibility of at least one of the compact objects comprising an eccentric binary being a neutron star. In order to effectively describe such a system, one must consider the effects of matter on these systems. This chapter will review some of the most recent findings regarding neutron stars in eccentric binaries.

Tidal Interactions and Resonances In [65], neutron stars with polytropic equations of state were considered in the context of highly eccentric black hole-neutron star binaries using numerical relativity simulations. Effectively, the compact objects were setup on unbound, nearly parabolic orbits with varying pericenter distance rp (or, alternatively, impact parameters). Due to the limitations of computational techniques at the time, the simulations were limited to rp < 15M. Three possible scenarios emerged depending on the value of the initial pericenter separation: a direct plunge, an initial burst leading into a highly eccentric orbit followed by a plunge at the next pericenter passage, or a single pericenter passage leading to a highly eccentric orbit that does not merge. The latter of these could not be tracked beyond the initial flyby due to computational limitations. A discussion of the end state of the plunge will be presented in section “Post-merger and Remnants”. In the latter two cases, extraction of the gravitational waves in the simulation found fundamental oscillations in the waveform after the burst associated with the initial flyby, which are not present in the binary black hole scenario (see the top panel of Fig. (3) in [65]). The oscillations are a result of the fundamental modes (f-modes) of the neutron star and indicate the presence of rich dynamics in such binaries. To get a better understanding, it is useful to consider analytic calculations of perturbations of neutron stars. In the context of binary systems, to leading order, the neutron star is acted upon by a tidal potential, generated by the second compact object, of the form Utidal =

1 Eij x i x j 2

(55)

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where Eij is the electric part of the tidal quadrupole moment [66]. The tidal potential acts to create a displacement of the fluid elements x i → x i + ξ i , which deforms the quadrupole moment of the star. In a spherical harmonic decomposition, the timedependent amplitude of the quadrupole moment obeys ¨m + γQ ˙ m + ω2 Qm = W2m K2m m2 e−imφ Q r3

(56)

where Wlm is given in Eq. (24) in [67], Klm is an equation of state-dependent parameter given in Eq. (38) of [67], m2 is the mass of the other body in the binary, r is the orbital separation, and φ is the orbital phase. The above equation describes the l = 2 f-mode of the neutron star, with damping coefficient γ and frequency ω. In the quasi-circular limit, the orbital separation r is a slowly evolving quantity under radiation reaction, and thus, the tidal perturbation of the neutron star effectively becomes fixed, with its adiabatic evolution becoming subdominant. This is the so-called static tide. However, this approximation does not hold for the entire coalescence, the last few orbit of which are known to be highly dynamical, leading the tide to become highly time dependent and to the excitation of f-modes. For eccentric systems, especially those with e ∼ 1, the situation is entirely reversed. The rapid nature and small orbital separation at pericenter passage lead to the tidal potential being its strongest during closest approach, causing the excitation of f-modes at each passage. Energy and angular momentum are deposited in the modes from the orbital energy and angular momentum, which in turn modifies the inspiral rate beyond that of the typical radiation reaction effects of point particles. Further, the f-mode leads to a time-dependent quadrupole moment for the neutron star, generating additional gravitational waves from the system through Eq. (14). Since the studies of [58, 65, 68], there has been much work on trying to understand these tidal effects in more detail, primarily through the use of analytic and numerical calculation in the post-Newtonian approximation. Yang et al. [69] performed detailed calculations of the energy and angular momentum deposited into the f-modes in eccentric binaries and compared the analytic post-Newtonian predictions to those of numerical relativity simulations. Both methods found that typically 10–20% of the energy and angular momentum in gravitational waves from a single flyby can be deposited into the modes, although there is a disparity between the two techniques, which is a result of the limited accuracy of the post-Newtonian approximation as well as numerical error in the simulations. These results were extended in [70] to create a model of the evolution of the orbit under tidal effects. One of the main challenges in developing such a model for arbitrary eccentricity is the lack of closed-form expressions for (r, φ) as functions of time (see section “Eccentric Dynamics”). However, this was solved by the introduction of Hansen coefficients, which were only able to be evaluated in the loweccentricity limit. Despite this, Yang found that the phase difference in the orbital

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evolution between including and neglecting tidal effects is only ∼0.06 radians for a (1.3, 1.3)M binary neutron star with e = 0.2 at orbital frequency of 50 Hz, which is only detectable for loud events with third-generation detectors. Yang [70] also explored the possibility of tidal resonances in binary neutron stars, where the f-mode frequency is an integer multiple of the orbital frequency. In such a scenario, the resonance causes a larger amount of orbital energy and angular momentum to be deposited into the mode, enhancing the phase shift in the orbital evolution. Generally, the f-mode frequency is significantly larger than the orbital frequency in such a binary, while the effects of resonances with large harmonic number are subdominant. However, in the late stages of the inspiral, it is possible for resonances to be achieved with the k = 2 and k = 3 harmonics, which Yang argued were the dominant contribution to the phase shift of the binary and the only ones of relevance to low-eccentricity binaries. For the same (1.3, 1.3)M binary neutron star, the phase shift due to a k = 3 resonance was found to be ∼0.5 radians. Despite the rough order of magnitude increase in the phase shift from the nonresonant case, such an effect is still of most relevance to third-generation detectors. These estimates, in both the resonant and nonresonant cases, do not take into account possible de-phasing at lower frequency, which would correspond to larger eccentricities. Vick and Lai [71, 72] have investigated these effects in the context of highly eccentric binaries using numerical integrations of the post-Newtonian equations of motion. Generically, in the nonresonant case, they found that the de-phasing in the gravitational waves has a nontrivial dependence on the eccentricity, which can be seen in Fig. 7 in [71]. Further, the cumulative phase shift in the gravitational waves was found to be larger than the results of Yang for smaller initial pericenter distances, regardless of the value of the eccentricity. In the resonant case, the phase shift due to the resonance was found to always be less than 0.1 radians, indicating that tidal resonances are not the dominant contribution to the de-phasing. Vick and Lai [72] also found a third plausible scenario of relevance to mode excitation in highly eccentric binaries. Recall that for parabolic orbits, the binding energy is zero, i.e.,  = 0, while for bound eccentric orbits,  < 0. Thus, for highly eccentric binaries with e ∼ 1,  ∼ 0. In such a scenario, it was found that the energy deposited into the mode is comparable to or larger than the binding energy, causing the mode to grow chaotically (see the bottom panel of Fig. 3 in [72] and Fig. 3 in [71]). From Fig. 2 in [72], the amount of energy extracted from chaotic modes over multiple orbits is typically several orders of magnitude larger than both the nonresonant and resonant cases. Under the effects of radiation reaction, Vick and Lai found that the chaotic modes will not grow continuously, but will eventually settle into a quasi-steady state as the binary loses eccentricity. The authors of these studies argue that these effects can be used to constrain the equation of state of neutron stars. To date, there are no studies that provide predictions for constraints on equations of state for highly eccentric neutron star binaries using parameter estimation with current and future ground-based detectors.

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Post-merger and Remnants Numerical relativity simulations of the merger phase and remnants of the coalescence for highly eccentric binaries with neutron stars have been considered in [58, 65, 68, 73–76]. The earliest study focused on black hole-neutron star binaries with a mass ratio of 4:1. The ultimate fate of the neutron star was found to be dependent on the initial pericenter separation of the dynamical capture and on the equation of state, which can be seen from Fig. 1 in [65]. The first possible scenario, for rp ∼ 5M, is a direct plunge of a highly deformed neutron star with little tidal disruption, resulting in less that 1% of the rest mass available to create an accretion disk post-merger. For slightly larger periapsis, rp ∼ 6.81M, the neutron star is disrupted into a long tidal tail, leading to an accretion disk with ∼12% of the rest mass at late times. At larger pericenter distances, mass transfer between the neutron star and black hole can occur during initial passage, with a subsequent highly distorted neutron star, ultimately leading to the creation of a nascent disk at late times. A follow-up to this study was performed in [58], including the effects of black hole spin. Generically, the innermost stable orbit (ISO) is dependent on whether the black hole spin is aligned or anti-aligned with the orbital angular momentum, leading to prograde and retrograde orbits, respectively. In the prograde (aligned) case, the ISO is closer to the black hole, and the neutron star experiences large tidal forces before merger, leading to the formation of accretion disks with larger rest mass compared to the non-spinning and retrograde cases. In the retrograde case, the ISO is located farther from the black hole, and tidal disruption is not as efficient, with accretion disks containing less than 1% of the neutron star’s rest mass. The equation of state also has a significant impact on the post-merger result, with more compact stars experiencing less tidal disruption and resulting in accretion disks with smaller rest mass. In [73], East and Pretorius considered the case of non-spinning binary neutron star mergers resulting from dynamical captures. The neutron stars were modeled using piecewise polytropic equations of state, and thus, magnetohydrodynamic and neutrino cooling effects were neglected. As with the black hole-neutron star case, the final remnant depends sensitively on the initial pericenter distance. For exceptionally small periapsis (rp ≤ 5M), the neutron stars directly collapse into a single black hole after collision, due to the lack of sufficient angular momentum to support a hypermassive remnant. The black hole mass is typically >0.98M with spin χ > 0.5. For larger periapsis, the collision of the neutron stars forms a deformed hypermassive neutron star supported by angular momentum. The hypermassive neutron star radiates gravitational waves due to its deformed quadrupole moment, losing angular momentum in the process and ultimately collapsing into a black hole. Full general relativistic hydrodynamic simulations of hypermassive neutron stars resulting from mergers in the dynamical capture scenario have been considered in [75–77]. Paschalidis et al. [77] found that the hypermassive neutron stars form a one-arm spiral instability, an m = 1 spherical harmonic mode, which was the first

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time the instability had been observed in binary neutron star mergers. In East et al. [75], it was found that the instability was most pronounced for binary neutron stars with small but non-negligible spin. Further, in [76], it was found that the equation of state strongly influences the relative amplitudes of m = 2 and m = 1 azimuthal modes, with stiffer equations of state having stronger m = 2 modes and weaker m = 1 modes. The instability is important since it contributes to the post-merger gravitational waves emitted by the hypermassive neutron star. Generically, the postmerger signal is dominated by the l = m = 2 modes of the neutron star, but these occur at high frequencies of a few kilohertz. The l = 2 and m = 1 mode generated by the instability has roughly half the frequency of the dominant mode and thus may be easier to detect by ground-based detectors. When considering the mergers of neutron stars, it is also important to characterize the amount of material that is ejected from the system, which can generate electromagnetic counterparts in the form of kilonovae. East and Pretorius [73] also considered the amount of ejecta created in the merger of their simulations and found that the mass is at most a few percent of a solar mass at mildly relativistic velocities, specifically 10–30% the speed of light. These results were later extended to the case of spinning neutron stars in [74], where it was found that the ejecta mass increases with increasing spin, up to a third of a solar mass for rapidly rotating neutron stars (see the bottom left panel of Fig. 4 therein).

Cross-References  Binary Neutron Stars  Black Hole-Neutron Star Mergers  Dynamical Formation of Merging Stellar-Mass Binary Black Holes  Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories  Formation Channels of Single and Binary Stellar-Mass Black Holes  Introduction to Gravitational Wave Astronomy  Numerical Relativity for Gravitational Wave Source Modeling  Post-Newtonian Templates for Gravitational Waves from Compact Binary

Inspirals  Principles of Gravitational-Wave Data Analysis  Testing General Relativity with Gravitational Waves

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Tianyu Zhao, Zhoujian Cao, Chun-Yu Lin, and Hwei-Jang Yo

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Numerical Relativity in Gravitational Wave Astronomy . . . . . . . . . . . . . . . . . . A Brief History and the Current Status of Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . The Core Difficulties of Numerical Relativity and Current Solutions . . . . . . . . . . . . . . . . . . The Partial Differential Equation Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Singularity Inside the Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Multiple Physical Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Coordinate Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Parallel Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem of Gravitational Waveform Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of Numerical Relativity to Gravitational Wave Source Modeling . . . . . . . . . . . Binary Black Hole (BBH) System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron Star-Black Hole Binary and Binary Neutron Star . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T. Zhao · Z. Cao () Department of Astronomy, Beijing Normal University, Beijing, China e-mail: [email protected]; [email protected] C-Y. Lin National Center for High-Performance Computing, Hsinchu, Taiwan e-mail: [email protected] H-J. Yo Department of Physics, National Cheng-Kung University, Tainan, Taiwan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_34

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Abstract

The first direct detection of gravitational wave has been realized by LIGO in 2015. It opens a brand new window to observe our universe and initiates the gravitational wave astronomy. The matched filtering technique is an optimal method to dig out the gravitational wave signal that is buried in the strong noise. The matched filtering technique is also essential to estimate the parameters of the gravitational wave source. Matched filtering requires a bank of accurate gravitational waveforms. Nowadays, two LIGO detectors and one Virgo are working. Other laser interferometric gravitational wave detectors such as KAGRA and the third LIGO in India, IndIGO, are developing. The space-borne detection projects including LISA, Taiji, and TianQin are also in progress. The pulsar timing approach with FAST, SKA, and other radio telescopes to detect gravitational wave is also in the rapid development. It is foreseeable the gravitational wave astronomy in the wide frequency band from 10−9 to 1000 Hz will be realized in the near future. Since the matched filtering plays an important role, the modeling of gravitational wave sources is urgent and important. For realistic objects without any symmetry, the analytical treatment of Einstein equation becomes nearly intractable. The numerical relativity is the most robust and reliable method and tool for solving the Einstein equation. We introduce the research of numerical relativity in the viewpoint of gravitational wave astronomy. Keywords

Numerical relativity · Gravitational wave astronomy · Einstein equation · Matched filtering

Introduction Gravitational wave is one of the most important theoretical predictions of general relativity. Gravitational wave detection is an unprecedented test of general relativity in the strong and dynamical space-time region. More importantly, it will open a new observation window to the universe–gravitational wave astronomy. But gravitational wave detection is extremely difficult. The gravitational wave detection relies on hardware sensitivity and the source modeling. Given a hardware sensitivity, a good source model can help to extract a weak signal buried in the strong Gaussian noise through matched filtering technique. The first detected gravitational wave event GW150914 results from a successful combination of hardware progress and theoretical research breakthroughs [1]. When treating the GW150914 data, the effective-one-body-numerical-relativity (EOBNR) model helps to dig out the gravitational wave signal from the strong noise. Such help results in confidence level 5.1σ in contrast to the 4.7σ without model help. At the same time, EOBNR model also helps to recognize that the source is a binary black hole (BBH), and also it gives out the mass, spin, and other information of the two black

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holes. This kind of source recognition realizes the astronomical observation of the source of gravitational wave. In addition to the mature LIGO-type ground-based gravitational wave detectors see  Chap. 2, “Terrestrial Laser Interferometers,” pulsar timing programs including FAST, SKA, and others are developing see  Chap. 4, “Pulsar Timing Array Experiments”. The space-based gravitational wave detectors LISA, Taiji, and TianQin will fly in 2034 [2–5] see chapter  “Space-Based Gravitational Wave Observatories”. The satellite LISA Pathfinder, which tested LISA technology, worked very well [6, 7]. All of these gravitational wave detection programs need gravitational wave source models. The generation, propagation, and measurement of gravitational wave can be studied separately. Regarding generation, the source of gravitational wave can be viewed as an isolated system. In order to avoid near-field effects and the gauge ambiguity of general relativity, people usually use null infinity to describe gravitational wave [8]. The Bondi-Sachs framework provides a theoretical basis for such descriptions [9–12]. The propagation of gravitational waves may be affected by the expansion of the universe [13], interstellar medium [14], and others. In the measurement phase, it is usually sufficient to describe the gravitational wave as perturbation of a Minkowski spacetime. For the propagation phase and the measurement phase, we need the generated wave by gravitational wave source as input. This article only focuses on the generation phase. In order to describe the gravitational wave generated by the wave source, we need to solve the Einstein equation. The methods to solve the Einstein equation corresponding to the gravitational wave source include analytical method [15], post-Newton approximation see  Chap. 31, “Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspirals,” spacetime perturbation see  Chap. 36, “Black Hole Perturbation Theory and Gravitational Self-Force,” and numerical calculation method [16]. This article only focuses on numerical calculation method, i.e., numerical relativity [17]. In the next section, we illustrate the role played by theoretical models of gravitational wave source in gravitational wave data analysis. After that, a brief history of the development of numerical relativity and its current research state will be introduced. The core difficulties of numerical relativity and the existing solutions will be introduced then. We also show several examples using numerical relativity to model the source of gravitational waves. Finally, we give a summary and outlook.

The Role of Numerical Relativity in Gravitational Wave Astronomy Gravitational wave detection is notoriously difficult. The reason is that the gravitational wave in the universe is extremely weak. The signal is usually buried in the noise [1, 18]. The matched filtering technique is a widely used method to treat the weak gravitational wave signal see  Chap. 41, “Principles of Gravitational-Wave Data Analysis.” The key step of matched filtering is the following inner product:

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 h(p, f ), s(f ) =

fmax fmin

h(p, f )¯s (f ) df Sn (f )

(1)

where f is the frequency, h is the model waveform, s is the detected data, s¯ is the complex conjugate of s, Sn is the noise power spectrum, and p is the parameter describing the configuration of the wave source. There are two key √ components in the above inner product. One is the division with denominator Sn , which is called whitening. The other is the integration range (fmin , fmax ) that corresponds to the sensitive frequency band of the detector. This operation is called bandpass filtering. Note that the model waveform depends on the parameter p. When you go over the parameter space of p, you get a function on the parameter space. One usually denotes such function as ρ(p). If the maximum value of this function exceeds a certain threshold, one can conclude that a gravitational wave signal is detected that is buried in the noise. The corresponding wave source can be described by the theoretical model with the configuration parameter p0 corresponding to the maximum value ρ(p0 ). Consequently, the measurement or observation of the celestial object through the gravitational wave is realized. In real data processing, due to the high dimension of the parameter space, it is impossible to calculate the whole ρ(p) function. How to find the maximum ρ(p0 ) is a problem of data processing, which is beyond the scope of this article. Readers interested in this topic could refer to relevant references [16, 18]. From the above description, we can see the importance of the theoretical model of gravitational wave source h(p, f ) for gravitational wave detection. Without an accurate theoretical model, it is impossible for us to extract signals buried in noise. After obtaining the gravitational wave signal, we also need the theoretical model h(p, f ) to invert the wave source parameter p0 from the gravitational wave signal, so as to achieve the purpose of astronomical observation. It can be seen that the theoretical model of the gravitational wave source is the foundation and premise of astronomical observation of gravitational wave astronomy. In order to establish the theoretical model corresponding to the source of gravitational wave, we need to solve the Einstein equation corresponding to the source of gravitational waves. Numerical relativity does not make any analytic approximation to Einstein equation; that means numerical relativity is a universal tool for the modeling of gravitational wave sources.

A Brief History and the Current Status of Numerical Relativity For usual problems of computational physics, people need to care about two key issues. The first one is constructing a good dynamical equation to describe the physical problem. To solve this issue, people need to understand well the mechanism of the physical problem. The second issue is solving the corresponding dynamical equation numerically with high accuracy and efficiency.

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Numerical relativity is very different to usual computational physics problems. Based on the general relativity, the involved dynamical equation is nothing but the Einstein equation. We do not care about the dynamical equation issue. At the same time, we do not have the freedom to make any changes to the Einstein equation except when one wants to consider alternative theories instead of general relativity. For numerical calculation, numerical relativity faces not only the problems of accuracy and efficiency but also a more fundamental problem of computational stability. The computational instability manifests itself as a rapid (exponential) increase of small errors during the calculation process. Afterward, this instability leads to the occurrence of NAN (not a number) when the numerical calculation goes on [19]. In the early days of the development of numerical relativity, scientists were tortured by stability issues for quite a long time, giving them no time to consider the issue of accuracy and efficiency and even no possibility to investigate physical problems. The history of numerical relativity is very long. As early as 1964, Hahn and Lindquist [20] made the first numerical calculation of binary black hole (BBH) collision. But their numerical code crashed only after tens of steps. No physical results were obtained. In the 1970s, Eppley [21] in Princeton University and Smarr [22] in the University of Texas at Austin made an in-depth research on the coordinate selection for numerical relativity. For some special physical situations, they made use of the inherent symmetries and developed special and delicate coordinates for numerical relativity. In the 1980s, Piran et al. [23, 24] gave preliminary numerical results for some simple modeling problems based on cylindrical symmetry or axisymmetric reduction. Afterward, people can calculate such special spacetime with symmetries, and many interesting physical results are obtained. The critical phenomena [25] is one of such decent physical results. The trick of special coordinate selection for spacetimes with symmetries cannot be generalized to the general spacetime without symmetry. The idea of symmetry automatically searching was proposed. But the numerical stability is not achieved. In the 1990s, people thought that the instability encountered in numerical relativity is possibly due to the inadequate grid resolution used in the calculations. Inspired by the LIGO project at that time [26], a bundle of US research institutes and universities jointly formed the “The Binary Black Hole Grand Challenge Alliance.” This alliance planned to build a completed numerical relativity software to realize the numerical simulation with high enough resolution for the BBH problem. Cactus (http://cactuscode.org/) was born consequently. Unfortunately, the research results of this alliance show that the instability of numerical relativity is not just a matter of computational resolution [27, 28]. After the resolution study, people also studied the influence of boundary conditions [29], numerical calculation methods [30], formalisms of the Einstein equation (such as Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [31, 32]), and others. People fond all of these factors have an impact on stability. But these factors cannot determine the stability individually. In 2000, the hardware construction of LIGO was completed, but the stability problem of numerical relativity had not been solved and even had no substantial progress. Witnessing this situation, Thorn said

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pessimistically: “It is likely that the success of gravitational wave detection will be realized earlier than the numerical calculation of binary black hole merger” [33]. In 2005, people were still struggling against the numerical stability. Unexpectedly, Pretorius [34] announced the first stable calculation of binary black hole merger. His numerical calculation covers the whole process including late inspiral, plunge, merger, and ringdown. About half a year later, NASA numerical relativity group [35] and the University of Texas at Brownsville (UTB) numerical relativity group (now the Rochester Institute of Technology (RIT) group) [36] also independently broke through the stability of numerical relativity. In the breakthrough work of Pretorius, two key factors determine the numerical stability. One is the adaptive mesh refinement (AMR) algorithm [28], and the other is the generalized harmonic (GH) formalism. The AMR guarantees the high resolution. The hyperbolic structure and the light speed character of the GH formalism guarantee the validity of excision treatment of black hole singularity. Differently, there are three key factors in the works of NASA group and UTB group. They are AMR [28], the BSSN formalism [31,32], and the moving puncture method. The AMR used by UTB group was an effective one [36]. Fisheye mesh grid was adopted. Afterward, more and more advanced AMR techniques and numerical codes are developed [35, 37–42]. There is a characteristic mode that admits speed faster than light in BSSN formalism. This makes people unable to use excision method to treat black hole singularity within the framework of BSSN formalism. Puncture method can work well with the faster-than-light mode. But if we fix the puncture, the highly dynamical behavior around the puncture point will distort the fields strongly and kill the numerical calculation [43]. The most important input of the two breakthrough works [35, 36] is letting the puncture move. Within moving puncture method, the computational domain is R 3 in contrast to the nontrivial topology involved in the excision method. Since then, BSSN formalism together with moving puncture method becomes more and more popular [44–48]. After the breakthrough of the stability problem, a major finding of numerical relativity is the recoil velocity produced by BBH merger [49–53]. After the stability of numerical relativity was solved, in order to establish the theoretical model of the gravitational waves produced by BBH systems, numerical relativity scientists shifted their focus to the accuracy and efficiency of numerical calculations. Compared with the finite difference method, the spectral method [54] has much better calculation accuracy. However, the difficulty of the stability when dealing with the coupling of the material field and the Einstein equation limits the further development of the spectral method in numerical relativity. Finite difference method is far more robust than the spectral method. At present, finite difference method is widely used in high-speed black hole collisions [55], multiple black hole problems [56], binary neutron star (BNS) mergers [57] see  Chap. 12, “Binary Neutron Stars,” modified gravity theory [58], high-dimensional gravity theory [59], negative mass black holes [60], and other issues. Finite difference method has also been greatly developed in recent years in terms of calculation accuracy. Z4c formalism and CCZ4 formalism were developed independently by two group. Compared to BSSN formalism. These two new formalisms improve the calculation accuracy by about 100 times. Based on the Z4c formalism, the constraint-preserving

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boundary condition can further improve the accuracy by about ten times [64]. Regarding to the high-spin BBH problem, suitable adjustment to BSSN formalism can increase the calculation accuracy of the spin angular momentum by about seven times [65]. On the computational efficiency, the global data exchange of the spectral method limits its ability for parallel computing. Finite difference method combined with the domain decomposition algorithm can achieve good parallel computing efficiency. In order to deal with the multi-scale characteristics in astrophysics, mesh refinement with multi-data level structure is indispensable nowadays. AMR is also very important in the stability issue. However, the data dependence between neighbor levels makes the number of grid points in a single data level limit the maximum number of cores for parallel computing. This fact consequently limits the strong parallel scalability [66]. In contrast, the finite element method (FEM) can use the spectral method within the element and use a similar method like finite difference between element and element. This way can combine the exponential convergence of spectral method and the high parallel scalability of finite difference. At the same time, since the mesh refinement in the FEM has no data level structure, all the elements can be treated uniformly. Therefore, it is expected that FEM can achieve good strong parallel scalability. However, the weak form of Einstein equation is difficult to construct [67], and its large-scale scientific calculation is also difficult to achieve. Due to these difficulties, the application of FEM in numerical relativity is quite limited so far [68–72]. Numerical calculation of Einstein equation is a typical computationally intensive problem. One can use graphics processing unit (GPU) to accelerate the numerical calculation. References [73–75] implemented a serial GPU program. Compared to central processing unit (CPU), the acceleration ratio is about 10. Among them, reference [75] used the ADM formalism, and references [73, 74] used the BSSN formalism. In order to apply GPU to actual numerical relativity calculations, we must make the GPU work with parallel AMR. Solving the conflicts between the large memory requirements of Einstein equation and GPU’s small hardware memory is not easy. At the same time, we must avoid affecting the data exchange between the parallel AMR data levels and between different CPUs. The authors of [66] achieved about ten times acceleration ratio compared to its CPU counter code. Concerning the 120 times hardware acceleration by GPU in gravitational wave data analysis [76], we expect that there is still room for further development of this issue [77].

The Core Difficulties of Numerical Relativity and Current Solutions The core difficulties of numerical relativity include the partial differential equation formalism, the black hole singularity, the problem of multiple scales, the problem of boundary conditions, the choice of coordinates, and the problem of parallel computation. These core difficulties and the current solutions are explained and introduced one by one below.

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The Partial Differential Equation Formalism The background manifold where the Einstein equation lives on is determined by the solution of the Einstein equation and consequently cannot be given in advance [8]. As a tensor equation, the Einstein equation itself is not an explicit partial differential equation (PDE). This fact makes it difficult to directly calculate the Einstein equation through numerical computation. Therefore, it is necessary to firstly reduce the Einstein equation to a suitable PDE for numerical calculation. There are a bundle of reduction methods. Different reduction methods result in different types of PDE. For example, some are elliptic equations and some are hyperbolic equations [78]. Different formalism holds different PDE property. Consequently, they have significant differences in calculation stability, convergence, accuracy, and calculation efficiency. This problem is called the formalism problem of Einstein equation in numerical relativity. There are two typical ways to reduce the Einstein equation in the field of numerical relativity at present. One is reducing the Einstein equation to a Cauchy problem that is an initial value problem. The other one is reducing the Einstein equation to a characteristic problem [9, 10, 79–81]. This way is called characteristic method. It depends on a null hypersurface to expand the Einstein equation. In strong gravity region, null hypersurface admits caustics in general except for highly symmetric spacetime. These caustics will make the characteristic method break down. Consequently, the characteristic method cannot be used alone to solve the Einstein equation. Due to this situation, people proposed Cauchy-characteristic matching method [82–86] and the idea of using the characteristic method to extract gravitational wave [87–92]. In this chapter, we only care about Cauchy problem reduction method. The first step in reducing the Einstein equation to the Cauchy initial value problem is performing a 3 + 1 decomposition of the spacetime manifold. The three-dimensional space-like hypersurface corresponds to the space of the Cauchy problem, and the remaining one-dimension corresponds to time. The second step of the reduction process is selecting the unknown functions. This step reflects the very special characteristics of the Einstein equation. Numerical relativity concerns the ten spacetime metric coefficients. There are ten unknown coefficients and ten Einstein equations. This seems the correspondence is perfect. However, due to the Bianchi identity [8], these ten equations are not independent. The Bianchi identity has f our components, resulting in only six independent equations. This means that f our degrees of freedom of the ten metric coefficients are not controlled by the equation, which corresponds to f our degrees of the diffeomorphism freedom. Hilbert ever used this geometrical property to correct the original form of Einstein equation proposed by Einstein. For the ten unknown functions, only six of them can be determined by Einstein equation. So choosing the six degrees of freedom to be solved by the Einstein equation is the second step. Different choices of six degrees of freedom can be a different combination of functions and their derivatives or integrals. The third step is determining the remaining f our degrees of freedom.

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From the perspective of diffeomorphism, these f our degrees of freedom can be specified arbitrarily. Therefore, these f our degrees of freedom are often referred to as gauge freedom. However, the different designation will greatly affect the properties of PDEs [93]. For example, some specified methods result in strong hyperbolic equations, while some result in weak hyperbolic equations [94]. The above three steps can be realized in different orders. The ADM formalism is implemented in the above order [95]. The ADM formalism first performs 3 + 1 decomposition on the spacetime manifold and then 3 + 1 decomposition of the metric. The metric can be written as ds 2 = −α 2 dt 2 + γij (dx i + β i dt)(dx j + β j dt).

(2)

In the ADM formalism, a total of six functions of γij are selected as unknown functions. Then, the four degrees of freedom α and β i are determined by choosing PDEs. In numerical relativity, people like more equations with derivatives first-order in time. For this purpose, people introduce auxiliary variables: Kij = −

1 (∂t − Lβ )γij . 2α

(3)

Under certain gauge selection α and β i , the ADM equation is only weakly hyperbolic. This implies its numerical instability. But why is the ADM formalism unstable? Is there any way to make it stable? These are still open questions. The current status is no one can use ADM formalism to realize stable calculation. The BSSN formalism [31, 32] employs the conformal decomposition of γij and the traceless decomposition of Kij . In order to result in strong hyperbolicity of PDEs, the auxiliary variable Γ˜ i is introduced. They correspond to the connection of the conformal 3−metric γ˜ij . Equations such as Z4c and CCZ4 are further improvements to the BSSN formalism [62–64]. These improvements effectively suppress the constraint violation during the numerical calculation. Explicitly, the BSSN formalism can be expressed as 1 1 ∂t φ = β i φ,i − αK + β,ii 6 6 2 k k ∂t γ˜ij = β k γ˜ij,k − 2α A˜ ij + 2γ˜k(i β,j ) − γ˜ij β,k 3   1 2 i 2 ij ˜ ˜ ∂t K = β K,i − D α + α Aij A + K + 4π(ρ + s) 3    T F ∂t A˜ ij = β k A˜ ij,k + e−4φ α Rij − 8π sij − Di Dj α

2 ˜ k k +α K A˜ ij − 2A˜ ik A˜ kj + 2A˜ k(i β,j ) − Aij β,k 3

(4) (5) (6)

(7)

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∂t Γ˜ i = β j Γ˜,ji − 2A˜ ij α,j  2 ij i ˜ kj ij ij ˜ ˜ +2α Γj k A − γ˜ K,j − 8π γ˜ sj + 6A φ,j 3 2 1 j i i + Γ˜ i β j , j + γ˜ ki β,j k + γ˜ kj β,kj −Γ˜ j β,j 3 3 ∂t α = −2αK + β i α,i 3 i i B + β j β,j 4 i ∂t B i = ∂t Γ˜ i − ηB i + β j B,j − β j Γ˜,ji ∂t β i =

(8) (9) (10) (11)

The spatial indexes range from 1 to 3. We have used the Einstein’s notation which means the repeated index represents a summation. φ is the conformal factor. K is the trace of Kij . And A˜ is the conformal transformation of the traceless part of Kij . The GH formalism firstly 3 + 1 decomposes the spacetime manifold. This operation is similar to the ADM formalism and the BSSN formalism. Secondly, the GH formalism specifies the gauge freedom. Combining the metric coefficient properly, we get

Hμ = −g αβ Γμαβ .

(12)

The GH formalism takes Hμ as the gauge freedom. Then in the Einstein equation, replace the term on the right-hand side of Equation (12) with Hμ . The ten Einstein equations treated through this way become independent and can be used to solve ten unknown functions gμν . So gμν are selected as the unknown function in the GH formalism [96–98]. GH formalism used by Pretorius in [34] can be specifically expressed as

g δγ gαβ,δγ +g δγ ,α gβδ,γ +g δγ ,β gαδ,γ +Hα,β +Hβ,α −2Hδ Γ γ αβ −2Γ γ δα Γ δ γβ = 0 (13) The Greek letters here represent the spacetime index, ranging from 0 to 3. The comma represents the partial derivative with respect to the coordinate. Γ is the Christoffel symbol corresponding to the spacetime metric. At present, numerical relativists use BSSN or GH formalism. Whether other formalisms for the Einstein equation are numerically stable is still an open question. The gauge freedom and the constraint equations are delicately entangled. The constraint equations are related to the solution of the Cauchy problem. These issues are not discussed in this article. Interested readers can check references [44, 46].

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The Singularity Inside the Black Hole A sufficiently strong gravitational field will result in a black hole. According to the cosmic censorship, a black hole is always surrounded by an event horizon. A singularity will appear at the center of the black hole [8]. Singularity may not exist in the real world. But theoretically this problem belongs to quantum gravity, which is beyond the scope of this article. In the framework of classical general relativity, the event horizon is the causal boundary of the black hole. The spacetime region outside the event horizon can influence the region inside the event horizon. But the region inside the event horizon does not influence the region outside the event horizon. When approaching the singularity, physical quantities such as spacetime curvature will diverge to infinity. In numerical calculations, such behaviors will cause NAN and kill the whole numerical calculation. So how to deal with black hole singularity is a tricky issue. There are currently two methods to deal with the singularity of black holes. Both methods make use of the causal interface property of the event horizon. One is the moving puncture method [35, 36], and the other is excision method [34]. The idea of the moving puncture method is to fill a metric field inside the event horizon and to make the singularity invisible. What and how to fill is the key problem of this method. Usually people fill a metric field in an asymptotically flat spacetime region at the initial moment of evolution. Squeeze such a metric field to a bounded area and fill it into the event horizon. Taking the Schwarzschild black hole as an example, its maximum extension spacetime includes two asymptotic flat regions. Taking a space-like hypersurface with a time symmetry, the two asymptotically flat regions are symmetric about the event horizon. We can take the isotropic coordinate to break this symmetry, where the infinity of an asymptotic flat region corresponds to the original point of the coordinate system. This idea can be generalized to any black hole [99]. Such an operation of squeezing the infinite region to a finite region compresses the infinity into a singularity. However, such a singularity is a weak singularity which is different to a black hole singularity. This weakly singular point is usually referred to as a puncture point. During the time evolution, the filled region will also evolve automatically. Consequently, one needs only to pay attention to the puncture point in the numerical calculation. The excision method is excluding the region within the event horizon directly from the numerical calculation. There are two problems involved in this approach. The first one is that the excision boundary becomes a boundary of numerical calculation. It is a new boundary introduced by the calculation problem instead of the physics problem itself. How to give the boundary condition is a delicate problem. In principle, no matter what kind of boundary condition is given there, it will not affect the time evolution of the region outside the event horizon. This is because the physical information will not run out of the event horizon of the black hole. But this conclusion only holds for physical degrees of freedom; non-physical degrees of freedom may be affected by this boundary condition. At the same time, this boundary condition will affect the stability of numerical calculations. The second problem is that the black hole will move during the evolution process, which will

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cause some grid points previously inside the event horizon to become grid points outside the event horizon. However, there are no data on grid points outside the calculation region. How to give the required data after they enter the calculation area is also a delicate problem. Generally, people place outgoing boundary conditions at the excision boundary and use extrapolation to fill the data [34,96,98]. But it seems this idea does not work for spectral method. So the SpEC code developed a delicate system to fix the excision boundary with respect to the calculation domain [54]. Although the BSSN formalism cannot work with the excision method due to the faster-than-light speed, GH formalism can work with the moving puncture method [100]. But note the work [100] used 3+1 form of GH formalism. And the equations involve derivatives first order in time but second order in space [101]. The most attracting feature of GH formalism is its wave-like equation form. It is interesting to ask whether such kind of GH formalism works together with the moving puncture method.

The Problem of Multiple Physical Scales The process of source-generating gravitational waves is a typical multiple physical scale problem. Taking the BBH system as an example, the involved scales include the size of the two black holes, the distance between the two black holes, the wavelength of the gravitational wave, and the size of the asymptotically flat spacetime region. All of these physical scales need to be taken into account in the numerical calculation. The smallest physical scale determines the spatial resolution of the calculation. The largest physical scale determines the entire calculation domain. If a uniform grid is used to handle all physical scales, memory and computing requirements are much beyond the support of the current computer hardware. This fact means mesh refinement technique is necessary in numerical relativity. Regarding mesh refinement, there are two important parameters. One is the number of mesh levels. The other is grid numbers on each data level. The setting of these two parameters should take the balance of numerical convergence and strong parallel scalability into consideration. For the Einstein equation, the computation amount of a single grid point is very massive. So people would like to set the grid number on each data level as small as possible. Then the total number of data levels is determined by the numerical resolution requirement. Generally, about 18 levels are used. As the physical parameters of the gravitational wave source change, the number of data level will also change. For example, when the mass ratio of two black holes becomes larger, the number of levels should increase. As we mentioned in the above section, one can use special coordinate transformation to realize mesh refinement [36]. Such method is less flexible as direct mesh refinement. In the following, we only care about direct mesh refinement. The idea of mesh refinement involves using meshes with different spatial resolutions to cover spatial regions with different physical scale. Use finer mesh to cover regions with small scale, and use coarser mesh to cover regions with larger scale. In specific

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implementation, one can let the coarsest mesh cover the entire computational domain. And use the finer mesh to cover the smaller regions in turn until the region with the smallest physical scale is covered by the finest mesh. In this way, a fine mesh is always nested in a coarse mesh. The meshes with different resolutions form different data levels. For those regions covered by different mesh levels, only the relatively finest mesh level plays a real role. That is to say, the data on the fine mesh will replace the data on the coarse mesh. At the boundaries of the fine mesh, since the fine mesh cannot provide the boundary condition itself, the coarse mesh can provide the boundary data for the fine mesh. Both operations of replacing the coarse grid data with the fine grid data and of providing the boundary conditions for the fine mesh with the coarse mesh data are realized by interpolation. For the time evolution, in order to guarantee the Courant-Friedrichs-Lewy (CFL) condition [102], smaller time step is needed by the finer level, while bigger time step is possible for coarser level. Of course, coarse levels can also use smaller time steps without affecting the CFL conditions [35], but this will waste the computation resources. So ideally shorter time steps are used for finer levels, and longer time steps are used for coarser levels. This kind of calculation method is called the Berger-Oliger scheme [103]. This will lead to the inconsistency of the time steps between the coarse and fine meshes. In order to deal with this inconsistency, an interpolation operation is needed in the time direction. So in the mesh refinement algorithm, the interpolation operation is used very frequently. The ratio of the step length of the coarse and fine grids is often 2 to 1. For the coarse and fine meshes, a unified Courant number is used, and the time step corresponding to the coarse and fine grids is also 2 to 1. If mesh refinement can be applied in the finite element framework, numerical calculations will be more flexible and convenient [104]. This idea method relies on the development of finite element numerical relativity [68–72].

The Problem of Boundary Condition Physically, the source of gravitational waves is usually described by an isolated system, which corresponds to an asymptotically flat spacetime. This means the source of gravitational waves is a system extending to infinity [8]. Mathematically, this corresponds to a well-defined initial value problem without boundary. However, there is no way to deal with the problem with infinite computational domain. There are two possible methods to deal with this problem. One is to use the method of conformal compression to transform infinity to finite with a conformal coordinate [34, 96, 98]. However, such a method will cause the gravitational waves to pile up at a finite region and consequently cause either instability or inaccuracy due to strong dissipation where gravitational waves pile up. The other method is to cut off at some place far enough. This will introduce an artificial boundary that is introduced purely by numerical calculations themselves. There is no corresponding physical boundary condition provided by original problem. That is to say, we get an initial-boundaryvalue problem instead of original pure initial value problem.

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How to set the boundary condition for this artificial boundary is an interweaving problem of three aspects including numerical calculation, mathematics, and physics. Regarding numerical calculation issue, improper boundary conditions will cause instability. Fortunately, the simple outgoing boundary condition used in the BSSN formalism can guarantee the stability [35, 36]. However, such boundary conditions used in the GH formalism cannot guarantee the stability. For the mathematical aspect, a suitable boundary condition is needed to result in a well-posed initial-boundary-value problem (IBVP) [105–107]. Now, such boundary conditions for the full nonlinear Einstein equation are unknown. For the linearized Einstein equation of flat spacetime perturbations, such boundary conditions have been proposed. The boundary conditions in [108] are for the GH formalism, and the boundary conditions in [64, 109] are for the BSSN formalism. At the physical level, we want a boundary condition that can recover the solution of the original initial value problem corresponding to the infinite space. It is believed that such a boundary condition must satisfy the mathematically well-posed initial-boundary-value problem. Moreover, this boundary condition can accurately describe the behavior of the physical field at the artificial boundary. In fact, this problem appears not only in numerical relativity. People have dealt with similar problems in electromagnetic waves [110], seismic waves [111], and others. But comparing these problems, the outstanding feature of Einstein equation is that when the gravitational wave propagates, it also disturbs the spacetime and results in a backscattering of the gravitational wave itself. In other problems, the wave is purely outgoing at the artificial boundary. So the problem is reduced to how to achieve zero reflection at the boundary. People need only to propose an ideal absorption boundary. The backscatter of the Einstein equation makes this boundary condition problem much more complicated. In order to obtain the boundary conditions at the physical level, we must understand the behavior of backscattering. At the physical level, this problem has no results yet.

The Problem of Coordinate Choice In order to numerically solve the Einstein equation, a certain coordinate system needs to be selected. Due to the property of the diffeomorphism invariance of general relativity, the coordinates themselves have no essential physical meaning, and consequently, the choice of coordinates can be arbitrary. A spacetime manifold often requires multiple coordinate patches to cover. The patches are connected by diffeomorphism in the overlapping areas [8]. The effects of coordinate singularity on the stability of numerical calculations are similar to those of the physical singularity. Therefore, there is no way to handle numerical calculations at the boundary of the patch domain. A natural idea is to transform to another patch by diffeomorphism before time evolves to the boundary of the patch domain. But two reasons make this unrealistic in practice. One reason is that there is no way to distinguish coordinate singularity and physical singularity in numerical calculations. So there is no way to alert when it is close to the boundary of the coordinate patch domain. The second

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reason is that the numerical calculation result is a discrete metric function. Based on the discrete metric function, it is difficult to calculate the Jacobian matrix required for the diffeomorphism transformation. In fact, the coordinate transformation from Cauchy coordinates to null coordinates is the essential difficulty that causes the ideal Cauchy-characteristic matching method to fail so far [80, 83]. On the problem of coordinate choice, the current method adopted by numerical relativity scientists is to choose a coordinate system as large as possible to cover the spacetime region in question. Taking the BBH merger problem as an example, the spacetime region physically concerned includes the space outside the apparent horizon of the black hole and the time range from inspiral phase to the ringdown phase. After a longtime research, people find that the coordinate condition consisting of the 1 + log slice condition and the Gamma driver gauge condition [49, 112] works for BSSN formalism. The coordinate condition consisting of the vanishing spatial component of the generalized harmonic function and harmonic driver for the time component of the generalized harmonic function works for GH formalism. Whether there are other coordinate conditions that can meet the above requirements is still an open question. For real numerical calculation, the coordinate conditions cannot be expressed in a form of known functions. People usually specify coordinate conditions by specifying PDEs. The two coordinate conditions mentioned above are realized in this way. So the numerical calculation problem of Einstein equation becomes a calculation problem of Einstein equation coupled to the coordinate condition equation. The overall property of the coupled equations affects the behavior of numerical calculations. Different formalism of the Einstein equation consequently requires different coordinate condition. For example, the former coordinate condition mentioned above is used for BSSN formalism [35, 36], and the latter one is used for GH formalism [34, 100]. The hyperbolicity of the PDEs of the entire coupled equation system gives a certain explanation to these stability issues [93]. In the numerical implementation of the SpEC version GH formalism, the above mentioned coordinate condition for GH formalism is unstable [113]. What causes this numerical instability is still an open question so far. Specifically, the 1 + log slice condition and the Gamma driver gauge condition for BSSN formalism can be expressed as ∂t α = −2αK + β i ∂i α 3 i B + β j ∂j β i 4 ∂t B i = ∂t Γ˜ i − β j ∂j Γ˜ i − ηB i + β j ∂j B i ∂t β i =

(14) (15) (16)

where B i is an auxiliary variable introduced in the coordinate conditions. With the 1 geometric units, η is a free parameter with dimension M . Its value will affect the stability of numerical calculations, which is generally taken as 1 divided by the mass of the black hole.

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The coordinate conditions used by Pretorius in [34] are Hi = 0 g μν H0,μν = −ξ1

(17) α−1 + ξ2 H0,ν nν αn

(18)

where α is the lapse function and nν is the future-oriented unit normal vector of the time slice. ξ1 , ξ2 , and η are three parameters for coordinate condition. With the 1 geometric units, the dimension of ξ1 and ξ2 is M , and η is dimensionless. The values given by Pretorius are ξ1 = 19/M, ξ2 = 2.5/M, and η = 5. Here, M is the mass of the black hole.

The Problem of Parallel Computation In order to cope with the huge computational requirements of numerical relativity, parallel computing plays an important role in numerical relativity. In the previous subsection, we have mentioned the importance of AMR in numerical relativity. So parallel AMR has become the basis of all numerical relativity software. From the perspective of computer software, the part of parallel AMR is a platform for the entire large-scale scientific computation of numerical relativity. This platform is responsible for the data structure management and process management of the entire program. It needs to deal with data exchange between different mesh levels and different computing processors. The platform has to handle these complex data exchanges and to interact well with other program modules. These abilities determine the parallel scalability of the corresponding computer software. Here, we take the GPU version of the AMSS-NCKU software [66] as an example to discuss the problem of parallel scalability. The GPU version of the AMSS-NCKU software aims to work with heterogeneous supercomputers composed of multi-core CPU and external GPU, such as Tianhe-1 in China, Titan and Blue Waters in the United States, and others. AMSS-NCKU uses a finite difference algorithm with AMR and an RungeKutta time evolution algorithm with a Berger-Oliger time refinement for hyperbolic PDEs. For parallel computation, a region decomposition algorithm is used. AMSSNCKU software decomposes the region on each data level. This algorithm takes into account the large amount of calculation on a single grid point and the relatively less number of grids in the numerical calculation of Einstein equation. Then we distribute the data to all of the computing nodes almost averagely. AMSS-NCKU sets the number of computing processes according to the number of GPUs for the same computing node. One process uses only one GPU to avoid resource contention and improve the efficiency of program execution. The data exchange inside the same data level in the same computing process is operated by direct copy. The data exchange between different data levels in the same calculation process is performed using interpolation method. Different computing processes

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exchange data for the same data level through the MPI directly. The data exchange between different data levels and different computing processes is operated in two steps. The first step is to complete the corresponding data interpolation on the computing process where the source data are located. The second step is to use MPI to complete the final data exchange. In the CPU hardware part, data storage and calculation are automatically allocated by the operation system. It should be noted that a computing node generally contains dozens of computing threads. These threads are included in the same process and can share memory. We use OpenMP to achieve multi-threaded parallel calculations. In the GPU part, we need to perform data exchange between CPU memory and GPU memory. The main calculation is performed on the GPU. A relatively small amount of calculations such as interpolation required for data exchange between different processes are completed by the CPU. We have designed an algorithm that allows the data exchange between the GPU and the CPU and the calculation of the GPU to be performed simultaneously. In this way, the calculation and the communication can overlap in time.

The Problem of Gravitational Waveform Extraction As Einstein emphasized, general relativity is a covariant theory that is independent of coordinate choice. As a physical quantity, gravitational wave should be independent of coordinate choice. But the numerical solution of spacetime metric coefficients gμν strongly depends on the coordinate we used in the numerical calculation. Then how to extract the gravitational waveform from the numerical solution we have obtained is another difficulty for numerical relativity to model gravitational wave source [114]. From the idea of gauge independent black hole perturbation technique, people constructed Regge-Wheeler-Zerilli quantities to extract the gravitational waveform [115]. One also calls this method Zerilli-Moncrief formalism [116]. Similar to the Zerilli-Moncrief formalism, people later propose to use Newman-Penrose formalism to count the spin effect of the spacetime [117]. In order to eliminate the near source effect on the waveform, one needs to calculate the waveform at a place far away enough. So ideally people extrapolate the waveform to null infinity [118]. The state-of-the-art technique for gravitational waveform extraction is the Cauchy-characteristic extraction method [119–123]. The only rigorous formalism for the global measurement of gravitational wave and its energy is the BondiMetzner-Sachs (BMS) theory [124–128]. The key content of BMS theory is the Bondi-Sachs (BS) coordinate that corresponds to the characteristic lines of Einstein equation. So the numerical formalism based on BS coordinate for Einstein equation is called characteristic formalism [85]. Consequently, the characteristic formalism can adopt BMS theory and touch null infinity where the gravitational wave can be defined rigorously. Due to these advantages, the Cauchy-characteristic extraction

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technique is very accurate for gravitational wave extraction in numerical relativity. Recently, characteristic extraction method has been applied to get gravitational wave memory [129–131] which cannot be directly calculated in numerical relativity otherwise. Since the characteristic calculates the spacetime along null direction, the gravitational wave emitted by the source can propagate “instantly” to the null infinity where the gravitational waveform is extracted. In contrast, Cauchy method has to calculate more time needed by the gravitational wave propagation from the source to the detector. In this sense, characteristic formalism is much more efficient than Cauchy method. But BS coordinate cannot cover general spacetime due to the coordinate singularity. This is why characteristic formalism cannot be used alone to solve Einstein equation. Consequently, the combination of Cauchy method and characteristic method, Cauchy-characteristic matching, is proposed [85]. But unfortunately Cauchy-characteristic matching has not been implemented stably [132, 133].

Application of Numerical Relativity to Gravitational Wave Source Modeling Binary Black Hole (BBH) System The BBH system is the most important and major source of gravitational waves for most gravitational wave detection experiments including pulsar timing observations, space-based gravitational wave detection programs, and ground-based gravitational wave detectors. The inspiraling BBHs surrounded by an accretion disk will wipe out the matter in the inner area of the accretion disk. An almost vacuum background is left for the BBH. So we need only the vacuum Einstein equation to describe the BBH systems. In order to determine the BBH system, we only need the masses (m1 , m2 ) of the two black holes, their spins (s1 , s2 ), and the orbital eccentricity e at a certain distance D of the two black holes. There are two points worth be pointed out about the physical meaning of eccentricity e. First, people usually use Newton’s law to define eccentricity e. Since gravitational waves take away energy, black hole orbits will shrink in the framework of a general relativity theory. How to define the eccentricity e is a delicate problem. Second, the orbit is the relative motion of two bodies. Or the orbit is the reduced one-body motion in the framework of Newtonian mechanics. Therefore, the motion corresponds to only one orbit. That is why we need only one eccentricity e to describe the orbit. Considering the scale invariance of Einstein equation with respect to the total mass of BBHs, the total mass of two black holes is always set to 1 as a unit in numerical relativity calculations [16]. Consequently, when dealing with the BBH problem with numerical relativity, we only need to care about the mass ratio q of BBHs, the dimensionless spin parameter (χ 1 , χ 2 ), the initial reduced distance d, and the initial eccentricity e.

34 Numerical Relativity for Gravitational Wave Source Modeling

q≡ χi ≡

m1 m2

d≡

si , i = 1, 2 m2i

D M

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(19)

M = m1 + m 2 .

(20)

In order to determine a BBH system in the universe, we also need its luminosity distance DL from the earth, angular position on the celestial sphere (θ, φ), the angle ι between the orbital plane and the direction from the earth to the BBH, the angle ψ between the line connecting the two black holes and the direction from the earth to the BBH, and the angle β between the major axis of the elliptical orbit and the direction from the earth to the BBH. The three angles – ι, ψ, and β – will change over time. So people usually use these three angles at initial time (some reference time t0 , or reference frequency f0 , or reference distance D0 equivalently) to describe the configuration of the system. (DL , θ, φ, ι, ψ, β) are often called external parameters of the BBH system. Numerical relativistic calculations can give information about the gravitational waves emitted in all directions by a BBH system with nine parameters (q, χ 1 , χ 2 , d, e). The scale transformation can transfer this result into information about the gravitational waves emitted in all directions by a BBH system with ten parameters (m1 , m2 , s1 , s2 , D, e). Combined with external parameters (DL , θ, φ, ι, ψ, β), we can obtain the gravitational wave model of the BBH system described by the specific 16 parameters. With respect to a detector, two more parameters, initial phase φ0 and arriving time t0 , will be involved. So altogether there are 18 parameters for BBH sources. For the case of 1 ≤ q ≤ 20, it has been fully studied by numerical relativity (excluding the cases of extremal spin) [134]. For example, the effective one-body numerical relativity based on the results of numerical relativity (EOBNR) model [16, 135] has played an important role in the data analysis of GW150914 event [1]. The relationship between the mass mf and the spin χ of the final Kerr black hole produced by BBH merger and the parameters of initial two black holes has been established [136–138]: mf = 1 + (m0 − 1)4η + m1 16η2 (χ1 + χ2 ) √ χ = χ 0 + χ 0 η(t4 χ 0 + t5 η + t0 ) + η(2 3 + t2 η + t3 η2 )

(22)

m0 = 0.9515 m1 = −0.013

(23)

χ1 +χ2 1+q 2

(24)

q=

m2 0 m1 , χ

=

q2

,η =

q (1+q)2

(21)

t0 = −2.8904, t2 = −3.5171, t3 = 2.5763,

(25)

t4 = −0.1229, t5 = 0.4537

(26)

The above expression is a phenomenological relation, where m0 , m1 , χ 0 , t0 , t2 , t3 , t4 , and t5 are only empirical coefficients and have no special physical meaning.

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According to the information about the quasi-normal modes of Kerr black holes [139], we can get the maximum frequency of gravitational waves in the merging process of corresponding BBH system. For a BBH system with q > 20, the large difference in scale of the two black holes leads to a huge amount of computation [41, 140, 141]. Such a BBH system is an important gravitational wave source for space-based gravitational wave detectors [142]. The current EOBNR model is based on the numerical relativistic results of 1 ≤ q ≤ 20. However, the mass ratio parameter used by the EOBNR model is the symmetric mass ratio η. Corresponding to the range of the parameters covered by numerical relativity is 0.045 ≤ η ≤ 0.25. The uncovered range is relatively small with respect to η. So it is speculated that the EOBNR model can be extended to the entire range of 0 ≤ η ≤ 0.25. But this speculation requires the verification of the accurate numerical calculation results of the large mass ratio BBH system [143], which is still missing. For the case where the spin of the black hole is not parallel or antiparallel to the orbital angular momentum, the spin of the black hole will strongly interact with the orbital angular momentum, leading to the precession of the orbital plane and the spin direction. The numerical relativity study of this case was first investigated by the numerical relativity group of the University of Texas at Brownsville (UTB) [144]. Later research showed that when the black hole spins pointing to a particular direction of the orbital plane, it will give the maximum recoil velocity among the circular orbital BBH mergers [145]. The SXS collaboration has widely investigated the precessing BBH systems [146]. In terms of waveforms of gravitational waves, post-Newtonian approximation and effective one-body approximation can describe the behavior of orbital plane precession to a certain extent. Later, EOBNR model has given a preliminary gravitational wave waveform model of orbital plane precession [147]. The problem of the BBH merger in elliptical orbits was first noticed by the group of PSU and the group of FSU [148, 149]. Combined with the results of numerical relativity research, the PSU group (now the Gatech group) in reference [150] proposed a post-Newton model (X model) to describe the elliptical orbital BBH problem. SEOBNR is the EOBNR waveform model for spin aligned BBHs. In references [135, 151], we extended the SEOBNRv1 model [152] to the SEOBNRE model that can describe BBH system with eccentricity. Other groups also developed different theoretical models for eccentric BBH systems [153–159].

Neutron Star-Black Hole Binary and Binary Neutron Star For neutron star-black hole see  Chap. 15, “Black Hole-Neutron Star Mergers” (NSBH) binary and binary neutron star (BNS) systems, the shape deformation of the neutron star caused by the interaction between the two components is ignorable when the distance between them is large. Therefore, the characteristics of gravitational waves radiated at this phase are consistent with the characteristics of gravitational waves radiated from the BBH system with corresponding mass. This is

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why the theoretical model of gravitational waves based on the BBH system is often used to process the data of inspiraling phase of the binary system [160]. The method of numerical relativity to deal with the NSBH system [161] and the BNS system [162] is to solve the coupling system of Einstein equation and material field dynamics equation numerically without any analytical approximation. The BNS system does not involve physical singularities. The earliest numerical simulation was completed in [163, 164], which is earlier than the successful numerical calculation of BBH systems [34–36]. In references [164–170], the authors used the equation of state (EOS) in the form of Gamma law: P = (Γ − 1)ρε,

(27)

where P , ρ, and ε are pressure, static mass density, and specific internal energy density and Γ is a constant parameter. The Gamma law EOS is a special case of the polytropic EOS (P ∝ ρ n ). The authors in [171] discussed the merger of a BNS system with a more realistic equation of state. In [171], the author divided the matter into a cold part and a hot part. The cold part uses an EOS fitted experimentally, and the hot part uses an EOS in the form of a Gamma law. The matter will deform under the influence of tidal forces, which in turn affects the evolution of the gravitational field and thus affects the gravitational wave radiation. The authors in [172] generalized EOB dynamics to include tidal force effects, which can describe the inspiral dynamics of BNSs to a certain extent. Later this work was extended to NRtidal waveform model for binary systems involving neutron star [173–176]. The authors of [177] investigated the spin precession effects on the BNS merger through numerical relativity simulations. Due to the strong gravitational field, matter is often extremely ionized and becomes a plasma. Consequently, a strong magnetic field maybe reduced. So magnetic fluid is another important issue involved in neutron stars. The University of Illinois Urbana-Champaign (UIUC) group and the Shibata group [166,167] firstly considered the role of magnetic fields in the BNS merger. They found that the magnetic field can delay the merger and extend the life of a supermassive neutron star formed after the merger. However, it has little effect on the mass of the accretion disks obtained after the merger. Also due to the strong gravitational field, some nuclear reactions will happen. Shibata’s group [178] did a systematic study of the effects of neutrino radiation. Shibata’s group studied the NSBH system in reference [179], where the black hole admits mass 3 ∼ 4M and the neutron star admits mass 1.4M . In that work, the neutron star is described by the Gamma law EOS with Γ = 2. They found that the mass of accretion disk formed after the merger was about 0.1M . The UIUC group studied an NSBH with mass ratios 3 and 5 in reference [180] and examined the effect of the spin of the black hole on the mass of accretion disks formed after the merger. They found that when the spin of the black hole is in the same direction as the orbital angular momentum, larger spin results in greater mass of the accretion disk. When the spin parameter is 0.75, about 0.2M accretion disks are obtained. The authors in [181] studied the effect of the spin direction of black

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holes on the merging dynamics. The more realistic NSBH system has a relatively larger mass ratio; this is because the characteristic space scales of black holes and neutron stars are quite different. Similar to the large-mass ratio BBH system, this problem requires a large amount of numerical calculations, and current research is not sufficient.

Summary and Outlook So far, tens of gravitational wave events have been directly detected. Along with the accumulation of gravitational wave signals by the ground-based gravitational wave detectors, gravitational wave astronomy has formed a new research field. In the meantime, the accuracy of pulsar timing array (PTA) in the 10−8 Hz frequency band will be soon close to ∼10−15 . When the SKA project begins to run, this accuracy will be improved more to ∼10−16 . Space-based gravitational wave detection programs LISA, Taiji, and TianQin are also smoothly developing. The data analysis in all of these gravitational wave detection plans needs the theoretical model of the gravitational wave source as a template to realize signal extraction and gravitational wave source parameter estimation. The generation of gravitational waves is a highly dynamic behavior of strong gravitational fields that is described by the Einstein equation. The theoretical study of gravitational wave sources involves solving Einstein equations. Various stars as gravitational wave sources often do not have any symmetry. Exact analytical solutions to their corresponding Einstein equations are almost impossible. Various approximation methods have limitations on the applicable objects. Numerical relativity does not make any approximation to Einstein equation. In this sense, numerical relativity is a universal tool in the modeling of gravitational wave source. The gravitational wave events about BBH have already given important information to the stellar evolution theory. The gravitational wave astronomy has provided decisive observational evidence on the mechanism of gamma-ray bursts based on the GW170817 event. BNS mergers and neutron star-black hole mergers have been confirmed to be the cause of short gamma-ray burst. The gravitational waves are generated at a few RS from the center of the wave source, which can reflect the physical process occurring at the center of the wave source. So we expect much more information will be obtained through gravitational wave detection [182]. For the space-based gravitational wave detection program, a binary system composed of a supermassive black hole and a small massive compact object is an important wave source. This wave source poses a challenge to the numerical calculation of the BBH system with large mass ratio for numerical relativity. This is essentially a problem of computational efficiency and accuracy. As long as a few typical systems can be calculated to meet the accuracy requirements of the spacebased detector, the effective-one-body-numerical-relativity model [135,151] can be verified or improved to meet the needs of gravitational wave detection. The metric is the intrinsic geometric quantity of the manifold. The problem of numerically solving the metric can be viewed as a computational geometry problem.

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According to the difference of the signs of the metric, the geometry is divided into Riemannian geometry, Lorentzian geometry, and other geometry. Numerical relativity is a computational Lorentzian geometry problem. The computational Riemannian geometry problem has been developed systematically in computational geometry [183]. In contrast, the mathematical basis for computational Lorentzian geometry is far from being systematic. Driven by the physical problem of gravitational wave astronomy, we hope that the study of computational Lorentzian geometry could have substantial development in the future.

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35

Black-Hole Superradiance: Searching for Ultralight Bosons with Gravitational Waves Richard Brito and Paolo Pani

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black-Hole Superradiance in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superradiant Instabilities in the Presence of Ultralight Fields . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Superradiant Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Superradiant Instabilities for Astrophysical BHs . . . . . . . . . . . . . . . . . . . . . . Bounds on Ultralight Bosons from BH Spin Measurements . . . . . . . . . . . . . . . . . . . . . . . GW Signatures of the Superradiant Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct GW Emission from Boson Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Emission from Level Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GW Bursts from Bosenova Explosions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signatures in Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Under certain circumstances, it is possible to extract angular momentum and energy from spinning bodies in a process known as (rotational) superradiance. This phenomenon is ubiquitous in many physical systems and also occurs for spinning black holes and stars, with far-reaching consequences for gravitationalwave astronomy and searches for new physics with astrophysical compact objects. In particular, ultralight bosonic fields (predicted in a variety of beyondstandard-model scenarios and compelling dark-matter candidates) can trigger superradiant instabilities in spinning black holes, forming macroscopic BoseEinstein condensates around them. Striking features of this phenomenon include

R. Brito · P. Pani () Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1, Rome, Italy e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_37

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gaps in the spin-mass distribution of astrophysical black holes and a continuous gravitational-wave signal emitted by the condensate. We give a concise and coherent overview of the phenomenology of the superradiant instability, summarizing few decades of work in the area as well as recent developments. Electromagnetic and gravitational-wave observations of spinning black holes in the mass range M ∈ (1, 1010 )M can be turned into a bound 10−10 eV/c2  mb  10−22 eV/c2 on the mass of a putative bosonic field. Keywords

Superradiance · Gravitational waves · Black holes · Neutron stars · Axions · Ultralight fields · Dark matter

Introduction In the last few years, few disciplines have experienced a revolution as dramatic and far-reaching as the one that gravitational physics is still undergoing. The advent of gravitational-wave (GW) astronomy, in parallel with its dramatic implications in several areas of astrophysics, is also reshaping tests of gravity, searches of beyondstandard-model physics, and fundamental physics at large [20, 29]. Indeed, a surprising connection between gravity in the strong field regime and particle physics has recently emerged in several contexts [14, 20, 29]. This chapter will be devoted to a specific but prominent example: the possibility to search for ultralight bosons with current [14–16, 37, 38, 90, 110] and future [17, 19, 90, 100, 123, 125] GW detectors and with electromagnetic observations of compact objects. This possibility is allowed by a phenomenon known as superradiant instability, which relies on the fact that, under certain circumstances, spinning bodies can transfer part of their angular momentum to their environment. This extraction of angular momentum is usually called (rotational) superradiance, although the same name had been used in the past for different (albeit related) phenomena, such as amplified coherent emission [60]. In 1971 Zeldovich showed that the scattering of radiation off a rotating absorbing surface results, under certain conditions, in an amplification of low-frequency waves, also suggesting that the same phenomenon should occur for spinning black holes (BHs) [150, 151]. This effect was soon confirmed by Press and Teukolsky with detailed numerical studies [116, 137] and can be considered the wave analog of the Penrose’s process occurring in the ergoregion of a Kerr BH [114]. Interestingly, the first experimental confirmations of superradiant scattering in some mechanical systems have been obtained only recently [52,139]. The interested reader can find a historical overview and a detailed discussion of the many faces of superradiance in the monograph [36]. In the rest of this chapter, we will give a very basic introduction to superradiant scattering off a BH and discuss the key ingredient necessary to understand the superradiant instability triggered by ultralight fields near the spinning BHs. We will then move to discuss the phenomenology of this instability, with particular focus on its GW signatures. We shall use G = c = 1 units throughout.

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Black-Hole Superradiance in a Nutshell Superradiant scattering off a rotating BH occurs for any low-frequency bosonic wave, including of course GWs. Remarkably, the physics of the process does not rely on the details of the Kerr metric and can be better understood by studying a test scalar field propagating on a stationary and axisymmetric (but otherwise generic) spinning BH geometry. By neglecting backreaction on the metric, the Klein-Gordon equation of this background can be written as a single, second-order, radial differential equation: d 2ψ + Veff (r∗ )ψ = 0 , dr∗2

(1)

where ψ is a function describing the radial behavior of the wave, r∗ is a suitable radial coordinate, and the BH horizon (asymptotic infinity) is located at r∗ → −∞ (r∗ → +∞). The potential Veff depends on r∗ , on the background’s curvature and also on the frequency ω and on the azimuthal number m of the wave, both quantities being conserved in a stationary and axisymmetric spacetime. Depending on the potential, an equation in the form (1) is valid also for electromagnetic perturbations and for GWs propagating on the Kerr geometry. The precise form of Veff depends on the details of the perturbation and of the background but is not important for our purposes; it is sufficient to know that the effective potential is real (or can be made such with a field redefinition [58,99]) and that it has the following asymptotic behavior:  Veff →

kH ≡ ω − mΩH r∗ → −∞ , ω r∗ → +∞

(2)

where ΩH is the angular velocity of the BH horizon. In other words, the potential asymptotes to a constant at the boundaries and, owing to the frame-dragging of the spinning spacetime, its asymptotic values are not the same. Let us consider the scattering of a monochromatic wave with frequency ω and azimuthal number m, incident from infinity with amplitude I . The solution to Eq. (1) near the boundaries is a superposition of plane waves, namely:  ψ∼

as r∗ → −∞ , T e−ikH r∗ iωr −iωr ∗ ∗ + Ie as r∗ → +∞ , Re

(3)

where T and R are the transmission and reflection coefficients of the wave, respectively. The presence of a horizon imposes that classical waves at r∗ → −∞ are purely ingoing. Since the effective potential is real, ψ ∗ (the complex conjugate of ψ) is still a solution to Eq. (1), which satisfies the boundary conditions that are the complex conjugate of those in Eq. (3). Furthermore, since the background is

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stationary, the field equations are invariant under the inversions t → −t, ΩH → −ΩH , and ω → −ω, which transform the conjugate boundary conditions into a form analog to Eq. (3). Therefore, the solutions ψ and ψ ∗ satisfy the same differential equation with the same form of boundary conditions and, in addition, dψ ∗ are linearly independent. This implies that their Wronskian, W = dr ψ − ∗ dψ ∗ dr∗ ψ,

is independent of r∗ . By evaluating W near the horizon and near infinity, using Eq. (3) and imposing that the two expressions be equal to each other, we obtain |R|2 = |I |2 −

kH ω − mΩH 2 |T |2 = |I |2 − |T | , ω ω

(4)

independently of the details of the potential in the wave equation. If the background is non-spinning (ΩH = 0), we get |R|2 = |I |2 − |T |2 < |I |2 for any frequency. This is consistent with the standard expectation that the BH horizon absorbs radiation and therefore the reflected energy, |R|2 , is smaller than the incident one, |I |2 . However, in the spinning case, low-frequency waves can satisfy the superradiant condition ω(ω − mΩH ) < 0 ,

(5)

which yields |R|2 > |I |2 , i.e., the reflected energy is larger than the incident one. The energy excess comes from the rotational energy of the BH, which necessarily spins down, as can be shown by computing the backreaction of the perturbations onto the metric [36]. The above argument relies on a frequency-domain analysis and seems to require ingoing boundary conditions at the horizon as a necessary ingredient for superradiant scattering. However, superradiance has been confirmed to occur also in time-domain analyses [10] and can occur also without horizons, for example, in the case of rotating perfect-fluid stars [45, 46, 120], provided the latter spin sufficiently fast as to develop an ergoregion [142], which provides the necessary friction (the frame-dragging in the above example). In order to study superradiant scattering off horizonless geometries, it is more convenient to perform a time-domain analysis. Indeed, in the case of horizonless geometries with an ergoregion, the superradiant scattering produces the so-called “ergoregion instability” [41, 42, 44, 50, 51, 74, 98, 99, 107, 111, 143, 145]. Low-frequency waves are still amplified, but, at the same time, the field grows exponentially in time. Finally, although most studies about BH superradiance focus on a linearized analysis and neglect backreaction, fully nonlinear studies of superradiant scattering [18, 66] confirm that low-frequency radiation does extract mass and spin from the BH. Overall, the nonlinear simulations performed so far agree quantitatively with linear predictions for small wave packet amplitudes. To summarize, although further studies would certainly be interesting, superradiance is confirmed at full nonlinear level for rotating BHs.

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Superradiant Instabilities in the Presence of Ultralight Fields As discussed in the previous section, superradiant amplification requires an ergoregion that “forces” the waves to be dragged along with the compact object, transfering energy and angular momentum from the object to the field. Right after Zeldovich’s work, it had been soon realized that superradiance can also produce an instability if the amplified radiation can be effectively (and coherently) confined near the spinning object [116]. Here, we shall focus on what is arguably the most interesting confining mechanism, namely, a simple mass term for the bosonic field. Indeed, owing to the Yukawa-like suppression, the field mass effectively confines low-frequency waves, therefore turning an energy-extraction mechanism such as superradiance into an instability mechanism [36, 53, 59], sometimes called a “BH bomb” [40, 116]. Although the superradiant instability of spinning BHs triggered by massive bosonic fields has been studied since the 1970s (See Ref. [127] for a mathematically rigorous construction of a superradiant-instability setup.), the topic has recently flourished with numerous developments in the last few years, due to its potential implications for searches of ultralight bosonic fields (such as the QCD axion, axionlike particles, dark photons, etc.), which could be a significant component of the dark matter [14, 70, 87, 101] and are predicted in a multitude of beyond-standardmodel scenarios [70, 87, 89, 91], including extra dimensions and string theories. Indeed, for a boson with mass mb ≡ μh, ¯ the superradiant instability is most effective when the boson’s Compton wavelength is comparable to the BH’s gravitational radius, i.e., when the gravitational coupling α ≡ Mμ = O(0.1), which requires −11

mb ∼ 10



M M

 eV .

(6)

As an order of magnitude, stellar-origin BHs (M ∼ [few, 65]M , the upper limit being set by the pair-instability supernova mass gap) can probe fields roughly with mb ∈ [10−11 , 10−13 ] eV, intermediate mass BHs (M ∼ [100, 104 ]M ) can probe fields with mb ∈ [10−13 , 10−15 ] eV, while supermassive BHs with M > 104 M can probe fields lighter than 10−15 M , down to mb = 10−21 eV, which corresponds to the heaviest BHs known to date (M ∼ 1010 M ). This last region (mb ∼ 10−21 eV) is particularly interesting since such ultralight bosons are also compelling darkmatter candidates [87]. Recent years have witnessed spectacular progress in understanding superradiant instabilities and their phenomenology, both for scalars (i.e., spin-0 fields) [15, 16, 37,38,53,59,61,155] and for vectors (i.e., spin-1 fields) [23,62–65,67,75,112,113, 128, 144] and, more recently, also for tensors (i.e., spin-2 fields) [34, 39]. Many of the qualitative aspects of the instability can be understood in the socalled Newtonian limit, which corresponds to α 1, in which case the field equations can be solved analytically. Overall, there exists a set of quasibound states whose (positive) frequency satisfies the superradiance condition ωR < mΩH

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[see Eq. (5)]. These modes are localized at a distance from the BH, which is governed by the “Bohr radius” ∼1/(μα), and decay exponentially at large distances. In the Newtonian limit, the spectrum of these modes resembles that of the hydrogen atom: ωR /μ ∼ 1 −

(Mμ)2 , 2(l + S + 1 + n)2

(7)

where l is the total angular momentum of the state with spin projections S = −s, −s + 1, . . . , s − 1, s, s being the spin of the field (i.e., s = 0, 1, 2 for scalar, vector, and tensor fields, respectively). To leading order in α and dimensionless BH spin χ ≡ J /M 2 , a detailed matched-asymptotic calculation yields [23, 25, 39, 59] 1 (s) Plm (χ ) 4l+2S+5 ωI = − ClS α (ωR − mΩH ) , 2 Plm (0)

(8)

where Plm (χ ) = (1 + Δ)Δ2l

l 

 1 + 4M 2

q=1



ωR − mΩH qκ

2  (9)

 1 − χ 2, is proportional to the BH absorption probability [130, 131], Δ = (s) Δ κ = 1+Δ , and ClS are constants (see, e.g., Ref. [36]). In the non-spinning case (ΩH = 0), the decay rate of these modes is ωI /μ ∝ −(Mμ)4l+2S+5 . For spinning geometries, ωI changes sign in the superradiant regime. Indeed, when ωR < mΩH , the imaginary part becomes positive and ωI corresponds to the growth rate of the field (τinst ≡ ωI−1 being the instability timescale). According to Eq. (8), the shortest instability timescale occurs for l = 1, S = −1 and for l = 2, S = −2. The only exception to the scaling (7) and (8) is given by the (non-hydrogenic) dipole polar mode of a spin-2 field, which has ωR /μ ≈ 0.72(1 − Mμ) and ωI ∝ (mΩH − ωR )(Mμ)3 [34]. (Since not much is known about the superradiant evolution and subsequent GW emission for this mode, we do not consider its phenomenological implications here. Therefore, the bounds on spin-2 fields described below should be seen as conservative.) Analytical results in the small-coupling limit are in excellent agreement with the exact numerical results [23, 25, 34, 39, 47, 59, 61, 62], although a full solution of the massive spin-2 case beyond the Mμ 1 limit is not yet available.

Evolution of the Superradiant Instability The results presented in the previous sections imply that quantum or classical fluctuations of any massive bosonic field can trigger a superradiant instability of the Kerr metric. The instability timescale τinst can be extremely short when compared

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to typical astrophysical timescales, especially in the presence of ultralight bosons satisfying Eq. (6), and therefore be of relevance for astrophysical BHs. This in turn implies that superradiant instabilities could have huge implications for searches of ultralight bosons, as we will discuss in the next two sections. However, before doing so, it is useful to briefly review how the instability is expected to evolve in realistic astrophysical systems. Consider a massive boson propagating in a BH spacetime, ignoring for the moment any coupling with other particles as well as possible self-interactions. To understand how the instability develops, one should first note that there are various timescales involved in this problem. As we just discussed, around a rotating BH bound states with oscillation frequency ωR , satisfying ωR < mΩH , will spontaneously grow by extracting angular momentum from the BH, exhibiting exponential growth with an e-folding timescale τinst M. This growth should continue until the BH spins down to a value such that superradiance shuts off, i.e., ωR ∼ mΩH , leading to the formation of a “cloud” of bosons corotating with the BH [14, 35]. If the bosonic field is real, the presence of a non-axisymmetric, oscillating boson cloud anchored to a rotating BH leads to the emission of GWs with frequency ∼ωR /π (see the next section). Due to the loss of energy in the form of GWs, the cloud slowly decays on a typical timescale τGW [13–15,35,37,64,65,147]. (Our discussion also applies to complex bosonic fields; however, for this case, GW emission is suppressed, and the evolution of the instability can give rise to rotating BHs with (metastable) complex bosonic hair [65, 82, 83].) Following the development of the instability in fully nonlinear simulations is extremely challenging because the timescales involved in the problem typically satisfy M τinst τGW [147]. However, this also makes the problem ideal for a quasi-adiabatic evolution [35]. In addition, since the superradiant instability is most effective when the gravitational coupling constant Mμ  O(0.1) [cf. Eq. (6)], the Bohr radius of the field satisfies ∼M/(Mμ)2 M, i.e., the bosonic cloud typically extends well beyond the horizon. Therefore, perturbations to the Kerr geometry are always small throughout the whole process [35]. Thus, we can assume that the BHcloud system moves through a sequence of Kerr spacetimes with decreasing mass M and angular momentum J , with the cloud being described by the hydrogen-like eigenstates of the Kerr background (see the previous section). During the instability phase, GW emission can be neglected since τinst τGW and the amplitude of the field increases until ΩH (Mf , Jf ) = ωR /m, where Mf and Jf are the BH parameters at the end of the instability phase, and the BH horizon frequency is given by ΩH (M, J ) = J /[2M(M 2 + (M 4 − J 2 )1/2 ]. The BH spin at the end of this phase can then be estimated to be Jf ≈

4mMf3 μ m2 + 4Mf2 μ2

,

(10)

where we used the fact that ωR ∼ μ [cf. Eq. (7)] in the small gravitational coupling limit Mμ 1. Assuming no other processes take place during the instability

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timescale, such as accretion of surrounding material, the mass of the cloud at the end of the superradiant growth is given by Mcsat = Mi − Mf = −δM,

δM =

ωR δJ m

(11)

where Mi is the initial BH mass and δJ = Jf − Ji is the variation of the BH angular momentum (δJ < 0 whenever superradiance occurs, and therefore, Mcsat > 0). Depending on the system’s parameters, the instability can extract up to ∼10% of the original BH’s mass [65, 84]. After the instability shuts off, GW emission from the cloud takes over and the cloud slowly decays. The GW power is proportional to the square of the cloud’s energy, E˙ GW ∝ Mc2 [15, 35, 147], and therefore, one finds that, after the BH spin reaches (10), the cloud starts decaying as [15] Mc (t) =

Mcsat , 1 + t/τGW

τGW =

Mcsat , E˙ GW (t = 0)

(12)

where here we define t = 0 as the time at which the cloud reaches the saturation mass Mcsat . For small couplings Mμ, the emitted GW flux for the l multipole and spin polarization S can be shown to scale as [23, 35, 39, 73, 128, 147] E˙ GW ≈ AlmS



Mc M

2 (μM)4l+4S+10 ,

(13)

where the exact value of the proportionality constants AlmS depends on the spin of the boson. This scaling is valid for small Mμ  0.1, whereas the exact value for the GW flux can be computed numerically by solving the Teukolsky equation [37, 68, 128, 147]. The simple picture we just gave has been confirmed through fully nonlinear simulations of the instability for massive vector fields [64, 65], although due to the long timescales involved in the problem, those simulations were only able to follow the evolution of the most unstable mode. However, from the quasi-adiabatic picture we described, we expect the same process to effectively occur for all unstable modes, although, during the finite lifetime of a BH, only the most unstable modes grow through superradiance.

Evolution of Superradiant Instabilities for Astrophysical BHs In a real astrophysical environment, BHs will also be surrounded by matter fields in the form of gas and plasma that can form accretion disks around BHs. Addition of mass and angular momentum to the BH via accretion competes with superradiant extraction but can also make a slowly rotating BH, which does not satisfy the superradiance condition, superradiantly unstable precisely because of

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angular momentum accretion. Accretion can also be important for light BHs, whose coupling parameter μM is initially too small for superradiance to play an important role, by increasing its mass until μM grows to a value where superradiance becomes important. The inclusion of accretion when trying to understand the evolution of the superradiant instability in an astrophysical scenario is therefore crucial. This can be done in the same quasi-adiabatic scheme described above as was done in Ref. [35]. In the quasi-adiabatic approximation, the evolution of the cloud is governed by a simple set of differential equations [35]. Energy and angular momentum conservation requires that M˙ + M˙ c = −E˙ GW + M˙ ACC ,

(14)

J˙ + J˙c = −J˙GW + J˙ACC ,

(15)

where Mc and Jc are the mass and angular momentum of the boson cloud, M˙ ACC and J˙ACC represent mass and angular momentum accretion, and E˙ GW and J˙GW = (m/ωR )E˙ GW are the energy and angular momentum lost due to GW emission. The system can be closed by two further equations: M˙ = −E˙ S + M˙ ACC ,

(16)

J˙ = −J˙S + J˙ACC ,

(17)

which describe the superradiant extraction of energy (E˙ S ) and angular momentum (J˙S = (m/ωR )E˙ S ) and the competitive effect of gas accretion. The energy flux extracted from the horizon through superradiance is given by E˙ S = 2Mc ωI ,

(18)

where ωI is given by Eq. (8) in the small gravitational coupling limit. In order to include accretion, Ref. [35] considered the simple model in which mass accretion occurs at a fraction of the Eddington rate (see, e.g., [21]): M(t) M yr−1 , M˙ ACC ≡ fEdd M˙ Edd ∼ 0.02fEdd 6 10 M

(19)

where one assumes an average value of the radiative efficiency η ≈ 0.1, as required by Soltan-type arguments [96, 129]. The Eddington ratio for mass accretion, fEdd , depends on the details of the accretion disk surrounding the BH. Typical values can range from order unity (or even larger for ultraluminous sources) for quasars and active galactic nuclei, from much smaller values for quiescent galactic nuclei (e.g., fEdd ∼ 10−9 for SgrA∗ ). For the evolution of the BH angular momentum through accretion, Ref. [35] made the conservative assumption that the disk lies on the equatorial plane and extends down to the innermost stable circular orbit (ISCO). Within this model, the evolution equation for the spin reads [22]

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L(M, J ) ˙ J˙ACC ≡ MACC , E(M, J )

(20)

√

√ where L(M, J ) = 2M/(3 3) 1 + 2 3rISCO /M − 2 and E(M, J ) = √ 1 − 2M/3rISCO are the angular momentum and energy per unit mass, respectively, of the ISCO of the Kerr metric, located at rISCO = rISCO (M, J ) in Boyer-Lindquist coordinates. In the absence of superradiance, Eqs. (19) and (20) predict that the BH mass and spin grow exponentially with an e-folding time given by a fraction 1/fEdd of the Salpeter timescale τSalpeter ∼ 4.5 × 107 yr. Therefore, the minimum timescale for accretion to be important is roughly given by τACC ∼ τSalpeter /fEdd . Let us now consider the evolution of the instability under this system of equations, for a massive scalar field and considering only the most unstable mode (l, m, n) = (1, 1, 0) [35]. Two representative cases, with different initial conditions for the BH mass and spin, are shown in Fig. 1 for a scalar field with mass mb = 10−18 eV and considering fEdd = 0.1 for the accretion rate. The two cases represent two different scenarios, one (left panel) in which superradiance is initially negligible because the gravitational coupling μM0 ∼ 10−4 is very small and the instability timescale τinst τACC ∼ 10 τSalpeter , and another scenario (right panel) where the BH starts already in a regime where τinst τACC . For the first scenario, the mass and spin of the BH grow through gas accretion, reaching extremality (J /M 2 ∼ 0.998) within the timescale τACC . Due to the mass growth after t ∼ 6 Gyr, the coupling μM becomes sufficiently large for superradiance to dominate the dynamics. At this point, the BH enters the region delimited by the dashed blue curve in Fig. 1, which corresponds to the region where superradiance dominates τinst < τACC , and the scalar cloud starts growing exponentially (left bottom panel), while the BH mass and angular momentum decrease (left top panel) until the condition (10) is satisfied. After this point, the scalar cloud dissipates through the emission of GWs, as can be seen in the left bottom panel of Fig. 1, whereas due to gas accretion the BH mass and spin start increasing again. However, because accretion restarts in a region in which μM is nonnegligible, the BH follows the “Regge trajectory” where ΩH ∼ ωR [13]. Assuming that accretion continues uninterrupted and neglecting modes with m > 1, the evolution continues until the spin reaches the critical value J /M 2 ∼ 0.998 and angular momentum accretion saturates. The evolution of the second case (right panel of Fig. 1) follows a similar pattern; however, because in this case the BH starts already in the instability regime, its spin grows only very little before superradiance becomes dominant, and the BH angular momentum is extracted after only 10 Myr. After superradiant extraction, the BH evolution follows the Regge trajectory, similarly to the first case. Thus, accretion can play an important role in the overall evolution of the BH mass and spin, whereas GW emission from the cloud is responsible for the decay of the boson condensate, but it is too weak to affect the evolution of the BH mass and spin significantly.

9

3×10

2

6×10

t [yr]

MS(t)/M(t)

critical

J(t)/M(t)

9

9

6

9×10 10

6Gyr

0.2

0.4

0.6

0.8

1.0

0.0 8 10

-1

9.5Gyr

eV, fEdd=10

-18

M/MO.

7

10

µ=10

6.8Gyr

6.7Gyr

6.6Gyr

0.00 5 10

0.05

0.10

0.15

0.0

0.2

0.4

0.6

0.8

1.0

2

10

6

7

t [yr]

10

MS(t)/M(t)

J(t)/M(t)

critical

10

8

9

10

7

M0=10 MO.

0.2

0.4

0.6

0.8

1.0

0.0 8 10

-1

eV, fEdd=10

0.5Gyr

-18

M/MO.

7

µ=10

20Myr

10Myr

10

t=0

1Gyr

J/M

2

J/M

2

Fig. 1 Evolution of the BH mass and spin and of the scalar cloud mass for a scalar field with mb = 10−18 eV, including the effect of superradiance, accretion of gas, and emission of GWs. Two different sets of initial conditions for the BH mass and spin are considered. Left: the initial BH mass is M0 = 104 M and the initial BH spin J0 /M02 = 0.5. In this case the BH enters the instability region after t ∼ 6 Gyr, when its mass M ∼ 107 M and its spin is quasi-extremal. Right: for this case M0 = 107 M and J0 /M02 = 0.8, such that the evolution starts already in the instability region for this scalar mass. For both cases, the left top panels show the dimensionless angular momentum J /M 2 and the critical superradiant threshold; the left bottom panels show the mass of the scalar cloud MS /M. The right panels show the trajectory of the BH in the Regge plane [13] during the evolution. The dashed blue line marks the threshold at which τ ∼ τACC . (From Ref. [35])

0.00

0.05

0.10

0.15

0.0

0.2

0.4

0.6

0.8

1.0

4

M0=10 MO.

35 Black-Hole Superradiance: Searching for Ultralight Bosons with. . . 1387

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Since the details of the evolution depend on the initial BH mass and spin, on the boson mass μ, and on the accretion rate, an obvious question is whether these results can be used to impose bounds on the existence of light bosons, as first proposed in Refs. [13, 14]. In particular, given the measurements of the mass and spin of an old BH, would these measurements be compatible with the evolution depicted in Fig. 1? This problem was addressed in Ref. [35] and is summarized in Fig. 2, which shows the final BH mass and spin in the Regge plane for N = 103 Monte Carlo evolutions of the instability, considering a scalar field with mass mb = 10−18 eV and different values of accretion rates fEdd . These results show that, independently on the accretion rate, a depleted region (a “hole”) in the Regge plane, where no old BHs can be found, always forms. Similar results would apply for different values of μ, as long as we consider BHs with masses around Eq. (6). This confirms that observations of massive BHs with various masses can be used to rule out a range of masses mb for the boson field, as discussed in Ref. [13, 14, 34, 112] and as we discuss below. The results also show that the holes in the Regge planes are slightly smaller with what naively predicted by the relation τinst ≈ τACC and, in fact, can be shown to be well approximated by [35, 73]. J  4μM M2

 ∪

M

96 μ10 τACC

1/9 .

(21)

These boundaries correspond to the threshold value J /M 2 (cf. Eq. (10), in the limit Mμ 1) for superradiance to occur and to a BH mass, which minimizes the spin for which τinst ≈ τACC , for a given μ [113]. Finally, we note that although the instability is strongly suppressed for higher multipoles and they are not included in the evolutions that we just discussed, the first few (l, m) modes (and not only the dipole with l = m = 1) can contribute to the depleted region in the Regge plane [13]. Because the superradiance condition depends on the azimuthal number m, for certain parameters it might occur that the modes with l = m = 1 are stable, whereas the modes with l = m = 2 are unstable, possibly with a superradiant extraction stronger than accretion. When this is the case, the depleted region of the Regge plane is the union of various holes [13, 14]. Typically, the presence of several modes does not alter the qualitative picture described above, unless the initial amplitude of some modes is large [73].

Bounds on Ultralight Bosons from BH Spin Measurements The results we just discussed show that a very generic and solid prediction of BH superradiant instabilities is the existence of holes in the Regge plane. Using estimates for the instability timescale, together with reliable spin measurements for BHs in various mass ranges, one can impose stringent constraints on the allowed mass of ultralight bosons [13–16,23,34,37,47,55,112,133,134]. In principle this can also be turned around: by comparing this prediction with the statistical distribution

35 Black-Hole Superradiance: Searching for Ultralight Bosons with. . .

J/M

2

J/M

2

J/M

2

4

10 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0

4

10

5

10

6

7

10

10

1389 8

9

10

10

-4

fEdd=10

fEdd=0.01

-18

10

eV fEdd=0.02

5

10

6

7

10

10

8

10

9

10

M/MO. Fig. 2 The final BH mass and spin in the Regge plane for N = 103 evolutions, considering a massive scalar field with mass mb = 10−18 eV and BHs with initial masses and spins randomly distributed between log10 M0 ∈ [4, 7.5] and J0 /M02 ∈ [0.001, 0.99]. The final BH mass and spin are extracted at t = tF , with tF following a Gaussian distribution centered around t¯F ∼ 2 × 109 yr with width σ = 0.1t¯F . The dashed blue line is the boundary where the superradiant instability timescale equals the accretion timescale, τinst ≈ τSalpeter /fEdd , whereas the solid green line denotes the region defined through Eq. (21). The experimental points with corresponding error bars refer to the supermassive BHs listed in Ref. [33]. (From Ref. [35])

of BHs in the Regge plane, one could possibly detect and measure the properties of ultralight bosons [16, 37, 71, 109]. A more detailed analysis of how constraints on the boson mass depend on the mass and spin of astrophysical BHs is shown in Fig. 3, where we consider spin0, spin-1, and spin-2 bosonic fields (from top to bottom). For each case, we plot contours corresponding to an instability timescale of the order of the Salpeter time for different masses of the bosonic field and considering the unstable modes with the smallest instability timescales. It is clear from Fig. 3 that precise BH spin measurements of astrophysical BHs in the range M  M  1010 M can exclude a considerable range of boson masses, with constraints generically stronger for bosonic fields with spin, because the instability timescales are typically shorter for those cases. The black data points in Fig. 3 denote estimates of stellar or supermassive BH spins obtained using either the Kα iron line or the continuum fitting method [33, 105], whereas red data points are the 90% confidence levels for the spins of the

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Fig. 3 Exclusion regions in the BH spin-mass diagram obtained from the superradiant instability of Kerr BHs against massive bosonic fields for the two most unstable modes (updated version of the exclusion plot shown in Ref. [36]). The top, middle, and bottom panels refer to scalar (spin0), vector (spin-1), and tensor (spin-2) fields, respectively. For each mass of the field (reported in units of eV), the separatrix corresponds to an instability timescale equal to the Salpeter time τSalpeter = 2×107 yr . Black data points denote electromagnetic estimates of stellar or supermassive BH spins obtained using either the Kα iron line or the continuum fitting method [33,105], whereas red data points are LIGO-Virgo 90% confidence levels for the spins of the primary and secondary BHs in the merger events detected to date [4–7], including the recent data from the second GW transient catalogue [8]. The range of the projected LISA measurements using three different population models for supermassive BH growth (popIII, Q3, and Q3-nod from [93]) is denoted by arrows. Green points are the 90% confidence levels for the mass-spin of a selection of the GW coalescence remnants [4–8] (which cannot be directly used to constrain the Regge plane but can identify targets of merger follow-up searches [15, 16, 23, 77, 90]). More direct (albeit slightly less stringent) constraints would come from comparing the instability timescale against the baseline (typically O (10 yr)) during which the spin of certain BH candidates is measured to be constant [47], as it is the case for LMC X-3 [132] and Cyg X-1 [78], shown by blue points

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primary and secondary BHs in the merger events detected by LIGO-Virgo in the first three observing runs [4–8]. Although LIGO-Virgo measurements are still not precise enough to give strong constraints, by combining information from multiple detections and with the expected improvements from future ground-based detectors [100, 123], constraints coming from the observation of merging stellar-mass BHs will improve significantly [16, 23, 109]. LISA, a future space-based GW detector, will complement these constraints by measuring BH masses and spins with a very good precision [93] in a range of masses falling right in between the ones currently coming from ground-based GW detectors and electromagnetic observations [37]. The range of possible BH masses that LISA will be able to observe is shown in Fig. 3 by arrows, for three representative population models for supermassive BH growth (popIII, Q3, and Q3-nod from [93]). Finally, the green points in Fig. 3 show the 90% confidence levels for the mass-spin of a selection of the GW coalescence remnants [4–8]. These measurements cannot be used to directly impose constraints on bosons, since they correspond to BHs that were just formed, but they identify possible targets for GW follow-up searches [15,16,23,77,90] (see the next section). We should note that instead of using the Salpeter time as a reference timescale, we can impose more direct constraints from comparing τinst against the baseline timescale during which the spin of BH candidates is measured to be constant [47] (typically O(10 yr)). This is the case for LMC X-3 [132] and Cyg X-1 [78], shown in the panels of Fig. 3 by blue points, that can be used to confidently exclude the range mb ∈ (10−11 , 10−13 ). In summary, by combining current and future BH observations in a wide range of BH masses, one will be able to constrain approximately the entire range: 10−21 eV  mb  10−10 eV .

(22)

where we assumed that spinning BHs exist in the entire mass range M  M  1010 M . The best upper bound would come from the lightest massive BH (M ≈ 5M [105] or even smaller if the lightest compact object in GW190814 is a BH [7]), whereas the best lower bound would come from the heaviest supermassive BHs for which spin measurements are reliable, such as the BH candidate Fairall 9 [124] or even the supermassive BH in M87∗ (single gray point in Fig. 3), if the claims that this BH has a large spin are confirmed [9, 55, 136]. Even more stringent bounds could be obtained, if the largest known supermassive BHs with M  2 × 1010 M [103, 104] were confirmed to have nonzero spin.

GW Signatures of the Superradiant Instability As mentioned in the previous section, the formation of a nonspherical oscillating boson cloud anchored to a spinning BH leads to the emission of GWs. Thus, a very unique signature from superradiant instabilities would be the direct detection of the GWs emitted by these sources. In fact, in the last few years, it has become clear that such systems are very promising GW sources for ground and space-based

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GW detectors [13–16, 23, 35, 37–39, 63, 64, 77, 90, 128, 147, 148, 154]. Furthermore, when BH-cloud systems are part of a binary, a wealth of novel effects can occur that lead to very distinct and potentially detectable signatures in the GWs emitted by the coalescing binary [11, 12, 24, 26, 28, 48, 72, 79, 152, 153]. Therefore, the detection of GWs provides an exciting new tool to search for ultralight bosons. Below we briefly review the different GW signatures through which one could detect or constrain ultralight bosons.

Direct GW Emission from Boson Clouds There are various mechanisms by which the boson cloud can emit gravitational radiation. Consider first the evolution discussed in the previous section, for a real massive bosonic field, where a single unstable mode dominates. (For a similar multimode analysis, see Ref. [73].) Since the stress-energy tensor is quadratic in the bosonic fields, it immediately follows that the GWs emitted by the cloud have an angular frequency ωGW = 2ωR , and the angular dependence has spheroidal harmonic components with azimuthal number m ˜ = ±2m, and are dominated by the ˜ = 2l contribution, although higher ˜ are also emitted and can be important in the relativistic regime [128, 147]. In a particle-like description, such GWs can be interpreted as arising from the annihilation of pairs of bosonic particles to produce gravitons [13, 15]. For example, considering spin-0 and spin-1 fields, the most unstable mode corresponds to the dipole mode l = m = 1 [cf. (8)], and the emitted GWs are mostly quadrupolar with ˜ = m ˜ = 2. On the other hand, for the spin-2 case, GW emission from the dominant hydrogenic mode is mostly hexadecapolar ˜ = m ˜ =4 because the signal is produced by a spinning quadrupolar field [39]. Since, in the small gravitational coupling limit Mμ 1, we have ωR ∼ μ [cf. (7)], this radiation is nearly monochromatic with frequency  fGW ∼ ωR /π ∼ 5 kHz

μh¯ 10−11 eV

 .

(23)

Due to relativistic effects and the small decay of the cloud, this frequency slowly changes over time although always staying very close to fGW ∼ μ/π [15, 23, 90, 154]. From Eq. (6), it also follows that ground-based detectors would be sensitive to the presence of bosonic clouds around stellar-mass BHs, whereas space-based detectors are sensitive to signatures of clouds around supermassive BHs (see Fig. 4 below). For a boson cloud formed solely through the superradiant instability, the backreaction of the boson field onto the spacetime can be shown to be small [35, 64, 65], and therefore, the GW emission process can be accurately computed using the Teukolsky equation, with the stress-energy tensor of the boson field acting as a source [37, 68, 128, 147]. In the Mμ 1 limit, the emitted GW flux scales as in Eq. (13), whereas the exact results valid for any value of Mμ and of the BH spin

35 Black-Hole Superradiance: Searching for Ultralight Bosons with. . .

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can be computed numerically, as was done in Refs. [37,68,147] for scalar fields and in Ref. [128] for vector fields. Since the dominant unstable modes for vector and tensor fields satisfy l = m = −S = 1 and l = m = −S = 2, respectively, one can infer from Eq. (13) that, for a fixed value of Mμ, the GW power for vector and tensor fields can be significantly larger when compared to the scalar field case, for which the most unstable mode occurs for l = m = 1, S = 0. This also implies that the typical half-life time τGW of the signal [cf. Eq. (12)] for the dominant mode is typically shorter for bosons with spin [15, 23, 37, 39, 128]:     0.1 15 0.5 M , ≈ 1.3 × 10 yr 10 M MμS χ i − χf  11   0.1 0.5 M V ,T τGW , ≈ 2 days 10 M MμV ,T χ i − χf 

5

S τGW

(24) (25)

where those expressions are valid in the limit Mμ 1; the subscripts S, V , T refer to the scalar, vector, and tensor case, respectively; and χi (χf ) is the dimensionless BH spin at the beginning (end) of the superradiant growth. Therefore, much like pulsars for LIGO or verification white dwarf binaries for LISA, BH-boson systems are continuous GW sources. However, depending on the value of Mμ, the GW emission timescale τGW for vectors and tensors can be significantly shorter than the observation time, resulting in an impulsive signal. In the Mμ 1 limit and considering only the dominant unstable hydrogenic modes, the GW strain amplitude at its peak (using the definition in [154]) can be approximated by [15, 23, 35, 37, 39, 128, 147]:      χ i − χf M MμS 7 Mpc h ≈ 5 × 10 , 10M 0.1 r 0.5       χ i − χf M MμV ,T 5 Mpc hV ,T ≈ 10−23 , 10M 0.1 r 0.5 S

−27



(26) (27)

for a source at luminosity distance r. Although strictly valid only when Mμ 1, these expressions give a reasonably good order of magnitude estimate also when Mμ  0.1 (see, e.g., Ref. [90] for an explicit comparison against numerical results for the scalar case). Since h ∝ Mc , as the cloud slowly dissipates through GW emission, the signal strength decreases from its peak following Eq. (12). To estimate the detectability of these sources for different GW detectors, it is useful to define the characteristic strain amplitude of the GW signal as [106] hc =

 Ncycles h ,

(28)

where the approximate number of GW cycles in a given detector’s frequency band can be estimated as Ncycles ∼ min[f Tobs , fGW τGW ] for a signal with detector’s

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frame frequency f = fGW /(1 + z) and considering an observation time Tobs for a source located at cosmological redshift z. In Fig. 4 we show the characteristic strain for χi = 0.9 and several values of the boson mass and cosmological redshift. Advanced LIGO is especially sensitive to bosons with masses ∼5 × [10−14 , 10−12 ] eV, whereas LISA will be mostly sensitive to bosons in the range ∼[10−19 , 10−15 ] eV. The small gap around mb ∼ 10−14 eV could be filled by planned third-generation ground-based detectors [90,100,123], such as the Einstein Telescope (ET) [85] and Cosmic Explorer [2, 69], or by decihertz detectors [125], such as DECIGO [92]. Quite interestingly, LISA and third-generation ground-based detectors have the potential to detect sources at very high cosmological redshifts, especially for vector and tensor fields [39]. This is better illustrated in the “waterfall” [17] plot in Fig. 5, where we show the typical angle-averaged signal-to-noise ratio for sources detected by ET and LISA, corresponding to BHs with masses in the range ∼[10, 104 ] M and ∼[104 , 109 ] M , respectively. For vector and tensor fields, the continuous GW signal could be detected even when z ≈ 20 or higher, if bosons in the right mass range exist, although the exact detection horizons will depend on the sensitivity of the search methods to these type signals [90]. In practice, the range of potentially detectable boson masses through direct GW detections depend on the formation rate of spinning BHs. This was studied in Refs. [15, 37, 38, 154] for scalar bosons and in [23] for vector bosons in the context of allsky searches with Advanced LIGO and with LISA. Depending on the boson mass (and to a lesser degree its spin) and the astrophysical BH population, up to O(104 ) signals could be detected, assuming that boson fields with masses in the optimal range exist. A detailed study of the ensemble of GW signals emitted by a population of isolated galactic BHs was recently done in [154], where it was shown that, with the sensitivity of the latest all-sky searches for continuous GWs [3, 56, 57, 110], a large number of signals could have already been detected, assuming realistic stellarmass BH population models. In principle, these null results already disfavor the existence of scalar bosons with masses in the range [2, 25] × 10−13 eV [110, 154]. However, as pointed out in [154], current methods do not take into account the fact that numerous signals could be simultaneously present in the data, in a very narrow range of frequencies, which could impact the sensitivity of these searches. Therefore, those constraints should be reassessed by properly taking the account the overall expected GW signal from the population of galactic BHs. Aside from blind all-sky searches, another promising avenue would be to perform targeted follow-up searches of known binary BH mergers remnants [16, 23, 77, 90] that could form a boson cloud after the merger. Such searches are especially promising for third-generation GW detectors [16, 23, 77, 90]. Another exciting prospect, put forward in Ref. [108], would be to do such targeted searches at binary stellar-mass BHs that would be detected in the LISA band in their early inspiral phase. Given that LISA would be able to measure the sky location of those systems with very good precision, such sources would be ideal targets for future groundbased GW detectors.

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Fig. 4 GW characteristic strain (thin lines), as defined by Eq. (28), for annihilation signals from scalar (spin-0), vector (spin-1), and tensor (spin-2) condensates (top, middle, and bottom, respectively). The GW strain amplitude is computed at its peak value, and we consider the emission from the dominant unstable mode, an initial BH spin χi = 0.9, and an observation time Tobs = 4 yr. Each (nearly vertical) line shows the characteristic strain for a given boson mass mb , computed at redshift z ∈ (0.001, 10) (from right to left, in steps of δz = 0.3), with Mμ/m varying in the superradiant range (0, MΩH ) along each line. Different colors correspond to different boson √ masses mb . For comparison we also show the characteristic noise strain amplitude (defined as f Sn (f ), with Sn (f ) the noise power spectral density) of Advanced LIGO at design sensitivity [1] and the sky-averaged characteristic noise strain of LISA [17, 121] (black thick curves). Thick colored lines show the stochastic GW background produced by the population of extragalactic BH-boson systems under optimistic assumptions [37, 38]. The characteristic noise strain of DECIGO [92] and the Einstein Telescope (ET) [85] (dashed lines) are also shown for reference. (See Refs. [38, 39] for more details)

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5

-12

-13

-14

-15

-16

-17

-18

-19

spin-0

4

z

3 2 1

SNR

0 20

1000

spin-1

500

z

15

200

10

100

5

50

0 20

20 spin-2

10

z

15 10 5 0

0

1

2

3

4 5 6 log10(M/M)

7

8

9

Fig. 5 Contours of constant angle-averaged signal-to-noise ratio (SNR) for annihilation GW signals detected with ET and LISA, for scalar (top), vector (middle), and tensor (bottom) condensates, as a function of the source-frame BH mass and cosmological redshift. We considered Mμ = 0.2, χi = 0.9, and an observation time Tobs = 4yr. The sources detectable by ET and LISA correspond to BHs with masses in the range ∼[10, 104 ]√ M and ∼[104 , 109 ] M , respectively. For simplicity we estimated the SNR using SNR ∝ hc / f Sn (f ) (with a proportionality factor taking into account the angle-averaged signal and triangular shape of LISA and ET, see, e.g., [27, 117]) and neglected the possible confusion noise from the stochastic GW background (see Fig. 4). We computed the GW amplitude and duration time using analytical estimates for the GW flux [see Eq. (13)] and assumed that the signal is detected starting from its peak amplitude. Notice the different scale for the redshift in the spin-0 case

In addition to GW sources that could be detected individually, one expects the existence of an even larger number of sources too faint to be extracted from the noise. The incoherent superposition of all those sources produces a stochastic GW background that was first computed in [37, 38] for extragalactic isolated BHs and

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considering only the most unstable mode (see also Ref. [154] for a characterization of the GW signal emitted by galactic sources as a whole). The background from extragalactic sources can be characterized in terms of its characteristic strain [138], which we show in Fig. 4 (thick solid curves) for the most optimistic astrophysical models of Refs. [37, 38]. The maximum frequency of the background is determined by the boson mass, whereas its amplitude is mostly determined by the astrophysical population of BHs, in particular by the BH spin distribution and to a smaller degree by the BH mass distribution (see Refs. [37, 38, 140, 141] for details). In the most optimistic scenario, this background could be detectable by LIGO at design sensitivity for bosons in the range ∼[10−13 , 10−12 ] eV, whereas LISA would be sensitive to bosons in the range ∼5 × [10−19 , 10−16 ] eV. Interestingly, the background could also be strong enough to produce a “confusion noise,” which could complicate the detection of individual sources. In the absence of a detection, this can be used to constrain the existence of such ultralight fields. Dedicated pipelines to search for individual GW signals from these sources in LIGO data are actively being developed and implemented for both blind all-sky [54, 56, 57, 110, 154] and targeted searches [90, 135], but no GW signal consistent with a boson annihilation has been detected so far. In addition, dedicated searches for the stochastic background in Advanced LIGO’s first observing runs found no evidence for such signal [140, 141], allowing them to exclude the range (95% credibility) 2.0×10−13 eV ≤ mb ≤ 3.8×10−13 eV for scalar fields [140] and 0.8×10−13 eV ≤ mb ≤ 6.0 × 10−13 eV for vector fields [141], under optimistic assumptions about the BH population.

GW Emission from Level Transitions GW emission can also happen through a second channel for cases in which modes with the same harmonic numbers (l, m) have overtones n > 0 with a larger growth rate than the fundamental n = 0 mode. This can happen for sufficiently high l [13–15, 128, 149]. In those cases, the overtone mode reaches saturation before the fundamental mode. However, because the fundamental mode is still unstable when this happens, it keeps growing, leading to the emission of a unique GW signal with multiple frequency components and a peculiar low-frequency beating pattern [64, 128, 144]. In particle physics terms, the beating component can be understood as arising due to level transitions between two states, similarly to photon emission through atomic transitions. The frequency of the emitted gravitational radiation is given by the frequency difference between the excited (n˜ e > n˜ g ) state and the ground state n˜ g , where n˜ denotes the principal quantum number n˜ = l + S + n + 1:

ωtrans

μ ∼ (Mμ)2 2



1 1 − 2 n˜ 2g n˜ e

 .

(29)

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Since the wavelength of this signal is usually much longer than the size of the system, the GW power emitted during the process can be computed using the quadrupole formula [13]. In particular, for the dominant transition in the scalar field case (i.e., a transition between l = m = 4, n = 1 and l = m = 4, n = 0 [15]) one finds [13, 15] E˙ GW ∼ O(10−6 ÷ 10−8 )

Mg Me (μM)8 , M2

(30)

where Mg and Me are the total mass in the ground and excited states, respectively. Using Eq. (29) one obtains that the typical frequency for the dominant transition for a scalar field is given by  ftrans ∼ 13 Hz

μS h¯ 10−11 eV

3 

M 5M

 ,

(31)

which falls in the sensitivity band of ground-based detectors for a boson with mass about 10−11 eV around a stellar-mass BH with mass M ∼ 5M , whereas, for a boson with mass about 10−15 eV around a supermassive BH with M ∼ 105 M , it would fall in the LISA band. Since this signal only occurs for subdominant (l, m) modes, prospects to observe such signals are much less optimistic than the “annihilation” GW signals we discussed above [15]. Nonetheless, a detection of transition GW signals could provide a unique opportunity to measure the spectral properties of boson clouds, much like measuring the emission spectrum of chemical elements.

GW Bursts from Bosenova Explosions The picture we gave so far mainly assumes a minimally coupled, massive bosonic field evolving around an isolated rotating BH. These fields might, however, have strong self-interactions or be coupled to other fields. In such cases one might expect that the evolution of the superradiant instability and the phenomenology associated with the process will be different when nonlinearities become important. For example, axionic couplings to photons can suppress the superradiant instability of massive scalar fields for sufficiently large couplings [32, 88, 122, 126], which would in turn affect the GW signal emitted by the cloud. Similarly, self-interaction terms can also affect the instability and lead to interesting signatures [15, 94, 146]. Consider, for example, the Lagrangian for an axion-like field Ψ minimally coupled to gravity with a periodic potential: L =

R 1 − g μν ∂μ Ψ ∂ν Ψ − U (Ψ ) , 16π 2

(32)

35 Black-Hole Superradiance: Searching for Ultralight Bosons with. . .

U (Ψ ) =

fa2 μ2

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   Ψ 1 − cos , fa

(33)

where the potential is parameterized by the mass μ term and the decay constant fa [94, 101, 146]. The axion decay constant fa depends on the model; in some models it can be of the order of the GUT scale, fa ≈ 1016 GeV, or it might explicitly depend on the mass μ, such as in the QCD axion case [101]. When Ψ/fa 1, the potential can be Taylor expanded, and one finds that the leading-order term is the usual potential for a minimally coupled massive scalar field, U ∼ μ2 Ψ 2 /2. Therefore, initially the superradiant instability proceeds as explained in the previous section. However, when Ψ ∼ fa , the Taylor expansion breaks down and new effects are expected to occur. In particular, nonlinear interactions might excite modes carrying angular momentum down the horizon and off the cloud, slowing or even suppressing the superradiant growth. Simulations of this process were done in Refs. [94, 146], where it was shown that, whenever nonlinear interactions are important, a “bosenova” can occur where a fraction of the cloud’s energy is absorbed by the BH, whereas the rest is emitted in the form of a GW burst. Since this reduces the size of the cloud, nonlinearities become weaker after each bosenova, allowing the cloud to be replenished again through superradiance until the next bosenova possibly occurs, with the process repeating itself until all the available BH spin is exhausted. Thus, at variance with the GW signals we discussed so far, the gravitational radiation expected from bosenova explosions is a periodic emission of bursts, whose separation depends on the superradiant instability timescale, the decay constant, and on the fraction of the cloud, which remains bound to the BH after each bosenova explosion. The typical frequency of a single bosenova burst was shown to be given by [15, 94, 146]: 

fbn

16rcloud ∼ 30 Hz tbn



μM 0.4l

2 

10M M

 ,

(34)

where tbn is the infall time and rcloud ∝ 1/(Mμ2 ) is the characteristic size of the cloud, with a GW strain given by [15, 94, 146]:

h ∼ 10−21



       kpc   16rcloud 2 μa M fa 2 M , d 0.05 tbn 0.4l 10M famax

(35)

with  is the fraction of the cloud falling into the BH (simulations indicate typical values  ≈ 5% [146]) and facrit is the critical coupling below which the bosenova occurs.

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Signatures in Binary Systems Our discussion so far assumed BHs for which perturbations from, e.g., surrounding matter or a companion object are small enough that they can be neglected. An interesting situation where such perturbations might be important concerns the case in which the BH-boson system is part of a binary. Since the presence of a companion object destroys the axisymmetry of the problem, it can affect the growth and development of superradiant instabilities by mixing unstable with stable modes [24]. Furthermore, even in cases where perturbations do not significantly affect the cloud, one may expect that the presence of a boson cloud could leave potentially detectable signatures in the gravitational waveform emitted by the binary. In a binary, the cloud feels the tidal field of the companion object, inducing tidal deformations that can lead to an efficient transfer of angular momentum between the cloud and the orbiting object [26, 152]. For weak tidal fields, the problem reduces to solving a wave equation (1) for the boson field, but with the effective potential Veff shifted by a small perturbation δV caused by the tidal field [24]. This problem can be solved perturbatively using standard techniques typically found in quantum mechanics [95]. In particular, the companion causes transition between levels to an extent which can be explicitly computed [24, 26, 28, 48, 152, 153]. These transitions can either excite higher overtones [24], causing the cloud to become spatially more extended since overtones have a larger spatial distribution, or cause mixing between angular modes, causing octupolar and higher modes to be excited, changing the cloud’s spatial profile, as was explicitly verified through numerical simulations [48]. For large orbital separations, mixing between modes is, in general, too small to significantly affect the orbital motion; however, the effect can be resonantly enhanced for orbital frequencies Ω close to the energy split Ω = (ωa − ωb )/(ma − mb ) between two modes Ψa and Ψb , for which transitions are allowed [24]. The tidally distorted cloud backreacts on the companion, similarly to how the Earth tides act on the moon, transfering angular momentum from the cloud to the companion [26, 152]. This mechanism would be especially important during resonances and, depending on the orientation of the orbit, can either make the orbit shrink much faster than in vacuum or even be strong enough to compensate for the angular momentum loss caused by GW emission and temporarily stall the orbit [26, 152] (see also [72] for a Newtonian analysis and [43] for a related effect for a companion with scalar charge). Considering a mixing between two levels with principal quantum number n˜ a and n˜ b , the GWs emitted by the binary at the resonances, where floating orbits can occur, have frequency [24, 26, 152]:

fres

Ω 1 = = 0.2 Hz π |Δm|



60M M



Mμ 0.07

 3  1   − 1,  n˜ 2 2 n˜ a  a

(36)

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where Δm = ma − mb is the difference between the azimuthal numbers of the two modes and M is the mass of the BH surrounded by the cloud. These resonances can lead to a significant GW dephasing when compared to a binary system in vacuum, especially for the case where a small BH perturbs a much more massive BH with a cloud [26], therefore being potentially detectable by LISA. Importantly, since the location of the resonances depends on the spectral properties of the cloud, one could in principle infer the intrinsic properties of the boson, such as their spin, if one is able to measure several resonances during the inspiral. In addition, the gravitational waveform emitted by the binary system would also encode information on the structure of the cloud through the multipole moments and tidal Love numbers of the cloud, which are expected to be significantly different from the vacuum BH case [24, 26]. In particular, the mode mixing discussed above would induce a strong time dependence on these quantities, which would be enhanced at frequencies close to the resonances [24, 26]. A detailed study of the detectability of these effects has not yet been done; however, this raises the exciting prospect to potentially use GW detections from binary BHs to detect and measure the intrinsic properties of ultralight bosons. The discussion so far assumes that the tidal field induced by the companion is sufficiently weak such that it can be treated perturbatively. However, when the tidal field becomes sufficiently large, the boson cloud can be tidally disrupted, just like a star when it passes too close to a very compact object. Detailed numerical simulations of this effect were done in Ref. [48], showing that, for scalar cloud and a companion with mass M∗ at a distance R, tidal disruption occurs when 

M∗ R3

 ≈ crit

(Mμ)6 . 250M 2

(37)

Such tidal disruptions could potentially leave a very clear signature in the GW waveform emitted by the binary (or it could even disrupt the binary), although a detailed computation of the GW signal emitted during the process has not been done so far. Besides tidal effects, the presence of a spatially extended boson cloud around astrophysical BHs can also influence the dynamics of a binary system through at least three additional effects: accretion, dynamical friction, and the self-gravity of the cloud [11,12,21]. Those effects can contribute to a potentially observable change in the phase of the GW signal emitted by the binary [11, 12, 72, 79, 87, 97, 153]. In particular, dynamical friction is expected to be especially important close to the peak of the bosonic configuration [11, 12, 72, 97, 153], where it can dominate over gravitational radiation reaction [153]. In particular, Ref. [153] showed that the effect of dynamical friction could be particularly important for stellar-mass BHs sweeping over the LISA band.

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Open Questions Over the last few years, there has been a dramatic improvement in the understanding of the BH superradiant instability. However, some outstanding open problems remain to be solved; we conclude with a list of them: • The case of ultralight massive spin-2 fields has been analyzed only for small spin [34] and recently in the Newtonian (Mμ 1) limit for any spin [39]. It would be important to study it in full generality, although this problem is challenging due to the apparent nonseparability of the field equations on a Kerr spacetime. • Furthermore, the coupling of a massive spin-2 field to gravity is highly nontrivial [80, 81, 86, 118, 119], and this increases the complexity of the problem. • The impact of nonlinear couplings and self-interactions on the bounds discussed in this chapter deserves further study. Nonlinearities might slow down or even saturate the superradiant growth of bosonic clouds [32, 88, 94, 146], thus making the constraints derived from BH superradiance less stringent [15, 31, 76, 102]. On the other hand, nonlinear effects, such as the bosenova [94, 146] or couplings to photons [30, 32, 49, 88, 115, 122], can also provide novel smoking guns for bosonic condensates around astrophysical BHs that could potentially be used to improve those bounds. • More work should be done to fully access the detectability of boson clouds in binary systems with current and future GW detectors. In particular, it would be important to build gravitational waveform templates incorporating the different effects discussed in this chapter and to estimate the event rates for this type of sources, in order to have a better understanding on the range of boson masses that could be probed by these systems [26]. Acknowledgments R.B. acknowledges financial support from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 792862. P.P. acknowledges financial support provided under the European Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480, and under the MIUR PRIN and FARE programs (GW-NEXT, CUP: B84I20000100001). The authors would like to acknowledge networking support by the COST Action CA16104 and support from the Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001).

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Black Hole Perturbation Theory and Gravitational Self-Force

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Adam Pound and Barry Wardell

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Theory in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolated, Stationary Black Hole Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Null Tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black Hole Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Teukolsky Formalism and Radiation Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric Perturbations of Schwarzschild Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regge-Wheeler Formalism and Regge-Wheeler Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorenz Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small Objects in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tools of Local Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Solution: Self-Field and an Effective External Metric . . . . . . . . . . . . . . . . . . . . . . . Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Skeleton Sources: Punctures and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbital Dynamics in Kerr Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geodesic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accelerated Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving the Einstein Equations with a Skeleton Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiscale Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode Decompositions of the Singular Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Pound () School of Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton, UK e-mail: [email protected] B. Wardell School of Mathematics and Statistics, University College Dublin, Dublin, Ireland e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_38

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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1519 1521 1521

Abstract

Much of the success of gravitational-wave astronomy rests on perturbation theory. Historically, perturbative analysis of gravitational-wave sources has largely focused on post-Newtonian theory. However, strong-field perturbation theory is essential in many cases such as the quasinormal ringdown following the merger of a binary system, tidally perturbed compact objects, and extreme-massratio inspirals. In this review, motivated primarily by small-mass-ratio binaries but not limited to them, we provide an overview of essential methods in (i) black hole perturbation theory, (ii) orbital mechanics in Kerr spacetime, and (iii) gravitational self-force theory. Our treatment of black hole perturbation theory covers most common methods, including the Teukolsky and Regge-Wheeler-Zerilli equations, methods of metric reconstruction, and Lorenz-gauge formulations, presenting them in a new consistent and self-contained form. Our treatment of orbital mechanics covers quasi-Keplerian and action-angle descriptions of bound geodesics and accelerated orbits, osculating geodesics, near-identity averaging transformations, multiscale expansions, and orbital resonances. Our summary of self-force theory’s foundations is brief, covering the main ideas and results of matched asymptotic expansions, local expansion methods, puncture schemes, and point particle descriptions. We conclude by combining the above methods in a multiscale expansion of the perturbative Einstein equations, leading to adiabatic and post-adiabatic evolution schemes. Our presentation is intended primarily as a reference for practitioners but includes a variety of new results. In particular, we present the first complete post-adiabatic waveform-generation framework for generic (nonresonant) orbits in Kerr. Keywords

Black holes · Multiscale methods · Perturbation theory · Self-force · Teukolsky

Introduction Black hole perturbation theory has a long and rich history, dating back at least as far as Regge and Wheeler’s study of odd parity perturbations of Schwarzschild spacetime in the late 1950s [1]. This was followed up in 1970 by Zerilli’s study of even parity perturbations [2, 3]. Soon afterward, Vishveshwara [4] identified quasinormal modes in perturbations of Schwarzschild spacetime, Press [5] studied the associated quasinormal mode frequencies, and Chandrasekhar and Detweiler [6] numerically computed the frequencies. Teukolsky’s success in deriving decoupled

36 Black Hole Perturbation Theory and Gravitational Self-Force

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and separable equations for perturbations of Kerr spacetime [7,8] paved the way for similar progress in the Kerr case. The idea of a self-force has an even longer history, having been studied by Dirac in 1938 in his relativistic generalization of the Abraham-Lorentz self-force to the context of an electric charge undergoing acceleration in flat spacetime [9]. In the 1960s this was extended by DeWitt and Brehme to the curved spacetime case [10]. The gravitational self-force acting on a point mass was studied in the late 1990s by Mino, Sasaki, and Tanaka [11] and by Quinn and Wald [12], leading to the MiSaTaQuWa equation that is named after the authors of those first papers. Subsequent work has put gravitational self-force theory on a very strong theoretical footing [13–15] and has extended the formalism to second order in perturbation theory [16–19]. The last 20 years have seen increasingly intense focus on the study of gravitational self-force in perturbations of black hole spacetimes. This has been motivated to a large extent by the European Space Agency’s LISA mission, which is scheduled for launch in the 2030s (https://www.lisamission.org, undated) and which will observe gravitational waves in the millihertz frequency band. One of the key sources for LISA will be extreme-mass-ratio inspirals (EMRIs), binary systems consisting of a compact solar-mass object orbiting a massive black hole. The presence of a small parameter (the mass ratio, which is expected to be in the region of 10−6 ) makes black hole perturbation theory an ideal tool for the development of theoretical models of the gravitational waveforms from EMRIs. Over the several year timescale that the LISA mission is expected to run, the smaller body in an EMRI will execute ∼104 –105 intricate orbits in the strong-field regime around the central black hole, acting as a precise probe and enabling high-precision measurements of the black hole’s parameters, tests of its Kerr nature, and tests of general relativity. Radiation reaction will cause the orbit to significantly evolve and possibly plunge into the black hole in that time, meaning that self-force effects will be important to include in waveform models. Indeed, in order to extract the maximum information from the observation of EMRIs by LISA, it has been established that it will be necessary to incorporate information at second order in perturbation theory by computing the second-order gravitational self-force [20–22]. Aside from EMRIs, gravitational self-force is also potentially highly accurate for intermediate-mass-ratio inspirals (IMRIs) [23], in which the mass ratio may be as large as ∼10−2 . This makes black hole perturbation theory and self-force also relevant for the current generation of ground-based gravitational wave detectors including LIGO (https://www.ligo.org, undated), Virgo (https://www.virgo-gw.eu, undated), and Kagra (https://gwcenter. icrr.u-tokyo.ac.jp, undated). There are already numerous reviews of these topics in the literature. The classic text by Chandrasekhar [24] provides a comprehensive introduction to black hole physics, linear black hole perturbation theory, and geodesic motion in black hole spacetimes. Reference [25] reviews linear black hole perturbation theory with an emphasis on analytical post-Newtonian expansions of the perturbation equations. Reference [26] provides a thorough introduction to quasinormal modes of black holes. Reference [27] offers a broad introduction to self-force calculations for

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nonexperts, including a survey of concrete physical results through 2018. References [28, 29] cover the foundations of self-force theory, and Ref. [30] provides a complementary view of the foundations from a fully nonlinear perspective. Finally, Refs. [31,32] provide detailed introductions to methods of computing the self-force. Our aim is to complement rather than reiterate these existing reviews. We keep our description of self-force theory brief, only summarizing the key ideas and methods, and we forgo a survey of physical results. Instead, we focus on detailing the main perturbative methods required to model waveforms from small-mass-ratio binaries, leading ultimately to a multiscale expansion of the Einstein equations with a small-body source. At the same time, we keep much of the material sufficiently general to apply to other scenarios of interest. Our aim is also not to provide detailed descriptions of the numerical approaches to solving the many equations detailed in this review. Open source codes implementing state-of-the-art numerical algorithms for solving the equations of black hole perturbation theory and self-force are available through the Black Hole Perturbation Toolkit [33]. The Black Hole Perturbation Toolkit also acts as a repository for collating data (typically in the form of numerical tables or analytical post-Newtonian series expansions) produced by the research community. Our discussion is divided into three distinct parts. Sections “Perturbation Theory in General Relativity” and “Isolated, Stationary Black Hole Spacetimes” briefly introduce relevant background material on perturbation theory in general relativity and the Kerr spacetime. Sections “Black Hole Perturbation Theory”, “Small Objects in General Relativity”, and “Orbital Dynamics in Kerr Spacetime” review three disjoint topics: black hole perturbation theory, geodesics and accelerated orbits in Kerr spacetime, and the foundations of the “local problem” in self-force theory. These three sections are written to be largely independent of one another, and they can be read in any order. Finally, in Section “Solving the Einstein Equations with a Skeleton Source”, we bring together all three topics in a description of black hole perturbation theory with a (skeletal) small-body source, focusing on the multiscale formulation. The multiscale expansion of the Einstein equation for generic (nonresonant) orbits in Kerr, and the post-adiabatic waveform-generation framework that comes along with it, appears here for the first time.

Perturbation Theory in General Relativity The overarching framework for our review is perturbation theory in general relativity. In self-force calculations, this is typically applied to the specific case of a small object in the spacetime of a Kerr black hole, and in much of the review, we specialize to that scenario. But to allow for generality in some sections, we first consider the more generic case of smooth perturbations of an arbitrary vacuum spacetime. We assume the metric can be expanded in powers of a small parameter ε, as:

36 Black Hole Perturbation Theory and Gravitational Self-Force exact 2 (2) 3 gμν = gμν + εh(1) μν + ε hμν + O(ε ),

1415

(1)

where gμν is a vacuum metric, and that the stress-energy can be similarly expanded as (1) (2) Tμν = εTμν + ε2 Tμν + O(ε3 ).

(2)

 (n) For later convenience, we define the total metric perturbation hμν = n>0 εn hμν . We also warn the reader that we will later treat ε as a formal counting parameter that can be set equal to 1. To expand the Einstein equations Gμν [g + h] = 8π Tμν in powers of ε, we first note that the Einstein tensor of a metric gμν + hμν can be expanded in powers of the (1) (2) exact perturbation hμν : Gμν [g + h] = Gμν [g] + Gμν [h] + Gμν [h, h] + O(|h|3 ). (n) The quantities Gμν are easily obtained from the exact Riemann tensor (e.g., see Ch. 7.5 of Ref. [34]). For a vacuum background, the first two terms are   (1) αβ α β 1 Rαβ , [h] = g g − g g G(1) μ ν μν μν 2   (2) αβ α β 1 G(2) Rαβ − [h, h] = g g − g g μ ν μν 2 μν

(3) 1 2



 (1) hμν g αβ − gμν hαβ Rαβ ,

(4)

where the linear and quadratic terms in the Ricci tensor are   (1) Rαβ [h] = − 12 hαβ + 2Rα μ β ν hμν − 2h¯ μ(α ;μ β) ,

(5)

    (2) Rαβ [h, h] = 14 hμν ;α hμν;β + 12 hμ β ;ν hμα;ν − hνα;μ − 12 h¯ μν ;ν 2hμ(α;β) − hαβ;μ   (6) − 12 hμν 2hμ(α;β)ν − hαβ;μν − hμν;αβ . Here we have defined the trace-reversed perturbation h¯ μν := hμν − 12 gμν g αβ hαβ and the d’Alembertian  := g μν ∇μ ∇ν . A semicolon and ∇ both denote the covariant derivative compatible with gμν . So, substituting the expansions (1) and (2) into the Einstein equations and equating powers of ε, we obtain (1) (1) G(1) μν [h ] = 8π Tμν ,

(7)

(2) (2) (2) (1) (1) G(1) μν [h ] = 8π Tμν − Gμν [h , h ].

(8)

This perturbative expansion comes with the freedom to perform gauge transformations: (1) h(1) μν → hμν + £ξ(1) gμν , (2) (1) 1 2 h(2) μν → hμν + £ξ(2) gμν + 2 £ξ(1) gμν + £ξ(1) hμν ,

(9) (10)

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α are freely chosen vector fields. In self-force where £ξ is a Lie derivative and ξ(n) theory, this freedom is commonly used to impose the Lorenz gauge condition,

∇α h¯ αβ = 0,

(11)

in which case the linearized Einstein tensor simplifies to   α β¯ 1 ¯ G(1) μν [h] = − 2 hμν + 2Rμ ν hαβ .

(12)

A perturbed metric will come hand in hand with a perturbed equation of motion for objects in the spacetime: D 2 zμ μ μ μ = f(0) + εf(1) + ε2 f(2) + O(ε3 ). dτ 2

(13)

Here zμ (τ ) is a perturbed worldline, τ is its proper time as measured in the 2 μ ν dzμ μ background gμν , Ddτz2 = dz dτ ∇ν dτ := a is its covariant acceleration with μ respect to gμν , and f(n) is the nth-order covariant force (per unit mass) driving the acceleration. In our review, we will consider the general case including a zerothμ μ order force, but we will focus primarily on cases with f(0) = 0. The forces f(n) will (n)

arise from (parts of) the metric perturbations hμν as well as from coupling of gμν to the matter that creates those perturbations. Here we have limited the treatment to first- and second-order perturbations, which are expected to be necessary and sufficient for modeling small-mass-ratio binaries. In some sections we will restrict the context to first, linearized order.

Isolated, Stationary Black Hole Spacetimes In most of our review, we take the background spacetime to be that of an isolated, stationary black hole. In this section we provide an overview of the properties of these spacetimes.

Metric The Schwarzschild spacetime is a static, spherically symmetric solution of the vacuum Einstein equations representing a nonrotating black hole with mass M. It has a line element given by   ds 2 = −f (r)dt 2 + f (r)−1 dr 2 + r 2 dθ 2 + sin2 θ dφ 2 ,

(14)

where f (r) := 1 − 2M r . The Schwarzschild spacetime may be generalized to allow the black hole to have a charge per unit mass, Q, resulting in the Reissner-Nordström

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solution of the Einstein-Maxwell equations, with line element 

Q2 2M + 2 ds = − 1 − r r 2





Q2 2M dt + 1 − + 2 r r 2

−1

  dr 2 +r 2 dθ 2 +sin2 θ dφ 2 .

(15) The spacetime of a spinning black hole is given by the Kerr metric with angular momentum per unit mass a. In Boyer-Lindquist coordinates, its line element is

2Mr ds = − 1 − Σ 2



Σ 2 4aMr sin2 θ dt dφ + dr Σ Δ

2Mr(r 2 + a 2 ) 2 sin2 θ dφ 2 , + Σ dθ + Δ + Σ

dt 2 −

(16)

2 2 2 2 2 where Σ := √ r + a cos θ and Δ := r − 2Mr + a = (r − r+ )(r − r− ) with 2 2 r± := M ± M − a the locations of the inner and outer horizons. As was the case with Schwarzschild spacetime, the Kerr spacetime may be generalized to allow the black hole to have a charge per unit mass, Q, giving the Kerr-Newman solution of the Einstein-Maxwell equations. In Boyer-Lindquist coordinates, the Kerr-Newman metric is

Σ 2Mr − Q2 2(2Mr − Q2 )a sin2 θ ds 2 = − 1 − dt 2 − dt dφ + dr 2 Σ Σ Δ + Q2

(2Mr − Q2 )(a 2 + r 2 ) sin2 θ dφ 2 . + Σ dθ 2 + Δ + Q2 + (17) Σ In astrophysical scenarios, a charged black hole will quickly be neutralized. For that reason, in later sections, we will restrict our attention to the Kerr spacetime. We will also later use Q to denote the Carter constant, associated with the Kerr metric’s third, hidden symmetry discussed below. However, we include the charged black hole metrics here for completeness.

Null Tetrads The black hole spacetimes above are all of Petrov type-D and thus have two nondegenerate principal null directions. This gives us a natural way to define a complex null tetrad by having two of the tetrad legs aligned with the principal α to align with the outward null direction and null directions. Choosing l α := e(1) α α n := e(2) to align with the inward null direction, there is still residual freedom in the choice of scaling of each tetrad leg and also in the relative orientation of the α and m α . The two most common remaining two tetrad legs, mα := e(3) ¯ α := e(4) choices in Kerr spacetime are Carter’s canonical tetrad [35],

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1  2 r + a 2 , Δ, 0, a , lα = √ 2ΔΣ i 1  mα = √ , ia sin θ, 0, 1, sin θ 2Σ

nα = √ m ¯α = √

1 2ΔΣ 1  2Σ

 r 2 + a 2 , −Δ, 0, a , − ia sin θ, 0, 1, −

i , sin θ (18)

and the Kinnersley tetrad

[36], which is related to Carter’s canonical tetrad by a ¯ Δ α α α α α √ζ simple rescaling: l = 2Σ lK , nα = 2Σ ¯ α = √ζ m ¯α, Δ nK , m = Σ mK , and m Σ K where ζ := r − ia cos θ

(19)

is an important quantity that we will encounter again later (note that Σ = ζ ζ¯ ). ¯ μ . Carter also Carter’s original tetrad had interchanged l μ ↔ nμ and mμ ↔ m ˜ worked in different coordinates (t˜ = t − aφ, r, q = a cos θ, φ = φ/a) which more fully reflect the inherent symmetries of Kerr. We deviate from that here and keep with the convention of having l α point outward and working in the more common Boyer-Lindquist coordinates. Irrespective of this choice, the Carter tetrad transforms as l↔−n, m↔m ¯ under {t, φ} → {−t, −φ}. Although the Kinnersley tetrad formed a crucial part of Teukolsky’s separability result for perturbations of the Weyl tensor [7], it has two unfortunate features that make it less than ideal for elucidating the symmetric structure of Kerr spacetime: (i) it violates the {t, φ} → {−t, −φ} symmetry; and (ii) it destroys a symmetry in {r, θ }. Carter’s canonical tetrad does not suffer from either of these deficiencies and is slightly preferable from that point of view. Note, however, that all of the results that follow can be derived using either tetrad.

Symmetries Much of the success in studying Kerr spacetime has arisen from the inherent symmetries it possesses. Two of these are associated with the existence of two Killing vectors, ξ α and ηα , which satisfy Killing’s equation: ξ(α;β) = 0 = η(α;β) .

(20)

Note that the Killing vector δφα = a1 ηα − aδtα is often used in place of ηα when working in Boyer-Lindquist coordinates. In Kerr spacetime these are related to the timelike and axial symmetries: ξ α = δtα ,

ηα = δφα˜ = a(δφα + aδtα ).

The spacetime also admits a conformal Killing-Yano tensor

(21)

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fαβ = (ζ + ζ¯ )n[α lβ] − (ζ − ζ¯ )m ¯ [α mβ] ,

(22)

fα(β;γ ) = gβγ ξα − gα(β ξγ ) .

(23)

which satisfies

Here, we have introduced the Killing spinor coefficient, ζ , which we previously encountered as a coordinate expression in Section “Null Tetrads”. Its appearance here can be considered more fundamental and does not depend on any particular coordinate choice. The divergence of this conformal Killing-Yano tensor is a Killing vector: ξα = 13 fαβ ;β ,

(24)

and its Hodge dual 

fαβ = 12 εαβ γ δ fγ δ = i(ζ − ζ¯ )n[α lβ] − i(ζ + ζ¯ )m ¯ [α mβ] ,

(25)

is a Killing-Yano tensor satisfying 

fα(β;γ ) = 0.

(26)

The products of these Killing-Yano tensors generate two conformal Killing tensors ¯ (α mβ) , Kαβ = fαγ fβ γ = 12 (ζ + ζ¯ )2 n(α lβ) − 12 (ζ − ζ¯ )2 m

(27)



K αβ = fαγ  fβ γ = 12 i(ζ 2 − ζ¯ 2 )(n(α lβ) + m ¯ (α mβ) ),

(28)

which satisfy K(αβ;γ ) = g(αβ Kγ ) ,





K (αβ;γ ) = g(αβ K γ ) , 

where Kα = 16 (2Kβα ;β + Kβ β ;α ) and K α = generate a Killing tensor

 1 ;β 6 (2K βα

(29) 

+ K β β ;α ). They also



K αβ =  fαγ  fβ γ = − 12 (ζ − ζ¯ )2 n(α lβ) + 12 (ζ + ζ¯ )2 m ¯ (α mβ) ,

(30)

satisfying 

K (αβ;γ ) = 0.

(31)

This Killing tensor gives a relationship between the two Killing vectors: 

ηα = −K αβ ξβ .

(32)

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Black Hole Perturbation Theory We now consider perturbations of isolated black hole spacetimes. We describe, in a unified notation, how to calculate metric perturbations in the most commonly used gauges: radiation gauges, Regge-Wheeler-Zerilli gauges, and the Lorenz gauge. Our focus is particularly on reconstruction methods, in which most or all of the metric perturbation is reconstructed from decoupled scalar variables. Since the left-hand sides of the perturbative Einstein equations (7) and (8) are the same at every order, we specialize to the first-order case. We refer the reader to Refs. [37, 38] for general discussions of second-order perturbation theory in Schwarzschild and Kerr spacetimes.

The Teukolsky Formalism and Radiation Gauge Teukolsky [7] showed that the equations governing perturbations of rotating black hole spacetimes can be recast into a form where they are given by decoupled equations. These equations further have the remarkable property of being separable, reducing the problem to the solution of a set of uncoupled ordinary differential equations. In the case of metric perturbations, Teukolsky’s results yield solutions for the spin-weight ±2 components of the perturbed Weyl tensor, but do not give a method for obtaining a corresponding metric perturbation. Subsequent results (and their equivalents for electromagnetic perturbations) [39–46] derived a method for reconstructing a metric perturbation from a Hertz potential, which in turn can be obtained from the spin-weight ±2 components of the Weyl tensor.

Geroch-Held-Penrose Formalism Our exposition makes use of the formalism of Geroch, Held, and Penrose (GHP) [47], which is a simplified and more explicitly coordinate independent version of the Newman-Penrose (NP) [48] formalism originally used by Teukolsky. Here we provide a concise review of the key features of the formalism needed to understand metric perturbations of black hole spacetimes; see Refs. [49–51] for more thorough treatments. The GHP formalism prioritizes the concepts of spin- and boost-weights; within the formalism, everything has a well-defined type {p, q}, which is related to its spinweight s = (p − q)/2 and its boost-weight b = (p + q)/2. Only objects of the same type can be added together, providing a useful consistency check on any equations. Multiplication of two quantities yields a resulting object with type given by the sum of the types of its constituents. α } = {l α , nα , mα , m We first introduce a null tetrad {e(a) ¯ α } with normalization l α nα = −1,

mα m ¯ α = 1,

(33)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1421

and with all other inner products vanishing. In terms of the tetrad vectors, the metric may be written as gαβ = −2l(α nβ) + 2m(α m ¯ β) .

(34)

There are three discrete transformations that reflect the inherent symmetry in the GHP formalism, corresponding to simultaneous interchange of the tetrad vectors: 1.  : l α ↔ nα and mα ↔ m ¯ α , {p, q} → {−p, −q}; α α 2. ¯ : m ↔ m ¯ , {p, q} → {q, p}; 3. ∗: l α → mα , nα → −m ¯ α , mα → −l α , m ¯ α → nα . We next introduce the spin coefficients, defined to be the 12 directional derivatives of the tetrad vectors. Of these, the eight with well-defined GHP type are: κ= − l μ mν ∇μ lν , σ = − mμ mν ∇μ lν ,

ρ= − m ¯ μ mν ∇μ lν , τ = − nμ mν ∇μ lν , (35)

along with their primed variants, κ  , σ  , ρ  , and τ  . These have GHP type given by: κ : {3, 1},

σ : {3, −1},

ρ : {1, 1},

τ : {1, −1}.

The remaining four spin coefficients are used to define the GHP derivative operators (that depend on the GHP type of the object on which they are acting) Þ := (l α ∇α − pε − q ε¯ ),

Þ := (nα ∇α + pε + q ε¯  ),

ð := (mα ∇α − pβ + q β¯  ),

¯ ð := (m ¯ α ∇α + pβ  − q β),

(36)

where β=

1 μ ν (m m ¯ ∇μ mν − mμ nν ∇μ lν ), 2

ε=

1 μ ν (l m ¯ ∇μ mν − l μ nν ∇μ lν ), 2 (37)

along with their primed variants, β  and ε . These spin coefficients have no welldefined GHP type and never appear explicitly in covariant equations. The action of a GHP derivative causes the type to change by an amount {p, q} → {p + r, q + s} where {r, s} for each of the operators is given by: Þ : {1, 1},

Þ : {−1, −1},

ð : {1, −1},

ð : {−1, 1}.

In this sense we interpret Þ and Þ as boost-raising and boost-lowering operators, respectively, while we interpret ð and ð as spin-raising and spin-lowering operators, respectively. The adjoints of the GHP operators are given by:

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Þ† := −(Þ − ρ − ρ), ¯

Þ† := −(Þ − ρ  − ρ¯  ),

ð† := −(ð − τ − τ¯  ),

ð† := −(ð − τ  − τ¯ ),

(38)

or, alternatively D† = −(Ψ2 Ψ¯ 2 )1/3 D(Ψ2 Ψ¯ 2 )−1/3 ,

D ∈ {Þ, Þ , ð, ð }.

(39)

In vacuum spacetimes, the only nonzero components of the Riemann tensor are given by the tetrad components of the Weyl tensor, which can be represented by five complex Weyl scalars Ψ0 = Clmlm ,

Ψ1 = Clnlm ,

Ψ2 = Clmmn ¯ ,

Ψ3 = Clnmn ¯ ,

Ψ4 = Cnmn ¯ m ¯, (40)

with types inherited from the tetrad vectors that appear in their definition Ψ0 : {4, 0},

Ψ1 : {2, 0},

Ψ2 : {0, 0},

Ψ3 : {−2, 0},

Ψ4 : {−4, 0}.

Many of the results that follow will be specialized to type-D spacetimes with l μ and nμ aligned to the two principal null directions, in which case the GoldbergSachs theorem implies that four of the spin coefficients vanish κ = κ  = σ = σ  = 0,

(41)

and also that most of the Weyl scalars vanish Ψ0 = Ψ1 = Ψ3 = Ψ4 = 0.

(42)

The GHP equations give relations between the Weyl scalars and the directional derivatives of the spin coefficients. For type-D spacetimes, they are given by Þρ = ρ 2 ,

Þτ = ρ(τ − τ¯  ),

ðτ = τ 2 ,

ðρ = τ (ρ − ρ), ¯

Þ ρ = ρ ρ¯  − τ τ¯ − Ψ2 + ð τ,

(43)

along with the Bianchi identity ÞΨ2 = 3ρΨ2 ,

ðΨ2 = 3τ Ψ2 ,

(44)

and the conjugate, prime, and prime conjugate of these equations. Similarly the commutator of any pair of directional derivatives can be written in terms of a linear combination of spin coefficients multiplying single directional derivatives. Again for type-D, they are given by

36 Black Hole Perturbation Theory and Gravitational Self-Force

1423

[Þ, Þ ] = (τ¯ − τ  )ð + (τ − τ¯  )ð − p(Ψ2 − τ τ  ) − q(Ψ¯ 2 − τ¯ τ¯  ),

(45a)

[Þ, ð] = ρð ¯ − τ¯  Þ + q ρ¯ τ¯  ,

(45b)

[ð, ð ] = (ρ¯ − ρ )Þ + (ρ − ρ)Þ ¯ + p(Ψ2 + ρρ ) − q(ρ¯ ρ¯ + Ψ¯ 2 ), 











(45c)

along with the conjugate, prime, and prime conjugate of these. 

If we further restrict to spacetimes that admit a Killing tensor, K αβ , the associated symmetries lead to additional identities relating the spin coefficients 1/3 ζ¯ ρ τ τ ρ M¯ 1/3 Ψ =  = −  = − = 1/3 21/3 = , ρ¯ ρ¯ τ¯ τ¯ ζ M Ψ¯

(46)

2

for some complex function M that is annihilated by Þ (In the case of Kerr spacetime, M is the mass of the spacetime as one might anticipate.). Here, we have used the fact that the Killing spinor coefficient is related to Ψ2 by −1/3

ζ = −M 1/3 Ψ2

.

(47)

These identities can be used along with the GHP equations to obtain a complementary set of identities Þτ  = 2ρτ  = ð ρ,

(48a)

ζ¯ 1 Ψ¯ 2 , Þ ρ = ρρ  + τ  (τ − τ¯  ) − Ψ2 − 2 2ζ

(48b)

1 ζ¯ ð τ = τ τ  + ρ(ρ  − ρ¯  ) + Ψ2 − Ψ¯ 2 , 2 2ζ

(48c)

along with the conjugate, prime, and prime conjugate of these equations. A consequence of these additional relations is that there is an operator  p q £ξ = −ζ − ρ  Þ + ρÞ + τ  ð − τ ð ) − ζ Ψ2 − ζ¯ Ψ¯ 2 , 2 2

(49)

associated with the Killing vector ξ α = −ζ (−ρ  l α + ρnα + τ  mα − τ m ¯ α ).

(50)

There is a second operator   £η = − ζ4 (ζ − ζ¯ )2 (ρ  Þ − ρÞ ) − (ζ + ζ¯ )2 (τ  ð − τ ð ) + p η h1 + q η h¯ 1

(51)

1424

A. Pound and B. Wardell

where η h1

= 18 ζ (ζ 2 + ζ¯ 2 )Ψ2 − 14 ζ ζ¯ 2 Ψ¯ 2 + 12 ρρ  ζ 2 (ζ¯ − ζ ) + 12 τ τ  ζ 2 (ζ¯ + ζ ).

(52)

This is associated with the second Killing vector   ηα = − ζ4 (ζ − ζ¯ )2 (ρ  l α − ρnα ) − (ζ + ζ¯ )2 (τ  mα − τ m ¯ α) .

(53)

Both £ξ and £η commute with all of the GHP operators and annihilate all of the spin coefficients and Ψ2 .

Teukolsky Equations We now consider perturbations of vacuum type-D spacetimes. Teukolsky [7] showed that the perturbations to Ψ0 and Ψ4 (which we will denote by ψ0 and ψ4 ) are gauge invariant and satisfy decoupled and fully separable second-order equations. These perturbations may be written in GHP form as (1)

ψ0 = Clmlm [h] = T0 h,

(1)

ψ4 = Cnmn ¯ m ¯ [h] = T4 h,

(54)

where the operators TI are given by T0 h = −

T4 h = −

1 (ð − τ¯  )(ð − τ¯  )hll + (Þ − ρ)(Þ ¯ − ρ)h ¯ mm 2   − (Þ − ρ)(ð ¯ − 2τ¯  ) + (ð − τ¯  )(Þ − 2ρ) ¯ h(lm) ,

(55a)

1  (ð − τ¯ )(ð − τ¯ )hnn + (Þ − ρ¯  )(Þ − ρ¯  )hm¯ m¯ 2   − (Þ − ρ¯  )(ð − 2τ¯ ) + (ð − τ¯ )(Þ − 2ρ¯  ) h(nm) ¯ .

(55b)

We will later also need the adjoints of these, which are given by 1 lα lβ (ð − τ )(ð − τ ) + mα mβ (Þ − ρ)(Þ − ρ) 2   ¯ − τ ) Ψ, − l(α mβ) (ð − τ + τ¯  )(Þ − ρ) + (Þ − ρ + ρ)(ð

(T0† Ψ )αβ = −

(56a)

1 nα nβ (ð − τ  )(ð − τ  ) + m ¯ αm ¯ β (Þ − ρ  )(Þ − ρ  ) 2   ¯ β) (ð − τ  + τ¯ )(Þ − ρ  ) + (Þ − ρ  + ρ¯  )(ð − τ  ) Ψ. − n(α m

(T4† Ψ )αβ = −

(56b) The scalars ψ0 and ψ4 satisfy the Teukolsky equations (Note that O  ψ4 = and Oψ0 = ζ −4 O  ζ 4 ψ0 .):

ζ −4 Oζ 4 ψ4

36 Black Hole Perturbation Theory and Gravitational Self-Force

Oψ0 = 8π S0 T ,

1425

O  ψ4 = 8π S4 T ,

(57)

where          O := Þ − 2 s ρ − ρ¯ Þ − ρ  − ð − 2 s τ − τ¯  ð − τ  + 12 6s − 2 − 4s 2 Ψ2 (58) is the spin-weight s Teukolsky operator (Some authors (e.g., [41,45]) define O to be the operator with s = +2. Then, the operator for the negative s fields is its adjoint O † .). The decoupling operators   S0 T = 12 (ð − τ¯  − 4τ ) (Þ − 2ρ)T ¯ (lm) − (ð − τ¯  )Tll   + 12 (Þ − 4ρ − ρ) ¯ (ð − 2τ¯  )T(lm) − (Þ − ρ)T ¯ mm ,    S4 T = 12 (ð − τ¯ − 4τ  ) (Þ − 2ρ¯  )T(nm) ¯ − (ð − τ¯ )Tnn     + 12 (Þ − 4ρ  − ρ¯  ) (ð − 2τ¯ )T(nm) ¯m ¯ , ¯ − (Þ − ρ¯ )Tm

(59a)

(59b)

allow the sources for the Teukolsky equations to be constructed from the stressenergy tensor. We will later also need the adjoints of these, which are given by (S0† Ψ )αβ = − 12 lα lβ (ð − τ )(ð + 3τ )Ψ − 12 mα mβ (Þ − ρ)(Þ + 3ρ)Ψ  ¯ + 3τ ) + (ð − τ + τ¯  )(Þ + 3ρ)]Ψ, + 12 l(α mβ) (Þ − ρ + ρ)(ð

(60a)

¯ αm ¯ β (Þ − ρ  )(Þ + 3ρ  )Ψ (S4† Ψ )αβ = − 12 nα nβ (ð − τ  )(ð + 3τ  )Ψ − 12 m  ¯ β) (Þ − ρ  + ρ¯  )(ð + 3τ  ) + (ð − τ  + τ¯ )(Þ + 3ρ  )]Ψ. + 12 n(α m (60b) (1)

Introducing the index-free linearized Einstein operator (E h)αβ := Gαβ [h], we see that Teukolsky’s result for decoupling the equations are a consequence of the operator identities S0 E = OT0 ,

S4 E = O  T4 .

(61)

In vacuum Kerr-NUT spacetimes, the Teukolsky operator may be written in manifestly separable form by rewriting it in terms of the commuting operators [50] R := ζ ζ¯ (Þ − ρ − ρ)(Þ ¯  − 2bρ  ) +

2b − 1 (ζ + ζ¯ )£ξ , 2

(62)

1426

A. Pound and B. Wardell

and S := ζ ζ¯ (ð − τ − τ¯  )(ð − 2sτ  ) +

2s − 1 (ζ − ζ¯ )£ξ . 2

(63)

Then, the Teukolsky operator is given by ζ ζ¯ O = R − S.

(64)

  The symmetry operators satisfy the commutation relations R, S = 0 when acting on a type {p, 0} object. We will see later that when written as a coordinate expression in Boyer-Lindquist coordinates in Kerr spacetime the operators R and S reduce to the radial Teukolsky and spin-weighted spheroidal operators (with a common eigenvalue).

Teukolsky-Starobinsky Identities As a consequence of the linearised Bianchi identities, the scalars ψ0 and ψ4 are not independent. Instead, they are related by the Teukolsky-Starobinsky identities, which are given in GHP form by Þ4 ζ 4 ψ4 = ð4 ζ 4 ψ0 − 3M£ξ ψ¯ 0 ,

(65a)

Þ4 ζ 4 ψ0 = ð4 ζ 4 ψ4 + 3M£ξ ψ¯ 4 ,

(65b)

where we recall that M = −ζ 3 ψ2 . From these, we can also derive eighth-order Teukolsky-Starobinsky identities that do not mix the scalars: Þ4 ζ¯ 4 Þ4 ζ 4 ψ0 = ð4 ζ¯ 4 ð4 ζ 4 ψ0 − 9M 2 £2ξ ψ0 ,

(66a)

Þ4 ζ¯ 4 Þ4 ζ 4 ψ4 = ð4 ζ¯ 4 ð4 ζ 4 ψ4 − 9M 2 £2ξ ψ4 .

(66b)

Reconstruction of a Metric Perturbation in Radiation Gauge Solutions of the Teukolsky equations can be related back to solutions for the metric perturbation hαβ by use of a Hertz potential [39–42, 52]. In fact, there are two different Hertz potentials: ψ IRG , which produces a metric perturbation in the ingoing radiation gauge, and ψ ORG , which produces a metric perturbation in the outgoing radiation gauge. In the ingoing radiation gauge (IRG), the metric perturbation may be reconstructed by applying a second-order differential operator to a scalar Hertz potential ψ IRG of type {−4, 0} (i.e., the same type as ψ4 ). In terms of this Hertz potential, the IRG metric perturbation is given explicitly by  † IRG  hIRG )αβ . αβ = 2 (S0 ψ

(67)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1427

where S0† is the operator given in Eq. (60a). The IRG Hertz potential satisfies Oψ IRG = ηIRG , where ηIRG satisfies 2 (T0† ηIRG )αβ = 8π Tαβ . In other words, ψ IRG is a solution of the equation satisfied by ζ 4 ψ4 (equivalently, the adjoint of the equation satisfied by ψ0 ), but with a different source. The IRG Hertz potential manifestly satisfies the gauge conditions l α hαβ = 0 and h = 0, and it necessarily requires that (E hIRG )ll = 0 = Tll . Computing the perturbed Weyl scalars from it, we find: 1 4 IRG Þ ψ 4 1 3 ψ4 = ð4 ψ IRG − Mζ −4 £ξ ψ IRG , 4 4 1  −2 2 + ζ Oζ + 2ζ −1 £ξ − 2(τ  τ − ρ  ρ − Ψ2 ) ηIRG . 4

ψ0 =

(68a)

(68b)

The IRG Hertz potential may therefore be obtained either by solving the sourced (adjoint) Teukolsky equation or by solving either of the fourth-order equations sourced by the perturbed Weyl scalars. The equations involving ψ0 and ψ4 are often referred to as the “radial” and “angular” inversion equations, respectively. Acting on the perturbed Weyl scalars with the Teukolsky operator and commuting operators, we find: 1 (Þ − ρ − ρ) ¯ 4 ηIRG 4 1 3 O  ψ4 = (ð − τ  − τ¯ )4 ηIRG − Mζ −4 £ξ ηIRG , 4 4 1   −2 2 + O ζ Oζ + 2ζ −1 £ξ − 2(τ  τ − ρ  ρ − Ψ2 ) ηIRG . 4 Oψ0 =

(69a)

(69b)

Thus, in regions where the Hertz potential satisfies the homogenous equation Oψ IRG = 0, the second line of Eq. (68b) vanishes, and the perturbed Weyl scalars satisfy the homogeneous Teukolsky equations. A similar procedure also works in the outgoing radiation gauge (ORG), the prime of the IRG. There, we have  † ORG  hORG )αβ , αβ = 2 (S4 ψ

(70)

where the ORG Hertz potential, ψ ORG , is of type {4, 0} (i.e., the same as ψ0 ). Some authors [53] define a slightly different ORG Hertz potential related to the one here by ψˆ ORG = ζ −4 ψ ORG and (Sˆ4† )αβ = (S4† ζ 4 )αβ . Both conventions yield the same metric perturbation, (Sˆ4† ψˆ ORG )αβ = (S4† ψ ORG )αβ ). The ORG Hertz potential satisfies O  ψ ORG = ηORG , where ηORG satisfies 2 (T4† ηORG )αβ = 8π Tαβ . In

1428

A. Pound and B. Wardell

other words, ψ ORG is a solution of the equation satisfied by ζ 4 ψ0 (equivalently, the adjoint of the equation satisfied by ψ4 ), but with a different source. The ORG Hertz potential manifestly satisfies the gauge conditions nα hαβ = 0 and h = 0, and it necessarily requires that (E hIRG )nn = 0 = Tnn . Computing the perturbed Weyl scalars from it, we find: 1 4 ORG 3 ð ψ + Mζ −4 £ξ ψ ORG 4 4 1  −2  2 + ζ O ζ − 2ζ −1 £ξ − 2(τ  τ − ρ  ρ − Ψ2 ) ηORG 4 1 4 ORG ψ4 = Þ ψ , 4 ψ0 =

(71a) (71b)

The ORG Hertz potential may therefore be obtained either by solving the sourced (adjoint) Teukolsky equation or by solving either of the fourth-order equations sourced by the perturbed Weyl scalars. The equations involving ψ0 and ψ4 are often referred to as the “angular” and “radial” inversion equations, respectively. Acting on the perturbed Weyl scalars with the Teukolsky operator and commuting operators, we find: 1 3 (ð − τ − τ¯  )4 ηORG + Mζ −4 £ξ ηORG 4 4 1  + O ζ −2 O  ζ 2 − 2ζ −1 £ξ − 2(τ  τ − ρ  ρ − Ψ2 ) ηORG 4 1 O  ψ4 = (Þ − ρ  − ρ¯  )4 ηORG . 4 Oψ0 =

(72a) (72b)

Thus, in regions where the Hertz potential satisfies the homogenous equation O  ψ ORG = 0, the second line of Eq. (71a) vanishes, and the perturbed Weyl scalars satisfy the homogeneous Teukolsky equations. As with the Weyl scalars, the IRG and ORG Hertz potentials are not independent. By demanding that they produce the same ψ0 and ψ4 , we obtain TeukolskyStarobinsky identities relating them in the homogeneous case: Þ4 ψ IRG = ð4 ψ ORG + 3Mζ −4 £ξ ψ ORG Þ4 ψ ORG = ð4 ψ IRG − 3Mζ −4 £ξ ψ IRG .

(73) (74)

The fact that the Hertz potentials yield solutions of the homogeneous linearized Einstein equations was succinctly summarized by Wald [41] using the method of adjoints: since the operators satisfy the identity S E = OT , by taking the adjoint and using the fact that E is self-adjoint, we find that E S † = T † O † , so we have a homogeneous solution of the linearized Einstein equations provided the Hertz potential satisfies the (adjoint) homogeneous Teukolsky equation.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1429

Finally, we note that in addition to imposing conditions on the stress-energy, the standard radiation gauge reconstruction procedure fails to reproduce certain “completion” portions of the metric perturbation associated with small shifts in the central mass and angular momentum, and with gauge. A more generally valid metric perturbation may be obtained by supplementing the reconstructed piece described here with completion pieces and with a “corrector” tensor xαβ that is designed to eliminate any restrictions on the stress-energy: hαβ = 2 (S † Ψ )αβ + xαβ + g˙ αβ + (£X g)αβ .

(75)

The interested reader may refer to [39–41] for the original derivations of the reconstruction procedure, to [54] for an analysis of the sourced equation satisfied by the Hertz potential, to [55, 56] for details of metric completion, and to [45, 46] for a thorough explanation of the corrector tensor approach.

Gravitational Waves In order to determine the gravitational wave strain, we require the metric perturbation far from the source. If we consider the metric perturbation reconstructed in radiation gauge, then to leading order in a large-distance expansion from the source the components hmm and hm¯ m¯ dominate, with both falling of as (distance)−1 . It is common to write these in terms of the two gravitational wave polarizations: hmm = h+ + ih× ,

hm¯ m¯ = h+ − ih× .

(76)

Furthermore, at large radius, the operator T4 of Eq. (55b) reduces to a second derivative along the l μ null direction, leading to a simple relationship between ψ4 and the second time derivative of the strain: 1 ψ4 ∼ − h¨ m¯ m¯ . 2

(77)

This gives us a straightforward way to determine the strain by computing two time integrals of ψ4 . Further mathematical details on the relationship between ψ4 and outgoing gravitational radiation are given in Refs. [48, 57, 58], on the equivalent relationship between ψ0 and incoming radiation in Ref. [59], and on numerical implementation considerations in Refs. [60, 61].

GHP Formalism in Kerr Spacetime We now give explicit expressions for the various quantities defined in the previous sections, specialized to Kerr spacetime. The spin coefficients are tetrad dependent. When working with the Carter tetrad, the nonzero spin coefficients have a particularly symmetric form given by:

1430

A. Pound and B. Wardell

1 ρ = −ρ = − ζ 

β = β = −



ia sin θ , τ = τ = − √ ζ 2Σ

Δ , 2Σ

i a + ir cos θ , √ ζ 2 sin θ 2Σ

ε = −ε =

Mr − a 2 − ia(r − M) cos θ , √ 2ζ 2ΣΔ (78)

while for the Kinnersley tetrad they are given by: 1 ρK = − , ζ cot θ βK = √ , 2 2ζ¯

 = ρK

Δ , 2ζ 2 ζ¯

ia sin θ τK = − √ , 2ζ ζ¯

ia sin θ cot θ  βK = √ − √ , 2 2ζ 2ζ 2

εK = 0,

ia sin θ τK = − √ , 2ζ 2  εK =

r −M Δ − . 2 ¯ 2ζ ζ 2ζ ζ¯ (79)

The commuting GHP operators have the same form in both tetrads: £ξ = ∂t ,

£η = a 2 ∂t + a∂φ = ∂φ˜ .

(80)

Mode-Decomposed Equations in Kerr Spacetime In addition to decoupling the equations, Teukolsky [7, 8] further showed that the Teukolsky equations are fully separable using a mode ansatz. The specific form of the ansatz depends on the choice of null tetrad. Teukolsky worked with the Kinnersley tetrad [36], in which case the Teukolsky equations are separable using the ansatz  ψ0 =

 

−∞ =2 m=−

 ζ 4 ψ4 =

∞ ∞ 

∞ ∞ 

 

−∞ =2 m=−

2 ψmω (r) 2 Sm (θ, φ; aω)e

−iωt

−2 ψmω (r) −2 Sm (θ, φ; aω)e

dω,

−iωt

dω,

(81)

(82)

with the functions s ψmω (r) and s Sm (θ, φ; aω) satisfying the spin-weighted spheroidal harmonic and Teukolsky radial equations, respectively:

 

2 d 2 d 2 2 2 (m + sχ ) (1−χ ) +a ω χ − −2asωχ +s +A s Sm = 0, dχ dχ 1 − χ2

(83)

and

  K 2 − 2is(r − M)K −s d s+1 d Δ Δ + + 4isωr − s λm s ψmω = s Tmω , dr dr Δ (84)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1431

where χ := cos θ , A := s λm + 2amω − a 2 ω2 and K := (r 2 + a 2 )ω − am and where the eigenvalue s λm depends on the value of aω. A similar separability result also holds when working with the Carter tetrad by replacing the left-hand sides as follows: ψ0 → ζ 2 Δ−1 ψ0 ,

ζ 4 ψ4 → ζ 2 Δψ4 ,

(S0 T ) → ζ 2 Δ−1 (S0 T ),

ζ 4 (S4 T ) → ζ 2 Δ(S4 T ). The factors of Δ here are not required for separability, but are included so that the radial functions are consistent with Teukolsky’s original radial functions. As with the standard spherical harmonics, the dependence of the spin-weighted spheroidal harmonics on the azimuthal coordinate is exclusively through an overall complex exponential factor: s Sm (θ, φ; aω)

= s Sm (θ, 0; aω)eimφ .

(85)

With this definition, the spin-weighted spheroidal harmonics are orthonormal: 

s Sm (θ, φ; aω)s S¯ m (θ, φ; aω)dΩ

= δ δmm ,

(86)

where dΩ = sin θ dθ dφ is the volume element on the two-sphere. They also satisfy two symmetry identities: s Sm (θ, φ; aω)

= (−1)+m −s Sm (π − θ, φ; aω),

(87a)

s Sm (θ, φ; aω)

= (−1)+s s S¯−m (π − θ, φ; −aω)

(87b)

which can be combined to obtain the useful identity s Sm (θ, φ; aω)

= (−1)m+s −s S¯−m (θ, φ; −aω),

(88)

which relates a (s, , m, aω) harmonic to the conjugate of a (−s, , −m, −aω) harmonic. Similarly, the eigenvalue satisfies the identities s λm (aω)

= −s λm (aω) − 2s,

(89a)

s λm (aω)

= s λ−m (−aω)

(89b)

which can be combined to obtain s λm (aω)

= −s λ−m (−aω) − 2s.

(90)

1432

A. Pound and B. Wardell

The sources for the radial Teukolsky equation are defined by:

8π(S0 T ) = −

1 2Σ

8π ζ 4 (S4 T ) = −

1 2Σ



∞ ∞ 

 

−∞ =2 m=−



∞ ∞ 

 

−∞ =2 m=−

2 Tmω (r)2 Sm (θ, ϕ; aω)e

−iωt

−2 Tmω (r)−2 Sm (θ, ϕ; aω)e

(91a)

dω,

−iωt

dω. (91b)

Finally, when acting on a single mode of the mode-decomposed Weyl scalars, the symmetry operators yield 1 |2| λm ψ0 , 2 1 S ψ4 = − |−2| λm ψ4 , 2 Sψ0 = −

1 |2| λm ψ0 + ζ ζ¯ S0 T , 2 1 R ψ4 = − |2| λm ψ4 + ζ ζ¯ S4 T , 2 Rψ0 = −

(92)

where |s| λmω := s λmω + |s| + s is independent of the sign of s (This is distinct from Chandrasekhar’s eigenvalue which is given in Eq. (104).). Solutions to the radial Teukolsky equation may be written in terms of a pair of homogeneous mode basis functions chosen according to their asymptotic behavior at the four null boundaries to the spacetime. For radiative (ω = 0) modes, the four common choices are denoted: • “in”: representing waves coming in from I − then partially falling into the horizon and partially scattering back out to I + ; these modes are purely ingoing into the horizon. • “up”: representing waves coming up from H − then partially traveling out to I + and partially scattering back into H + ; these modes are purely outgoing at infinity. • “out”: representing waves coming from I − and H − then traveling out to I + ; these modes are purely outgoing from the horizon. • “down”: representing waves coming from I − and H − then traveling down to H + ; these modes are purely incoming at infinity. These have asymptotic behavior given by:

in s Rmω (r)

∼



0 in,ref −1−2s +iωr∗ e s Rmω r

in,trans −s −ikr∗ + s Rmω Δ e in,inc −1 −iωr∗ + s Rmω r e

r → r+ r→∞ (93a)

36 Black Hole Perturbation Theory and Gravitational Self-Force

up s Rmω (r)

∼

out s Rmω (r)

∼

down s Rmω (r)

∼





up,inc +ikr∗ s Rmω e up,trans −1−2s +iωr∗ e s Rmω r

up,ref −s −ikr∗ + s Rmω Δ e + 0

r → r+ r→∞ (93b)

out,trans +ikr∗ s Rmω e out,inc −1−2s +iωr∗ e s Rmω r

0 + out,ref −1 −iωr∗ + s Rmω r e

r → r+ r→∞ (93c)

down,inc −s −ikr∗ + s Rmω Δ e down,trans −1 −iωr∗ + s Rmω r e

r → r+ r→∞ (93d)



down,ref +ikr∗ e s Rmω

0

where k := ω − mΩ+ with Ω+ := where r∗ := r +

1433

1 2κ+

ln

r−r+ 2M

+

a 2Mr+ r−r− 1 2κ− ln 2M

the angular velocity of the horizon and with κ± :=

r± −r∓ 2 +a 2 ) 2(r±

the surface gravity

on the outer/inner horizon. This behavior is depicted graphically in Fig. 1. Inhomogeneous solutions of the radial Teukolsky equation can then be written in terms of a linear combination of the basis functions: s ψmω (r)

up

up

in in = s Cmω (r)s Rmω (r) + s Cmω (r)s Rmω (r),

(94)

where the weighting coefficients are determined by variation of parameters:  in s Cmω (r)

=

up s Cmω (r)

=

∞

r



r

r+ up

up  s Rmω (r )   s Tmω (r )dr , W (r  )Δ

in  s Rmω (r )   s Tmω (r )dr , W (r  )Δ

(95a) (95b)

up

in (r)∂ [ R in with W (r) = s Rmω r s mω (r)] − s Rmω (r)∂r [s Rmω (r)] the Wronskian [in s+1 practice, it is convenient to use the fact that Δ W (r) = const]. If one computes the “in” and “up” mode functions with normalization such that up,trans in,trans transmission coefficients are unity, s Rmω = 1 = s Rmω , then the gravitational wave strain can be determined directly from ψ4 using Eq. (77) to give:

Fig. 1 Left to right: boundary conditions satisfied by the “in,” “up,” “out,” and “down” solutions

1434

A. Pound and B. Wardell

 lim r(h+ − ih× ) = 2

r→∞

∞ ∞ 

 

−∞ =2 m=−

up −2 Cmω −iω(t−r∗ ) dω, −2 Sm (θ, φ; aω)e ω2

(96) up where the weighting coefficient −2 Cmω is to be evaluated in the limit r → ∞. Similarly, the time-averaged flux of energy carried by gravitational waves passing through infinity and the horizon can be computed from the “in” and “up” normalization coefficients [63]: FEH = lim

r→r+

 2π αmω in |−2 Cmω |2 , ω2

(97)

mω

 2π up | −2 Cmω |2 , r→∞ ω2

FEI = lim

(98)

mω

+4ε )(k +16ε )ω with ε := where αmω := 256(2Mr+ ) k(k |Cmω |2 Similarly, the flux of angular momentum is given by: 5

2

FLHz = lim

r→r+

2

2

2

3

√

M 2 − a 2 /(4Mr+ ).

 2π mαmω in |−2 Cmω |2 , ω3

(99)

mω

 2π m up | −2 Cmω |2 . r→∞ ω3

FLIz = lim

(100)

mω

Similar expressions can be obtained in terms of the modes 2 ψmω of ψ0 by using up up the Teukolsky-Starobinsky identities to relate −2 Cmω to 2 Cmω . The necessary details of how these asymptotic amplitudes are related can be found in Refs. [53,64]. Strictly speaking, the horizon fluxes given here have been derived from the rates of change of the black hole parameters due to shear of the horizon generators [62]. It is generally assumed that these are equivalent to the gravitational wave fluxes, although this has not, to our knowledge, been shown explicitly. When decomposed into modes, each of the Teukolsky-Starobinsky identities separate to yield identities relating the positive spin-weight spheroidal and radial functions to the negative spin-weight ones: D04 (−2 ψmω ) = 14 Cmω 2 ψmω , Δ2 (D0† )4 (Δ2 2 ψmω )

= 4C¯mω −2 ψmω ,

L−1 L0 L1 L2 (2 Smω ) = D −2 Smω , † L0† L1† L2† (−2 Smω ) = D 2 Smω , L−1

(101a) (101b) (101c) (101d)

where Dn := ∂r −

iK r −M + 2n , Δ Δ

Dn† := ∂r +

iK r −M + 2n , Δ Δ

(102a)

36 Black Hole Perturbation Theory and Gravitational Self-Force

Ln := ∂θ + Q + n cot θ,

1435

Ln† := ∂θ − Q + n cot θ,

(102b)

(with K defined above and Q := −aω sin θ +m csc θ ) are essentially mode versions of the GHP differential operators. The constants of proportionality are given by: Cmω = D + (−1)+m 12iMω, D = 2

− 2)   2 + 48(aω)2 2(s λCh m − 2) + 3(m − aω) ,

2 Ch (s λCh m ) (s λm

2

+ 8aω(m − aω)(s λCh m

(103a) − 2)(5s λCh m

− 4) (103b)

where Ch s λmω

:= s λmω + s 2 + s

(104)

is the eigenvalue used by Chandrasekhar [24]. This particular choice of Cmω ensures that the s = +2 and s = −2 modes represent the same physical perturbation. An alternative proportionality constant can be derived such that the s = +2 and s = −2 modes have the same transmission coefficient; see [64] for details. Finally, when written in terms of modes, the homogeneous radiation gauge angular inversion equations can be algebraically inverted to give the modes of the Hertz potentials in terms of the modes of the Weyl scalar ORG ψmω = 16

(−1)m D 2 ψ¯ −ω−m + 12iMω 2 ψmω , |Cmω |2

(105)

IRG ψmω = 16

(−1)m D −2 ψ¯ −ω−m − 12iMω −2 ψmω . |Cmω |2

(106)

where the separability ansatz for the Hertz potentials differs by a factor of ζ −4 from that of the Weyl scalars:

ζ

−4

 ψ

ORG

=

ψ

=

 

−∞ =2 m=−

 IRG

∞ ∞ 

∞ ∞ 

 

−∞ =2 m=−

ORG ψmω (r) 2 Sm (θ, φ; aω)e−iωt dω,

(107)

IRG ψmω (r) −2 Sm (θ, φ; aω)e−iωt dω.

(108)

Alternatively, one can use the radial inversion equations to relate the asymptotic IRG to the asymptotic amplitudes of ψ amplitudes of ψmω 2 mω and to relate the ORG to the asymptotic amplitudes of asymptotic amplitudes of ψmω −2 ψmω . Further details are given in [64] for the IRG case and in [53] for the ORG case.

1436

A. Pound and B. Wardell

Note that in order to transform back to the time-domain solution, as a final step we must perform an inverse Fourier transform. This poses a challenge in gravitational self-force calculations, where non-smoothness of the solutions in the vicinity of the worldline leads to the Gibbs phenomenon of non-convergence of the inverse Fourier transform. Resolutions to this problem typically rely on avoiding directly transforming the inhomogeneous solution by using the methods of extended homogeneous or extended particular solutions. For further details, see [65, 66].

Sasaki-Nakamura Transformation In numerical implementations, the Teukolsky equation can be problematic to work with due to the presence of a long-ranged potential. One approach to this problem is to transform to an alternative master function that satisfies an equation with a more short-ranged potential. The Sasaki-Nakamura transformation is designed to do exactly this. It introduces a new function of the form  ζ¯ 2 X∼

(r 2 + a 2 )1/2 r 2 Þ Þ r12 ζ 4 ψ0

ζ2 (r 2

+ a 2 )1/2 r 2 ÞÞ r12 ζ 4 ψ4

,

(109)

where the factors of ζ ensure that these are purely radial operators (This expression is appropriate when working with the Kinnersley tetrad; for the Carter tetrad, both ¯ definitions for X need to be scaled by a common factor of ζζ to obtain a radial operator.). There is considerable freedom to rescale these expressions by inserting appropriate functions of r; for √ more details, see Ref. [67] in which case the X given here corresponds to r 2 + a 2 r 2 J− J− r12 R in the s = −2 case and to √ 1 Δ2 2 2 2 4 r + a r J+ J+ r 2 R in the s = +2 case.

Metric Perturbations of Schwarzschild Spacetime On a Schwarzschild background spacetime, separability is readily achieved without having to rely on the Teukolsky formalism. Writing the metric perturbation in terms of its null tetrad components, they have GHP type: s=0:

hln : {0, 0},

hmm¯ : {0, 0},

hll : {2, 2},

s = ±1 :

hlm : {2, 0},

hl m¯ : {0, 2},

hnm : {0, −2},

s = ±2 :

hmm : {2, −2},

hm¯ m¯ : {−2, 2}.

hnn : {−2, −2} hnm¯ : {−2, 0}

Here we have gathered the components into scalar (s = 0), vector (s = ±1), and tensor (s = ±2) sectors. In some instances, it is convenient to work with the trace-reversed metric perturbation, h¯ αβ = hαβ − 12 h gαβ . In terms of null tetrad components, the trace is given by h = −2(hln − hmm¯ ), so a trace reversal simply corresponds to the

36 Black Hole Perturbation Theory and Gravitational Self-Force

1437

interchange hln ↔ hmm¯ : h¯ ln = hmm¯ and h¯ mm¯ = hln , with all other components unchanged. The tetrad components may be decomposed into a basis of spin-weighted spherical harmonics:

hab =

∞   

hm ab (t, r) s Ym (θ, φ)

(110)

=|s| m=−

where s = 0 for hln , hmm¯ , hll , and hnn , s = +1 for hlm and hnm , s = −1 for hl m¯ and hnm¯ , s = +2 for hmm , and s = −2 for hm¯ m¯ . Here, we have introduced the spinweighted spherical harmonics s Ym (θ, φ) = s Smω (θ, φ; 0) with the associated eigenvalue s λ := s λm (aω = 0) = ( + 1) − s(s + 1). In the Schwarzschild case, the GHP derivative operators split into operators that (up to an overall factor of 1r ) act only on the two-sphere ð=

√1 (∂θ 2r

+ i csc θ ∂φ − s cot θ ),

ð =

√1 (∂θ 2r

− i csc θ ∂φ + s cot θ ).

(111)

and operators that act only in the t − r subspace

bM 1 ∂ t + f ∂r − 2 , Þ= √ r 2f

1 bM Þ = √ ∂t − f ∂ r − 2 . r 2f

(112)

These expressions are obtained when working with the Carter tetrad. The equivalent operators for the Kinnersley tetrad are Þ = f −1 ∂t + ∂r ,

Þ = 12 (∂t − f ∂r − 2bM/r 2 ).

(113)

The two-sphere operators act as spin-raising and spin-lowering operators to relate spin-weighted spherical harmonics of different spin-weight: √

   1/2 2r ð s Ym (θ, φ) = − ( + 1) − s(s + 1) s+1 Ym (θ, φ), √     1/2 2r ð s Ym (θ, φ) = ( + 1) − s(s − 1) s−1 Ym (θ, φ).

(114a) (114b)

In particular, this provides a relationship between the spin-weighted spherical harmonics and the scalar spherical harmonics. It is convenient to split the six vector and tensor sector components of the metric perturbation into real (even parity) and imaginary (odd parity) parts, representing whether they are even or odd under the transformation (θ, φ) → (π − θ, φ + π ):

1438

A. Pound and B. Wardell m m hm lm = hl,even + i hl,odd ,

m m hm lm ¯ = −hl,even + i hl,odd ,

m m hm nm = hn,even + i hn,odd ,

m m hm nm ¯ = −hn,even + i hn,odd ,

m m hm mm = h2,even + i h2,odd ,

m m hm m ¯m ¯ = h2,even − i h2,odd .

The four scalar sector components are necessarily even parity, so we therefore have seven fields in the even parity sector and three in the odd parity sector. The even and odd parity sectors decouple, meaning that they can be solved for independently. In instances where there is symmetry under reflection about the equatorial plane, this decoupling is more explicit in that the even parity sector only contributes for  + m even and the odd parity sector only contributes for  + m odd. Finally, we can also optionally further decompose into the frequency domain  hm ab (t, r) =

∞ −∞

−iωt hmω dω ab (r)e

(115)

in order to obtain functions of r only. This has the advantage of reducing the problem of computing the metric perturbation to that of solving systems of 7 + 3 coupled ordinary differential equations, one for each (, m, ω).

Alternative Tensor Bases There is some freedom in the specific choice of basis into which tensors are decomposed. In particular, the relative scaling of the l μ and nμ tetrad vectors leads to a slightly different basis if one works with the Kinnersley tetrad rather than the Carter tetrad. It is also possible to work with alternative basis vectors spanning the t − r space. In some instances it is convenient to work with coordinate basis vectors μ μ δt and δr rather than null vectors. One can also choose to omit the factor of 1r in the definition of mμ and m ¯ μ . The choice of basis does not have a fundamental impact, but some choices lead to more straightforward or natural interpretations of the resulting equations. Additionally, as an alternative to a spin-weighted harmonic basis, one could equivalently work with an orthonormal basis of vector and tensor spherical harmonics, which are related to the spin-weighted spherical harmonics by:   1  m ZA := ( + 1)]−1/2 DA Y m = √ −1 Ym mA − 1 Ym m ¯A , 2

1/2   ( − 2)! m ZAB DA DB + 12 ( + 1)ΩAB Y m := 2 ( + 2)!  1  = √ −2 Ym mA mB + 2 Ym m ¯ Am ¯B , 2 for the even parity sector and

(116a)

(116b)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1439

  i  m XA := − ( + 1)]−1/2 εA B DB Y m = − √ −1 Ym mA + 1 Ym m ¯A , 2 (117a)

 ( − 2)! 1/2 C i  m XAB := − 2 ε(A DB) DC Y m =− √ −2 Ym mA mB −2 Ym m ¯ Am ¯B , (+2)! 2 (117b) for the odd parity sector. Here, mA = √1 [1, i sin θ ] and m ¯ A form a complex 2 orthonormal basis on the two-sphere and are related to the two-sphere components of the tetrad vectors mα and m ¯ α by a factor of r. The differential operator D is the covariant derivate on the two-sphere with metric ΩAB = diag(1, sin2 θ ). Note that the definitions of the vector and tensor harmonics given here differs from those of Martel and Poisson [68] in their overall normalization but are otherwise the same. The choice here ensures that the harmonics are unit-normalized:  A  m Zm (θ, φ)Z¯ A (θ, φ)dΩ = δ δmm , (118a)   

A  m Xm (θ, φ)X¯ A (θ, φ)dΩ = δ δmm ,

(118b)

AB  m Zm (θ, φ)Z¯ AB (θ, φ)dΩ = δ δmm ,

(118c)





AB m Xm (θ, φ)X¯ AB (θ, φ)dΩ = δ δmm .

(118d)

For simplicity we opt to work exclusively with a spin-weighted spherical harmonic basis but point out that equivalent results hold for other choices of basis. In particular, the expressions that follow can be transformed to the commonly used Barack-Lousto-Sago [69–71] basis, h(i) m , the A–K basis [72, 73], the MartelPoisson basis [68], and the Berndtson basis [74] using the relations given in Table 1. The table also gives the translation between a Carter null tetrad basis (l α , nα , mα , m ¯ α ) and a t–r coordinate basis (δtα , δ α , mα , m ¯ α ). Note that the BarackLousto-Sago expressions involve the non-trace-reversed metric. A trace reversal (3) (6) in the Barack-Lousto-Sago basis corresponds to the interchange hm ↔ hm , consistent with the trace reversal in the null tetrad basis corresponding to the m . interchange hm ¯ ln ↔ hmm

Regge-Wheeler Formalism and Regge-Wheeler Gauge The Regge-Wheeler formalism is based on the idea of constructing solutions to the linearized Einstein equations from solutions to the scalar wave equation with a potential. In the case of the Regge-Wheeler master function, it is a solution of

1440

A. Pound and B. Wardell

Table 1 Relationship between choices of basis for perturbations of Schwarzschild spacetime Barack-LoustoTetrad Sago A–K     (1) (3) f m m 1 h m Am m +f hm 2 hll + hnn + 2hln 2r     (1) (3) 1 1 m m m hm −f hm K m 2f hll +hnn −2hln 2rf 2   1 1 (2) m m −D m 2 hll − hnn 2rf hm hm mm ¯





f 2

1 (6) 2r hm

E m

(4) hm √

m hm − √ l,even + hn,even 2r 2 (+1)   (5) hm 1 m m √ √ 2f hl,even − hn,even − 2rf 2 (+1)

  (8) h f m m √ √m 2 hl,odd + hn,odd 2r 2 (+1)

  (9) hm 1 m m √ √ 2f hl,odd − hn,odd 2rf 2 (+1)

(−2)! (7) 1 hm 2,even 2r (+2)! hm

(10) hm − 2r1 (−2)! 2,odd (+2)! hm

+

Martel-Poisson Berndtson

Coord.

hm tt

f H0

hm tt

hm rr

1 f H2

hm rr

hm tr

H1

hm tr

K m

K

hm mm ¯

√

√

(+1) √ √ B m − (+1) jtm 2 r 2 √ √ √ √ − (+1) H m − (+1) jrm 2 r 2 √ √ (+1) √ √ C m − (+1) hm t 2 r 2 √ √ √ √ − (+1) J m − (+1) hm r 2 r 2 1 2 1 2



(+2)! m 1 (−2)! F 2

(+2)! m 1 (−2)! G 2r 2

(+2)! m (−2)! G

√

−

√

− √ √

(+1) √ h0 r 2

hm t,even

(+1) √ h1 r 2

hm r,even

(+1) √ h0 r 2

hm t,odd

(+1) √ h1 r 2

hm r,odd

(+2)! m (−2)! G

hm 2,even

(+2)! m 1 (−2)! h2 r 2

(+2)! (−2)! h2

2Ms 2 ψsRW = Ss , r3

hm 2,odd

(119)

where s is the spin of the field (s = 0 for scalar fields, s = 1 for electromagnetic fields, and s = 2 for gravitational fields). Equation (119) is separable in Schwarzschild spacetime using the ansatz ψsRW =

 ∞   1 RW ψ (t, r) 0 Ym (θ, φ), r sm

(120)

=0 m=−

RW (t, r) satisfying the Regge-Wheeler equation with ψsm





   ∂ ∂ 1 ∂ ( + 1) 2M(1 − s 2 ) RW RW ψsm f − − + = Ssm . ∂r ∂r f ∂t 2 r2 r3

(121)

In order to study metric perturbations of Schwarzschild spacetime, we consider the s = 2 case. The Regge-Wheeler master function is then defined in terms of the metric perturbation by: RW ψ2m

√

r 2 ∂r hm 2r hm f r,odd 2,odd +√ . := − √ r ( + 1) ( − 1)( + 1)( + 2)

(122)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1441

It satisfies the s = 2 Regge-Wheeler equation with source derived from the modedecomposed stress-energy tensor: RW S2m

√ m m ) r ∂r (f T2,odd 2f Tr,odd +√ . = 16π √ ( + 1) 2( − 1)( + 1)( + 2)

(123)

Different conventions for the source exist in the literature. For example, the source given in Ref. [65] differs from that given here by a factor of f ; this is a consequence of the left-hand side of their Regge-Wheeler equation (Eq. (2.13) in Ref. [65]) also differing from our Eq. (121) by a factor of f . Rather than working with the Regge-Wheeler master function itself, it is often preferable to introduce two closely related functions: the Cunningham-PriceMoncrief (CPM) master function defined by √

2r 2 m m ∂r (rhm := − √ ) − ∂ (rh ) − 2h t t,odd r,odd t,odd , ( + 1) ( − 1)( + 2) (124) and the Zerilli-Moncrief (ZM) master function defined by CPM ψm

ZM ψm :=



2r 2 m ˜ − rf ∂ ) , K K˜ m + (f 2 h˜ m r rr ( + 1) Λ

where Λ := ( − 1)( + 2) +

K˜ m h˜ m rr

6M r

(125)

and

 m ( + 1) − 2r f ∂r hm 2f h 2,even r,even + √ , := hm mm ¯ − √ ( + 1) 2( − 1)( + 1)( + 2)   2∂r (r hm ∂r r 2 ∂r heven r,even ) 2 m +√ := hrr − √ ( + 1) 2( − 1)( + 1)( + 2)

(126a) (126b)

are gauge invariant fields. The CPM master function satisfies the same s = 2 Regge-Wheeler equation as the Regge-Wheeler master function, but with a different source given by: √

CPM Sm



2r 2 m m ∂r (r Tt,odd ) − ∂t (r Tr,odd ) . = 16π √ ( + 1) ( − 1)( + 2)

(127)

The RW and CPM master functions are related by a time derivative (plus source terms): RW ψ2m

√ 2 1 16π r 2 f CPM T m . = ∂t ψm − √ 2 ( − 1)( + 2) ( + 1) r,odd

(128)

1442

A. Pound and B. Wardell

The ZM master function satisfies the Zerilli equation (the Regge-Wheeler equation with a different potential)

  ∂ ∂ 1 ∂ ZM ZM ZM ψm f − − V = Sm , ∂r ∂r f ∂t 2

(129)

( + 1) 6M 72M 2 f 24M(r − 3M) − + − , r2 r3 Λ2 r 4 Λr 4

(130)

where V ZM =

and where the ZM source is ZM Sm

√ 16π 4f 16π 2 m T r T m = −√ √ Λ ( + 1) r,even ( − 1)( + 1)( + 2) 2,even  r  M2  2 M ( − 1)( + 2)(l 2 + l − 4)+12(2 +  − 5) +84 2 + ( + 1)Λ Λ r r  24M f 2 Trrm + 2r f Tmm − 2r 2 f ∂r (f Trrm − f −1 Tttm ) + m ¯ . Λ (131)

Regge-Wheeler Formalism in the Frequency Domain Transforming to the frequency domain, the Regge-Wheeler and Zerilli equations become a set of ordinary differential equations, one for each (, m, ω) mode. Solutions to these equations may be written in terms of a pair of homogeneous mode basis functions chosen according to their asymptotic behavior at the four null boundaries to the spacetime. For radiative (ω = 0) modes, the four common choices are denoted “in,” “up,” “out,” and “down”, with the same interpretation as described in Section “Mode-Decomposed Equations in Kerr Spacetime” for the Teukolsky equation. These have asymptotic behavior given by: in s Xmω (r)

∼

up s Xmω (r)

∼

out s Xmω (r)

∼







in,trans −iωr∗ 0 + s Xmω e in,ref +iωr∗ in,inc −iωr∗ + s Xmω e s Xmω e

r → 2M r→∞ (132a)

up,inc +iωr∗ s Xmω e up,trans +iωr∗ s Xmω e

up,ref −iωr∗ + s Xmω e + 0

r → 2M r→∞ (132b)

out,trans +iωr∗ s Xmω e out,inc +iωr∗ s Xmω e

0 + out,ref −iωr∗ + s Xmω e

r → 2M r→∞ (132c)

36 Black Hole Perturbation Theory and Gravitational Self-Force

down s Xmω (r)

∼



down,ref +iωr∗ e s Xmω

0

1443

down,inc −iωr∗ + s Xmω e down,trans −iωr∗ + s Xmω e

r → 2M r→∞ (132d)

r − 1) is the Regge-Wheeler tortoise coordinate. where r∗ = r + 2M ln( 2M Inhomogeneous solutions of the Regge-Wheeler equation can then be written in terms of a linear combination of the basis functions: s ψmω (r)

up

up

in in = s Cmω (r)s Xmω (r) + s Cmω (r)s Xmω (r),

(133)

where the weighting coefficients are determined by variation of parameters: 

∞

=

in s Cmω (r)



up s Cmω (r)

=

r r 2M

up  s Xmω (r )   s Smω (r )dr , W (r  )f

(134a)

in  s Xmω (r )   s Smω (r )dr , W (r  )f

(134b)

up

up

in (r)∂ [ X in with W (r) = s Xmω r s mω (r)] − s Xmω (r)∂r [s Xmω (r)] the Wronskian [in practice, it is convenient to use the fact that f (r)W (r) = const].

Transformation Between Regge-Wheeler and Zerilli Solutions Homogeneous solutions to the Zerilli equation can be obtained from homogeneous solutions to the Regge-Wheeler equation by applying differential operators:

ZM,up Xmω

=

 ( − 1)( + 1)( + 2) +



RW,up 72M 2 f X r 2 Λ 2 mω

+ 3Mf

RW,up

d 2 Xmω dr

,

( − 1)( + 1)( + 2) + 12iωM

(135a) ZM,in Xmω =

 ( − 1)( + 1)( + 2) +



72M 2 f XRW,in r 2 Λ 2 mω

+ 3Mf

( − 1)( + 1)( + 2) − 12iωM

RW,in d 2 Xmω dr

.

(135b)

The constant of proportionality here is such that the transmission coefficients of the two Zerilli solutions is the same as that of the Regge-Wheeler solution.

Transformation Between Regge-Wheeler and Teukolsky Formalism The modes of the CPM master function are related to the modes of the Teukolsky radial function by the Chandrasekhar transformation 2 ψmω −2 ψmω

√  CPM  = − i4rD 2 DD rψmω , √  CPM  , = − i 16D r 2 f 2 D † D † rψmω

(136a) (136b)

1444

A. Pound and B. Wardell

where D = ( − 1)( + 1)( + 2) is the Schwarzschild limit of the constant that appears in the Teukolsky-Starobinsky identities, Eq. (103b). In the absence of sources, this can be inverted to give  ,   r 2 f −1 D † r 2 f 2 D † rf 2 ψmω , 

CPM = ψmω

√ 1 r 3 DD r12 −2 ψmω D Cmω

(137)

CPM ψmω =

√ 1 † 4 D Cmω

(138)

where Cmω is the Schwarzschild limit of the second constant that appears in the Teukolsky-Starobinsky identities, Eq. (103a).

Gravitational Waves As in the radiation gauge case, the gravitational wave strain can be determined directly from ψ ZM and ψ CPM . There is a slight subtlety in that the Regge-WheelerZerilli gauge in which the metric is typically reconstructed is not compatible with the transverse-traceless gauge in which gravitational waves are normally defined (it is easy to see this since hmm = 0 = hm¯ m¯ in the Regge-Wheeler-Zerilli gauge). Instead, we can use the Chandrasekhar transformation in Eq. (136) to first transform to ψ4 and then compute the strain from that as we did in radiation gauge. Doing so we have r(h+ − ih× ) =

 ∞   =2 m=−

√ D ZM CPM (ψm − iψm )−2 Ym (θ, φ), 2

(139)

where it is understood that equality holds in the limit r → ∞ (at fixed u = t − r ∗ ). If we work in the frequency domain and compute the “in” and “up” mode functions in,trans = 1 = with normalization such that transmission coefficients are unity, s Xmω up,trans ZM and ψ CPM are given by the “up” weighting coefficients C ZM,up X , then ψ s mω mω m m CPM,up and Cmω evaluated in the limit r → ∞. Similarly, the time-averaged flux of energy carried by gravitational waves passing through infinity and the horizon can be computed from the “in” and “up” weighting coefficients:

 D ZM,in 2 CPM,in 2 (140a) ω2 |Cmω FEH = lim | + |Cmω | , r→2M 64π mω

 D ZM,up 2 CPM,up 2 I 2 FE = lim (140b) ω |Cmω | + |Cmω | . r→∞ 64π mω

Likewise, the flux of angular momentum through infinity and the horizon can be computed from the “in” and “up” normalization coefficients: FLHz = lim

r→2M



 D ZM,in 2 CPM,in 2 mω |Cmω | + |Cmω | , 64π

mω

(141a)

36 Black Hole Perturbation Theory and Gravitational Self-Force

FLIz



 D ZM,up 2 CPM,up 2 mω |Cmω | + |Cmω | . = lim r→∞ 64π

1445

(141b)

mω

Metric Reconstruction in Regge-Wheeler Gauge Much like in the Teukolsky formalism, the CPM master function is gauge invariant and may be used to reconstruct the metric perturbation in a chosen gauge. In the m m Regge-Wheeler gauge, defined by the choice hm a,even = h2,even = h2,odd = 0, the odd parity metric perturbation is given by: √ √  CPM  f ( + 1)  16π r 2 T m , ∂r + f −1 ∂t rψm + √ ( − 1)( + 2) l,odd 2 2r √ √  CPM  f ( + 1)  16π r 2 T m , ∂r − f −1 ∂t rψm + = √ ( − 1)( + 2) n,odd 2 2r

hm l,odd =

(142)

hm n,odd

(143)

and the even parity metric perturbation is given by [65]: 32π r 2 T m , ( + 1)Λ tt

r Λ ( + 1) ZM m = − h ψ ∂r hm m mm ¯ + mm ¯, 2r f 2f 2

16π r 2 2r ZM ZM Ttrm − ∂t Tttm , = r∂t ∂r ψm + rB ∂t ψm + ( + 1) fΛ

ZM ZM hm mm ¯ = f ∂r ψm + Aψm −

hm rr hm tr

2 hm tt = f hrr + √

(144)

8πf T m , 2( − 1)( + 1)( + 2) 2,even

where

2 1 3M  4M  A(r) := ( − 1)( + 1)( + 2) + ( − 1)( + 2) + , rΛ 4 2r r (145)

  ( − 1)( + 2) 3M 3M 2 2 (146) 1− − 2 . B(r) := rf Λ 2 r r As in the Teukolsky case, in order to transform back to the time-domain solution, as a final step, we must perform an inverse Fourier transform. This poses a challenge in gravitational self-force calculations, where non-smoothness of the solutions in the vicinity of the worldline leads to the Gibbs phenomenon of non-convergence of the inverse Fourier transform. Resolutions to this problem typically rely on avoiding directly transforming the inhomogeneous solution by using the methods of extended homogeneous or extended particular solutions. For further details, see [65, 66].

1446

A. Pound and B. Wardell

Lorenz Gauge In the case of perturbations of a Schwarzschild black hole, the equations for the metric perturbation itself are separable. This makes it practical to work in the Lorenz gauge and to directly solve the Lorenz gauge field equations for the metric perturbation. Rewriting the Lorenz gauge condition, Eq. (11), in terms of null tetrad components, we have four gauge equations: (Þ − 2ρ  )hll + (Þ − 2ρ)hmm¯ − 2ρhln − (ðhl m¯ + ð hlm ) = 0, 







(147a)

(Þ − 2ρ)hnn + (Þ − 2ρ )hmm¯ − 2ρ hln − (ð hnm + ðhnm¯ ) = 0,

(147b)

(Þ − 3ρ  )hlm + (Þ − 3ρ)hnm − ðhln − ð hmm = 0,

(147c)







(Þ − 3ρ )hl m¯ + (Þ − 3ρ)hnm¯ − ð hln − ðhm¯ m¯ = 0.

(147d)

These decouple into three even parity equations (the first two and the real part of either the third or fourth) and one odd parity equation (the imaginary part of either the third or fourth equation). Similarly, the Lorenz gauge-linearized Einstein equation, Eq. (12), yields ten field equations (7 even and 3 odd) given by: ˆ mm¯ − hln ) = 8π T , (h

(148a)

ˆ − 8ψ2 + 8ρρ  )(hln + hmm¯ ) + 4ρ 2 hnn + 4ρ 2 hll ( + 4ρ(ðhnm¯ + ð hnm ) + 4ρ  (ðhl m¯ + ð hlm ) = −16π(Tln + Tmm¯ ), (148b) ˆ + 4ρρ  )hll + 4ρ 2 (hln + hmm¯ ) + 4ρ(ðhl m¯ + ð hlm ) = −16π Tll , (

(148c)

ˆ  + 4ρρ  )hnn + 4ρ 2 (hln + hmm¯ ) + 4ρ  (ð hnm + ðhnm¯ ) = −16π Tnn , ( (148d) ˆ − 6ψ2 + 4ρρ  )hlm + 4ρ 2 hnm + 2ρð(hln + hmm¯ ) ( + 2ρ  ðhll + 2ρð hmm = −16π Tlm ,

(148e)

¯ˆ ( − 6ψ2 + 4ρρ  )hl m¯ + 4ρ 2 hnm¯ + 2ρð (hln + hmm¯ ) + 2ρ  ð hll + 2ρðhm¯ m¯ = −16π Tl m¯ ,

(148f)

¯ˆ  ( − 6ψ2 + 4ρρ  )hnm + 4ρ 2 hlm + 2ρ  ð(hln + hmm¯ ) + 2ρðhnn + 2ρ  ð hmm = −16π Tnm ,

(148g)

ˆ  − 6ψ2 + 4ρρ  )hnm¯ + 4ρ 2 hl m¯ + 2ρ  ð (hln + hmm¯ ) ( + 2ρð hnn + 2ρ  ðhm¯ m¯ = −16π Tnm¯ , ˆ mm + 4ρðhnm + 4ρ  ðhlm = −16π Tmm , h

(148h) (148i)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1447

¯ˆ    h m ¯m ¯ + 4ρð hnm ¯ + 4ρ ð hl m ¯ = −16π Tm ¯m ¯

(148j)

where the operators ˆ := −2ÞÞ + 2ρ  Þ + 2ρÞ + 2ðð , 

ˆ  := −2Þ Þ + 2ρÞ + 2ρ  Þ + 2ð ð, 

¯ˆ  := −2ÞÞ + 2ρ  Þ + 2ρÞ + 2ð ð,

¯ˆ   := −2Þ Þ + 2ρÞ + 2ρ  Þ + 2ðð ,

all coincide with the scalar wave operator when acting on type {0, 0} objects (but differ when acting on objects of generic GHP type). Note that we have chosen here to work with the non-trace-reversed metric perturbation; equivalent equations for the trace-reversed perturbation can be obtained by noting that a trace-reversal corresponds to the interchange hln ↔ hmm¯ . The Lorenz gauge equations can be decomposed into the same basis of spinweighted spherical harmonics as for the metric perturbation itself. The modedecomposed equations follow immediately from the above GHP expressions along with Eqs. (114) and either (113) or (112) for the GHP derivative operators (the specific form for the mode-decomposed equations depends on the choice of tetrad).

Lorenz Gauge Formalism in the Frequency Domain Following a procedure much like in the Regge-Wheeler and Teukolsky cases, one can construct solutions to the Lorenz gauge equations by working in the frequency domain and solving ordinary differential equations [71, 75, 76]. The only additional complexity is that for each (, m, ω) mode we must now work with a system of k coupled second-order radial equations with 2k linearly independent homogeneous solutions. There are k = 7 even parity equations and k = 3 odd parity equations in general, although these can be reduced to 4 + 2 equations using the 3 + 1 gauge conditions. The number of equations is also further reduced in certain special cases such as static or low multipole modes. As we did in the Regge-Wheeler and Teukolsky cases, it is natural to divide these into k “in” solutions and k “up” solutions satisfying appropriate boundary conditions at the horizon or radial infinity. Then, using variation of parameters, the inhomogeneous solutions are given by: (i)

up

(i),in

(i),up

hmω (r) = Cin mω (r) · hmω (r) + Cmω (r) · hmω (r)

(149)

where i = 1, . . . , k represent the k components of the metric perturbation and where (i),in hmω (r) are vectors of k linearly independent homogeneous solutions for a given i. in/up To compute the weighting coefficient vectors Cmω (r), we define a 2k × 2k matrix of homogeneous solutions by:  Φ(r) =

(i),in

(i),up

−hmω hmω (i),up −∂r h(i),in mω ∂r hmω

 .

(150)

1448

A. Pound and B. Wardell

The vectors of weighting coefficients are then obtained with the standard variation of parameter prescription 

Cin (r) Cup (r)



 =

Φ −1 (r  )



0 T(r  )



dr  ,

(151)

where T(r  ) represents the vector of k sources constructed from the components of the stress-energy tensor projected onto the basis and decomposed into modes. The limits on the integral depend upon whether the “in” or “up” weighting coefficient are being solved for, in the same way as for the Regge-Wheeler and Teukolsky cases. As in the Regge-Wheeler and Teukolsky cases, in order to transform back to the time-domain solution, as a final step, we must perform an inverse Fourier transform. This poses a challenge in gravitational self-force calculations, where non-smoothness of the solutions in the vicinity of the worldline leads to the Gibbs phenomenon of non-convergence of the inverse Fourier transform. Resolutions to this problem typically rely on avoiding directly transforming the inhomogeneous solution by using the methods of extended homogeneous or extended particular solutions. For further details, see [65, 66].

Lorenz Gauge Metric Reconstruction from Regge-Wheeler Master Functions As an alternative to directly solving the 7 + 3 coupled Lorenz gauge field equations, Berndtson [74] showed that the solutions could instead be reconstructed from particular solutions to the s = 0, 1, and 2 Regge-Wheeler-Zerilli equations, along with a fourth field obtained by solving the s = 0 Regge-Wheeler equation sourced by the other s = 0 field. The explicit expressions are quite unwieldy, particularly when sources are included. Focusing only on the relatively simple odd sector and ignoring special cases such as low multipoles or ω = 0 modes, Berndtson’s expressions may be written as: √ √  RW 

ψ1m 2λ  RW  f (+1) 2 8πf + D r rψ D (T m −T m ), 0 0 2m + 3r 2(iω)2 r r2 (iω)2 l,odd n,odd (152) √ √  RW 

2λ  RW  f ( + 1) 2 † ψ1m 8πf + D0† rψ2m r D0 − (T m −T m ), hm n,odd = 2 2 3r 2(iω) r r (iω)2 l,odd n,odd (153) √   RW  2λ RW ( − 1)( + 1)( + 2) RW 16πf m ψ1m +f ∂r rψ2m T , + ψ2m + hm 2,odd = 2 2 3 (iω) r (iω)2 2,odd (154) hm l,odd =−

where D0 and D0† are the operators defined in Eq. (102) specialized to the Schwarzschild (a = 0) case. Equivalent expressions for the even sector are

36 Black Hole Perturbation Theory and Gravitational Self-Force

1449

significantly more complicated and are given in Appendix A of Ref. [74], while expressions for low multipoles and ω = 0 modes are given elsewhere in the same reference.

Gravitational Waves As in the Regge-Wheeler-Zerilli and Teukolsky cases, the flux of gravitational wave energy and angular momentum may be computed from the asymptotic values of the fields. In the Lorenz gauge case where one solves for the metric perturbation directly, the gravitational wave strain is simply given by hmm as in Eq. (76): r(h+ + ih× ) = r hmm =

  ∞  

∞

=2 m=− −∞

−iω(t−r∗ ) r hmω dω, mm 2 Ym (θ, φ)e

(155) where it is understood that the equality holds in the limit r → ∞. Similarly, the energy fluxes are given explicitly by [70]:  ω2 r 2 |hmω |2 , r→∞ 16π mm

FEI = lim

(156a)

mω



FEH = lim

r→2M

mω

1 × 256π M 2 (1 + 16M 2 ω2 )

   ( − 1)( + 1)( + 2)rf hmω ll    − 2 ( − 1)( + 2)(1 + 4iMω)r f hmω lm 2   + 4iMω(1 + 4iMω)rhmω mm  ,

(156b)

and the angular momentum fluxes are given by:  mωr 2 2 |hmω mm | , r→∞ 16π

FLIz = lim

(157a)

mω

FLHz = lim

r→2M



m

mω

256π M 2 ω(1 + 16M 2 ω2 )

×

   ( − 1)( + 1)( + 2)rf hmω ll    − 2 ( − 1)( + 2)(1 + 4iMω)r f hmω lm 2   + 4iMω(1 + 4iMω)rhmω mm  .

(157b)

1450

A. Pound and B. Wardell

Small Objects in General Relativity In the previous section, we reviewed black hole perturbation theory with a generic source term. In this section, we consider how to formulate the source describing a small object. This is the local problem in self-force theory: in a spacetime perturbed by a small body, what are the sources in the field equations (7) and (8)? Moreover, if the body’s bulk motion is described by an equation of motion (13), what are the forces on the right-hand side? The result of the analysis is (i) a skeletonization of the small body, in which the body is reduced to a singularity equipped with the body’s multipole moments, together with (ii) an equation of motion governing the singularity’s trajectory. The setting here is very general: the background can be any vacuum spacetime. Our coverage of the subject is terse, and we refer to Refs. [28, 29] for detailed reviews or to Ref. [27] for a nonexpert introduction.

Matched Asymptotic Expansions For simplicity, we assume that, outside of the small object, the spacetime is vacuum and that the perturbations are due solely to the object. Over most of the spacetime, the metric is well described by the external background metric gαβ . However, very near the object, in a region comparable to the object’s own size, the object’s gravity dominates. In this region, which we call the body zone, the approximation (1) breaks down. This problem is usually overcome in one of two ways: using effective field theory [77] (common in post-Newtonian and post-Minkowskian theory [78]) or using the method of matched asymptotic expansions (e.g., see Refs. [79, 80] for broad introductions, Refs. [81–83] for applications in post-Newtonian theory, and Refs. [11, 13, 14, 17–19, 84–91] and the reviews [28, 29] for the work most relevant here). Here we adopt the latter approach. We let ε = m/R, where m is the small object’s mass and R is a characteristic length scale of the external universe; in a small-mass-ratio binary, R will be the mass M of the primary, while in a weakfield binary it can be the orbital separation. We then assume Eq. (1), which we dub the outer expansion, is accurate outside the body zone. Near the object, we assume the metric is well approximated by a second expansion, called an inner expansion, that effectively zooms in on the body zone. To make this “zooming in” precise, we first choose some measure, r, of radial distance from the object, with r an order-1 function of the external coordinates x α . We then define the scaled distance r˜ := r/ε. The body zone corresponds to r ∼ εR but to r˜ ∼ R. The outer expansion (1) is an approximation in the limit ε → 0 at fixed coordinate values and therefore at fixed r. The inner expansion is instead an approximation in the limit ε → 0 at fixed r: ˜ obj

exact (1) (2) gμν (r, ˜ ε) = gμν (r) ˜ + εHμν (r) ˜ + ε2 Hμν (r) ˜ + O(ε3 ).

(158)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1451 obj

(We suppress other coordinate dependence.) In the body zone, the coefficients gμν obj (n) and Hμν are order unity. The background metric gμν represents the metric of the (n) arise from small object’s spacetime as if it were isolated, and the perturbations Hμν the tidal fields of the external universe and nonlinear interactions between those tidal fields and the body’s own gravity. In our construction of the inner expansion, we have assumed that there is only one scale that sets the size of the body zone: the object’s mass m. This implicitly assumes that the object is compact, such that its typical diameter d is comparable to m. That in turn implies that the object’s th multipole moment scales as

md  ∼ m+1 = ε+1 R +1 .

(159)

For a noncompact object, we would need to introduce additional perturbation parameters in the outer expansion and additional scales in the inner expansion. Our inner expansion also assumes that while there is a small length scale associated with the object, there is no analogous timescale; in other words, the object is not undergoing changes on its own internal timescale ∼m. This is equivalent to assuming the object is in quasi-equilibrium with its surroundings. In practice it corresponds to a spatial derivative near the object dominating over a time derivative by one power of r (in the outer expansion) or by one power of ε (in the inner expansion). To date, inner expansions have been calculated for tidally perturbed Schwarzschild and Kerr black holes as well as nonrotating or slowly rotating neutron stars; see Refs. [83, 92–100] for recent examples of such work. (Ref. [101] alerts readers to a significant error in some of the work on slowly rotating bodies.) These calculations represent one of the major applications of the methods of black hole perturbation theory reviewed in the previous section, and they form part of an ongoing endeavor to include tidal effects in gravitational wave templates and to infer properties of neutron stars from observed signals [102, 103]. However, in self-force applications, we require only a minimal amount of information from the inner expansion, often much less than is provided in the above references. The necessary information is extracted from a matching condition: because the two expansions are expansions of the same metric, they must match one another when appropriately compared. The most pragmatic formulation of this condition is that the inner and outer expansions must commute. If we perform an outer expansion of the inner expansion (or equivalently, re-expand it for r  εR) and if we perform an inner expansion of the outer expansion (or equivalently, expand for r  R) and express the end results as functions of r, then both procedures yield a double series for small ε and small r. We assume that these two double expansions agree with one another, order by order in ε and r. A primary consequence of this matching condition is that near the small object, the metric perturbations in the outer expansions must behave as

1452

A. Pound and B. Wardell (n,−n)

h(n) μν =

hμν rn

(n,−n+1)

+

hμν r n−1

(n,−n+2)

+

hμν r n−2

+ ...,

(160)

(n) 1 growing large at small r. If h(n) μν grew more rapidly (e.g., if hμν ∼ r n+1 ), then the outer expansion could not match an inner expansion. Moreover, the coefficient of obj 1 r n matches a term in the r  εR expansion of gμν :

obj

gμν = ημν +

(1,−1)

εhμν r

(2,−2)

+

ε2 hμν r2

(3,−3)

+

ε3 hμν r3

+ ...,

(161)

where ημν is the metric of flat spacetime. The terms in this series are in one-to-one obj correspondence with the multipole moments of gμν , which in turn can be interpreted (n,−n) in terms as the multipole moments of the object itself. This allows us to write hμν of the object’s first n moments; one new moment arises at each new order in ε, just as one would expect from the scaling (159). The moments, together with the general form (160), are all we require from the inner expansion. After it is obtained, we can effectively “integrate out” the body zone from the problem, as described below. To intuitively understand the meanings of the double expansions and of expressions such as (160) and (161), we can interpret them as being valid in the buffer region εR  r  R. This region is the large-r˜ limit of the body zone but the small-r limit of the external universe.

Tools of Local Analysis To determine more than just the general form of the perturbations, we substitute Eq. (160) into the Einstein equations (7)–(8) and then solve order by order in ε and r. These types of local calculations are carried out using two tools: covariant nearcoincidence expansions and expansions in local coordinate systems. Reference [28] contains a thorough, pedagogical introduction to both methods. Here we summarize only the basic ingredients. Covariant expansions are based on Synge’s world function: 

σ (x α , x α ) =

1 2



2 ds

,

(162)

β

which is equal to 1/2 the square of the proper distance s (as measured in gμν )  between the points x α and x α along the unique geodesic β connecting the two  points; for a given x α , this is a well-defined function of x α so long as x α is within  the convex normal neighborhood of x α . The other necessary tool is the bitensor    μ gμ (x α , x α ), which parallel propagates vectors from x α to x α . A smooth tensor  field Aμ ν at x α can be expanded around x α as

36 Black Hole Perturbation Theory and Gravitational Self-Force

1453

        Aμ ν (x α ) = gμμ gνν Aμ ν − Aμ ν ;α  σ α + 12 Aμ ν ;α  β  σ α σ β + O(λ3 ) , (163)  where we use λ := 1 to count powers of distance√between x α and x α . The vector σα  := ∇α  σ is tangent to β and has a magnitude 2σ equal to the proper distance  between x α and x α . The (perhaps unexpected) minus sign in Eq. (163) arises because σα  points away from x α rather than toward it.  When a derivative, either at x α or at x α , acts on an expansion like (163), μ it involves derivatives of gμ and σα  . These can then be re-expanded using, for example, μ

gμ ;ν =

1 μ ν ρ  gρ  gν R μ ν  δ  σ δ + O(λ2 ) 2

(164)

and      σ;μμ = −gμν gμ ν  + 16 Rμ α  ν  β  σ α σ β + O(λ3 );

(165)

see Eqs. (6.7)–(6.11) of Ref. [28]. To make use of these tools, we install a curve γ , with coordinates zα (τ ), in the spacetime of gμν , which will be a representative worldline for the small object. Recall that τ is proper time as measured in gμν . If the object is a material body, the worldline will be in its physical interior. If the object is a black hole, the worldline will only serve as a reference point for the field outside the black hole; mathematically, γ resides in the manifold on which gμν lives, not the manifold on obj which gμν lives. In either case, we only analyze the metric in the object’s exterior, never in its interior. A suitable measure of distance from γ is 

s(x α , x α ) :=





Pμ ν  σ μ σ ν ,

(166)



where x α = zα (τ  ) is a point on γ near x α and Pμν := gμν + uμ uν projects μ orthogonally to γ . Here we have introduced γ ’s four-velocity uμ = dz dτ , normalized  to gμν uμ uν = −1. Note that s remains positive regardless of whether x α and x α are connected by a spacelike, timelike, or null geodesic. In terms of these covariant quantities, the expansion (160) can be written more concretely as ⎡ μ

ν

α ⎣ h(n) μν (x ) = gμ gν

(n,−n)





hμ ν  (x α , σ α /s) sn

(n,−n+1)

+

hμ ν 





(x α , σ α /s)

sn−1

⎤ + O(λ−n+2 )⎦.

(167)  s represents the distance from x α to x α [playing the role of r in (160)], and   the vector σ α /s represents the direction of the geodesic connecting x α to x α . Generically, log(s) terms also appear [104], but we suppress them for simplicity.

1454

A. Pound and B. Wardell

Rather than directly substituting an ansatz of the form (167) into the vacuum field (n,p) equations and solving for the coefficients hμ ν  , it is typically more convenient to adopt a local coordinate system centered on γ and afterward recover (167). Here we adopt Fermi-Walker coordinates (τ, x a ), which are quasi-Cartesian coordinates constructed from a tetrad (uα , eaα ) on γ . The spatial triad eaα is Fermi-Walker transported along the worldline according to Deaα = aa uα , dτ

(168)

μ

D := uμ ∇μ . aa := aμ ea is a spatial component of the covariant acceleration where dτ Duμ μ a := dτ ; this will eventually become the left-hand side of Eq. (13). At each value of proper time τ , we send a space-filling family of geodesics orthogonally outward from x¯ α = zα (τ ), generating a spatial hypersurface Στ . Each such surface is labeled with a coordinate time τ , and each point on the surface is labeled with spatial coordinates

x a = −eαa¯ σ α¯ ,

(169)

α¯ where σα¯ := ∇α¯ σ is tangent to Στ¯ , satisfying

σα¯ u = 0. The magnitude of these  coordinates, given by s := δab x a x b = gα¯ β¯ σ α¯ σ β¯ , is the proper distance from 

x¯ α to x α . In the special case that x α = x¯ α , s and s are identical. The analog of Eq. (163) is the coordinate Taylor series Aμ ν (τ, x a ) = Aμ ν (τ, 0) + Aμ ν ,a (τ, 0)x a + 12 Aμ ν ,ab (τ, 0)x a x b + O(s 3 ).

(170)

In these coordinates, the four-velocity reduces to uμ = (1, 0, 0, 0), and the acceleration to a μ = (0, a i ). The external background metric, which is smooth at x a = 0, is given by   gτ τ = −1 − 2ai x i − Rτ iτj + ai aj x i x j + O(s 3 ),

(171a)

gτ a = − 23 Rτ iaj x i x j + O(s 3 ),

(171b)

gab = δab − 13 Raibj x i x j + O(s 3 ),

(171c)

reducing to the Minkowski metric on γ , and the only nonzero Christoffel symbols on γ are Γτaτ = a a and Γττa = aa . If the worldline is not accelerating, the coordinates become inertial along γ . The Riemann tensor components in Eq. (171) are evaluated on the worldline. Higher powers of x a in the expansion come with higher powers of the acceleration, derivatives of the Riemann tensor, and nonlinear combinations of the Riemann tensor. In a vacuum background, the Riemann tensor on the worldline is commonly decomposed into tidal moments. The quadrupolar moments are defined as

36 Black Hole Perturbation Theory and Gravitational Self-Force

1455

Eab := Rτ aτ b ,

(172)

Bab := 12 εpq (a Rb)τpq .

(173)

Higher moments involve derivatives of the Riemann tensor. Equations (44)–(48) of Ref. [105] display the background metric (171) through order s 3 and the octupolar tidal moments. Reference [106] presents the background metric in an alternative, light-cone-based coordinate system through order λ4 and the hexadecapolar moments. Given the local Fermi-Walker coordinates, one can adopt a coordinate analog of Eq. (167): (n,−n)

h(n) μν =

hμν

(τ, na )

sn

(n,−n+1)

+

hμν

(τ, na )

s n−1

(n,−n+1)

+

hμν

(τ, na )

s n−2

+O(s −n+3 ).

(174)

a

Here na = xs = δ ab ∂b s is a radial unit vector. To facilitate solving the field equations, we can expand the coefficients in angular harmonics: a h(n,p) μν (τ, n ) =



(n,p,)

hμνL (τ )nˆ L ,

(175)

≥0

where L := i1 · · · i is a multi-index, and nˆ L := nL , where nL := ni1 · · · ni . The angular brackets denote the symmetric, trace-free (STF) combination of indices, where the trace is defined with δab . This is equivalent to expanding the coefficients (n,p) hμν in scalar spherical harmonics: a h(n,p) μν (τ, n ) =

 ∞  

h(n,p,m) (τ )Y m (ϑ, ϕ), μν

(176)

=0 m=−

where the angles (ϑ, ϕ) are defined in the natural way from na = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ).

(177)

Like spherical harmonics, nˆ L is an eigenfunction of the flat-space Laplacian, (n,p,) nˆ L . One can further decompose hμνL into satisfying δ ab ∂a ∂b nˆ L = − (+1) s2 irreducible STF pieces that are in one-to-one correspondence with the coefficients in a tensor spherical harmonic decomposition. We refer the reader to Appendix A of Ref. [107] for a detailed introduction to such expansions and a collection of useful identities. The general local solution in the buffer region can be found by substituting the expansions (171) and (174) into the vacuum field equations and working order by order in ε and s. Because spatial derivatives increase the power of 1/s, dominating

1456

A. Pound and B. Wardell

over τ derivatives, this process reduces to solving a sequence of stationary field equations. (n) An alternative approach is to instead solve for the perturbations Hμν in the inner expansion, starting with a large-r˜ ansatz complementary to Eq. (160), and then (n) translate the results into the small-r expansions for hμν . This approach can draw on existing, high-order inner expansions (e.g., Refs. [83, 99, 106]), though doing so often requires transformations of the coordinates and of the perturbative gauge to arrive at a practical form for the outer expansion (e.g., see Ref. [91]).

Local Solution: Self-Field and an Effective External Metric (1)

(2)

The general solutions for hμν and hμν in the buffer region are known to varying orders in ε and r in a variety of gauges, including classes of “rest gauges” (terminology from Ref. [91]), “P smooth” gauges [19], “highly regular” gauges [91] (in which no 1/s 2 term appears in h(2) μν ), radiation gauges [43], and the Regge-WheelerZerilli gauge [73]. (In the last two cases, the gauge choices are restricted to particular classes of external backgrounds.) However, nearly all covariant expressions, and expansions to the highest order in r, are in the Lorenz gauge. Reference [104] provides an algorithm for generating the local solution in the Lorenz gauge, and a large class of similar gauges, to arbitrary order in ε. In all gauges, the general solution is typically divided into two pieces: S(n) R(n) h(n) μν = hμν + hμν .

(178)

This is akin to the usual split of a general solution into a particular and a  n S(n) homogeneous solution. hSμν = n ε hμν is the object’s self-field, encoding all the local information about the object’s multipole structure (including the entirety obj of gμν ). Although this field is defined only outside the object, it would be singular at s = 0 if the expansion (174) was taken to hold for all R  s > 0; it contains all the negative powers of s in (174), as well as all nonnegative powers of s with finite differentiability (e.g., all terms proportional to s p nL with p ≥ 0 but p = ). For that reason, it is also known as the singular field.  n R(n) The second piece of the general solution, hR μν = n ε hμν , encodes effectively external information linked to global boundary conditions. It takes the form of a power series: hR(n) μν =



(n)

cμνL (τ )x L ;

(179)



unlike hSμν , which involves the locally determined multipole moments, every (n)

coefficient cμνL is an unknown that can only be determined when external boundary conditions are imposed. Although, once again, the field is defined outside the

36 Black Hole Perturbation Theory and Gravitational Self-Force

object, we can identify



(n) L  cμνL (τ )x

R(n) 1 ! ∂L hμν (τ, 0)

(n) cμνL (τ )

1457

with a Taylor series, where the coefficients

R(n) hμν

= define and its derivatives on the worldline. Moreover, hR can be combined with the external background to form an effective μν metric g˘ μν := gμν + hR μν

(180)

that is a vacuum solution, satisfying Gμν [g] ˘ =0

(181)

even on γ . g˘ μν characterizes the object’s rest frame and local tidal environment. a Because hR μν is smooth at x = 0, it is also referred to as the regular field. This type of division of the local solution into hSμν and hR μν was first emphasized by Detweiler and Whiting at first order in ε [88,108]. There is considerable freedom in the specific division, as smooth vacuum perturbations can be interchanged between the two pieces, and multiple distinct choices have been made in practice, particularly beyond linear order [18, 19, 109, 110]. However, one can always choose the division such that (i) g˘ μν is a smooth vacuum metric and (ii) g˘ μν is effectively the “external” metric, in the sense that the object moves as a test body in it, as described in the next section. Here for concreteness we adopt the choice introduced in Ref. [18] (see also Refs. [29, 104, 105]), and we provide the explicit forms of the first- and second-order self-fields in the Lorenz gauge, as presented in Ref. [105]. For the purpose of explicitly displaying factors of the object’s multipole moments, from this point forward, we take ε to be a formal counting parameter that can be set equal to unity. At first order, the self-field is determined by the object’s mass. It is given in Fermi-Walker coordinates by 2m + 3mai ni + 53 msEab nˆ ab + O(s 2 ), s   = 2ms 13 B bc εacd nˆ b d − a˙ a + O(s 2 ),

hS(1) ττ =

(182a)

hS(1) τa

(182b)

S(1)

hab =

 2mδab − mδab ai ni + ms 43 E(a c nˆ b)c − s

38 9 Eab

 − Ecd δab nˆ cd + O(s 2 ) (182c)

and in covariant form by hS(1) μν =

  mλ0    2m α  β   gμ gν gα  β  + 2uα  uβ  + 3 gμα gνβ s2 − r2 aσ (gα  β  + 2uα  uβ  ) λs s +8rs a 2

(α 

u

β)



 β  mgμα gν  2  2 +λ r gα  β  + 2uα  uβ  Ruσ uσ − s 3s3

1458

A. Pound and B. Wardell

− 12s4 Rα  uβ  u − 12rs2 u(α  Rβ  )uσ u + 12s2 (r2 + s2 )a˙ (α  uβ  ) + r(3s2 − r2 )a˙ σ (gα  β  + 2uα  uβ  ) + O(λ2 ),

(183)



where x α is an arbitrary point on γ near the field point x α . In the covariant expres     sions, we have adopted the notation aσ := aα  σ α , Ruσ uσ := Rμ α  ν  β  uμ σ α uν σ β ,   etc. The quantity r := uμ σ μ is a measure of the proper time between x α and x α . Equations (182) and (183) are given in Ref. [105] through order λ2 . Equation (4.7) S(1) of Ref. [111] presents the covariant expansion of hμν through order λ4 (omitting acceleration terms). At second order, the self-field involves both the mass and spin of the object. It can be written as the sum of three pieces (Ref. [105] further divides hSR μν into two SR δm pieces, labeled hμν and hμν ): spin

SS SR hS(2) μν = hμν + hμν + hμν .

(184)

The spin contribution is spin

hτ a =

2Sai ni + O(s 0 ), s2

(185)

where other components are O(s 0 ) and where Sab = εabi S i is the spin tensor and S i the spin vector. The other two pieces are either quadratic in the mass: hSS ττ = −

2m2 − 73 m2 Eab nˆ ab + O(s ln s), s2

10 2 bc hSS ˆ b d + O(s ln s), τ a = − 3 m B εacd n

hSS ab =

8 2 3 m δab

−

(186a) (186b)

  − 7m2 nˆ ab c 2 cd cd 4 7 4E + m n ˆ − E δ n ˆ + E n ˆ c(a b) 3 cd ab 5 cd ab s2

16 2 15 m Eab

ln s + O(s ln s),

(186c)

or involve products of the mass with the regular field:  m  R1 ab 1 R1 ab 0 hab nˆ + 3 hab δ + 2hR1 τ τ + O(s ), s m  R1 b 4 R1  hτ b nˆ a + 3 hτ a + O(s 0 ), =− s   m  R1 c ij R1 2hc(a nˆ b) − δab hR1 ˆ ab = ˆ cd − hR1 cd n ij δ + hτ τ n s R1 cd R1 0 1 2 + 23 hR1 ab + 3 δab hcd δ + 3 δab hτ τ + O(s ).

hSR ττ = −

(187a)

hSR τa

(187b)

hSR ab

(187c)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1459

0 SR On the right, the components of hR1 μν are evaluated at s = 0. At order s , hμν also depends on first derivatives of hR1 μν evaluated at s = 0; at order s, it depends on S(2)

second derivatives of hR1 μν evaluated at s = 0; and so on. hαβ is given in Fermi coordinates through order s in Appendix D of Ref. [104]. In covariant form, these fields are 

spin hμν

=

β

4gμα gν u(α  Sβ  )γ  σ γ λ2 s3



+ O(λ0 ),

(188)

with Sα  β  := eαa  eβb  Sab , hSS μν =

 m2 α  β   2 2 2  β  − 7σα  σβ  − 14rσ(α  uβ  ) − (7r − 3s )uα  uβ  5s g g g α μ ν λ2 s4    16 m2 λ0 α  β    2 2 2 − m2 gμα gνβ ln(λs)Rα  uβ  u + Rσ uσ u g g μ ν 10s gα  β  25r + s 6 15 150s    + 20rs2 35rσ(α  Rβ  )uσ u + 35r2 − 31s2 u(α  Rβ  )uσ u − s2 Rσ (α  β  )u   + 10s4 Rα  σβ  σ − 350rs2 σ(α  Rβ  )σ uσ − 10s2 35r2 − 17s2 u(α  Rβ  )σ uσ     + 2s4 5r2 + 26s2 Rα  uβ  u − 70 10r2 − 3s2 σα  σβ        + 4r 5r2 − 4s2 u(α  σβ  ) Rσ uσ u − 20 35r4 −53r2 s2 − 6s4 uα  uβ  Rσ uσ u + O (λ ln λ) ,

(189)

and hSR μν



 2 2 R1 μ ν   2 m α β  2 R1 R1 R1 s hμ ν  g = 3 gμ gν gα  β  − r − s huu − hσ σ − 2rhuσ 3 λs 2 2 R1 R1 R1 + s2 δmα  β  − hR1   s + 2hσ (α  σβ  ) + 2rhσ (α  uβ  ) − 2hσ σ uα  uβ  3 αβ  μ ν  2 2 R1  σβ  + 2rσ(α  uβ  ) + (r − s )uα  uβ  + 2rh σ − hR1 g   α μν u(α  σβ  )    R1 R1      + O λ0 , + 2(r2 − s2 )hR1 u + 4h σ u − 2h σ σ  uσ (α β ) uu α β u(α β )

(190)

where δmαβ =

 1  R1 R1 μ m 2hαβ + gαβ g μν hR1 μν + 4mu(α hβ)μ u 3 + m(gαβ + 2uα uβ )uμ uν hR1 μν .

(191)

1460

A. Pound and B. Wardell

SR The covariant expressions for hSS μν and hμν are known through order λ [105] and spin

are available upon request to the authors. The covariant expansion of hμν appears explicitly here for the first time, but it is known to higher order in λ [112]. In this section, we have stated results from the so-called self-consistent expansion of the metric [14]. In this framework, the metric is not expanded in an ordinary Taylor series in ε. Instead, it takes the form exact α α 2 (2) α 3 gμν (x , ε) = gμν (x α ) + εh(1) μν (x , P) + ε hμν (x , P) + O(ε ),

(192)

where P represents a list of system parameters: the worldline γ and multipole moments of the small object, along with any evolving external parameters. If the small object is orbiting a black hole that is approximately Kerr, the external parameters will consist of small, slowly evolving corrections to the black hole’s mass and spin [114]. These parameters all evolve with time in an ε-dependent way, meaning that Eq. (192) is not a Taylor series; this allowance for ε-dependent coefficients is a hallmark of singular perturbation theory [80]. In the self-force problem, it must be allowed in order to construct a uniformly accurate approximation on large timescales [115]. It will lead naturally into the multiscale expansion described in the later sections of this review. If we use an ordinary Taylor series in place of Eq. (192), then zμ is replaced μ μ with the series expansion zμ (τ, ε) = z0 (τ ) + εz1 (τ ) + . . . (referred to as a Grallaμ Wald expansion after the authors of Ref. [13]). Here z0 is a geodesic of the external background spacetime, and the local analysis described above is carried out with series expansions in powers of distance from this geodesic. The acceleration a μ in this approach is thus set to zero in all the above formulas. The ε dependence of zμ S(2) then manifests itself in hμν through an additional term, dipole

hμν

=

2mi ni (gμν + 2uμ uν ) + O(1/s) s2 

= gμα gνβ





−

(193) !

2mμ σ μ (gα  β  + 2uα  uβ  ) + O(1/λ) , λ2 s3

(194)

proportional to a mass dipole moment mα = eaα ma = mz1α . mα describes the μ position of the object’s center of mass relative to z0 . It appears in the secondorder metric perturbation in the outer expansion but in the zeroth-order inner metric, obj gμν . By setting mi to zero in the self-consistent expansion, one defines γ to be the (3) center of mass at this order. A correction to mi generically appears in hμν and in obj (1) gμν + εHμν , and it is likewise set to zero in a self-consistent expansion [18, 91]. In a Gralla-Wald expansion, mi and corrections to it are allowed to be nonzero; dipole for that case, hμν is given through order λ0 in Fermi coordinates in Sec. IVC of Ref. [14] (where mi is denoted Mi ). Explicit expressions through order λ, in

36 Black Hole Perturbation Theory and Gravitational Self-Force

1461

both Fermi coordinate and covariant form, are known through order λ [105] and are available upon request. In the context of a binary, the small object inspirals, eventually moving very μ far from any initially nearby background geodesic. This causes z1 and higher corrections to grow large with time, spelling the breakdown of the Gralla-Wald expansion. For this reason, we have focused on the self-consistent formulation in this review. References [14,29,116] provide detailed explications of the relationship between the two types of expansions.

Equations of Motion Along with the local form of the metric perturbations, the Einstein equations determine the motion of the small object and the evolution of its mass and spin. Specifically, if we let γ be the object’s center of mass (by setting the mass dipole (2) moment in hμν to zero), then the vacuum field equations uniquely determine the first-order equations of motion [11, 13, 14]:  D 2 zα 1 αδ  R(1) 1 α R(1) P R βγ δ uβ S γ δ + O(ε2 ) 2h uβ uγ − = − − h δβ;γ βγ ;δ 2 2 2m dτ

(195)

and dm = O(ε2 ) dτ

and

DS αβ = O(ε3 ). dτ

(196)

The first term on the right of Eq. (195) is referred to as the first-order gravitational self-force (per unit mass) or as the MiSaTaQuWa force (after the authors of Refs. [11, 12]); the second term on the right is the Mathisson-Papapetrou spin force [117, 118]. Equation (195) represents the leading correction to geodesic motion for a gravitating, extended, compact object. (For a non-compact object, finite-size effects from higher multipole moments will dominate over self-force effects.) However, these equations are equivalent to those of a test body, not in the background or in the physical spacetime but in the effective metric g˘ μν . In particular, Eq. (195) can be rewritten as: D˘ 2 zα 1 ˘α =− R βγ δ u˘ β S γ δ + O(ε2 ), 2 2m d τ˘

(197)

˘ where τ˘ is proper time in g˘ μν , dDτ˘ := u˘ α ∇˘α , ∇˘ is a covariant derivative compatible μ with g˘ μν , and u˘ μ = dz d τ˘ . This is the equation of motion of a spinning test particle. Similarly, the evolution equations (196) are the equations of a test mass and spin, which are constant and parallel propagated, respectively.

1462

A. Pound and B. Wardell

If we specialize to a spherical, nonspinning object (and set the subleading mass dipole moment to zero), the field equations determine the second-order equation of motion [18, 91]:   D 2 zα 1 αμ  δ Rδ R R β γ 3 g 2h = − − h − h P μ μ δβ;γ βγ ;δ u u + O(ε ) 2 dτ 2 and

dm dτ

(198)

= O(ε3 ). This can be rewritten as the geodesic equation in g˘ μν : D˘ 2 zμ = O(ε3 ). d τ˘ 2

(199)

See Sec. IIIA of Ref. [116] for the (simple) steps involved in rewriting Eq. (198) as Eq. (199). For a generic compact object, the spin and quadrupole moments will both appear in Eq. (198). Although the second-order equations of motion have not been derived directly from the field equations in that case, it is known that at least through this order, the motion remains that of a test body in some effective metric [86]. At least for a material body, this remains true even in the fully nonlinear setting [110]. The spin’s evolution and its contribution to the acceleration through second order (but omitting quadratic-in-spin terms), extracted from the nonlinear results for a material body, are given in Eq. (6) of Ref. [113] this corrects Eq. (2.11) of Ref. [119]. In this section we have again presented results for the self-consistent expansion. In the Gralla-Wald approach, one instead obtains evolution equations for the mass dipole moment. Such equations are derived at first order in Refs. [13, 14, 120] and at second order in Ref. [19] (see also Ref. [116], which derives such second-order equations in a more compact, parametrization invariant form). We stress that the equations in this section follow directly from the vacuum Einstein equations, together with a center-of-mass condition, outside the small object. There is no assumption about the object’s internal composition, nor is there any regularization of singular quantities. We refer to Refs. [16, 17, 88] for variants of the approach described here and to Refs. [12, 77, 110] for alternatives to the matched-expansion approach.

Skeleton Sources: Punctures and Particles After having derived the local form of the metric and the equations of motion, we can effectively remove the body zone from the problem. We do so by allowing the local forms (179)–(190) to hold all the way down to γ . This causes the self-field to diverge at γ , artificially introducing a singular field. However, this does not alter the physics in the buffer region or external universe, and the singularity is more easily handled than the small-scale physics of the small object.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1463

Once the fields have been extended to γ , one can solve the field equations throughout the spacetime using either a puncture scheme or point-particle methods. The puncture scheme is the more general of the two approaches. We define the puncture field (n) hP := hS(n) μν μν W

(200)

k as the local expansion of hS(n) μν truncated at some order λ , multiplied by a window α function W that is equal to 1 in a neighborhood of z and transitions to zero at some P (n) S(n) finite distance from zα . This implies that hμν = hμν + O(λk+1 ). We then define the residual field: (n) P (n) hR := h(n) μν μν − hμν ,

R (n)

R(n)

(201) R (n)

which satisfies hμν = hμν + O(λk+1 ), making hμν a C k field at γ . Outside P (n) R (n) (n) the support of hμν , hμν becomes identical to the full-field hμν . (1) Moving hP to the right-hand side of the vacuum field equations, we obtain μν (n) field equations for hR μν : R (1) P (1) eff(1) G(1) ] = −G(1) ] := Sμν , μν [h μν [h

(202)

R (2) (1) (1) (1) P (2) eff(2) G(1) ] = −G(2) ] := Sμν . μν [h μν [h , h ] − Gμν [h

(203)

These equations hold at all points off γ . The C k behavior of the solution is then eff(n) enforced by defining the effective sources Sμν as ordinary integrable functions at (1) γ , rather than treating Gμν [hP (n) ] in the distributional sense of a linear operator acting on an integrable function; this distinction is important to rule out delta functions in the source, which would create spurious singularities in the residual field. In the self-consistent approach, some care is required in formulating Eqs. (202) and (203). Specifically, they can only be split into a sequence of equations, one at each order in ε, after imposing a gauge condition [14]; this is required in order to allow the puncture to move on an accelerated trajectory. We do not belabor this point because we ultimately formulate the equations in a somewhat different, multiscale R (n) form tailored to binary inspirals. If k ≥ 1, then we can replace hR(n) in μν with hμν (n) R (n) P (n) the equations of motion (195) and (198). The total field hμν = hμν + hμν is also guaranteed to satisfy the physical boundary condition in the buffer region (i.e., the matching condition) and at the outer boundaries of the problem. Our description may seem (incorrectly) to imply that the puncture field is only defined in a convex normal neighborhood of the body. For numerical purposes, the puncture is extended over a region of any convenient size. Typically this is done by converting the local, covariant expressions in terms of Synge’s world function into expansions in coordinate distance, using, e.g., the Boyer-Lindquist coordinates of

1464

A. Pound and B. Wardell

the background spacetime. The punctures can then be extended as these coordinate R (n) P (n) functions. The end result for the combined field h(n) is insensitive μν = hμν + hμν to the choice of extension. An alternative to the puncture scheme is to solve directly (n) for the total fields hμν . Once extended to γ , they satisfy (1) (1) (1) 3 + ε2 h(2) ] + ε2 G(2) G(1) μν [εh μν [h , h ] = 8π Tμν + O(ε ),

(204)

where here we do interpret each term on the left-hand side in a distributional sense. The stress-energy tensor is then defined by the left-hand side. Through second order, it can be shown to be the stress-energy of a spinning particle in the effective metric [13, 14, 84, 104, 121]: 

˘ z(τ˘ ))d τ˘ + u˘ μ u˘ ν δ(x,

Tμν = m γ

˘ x) = where δ(x,



˘ z(τ˘ ))d τ˘ , u˘ (μ Sν) α ∇˘α δ(x,

(205)

γ α α δ 4 (x √ −x ) −g˘

and u˘ μ := g˘ μν u˘ ν . (At second order, this is true

in a class of highly regular gauges. In other gauges, it requires a direct use of the puncture via a particular distributional definition of the nonlinear quantity (2) Gμν [h(1) , h(1) ] [121].) We refer to this point-particle stress-energy as the Detweiler stress-energy after the author of Ref. [17]. Like the equations of motion, the pointparticle approximation is a derived consequence of the vacuum Einstein equations and the matching condition, rather than an input. In cases where the point-particle method is well defined, it and the puncture (n) scheme yield identical full fields hμν . However, unlike a puncture scheme, a pointR(n) particle method does not yield the regular fields hμν as output. The regular fields, (n) and self-forces, must instead be extracted from hμν . This is most often done using the method of mode-sum regularization [122, 123] reviewed in detail in Refs. [31, 32] and sketched in Section “Mode Decompositions of the Singular Field.” eff(n) We will refer to both the effective sources Sμν and the point-particle source Tμν as skeleton sources. This terminology follows Mathisson’s notion [117, 124] of a “gravitational skeleton” (see also Refs. [125–127]): an extended body can be represented by a singularity equipped with an infinite set of multipole moments. Punctures provide a generalization of this concept to settings where the singularities are too strong to be represented by distributions. For that reason, although the nomenclature of punctures and effective sources originated from methods of solving the first-order field equations in Refs. [128,129], punctures have a more fundamental role at second and higher orders [16, 17, 19, 29, 104, 109]. For the same reason, we have presented punctures as a more primitive concept than the point-particle stressenergy. In either approach, the skeleton sources presented here apply equally for all compact objects, whether black holes or material bodies. The only distinguishing feature of a material body would be a spin that surpasses the Kerr bound (i.e., |S i | > m2 ). However, at third order in perturbation theory, the quadrupole moment will

36 Black Hole Perturbation Theory and Gravitational Self-Force

1465

(3)

appear in the perturbation hμν . Unlike the mass and dipole moments, the quadrupole moment is not governed by the Einstein equation [30, 110, 127], and its evolution must be determined from the object’s equation of state. Hence, at third order the interior composition of the object begins to influence the external metric, and we can begin to distinguish between black holes and material bodies. But note that the quadrupole moments of compact objects differ primarily due to their differing tidal deformability, and this difference is suppressed by an additional five powers of ε [93], suggesting it is almost certainly irrelevant for small-mass-ratio binaries.

Orbital Dynamics in Kerr Spacetime The previous section summarized the local problem in self-force theory: the reduction of an extended body to a skeleton source in the Einstein equations, along with an equation of motion for that source. In the remaining sections, we turn to the global problem: solving the perturbative Einstein equations, coupled to the equation of motion (195) or (198), globally in a specific background metric. In the context of a small-mass-ratio binary, the background geometry is the Kerr spacetime of the central black hole. According to the equations of motion, the small body in the binary is only slightly accelerated away from geodesic motion in that background. This section summarizes (i) properties of bound geodesic motion in Kerr spacetime and (ii) how to exploit those properties to analyze accelerated orbits. We emphasize action-angle methods that mesh specifically with our treatment of the Einstein equations in the final section of this review. However, much of our treatment is valid for a more generic acceleration. We warn the reader that the notation in this section differs in several ways from that of the preceding section. The differences are noted in the first subsubsection below.

Geodesic Motion Constants of Motion, Separable Geodesic Equation, and Conventions Geodesics in Kerr spacetime are integrable, with three constants of motion associated with the spacetime’s three Killing symmetries: (specific) energy E = −uα ξ α , (specific) azimuthal angular momentum Lz = uα δφα , and the Carter constant 



Q = uα uβ (K αβ − a12 ηα ηβ ). (The constant K = uα uβ K αβ is also sometimes referred to as Carter’s constant.) Inverting these three equations, together with g αβ uα uβ = −1, for the fourvelocity components, we obtain [130]  Σ2

dr dτ

2 = R(r),

(206)

1466

A. Pound and B. Wardell

 Σ2

dz dτ

2 = Z(z),

dt = Tr (r) + Tz (z) + aLz := ft , dτ dφ Σ = Φr (r) + Φz (z) − aE := fφ . dτ Σ

(207) (208) (209)

Here (t, r, z := cos θ, φ) refer to Boyer-Lindquist coordinates, and  R(r) := [P (r)]2 − Δ r 2 + (aE − Lz )2 + Q ,   Z(z) := Q − Q + a 2 γ + L2z z2 + a 2 γ z4 , Tr (r) :=

r 2 + a2 P (r), Δ

Tz (z) := −a 2 E(1 − z2 ), a Φr (r) := P (r), Δ Lz Φz (z) := , 1 − z2

(210) (211) (212) (213) (214) (215)

with P (r) := E(r 2 + a 2 ) − aLz and γ := 1 − E 2 . We opt to use z rather than θ throughout this section. Refs. [130, 131] and many other references instead define z as cos2 θ , with analogous differences in their definitions of the roots zn defined below. The equations for r(τ ) and z(τ ) are coupled, but they are immediately decoupled by adopting a new parameter λ, called Mino time [132], that satisfies: dλ = Σ −1 . dτ

(216)

(This is not to be confused with the bookkeeping parameter used in the local expansions of the previous section.) The equations also take a hierarchical form: once r(λ) and z(λ) are known, Eqs. (208) and (209) can be straightforwardly integrated to obtain t (λ) and φ(λ). Given this hierarchical form, we will focus on the r–z dynamics. In Eq. (206), R(r) is a fourth-order polynomial in r, meaning it can also be written as R(r) = −γ (r − r1 )(r − r2 )(r − r3 )(r − r4 ), with r1 ≥ r2 ≥ r3 ≥ r4 . Similarly, in Eq. (207), Z(z) = a 2 γ (z2 − z12 )(z2 − z22 ), with |z1 | > |z2 |. For bound orbits, the radial motion oscillates between the turning points ra = r1 (apoapsis) and rp = r2 (periapsis) and the polar motion between zmax = |z2 | and zmin = −|z2 |. The other roots (r3 , r4 , and z1 ) do not correspond to physical turning points. In particular, |z1 | > 1. Hence, the geodesic is confined to a torus-like region rp ≤ r ≤ ra , |z| < zmax .

36 Black Hole Perturbation Theory and Gravitational Self-Force

1467

If Q = 0, the motion is confined to the equatorial plane z = 0. If a = 0 (i.e., in Schwarzschild spacetime), the geodesic is likewise confined to a plane, which, due to Schwarzschild’s spherical symmetry, can be freely chosen as z = 0. However, a generic orbit ergodically fills the torus-like region. For convenience in the remaining sections, we use lowercase Latin indices from the beginning of the alphabet (a, b, c) to denote r or z and define x = (r, z). However, repeated indices, as in an expression such as fa x a , are not summed over; instead, such sums will be written as f · x := fr x r + fz x z . An expression such as fa (x a ) will denote either one of fr (r) or fz (z), while an expression such as fa (x) will denote either one of fr (r, z) or fz (r, z). fα (x β ) will denote fα (t, r, z, φ). We use lowercase Latin indices from the middle of the alphabet (i, j, k) to label elements of a set of orbital parameters. For example, P i = (E, Lz , Q). For these indices, unlike a, b, c, we use Einstein summation. We use f throughout this section to denote a generic function, not the specific function f (r) that appears in the Schwarzschild metric (14). An overdot will denote a derivative with respect to λ. Finally, we preemptively refer the reader to Refs. [130–136] for additional details about geodesic orbits in Kerr.

Quasi-Keplerian Parametrization Unlike Keplerian orbits, generic geodesics in Kerr do not close; the periods of radial, polar, and azimuthal motion are all, generically, incommensurate. Nevertheless, because of their doubly oscillatory nature, it is often useful in applications to express the geodesic trajectories in a quasi-Keplerian form, replacing the constants {E, Lz , Q} with an alternative set {p, e, zmax }. In terms of these, r and z can be written in the manifestly periodic form [131]: r(ψr ) =

pM , 1 + e cos ψr

z(ψz ) = zmax cos ψz ,

(217) (218)

where, for a bound orbit, 0 ≤ e < 1. The phases (ψr , ψz ) are multiples of 2π at periapsis and at z = zmax , respectively. Unlike r and z, which change direction every half cycle, ψr and ψz grow monotonically, leading to better numerical behavior at the turning points. Because none of the periods are commensurate, ψr and ψz evolve independently dx a dx a a (of each other and of φ). Using dψ dλ = dλ / dψa , one finds [131] √ M γ [(p − p3 ) − e(p + p3 cos ψr )][(p − p4 ) + e(p − p4 cos ψr )] dψr = dλ 1 − e2 := fr ,

dψz 2 cos2 ψ ) := f , = a 2 γ (z12 − zmax z z dλ

(219) (220)

1468

A. Pound and B. Wardell

where p3 := r3 (1 − e)/M and p4 := r4 (1 + e)/M. These can be integrated subject to arbitrary choices of initial phase ψa (0) = ψa0 . The parameters {p, e, zmax }, unlike {E, Lz , Q}, are related directly to the coordinate shape of the orbit, specifically to its turning points. Equation (217) is the formula for an ellipse, and it implicitly defines p and e to be the semi-latus rectum and eccentricity of that ellipse, related to the periapsis and apoapsis by rp =

pM 1+e

and

ra =

pM . 1−e

(221)

As stated above, zmax = z2 , but to further the analogy with Keplerian orbits, we can also define an inclination angle ι such that z2 = zmax = sin ι.

(222)

Ref. [131] and some other authors use the alternative, inequivalent definition cos ι = √ L2z . This does not describe the maximum coordinate inclination angle Lz +Q

but has other useful properties [137]. The remaining roots of R(r) and Z(z) are also compactly expressed in terms of these parameters [130]: r3 =

 

1 α + α 2 − 4β 2

and

r4 = β/r3 ,

(223)

where α := 2M/γ − (ra + rp ) and β := a 2 Q/(γ ra rp ), and " z1 =

Q . 2 a 2 γ zmax

(224)

These expressions are in a “mixed” form that involves both sets of constants. However, {E, Lz , Q} can be written in terms of {p, e, ι} as [134]:  2|d, g, h| − |d, h, f | − 2χ |d, g, h|2 + |h, d, g, h, f | + |h, d, h, g, f | E = , |f, h|2 + 4|f, g, h| (225) " gp2 E 2 gp ME fp E 2 − dp Lz = − + Mχ + , (226) 2 hp hp hp   L2z 2 2 Q = zmax a γ + , (227) cos2 ι 2

where χ := sgn(Lz ) is +1 for prograde orbits and −1 for retrograde:

36 Black Hole Perturbation Theory and Gravitational Self-Force

1469

2 d(r) := Δ(r 2 + zmax a 2 )/M 4 ,

(228)

2 f (r) := (r/M)4 + a 2 [r(r + 2M) + zmax Δ]/M 4 ,

(229)

g(r) := 2ar/M 2 ,

(230)

h(r) := [r(r − 2M) + Δ tan2 ι]/M 2 ,

(231)

and a subscript a or p indicates evaluation at ra or rp . The quantities | · | appearing in E 2 are determinants or products of determinants that we define recursively as |f, g| := fp ga − fa gp and |f, g, . . . | := |f, g||g, . . . |. Note that r1 and r2 have the opposite meaning in Ref. [134] than their meaning here. Our notation for the roots rn follows Ref. [130]. Given the parametrizations (217) and (218) and the equations of motion (208) and (209), t (λ) and φ(λ) can be written as t (λ) = t0 + tr (ψr (λ)) + tz (ψz (λ)) + aLz λ,

(232)

φ(λ) = φ0 + φr (ψr (λ)) + φz (ψz (λ)) − aEλ,

(233)

with:  ta (ψa ) =

ψa

ψa0

Ta (ψa ) dψ  fa (ψa ) a

 and

φa (ψa ) =

ψa ψa0

Φa (ψa )  dψ . fa (ψa ) a

(234)

Here t0 and φ0 are integration constants. This completes the quasi-Keplerian description of geodesic orbital motion. Trajectories are described by the three constants of motion pi := (p, e, ι) and the four secularly growing phase variables ψα := (t, ψr , ψz , φ). A given trajectory is uniquely specified by the set of seven constants {p, e, ι, t0 , ψr0 , ψz0 , φ0 }, called orbital elements. The solution to the geodesic equation can also be put in closed, analytical form [130] by expressing ψα (λ) in terms of elliptic integrals and their inverses (the Jacobi elliptic functions).

Fundamental Mino Frequencies and Action Angles It is often essential to decompose quantities on the worldline into Fourier series, particularly when solving the perturbative Einstein equations in the frequency domain. This procedure is expedited by knowing the orbit’s fundamental frequencies. In this section, we summarize the calculation of frequencies and of phase variables (action angles) associated with those frequencies. Unlike the phases ψα , the angle variables are strictly linear in λ, facilitating Fourier expansions in that time variable. In the right-hand sides of Eqs. (208), (209), (219), and (220), we have defined the “frequencies” fα (ψ) as the rates of change of ψα : dψα = fα (ψ). dλ

(235)

1470

A. Pound and B. Wardell

The true frequencies Υα associated with λ are the average rates of change of ψα , Υα = fα λ ,

(236)

and the corresponding action angles are qα = Υα λ + qα0 ,

(237)

with arbitrary constants qα0 . For a function f [r(λ), z(λ)] on the worldline, the average is defined as 1 Λ→∞ 2Λ



f λ := lim

Λ −Λ

f dλ.

(238)

For a generic, nonresonant orbit, this average agrees with the torus average: f q =

1 (2π )2

# f d 2 q.

(239)

$ % 2π % 2π $ We use d 2 q to denote 0 dqr 0 dqz and d 2 ψ for the analogous integral over ψ. We focus only on functions of r and z, which are automatically periodic functions of the intrinsic phases ψ and q. The averaging operation immediately generalizes in the natural way to functions f [zα (λ)] that are periodic in t and φ. To simplify the analysis, we choose our phase space coordinates q such that qr vanishes at some periapsis and qz vanishes at some z = zmax . We furthermore choose qt , qφ , our spacetime coordinates t and φ, and our parameter λ such that they all vanish at some particular passage through periapsis. These choices, which do not represent any loss of generality, imply qα0 = ψα0 = 0

for α = t, r, φ,

qz0 = −Υz λ0z ,

(240a) (240b)

where λ0z is the first value of λ at which z = zmax . ψz0 can be inferred from qz0 . One can easily do without these specifications if desired. With our choices, qa represents the mean growth of ψa from the first radial or polar turning point, and we can express it in terms of ψa as 

ψa

qa (ψa ) = Υa 0

dψa . fa (ψa )

(241)

This allows us to straightforwardly write the torus average as an integral over ψ: f q =

1 Λr Λz

#

f d 2ψ , fr (ψr )fz (ψz )

(242)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1471

% 2π a where Λa = 0 fadψ (ψa ) is the radial or polar period with respect to λ. Although they agree generically, f λ and f q differ in the special case of resonant orbits, discussed in later sections. For the r and z motion, the frequencies reduce to Υa = 2π/Λa , which can be analytically evaluated to [130]  π γ (ra − r3 )(rp − r4 ) Υr = , 2K(kr )  π a 2 γ z1 , Υz = 2K(kz )

(243) (244)

where 

π/2

K(x) := 0



dθ 1 − x sin2 θ

(245)

is the complete elliptic integral of the first kind, and its arguments are kr := ra −rp r3 −r4 2 ra −r3 rp −r4 and kz := (zmax /z1 ) . The frequencies of t and φ motion can also be found analytically. Because of the dt additive forms of dλ and dφ dλ in (208) and (209), the averages reduce to a sum of one-dimensional integrals. Evaluating those integrals leads to [135] Υt =

 E r3 (ra + rp + r3 ) − ra rp +(ra + rp + r3 + r4 )Fr +(ra − r3 )(rp − r4 )Gr 2  

(4M 2 E − aLz )r+ − 2Ma 2 E F+ 2M 1− − (+ ↔ −) + r+ − r− r3 − r+ rp − r+

EQ(1 − Gz ) + 2ME(r3 + Fr ), 2 γ zmax  

2MEr+ − aLz F+ a 1− − (+ ↔ −) Υφ = r+ − r− r3 − r+ rp − r+ + 4M 2 E +

+

2 ,k ) Lz Π (zmax z , K(kz )

(246)

(247) E(ka ) K(ka ) and FA := (r −r )(r −r ) h± := (raa −rp3 )(rp3 −r±± ) .

A ,kr ) (rp − r3 ) Π(h for A = K(kr ) √ Here r± = M ± M 2 − a 2

 dθ 1 − x sin2 θ ,

(248)

where we have introduced Ga := r −r

{r, +, −}, with hr = raa −rp3 and denote the inner and outer horizon radii, and 

π/2

E(x) := 0

1472

A. Pound and B. Wardell



π/2

Π (x, y) := 0

dθ

(1 − x sin2 θ ) 1 − y sin2 θ

(249)

are the complete elliptic integrals of the second and third kind, respectively. In terms of the angle variables, a quantity f (r, z) on the worldline can be expanded in the Fourier series 

f [r(λ), z(λ)] =

fk e−iqk (λ) , (q)

(250)

k

where qk := k · q = kr qr + kz qz , and unless stated otherwise, sums range over k ∈ Z2 . The coefficients are given by (q)

=

fk

1 (2π )2

# f eiqk d 2 q,

(251)

f eiqk (ψ) d 2ψ fr (ψr )fz (ψz )

(252)

which can also be calculated as (q) fk

Υr Υz = (2π )2

#

with qk (ψ) = kr qr (ψr ) + ikz qz (ψz ) given by Eq. (241). The torus average of the function (and infinite λ average for nonresonant orbits) is the zero mode in the (q) Fourier series: f q = f00 . Using such Fourier expansions, we can invert Eq. (241) to write the phases ψα in terms of the angle variables. The transformation qα → ψα (qβ ) must satisfy ∂ψα dψα ∂qβ Υβ = dλ = fα together with our choice ψα (qβ = 0) = 0. The solution is the sum of a secular and a purely oscillatory piece: ψα (qβ ) = qα − Δψα (0) + Δψα (q),

(253)

where Δψa =

 f k e−ikqa a

k =0

−ikΥa

(254)

,

   T k e−ikqr Tzk e−ikqz r + Δt = Δtr + Δtz := , −ikΥr −ikΥz

(255)

k =0

  k e−ikqz  Φ k e−ikqr Φ z r + Δφ = Δφr + Δφz := . −ikΥr −ikΥz k =0

(256)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1473

Ta and Φa are given in Eqs. (212), (213), and (215), and we have written  them, along with fa (qa ), as one-dimensional Fourier series in qa ; e.g., Ta = k Tak e−ikqa . We will consistently use Δ to indicate that a quantity is periodic in q with zero average. We can conveniently write the coordinate trajectory in terms of the action angles as the sum of a secular term and an oscillatory term: α zα (λ) = zsec [qβ (λ)] + Δzα [q(λ)],

(257)

where the secular piece is  (q) (q) α (qβ ) = (qt , 0, 0, qφ ) + −Δt (0), r0 , z0 , −Δφ(0) , zsec

(258)

and the purely oscillatory pieces are given by Eqs. (255), (256), and Δx a (qa ) =



a x(q)k e−ikqa ,

(259)

k =0

with coefficients readily calculated from Eq. (252). Reference [130] gives x a (qa ) in closed form in their Eqs. (26) and (38), Δt as the sum of their t (r) and t (θ) in their Eqs. (28) and (39), and Δφ as the sum of their φ (r) and φ (θ) in those same equations. (We caution the reader that the notation in Ref. [130] differs from ours in several ways.) Our description here has followed the constructive approach of Refs. [130, 131, 134], finding the frequencies and angle variables by directly solving the geodesic equation. There is an alternative, historically prior approach [133] based on the Hamiltonian description of geodesics, which builds on Carter’s original proof [138] of integrability using the Hamilton-Jacobi equation. That approach derives action angles and their associated fundamental frequencies from a canonical transformation (zα , uα ) → (q α , Jα ), where the actions Jα are the canonical momenta conjugate to the action angles q α .

Fundamental Boyer-Lindquist Frequencies and Action Angles For the purpose of decomposing fields, such as the metric perturbation, into Fourier modes, it is more useful to know the frequencies with respect to coordinate time t. These are the frequencies observed at infinity and that appear in the gravitational waveform. They are given by Ωα =

Υα . Υt

(260)

The angle variables associated with them are ϕα = Ωα t + ϕα0

(261)

1474

A. Pound and B. Wardell

with Ωt = 1. We choose the origin of this phase space in analogy with Eq. (240): ϕα0 = 0 for α = t, r, φ,

and ϕz0 = −Ωz tz0 ,

(262)

where tz0 is the first value of t at which z = zmax . These new angle variables are related to qα by a transformation that must satisfy ∂ϕα dϕα ∂qβ Υβ = dλ = Ωα ft (q). Such a transformation, with our choice of origin ϕα (qβ = 0) = 0, is ϕα (qβ ) = qα − Ωα Δt (0) + Ωα Δt (q)

(263)

with Δt given by Eq. (255). In analogy with Eq. (250), a function of r and z on the worldline can be expanded in a Fourier series: f [r(t), z(t)] =



fk e−iϕk (t) , (ϕ)

(264)

k

with ϕk := k · ϕ = kr ϕr + kzϕz and  with coefficients given by the analog of ∂ϕa Eq. (251). Using the Jacobian det ∂qb = ft /Υt , we can also write the coefficients as integrals over q: (ϕ)

fk

=

e−iΩk Δt (0) (2π )2 Υt

#

ft eiqk +iΩk Δt (q) f d 2 q,

(265)

where Ωk := kr Ωr + kz Ωz . Or we can write them as integrals over ψ: (ϕ)

fk

=

Υr Υz (2π )2 Υt

#

ft (ψ) eiqk (ψ)+iΩk [δtr (ψr )+δtz (ψz )] f 2 d ψ, fr (ψr )fz (ψz )

(266)

where qa (ψa ) is given by Eq. (241) and 

ψa

δta (ψa ) := Δta [qa (ψa )] − Δta (0) = 0

Ta (ψa ) − Ta λ dψa , fa (ψa )

(267)

with Ta given by Eqs. (212) and (213). If f is separable [i.e., if it can be written as a sum of products of the form fr (r)fz (z)], then expressing the integrals in terms of q or ψ allows one to evaluate the two-dimensional integral as a product of onedimensional integrals. We can further define an average over t, 1 T →∞ 2T

f [r(t), z(t)]t := lim



∞ −∞

f dt,

(268)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1475

which for nonresonant orbits is equal to the torus average f ϕ :=

1 (2π )2

#

(ϕ)

f d 2 ϕ = f00 .

(269)

Note that the meaning of a time average (and associated torus average) inherently depends on one’s choice of time parameter [139] and that f t differs from f λ : f t =

1 1  (q) (q) ft f λ = f λ + ftk f−k . Υt Υt

(270)

k =0

The relevance of each average depends on context. Using these Fourier expansions, we can express the phases ψα in terms of ϕα . dψα α The two are related by a transformation satisfying ∂ψ ∂ϕβ Ωβ = dt . With our choice of origin ψα (ϕβ = 0) = 0, the solution is ψα (ϕβ ) = ϕα − Δϕ ψα (0) + Δϕ ψα (ϕ).

(271)

The oscillatory terms are Δϕ ψt = 0 and

Δϕ ψα =

 k =0



(ϕ) dψα dt k −iϕk

−iΩk

e

for α = r, z, φ.

(272)

Here we use Δϕ rather than Δ to indicate that a quantity is purely oscillatory (ϕ) (i.e., periodic with zero mean) with respect to ϕ rather than q. (dψα /dt)k can be calculated using Eq. (266) with dψα /dt = fα /ft . Just as we did with qα , we can express the coordinate trajectory in terms of ϕα as the sum of a secular and an oscillatory piece: α zα (t) = z(ϕ)sec [ϕβ (t)] + Δϕ zα [ϕ(t)],

(273)

where the secular piece is   (ϕ) (ϕ)  α z(ϕ)sec (ϕβ ) = ϕt , 0, 0, ϕφ − 0, r00 , z00 , −Δϕ φ(0) ,

(274)

and the oscillatory pieces are Δϕ t = 0, Δϕ φ given by Eq. (272) (recalling ψφ := φ), and Δϕ x a (ϕ) =

 k =0

with coefficients calculated from Eq. (266).

a x(ϕ)k e−iϕk ,

(275)

1476

A. Pound and B. Wardell

Resonant Orbits Recall that the radial and polar motions are restricted to a torus-like region rp ≤ r ≤ ra and |z| ≤ zmax in physical space. If the periods of radial and polar motion are incommensurate, then the orbit ergodically fills this region. The transformation x a → q a maps the r–z motion onto the surface of a torus in phase space, which the angles q a ergodically cover. However, for some values of the orbital parameters, the periods are commensurate, meaning krres Υr + kzres Υz = 0 for some nonzero integers (krres , kzres ). [Since integer multiples of (krres , kzres ) will also satisfy this condition, we take (krres , kzres ) to be the smallest two such integers.] In these cases, rather than having two independent frequencies, the r–z motion has a single frequency, Υ = Υz /|krres | = Υr /|kzres |, and rather than ergodically covering the torus, the orbit closes on the torus and in the r–z plane. The actual shape of this closed orbit is not uniquely specified by its frequencies but depends strongly on the relative initial phase ψr0 − ψz0 . Such orbits are referred to as resonant [140]. For resonant orbits, the average over the torus no longer represents a meaningful average over the orbit. Rather than (q) having the single stationary mode f00 , a function f (r, z) on the worldline has an infinite set of stationary modes corresponding to all integer multiples of k res . The infinite Mino-time average in Eq. (238) is then

1 Λ→∞ 2Λ

f [r(λ), z(λ)]λ = lim



Λ

−Λ

f dλ =

∞  N =−∞

(q)

fN k res ;

(276)

for a resonant orbit, the infinite λ average does not, generically, agree with the torus (q) average f00 . More broadly, the Fourier series (250) becomes degenerate: qk (λ) = qk+N k res (λ) for all integers N . However, since there is a common period, we can replace the two action angles qa with a single angle q(λ) = Υ λ and rewrite the two-dimensional Fourier series (250) as a nondegenerate one-dimensional one:

f [r(λ), z(λ)] =



fk e−kq(λ) . (q)

(277)

k∈Z

 (q) (q) The coefficients are related to those in Eq. (250) by fk = k fk , where the sum (q) ranges over all (kr , kz ) satisfying kr |kzres | + kz |krres | = k. We then have f λ = f0 . The set of resonant orbits is dense in the space of frequencies, though it is of measure zero. A given resonant ratio Υr /Υz = |kzres |/|krres | describes a surface in the parameter space spanned by pi . We refer to Ref. [141] for the characterization of the locations of these surfaces and to Refs. [142–145] for further discussion of resonant geodesic orbits.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1477

Accelerated Motion Evolution of Orbital Parameters We now consider an accelerated orbit satisfying the equation of motion (13), which we write compactly as D 2 zα = f α. dτ 2

(278)

The normalization uα uα = −1 implies that f α is orthogonal to the worldline: f α uα = 0. 

If we continue to define E = −ut , Lz = uφ , and Q = uα uβ K αβ − the accelerated orbit, then dE = −ft , dτ

dLz = fφ , dτ

 dQ 2 = 2K αβ uα fβ − 2 uφ˜ fφ˜ , dτ a

1 (u )2 a 2 φ˜

on

(279)

where fφ˜ = a(fφ + aft ) and uφ˜ = a(Lz − aE). In other words, the “constants” of motion are no longer constant. However, if f α is small, each parameter will change only slowly or oscillate slightly around a slowly varying mean. Our treatment of accelerated orbits mirrors that of geodesics, beginning with quasi-Keplerian methods and then describing the calculation of fundamental frequencies and perturbed angle variables. In the quasi-Keplerian treatment, we place no restriction on f α , and in particular, we do not assume it is small. In the treatment of perturbed angle variables, we restrict the analysis to a small perturbing force, μ setting f(0) = 0 in Eq. (13).

Method of Osculating Geodesics In celestial mechanics, perturbed Keplerian orbits have historically been described using the method of osculating orbits. The idea in this method is, given an exact solution to the unperturbed problem in terms of a set of orbital elements pi = {p, e, ι, . . .}, to write the perturbed orbit in precisely the same form but to promote the orbital elements to functions of time. At each instant t, the perturbed orbit with elements {p(t), e(t), ι(t), . . .} is tangent to a Keplerian ellipse (called the osculating orbit) with those same values of orbital elements. In general relativity, this idea is referred to as the method of osculating geodesics [132, 146–148]. Our geodesics in Kerr are described by Eqs. (217), (218), (232), and (233), which involve the seven orbital elements I A = {p, e, ι, t0 , ψr0 , ψz0 , φ0 }. α (I A , λ) denote a geodesic with these orbital elements and z˙ α (I A , λ) = If we let zG G α ∂zG /∂λ its tangent vector, then the osculation conditions are α A zα (λ) = zG [I (λ), λ]

and

dzα α A (λ) = z˙ G [I (λ), λ]. dλ

(280)

1478

A. Pound and B. Wardell

These conditions define a one-to-one transformation (zα , z˙ α ) → I A . Such a transformation is possible because the number of orbital elements is equal to the number of degrees of freedom on the orbit: the eight degrees of freedom {zα , z˙ α } minus the constraint z˙ α f α = 0. The osculation conditions can be used to transform the equation of motion (278) α α α ∂zG dI A ∂zG into evolution equations for I A (λ). Appealing to the chain rule dz dλ = ∂I A dλ + ∂λ , α (in terms of the non-affine parameter λ), and to the to the geodesic equation for zG equation of motion (278) for zα (converted to the non-affine parametrization), we find [146] α ∂zG dI A = 0, ∂I A dλ α ∂ z˙ G dI A = fα ∂I A dλ

(281) 

dτ dλ

2 α + [κ(λ) − κG (λ)]˙zG ,

(282)

 −1 d 2 τ where κ = dτ . If we define λ to satisfy Eq. (216) on both the geodesic and dλ dλ2 accelerated orbit, then κ = κG = Σ −1 dΣ dλ , simplifying Eq. (282) to α ∂ z˙ G dI A = Σ 2f α . ∂I A dλ

(283)

These equations are exact, and f α need not be small. A Equations (281) and (283) can be straightforwardly inverted to solve for dI dλ , providing a system of first-order ordinary differential equations for the orbital elements. However, working with the initial phases {t0 , ψr0 , ψz0 , φ0 } is cumbersome in practice. In the above evolution equations, the phases ψa are given by their geodesic values, meaning the solutions to Eqs. (219) and (220) with fixed I A . That is, at each value of λ, in Eqs. (219) and (220), we replace dψa /dλ with dψa /dλ and then integrate from λ = 0, with initial values ψa0 (λ), up to λ = λ. Similarly, in Eqs. (234), the integrals are evaluated with fixed orbital elements in the integrands. The evolution equations also involve derivatives of these integrals with respect to the orbital elements. Evaluating all these integrals at every time step would be computationally expensive. In applications, it is therefore preferable to work with the variables {p, e, ι, ψα } instead of I A . We write a geodesic trajectory and its tangent vector α α [p i , ψ (λ)] and z˙ α [p i , ψ (λ)] = ψ ˙ G ∂zG , where ψ˙ αG = fα (pi , ψ) are the as zG β β β ∂ψβ G geodesic “frequencies” given by Eqs. (208), (209), (219), and (220). The osculation conditions then read α zα (λ) = zG [pi (λ), ψβ (λ)] and

dzα α (λ) = z˙ G [pi (λ), ψβ (λ)]. dλ

(284)

36 Black Hole Perturbation Theory and Gravitational Self-Force dψ ∂zα

α

i

1479

∂zα

dp β G G Appealing to the chain rule dz dλ = dλ ∂ψβ + dλ ∂pi , to the geodesic equation α for zG (in terms of λ), and to the equation of motion (278) for zα (in terms of λ), we find that the osculation conditions imply

Here

α ∂zG ∂ψ

· δf =

α ∂zG ∂ψr δfr

α α ∂zG ∂zG dpi + · δf = 0, ∂pi dλ ∂ψ

(285)

α α ∂ z˙ G ∂ z˙ G dpi + · δf = Σ 2 f α . ∂pi dλ ∂ψ

(286)

+

α ∂zG ∂ψz δfz ,

and we have defined

δfa :=

dψa − ψ˙ aG . dλ

(287)

Equations (285) and (286) provide eight equations for the seven derivatives dpi

dψα dλ

and dλ ; any one of the four equations represented by (286) may be eliminated using f α uα = 0. The t and φ components of Eq. (285) are simply the osculation conditions: dψα = fα (ψ, pi ) for α = t, φ. dλ

(288)

The r and z components of Eq. (285) can be rearranged to obtain δfa = −

a /∂p i ∂zG dpi , a ∂zG /∂ψa dλ

(289)

where we have used the fact that r is independent of ψz and that z is independent of ψr . Substituting this into Eq. (286) yields dp i α Li (zG ) = Σ 2f α , dλ

(290)

where Li (x) :=

∂ x˙ ∂r/∂pi ∂ x˙ ∂z/∂pi ∂ x˙ − − . ∂pi ∂r/∂ψr ∂ψr ∂z/∂ψz ∂ψz

One can easily invert Eq. (290) to obtain equations for

(291)

dpi dλ :

' Σ2 & dp = [Le (z), Lι (φ)] f r + [Le (φ), Lι (r)] f z + [Le (r), Lι (z)] f φ , dλ D (292)

1480

A. Pound and B. Wardell

     ' Σ 2 & de = Lι (z), Lp (φ) f r + Lι (φ), Lp (r) f z + Lι (r), Lp (z) f φ , dλ D (293)      ' dι Σ 2 & = Lp (z), Le (φ) f r + Lp (φ), Le (r) f z + Lp (r), Le (z) f φ , dλ D (294) with [Li (x), Lj (y)] := Li (x)Lj (y) − Lj (x)Li (y) and D := Lp (r)[Le (z), Lι (φ)] + Le (r)[Lι (z), Lp (φ)] + Lι (r)[Lp (z), Le (φ)]. (295) Finally, the evolution equations for ψa are obtained by substituting Eqs. (292), (293), and (294) into Eq. (289), yielding dψa = fa (pi , ψ) + δfa (pi , ψ), dλ

(296)



de 1 dp (1 + e cos ψr ) −p cos ψr , pe sin ψr dλ dλ

(297)

where: δfr = − δfz =

dι cot ι cot ψz . dλ

(298)

There are superficial singularities in these formulas when ψa is an integer multiple of π . However, the divergences are analytically eliminated when the formulas are explicitly evaluated. The full set of evolution equations is given by Eqs. (292), (293), (294), (296), a [p i (λ), ψ (λ)] is given by Eqs. (217)– and (288). In these equations, x a (λ) = xG a a a i (218), x˙ (λ) = x˙G [p (λ), ψa (λ)] by r˙ =

pMefr sin ψr (1 + e cos ψr )2

and

z˙ = −zmax fz sin ψz ,

(299)

˙ = (ft , fφ ) by Eqs. (208) and (209). with Eqs. (219)–(220) for fa , and (t˙, φ) Wherever E, Lz , and Q appear, they are given in terms of pi by their geodesic expressions (225), (226), and (227). The quantities [Li (x), Lj (y)], when explicitly evaluated, constitute lengthy analytical formulas in terms of pi and ψ. However, for several Li (x), the second and third term vanish in Eq. (291). Specifically, ∂r ∂fr ∂ z˙ ∂z ∂fz Lι (r) = ∂∂ιr˙ = ∂ψ , and Lj (z) = ∂p j = ∂ψz ∂j for j = p, e. r ∂ι The evolution can be slightly simplified by adopting ψr or ψz as the parameter i dpi = dψa1/dλ dp along the trajectory. That is easily done by using, e.g., dψ dλ . However, a

36 Black Hole Perturbation Theory and Gravitational Self-Force

1481

a for a sufficiently large perturbing force, dψ dλ can vanish at some points in the evolution, making ψa an invalid parameter. In that case, we can split ψa into ψa = ψaG − ψa0 , where dψaG /dλ = fa and dψa0 /dλ = −δfa . ψaG is then a convenient, monotonic parameter along the worldline. Alternatively, t can be used, i −1 dpi applying, e.g., dp dt = ft dλ . The evolution equations simplify more dramatically in the special case of equatorial orbits, for which z = f z = 0. In this case, ι and ψz do not appear, and Eqs. (292)–(293) reduce to

  r 4 Le (φ)f r − Le (r)f φ dp = , dλ Lp (r)Le (φ) − Le (r)Lp (φ)   r 4 Lp (r)f φ − Lp (φ)f r de = , dλ Lp (r)Le (φ) − Le (r)Lp (φ)

(300) (301)

If ψrG is used as the independent parameter along the orbit, then the other three evolution equations are dψr0 /dψrG = −δfr /fr and Eq. (288) for t (ψrG ) and φ(ψrG ). In our treatment we have left the evolution equations in a highly inexplicit form even in the relatively simple equatorial case. References [146] and [148] provide explicit formulas in the cases of planar and nonplanar orbits in Schwarzschild spacetime. Reference [147] details the generic case in Kerr spacetime and describes several alternative formulations of the osculating evolution. Before proceeding, we note again that the equations in this section are valid for arbitrary forces, though the orbital elements are most meaningful when the force is small and the orbit is close to a geodesic. However, the method has most often been applied [148–151] in the context of an approximation in which the selfforce at each instant is approximated by the value it would take if the particle had spent its entire prior history moving on the osculating geodesic. Since the force is then constructed from the field generated by the osculating geodesic particle, this approximation might more properly be dubbed the method of osculating sources. In the next section, we restrict to the case of a small perturbing force.

Perturbed Mino Frequencies and Action Angles In the unperturbed case, the equations of geodesic motion could be written in terms of the orbital parameters and angle variables as dqα = Υα (pi ), dλ

(302)

dpi = 0. dλ

(303)

If the perturbing force is small, with an expansion

1482

A. Pound and B. Wardell α α f α = εf(1) (zμ , z˙ μ ) + ε2 f(2) (zμ , z˙ μ ) + O(ε3 ),

(304)

and is periodic in t and φ, then the equations of forced motion can still be written in terms of orbital parameters and angle variables: dqα j j = Υα(0) (pq ) + εΥα(1) (pq ) + O(ε2 ), dλ dpqi dλ

j

j

= εGi(1) (pq ) + ε2 Gi(2) (pq ) + O(ε3 ).

(305) (306)

Note that the subscript q on the orbital parameters pqi = (pq , eq , ιq ) is a label, not an index. The form (304) is mildly restrictive, and it does not include the Mathisson-Papapetrou spin force, for example; for a spinning body, we must introduce additional parameters and action angles describing the spin’s magnitude and direction [152, 153]. For our purposes we adopt a more restrictive form: α α (x, z˙ μ ) + ε2 f(2) (x, z˙ μ ) + O(ε3 ), f α = εf(1)

(307)

which assumes that the force inherits the background spacetime’s symmetries. We explain in Section “Multiscale Expansion of Source Terms and Driving Forces” that the form (307) still needs further, minor alteration to describe the self-force, but it is sufficiently general as a starting point. Unlike in the unperturbed case, the orbital parameters pqi and frequencies are no longer constant; they evolve slowly with time. However, the variables (qα , pqi ) cleanly separate the two scales in the orbit’s evolution: the variables pqi only change slowly, over the long timescale ∼1/ε, while the angle variables qα change on the (0) orbital timescale ∼2π/Υα . In the context of a small-mass-ratio binary, where the inspiral is driven by gravitational wave emission, the long timescale ∼1/ε is referred to as the radiation-reaction time. The division of the orbital dynamics into slowly and rapidly varying functions has the same utility as in the geodesic case: it enables convenient Fourier expansions of functions on the worldline, which mesh with a Fourier expansion of the field equations (described in the final section of this chapter). Functions f (r, z) on the accelerated worldline can be expanded in the Fourier series f [r(λ), z(λ)] =



fk (pq )e−iqk (λ) , (q)

j

(308)

k

with a clean separation between slowly varying amplitudes and rapidly varying phases. The coefficients remain given by Eq. (251). By eliminating oscillatory driving terms in the orbital evolution equations, the transformation to (qα , pqi ) also facilitates more rapid numerical evolutions [151] and, ultimately, more rapid generation of waveforms [154]. In this and the next section, for visual simplicity, we shall omit the label “(q)” from mode coefficients associated with q.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1483

Now, to put the equations of motion in the form (305)–(306), we begin with the evolution equations (288), (292), (293), (294), and (296). Given the expansion (307), these equations take the form dψα = fα(0) (ψ, pj ) + εfα(1) (ψ, pj ) + O(ε2 ), dλ dpi i i = εg(1) (ψ, pj ) + ε2 g(2) (ψ, pj ) + O(ε3 ). dλ

(309) (310)

i is given by Eqs. (292), (293), and (294) with f α → f α . We have renamed Here g(n) (n) (0)

(1)

(1)

α , and f fα to fα , fa is given by δfa with f α → f(1) α = 0 for α = t, φ. In this form of the equations, every term on the right is a periodic, oscillatory function of the phases. However, one can transform to the new variables (qα , pqi ), which have no oscillatory driving terms, using an averaging transformation [80, 151] (here we combine a near-identity averaging transformation with a zeroth-order one): j

j

j

ψα (qβ , pq , ε) = ψα(0) (qβ , pq ) + εψα(1) (qβ , pq ) + O(ε2 ), j

j

j

i i pi (qβ , pq , ε) = pqi + εp(1) (q, pq ) + ε2 p(2) (q, pq ) + O(ε3 ),

(311) (312)

where j

j

ψα(0) (qβ , pq ) = qα − Δψα(0) (0, pqi ) + Δψα(0) (q, pq )

(313)

(n)

i for n > 0 are 2π is the geodesic relationship and the corrections ψα and p(n) periodic in each qa (with a potentially nonzero mean). In analogy with the geodesic (0) case, we have chosen the origin of phase space such that ψα (qβ = 0) = 0. (0) This choice will ensure that at fixed pqi , ψα and qα satisfy all the relationships in Section “Fundamental Mino Frequencies and Action Angles”. Note that we could (0) replace Δψα (0, pqi ) with any other qα -independent function of pqi ; this would still (0)

represent a geodesic relationship between ψα and qα , but with different choices of initial phases for different values of pqi . For convenience in later expressions, we define Aα (pqi ) := −Δψα(0) (0, pqi ).

(314)

By substituting the expansions (311) and (312) into Eqs. (309) and (310), appealing to (305) and (306), and equating coefficients of powers of ε, one can solve for the frequencies Υα(n) and driving forces Gi(n) , as well as for the functions in the averaging transformation. Explicitly, the leading-order terms in Eqs. (309) and (310) are

1484

A. Pound and B. Wardell (0)

(0)

∂ψα ∂Δψα (0) Υ = Υα(0) + Υ (0) · ∂qβ β ∂q Gi(1) + Υ (0) ·

j

(315)

j

(316)

= fα(0) (ψ (0) , pq ),

i ∂p(1)

i = g(1) (ψ (0) , pq ).

∂q

Equation (315) is simply the geodesic relationship between ψα and qα . It follows (0) that we can use the geodesic solution (241) for qa (ψa , pqi ). Concretely, we may write  qa (ψa(0) , pqi )

=

Υa(0) (pqi )

(0)

dψa

ψa

fa(0) (ψa , pqi )

0

,

(317)

implying that the Fourier coefficients in Eq. (308) can be computed as the integrals (0) (0) over ψ (0) in Eq. (251), with the replacements Υa → Υa and ψa → ψa . This relies on our particular choice of Aα in Eq. (314); different choices would lead to different pqi -dependent lower limits of integration in Eq. (317), which in turn would lead to pqi -dependent phase factors appearing in Eq. (251). Using either of the forms (251) or (252), we can easily decompose Eqs. (315)  (0) (0,k) j i = and (316) into Fourier series, with Δψα = k =0 Δψα (pq )e−iqk and p(1)  i j −iqk . From the k = 0 terms in the equations, we find k p(1,k) (pq )e ( ) j j Υα(0) (pq ) = fα(0) (ψ (0) , pq ) ,

(318)

( ) j j i (ψ (0) , pq ) , Gi(1) (pq ) = g(1)

(319)

q

q

and from the k = 0 terms, we find (0,k)

j

Δψα(0,k) (pq ) = −

fα

(0)

j

(pq ) j

,

(320)

,

(321)

iΥk (pq ) j

j i p(1,k) (pq )

=−

i (pq ) g(1,k) (0)

j

iΥk (pq )

i where Υk(0) := k · Υ (0) = kr Υr(0) + kz Υz(0) . Note that these equations leave p(1,00) arbitrary. As foreshadowed above, Eqs. (318) and (320) are precisely the same as the geodesic formulas (236) and (253) (with the replacement pi → pqi ). The only change is that the orbital parameters pqi , which determine the frequencies and amplitudes, now adiabatically evolve with time.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1485

(0)

Importantly, Eq. (321) requires Υk = 0. This condition fails at resonances, (0) where Υkres = 0. Therefore, the averaging transformation is impossible when there is a resonance. We discuss this resonant case in the next section. Equation (320) also superficially appears to encounter a singularity at resonance, but this is skirted by (0,k) the particular form of fα , as we see from the more explicit formula (253). Moving onto the first subleading order in Eqs. (309) and (310), we have (0)

j

Υα(1) + G(1)

∂ψα

(0)

+ Υ (1) ·

j

∂pq

(1)

∂Δψα ∂ψα + Υ (0) · ∂q ∂q (0)

∂fα

j

= fα(1) + p(1) j

Gi(2) + G(1)

i ∂p(1) j

∂pq

+ Υ (1) ·

i ∂p(1)

∂q

j

∂pq

+ Υ (0) · j

i + p(1) = g(2)

(0)

∂fα

+ ψ (1) ·

∂ψ (0)

(322)

,

i ∂p(2)

∂q

i ∂g(1)

+ ψ (1) ·

j

∂pq

i ∂g(1)

(323)

,

∂ψ (0)

j

where all quantities on the left are functions of (q, pq ) and all those on the right are j functions of (ψ (0) , pq ). Taking the average of these equations yields Υα(1) (pqi ) j Gi(2) (pq )

=

=

(

fα(1)

(

i g(2)

) q

) q

*

(0) i ∂fα p(1) ∂pqi

+ *

j

+ p(1)

i ∂g(1) j

∂pq

+ψ

(1)

+ ψ (1) ·

·

∂fα(0)

+ j

∂ψ (0) q + i ∂g(1) ∂ψ (0)

q

− G(1) j

− G(1)

∂Aα j

(324)

,

∂pq

i ∂p(1,00) j

.

(325)

∂pq

i allows us to set We see from Eq. (324) that a judicious choice of p(1,00)

Υα(1) = 0 for α = r, z, φ.

(326)

i is determined from Such a p(1,00)

* + (0) (0,−k) (0) ) (  ∂Υ ∂f ∂f α α α j ∂Aα i i = − fα(1) − p(1,k) − ψ (1) · + G(1) j p(1,00) (0) q ∂pqi ∂pqi ∂ψ ∂pq k =0 q (327) for α = r, z, φ. We could alternatively set Υα(1) = 0 for a different trio of components, but this choice will be particularly useful in the final section of this review. This freedom is in addition to the freedom discussed above regarding the

1486

A. Pound and B. Wardell

i choice of Aα ; i.e., the functions Aα and p(1,00) in the averaging transformation are (1)

degenerate with Υα . Reference [151] provides a more thorough discussion of the freedom within near-identity averaging transformations. (1) The averages in Eqs. (324)–(325) involve ψa , which can be obtained from Eq. (322). A 2π -biperiodic solution to that equation is 

1

j

ψa(1) (q, pq ) =

j

Ya (qa , pq )

j

(0)

k

k

j

Sak (pq )Yak (pq ) −iΥk

(0)

− ikΥa

− fa q

e−iqk −ikqa , (328)

 k −ikq j j (0) j a, F where Ya (qa , pq ) := exp[−Fa (qa , pq )/Υa (pq )] = a := k Ya e  k  fa −ikqa is the antiderivative of the purely oscillatory part of fa := k =0 −ik e ∂fa(0) /∂ψa(0) , and

j

Sa (q, pqi ) := −G(1)

(0)

∂ψa

j ∂pq

j

(0)

+ fa(1) + p(1)

∂fa

j ∂pq

=



Sak (pqi )e−iqk .

(329)

k

Equation (328) seems to be the unique 2π -biperiodic solution. Any other solution can only differ by a homogeneous solution to Eq. (322), which must take the form % exp( fa dqa /Υa(0) )f (qb − qa Υb(0) /Υa(0) ) for some function f , with b = a. It appears that such a function cannot simultaneously be 2π periodic in both qa and qb . The remaining pieces of Eqs. (322) and (323) determine the purely oscillatory (1) (1) i . Specifically parts of ψt , ψφ , and p(2)

ψα(1,k) = i p(2,k)

=



1 (0)

−iΥk

(0,k) j ∂Δψα fα(1,k) + Pαk − G(1) j ∂pq



1 (0)

−iΥk

j

i g(2,k)

+ Qik

i j ∂p(1,k) − G(1) j ∂pq

(0)

(0)





(331)

i j ∂g(1)

for α = t, φ, where Pα := p(1) ∂fαj + ψ (1) · ∂fα(0) and Qi := p(1) ∂pq

∂ψ

(330)

,

j ∂pq

+ ψ (1) ·

i ∂g(1)

∂ψ (0)

.

This averaging transformation can be carried to any order. Analogous calculations also apply if we use P i = (E, Lz , Q) rather than pi = (p, e, ι). Ultimately, the coordinate trajectory zα can be expressed in terms of (qα , pqi ) as α α zα (qβ , pqi ) = z(0) (qβ , pqi ) + εz(1) (q, pqi ) + O(ε2 ).

(332)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1487

The leading-order trajectory has the same dependence on qα and pqi as a geodesic. α (q , p i ), given by Eq. (257), then zα (q , p i ) = That is, if we write a geodesic as zG β q (0) β α i zG (qβ , pq ). Wherever the geodesic expressions involve P i , they are here evaluated at Pqi = (Eq , Lq , Qq ), which are related to pqi by the geodesic relationships. (We suppress the subscript z on Lz .) The difference between the geodesic and the accelerated trajectory lies entirely in the evolution of their arguments: rather than evolving according to Eqs. (302) and (303), qα and pqi now evolve according to Eqs. (305) and (306). α (q, p i ) to the trajectory In the context of a binary, the small corrections εz(1) q remain uniformly small over the entire inspiral until the transition to plunge [155]; because they are periodic functions of q, they have no large secular terms. t (1) and φ (1) are given by Eq. (330), with t (1,00) and φ (1,00) left arbitrary. r (1) and z(1) are given by a x(1) = ψ (1) ·

a ∂xG

∂ψ (0)

i + p(1)

a ∂xG , ∂pqi

(333)

(1)

i by Eq. (321), and p i with ψa given by Eq. (328), the oscillatory part of p(1) (1,00) by Eq. (327). References [20,144,151,156,157] contain more detailed action-angle treatments of perturbed orbits. With the exception of Ref. [151], these treatments have not begun with equations of the form (309) and (310). Instead, they began with approximate angle variables, which we will denote qˆα and which satisfy qˆα = qα + O(ε). The equations of motion then take the form

d qˆα ˆ pj ) + O(ε2 ), = Υα(0) (pj ) + εUα(1) (q, dλ

(334)

dpi i i ˆ pj ) + ε2 F(2) ˆ pj ) + O(ε3 ). = εF(1) (q, (q, dλ

(335)

Reference [20] derives concrete equations of this form in the case that proper time τ is used instead of Mino time and that {E, Lz , Q} are used instead of {p, e, ι}. However, note that the notation in Ref. [20] differs in several significant ways from ours. In particular, Ref. [20] uses λ to denote a rescaled τ , qα to denote an analog of our qˆα (and associated with τ rather than Mino time), and ψα to denote an analog of our qα (again associated with τ ). i in Eq. (335) are given by Eq. (279). The averaging The driving forces F(n) transformation (qˆα , pi ) → (qα , pqi ) can be found as we did above, with substantial

qˆα simplifications arising from the fact that ddλ is constant at leading order; the transformation is given by Eqs. (383) and (384) (without the restriction k = Nk res in the nonresonant case). Our particular construction in this section and the next is instead designed to link the action-angle description with the quasi-Keplerian one. It appears here for the

1488

A. Pound and B. Wardell

first time. However, Ref. [151] considers more general sets of coupled differential equations that involve variables analogous to our ψα as well as variables analogous to qˆα , though without providing a solution analogous to our (328).

Perturbed Boyer-Lindquist Frequencies and Action Angles Because we solve field equations and extract waveforms using Boyer-Lindquist time t, it is once again useful to construct variables (ϕα , pϕi ) associated with t, where ϕt = ψt = t and pϕi = (pϕ , eϕ , ιϕ ). The construction of the variables (ϕα , pϕi ) (and of their evolution equations) is analogous to the construction based on λ: the osculating geodesic equations for dψα /dt and dpi /dt have the same form as (n) (n) (0) i i /f (0) , and Eqs. (309) and (310), simply with fα → fα /ft and g(n) → g(n) t after a near-identity averaging transformation, we arrive at the equations of motion dϕα = Ωα(0) (pϕj ), dt dpϕi dt (0)

(336)

i i = εΓ(1) (pϕj ) + ε2 Γ(2) (pϕj ) + O(ε3 ). (0)

(337) (n)

Ωα are the geodesic frequencies, and Ωt = 1. The corrections Ωα for α = (n) r, z, φ and n > 0 are eliminated just as in the previous section, while Ωt = 0 trivially for n > 0 because dϕt /dt = 1. The two sets of variables (ϕα , pϕi ) and (qα , pqi ) are related by a transformation j

j

ϕα (qβ , pq , ε) = ϕα(0) (qα , pqi ) + εΦα(1) (q, pq ) + O(ε2 ), j

j

i pϕi (qβ , pq , ε) = pqi + επ(1) (q, pq ) + O(ε2 ),

(338) (339)

where the leading term in ϕα is given by the geodesic relationship (263), which we restate as j

ϕ (0) (qα , pqi ) := qα + Bα (pqi ) + Ωα(0) (pqi )Δt (0) (q, pq ),

(340)

defining j

Bα (pqi ) := −Ωα(0) (pqi )Δt (0) (0, pq ).

(341) (0)

Like in the geodesic case, this value for Bα imposes that ϕα and qα (and ψα ) have the same origin in phase space. As discussed around Eqs. (314) and (317), this means that we can immediately utilize all the relationships from Section “Funda(1) mental Boyer-Lindquist Frequencies and Action Angles”. The corrections Φα and i π(1) are 2π -biperiodic in q. i , can be derived The terms in this transformation, as well as the driving forces Γ(n) by substituting Eqs. (338) and (339) into Eqs. (336) and (337). Taking the average

36 Black Hole Perturbation Theory and Gravitational Self-Force

1489

of the resulting equations and appealing to Eqs. (305) and (306), we obtain (0)

Ωα(0) =

Υα

i Γ(1) =

and

(0)

Υt

Gi(1)

(342)

(0)

Υt

at leading order and Ωα(1) = 0 = − i Γ(2) =

 1  j i (0) (0) G , ∂ B + R  ∂ Ω + P  Ω j i q p t q α α (1) p α (0)

(343)

Υt

 1  i j i i i G(2) + G(1) ∂pj π(1) q − R j q ∂pj Γ(1) − Pt q Γ(1) (0)

(344)

Υt

(0)

(0)

i and P = ψ (1) · ∂ f at the first subleading order, where R i := ft π(1) t ψ t i ∂ f (0) . p(1) pi t

i  π(1) q

The average is chosen to enforce oscillatory parts of the equations yield Δt (0,k) =

ft(0,k) (0)

−iΥk

(1) Ωα

+

= 0 in Eq. (343). The

,

(345)

i Γ(1)

(346)

(0,k)

i π(1,k) =

ft

(0)

−iΥk

at leading order and Φα(1,k) =



1 (0) −iΥk

Rki ∂pi Ωα(0) + Ptk Ωα(0) − Gi(1) ∂i Δϕα(0,k)

 (347)

at the first subleading order. In all of the above expressions, k refers to a Fourier decomposition into e−iqk modes. All functions of pi are evaluated at pqi , and inside the integrals (251), all functions of ψ are evaluated at ψ (0) (q, pqi ). When solving the field equations, we shall require Fourier decompositions with respect to ϕ: f [r(t), z(t)] =



fk (pϕj )e−iϕk . (ϕ)

(348)

k

We can calculate the coefficients as integrals over q using the transformation (338). However, it is simpler to use the geodesic change of variables defined by the leading term in the transformation. The coefficients are then given by Eq. (265) with the (0) (0) replacements pi → pϕi , Υα → Υα , and fα → fα or by Eq. (266) with the additional replacement ψ → ψ (0) .

1490

A. Pound and B. Wardell

We will also require the transformation from (ψα , pi ) to (ϕα , pϕi ): ψα (ϕβ , pϕi , ε) = ψα(0) (ϕβ , pϕi ) + εψα(ϕ,1) (ϕ, pϕi ) + O(ε2 ),

(349)

i pi (ϕα , pϕj , ε) = pϕi + εp(ϕ,1) (ϕ, pϕj ) + O(ε2 ).

(350)

Following the same steps as in the previous section, at leading order, we recover (0) the geodesic frequencies and find ψα (ϕβ , pϕi ) is given by the geodesic relationship (271). Solving the subleading-order equations is made difficult because the analog of Eq. (322) has the form (ϕ,1)

Ω

(0)

∂ψα · ∂ϕ

−ψ

(ϕ,1)

·



∂ ∂ψ (0)

(0)

fα

 = ...

(0)

ft

(351)

The α = r, z components of this equation, unlike those of Eq. (322), are coupled partial differential equations for ψ (ϕ,1) , which do not have a solution of the form (328). However, we can find the subleading terms in Eqs. (349) and (350) by combining our knowledge of (ϕα , pϕi ) and (ψα , pi ) as functions of (qα , pqi ). Substituting the expansions (338) and (339) into the right-hand sides of Eqs. (349) and (350) and equating the results with Eqs. (311) and (312), we find (0) (1) ∂ψα

(0)

ψα(ϕ,1) (ϕβ , pϕi ) = ψα(1) (qβ , pϕi ) − ϕβ

∂ϕβ

(0)

i (ϕγ(0) , pϕi ) − π(1)

∂ψα (ϕγ(0) , pϕi ), ∂pϕi (352)

i i i (ϕ (0) , pϕj ) = p(1) (q, pϕi ) − π(1) (ϕ (0) , pϕj ), p(ϕ,1)

(353)

(0)

where ϕα is given by Eq. (340) with pqi → pϕi . The inverse transformation, which we will also need, is (0) (1) ∂ϕα

ϕα (ψβ , pi , ε) = ϕα(0) (ψβ , pi ) + εΦα(1) (q (0) , pi ) − εψβ

∂ψβ

(0)

i − εp(1)

i i (ϕ (0) , pj ) − εp(1) (q (0) , pj ), pϕi (ψβ , pi , ε) = pi + επ(1) (0)

∂ϕα , ∂pi (354) (355)

(0)

where ϕα and qα are the geodesic functions of ψα and pi . Finally, we can reconstruct the coordinate trajectory zα in the form α α zα (ϕβ , pϕi ) = z(0) (ϕβ , pϕi ) + εz(ϕ,1) (ϕ, pϕi ) + O(ε2 ).

(356)

The zeroth-order trajectory has the same functional dependence as a geodesic; that α (ϕ , p i ) = zα (ϕ , p i ), where zα (ϕ , p i ) is given in Eq. (273). In analogy is, z(0) β ϕ ϕ G β G β

36 Black Hole Perturbation Theory and Gravitational Self-Force

1491

with the Mino-time solution, wherever the geodesic expressions involve P i , they are here evaluated at Pϕi = (Eϕ , Lϕ , Qϕ ), which are related to pϕi by the geodesic (ϕ,1)

α are t(ϕ,1) = 0, φ(ϕ,1) = ψφ relationships. The first-order corrections z(ϕ,1) by Eq. (352), and a x(ϕ,1) = ψ (ϕ,1) ·

a ∂xG

∂ψ (0)

i + p(ϕ,1)

a ∂xG , ∂pϕi

given

(357)

i given by Eqs. (352) and (353). with ψ (ϕ,1) and p(ϕ,1)

Multiscale Expansions, Adiabatic Approximation, and Post-Adiabatic Approximations Self-accelerated orbits are often described with a multiscale (or two-timescale) expansion [15,20,114,139,158–161]. This is essentially equivalent to the averaging transformation described above. To illustrate the method, we return to Eqs. (309) and (310). We introduce a slow time variable λ˜ := ελ; this changes by an amount ∼ ε0 on the timescale ∼ 1/ε. In place of the transformations (311) and (312), we adopt expansions ˜ ε) = qα + A˜ α (λ) ˜ + Δψ˜ α(0) (q, λ) ˜ + εψ˜ α(1) (q, λ) ˜ + O(ε2 ), ψα (qβ , λ, i i pi (qα , λ˜ , ε) = p˜ (0) (λ˜ ) + εp˜ (1) (q, λ˜ ) + O(ε2 ),

(358) (359)

where qα satisfies dqα = Υ˜α(0) (ελ) + εΥ˜α(1) (ελ) + O(ε2 ) := Υ˜α (ελ, ε). dλ

(360)

We then substitute these expansions into Eqs. (309) and (310), applying the chain rule d ∂ ∂ = Υ˜α +ε . dλ ∂qα ∂ λ˜

(361)

qα and λ˜ are then treated as independent variables, making Eqs. (309) and (310) into a sequence of equations, one set at each order in ε. These equations are essentially equivalent to (315) and (316) at leading order and to Eqs. (322) and (323) at first subleading order, with tildes placed over all quantities and the (0) ˜ Gi ∂ψαj → d(A˜ α + following replacements: pqi → p˜ i , Gi → d p˜ i /d λ, (0)

Δψ˜ α )/d λ˜ , and Gi(1) (0)

i ∂p(1) j

∂pq

(n)

(n−1)

(1) ∂p q

→ 0. These equations can be solved just as we solved

Eqs. (315), (316), (322), and (323). The only difference between this expansion and Eqs. (311) and (312) is how each parameterizes the orbit’s slow evolution, whether with slowly evolving parameters

1492

A. Pound and B. Wardell

pqi or with slow time λ˜ . Indeed, the solutions are easily related. The solutions to Eqs. (305) and (306) can be expanded as 1  (0) ˜ + εq˜α(1) (λ) ˜ + O(ε2 ) , q˜α (λ) ε i ˜ i i (λ˜ ) + εp˜ q(1) (λ˜ ) + O(ε2 ), pq (λ, ε) = p˜ q(0) ˜ ε) = qα (λ,

˜ = where q˜α (λ) (n)

% λ˜ 0

(362) (363)

(n) (n) Υ˜α (λ˜  )d λ˜  + q˜α (0) with

˜ = Υα(0) , Υ˜α(0) (λ)

(364)

˜ = Υα(1) + p˜ i (λ)∂ ˜ pi Υα(0) . Υ˜α(1) (λ) q(1)

(365)

(n) i ˜ Substituting these On the right, Υα and its derivatives are evaluated at p˜ q(0) (λ). expansions into Eqs. (311) and (312) and comparing to Eqs. (358) and (359), we read off

Δψ˜ α(0) (q, λ˜ ) = Δψα(0) ,

(366)

i ˜ = p˜ i (λ), ˜ p˜ (0) (λ) q(0)

(367)

˜ = ψα(1) + p˜ j (λ)∂ ˜ pj ψα(0) , ψ˜ α(1) (q, λ) q(1)

(368)

i ˜ = pi + p˜ i (λ). ˜ p˜ (1) (q, λ) (1) q(1)

(369)

i , ψ , and their derivatives are evaluated at [q, p˜ ˜ Here ψα , p(1) α q(0) (λ)]. These i ˜ = Aα [p˜ (λ)]. ˜ Just as the particular relationships rely on choosing A˜ α (λ) q(0) averaging transformation did, the multiscale expansion has considerable degeneracy (1) i . If different choices are made, then we cannot identify between A˜ α , Υ˜α , and p˜ (1) qα between the two methods. However, regardless of choices, both methods will ˜ ε) and pi (λ, ˜ ε) (assuming ultimately output identical solutions of the form ψα (λ, identical initial conditions), and when written in that form, they can be unambiguously related. All the same relationships apply if we instead use t-based variables with a slow time t˜ := εt. When considering the multiscale expansion of the Einstein equation, it will be useful to have at hand the expansions (0)

j

(1)

ϕα (t˜, ε) =

1  (0) ϕ˜α (t˜) + εϕ˜α(1) (t˜) + O(ε2 ) , ε

i i (t˜) + εp˜ ϕ(1) (t˜) + O(ε2 ). pϕi (t˜, ε) = p˜ ϕ(0)

(370) (371)

It follows from Eqs. (336) and (337) that the coefficients in these expansions satisfy

36 Black Hole Perturbation Theory and Gravitational Self-Force

1493

(0)

d ϕ˜α j = Ωα(0) (p˜ ϕ(0) ), d t˜ i d p˜ ϕ(0)

d t˜

(372)

j

i (p˜ ϕ(0) ), = Γ(1)

(373)

(1)

d ϕ˜α j j = p˜ ϕ(1) ∂j Ωα(0) (p˜ ϕ(0) ), d t˜ i d p˜ ϕ(1)

d t˜

j

j

(374) j

i i (p˜ ϕ(0) ) + p˜ ϕ(1) ∂j Γ(1) (p˜ ϕ(0) ). = Γ(2)

(375)

We can also write Eq. (370) as 1 ε



t˜

Ωα (t˜ , ε)d t˜ + ϕα (0, ε),

(376)

Ωα (t˜, ε) = Ω˜ α(0) (t˜) + εΩ˜ α(1) (t˜) + O(ε2 ),

(377)

ϕα (t˜, ε) =

0

with

j (0) (0) i (1) (0) j where Ω˜ α (t˜) = Ωα [p˜ ϕ(0) (t˜)] and Ω˜ α (t˜) = p˜ ϕ(1) (t˜)∂j Ωα [p˜ ϕ(0) (t˜)]. There is a trade-off in solving Eqs. (372), (373), (374), and (375) rather than Eqs. (336) and (337): Eqs. (372), (373), (374), and (375) double the number of numerical variables, but they are independent of ε, meaning they can be solved for all values of ε simultaneously. Equations (336) and (337) have half as many variables, but they cannot be solved without first specifying a value of ε. Since the waveform phase in a binary is directly related to the orbital phase, the expansion (370) provides a simple means of assessing the level of accuracy of a given approximation. The approximation that includes only the first term, ϕ˜α(0) , is called the adiabatic approximation (denoted 0PA); it consists of the coupled equations (372) and (373), which describe a slow evolution of the geodesic frequencies. (n) An approximation that includes all terms through ϕ˜α is called an nth post-adiabatic approximation (denoted nPA); it consists of the coupled equations (372), (373), (374), and (375). We return to the efficacy of 0PA and 1PA approximations in the final section of this review. Reference [20] determined what inputs are required for a 0PA or 1PA approximation. To describe these inputs, we define the time-reversal ψα → −ψα , f α (ψ) → εα f α (−ψ), where f α is the accelerating force, εα := (−1, 1, 1, −1), and there is no summation over α. We then define the dissipative and conservative pieces of the force: α fdiss =

1 α 1 f (ψ) − εα f α (−ψ), 2 2

(378)

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A. Pound and B. Wardell

α fcon =

1 α 1 f (ψ) + εα f α (−ψ). 2 2

(379)

α → −f α and f α → +f α . These definitions imply that under time reversal, fdiss con con diss It is straightforward to see from Eqs. (292), (293), (294), (297), (298), (299), and (298), and the definition of Li that dpi /dt only receives a direct contribution α , while dψ /dt only receives a direct contribution from f α . At 0PA from fdiss a con order, f α enters the evolution through Eq. (373), in the quantity * + * +   i i dp 1 dp i   = = (0) . (380) Γ(1)  α α dt f α →f α dλ Υ f →f(1) t (1) ϕ q α Hence, the 0PA approximation only requires f(1)diss . At 1PA, f α enters the i i i is given by Eq. (344) evolution through both Γ(1) and Γ(2) in Eq. (375), where Γ(2) (1)

i chosen such that Υα with (325), (321), (328), and with p(1,00)

quantities involve (1)

α f(2)diss

(via

i  g(2) q

=

i α α  dp dλ |f →f(2) q

= 0. These

α in Eq. (325)), f(1)con (via (1)

i α ), and , which are both partially determined by fa = δfa |f α →f(1) ψα and p(1,00) α α α f(1)diss . Hence, the 1PA approximation requires the entirety of f(1) as well asf(2)diss . The fact that dissipative effects dominate over conservative ones on the long timescale of an inspiral is important in practical simulations of binaries. At least at first order, the dissipative self-force is substantially easier to compute than the conservative self-force. We discuss this in the final section of this review. We refer to Ref. [80] for a pedagogical introduction to multiscale expansions in more general contexts. Reference [20] contains a detailed discussion of the multiscale approximation for self-accelerated orbits in Kerr spacetime. References [15, 139] present variants of the method in simpler binary scenarios.

Transient Resonances Given that the orbital frequencies slowly evolve, they will occasionally encounter a resonance. Typically [144], the frequencies will continue to evolve, transitioning out of the resonance. These transient resonances have significant impact on orbital evolution. The near-identity averaging transformation (ψα , pi ) → (qα , pqi ) becomes singular at a resonance, as described below Eq. (321). Specifically, it becomes singular for the mode numbers Nk res for which ΥN kres = 0. To assess the effect of a resonance, we start from the equations of motion in the form (334)–(335). i and “frequencies” U (1) can be expanded in Fourier series The driving forces F(n) α  i = i j )e−i qˆk . However, near a resonance, a set of apparently such as F(n) F (p k (n,k) oscillatory terms becomes approximately stationary. Specifically, near a resonance (0) ˆ and all integer where Υres := krres Υr(0) + kzres Υz(0) = 0, the phase qres = k res · q, multiples of it, ceases to evolve on the orbital timescale. To see this, suppose the resonance occurs at a time λres . Near that time, qres can be expanded in a Taylor series:

36 Black Hole Perturbation Theory and Gravitational Self-Force

1495

1 qres (λ) = qres (λres ) + q˙res (λres )(λ − λres ) + q¨res (λres )(λ − λres )2 + . . . 2 (0)

(381)

(0)

Since q˙res (λres ) ≈ Υres (λres ) = 0 and q¨res ≈ dΥres /dλ, we see that qres changes on the timescale " δλ =

1 (0) dΥres /dλ

1 ∼√ , ε

(382)

which is much longer than the orbital timescale (but much shorter than the radiationreaction time). During the passage through a resonance, these additional quasi-stationary driving forces cause secular changes to the orbital parameters. We isolate these effects by performing a partial near-identity averaging transformation that eliminates all oscillations from the evolution equations except those depending on the resonant angle variable qres . An appropriate transformation, through 1PA order, is given by 

j

qˆα (q, pq , ε) = qα + εBα β (pqi )qβ + ε

k =N k res

p

i

j (q, pq , ε)

=

pqi

+ εC

i

j (pq ) + ε



i F(1,k)

k =N k res

(0) −iΥres

j

F(1,k) ∂Υα(0) j (0) iΥk ∂pq −iq e k, (0) j −iΥres (pq )

(1,k)

Uα

−

e−iqk ,

(383)

(384)

j

where functions of pj inside the sums are evaluated at pq and Bα β and C i are any functions satisfying (0)

Bα β Υ β

= C j ∂pj Υα(0) + Uα(1) q . q

(385)

These transformations satisfy the analogs of Eqs. (316) and (322) but with resonant modes excluded, with Bα β and C i chosen to eliminate the frequency corrections Υα(1) , and with the simplifications that fa(0) is replaced by Υa(0) and Δψa(0) by 0 (consequences of qα(0) already being a leading-order action angle). Together, the transformations (383) and (384) bring the equations of motion to the form dqα j = Υα(0) (pq ) + O(ε), dλ  dpqi j j −iN qres i F(1,N + O(ε2 ), = εGi(1) (pq ) + ε k res ) (pq )e dλ N =0

i  . For simplicity, we suppress 1PA terms. with Gi(1) = F(1) q

(386) (387)

1496

A. Pound and B. Wardell

How much does the second term in Eq. (387) contribute to the evolution of pqi ? Far from resonance, the additional term averages to zero. If we%denote the term as εδGi , then across resonance, it contributes an amount δpqi = ε δGi dλ. Applying the stationary phase approximation to the integral, we find

δp = i

 N =0

" i F(1,N k res )



  √ 2π ε (0) iπ ˙ exp sgn N Υres + iNqres (λres ) + o( ε), (0) 4 |N Υ˙res | (388)

(0) where we use a dot to denote a derivative with respect to λ˜ , Υ˙res := (0) i ∂Υres (1) ∂pqi

(0)

dΥres d λ˜

=

i , and both Υ˙res and F(1,N k res ) are evaluated at pq (λres ). The magnitude of √ δpi is ∼ ε; intuitively, this corresponds to a quasi-stationary driving force of size √ ∼ε multiplied by the resonance-crossing time δλ ∼ 1/ ε. But δpi is not a simple product of the two; each quasi-stationary driving force is weighted by a phase factor, such that δpi depends sensitively on the value of the resonant phase at resonance, qres (λres ). This implies that in order to determine δpi at leading order, one must know the 1PA-phase evolution prior to resonance. A proper accounting of the passage through resonance requires matching a near-resonance expansion to an off-resonance, multiscale expansion; see Sec. III of Ref. [144] or Appendix B of Ref. [162] for demonstrations of this matching √ procedure. Because a resonance shifts the orbital parameters by an amount ∼ ε and the shifted parameters subsequently evolve over the long timescale ∼1/ε, the resonance introduces half-integer powers into the multiscale expansion. For example, after a resonance, Eqs. (370)–(371) become

G

ϕα (t˜, ε) =

(0)

j

1  (0) ϕ˜α (t˜) + ε1/2 ϕ˜α(1/2) (t˜) + εϕ˜α(1) (t˜) + O(ε3/2 ) , ε

i i i (t˜) + ε1/2 p˜ ϕ(1/2) (t˜) + εp˜ ϕ(1) (t˜) + O(ε3/2 ). pϕi (t˜, ε) = p˜ ϕ(0)

(389) (390)

The effect of a single resonance therefore dominates over all other post-adiabatic (1/2) i effects. However, determining the resonant corrections ϕ˜α and pϕ(1/2) requires the shifts (388), which in turn require the resonant phase qres through 1PA order. This means that the 1/2-post-adiabatic-order corrections can be thought of as outsize 1PA corrections. Further discussions of transient resonances can be found in Refs. [140, 143, 144, 157, 162–166]. Because resonances are dense in the parameter space, an inspiraling body will pass through an infinite number of them. However, because the forcing i coefficients F(1,k) decay with increasing k, the only resonances with significant impact are “low-order” resonances, such as Υr /Υz = 1/2. A large fraction of inspiraling orbits will encounter such a resonance in the late inspiral [162, 164], but neglecting the effect of resonance in EMRIs may lead to only a small loss of detectable signals [162].

36 Black Hole Perturbation Theory and Gravitational Self-Force

1497

In addition to the intrinsic r–z orbital resonances discussed here, resonances can also occur due to a variety of other effects. There can be extrinsic resonances in which krres Ωr + kzres Ωz + kφres Ωφ = 0 for some triplet (krres , kzres , kφres ); these lead to non-isotropic emission of gravitational waves, causing possibly observable kicks to the system’s center of mass [167, 168], but their effects are subdominant relative to r–z resonances. If the secondary is spinning, its spin can also create resonances [169], as can the presence of external matter source such as a third body [161, 170].

Solving the Einstein Equations with a Skeleton Source In this section we describe how to combine the methods of the previous sections to model small-mass-ratio binaries. This consists of solving the global problem in a Kerr background: the perturbative Einstein equations with a skeleton source (i.e., a point particle or effective source) moving on a trajectory governed by Eq. (195) or (198). The first part of the section summarizes a multiscale expansion of the field equations, building directly on our treatment of orbital dynamics. At adiabatic order, waveforms can be generated by solving the linearized Einstein or Teukolsky equation with a point particle source and calculating the dissipative first-order selfforce. At 1PA order, one must solve the second-order Einstein equation and compute the first-order conservative self-force and second-order dissipative self-force. These 1PA calculations require, as a central ingredient, a mode decomposition of the singular field; this is the subject of the second part of the section. For simplicity, we assume the small object is spherical and nonspinning and that it does not encounter any significant orbital resonances.

Multiscale Expansion Structure of the Expansion Like the orbital dynamics, the metric in a binary has two distinct timescales: the orbital periods Tα = 2π/Ωα and the long radiation-reaction time ∼1/(εTα ). The evolution on the orbital timescale is characterized by periodic dependence on the orbital action angles ϕα , which satisfy Eq. (336). The evolution on the radiationreaction time is characterized by a slow change of the orbital parameters pϕi , governed by Eqs. (337), and of the central black hole parameters (MBH , JBH ), which evolve due to absorption of energy and angular momentum according to Eqs. (97) and (99). If we did not neglect the small object’s spin and higher moments, they would come with additional parameters and phases [152, 153]. The black hole parameters change at a rate F H ∝ |h|2 ∼ ε2 . Over the radiationreaction time, this accumulates to a change ∼ε, allowing us to write the evolving parameters as MBH = M + εδM and JBH = J + εδJ , where M and J are constant and MA := (δM, δJ ) evolve on the radiation-reaction time. We then work on the

1498

A. Pound and B. Wardell

fixed Kerr background with parameters M and a = J /M, with a set of slowly evolving system parameters P α = {pϕi , MA }. We will use the split into action angles ϕα and system parameters P α to expand the metric perturbation and stress-energy as hμν =

2  

εn h(nmk) (P α , x)eimφ−iϕmk + O(ε3 ), μν

(391)

(nmk) εn Tμν (P α , x)eimφ−iϕmk + O(ε3 ),

(392)

n=1 m,k

Tμν =

2   n=1 m,k

where m, kr , kz all run from −∞ to ∞, x = (r, z), and ϕmk := mϕφ +kr ϕr +kz ϕz . Here ϕα and P α are functions of t and ε governed by Eqs. (336), (337), and dMA = εFA(1) (pϕi ) + O(ε2 ), dt

(393)

where FA = (FEH /ε2 , FLHz /ε2 ) is given by any of Eqs. (97) and (99), Eqs. (140a) and (141a), or Eqs. (156b) and (157b); the reason this depends only on pϕi at leading order is explained in Section “Adiabatic Approximation”. The decomposition into azimuthal modes eimφ is not strictly necessary here, but it simplifies the analysis of the stress-energy in the next subsection, and it dovetails with the decompositions into angular harmonics in Sec. 4, as all the bases of harmonics involve φ only through the factor eimφ . The expansions (391) and (392) differ from the “self-consistent expansion” (192) in that the parameters P in the self-consistent expansion include the complete trajectory zμ and its derivatives. We can therefore move from Eqs. (192) and (205) to Eqs. (391) and (392) by substituting the expansion of zα (t) from Eq. (356). To fully motivate our multiscale expansion, we work through this expansion of Tμν in the next subsection. But first, we focus on the overall structure and efficacy of the multiscale expansion. Given Eqs. (391) and (392), the perturbative field equations become (nmk) equations for the Fourier coefficients hμν . These are identical, at leading order, to the usual frequency-domain field equations of black hole perturbation theory, with discrete frequencies: (1)

dϕmk (0) = ωmk (pϕi ) := kr Ωr(0) (pϕi ) + kz Ωz(0) (pϕi ) + mΩφ (pϕi ). dt

(394)

More concretely, if we substitute the expansions (391) and (392) into the Einstein equations, then t derivatives act as ∂ dP α ∂ ∂ = Ωα(0) (pϕj ) + ∂t ∂ϕα dt ∂P α

(395)

36 Black Hole Perturbation Theory and Gravitational Self-Force

→

−iωmk (pϕj ) + ε

i Γ(1) (pϕj )

∂ ∂ (1) + FA (pϕj ) ∂pϕi ∂MA

1499

! + O(ε2 ).

(396)

Using this, we can write covariant derivatives as ∇α → ∇˜ α0mk + εδαt ∂˜t1mk + O(ε2 ),

(397)

where ∇˜ α0mk is an ordinary covariant derivative with ∂φ → im and ∂t → −iωmk and ∂˜t1mk is the operator in square brackets in Eq. (396). If we then treat ϕα and P α as independent variables, we can split the field equations into coefficients of eimφ−iϕmk and of explicit powers of ε. This results in a sequence of differential (nmk) equations in (r, z) for the coefficients hμν : (1mk) [h(1mk) ] = 8π Tμν , G(1mk) μν (2mk) G(1mk) [h(2mk) ] = 8π Tμν − μν

(398)  

 

 k  )

G(2mk) [h(1m k ) , h(1m μν

]

m m k  k  (1) ˙ (1mk) i ˙ (1mk) − Γ(1) [∂MA h(1mk) ]. Gμν [∂pϕi h(1mk) ] − FA G μν

(399) (1mk)

(2mk)

and Gμν are the linearized and quadratic Einstein tensors (3) Here Gμν (1) ˙ (1mk) and (4) with the replacement ∇α → ∇˜ α0mk . G is the piece of Gμν that, μν after applying the rule (397), is linear in ∂˜t1mk . Explicit expressions for these quantities can be found in Sec. VC of Ref. [114] in a Schwarzschild background in the Lorenz gauge. The field equations in Ref. [114] are further specialized to quasicircular orbits, with frequencies ωm = mΩφ(0) , but they remain valid under the replacement ωm → ωmk . In Sec. VC of Ref. [114], they also include (1) frequency corrections Ωφ , which we have eliminated here with our choice of averaged variables (ϕα , pϕi ); the analog of our choice is described in their Appendix A. Beyond these minor differences, they more substantially differ by allowing the phases and system variables to depend on r in addition to t. We discuss the reason for this in Section “Snapshot Solutions and Evolving Waveforms”. The left-hand side of the field equations (398) and (399) is identical to what it would be if we expanded hμν in Fourier modes eimφ−iωmk t . Such a Fourier expansion is what has been implemented historically in first-order frequencydomain calculations with geodesic sources (e.g., [53, 63, 65, 66, 75, 76, 134, 143, 171–179]), and we can now immediately reinterpret those computations as leadingorder implementations of the expansion (391). This is a principal advantage of using the variables (ϕα , pϕi ) instead of (qα , pqi ). First-order implementations in the time domain [54, 69, 70, 180–183] do not mesh quite so readily with a multiscale expansion. We discuss their utility within a multiscale expansion in later subsections.

1500

A. Pound and B. Wardell

Importantly, Eqs. (398) and (399) can be solved for any values of the parameters P α , without having to simulate complete inspirals. At each point in the parameter (nmk) space, the solution, comprising the set of amplitudes hμν , can loosely be thought of as a “snapshot” of the spacetime in the frequency domain. These solutions can be used to calculate the driving forces in the evolution equations (337) and (393) for dP α /dt. After populating the space of snapshots, one can then use these evolution equations, together with the phase evolution equation (336), to evolve any particular binary spacetime through the space. Note that even though each snapshot is determined by an “instantaneous” value P α , each snapshot fully accounts for dissipation and for the nongeodesic past history of the binary: because the evolution is slow compared to the orbital timescale, these effects are suppressed by a power ˙ (1mk) of ε and are incorporated through the G source terms in Eq. (399). μν What would go wrong if, rather than using this multiscale expansion, we were  (1) (1mk) to actually use hμν = (r, z)eimφ−iωmk t as our first-order metric mk hμν perturbation? This would be approximating the trajectory of the companion as a geodesic of the background black hole spacetime. As explained in the discussion around Eq. (193), such an approximation would accumulate large errors with time: the “small” corrections to the trajectory would grow large as the object spirals inward. The growing correction, represented by z1α = mα /m in Eq. (193), would (2) manifest itself as a dipole term in h(2) μν that would grow until hμν became larger than h(1) μν , spelling the breakdown of regular perturbation theory. We can now understand this behavior directly from the orbital phases. If we were to use the geodesic phases ωmk t, we would be implicitly expanding the phase 

t

ϕα (t, ε) = 0

Ωα(0) [pϕj (εt  , ε)]dt  + ϕα0

(400)

in powers of ε, as ϕα (t, ε) = Ω˜ α(0) (0)t + ε

! (0) 1 d Ω˜ α 2 (1) ˜ (0)t + Ωα (0)t + O(ε2 ) + ϕα0 , 2 d t˜

(401)

where we have used Eq. (377). Such an expansion would be accurate on √ the orbital timescale but would accumulate large errors on the dephasing time ∼ 1/ ε, which is much shorter than the radiation reaction time. Moreover, the order-ε terms in this expansion would appear as non-oscillatory, linear- and quadratic-in-t terms in (2) (2) hμν , implying that hμν would not admit a discrete Fourier expansion or correctly describe the system’s approximate triperiodicity. The multiscale expansion avoids these errors and maintains uniform accuracy prior to the transition to plunge (and excluding resonances). The basic idea of this multiscale expansion of the field equations was first put forward in Ref. [20]. It is described in detail in Ref. [114] for the special case of quasicircular orbits in Schwarzschild spacetime. Our presentation here, building on our particular treatment of orbital motion in the preceding section, is the most

36 Black Hole Perturbation Theory and Gravitational Self-Force

1501

complete description to date of the generic case. We provide additional details below. A thorough description is in preparation [184].

Multiscale Expansion of Source Terms and Driving Forces We illustrate, and further motivate, the multiscale expansion by examining the multiscale form of the source terms in the coupled equations (204) and (198): the Detweiler stress-energy and the self-force. We start with the stress-energy (205). Writing the trajectory as zα (t) = [t, ro (t), zo (t), φo (t)]

(402)

(where the subscript stands for “object’s orbit”), setting the spin  to zero, using the δ function to evaluate the integral, and expanding the factors of −g˘ and dτ d τ˘ , we express Tμν as Tμν =

 R(1) mg˘ μα g˘ νβ uα uβ  ε γ δ 1+ u u − g γ δ hγ δ t uΣ 2 × δ 2 [x − x o (t)]δ[φ − φo (t)] + O(ε3 ).

(403)

We now take as a given our multiscale expansion (356) of zα (t); this assumed the form (307) for the force, which we return to below. Substituting (356) and using uμ = z˙ μ /Σ, we obtain the coefficients in the expansion: (1) (2) Tμν = εTμν (ϕα , pϕi ) + ε2 Tμν (ϕα , pϕi ) + O(ε3 ).

(404)

The leading term is (0) (0)

(1) Tμν (ϕα , pϕi ) =

m˙zμ z˙ ν

2 ft(0) Σ(0)

δ 2 [x − x (0) (ϕ, pϕi )]δ[φ − φ(0) (ϕα , pϕi )],

(405)

μ

(0) (0) ν 2 := r 2 + a 2 z2 , z˙ (0) := g (x )˙ i where Σ(0) μν (0) z(0) , z˙ (0) = fμ (ψ , pϕ ) for μ = (0) (0) μ (0) t, φ, and x˙ (0) is given by Eq. (299) with pi → pϕi and ψa → ψa . The secondorder term is

(2) (ϕα , pϕi ) Tμν

=

m (0)

2 ft Σ(0)

(0) R(1) β 2˙z(μ hν)β z˙ (0)

 1 (0) (0)  γ δ γδ R(1) + z˙ μ z˙ ν u(0) u(0) − g(0) hγ δ 2



(1)

α × δ 2 [x − x (0) (ϕ, pϕi )]δ[φ − φ(0) (ϕα , pϕi )] + z(ϕ,1)

∂Tμν α , ∂x(0) (406)

1502

A. Pound and B. Wardell μ

γδ

μ

α where u(0) = z˙ (0) /Σ(0) , g(0) := g γ δ (x (0) ), and hγR(1) δ is evaluated at z(0) . The last μ ∂ (0) α i term in Eq. (406) involves the action of z(ϕ,1) ∂x α on z˙ (0) (ψ , pϕ ); this can be (0)

evaluated using x (ϕ,1) ·

∂ ∂ ∂ i = ψ (ϕ,1) · + p(ϕ,1) , i (0) ∂x (0) ∂p ∂ψ ϕ

(407)

i given by Eqs. (352) and (353). with ψ (ϕ,1) and p(ϕ,1) Next, we consider the mode decomposition of the expanded stress-energy. We first define the mode coefficients # 1 (nmm k) (n) iϕmk −im φ 2 := e d ϕdϕφ dφ, (408) Tμν Tμν (2π )4

which assume no relationship between the dependence on φ and ϕφ . Substituting (1) Tμν from Eq. (405), using the azimuthal δ function to evaluate the integral over φ, $  inserting Eq. (271) for φ(0) , and using ei(m−m )ϕφ dϕφ = 2π δmm , we obtain (1mm k) Tμν

δmm = (2π )3

#

(0) (0)

m˙zμ z˙ ν

2 ft(0) Σ(0)

δ 2 (x − x (0) )eiϕk −imΔϕ φ d 2 ϕ. (0)

(409)

This enforces m = m, establishing that the stress-energy only depends on φ and ϕφ in the combination eim(φ−ϕφ ) . We can now do away with the m label and evaluate the integral in Eq. (409) in the form (265) or (266). The result is (1mk) = Tμν

mΥr(0) Υz(0)  (0)

(2π )3 Υt

(0)

(0)

z˙ μ (ψ σ )˙zν (ψ σ ) iϕk (ψ σ )−imΔϕ φ (0) (ψ σ ) σz e σr 2 Σ (x)˙r(0) (ψr )˙z(0) (ψz ) =±

σr σz =±

× [θ (r − rp ) − θ (r − ra )]θ (zmax − |z|),

(410)

where the various quantities have been defined as functions of the field point x = (r, z) and σa = ± refers to a portion of the orbit in which x a is increasing (σa = +) or decreasing (σa = −). ψr± (r) is the value of ψr satisfying Eq. (217) (with pi → pϕi ) on an outgoing (+) or ingoing (−) leg of the radial motion; ψz± (z) is defined analogously from Eq. (218). ϕk (ψ σ ) is given by (0)

σ

ϕk (ψ σ ) = qk (ψ σ ) + Ωk · [δtr (ψrσr ) + δtz (ψz z )],

(411)

with qa (ψa ) by Eq. (241) and δta (ψa ) by Eq. (267). Δϕ φ (0) (ψ σ ) is given by (0) ν (ψ σ ) with x˙ a (ψ) (ψ σ ) = gμν (x)˙z(0) Eq. (272). The Mino-time velocities are z˙ μ (0) given by Eq. (299) [or (206) and (207)] and t˙ and φ˙ by Eqs. (208) and (209). We

36 Black Hole Perturbation Theory and Gravitational Self-Force (0)

1503

(0)

can also use z˙ t = −Eϕ Σ and z˙ φ = Lϕ Σ; recall that we suppress the subscript z on Lz in Pϕi = (Eϕ , Lϕ , Qϕ ). (1)

This calculation demonstrates how Tμν inherits the form (392) from the trajec(1) tory zμ . Given this form of Tμν , the linearized Einstein equation preserves it (in an (1) (1) appropriate class of gauges), justifying our ansatz for hμν . Given that form of hμν , the second-order stress-energy (406) inherits the same form, as do the other sources in Eq. (399) and so, finally, does h(2) μν . All of this relies on the presumed form (307) for the force, from which we derived the form (356) for zμ . Our force on the right-hand side of Eq. (198) is not quite of that form. To derive its form, first note that, assuming Eq. (356), the puncture R field hP μν has a form analogous to Eq. (391), and therefore hμν does as well. If we α write this as hR μν (P , x, ϕφ −φ, ϕ, ε), apply a covariant derivative using (397), and evaluate the result on the trajectory zμ (t), and then the right-hand side of Eq. (198) takes the form f μ (P α , x o , z˙ α , ϕ, ε),

(412)

where we have used ϕφ − φo = −Δϕ φ (0) (ϕ) − εφ (ϕ,1) (ϕ) to eliminate dependence on ϕφ . This differs from Eq. (307) in two ways: it depends explicitly on (ϕ, pϕi ), and it depends on the additional parameters MA . With respect to the first difference, we can use Eqs. (354) and (355) to write the force in the form μ

μ

f μ = εf(1) (ψ, pi , MA ) + ε2 f(2) (ψ, pi , MA ) + O(ε3 ).

(413)

The system of equations (309) and (310) thus becomes dψα = fα(0) (ψ, pj ) + εfα(1) (ψ, pj , MA ) + O(ε2 ), dλ dpi i i = εg(1) (ψ, pj , MA ) + ε2 g(2) (ψ, pj , MA ) + O(ε3 ), dλ dMA (0) (1) = εft (ψ, pi )FA (pi ) + O(ε2 ). dλ

(414) (415) (416)

The analysis of these equations then follows essentially without change as in Secs. 6.2.3–6.2.5. To see why the use of Eqs. (309) and (310) does not lead to vicious μ circularity, note that their subleading terms only affect f(2) , which only enters into the dynamics in Eq. (325). The nongeodesic functions appearing in Eqs. (309) and (310) are therefore determined from lower-order equations prior to requiring μ f(2) . Finally, how does MA influence the orbital dynamics? It enters into the driving i and f (1) . However, it does not enter into g (1) . This follows from the fact forces g(n) α i

1504

A. Pound and B. Wardell μ

μ

that f(1)con depends on MA but f(1)diss does not, as explained in Section “Adiabatic Approximation”. MA therefore contributes to the action-angle dynamics at 1PA μ order via Eq. (325), as well as to the coordinate trajectory correction z(ϕ,1) at 1PA (ϕ,1)

i order through ψα and p(ϕ,1) . This is the only material change to our treatment of the orbital dynamics in Sections “Perturbed Mino Frequencies and Action Angles”, “Perturbed Boyer-Lindquist Frequencies and Action Angles”, and “Multiscale Expansions, Adiabatic Approximation, and Post-Adiabatic Approximations”. Together, the analyses of this section establish the consistency of our multiscale treatments of the field equations and orbital motion. In the following sections, we describe more concretely how to utilize these treatments.

Snapshot Solutions and Evolving Waveforms α Snapshot solutions, consisting of the mode amplitudes hnmk μν (P , x), can be computed using any of the frequency-domain methods reviewed in Section “Black Hole Perturbation Theory”. As an example, in this section, we sketch how this is done at first order using the method of metric reconstruction in the radiation gauge, starting from the Teukolsky equation. This summarizes work from Ref. [179], which provided the first calculation of the full first-order self-force for generic bound orbits in Kerr spacetime. Our summary also appeals to methods and results from Refs. [43, 55, 56, 64, 134]. (1) We first define leading-order Weyl scalars ψ0 and ψ4 related to the hμν of Eq. (391) by Eqs. (54)–(55b) with the replacements ∂t → −iωmk and ∂φ → im. For concreteness, we use ψ0 . In analogy with Eqs. (81) and (82), it can be written as ψ0 =

 ∞   

i −iϕmk . 2 ψmk (pϕ , r)2 Sm (θ, φ; aωmk )e

(417)

=2 m=− k

Note that the radial coefficients depend on pϕi but not on MA ; this is because the linearized ψ0 and ψ4 are insensitive to linear perturbations of the central black hole’s mass or spin [185]. The coefficients 2 ψmk (pϕi , r) satisfy the radial Teukolsky equation (84) with ω → ωmk and s ψmω → s ψmk . The source in that equation is constructed from the stress-energy (405) or its modes (410) using the analog of Eq. (91a),  2 2 Tmk = −32π Σ

zmax

−zmax

(S˜0mk T )(r, z)2 Sm (θ, 0; aωmk )dz,

(418)

(1mk)

where the integral ranges over the support of Tμν , θ is related to z by z = cos θ , and we have suppressed the dependence on pϕi . The source S˜0mk T in the integrand is given by Eq. (59) with Tll → Tll(1mk) (and the same for other tetrad components), ∂t → −iωmk , and ∂φ → im. What may appear to be an extra factor of 2π in Eq. (418) accounts for the factor of 1/(2π ) introduced in the integration over φ in Eq. (408).

36 Black Hole Perturbation Theory and Gravitational Self-Force

1505

The retarded solution to the Teukolsky equation, as given in the variation-ofparameters form (94), is 2 ψmk (r)

up

up

in in = 2 Cmk (r)2 Rmk (r) + 2 Cmk (r)2 Rmk (r), in/up

(419) in/up

where we have defined the homogeneous solutions 2 Rmk (r) := 2 Rmωmk (r). The weighting coefficients are given by Eq. (95), which we restate here as  in 2 Cmk (r) :=

r



up

ra

2 Cmk (r) :=

r

rp

up  2 Rmk (r )   2 Tmk (r )dr , W (r  )Δ in  2 Rmk (r )   2 Tmk (r )dr .  W (r )Δ

(420a) (420b)

In the vacuum regions r > ra and r < rp , outside the support of 2 Tmk , the weighting coefficients become constants, ˆ in = 2C mk ˆ up = 2C mk



ra rp



up  2 Rmk (r )   2 Tmk (r )dr , W (r  )Δ

 ra 2 R in mωmk (r ) rp

W (r  )Δ

2 Tmk (r



)dr  ,

(421a)

(421b)

and in those regions, the solution becomes  2 ψmk (r)

=

ˆ in 2 R in (r) 2C mk mk up up ˆ R C 2 mk 2 mk (r) in/up

for r < rp for r > ra .

(422)

We can evaluate the r and z integrals in 2 Cmk as integrals over ψr and ψz by using appropriate changes of variables for each value of σa in% Eq. (410). r For σr = + and a generic function f (r), the radial integrals are rp f dr  = % ψr+ (r) dr %r %π dr f dψr dψr and r a f dr  = ψr+ (r) f dψ dψr ; for σr = −, they are 0 r %r % % % ψr− (r) dr 2π r dr a   f dψr dψr . The transforrp f dr = − ψr− (r) f dψr dψr and r f dr = − π mations for σz = ± are analogous. We can also write the r and z derivatives in a ∂ S˜0mk T as ∂x∂ a = ∂ψ ∂x a ∂ψa (with no sum over a). For more explicit formulas for the integrands, see Sec. 3B of Ref. [134]. See also, e.g., Refs. [174, 178] for discussion of practical methods of numerically evaluating such integrals. The modes of ψ0 (or ψ4 ) are by themselves sufficient to calculate many quantities, such as gravitational wave fluxes. But for other purposes, such as the calculation of the self-force and the needed input for the second-order field equations, one must compute the entire metric perturbation. Starting from the modes of ψ0 or ψ4 , this can be done using the method of metric reconstruction reviewed

1506

A. Pound and B. Wardell

in Section “Reconstruction of a Metric Perturbation in Radiation Gauge”. In the presence of a source, metric reconstruction typically yields a metric perturbation that has a gauge singularity extending in a “shadow” from the matter source to the black hole horizon or from the matter to infinity [45, 64]. In the case of a point particle, this shadow becomes a string singularity. However, we can more usefully reconstruct the metric perturbation in a “no-string” radiation gauge [43], in which it has no string but does have a jump discontinuity and radial δ function on a sphere of varying radius r = r(0) (t). To construct the no-string solution in practice, we first find a Hertz potential ψ I RG satisfying Eq. (68a) (at fixed pϕi ) in the disjoint vacuum regions r < rp and r > ra , subject to regularity at infinity and the horizon. The appropriate solution in each region is given by Eq. (15) of Ref. [64]. In the libration region rp < r < ra , the radial source 2 Tmk is nonzero, as the Fourier decomposition smears the point particle source over the entire toroidal region {rp < r < ra , |z| < zmax }. The solution (15) of Ref. [64] therefore cannot be used in the libration region. However, it can be analytically extended into that region, using Eq. (422) in place of Eq. (419). Because the time-domain solution is analytic everywhere except on the sphere at r(0) (t), the sum over mk of the analytically continued functions from r < rp yields the correct result for all r ≤ r(0) (t), and the sum over mk of the analytically continued functions from r > ra yields the correct result for all r ≥ r(0) (t) [53, 179]; this is the method of extended homogeneous solutions [65, 186]. (See also Ref. [66] for a generalization of this method to problems with sources that are nowhere vanishing.) As alluded to in Section “Mode-Decomposed Equations in Kerr Spacetime”, this method was originally devised to alleviate another problem that arises in frequency-domain calculations for eccentric orbits: the sum over k modes of the inhomogeneous solution converges slowly within the libration region. In the context of metric reconstruction, the method allows one to avoid the complexities of nonvacuum reconstruction. From the extended modes of the Hertz potential, we can reconstruct modes of (1mk)rec an incomplete metric perturbation, hμν , using Eq. (67) (as ever, with ∂t → −iωmk and ∂φ → im). To complete this perturbation, in the region r > r(0) (t), ∂gμν ∂g we add mass and spin perturbations, Eϕ ∂M and Lϕ ∂Jμν , where the M derivative is taken at fixed J = Ma and the J derivative at fixed M; these account for the mass and spin that the particle contributes to the spacetime [55, 56]. In general we must ∂gμν ∂g also add mass and spin perturbations δM ∂M and δJ ∂Jμν throughout the spacetime (at fixed pϕi ); these account for the slowly evolving corrections to the central black hole’s mass and spin. (These corrections proportional to MA have not been added historically because for any specific snapshot with parameters P α , call them P0α , they can be absorbed with a redefinition M → M +εδM0 and J → J +εδJ0 , setting MA0 = 0. However, in the context of an evolution, which moves through the space of P α values, they must always be included at 1PA order. Even at a single value of P α where MA0 = 0, their time derivatives must be included in Eq. (399). See Ref. [114] for a discussion.) Finally, in the region r < r(0) (t), we must add gauge perturbations that ensure the coordinates t and φ, and therefore the frequencies Ωα(0) ,

36 Black Hole Perturbation Theory and Gravitational Self-Force

1507

have the same meaning in the two regions r < r(0) (t) and r > r(0) (t) [187] (see also [188]). (P α , x) in hand, one can calculate any With the completed modes h(1mk) μν quantity of interest on the orbital timescale with fixed P α . In particular, one can calculate the first-order self-force and its dynamical effects using the mode-sum regularization formula derived in Ref. [43]; the formula in the no-string gauge is given by Eq. (125) in that reference. To date, Ref. [179] is the only work to carry out the entire calculation we have just described for generic bound orbits in Kerr spacetime. However, for orbits in Schwarzschild spacetime and for equatorial orbits in Kerr, snapshot frequencyμ (1) domain calculations of the complete hμν and f(1) are now routine, whether in the Lorenz gauge, Regge-Wheeler-Zerilli gauge, or no-string radiation gauge [53,71,75,76,173,175,177]. Numerical implementations at second order, which are necessary for post-adiabatic accuracy, are still in an early stage but have computed some physical quantities for quasicircular orbits in Schwarzschild spacetime [189]. We can use the output of these snapshot calculations to obtain the true, evolving gravitational waveforms. Once the snapshot mode amplitudes are calculated, from them we can calculate the inputs for the evolution equations (336), (337), and (393). In analogy with Eqs. (96), (139), and (155), the waveforms are then given by any of

h+ − ih× = 2

ˆ up [pϕi (u, ˜ ε)] −2 C 1mk −iϕmk (u,ε) ˜ −2 Sm (θ, φ; aωmk )e 2 ω mk mk 

√   D  ZM,up CPM,up Cˆ 1mk [pϕi (u, ˜ ε)] − i Cˆ 1mk [pϕi (u, ˜ ε)] = 2

+ O(ε2 ), (423a)

mk

=



˜ + O(ε2 ), × −2 Ym (θ, φ)e−iϕmk (u,ε)

(423b)

1mk i ˜ [pϕ (u, ˜ ε)]2 Ym (θ, φ)e−iϕmk (u,ε) + O(ε2 ), Cˆ mm

(423c)

mk

where D = ( − 1)( + 1)( + 2) and ωmk = ωmk [pϕi (u, ˜ ε)]. Here we have written the waveform in terms of u˜ := ε(t − r ∗ ); we return to this point below. We also note that we have given the waveform in terms of modes of ψ4 rather than the less natural (for this purpose) ψ0 . In analogy with Eq. (421), ZM,up CPM,up we have defined the amplitudes Cˆ 1mk and Cˆ 1mk as the relevant weighting coefficients for r > rp , and we have defined the Lorenz gauge amplitudes 1mk := lim iω r ∗ 1mk Cˆ mm r→∞ (re mk hmm ). We have also intentionally inserted a label “1” onto the mode amplitudes and omitted O(ε2 ) amplitudes. This is because even if we determine the phase ϕmk (u, ˜ ε) through 1PA order, the second-order amplitudes do not increase the waveform’s order of accuracy; an order-ε2 amplitude in the waveform is indistinguishable from a 2PA (order-ε) correction to the phase.

1508

A. Pound and B. Wardell

The waveform (423) is in the time domain, but it is% almost trivially related to ∞ 1 iωu du and the frequency-domain waveform. Defining h(ω) := 2π −∞ (h+ − ih× )e applying the stationary-phase approximation, we obtain, e.g., " 1  2π ε h(ω) = Cˆ 1mk [t˜mk (ω)]2 Ym 2π |dωmk /d t˜| mm mk     iπ √ + o( ε). × exp i[ω t˜mk (ω) − ϕmk (ω)] + sgn dωmk /d t˜ 4 (424) Here t˜mk (ω) is the solution to ω = ωmk (t˜), and the phase as a function of ω is ϕmk (ω) = ϕmk [t˜mk (ω)]. Before proceeding, we return to the dependence on u˜ rather than t˜. In Eq. (423), all functions of u˜ are the functions obtained by solving (336), (337), and (393), simply evaluated as a function of u. ˜ For example, from Eq. (400),  1 u˜ (0) j  ϕα (u, ˜ ε) = Ωα [pϕ (t˜ , ε)]d t˜ + ϕα0 . (425) ε 0 This dependence on u is not a trivial consequence of the multiscale expansion (391). To justify it, one must adopt a hyperboloidal choice of time that asymptotes to u at I + or perform a matched-expansion calculation, matching the solution (391) to an outgoing wave solution near I + . Reference [114] discusses these points along with several additional advantages of using a hyperboloidal slicing. To see why the replacement t → u is intuitively sensible, note that with it, Eq. (423) correctly reduces to a snapshot waveform on the orbital timescale if we fix pϕi and replace (0)

(0)

ϕmk (u, ˜ ε) with its geodesic approximation Ωmk u (with fixed Ωmk ); without the replacement, the multiscale waveform would not correctly reduce in this way. In the next two subsections, we outline the steps required to generate multiscale waveforms at adiabatic (0PA) and 1PA order, whether in the time or frequency domain.

Adiabatic Approximation At this stage we consider the evolution equations in the form (372), (373), (374), and (375). There is no difference between that form and Eqs. (336)–(337) at adiabatic order, but we adopt the notation of Eqs. (372), (373), (374), and (375) here for consistency with our discussion of the 1PA approximation in the next subsection. For convenience, we transcribe the adiabatic evolution equations (372) and (373): (0)

d ϕ˜α j = Ωα(0) (p˜ ϕ(0) ), d t˜ i d p˜ ϕ(0)

d t˜

j

i (p˜ ϕ(0) ). = Γ(1)

(426) (427)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1509

An adiabatic waveform-generation scheme consists of the following steps: 1. Solve the field equation (398) or the associated Teukolsky equation or ReggeWheeler-Zerilli equations, on a grid of pϕi values. At each grid point in parameter i (p i ) in Eq. (373) and space, compute and store two things: the driving forces Γ(1) ϕ up the asymptotic mode amplitudes at I + [e.g., ±2 Cˆ 1mk (pϕi ) in the Teukolsky case]. i , evolve through the parameter space by solving 2. Using the stored values of Γ(1) i the coupled equations (426) and (427) to obtain the adiabatic parameters p˜ ϕ(0) (0) (0) (0) and phases ϕ˜r , ϕ˜z , and ϕ˜φ as functions of t˜ = εt. 3. Construct the adiabatic waveform using, e.g.,

h+ − ih× = 2



ˆ up [p˜ i (u)] −2 C 1mk ϕ(0) ˜

mk

2 ωmk

−2 Sm (θ, φ; aωmk )e

(0)

−iϕmk (u)/ε ˜

,

(428) i (u)]. ˜ where ωmk = ωmk [p˜ ϕ(0)

Starting from seminal work in Refs. [132, 190], two groups of authors have developed practical implementations of this scheme [63, 157, 160, 172, 191–194]. One of the convenient aspects of the adiabatic approximation is that it can be implemented entirely in terms of the Teukolsky equation with a point-particle source, with no requirement to calculate a reconstructed and completed metric or to R(n) extract the regular fields hμν . The reason is that, as explained around Eqs. (378) μ and (379), only the first-order dissipative force f(1)diss is needed to calculate the i . This force is entirely due to the half-retarded minus halfdriving force Γ(1) (1)

advanced piece of hμν [132]: (1)rad hμν =

1 (1)ret 1 (1)adv h − hμν . 2 μν 2

(429)

(1)rad Because hμν is a vacuum solution to the linearized Einstein equation, it can be reconstructed from the half-retarded minus half-advanced piece of ψ0 or ψ4 , using the radiation-gauge reconstruction method reviewed in Section “Reconstruction of a Metric Perturbation in Radiation Gauge” (as translated to the multiscale expansion in the previous section). Again because it is a vacuum solution, it is smooth at μ R(1) the particle, and it is equal there to the relevant part of hμν that creates f(1)diss . (1)rad

can contain no stationary perturbations, implying it cannot Furthermore, hμν contain any contribution from the mass and spin perturbations MA , so it needs no completion. Hence, one can evolve the orbit and generate the waveform entirely from mode amplitudes of ψ0 or ψ4 . Concrete formulas for adiabatic driving forces in terms of Teukolsky amplitudes were first derived in Ref. [190], which showed that the average rates of change of

1510

A. Pound and B. Wardell

α E and Lz due to f(1)diss satisfy a balance law:

d E˜ ϕ(0) = −FEH − FEI , dt

(430)

d L˜ ϕ = −FLHz − FLIz , dt

(431)

(0)

(0) (0) (0) i i where P˜ϕ(0) = (E˜ ϕ , L˜ ϕ , Q˜ ϕ ) are related to p˜ ϕ(0) by the geodesic relationships (225), (226), and (227) between P i and pi . The fluxes are those due to the retarded field, which we can translate from Eqs. (97), (98), (99), and (100) as

FEH =

 2π αmω  mk in |−2 Cˆ 1mk |2 := FEH mk , 2 ω mk mk mk

(432)

FEI =

 2π  ˆ up |2 := | FEI mk , C −2 1mk 2 ω mk mk mk

(433)

and similarly for FLHz and FLIz . Equation (430) states that the change in the particle’s orbital energy is equal at leading order to the sum total of energy carried out of the system (into the black hole and out to infinity). Equation (431) states the analog about the particle’s angular momentum. Some time later, Ref. [192] derived a similar formula for the average rate of α change of the Carter constant due to f(1)diss :     ˜ (0) dQ dQ H dQ I ϕ =− − , dt dt dt

(434)

where 

dQ dt

 =2

 Lmk + kz Υ˜z(0) FE mk ωmk

(435)

mk

with 2 2 (0) Lmk = mcot2 θ(0) λ L(0) ϕ − a ωmk cos θ(0) λ Eϕ .

(436)

Note that Ref. [192]  uses  C to denote our Q and Q to denote our K. We give here the expression for dQ as presented in Ref. [143]. In all cases in the literature, dt expressions such as these are written in terms of averages ·, which we can omit because we work with already averaged orbital variables.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1511

While the evolution equation (434) for Q superficially resembles those for E and Lz , it is of fundamentally different character. The quantities FE and FLz are true physical fluxes across the horizon and out to infinity; they are defined   entirely 

in terms of the metric on the surfaces H + and I + . The quantities dQ , on dt the other hand, directly involve orbital parameters; they are not locally measurable fluxes. Thus, although Eq. (434) is sometimes referred to as a flux-balance law, there is no known sense in which it can be meaningfully described as such. However, the evolution equations for E, Lz , and Q all share the same practical advantage: they can be evaluated directly from the retarded solution to the Teukolsky equation with a point-particle source, with no need to reconstruct the complete metric perturbation or to extract the regular field. Combining Eqs. (430), (431), and (434), we can compute the adiabatic driving forces i d P˜ϕ(0) ∂ p˜ ϕ(0) j

i i Γ(1) (p˜ ϕ(0) )=

j ∂ P˜ϕ(0)

dt

(437)

in/up from the Teukolsky amplitudes −2 Cˆ 1mk given by Eq. (421). We can then follow the prescription outlined at the beginning of the section. Alternatively, we can i ˜ (0) express the geodesic frequencies in terms of P˜ϕ(0) = (E˜ ϕ(0) , L˜ (0) ϕ ,Q ϕ ) and work i i ˜ directly with those variables, treating p˜ ϕ(0) as a function of Pϕ(0) by inverting the relationships (225), (226), and (227). The adiabatic approximation has been used to evolve equatorial orbits in Kerr spacetime [195] and to generate waveforms in Schwarzschild spacetime [154]. Yet, despite the approximation’s efficient formulation, to date no adiabatic waveforms have been generated for orbits in Kerr spacetime, nor have orbital evolutions been performed for generic (eccentric and inclined) orbits. There are two main obstacles. One is generating sufficiently dense data on the pϕi space to perform accurate interpolation or fitting. The second is the very large (∼104 ) number of mode amplitudes that are required to achieve an accurate waveform. Both obstacles are expected to be soon overcome [154, 195], but as of this writing, the gold standard for generic orbits remains snapshot waveforms [193] that use geodesic phases.

First Post-Adiabatic Approximation The 1PA evolution equations (372), (373), (374), and (375), as extended following the discussion around Eqs. (414), (415), and (416), are (0)

d ϕ˜α j = Ωα(0) (p˜ ϕ(0) ), d t˜ i d p˜ ϕ(0)

d t˜

j

i (p˜ ϕ(0) ), = Γ(1)

(438) (439)

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A. Pound and B. Wardell (1)

d ϕ˜α j j = p˜ ϕ(1) ∂j Ωα(0) (p˜ ϕ(0) ), d t˜ i d p˜ ϕ(1)

d t˜

(440)

i i (p˜ ϕ(0) , M˜ A ) + p˜ ϕ(1) ∂j Γ(1) (p˜ ϕ(0) ), = Γ(2) j

(1)

j

j

(1) d M˜ A (1) j = FA (p˜ ϕ(0) ). d t˜

(441) (442)

(1) Here we have assumed MA = M˜ A (t˜) + O(ε). Because (i) MA only contributes stationary modes to h(1mk) , (ii) any source term for h(2mk) that is quadratic in μν μν these modes will also be stationary, and (iii) a stationary mode of h(2mk) will not μν i , it follows that Γ i is linear in M , implying we can write it in contribute to Γ(2) A (2) the form i i Γ(2) (p˜ ϕ(0) , M˜ A ) = Γ(2) (p˜ ϕ(0) , 0) + M˜ A γAi (p˜ ϕ(0) ), j

j

(1)

(1)

j

(443)

j (1) where A is summed over. γAi (p˜ ϕ(0) ) here is defined as the coefficient of M˜ A in i (p˜ j , M ˜ (1) ). Γ(2) A ϕ(0) A 1PA waveform-generation scheme then consists of the following steps:

1. Solve the field equations (398) and (399) on a grid of pϕi values. At each i (p i ), grid point, compute and store the following: (i) the driving forces Γ(1) ϕ i i i i Γ(2) (pϕ , 0), and γA (pϕ ), (ii) the asymptotic first-order mode amplitudes at I + up [e.g., −2 Cˆ 1mk (pϕi )]. 2. Using the stored values of the driving forces, evolve through the parameter space by solving the coupled equations (438), (439), (440), (441), and (442) to obtain i p˜ ϕ(0) and the phases ϕ˜α(0) and ϕ˜α(1) as functions of t˜ = εt. 3. Construct the 1PA waveform: h+ −ih× =2



ˆ up [p˜ i (u)] −2 C 1mk ϕ(0) ˜

mk

2 ωmk

−2 Sm (θ, φ; aωmk )e

 (0) (1) −i ϕmk (u)+εϕ ˜ ˜ /ε mk (u)

,

(444) i (u)]. ˜ where ωmk = ωmk [p˜ ϕ(0)

We make two potentially clarifying remarks about these steps. First, even though the 1PA dynamics depend on the black hole parameters MA , we need not include these parameters in our storage grid. This is because the 1PA effect of MA is linear in MA , allowing us to only store its coefficient. However, note that the background spin parameter a must be included in the grid (the background parameter M need not be, as we can measure all lengths in units of M). Our second remark is that though i pϕi = p˜ ϕ(0) at a given value of t˜ and ε, we can still freely solve (398) and (399),

36 Black Hole Perturbation Theory and Gravitational Self-Force

1513

working with pϕi , in order to determine the driving forces as functions; it is precisely i those functions, simply with pϕi → p˜ ϕ(0) , that appear in Eqs. (438), (439), (440), (441), and (442). A scheme of this sort was first sketched in Ref. [189] and detailed in Ref. [114] for the special case of quasicircular orbits into a Schwarzschild black hole. Figure 3 of Ref. [114] gives a more thorough breakdown, though the structure of the multiscale expansion differs slightly from our formulation here. The general case for generic bound orbits in Kerr appears here for the first time. At its core, the scheme requires three key ingredients for each set of orbital parameter values: the full first-order self-force, the asymptotic mode amplitudes of the first-order waveform, and the second-order dissipative self-force. As we summarized in Section “Multiscale Expansion of Source Terms and Driving Forces”, the first two ingredients have been calculated for generic bound orbits in Kerr spacetime [179] and are routinely calculated for less generic configurations. The main obstacle to including these ingredients in an evolution scheme is the computational cost and runtime of sufficiently covering the parameter space. The third ingredient has not yet been calculated in even the simplest configurations, though there is ongoing development of a practical implementation [114, 159, 196], which led to the recent calculation of a second-order conservative effect [189].

Mode Decompositions of the Singular Field In our description so far, we have largely glossed over what is the pivotal step in almost all self-force calculations beyond the adiabatic approximation: the R(n) calculation of hμν and its derivatives, which are required for the conservative first-order self-force, and the second-order self-force, as inputs for the second-order sources (whether the Detweiler stress-energy, the effective source, or the secondorder Einsten tensor [196]) and as the essential ingredient in most dynamical quantities of interest. R(n) In order to compute hμν (either using a puncture or the point-particle method with regularization) in a mode-decomposed calculation, a crucial component is a mode-decomposed form for the puncture field. This can be obtained by expanding the puncture field into the same basis as is used in the calculation of the retarded field and can typically be done analytically, or at least semi-analytically. The specific details depend on the context (e.g., choice of gauge, whether the mode decomposition needs to be exact or if it can be an approximation, whether the harmonics are spheroidal or spherical and scalar, vector, tensor, or spin-weighted). However, the essential ingredients in the method are common to all cases: 1. Introduce a rotated angular coordinate system (θ  , φ  ) such that the worldline is instantaneously at the pole, θ  = 0. This makes the mode decomposition integrals analytically tractable and in some instances reduces the number of spherical harmonic m modes that need to be considered.

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2. Expand the relevant quantity in a coordinate series about the worldline. In doing so, it is important to ensure that the series approximation is well behaved away from the worldline, in particular at θ  = π . This can be achieved by multiplying by an appropriate window function in the θ  direction [71]. The resulting expansion can always be algebraically manipulated into the form of a power series (including log terms in some cases) in  1/2 . ρ := k1 χ 1/2 δ 2 + 1 − cos θ 

(445)

2

2  2 Here, δ 2 = k2 Δr χ , Δr := r − r0 and χ := 1 − k3 sin φ , and k1 , k2 and k3 depend on the orbital parameters and can be treated as constants in the mode decomposition. In some instances (e.g., eccentric orbits) obtaining an expression for ρ in this form requires the definition of the rotated coordinates to include a dependence not just on the unrotated angular coordinates, (θ, φ), but also on the other coordinates (e.g., Δr for the eccentric case). In all cases, the coefficients in the power series contain powers of Δr and χ and also depend on the orbital parameters. Apart from that, the dependence on φ  is only via one of four possibilities: a. independent of φ  , b. cos φ  sin φ  , c. cos φ  , d. sin φ  . The   resulting dependence on φ  will then combine in the next step with the e−im φ  factor from the harmonic to produce a dependence on φ only via powers of χ . When decomposing tensors, certain tensor components may also include an overall factor of sin θ  , but only ever in such a way that it cancels a singularity in the harmonic at θ  = 0 so that the final integrand is nonsingular away from Δr = 0. 3. Integrate against (the conjugate of) the relevant harmonic to obtain a mode decomposition in (, m ) modes with respect to the rotated coordinate system. In the case of spin-weighted or vector and tensor harmonics, we must also be careful to account for the rotation, R, of the frame, either by including the    appropriate factor of eisγ (θ ,φ ,R ) in the spin-weighted case [197] or by including the appropriate tensor transformation in the case of vector and tensor harmonics [71]. In performing the integrals, we can exploit the fact that only certain integrals over φ  are nonvanishing. In particular for the four possibilities listed in the previous step:   a. only contributes for m even and only for the real part of eim φ .    b. only contributes for m even and only for the imaginary part of eim φ .  φ  im . c. only contributes for m odd and only for the real part of e   d. only contributes for m odd and only for the imaginary part of eim φ .  The integrals over θ can all be done analytically and result in expressions of the form − n+4 2

δ

n+2

(δ + 2) 2

(n+2)/2

 i=0

n+2

ai δ + log 2i

 δ 2 + 2  + 2 δ2

bi δ 2i

n even

i=0

(446)

36 Black Hole Perturbation Theory and Gravitational Self-Force

(δ 2 + 2)(n+2)/2

  i=0

ci δ 2i + |δ|δ n+1

 

di δ 2i

1515

n odd

i=m

(447) where n is the power of ρ and where the coefficients ai , bi , ci , and di are dependent rational numbers. The specific limits on the sums given here is for the m = 0 scalar harmonic case. Structurally similar expressions also appear for log ρ terms, for m = 0, and for spin-weighted harmonics, but with the sums running over different ranges of i. The integrals over φ  can also be done analytically and result in power series (for integer powers of χ ), elliptic integrals (for half-integer powers), or the derivative of a hypergeometric function with respect to its argument (for log terms). In all three cases, they are functions of k3 and potentially also Δr. 4. Transform back to the (, m) modes with respect to the unrotated (θ, φ)  coordinate system using the Wigner D-matrix, Dmm  (R). With a moving worldline, the rotation is time-dependent, but this complication is not relevant in many cases, the notable exception being in the effective source method where it is necessary to take time derivatives when computing the source from the puncture [196, 198]. In many practical applications, an exact mode decomposition is not necessary, and an approximation is sufficient. For example, in the mode-sum regularization scheme, one is only interested in the modes of the puncture (or its radial derivative in the case of the self-force) evaluated in the limit Δr → 0. Similarly, in the effective source scheme, a series expansion to some power in Δr suffices. Then, the exact expression for the mode-decomposed puncture field has a series expansion in Δr of the form  c1,i Δr i + c2,j Δr j |Δr| + c3,k Δkr log Δr (448) m ,ij k

where the coefficients c1,i , c2,j , and c3,k depend on the orbital parameters. In those cases, the mode decomposition procedure simplifies significantly and one need only compute up to some maximum value for i, j , and k. Similarly, another simplification arises from the fact that one may only be interested in the decomposition of the puncture accurate to some order in distance from the worldline in the angular directions. This is reflected in the number of m modes that must be included: for a puncture accurate to n derivatives, one must include up to |m | = |s| ± n for the spin-weighted case (the vector and tensor cases similarly follow from their relation to the spin-weighed harmonics: |s| = 1 for the vector case and |s| = 2 for the tensor case). One particularly important special case is that of mode-sum regularization, where one is only interested in the result for a given quantity summed over m and with Δr = 0. This leads to the so-called mode-sum regularization formulas. For example, in the case of the first-order gravitational self-force, this is given by

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A. Pound and B. Wardell

Fα =

 α (Fret − Aα± (2 + 1) − B α ) + D α

(449)



where Aα± , B α , and D α are “regularization parameters” that depend on the orbital parameters. Here, the value of Aα± depends on whether the limit Δr → 0 is taken from above or below; this is because it comes from the derivative of the |Δr| piece of the puncture. The parameter B α does not have this property as it comes from the piece of the puncture that does not involve |Δr| (in particular, for the self-force, it comes from the derivative of the Δr 1 piece of the puncture). The parameter D α accounts for the possibility that the subtraction does not exactly capture the behavior of the contribution from the singular field (and only the singular field) and can often be set to 0 by appropriately defining the subtraction [43, 71].

Example: Leading Order Puncture for Circular Orbits in Schwarzschild Spacetime As a simple representative example, consider the problem of decomposing the leading-order piece of the Lorenz gauge puncture (i.e., the first term in Eq. (183)) into the spin-weighted spherical harmonic basis introduced in Section “Metric Perturbations of Schwarzschild Spacetime”. For concreteness, we consider a circular

geodesic orbit of radius ro with four-velocity uα = ut [1, 0, 0, Ω], where Ω = M ro3

ro t and u = ro −3M . As a first step, we expand the covariant expression in a coordinate series. Keeping  only the leading term in the coordinate expansion, we have gμα = δμα + O(Δx) and s = ρ + O(Δx 2 ) where ρ 2 := B 2 (δ 2 + 1 − cos θ  ), δ 2 := 2ro2 (ro −2M)χ (ro −3M) ,

ro Δr 2 , B 2 (ro −2M)

χ :=

M sin2 φ  , B 2 := and Δr = r − ro . Here, we have made the 1 − ro −2M standard choice of identifying a point on the worldline with the point where the puncture is evaluated by setting Δt = t − to = 0. Then, working with the Carter tetrad, the tetrad components of the puncture are:

hll = hnn =

2 ro − 2M , ρ ro − 3M

M 1 2M hmm¯ = , ro − 2M ρ ro − 3M   cos2 θ2 2i(ro − 2M)ro Ω , = −hl m¯ = −hnm¯ = − √ ρ fo (ro − 3M)   cos4 θ2 2M . hmm = hm¯ m¯ = − ρ ro − 3M

hln = hlm = hnm

(450)

36 Black Hole Perturbation Theory and Gravitational Self-Force

1517

    Note that we have included factors of cos2 θ2 and cos4 θ2 to ensure that the puncture is sufficiently regular at θ  = π while not altering its leading-order behavior near the worldline at θ  = 0. We now integrate these against the appropriate spin-weighted spherical harmonic to obtain mode-decomposed versions. In doing so, we must take account of the fact that our integration is with respect to a rotated coordinate system by including a     factor of eisγ (θ ,φ ,R ) ≈ i s eisφ + O(θ 2 ). Since we are only interested in the leading-order behavior near the worldline, we will only consider the modes |m | = |s| series expanded through O(Δr 1 ). Then, we encounter the following integrals over θ  : 

π

0





√ 1 1 ¯ 1    2 − (θ , 0) sin θ dθ ≈ 2(2 + 1)|δ| , Y √ 0 0 ρ B 2π(2 + 1)

π

0



π

0



π

0

 0

π

(451a)

     π cos2 θ2 cos2 θ2       ¯ 1 Y,−1 (θ , 0) sin θ dθ = −1 Y¯1 (θ , 0) sin θ dθ ρ ρ 0

√ 1 1 8Λ1 − 2(2 + 1)|δ| , ≈− √ (451b) B 2π(2 + 1)      π cos2 θ2 cos2 θ2       1 Y¯1 (θ , 0) sin θ dθ = −1 Y¯,−1 (θ , 0) sin θ dθ ρ ρ 0

1 1 12 , (451c) 8Λ1 − ≈− √ B 2π(2 + 1) (2 − 1)(2 + 3)      π cos4 θ2 cos4 θ2       2 Y¯,−2 (θ , 0) sin θ dθ = −2 Y¯2 (θ , 0) sin θ dθ ρ ρ 0

√ 1 1 32Λ2 − 2(2 + 1)|δ| , ≈ √ (451d) B 2π(2 + 1)      π cos4 θ2 cos4 θ2       ¯ 2 Y2 (θ , 0) sin θ dθ = −2 Y¯,−2 (θ , 0) sin θ dθ ρ ρ 0

1 1 80 , (451e) 32Λ2 − ≈ √ B 2π(2 + 1) (2 − 1)(2 + 3)

(+1) (−1)(+1)(+2) where Λ1 := (2−1)(2+3) and Λ2 := (2−3)(2−1)(2+3)(2+5) . Next, performing the  integrals over φ , the integrands all involve integer (for the |δ| terms) and half-integer

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A. Pound and B. Wardell

(for the δ 0 terms) powers of χ , producing elliptic integrals or polynomial functions M of ro −2M , respectively. Putting everything together, transforming to the frequency domain (which in this case amounts to simply dividing by 2π ) and transforming back to the unrotated frame using the Wigner D-matrix, we then obtain expressions for the mode-decomposed punctures: hmω ll

"

 (R) Dm,0 (2 + 1) 4K ro − 2M − 3/2 √ =√ |Δr| , (2 + 1)π π ro ro − 3M ro ro − 3M

(452a)

 (R) Dm,0 4MK (2 + 1)M − hmω = |Δr| , √ √ √ √ ln (2 + 1)π π ro ro − 2M ro − 3M ro3/2 ro − 3M(ro − 2M) (452b) hmω lm

  Dm,−1 (R) (2 + 1)Ω ro 16Λ1 ΩK +√ |Δr| =√ − √ π ro − 3M (2 + 1)π ro − 3M ro − 2M

 (R) Dm,1 4[(2ro − 5M)K − 2(ro − 2M)E ] 6 +√ 4Λ1 − (2−1)(2+3) , √ (2 + 1)π M 1/2 ro π ro − 3M (452c)

 Dm,−2 (R) hmω mm = √



64MΛ2 K (2 + 1)M − |Δr| √ √ √ (2 + 1)π π ro ro − 2M ro − 3M ro3/2 ro − 3M(ro − 2M)

 (R) Dm,2 40 × 16Λ2 − (2−1)(2+3) +√ (2 + 1)π



4[(4ro − 9M)(4ro − 11M)K − 8(ro − 2M)(2ro − 5M)E ] , √ √ 3Mπ ro ro − 3M ro − 2M (452d) mω = with the other components given either by hmω = hmω nn , hmm ¯ ll mω hmω nm = hlm , or by complex conjugation. Here:

−1/2 M 2  1− sin φ dφ  , ro − 2M 0  1/2  M 1 2π 1− sin2 φ  E := dφ  , 4 0 ro − 2M

1 K := 4



2π

ro −2M mω M hln ,



(453a) (453b)

are complete elliptic integrals of the first and second kind, respectively. Higher-order circular-orbit punctures including the contribution at O(λ0 ) are available in Ref. [71]. Yet higher orders and punctures for more generic cases are available upon request to the authors.

36 Black Hole Perturbation Theory and Gravitational Self-Force

1519

Conclusion We stated in the introduction to this review that we aimed to summarize the key methods of black hole perturbation theory and self-force theory rather than summarizing the status of the field, leaving that task to existing reviews. However, it is worth putting this review in the context of the field’s current state, and it is worth mentioning key topics that we did not cover. Regarding topics we neglected, we first state the obvious: we did not cover any applications of black hole perturbation theory other than small-mass-ratio binaries. Although the bulk of the review is intended to provide general treatments of black hole perturbation theory, orbital dynamics in black hole spacetimes, and self-force theory in generic spacetimes, without specializing to binaries, it is undoubtedly slanted toward our application of interest. For that reason, we will also focus exclusively on the state of small-massratio binary modeling. It is well established that 1PA waveforms are almost certainly required to perform high-precision measurements of these binaries. Such measurements will require phase errors much smaller than 1√radian, while the errors in 0PA waveforms will have errors of O(ε0 ) (or O(1/ ε) in the case of a resonance), which could be 1 or many more radians. Reference [23] has recently provided strong numerical evidence that a 0PA waveform will have significant errors for all mass ratios. Conversely, the same reference shows that a 1PA waveform should be not only highly accurate for EMRIs and IMRIs but reasonably accurate even for comparable-mass binaries. This bolsters a long line of evidence that perturbative self-force theory is surprisingly accurate well outside its expected domain of validity; see Ref. [199] for other recent evidence, as well as the reviews [27, 200]. There are two main hurdles to overcome on the way to generating 1PA waveforms. One is the difficulty of efficiently covering the parameter space. Once a region is well covered by snapshots, recent advances make it possible to generate long, accurate waveforms extremely rapidly in that region, with generation times of a few tens of milliseconds for eccentric orbits in Schwarzschild spacetime [154]. However, covering the parameter space of generic orbits in Kerr is highly expensive even for adiabatic codes, let alone calculations of the first-order self-force. The second main hurdle is calculating the necessary second-order inputs for the 1PA evolution. There has been steady progress in developing practical methods of computing these inputs, but only recently have results begun to materialize [189]. To date, these calculations have been restricted to quasicircular orbits in Schwarzschild spacetime; they must be extended to Kerr and to generic orbits. In lieu of accurate evolving waveforms, the development of data analysis methods has so far been based on “kludge” waveforms constructed using a host of additional approximations (primarily, post-Newtonian approximations for the fluxes) [201–206]. These kludges will be very far from accurate enough to enable precise parameter estimation, but they are sufficiently similar to accurate waveforms to serve as testbeds for analysis methods. They may also be sufficiently accurate for detection of loud signals. There is also ongoing work to improve the accuracy of

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A. Pound and B. Wardell

post-Newtonian 0PA approximations to enable them to accurately fill out the weakfield region of the small-mass-ratio parameter space [207]. Our summary of multiscale evolution has also omitted some important ingredients in an accurate model. We must correctly account for passages through resonance, and we may need to include the transition to plunge for mass ratios ∼1 : 50. We must also account for the secondary’s spin, which enters into the 1PA dynamics in three ways: (i) through the Mathisson-Papapetrou spin force (195), μ (2) which contributes to f(1)con ; (ii) through the spin’s contribution to Tμν in Eq. (205), μ which generates a perturbation that contributes to f(2)diss ; and (iii) through a couR(1) and the spin, which again contributes a second-order dissipative pling between hμν effect. We refer to Refs. [119, 148, 153, 169] for a sample of recent work on calculating these effects and incorporating them into waveform-generation schemes. Specifically, Ref. [148] generated waveforms from inspirals into a Schwarzschild black hole including first-order conservative (but not second-order dissipative) spin effects; Ref. [119] derived balance laws incorporating spin; Ref. [153] derived the spin correction to the fundamental frequencies; and Ref. [169] computed the spin’s contribution to fluxes from spinning particles on generic orbits in Schwarzschild spacetime. We also note that while we have focused on a multiscale expansion built on frequency-domain methods, there has been considerable development of timeμ (1) domain snapshot calculations of hμν and f(1) using fixed geodesic sources [54, (1)

μ

69,70,181,182]. The quantities hμν and f(1) output from such computations cannot be directly fed into the second-order field equations (399) or into the multiscale evolution scheme. However, if we decompose the outputs into Fourier modes, as in  (1) (1mk) (1mk) hμν = mk hμν eimφ−ωmk t , then the coefficients hμν are identical to those in a multiscale expansion, and these can be used as inputs for the multiscale scheme. Moreover, any first-order quantity that depends only on P α will be identical in the time domain with a geodesic source as in the multiscale expansion; this includes any quantity constructed as an average over the orbit, which includes most physical quantities of interest [27]. Because time-domain methods are typically more efficient than frequency-domain ones for highly eccentric orbits, certain dynamical quantities entering into the evolution may be more usefully computed in the time domain. Time-domain calculations also offer an alternative framework for waveform generation: rather than using Eq. (423), one can perform a multiscale evolution of P α to generate a self-accelerated trajectory and then solve the Teukolsky equation in the time domain with an accelerated point-particle source [180, 183]. This may seem redundant, given that in the process of generating the multiscale evolution one must already compute all the inputs for Eq. (423). However, it offers significant flexibility, in that it can take as input trajectories generated with any method, such as inspirals which have been produced that include the full first-order self-force but omit second-order dissipative effects [149–151]. This gives it the additional advantage of being able to easily evolve through different dynamical regimes, such as the evolution from the adiabatic inspiral to the transition to plunge [199, 208].

36 Black Hole Perturbation Theory and Gravitational Self-Force

1521

Beyond these alternative methods of wave generation, we have also passed over what has been the main application of self-force calculations. Although such calculations were originally motivated by modeling EMRI waveforms (and more recently, the prospect of using them to model IMRIs), they have also enabled the calculation of numerous physical effects in binaries. These, in turn, have facilitated a rich interaction with other binary models: post-Newtonian and post-Minkowskian theory, effective one body theory, and fully nonlinear numerical relativity [200]. Sections 7 and 8 of Ref. [27] provide a summary of the physical effects that have been computed and the synergies with other models. We highlight Refs. [209–211] for more recent discussions of the power of such synergies and of the potential future impact of self-force calculations.

Cross-References  Introduction to Gravitational Wave Astronomy  Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves  Post-Newtonian Templates for Gravitational Waves from Compact Binary Inspi-

rals  Space-Based Gravitational Wave Observatories  The Gravitational Capture of Compact Objects by Massive Black Holes Acknowledgments The acknowledgement of a Royal Society University Research Fellowship was a funding acknowledgement. Please reinstate it.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Siren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Redshift Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Astrophysical Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strong Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Effect of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep Gravitational Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peculiar Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Effect of Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Many objects discovered by LIGO and Virgo are peculiar because they fall in a mass range which in the past was considered unpopulated by compact objects. Given the significance of the astrophysical implications, it is important to first understand how their masses are measured from gravitational-wave signals. How accurate is the measurement? Are there elements missing in our current model which may result in a bias? This chapter is dedicated to these questions. In particular, we will highlight several astrophysical factors which are not included in the standard model of GW sources but could result in a significant bias in the estimation of the mass. These factors include strong gravitational lensing, a relative motion of the source, a nearby massive object, and a gaseous background.

X. Chen () Astronomy Department, Peking University, Beijing, P. R. China Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing, P. R. China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_39

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Keywords

Gravitational waves · Chirp mass · Redshift · Acceleration · Hydrodynamical friction

Introduction The detection of gravitational waves (GWs) by LIGO and Virgo has reshaped our understanding of stellar-mass black holes (BHs). Before the detection, stellar-mass BHs were exclusively found in X-ray binaries. Their masses fall in a range of about 5–20 M . Below or above this range, no (stellar-mass) BH was detected. There were also speculations based on theoretical grounds that stellar evolution can hardly produce a BH between 3 and ∼5 M (known as the “lower mass gap”) or between 50 and ∼120 M (the “upper mass gap”). Nevertheless, the very first GW source detected by LIGO, GW150914, had set a new record for the mass of BHs: GW150914 is formed by two BHs each of which is about 30 M [1]. Later, during the first and second observing runs, LIGO and Virgo had detected ten mergers of binary black holes (BBHs). The majority of them contained BHs heavier than 20 M prior to the coalescence [2]. More recently, LIGO and Virgo have detected a merger product of 3.4 M , clearly inside the lower mass gap [3], as well as a merger of two BHs both above 60 M , well inside the upper mass gap [4]. The masses of these newly found BHs deserve an explanation. While many astrophysicists believe that our model of BH formation and evolution needs a modification, a few are pondering over a rather different question. Is it possible that somehow, we have measured the masses inappropriately? This is a valid question because mass is not a direct observable in GW astronomy. The observable is a wave signal, which can be characterized by a frequency f and an amplitude h; both could evolve with time. To derive from these observables a mass for the source, a model is needed. Needless to say, different models could yield different masses. What is the current standard model? Take BBHs for example. The running assumption is that the orbital dynamics is determined only by GW radiation and the detected waveform is different from the emitted one only by a cosmological redshift. More specifically, this assumption implies that the BBH is isolated from other astrophysical objects or matter and is not moving relative to the observer. Under these circumstances, the orbit of the binary shrinks according to the GW power predicted by the theory of general relativity. The waveform is also straightforward to calculate. It is dictated mainly by the acceleration of the mass quadrupole moment. Given the simplicity of the physical picture of a merging BBH, many people believe that the precision in the measurement of the mass is limited not by our theory, but by the observation. However, the universe is more sophisticated than what the standard model depicts. First, all astrophysical objects are moving relative to us, including BBHs. The motion induces a Doppler shift to the frequency of the GWs, but the frequency, as is mentioned earlier, is an observable essential to the measurement of the mass.

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Second, BBHs do not necessarily form in isolation. They could form, for example, in triple systems (e.g., [5] and references therein). The interaction with the tertiary may alter the orbital dynamics of a BBH and hence affects our interpretation of the GW signal. Note that the tertiary could be a supermassive BH (SMBH) of more than 106 M , according to several recent theoretical studies (pointed out by [6] and see a brief summary in [7]). In this case, the SMBH could induce not only a relatively high speed to the BBH but also a significant gravitational redshift to the GW signal, if the distance between the BBH and the SMBH is small. Third, the immediate vicinity of a BBH is not vacuum either. For example, BBHs may form in a gaseous environment as several major formation channels would suggest (see a brief review in [8]). The interaction with gas could affect the evolution of the binary too, so that the GW signal may not faithfully reflect the orbital dynamics in a vacuum. Last but not least, GWs do not propagate freely from the source to the observer. In between lies structures of a variety of scales and masses. One well-known effect related to the propagation of GWs in a nonuniform background is gravitational lensing [9–12]. It would distort our perception of the distance and power of a GW source. What would happen if one, unaware of the above astrophysical factors, insists on using the vacuum solution of isolated BBH mergers to model the GW signals? Could the observer still detect a signal? If a signal is detected, how accurately could the physical parameters such as mass be retrieved? This chapter is dedicated to addressing these questions.

Standard Siren In our daily lives, we are accustomed to perceiving the distance of a source not only by its emitted light but also by its sound. For example, most of us have heard the buzz of a bee. We could tell roughly how far it is when we hear one. We are able to do this because knowing what a bee sounds like allows us to calibrate the distance based on the loudness of the buzz. More experienced people can even infer the size of the source from the tone of its sound. Many of us can distinguish the buzz of a bee from the whining sound of a mosquito because we know the trick that mosquitoes make higher tones. Whenever we hear mosquitoes, we know immediately that they are “dangerously” close to us. We know it because mosquitoes are not as “powerful” as bees and hence should be relatively close by once heard. Analyzing GW signals is strikingly similar to perceiving sound. First, lower GW frequencies normally correlate with more massive objects. Second, once the mass could be estimated, e.g., from the frequency, we could further infer the distance according to the loudness, or “amplitude” in more physical terms, of the GW signal. Among the long list of GW sources, BBHs, binary neutron stars (BNSs), and double white dwarfs (DWDs) are particularly useful for distance measurement. This is because their orbital dynamics is simple enough so that we can have a good theoretical understanding of what their GW radiation is like. In other words, we know how they “sound.” Such a binary is often referred to as a “standard siren [13].”

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Fig. 1 Evolution of the amplitude (upper panel) and frequency (lower panel) of the GWs produced by a GW150914-like BBH. In the calculation, the two BH masses are chosen to be m1 = 40 M and m2 = 30 M , and the distance is set to d = 400 Mpc

To have an idea of what a standard siren is like, we show in Fig. 1 the GW signal of a BBH merger similar to GW150914 (adopted from the GW Open Science Center (https://www.gw-openscience.org/tutorials/)). The upper panel shows the evolution of the amplitude of the space-time distortion, called “strain.” The lower panel shows the corresponding GW frequency. The merger of the two BHs happens at the time t = 0. We can roughly divide the signal into three parts. (i) Long before the merger, both the amplitude and the frequency increase with time. Such a behavior is called “chirp.” The physical reason is that GW radiation causes the orbit to decay, so that the orbital frequency rises and the acceleration of the mass quadrupole moment intensifies. During this phase, the orbital semimajor axis is much greater than the gravitational radii of the BHs. Therefore, the orbit can be approximated by a Keplerian motion, and the GW radiation can be computed analytically. Since the shrinkage of the orbit happens on a timescale much longer than the orbital period, the two BHs spiral inward gradually. For this reason, this phase is known as the “inspiral phase.” (ii) As the separation of the two BHs becomes comparable to a few gravitational radii, the Keplerian approximation breaks down, and the radial motion of the two BHs becomes more prominent. We can no longer solve the

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evolution analytically but have to resort to numerical relativity. This phase is called the “merger phase.” It is marked by the cyan stripes in Fig. 1. (iii) Immediately after the coalescence of the horizons of the two BHs, the remnant is highly perturbed relative to a stationary Kerr metric. To get rid of the excessive energy, the BH will emit GWs at a series of characteristic frequencies (not shown in the plot) determined by the mass and the spin parameters. This process is called “ringdown.” Eventually, a single spinning BH is left. Now that we know what the signal is like after the parameters are specified, can we reverse the process and estimate the parameters based on the signal? In the canonical scenario, i.e., the BBH is isolated in a vacuum background and not moving relative to us, the answer is yes. Take the inspiral phase for example (since the chirp signal can be computed analytically). We can derive from the signal at least two measurable quantities, the amplitude h and the frequency f . From the evolution of the frequency, we can further derive the time derivative f˙. These three observables, h, f , and f˙, encode the mass and distance of the source. Since both f and f˙ are functions of the masses of the two BHs, m1 and m2 , and the semimajor axis of the orbit, a, we can combine them to eliminate the dependence on a. We will show that the result is a quantity characterizing the mass of the system. First, we follow the Keplerian approximation, normally acceptable for the inspiral phase, and derive the GW frequency using twice the orbital frequency, 1 f = π



G(m1 + m2 ) a3

1/2 ,

(1)

where G is the gravitational constant. Here, we have assumed a circular orbit for simplicity. Elliptical orbits will lead to additional harmonics of different frequencies. Because of the GW radiation, the semimajor axis a decays as a˙ gw := −

64 G3 m1 m2 (m1 + m2 ) , 5 c5 a 3

(2)

where c is the speed of light (as is derived in [14]). Combining the previous two equations, we can derive 96G7/2 m1 m2 (m1 + m2 )3/2 , f˙ = f˙gw = 5π c5 a 11/2

(3)

where f˙gw denotes the contribution from GW radiation only. Now, we can eliminate the a in Equation (3) using Equation (1). The result is c3 G



3/5 5f −11/3 f˙ (m1 m2 )3/5 = . 96π 8/3 (m1 + m2 )1/5

(4)

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Fig. 2 Chirp signals of three BBHs with the same chirp mass, M  8.7, but different mass ratios, q = 1, 1/3, and 1/10. The effect of different mass ratios starts to appear when f exceeds about 20 Hz

It is now clear that from the two observables f and f˙, one can derive a quantity of the dimension of mass, M =

(m1 m2 )3/5 . (m1 + m2 )1/5

(5)

This mass is known at the “chirp mass” because, as Equation (4) indicates, it determines how f increases with time. Besides the chirp mass, can we derive m1 and m2 separately? It is difficult if the orbit is nearly Keplerian. In this case, we have seen that f˙ is determined only by M . The mass ratio of the two BHs, q := m2 /m1 (assuming m2 ≤ m1 ), plays no role as long as M is fixed. However, as the binary shrinks to a size of about dozens of the gravitational radii, GR effect becomes significant, and the orbit starts to deviate from Keplerian motion. In fact, the deviation is q-dependent. Therefore, we can use it to disentangle the masses of the two BHs. To illustrate this idea, we show in Fig. 2 the chirp signals of three BBHs with the same chirp mass but different mass ratios. To include the GR effects, the waveforms are computed to the 3.5 post-Newtonian order [15]. Since the chirp masses are the same, the three signals are almost identical when f  15 Hz. At the later evolutionary stage when f  20 Hz, we start to see a divergence. The most prominent difference is the final frequency. We can see that higher frequency is correlated with larger mass ratio (q approaches 1). This is because as q increases, the mass of the primary BH (m1 ) decreases, so that the innermost stable circular orbit (ISCO) is smaller, and the corresponding frequency is higher. This general result suggests that one has to detect the final merger to be able to measure the mass ratio of a BBH. This is the reason that LIGO and Virgo, by detecting GWs in the 10–102 band, are capable of estimating the masses of

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individual BHs in binaries. The Laser Interferometer Space Antenna (LISA; see [16]), on the other hand, can detect only the early inspiral phase of (stellar-mass) BBHs because LISA is tuned to be sensitive to millihertz (mHz) GWs. As a result, it is generally more difficult for LISA to measure the individual masses of (stellarmass) BHs. It is worth noting that LISA is capable of measuring the masses of intermediate-massive and supermassive BHs between 103 and 1010 M , because their final mergers produce mHz GWs. How about measuring the distance? So far, we have not yet used the third observable h. In principle, it is inversely proportional to the distance because larger distance should lead to smaller GW amplitude. In fact, in the Keplerian approximation, the distance d depends on h as 4G M d= 2 c h



G πfM c3

2/3 .

(6)

Here, we have omitted the uncertainty induced by the inclination of the binary orbit because in principle it can be eliminated by measuring the GW polarizations. The important point is that all the quantities on the right-hand side of the last equation can be derived from the three observables f , f˙, and h. This completes the theory of measuring the mass and distance of a BBH using the chirp signal.

Mass-Redshift Degeneracy In the last section, we have assumed a flat Minkowski space. Because of this assumption, the signal detected is identical to the signal emitted from the source. However, the expansion of the universe will complicate this relationship. The cosmological expansion effectively stretches the signal. Consequently, the detected GWs will appear redshifted. Figure 3 illustrates this effect. From such a distorted signal, can we still retrieve the correct mass and distance of the source? As a result of the cosmological redshift, the apparent frequency fo will be lower than the intrinsic frequency f measured in the rest frame of the source. The difference can be calculated with fo = f (1 + zcos )−1 ,

(7)

where zcos denotes the cosmological redshift. Moreover, to the observer, the chirp rate appears to be f˙o = f˙gw (1 + zcos )−2 ,

(8)

where the additional factor of (1 + zcos )−1 is due to time dilation. The students can verify the above equation by noticing that f˙o = dfo /dto and dt/dto = (1 + zcos )−1 .

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Fig. 3 Effects of redshift on GW signals. Upper panel: Redshift stretches a waveform. Lower panel: An illustration of the problem of mass-redshift degeneracy. The intrinsic chirp signal of a 10–10 M binary (blue solid curve) gets redshifted if the binary is placed at a redshift of z = 2 (red solid curve). The redshifted signal is identical to that of a more massive 30–30 M binary residing at z = 0 (gray dashed curve)

If we wish to derive a chirp mass from the observed, redshifted signal, we can only get c3 Mo = G



−11/3

5fo f˙o 8/3 96π

3/5 = M (1 + zcos ).

(9)

This new mass is greater than the intrinsic chirp mass by a redshift factor 1 + zcos . For this reason, it is called the “redshifted chirp mass.” Now, we have a problem. The mass appears greater than the intrinsic one. More seriously, without knowing the cosmological redshift of the source, we would not know the real mass. This famous problem in GW astronomy is called “the massredshift degeneracy.” Given such a problem, should we trust the BH masses derived from GWs, especially when the detected BHs seem a bit overweight? There are two possibilities of breaking this degeneracy. One possibility is to first measure the distance using GWs and then infer the redshift based on a chosen cosmological model. But what kind of distance is measurable from GWs in an expanding universe? Let us take the ΛCDM cosmology for example (h = 0.7, ΩM = 0.29, and ΩΛ = 0.71). In this cosmology, the GW amplitude an observer detects is related to the transverse comoving distance dC as

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Fig. 4 The effect of cosmological redshift on the apparent mass and distance of GW150914. The blue dot marks the location of the real chirp mass and (luminosity) distance of the source. The red dot shows the redshifted chirp mass and apparent distance, measurable from GW signal

ho =

4G M c2 dC



G πfM c3

2/3 (10)

,

where f and M refer to the intrinsic, non-redshifted quantities. Using this apparent GW amplitude, we can only infer a distance of do =

4G Mo c 2 ho



G π fo Mo c3

2/3 .

(11)

Substituting the observed quantities on the right-hand side (those with the subscript o) using Equations (7), (9), and (10), one will find that do = dC (1 + zcos ) = dL ,

(12)

where dL is the luminosity distance. Therefore, what we derive from a chirp signal is in fact the luminosity distance. From the luminosity distance, we can calculate the corresponding redshift based on the chosen cosmological model. Finally, with this redshift, we can break the mass-redshift degeneracy. To illustrate this idea, we show in Fig. 4 the effect of cosmological expansion on the apparent chirp mass and distance of the first GW event GW150914. In the ΛCDM cosmology we have chosen, the apparent distance do is identical to the luminosity distance, which corresponds to a redshift of zcos  0.09. Therefore, we can infer that the intrinsic chirp mass M is smaller than the apparent one, Mo , by only a factor of about 1.1. For this reason, it is believed that the BHs in GW150914 are intrinsically massive.

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Another possibility is that an electromagnetic counterpart associated with a GW event is also detected. Then, the redshift can be measured independently, e.g., from the spectrum. Detecting the electromagnetic counterpart will greatly enhance the scientific payback of GW observations. With the redshift zcos derived from the counterpart and the luminosity distance dL from the GW signal, one can measure the Hubble constant and, in turn, constrain cosmological models. Unfortunately, detecting electromagnetic counterparts is challenging with the current telescopes. Moreover, some GW events, such as BBHs merging in a vacuum background, are not expected to produce strong electromagnetic radiation.

Effects of Astrophysical Environments The standard model of a BBH merger does not take into account the motion of the binary relative to the observer. Neither does it include other astrophysical objects nor matter around the BHs or along the path of the propagation of GWs. Now, we will add these new ingredients into our model and study the potential impact on our measurement of the physical parameters of the source.

Strong Gravitational Lensing Strong gravitational lensing by foreground galaxies or galaxy clusters has been observed for transient light sources, such as supernovae and gamma ray bursts. It is reasonable to speculate that some LIGO/Virgo events may also be strongly lensed [10]. The effect of strong lensing is to magnify the amplitude of GWs. The significance can be characterized by the magnification factor A , such that  ho = A

4G M c2 dC



G πfM c3

2/3 .

(13)

According to Equation (11), the distance of a lensed GW sources will appear to be do = dL /A ,

(14)

which is smaller than the real luminosity distance. The apparent mass, which one would derive from the GW signal, remains Mo = M (1 + zcos ).

(15)

Figure 5 illustrates this effect using GW150914 as an example. It shows that a low-mass BBH residing at a high redshift, if getting gravitationally lensed, would appear as a high-mass binary residing at a relatively low redshift (as has been pointed out in [17, 18]). Now, we cannot use the apparent distance do to break the

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Mo [M]

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30 20 10 0 2 10

?

Strong lensing:

zcos = 2,

103 do [Mpc]

≃ 37

104

Fig. 5 The lensing model of GW150914. For illustrative purposes, the source (blue dot) is placed at a cosmological redshift of zcos = 2, corresponding to a luminosity distance of 16 Gpc in the ΛCDM cosmology. After lensing, the mass and distance appear to coincide with those of GW150914 (red dot). The corresponding magnification factor is A  37

mass-redshift degeneracy, because it does not reflect the real luminosity distance of the source. Could strong lensing explain all the massive BBHs detected by LIGO and Virgo? It is difficult because from the statistical point of view, lensing events should be uncommon. Moreover, as is indicated by the direction of the arrow in Fig. 5, the lensing effect tends to produce an anticorrelation between the observed quantities do and Mo [17]. The current observations do not favor such a relation [19]. However, as the number of LIGO/Virgo BBHs increases and the detection horizon expands, the possibility of catching a lensed event increases as well. Therefore, watch out for massive BBHs at relatively low redshift! It is worth noting that the above analysis of GW lensing assumes that the intervening matter distribution is static. In the real universe, however, the structure is evolving with time. As has been shown in [20], the effect of an evolving matter distribution on GWs is analogy to the integrated Sachs-Wolfe effect on cosmic microwave photons, for which the evolution of the large-scale structure induces an addition variation in the temperature anisotropy. Calculations have shown that such an effect could affect most significantly the GW amplitude and lead to an systematic error of (1–10)% in the measurement of the distance if the source resides at a redshift of zcos  1.

The Effect of Motion Almost all astrophysical objects are moving relative to us. The general effect of motion is to induce a Doppler shift to the frequency of a wave signal. This shift can

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be characterized by the relativistic Doppler factor, 1 + zdop = γ (1 + v · n/c),

(16)

where v is the velocity of the source, n is a unit vector coinciding with the line of  sight, and γ = 1/ 1 − (v/c)2 is the Lorentz factor. Using this factor and taking the cosmological redshift into account, the apparent frequency can be written as fo = f (1 + zcos )−1 (1 + zdop )−1 ,

(17)

and the chirp rate changes to f˙o = f˙gw (1 + zcos )−2 (1 + zdop )−2 .

(18)

Applying Equation (9) again, we find that the apparent chirp mass is Mo = M (1 + zcos )(1 + zdop ).

(19)

As for the distance, we notice that motion does not significantly affect the amplitude of GWs [21]. Therefore, we can follow Equation (10) to calculate ho and further derive that do = d(1 + zcos )(1 + zdop ).

(20)

The last two equations suggest that Doppler shift can also affect the measurement of the mass and distance of BBH. It introduces a systematic error if not appropriately accounted for in GW data analysis. In particular, when the relative velocity and the line of sight are aligned, i.e., v · n > 0, the Doppler effect would make the mass and distance appear even greater than their intrinsic values. We note one difference between the Doppler effect and the cosmological redshift. While cosmological redshift always makes the mass and distance appear bigger, Doppler effect could make them appear smaller as well. This happens when the source is moving toward the observer, i.e., when v · n < 0. In this case, the Doppler factor, as well as the effective redshift factor 1 + zdop , becomes smaller than 1. How large could the factor 1 + zdop be? In the conventional scenario, BBHs form either in isolated binaries in galaxy bulges and disks or in star clusters [22]. In our Milky Way, isolated binaries and star clusters are moving at a typical velocity of 200 km s−1 relative to the earth. Therefore, their Doppler effect can be neglected. For the BBHs in external galaxies, their velocities relative to the Milky Way could be much greater. For example, in the most massive galaxy clusters, the velocities of the galaxy members can reach thousands of km s−1 . Even this velocity is much smaller than the speed of light. For these reasons, Doppler effect is neglected in the standard procedure of GW data analysis. However, the conventional picture of BBH formation is incomplete. Recent studies suggest that in galaxy centers, especially where there are SMBHs, the

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merger rate of BBHs can be enhanced. The reasons are multifold [7]. (i) Close to an SMBH, the escape velocity is large. This is a place where newly formed compact objects are likely to stay. (ii) SMBHs are normally surrounded by a cluster of stars, known as the “nuclear star cluster.” Due to a net loss of energy during the gravitational interaction with other stars, massive objects such as BHs and neutron stars will gradually segregate toward the center of a nuclear star cluster. This “mass segregation effect” increases even higher the concentration of compact objects near SMBHs. Such a condition is favorable to the formation of BBHs. (iii) The tidal force exerted by SMBHs could excite the eccentricities of the nearby BBHs through the so-called “Lidov-Kozai” mechanism. A higher eccentricity normally results in a stronger GW radiation and hence leads to a faster merger. (iv) If an SMBH is surrounded by gas, as would be the case in an active galactic nucleus (AGN) due to the presence of a gaseous accretion disk, interaction with the gas could also dissipate the internal energy of a BBH and lead to its faster merger. Therefore, it is possible that a BBH may merge in the vicinity of an SMBH. If such a merger happens, the binary is likely to obtain a large velocity due to the orbital motion around the SMBH. For a BBH detected by LIGO/Virgo, the velocity is almost constant because the event lasts normally less than a second, but the orbital period around the SMBH is orders of magnitude longer. The tidal force of the SMBH does not significantly affect the shrinkage of the binary because GW radiation predominates at this stage. Under these conditions, the theory developed in this section applies. It is clear that the effect is more significant if the merger happens closer to the SMBH. We can use the velocities at the last stable orbits to estimate the order of magnitude of the maximum effect. If the binary is on a circular orbit around the SMBH, as we would expect for the BBHs in AGN accretion disks, the largest velocity appears at the ISCO. For non-spinning Schwarzschild BHs, the orbit is at a radius of R = 6GM/c2 , where M is the mass of the SMBH. In this case, the √ velocity is approximately c/ 6  0.408c. The corresponding Doppler factor is 1 + zdop  1.54. On the other hand, if the orbit of the BBH is nearly parabolic, a more likely configuration if the binary forms far away and later gets captured by the SMBH, the last orbit is the innermost bound orbit (IBO). In this case, the √ pericenter is at 4GM/c2 . The corresponding velocity is c/ 2 = 0.707c, and the Doppler factor becomes 1 + zdop  2.41. Of course, even higher values are possible if the distance is smaller, but such events are rarer. The values derived above suggest that the Doppler effect could significantly affect our measurement of the mass and distance of a GW source, if the source is close to an SMBH. The event rate, therefore, deserves careful calculation.

Deep Gravitational Potential Close to an SMBH, gravitational redshift is also significant. This additional redshift would further distort a GW signal and make the measurement of the physical parameters even more biased.

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Suppose the gravitational redshift is zgra , the observed GW frequency would be lowered relative to the emitted one by a factor of 1+zgra . Similar to the cosmological redshift and the Doppler effect, the chirp rate would be reduced by a factor of (1 + zgra )2 due to time dilation. Using these relations, it is straightforward to show that the chirp mass appears to be Mo = M (1 + zcos )(1 + zdop )(1 + zgra )

(21)

to an observer. To derive the apparent distance, we notice that the GW amplitude is not significantly affected by the gravitational redshift as long as the merger does not happen extremely close to the event horizon (see [7] for a more quantitative description). As a result, Equation (10) remains a good approximation, from which we can derive that do = dC (1 + zcos )(1 + zdop )(1 + zgra ).

(22)

We can see that the apparent mass and distance of a BBH are determined by three types of redshift. Nevertheless, LIGO/Virgo only considered the cosmological redshift in their analysis. Such a treatment is not suitable for the BBHs forming in the vicinity of SMBHs. We can again evaluate how extreme the value of (1 + zdop )(1 + zgra ) could be by investigating the last stable orbits. Assuming a non-spinning SMBH, we can calculate the gravitational redshift by 1 + zgra = (1 − RS /R)−1/2 ,

(23)

where RS = 2GM/c2 is the Schwarzschild radius. Using this equation, we find that for ISCO, the value of (1 + zdop )(1 + zgra ) is approximately 1.89, while for IBO, it is about 3.41. Interestingly, the values of these redshift factors coincide with the difference between the BH masses in GW150914 (∼30 M ) and those in X-ray binaries (∼10 M ). Could the Doppler+gravitational redshift explain the high-mass BHs detected by LIGO and Virgo (see Fig. 6)? The key to answer this question is, again, the event rate. More works along this line of research are deserved.

Peculiar Acceleration For a binary orbiting an SMBH or any other tertiary object, the velocity of the center of mass of the binary in principle is not constant. This acceleration may not be a problem for LIGO/Virgo binaries because the event lasts only a fraction of a second, too short compared to the orbital period of the tertiary to cause any noticeable changes of the velocity. However, for LISA BBHs, the acceleration is no longer negligible. This is because LISA sources could dwell in the band for years.

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Mo [M]

40

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30 20 10 0

?

Doppler & gravitational redshift

(1 + zdop)(1 + zgra) = 3

102

do [Mpc]

103

Fig. 6 The Doppler+gravitational redshift model for GW150914. The location of the intrinsic chirp mass and luminosity distance is marked by the blue dot. The location of the apparent mass and distance are shown by the red dot. In the plot, a total redshift factor of (1 + zdop )(1 + zgra ) = 3 is assumed, which corresponds to a place close to the pericenter of the IBO around a Schwarzschild SMBH

During this period, the velocity of the binaries may change significantly. This in turn changes the Doppler redshift [23, 24]. Let us study the effect of this peculiar acceleration (different from the acceleration of the cosmological expansion) on our measurement of the mass and distance of binary compact objects. Starting from Equation (17) and neglecting the cosmological redshift for the time being, we can rewrite the relationship between the apparent frequency and the center-of-mass velocity of the binary as 

fo = f

1 − β2 , 1 + v · n/c

(24)

where β = v/c and f remain twice the orbital frequency. Differentiating it relative to the observer’s time, to , we get   

2 f˙gw 1 df 1 − β df dt d o o = f˙o = = +f dto dto dt 1 + zdop 1 + zdop dt 1 + v · n/c    f˙gw d 1 − β2 + fo = . 2 dt 1 + v · n/c (1 + zdop )

(25)

(26)

If we compare the last equation with Equation (18) and notice that zcos = 0 for now, we will see that the second term on the right-hand side of the last equation is caused by the variation of v. This term could be either positive or negative, depending on the

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orbital dynamics. Therefore, the apparent chirp mass, which should be calculated from fo and f˙o , could also be bigger or smaller than the real value, and so does the apparent distance. To further quantify this effect, we define Γacc to be the ratio between the second and the first term in Equation (26). This treatment allows us to write Mo and do in the following compact forms Mo = M (1 + zdop )(1 + Γacc )3/5

(27)

do = d(1 + zdop )(1 + Γacc ).

(28)

and

In particular, Equation (27) indicates that mass is degenerate with not only redshift but also acceleration. This is a new type of degeneracy in GW astronomy. Notice that Γacc could be negative so that 1 + Γacc may be smaller than 0. In this case, we see an inverse chirp, i.e., the GW frequency decreases with time. Such a signal immediately indicates a nonstandard condition for the binary. These events could be singled out in data analysis without causing further confusion. Therefore, in the following, we focus on the case in which 1 + Γacc > 0. For those binaries in triple systems, when will Γacc significantly exceed unity? Since f˙gw ∝ f 11/3 , the value of Γacc is higher for lower f . This is another reason that we focus on LISA binaries in this section. To derive a value for Γacc , we must specify the parameters of the tertiary. For simplicity, we assume circular orbits for the tertiary. In this case, the second term on the right-hand side of Equation (26) is of the order of Gm3 fo /(r 2 c), where m3 is the mass of the tertiary and r is the distance between thetertiary and the center of mass of the binary. Here, we have omitted the term 1 − β 2 because we only consider the case in which the center-of-mass velocity is much smaller than the speed of light. This consideration is acceptable for LISA binaries because they cannot reside at several Schwarzschild radii of an SMBH. Otherwise, the binaries would be tidally disrupted given their large semimajor axes. Using this approximation for the acceleration, we find that  Γacc  4.4

f 2 mHz

−8/3 

M 0.3 M

−5/3 

m3 1 M



r −2 . 30 AU

(29)

This result suggests that the acceleration affects more seriously the low-mass binaries (small M ) in the LISA band. Let us take DWDs for example since they are the most numerous binaries in band [16]. If we assume M = 0.3 M and set f = 2 mHz, the most sensitive band of LISA, we find that Γacc = 4.4 for a tertiary of m3 = 1 M and r = 20 AU. Correspondingly, the chirp mass appears bigger, due to the term (1 + Γacc )3/5 , by a factor of 2.8. The distance, on the other hand, appears bigger by a factor of about 5.4. In this case, a DWD inside the Milky Way would appear in a nearby galaxy! If, on the other hand, we reverse the orbital velocity of the tertiary and increase the chirp mass to 0.8 M , we will get Γacc = −0.86. In this case, the DWD will appear lighter by a factor of 3.2 and closer by a factor

37 Distortion of Gravitational-Wave Signals by Astrophysical Environments

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Frequency (mHz)

No acceleration, increase ℳ

Fix

ℳ,

inc

rea

se

Γa

cc

Time (~5 years) Fig. 7 Illustration of the degeneracy between mass and acceleration. The evolution of the apparent frequency is effectively a straight line if the chirp mass of the source is intrinsically small, such as in the case of DWDs. The gray dashed lines show the dependence of the chirp signal on the chirp mass of the system. The red lines show the dependence on the acceleration. It is clear that the effect of increasing the acceleration is identical to the effect of increasing the chirp mass

of 7.0! Similar cases could potentially impede our understanding of the formation and distribution of DWDs in the Milky Way. If the tertiary is an SMBH, as would be the case for those DWDs forming in the Galactic Center, Γacc would be even greater. For example, if we assume m3 = 4 × 106 M and r = 0.1 pc, a DWD with M = 0.3 M and f = 2 mHz will have Γacc = 38. The apparent chirp mass will increase to Mo  2.7 M . Now, the white dwarfs will masquerade as neutron stars or BHs! So far, the estimation of the chirp mass is based on only two observables, f and f˙. One may wonder that given the long observational period of LISA (∼5 years), could we derive also f¨ from the waveform? This is an important question because if we could measure f¨, maybe we could use it to break the degeneracy between mass and acceleration. In fact, for BBHs, more advanced data analysis does show that it is possible [25]. For DWDs, however, it is more difficult because the value of f¨ induced by either GW radiation or peculiar acceleration is small. As Fig. 7 shows, such a chirp signal is featureless. The evolution of fo is effectively a straight line. Since we use the slope of the straight line to infer the chirp mass and small BBHs with large acceleration lead to the same slopes as those big BBHs without acceleration, it is difficult to distinguish these two scenarios [26].

The Effect of Gas Gas is another important factor for the formation and evolution of BBHs. For example, in the model of binary-star evolution, there is a phase when both binary

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members are immersed in a common gaseous envelope. This phase is considered to be essential to the formation of close binaries. Moreover, as has been mentioned before, AGN accretion disk is also a breeding ground for BBHs. In general, gas will change the orbital dynamics of a BBH and in this way affect the GW signal. Without knowing of such an effect, would a GW observer retrieve correctly the physical parameters of a BBH? We now know that to address this question, we need to investigate the effect of gas on f and f˙ and use them to derive the apparent chirp mass and distance. In fact, gas does not significantly affect f because, in the astrophysical scenarios mentioned above, the total mass of the gas is negligible relative to the total mass of the BHs. The effect is more prominent for f˙. Depending on the thermal dynamical properties and the distribution, gas could either accelerate or decelerate the binary shrinkage and hence increase or decrease f˙ relative to f˙gw . For example, hydrodynamical friction makes the binary orbit shrink faster, but tidal torque and accretion could work in the opposite direction (see a brief discussion in [8]). To keep the following analysis general, we can characterize the gas effect by rewriting the apparent chirp rate as f˙o = f˙gw + f˙gas = f˙gw (1 + Γgas ),

(30)

where f˙gas is the chirp rate due to gas dynamics and Γgas := f˙gas /f˙gw . The redshift effects are neglected here for simplicity. Similar to the analysis of peculiar acceleration, we can derive that Mo = M (1 + Γgas )3/5 , do = d(1 + Γgas ).

(31) (32)

Since Γgas can be either positive or negative, the apparent mass and distance could be bigger or smaller than their intrinsic values. Figure 8 shows that the effect of increasing the efficiency of gas dynamics (increasing Γgas ) is identical to the effect of increasing the chirp mass. Like the effect of peculiar acceleration, gas is more important for those binaries in the mHz. In this band, f˙gw is small so that the gas-induced f˙gas could be relatively large. Recent models show that in AGN accretion disks and common envelopes, Γgas could be much greater than 1 [8, 27, 28]. Therefore, in these environments, BBHs, as well DNSs and DWDs, could appear more massive than they really are.

Summary In this chapter, we have seen that astrophysical factors could affect the measurement of the masses of GW sources. In particular, strong gravitational lensing, the Doppler and gravitational redshift around an SMBH, the peculiar acceleration induced by a tertiary star or SMBH, and gas dynamics could all increase the apparent chirp mass

37 Distortion of Gravitational-Wave Signals by Astrophysical Environments

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Frequency (mHz)

No gas, increase ℳ

Fix

ℳ

, in

cre

as

eΓ

gas

Time (~5 years) Fig. 8 Same as Fig. 7 but showing the effect of gas on the slope of the chirp signal

of a binary. In suitable conditions, the Doppler blueshift in the vicinity of an SMBH, the peculiar acceleration, and gas dynamics could also work in the opposite way, reducing the apparent mass. These results have important implications for the peculiar compact objects appearing in the bands of LIGO/Virgo and the future LISA. Such peculiar objects include compact objects in the lower mass gap (between 3 and 5 M ) and BHs heavier than 20 M . They had never been detected in the past, before the GW era. Therefore, their nature deserves a thorough scrutiny. Now, we can revisit the question raised at the beginning of this chapter. Once LIGO/Virgo or LISA detect a peculiar object, could we readily accept the apparent mass as the real one, or what are the other possibilities? Table 1 summarizes the alternative possibilities. The following is a brief discussion of each case. Interested readers could refer to the previous sections for more detailed explanations: (i) In the LIGO/Virgo band (10–102 Hz), the BHs with an apparent mass higher than 20 M could be mimicked by the BHs less massive than 20 M , due to strong lensing, or the Doppler and gravitational redshift around SMBHs. For the BBHs in this band, peculiar acceleration and gas dynamics are too weak relative to GW radiation to cause such a strong effect. (ii) BNSs are unlikely imposters of the massive BBHs with m1 , m2 > 20 M in the LIGO/Virgo band. In the scenarios of strong lensing, Doppler redshift, and gravitational redshift, the BNSs should have a redshift of z  10 to meet the requirement. The corresponding magnification factor (A ) would be too extreme and the distance to SMBH too small. Such events are rare. Peculiar acceleration and gas dynamics are unlikely to take effect either, because GW radiation would predominate the evolution of f˙o as soon as the BNSs enter the LIGO/Virgo band.

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Table 1 Alternative models for the peculiar objects appearing in the LIGO/Virgo/LISA band Detector LIGO/Virgo

Peculiar objects 20 M

(3, 5) M

LISA

20 M

(3, 5) M

Imposters BBH m1 , m2 BNS DWD BBH m1 , m2 BNS DWD BBH m1 , m2 BNS DWD BBH m1 , m2 BNS DWD

< 20 M

≥ 5 M

< 20 M

≥ 5 M

Models 1, 2, 4 – – 3 2, 4 – 6 6 6 5, 6 5, 6 5, 6

Notes i ii iii iv v vi vii viii ix x xi xii

1. Gravitational lensing; 2. Doppler redshift (SMBH); 3. Doppler blueshift (SMBH); 4. gravitational redshift (SMBH); 5. peculiar acceleration; 6. gas dynamics

(iii) DWDs cannot masquerade as massive BBHs in the LIGO/Virgo band either. Because of the large sizes of white dwarfs, the coalescence happens at a frequency of about 0.1 Hz. Therefore, DWDs do not enter the LIGO/Virgo band. (iv) The Doppler blueshift could also make an ordinary BH slightly more massive than 5 M appear in the lower mass gap in LIGO/Virgo observations. For example, a BBH at the ISCO of a Schwarzschild SMBH could be blueshifted. Combined with the gravitational redshift, the apparent mass could be a factor of (1 + zdop )(1 + zgra )  0.79 of the real mass. For the IBO, the effect is more significant, and the fact could reduce to 0.58. Around Kerr SMBHs, the factor could be even smaller. Peculiar acceleration and gas dynamics could hardly induce such an effect because, as is mentioned in (i), they are too weak compared to GW radiation. (v) BNSs merging in the vicinity of an SMBH could also appear in the lower mass gap in LIGO/Virgo observations, because of the Doppler and gravitational redshift. Strong lensing, on the other hand, is unlikely to cause such an effect, because BNSs have to be relatively nearby to be detected by LIGO/Virgo. Therefore, the probability of being lensed is low. For the same reason given in (ii), peculiar acceleration and gas dynamics are unlikely to move BNSs into the lower mass gap either. (vi) Same as (iii). (vii) For LISA, the GW frequency is low. The corresponding BBHs are at an early evolutionary stage, and the GW power is relatively weak. In this case, gas dynamics could compete with GW radiation and make less massive BH appear in the mass range of 20 M , while the effect of peculiar acceleration is still too weak to significantly affect f˙o . Strong lensing is unlikely because LISA BBHs are at relatively small luminosity distances.

37 Distortion of Gravitational-Wave Signals by Astrophysical Environments

(viii) (ix) (x)

(xi)

(xii)

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Doppler and gravitational redshift induced by a nearby SMBH are unlikely, because LISA BBHs, due to their large semimajor axis, are likely to be tidally disrupted if they approach the Schwarzschild radius of an SMBH. BNSs could masquerade as BBHs in the LISA band if there is gas around. The other models are unlikely to work here for the same reasons given in (vii). The same as (vii) and (viii). Since gas dynamics could also reduce f˙o , it could make the BHs more massive than 5 M appear in the low mass gap in LISA observations. Peculiar acceleration could work in the same way but only when the BHs are slightly more massive than 5 M and the tertiary is an SMBH. Doppler blueshift around an SMBH could not induced such an effect because it would require BBHs to approach several Schwarzschild radii of the SMBH. LISA BBHs would be tidally disrupted at such a small distance. BNSs could masquerade as lower-mass-gap objects in the LISA band due to the peculiar acceleration and gas dynamical effects. Strong gravitational lensing has difficulty inducing such an effect because LISA BNSs are relatively close by. Doppler and gravitational redshift due to a nearby SMBH are unlikely to work here, because of the strong tidal force close to the SMBH. The same as (xi), but the peculiar acceleration works only when the tertiary is an SMBH.

Estimating the corresponding event rate could help us evaluate the likelihood of the above models and, maybe, reject some of them [2, 7]. But the estimation is subject to uncertainties because of our incomplete understanding of the astrophysical environments around GW sources. A more rigorous approach is to study the distortion of the wave front by those astrophysical factors and look for a detectable signature (e.g., [21, 29, 30]). Both directions deserve further exploration.

References 1. Abbott BP, Abbott R, Abbott TD, Abernathy MR, Acernese F, Ackley K, Adams C, Adams T, Addesso P, Adhikari RX, et al (2016) Observation of gravitational waves from a binary black hole merger. Phys Rev Lett 116:061102 2. LIGO Scientific Collaboration and Virgo Collaboration (2019) Binary black hole population properties inferred from the first and second observing runs of advanced LIGO and advanced Virgo. Astrophys J Lett 882:L24 3. LIGO Scientific Collaboration and Virgo Collaboration (2020) GW190425: observation of a compact binary coalescence with total mass ∼3.4 m . Astrophys J 892:L3 4. L. S. Collaboration and V. Collaboration (2020) Gw190521: a binary black hole merger with a total mass of 150 M . Phys Rev Lett 125:101102 5. Miller MC, Hamilton DP (2002) Four-body effects in globular cluster black hole coalescence. Astrophys J 576:894–898 6. Antonini F, Perets HB (2012) Secular evolution of compact binaries near massive black holes: gravitational Wave Sources and Other Exotica. Astrophys J 757:27 7. Chen X, Li S, Cao Z (2019) Mass-redshift degeneracy for the gravitational-wave sources in the vicinity of supermassive black holes. Mon Not R Astron Soc 485:L141–L145

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8. Chen X, Xuan Z-Y, Peng P (2020) Fake massive black holes in the milli-hertz gravitationalwave band. Astrophys J 896:171 9. Markovi´c D (1993) Possibility of determining cosmological parameters from measurements of gravitational waves emitted by coalescing, compact binaries. Phys Rev D 48:4738–4756 10. Wang Y, Stebbins A, Turner EL (1996) Gravitational lensing of gravitational waves from merging neutron star binaries. Phys Rev Lett 77:2875–2878 11. Nakamura TT (1998) Gravitational lensing of gravitational waves from inspiraling binaries by a point mass lens. Phys Rev Lett 80:1138–1141 12. Takahashi R, Nakamura T (2003) Wave effects in the gravitational lensing of gravitational waves from chirping binaries. Astrophys J 595:1039–1051 13. Schutz BF (1986) Determining the Hubble constant from gravitational wave observations. Nature 323:310 14. Peters PC (1964) Gravitational radiation and the motion of two point masses. Phys Rev 136:1224–1232 15. Sathyaprakash BS, Schutz BF (2009) Physics, astrophysics and cosmology with gravitational waves. Living Rev Relativ 12:2 16. Amaro-Seoane P, Audley H, Babak S, Baker J, Barausse E, Bender P, Berti E, Binetruy P, Born M, Bortoluzzi DEA (2017) Laser interferometer space antenna. ArXiv e-prints 17. Broadhurst T, Diego JM, Smoot GI (2018) Reinterpreting low frequency LIGO/Virgo events as magnified stellar-mass black holes at cosmological distances. arXiv e-prints, arXiv:1802.05273 18. Smith GP, Jauzac M, Veitch J, Farr WM, Massey R, Richard J (2018) What if LIGO’s gravitational wave detections are strongly lensed by massive galaxy clusters? Mon Not R Astron Soc 475:3823–3828 19. LIGO Scientific Collaboration and Virgo Collaboration (2019) GWTC-1: a gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. Phys Rev X 9:031040 20. Laguna P, Larson SL, Spergel D, Yunes N (2010) Integrated Sachs-Wolfe effect for gravitational radiation. Astrophys J Lett 715:L12–L15 21. Torres-Orjuela A, Chen X, Cao Z, Amaro-Seoane P, Peng P (2019) Detecting the beaming effect of gravitational waves. Phys Rev D 100:063012 22. Abbott BP, Abbott R, Abbott TD, Abernathy MR, Acernese F, Ackley K, Adams C, Adams T, Addesso P, Adhikari RX, et al (2016) Astrophysical implications of the binary black-hole merger GW150914. Astrophys J Lett 818:L22 23. Meiron Y, Kocsis B, Loeb A (2017) Detecting triple systems with gravitational wave observations. Astrophys J 834:200 24. Inayoshi K, Tamanini N, Caprini C, Haiman Z (2017) Probing stellar binary black hole formation in galactic nuclei via the imprint of their center of mass acceleration on their gravitational wave signal. Phys Rev D 96:063014 25. Tamanini N, Klein A, Bonvin C, Barausse E, Caprini C (2020) Peculiar acceleration of stellarorigin black hole binaries: measurement and biases with LISA. Phys Rev D 101:063002 26. Robson T, Cornish NJ, Tamanini N, Toonen S (2018) Detecting hierarchical stellar systems with LISA. Phys Rev D 98:064012 27. Chen X, Shen Z (2019) Retrieving the true masses of gravitational wave sources. Proceedings 17(1):4 28. Caputo A, Sberna L, Toubiana A, Babak S, Barausse E, Marsat S, Pani P (2020) Gravitationalwave detection and parameter estimation for accreting black-hole binaries and their electromagnetic counterpart. Astrophys J 892:90 29. Hannuksela OA, Haris K, Ng KKY, Kumar S, Mehta AK, Keitel D, Li TGF, Ajith P (2019) Search for gravitational lensing signatures in LIGO-virgo binary black hole events. Astrophys J 874:L2 30. Torres-Orjuela A, Chen X, Amaro-Seoane P (2020) Phase shift of gravitational waves induced by aberration. Phys Rev D 101:083028

38

Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories Mariafelicia De Laurentis and Ivan De Martino

Contents Theory of Gravity and Its Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f (R)-Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Chameleon Mechanism in f (R)-Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gravitational Wave Stress-Energy (Pseudo) Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application to Binary Systems and Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . Time Variation of the Orbital Period in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . Periastron Advance in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Post-Keplerian Parameters and Observational Constraints on General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First-Time Derivative of the Orbital Period in f (R)-Gravity . . . . . . . . . . . . . . . . . . . . . . . Periastron Advance in f (R)-Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraining Alternative Theories of Gravity Using GW150914 and GW151226 . . . . . . . . Constraints from the Shapiro Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. De Laurentis () Dipartimento di Fisica, Università di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Napoli, Italy e-mail: [email protected] I. De Martino Universidad de Salamanca, Departamento de Fisica Fundamental, P. de la Merced, Salamanca, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_40

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Abstract

The recent detections of gravitational waves by the advanced LIGO and Virgo detectors open a formidable way to set constraints on alternative metric theories of gravity in the strong field regime. Such tests rely sensitively on the phase evolution of the gravitational waves, which is controlled by the energy momentum carried by such waves out of the system. The weak-field limit of alternative metric theories shows new aspects of gravitation which are not present in general relativity and exhibits new gravitational field modes which can easily be interpreted as massive gravitons. The study of the generation, propagation, and detection of gravitational waves in the weak-field limit of a given relativistic theory of gravity is an important part of astrophysics. Keywords

Gravitation · Gravitational waves · Binary systems · Scalar tensor gravity · f(R) gravity

Theory of Gravity and Its Extensions General relativity (GR) explains successfully gravitational phenomena at scales ranging from kilometers, such as the ones of compact objects like white dwarfs and neutron stars, to cosmological ones. Nevertheless, several attempts to modify or extend GR have been pursued in order to avoid the need of introducing dark matter and dark energy in the total energy/density budget of the Universe to explain the emergence of the large-scale structure and the current accelerated expansion of the space-time. Therefore, an important question is whether there is any need to pursue this road. Back in the 1920s, modifications of GR were proposed in order to unify it with other interactions (see, for instance, [39,68,88]). In fact, despite its many successes, GR is not a quantum theory, and it cannot provide a description of the Universe at the quantum scales. Many efforts have been devoted to unify the quantum field theory and GR with little success. Furthermore, our knowledge about dark matter and dark energy comes from to their dynamical effects, but their fundamental nature, whether particles or scalar fields, is completely unknown [11, 12, 41]. The need for requiring two unknown matter/energy components to fully explain the observations within a GR framework has been interpreted as breakdown of the theory at astrophysical and cosmological scales [19, 20, 70]. Further, whatever is the unification scheme one has in mind, let it be superstrings, supergravity, or grand unified theories, they all consider non-minimal couplings to the geometry and higher-order terms in the curvature invariants to be present in the effective Lagrangian. Thus, it is important to explore if those extensions of GR overcome the previous shortcomings. In the most general approach, extended theories of gravity (ETGs) are extensions of GR obtained by including in the Lagrangian higher-order curvature invariants

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

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(such as R 2 , Rμν R μν , R μναβ Rμναβ , R R, or R k R) and minimally or nonminimally coupled terms between scalar fields and geometry (such as φ 2 R) [19]. ETGs can be roughly classified in (A) scalar-tensor theories if the geometry is non-minimally coupled to some scalar field and (B) higher-order theories if the action contains derivatives of the metric components of order higher than two. Combinations of both types give rise higher-order/scalar-tensor gravity theories. In the most general case, the action can be written as  A =

  √  d 4 x −g f (R, R, 2 R, ..k R, φ) − g μν φ;μ φ;ν + 2κL(m) , (1) 2

where f represents a generic function of curvature invariants, κ is the coupling constant, and φ is a scalar field whose nature and dynamics (standard, phantom, or no-dynamical field) are specified by  = 0, ±1, respectively [40, 67, 71]. L(m) is the standard matter Lagrangian. In the metric approach, the field equations are obtained by varying the action in Eq. (1) with respect to the metric tensor gμν . Thus, one gets the following (2k + 4)order differential equations: Gμν =

 1 1 κT μν + g μν (F − G R) + (g μλ g νσ − g μν g λσ )G;λσ G 2   i k ∂F 1   μν λσ + (g g + g μλ g νσ )(j −i );σ i−j 2 ∂i R ;λ i=1 j =1

−g

μν λσ

g

  j −1 i−j ∂F ( R);σ  , ∂i R ;λ

where Gμν is the above Einstein tensor and   n  ∂F . G ≡ j ∂j R

(2)

(3)

j =0

The stress-energy tensor is due to the kinetic part of the scalar field and to the ordinary matter:  1 (m) Tμν = Tμν + [φ;μ φ;ν − φ;α φ;α ] . 2 2

(4)

The (eventual) contribution of a potential V (φ) is contained in the definition of f . ETGs make very concrete predictions that differ from GR and therefore are testable even with current instruments. One important feature of the ETGs is that the conservation laws, derived from gauge invariance, are well defined only at low energy limit. As a consequence, the fundamental physical constants can vary [9,80]. These theories are also important from a cosmological point of view, because they

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can directly explain the accelerated expansion of the Universe without requiring additional energy components, avoiding the fine-tune problems on the nature of dark energy [5].

Generation of Gravitational Waves In the language of field theory, the gravitational waves (GW) as generated in GR correspond to a massless spin-two graviton field propagating at the speed of light with two independent polarizations. In alternative theories of gravity, there are additional degrees of freedom which give rise to extra-polarization modes of various spin, massless or massive. For example, in scalar-tensor and metric f (R) theories of gravity, there is an extra-scalar mode, while in vector-tensor-scalar theories, a richer spectrum of modes appears. Indeed, GW in ETGs can be classified according to the effect of their transverse and longitudinal modes on a sphere of test particles at rest before the wave arrives (see [37, 38, 89] for a more comprehensive description). Let us start to review the form of the field equations of GR linearized around a Minkowski background. The starting point is the Einstein-Hilbert action [19]:  A =



√ d 4 x −g R + L(m) .

(5)

In the weak field approximation, the space-time metric, gμν , is assumed to deviate only slightly from the Minkowskian metric, ημν , so that gμν = ημν + hμν , where hμν is the perturbation term (for a comprehensive review on the field, refer to [61, 84]). Thus, the linearized field equations are η hμν + ∂μ ∂ν h − ∂μ ∂α hαν − ∂ν ∂α hαμ = 0 ,

(6)

where η is the d’Alembertian of the Minkowski metric, and h ≡ ημν hμν is the trace of the metric perturbation hμν (indices are raised and lowered with ημν and ημν ). By changing coordinates, it is possible to choose the Lorentz gauge ∂α hαμ −

1 ∂μ h = 0 . 2

(7)

Therefore, the linearized equations of motion for the metric perturbations in this gauge reduce to η hμν = 0 .

(8)

The solutions can be expanded in plane waves as α

α

(+) ikα x ikα x hμν = Aμν e + A(×) , μν e

(9)

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1557

Fig. 1 Motion of point-like particles distributed on ring and due to a GW that propagates perpendicular to the plane of the ring. The left panel depicts a wave with a polarization mode +, and the right panel shows a wave with a polarization mode ×

(+,×)

where Aμν are constant polarization tensors and ημν k μ k ν = 0. If the wave propagates along the z-axis, the transverse-traceless tensor can be recast as ⎛

⎞ h+ h× 0    ⎝ h× −h+ 0 ⎠ cos ω t − z , hTT ij (t, z) = c 0 0 0

(10)

where h+ and h× denote the two independent polarization modes. These two modes, plus (+) and cross (×), differ by a rotation of 45o . Figure 1 depicts the motion of point-like particles distributed on a ring due to a GW propagating perpendicularly to the plane of the ring. The solid lines represent the force lines associated with the two polarization modes. To show the existence of the additional polarization modes in ETGs, let us now review, as an example, the main equations describing the GWs in scalar-tensor gravity. The starting point is again the action which is obtained from Eq. (1) by setting f (R, φ) = f (φ)R − V (φ) and  = −1:  A =

  1 μν d x −g f (φ)R + g φ;μ φ;ν − V (φ) + L(m) . 2 4

√

(11)

GWs are generated throughout small perturbations of of the space-time: gμν = ημν + hμν ,

(12)

φ = φ0 + ϕ ,

(13)

where the scalar and tensor modes have the same order of magnitude in terms of  a smallness parameter  O (ϕ/φ0 ) = O hμν = O (). Thus, the linearized field equations are [82, 84, 89]: η hμν + ∂μ ∂ν h − ∂μ ∂α hαν − ∂ν ∂α hαμ = −  η ϕ = m2 ϕ ,

∂ μ ∂ν ϕ , φ0

(14) (15)

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M. De Laurentis and I. De Martino

where, being V (φ) the potential of the scalar field, one gets m2 =

V  (φ0 ) − φ0 V  (φ0 ) ϕ. 2ω + 3

(16)

The solutions can be expanded in plane waves as follows: α

α

(+) ikα x ikα x hμν = Aμν e + A(×) , μν e

ϕ = ϕ0 eilα

xα

,

(17) (18)

(+,×)

where Aμν and ϕ0 are constants and ημν k μ k ν = ημν l μ l ν = 0. Scalar modes accompany the spin-two modes. If the potential V (φ) is zero in the action, l μ is a null vector, and the scalar ϕ-waves are massless and propagate at light speed. On the contrary, if V (φ) = 0, then ϕ-waves acquire mass. This means that, in scalar-tensor gravity, the scalar field generates a third polarization mode of the GWs. This happens because in scalar-tensor gravity, there are three different degrees of freedom (see [24]), while in GR there are only two. Finally, one gets ⎛

⎞ h+ h× 0   z  ⎝ ⎠ , hTT (t, z) = cos ω t − −h 0 h × + ij c 0 0 hs

(19)

where hs is the additional polarization mode that is depicted in Fig. 2. It is widely known that f (R)-gravity is equivalent to a scalar-tensor theory [15]. In fact, with the identification φ = f  (R) (where the prime indicates  the derivative with respect to the Ricci scalar) and V  (φ) = 2f (R) − Rf  (R) /3 (which also means putting f = f (R) and  = 0 in Eq. (1)), and linearizing the field equation of f (R)-gravity [19], one obtains that the function f  (R) generates a third massive polarization mode [15].

Fig. 2 Motion of point-like particles distributed on ring and due to a GW that propagates perpendicular to the plane of the ring. The scalar polarization mode hs is shown

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1559

More in general, higher-order theories predict the existence of the massless spintwo field (the standard graviton) and also of spin-0 and spin-2 massive modes with the latter being, in general, ghost modes [15]. It is important to notice that in Eq. (18)  → p ) ωmspin2 = m2spin2 + p2 kα ≡ (ωmspin2 , − (20)  → l ≡ (ω , − p) ω = m2 + p 2 . α

ms

ms

s

where mspin2 is zero (nonzero) in the case of massless (massive) spin-two mode and the polarization tensors can be found in Ref. [81] (see Equations (21)–(23)). The fact that the dispersion law for the modes of the massive/scalar field hs is not linear has to be emphasized. The velocity of every “ordinary” (i.e., which arises from GR) mode hμν is the speed of light, but the dispersion law (the second of Eq. (20)) for the modes of hs is that of a massive field which can be discussed like a wave packet [34]. The group velocity of such a wave packet centered in p is vG =

p , ω

(21)

which is exactly the velocity of a massive particle with mass m and momentum p. From Eqs. (20) and (21), it is simple to obtain √ ω 2 − m2 vG = . (22) ω Then, to have a constant speed of the wave packet, the following relation must hold [34]:  2 )ω. m = (1 − vG (23) Taking into account the frequency range at which GWs can be detected by both of space-based and ground-based gravitational wave detectors, that is, the interval 10−4 Hz ≤ f ≤ 10 kHz, a quite strong limitation will arise. For a massive GW, the following holds [34]:  2πf = ω = m2 + p2 , (24) where p is the momentum. Thus, it needs 0 ≤ m(eV ) ≤ 10−11 .

(25)

An even stronger bound is found once one imposes that ETGs must match the cosmological and solar system constraints: 0eV ≤ m ≤ 10−33 eV . For these light scalars, their effect can still be discussed as a coherent GW.

(26)

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M. De Laurentis and I. De Martino

Table 1 f (R) models having viable cosmological solutions f (R)-model f0 R n R − m2

c1 (R/m2 )n c2 (R/m2 )n + 1  R q

R + λRc

Rc

 R R + R∗ ln 1 + R∗ R + λRc



1+

 R 2 −p − 1 Rc2

Features Fits low surface brightness galaxy rotation curves

Ref. [18]

Satisfies both cosmological and solar system tests

[45]

Satisfies both cosmological and solar system tests Free of singularities

[59] [6]

Cosmological viable solutions distinguishable from ΛCDM; compatible with relativistic stars; free of singularities

[62]

Satisfies both cosmological and solar system tests

[79]

f (R)-Models The first attempt to describe the acceleration of the Universe extending GR is dated back to the 1980s [78]. Models such as f (R) = R + αR n , which include for n = 2 the Starobinsky’s model, give rise to quadratic corrections in the Ricci scalar that are particularly relevant in cosmology since they allow to explain the early acceleration of the Universe. Afterward, many models have been considered and tested. Some examples of models having viable cosmological solutions are listed in Table 1. These models are also studied as viable f (R)-gravity models to address the solar system tests and to generate the stochastic gravitational waves background [21]. Nevertheless, as an alternative approach, instead of fixing the Lagrangian, one may study f (R) theories that are analytic functions of the Ricci scalar, i.e., are expandable in Taylor’s series, as

f (R) =

+∞ n  f (R0 ) n=0

n!

(R − R0 )n = f0 + f0 R + f0 R 2 + f0 R 3 + ... ,

(27)

d n f (R)  where f n (R) = . This approach is more general with respect to fixing  dR n R=0 a particular form of the Lagrangian; indeed it allows to confirm or rule out large classes of models using data. In analytic f (R) models, the existence of stable cosmological solutions is guaranteed, for perfect fluids, only if f0 < 0 [23]. Moreover, in such a model, the graviton massive states can be defined as

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

  f0 mg = −  , 3f0

1561

(28)

and they are positively defined under the condition f0 < 0 discussed above.

The Chameleon Mechanism in f (R)-Gravity To make compatible f (R) models with local gravity constraints, a “chameleon mechanism” is usually required. When considering theories with a non-minimally coupled scalar field, one has to impose strong conditions on the effective mass of the scalar field so the tight constraints on the solar system are verified [55, 56]. Those f (R) models need to require mr 103 eV corresponds to a range λ < 0.2 mm) at solar system densities (ρ 1 − 10 g/cm−3 ), escaping the experimental constraints and have (0) a long range at cosmological densities (ρc 10−29 g/cm−3 ), and it can propagate freely affecting the cosmological dynamic and driving the accelerated expansion (see for a comprehensive review [28]). Similar mechanisms have been proposed in the literature from symmetron to braneworld models with the same purpose [36, 44, 65].

The Gravitational Wave Stress-Energy (Pseudo) Tensor The detection of gravitational waves and the information that can be extracted from them depend sensitively on the models used to extract these waves from the noise. Because of the way GW detectors work, the extraction relies on an accurate modeling of the rate of change of the GW phase and frequency. In binary systems, this is calculated from the balance law between the rate of change of the binary’s binding energy and the energy and momentum extracted by all propagating degrees of freedom. In turn, the latter is obtained from the GW stress-energy (pseudo) tensor, which can be calculated in a variety of ways. The GW stress-energy tensor was first found in GR by Isaacson in the late 1960s [48, 49] using the perturbed field equations method. This method consists of perturbing the Einstein field equations to second order in the metric perturbation with respect to a generic background. The first-order field equations describe the evolution of the gravitational radiation. The right-hand side of the second-order field equations yield the GW stress-energy tensor (see [72] for a detailed explanation). Landau and Lifshitz developed a method that consists of constructing a pseudo-tensor from tensor densities with certain symmetries such that its partial divergence vanishes, leading to a conservation law [58]. The final method to investigate is one that makes use of Noether’s theorem,

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M. De Laurentis and I. De Martino

which asserts that the diffeomorphism invariance of a theory automatically leads to the conservation of a tensor [66]. These methods give the same GW stress-energy tensor in GR, but not in other gravity theories. Scalar-tensor theories, originally proposed by Jordan [52], Fierz [42], and Brans and Dicke [16], and certain vectortensor theories, such as Einstein-Æther theory [50,51], or f (R)-theory [74,75], are two examples where a priori it is not obvious to obtain the same result. Adopting the general definition given by Landau and Lifshitz [29, 58], and imposing that the energy momentum tensor satisfies a conservation law as required by the Bianchi identities, one can write 1 tαλ = √ −g



∂L ∂L − ∂ξ ∂gρσ,λ ∂gρσ,λξ

 gρσ,α +

∂L gρσ,ξ α − δαλ L ∂gρσ,λξ

 . (29)

In GR, after short-wavelength averaging, and using the TT gauge, one obtains tβα GR =

1  α TT β γδ  ∂ hγ δ ∂ hT T . 32π G

(30)

Following the same procedure, in scalar-tensor theories, one finds ST tαβ

  φ0 1 TT γδ = ∂α θγ δ ∂β θTT + 2 (6 + 4ω) ∂α ϕ ∂β ϕ , 32π G φ0

(31)

where the “reduced field” was introduced (see [89] for derivation of this quantity) 1 1 θαβ = hαβ − g˜ αβ h − g˜ αβ ϕ . 2 φ˜

(32)

Here, g˜ αβ is the background metric, and φ˜ is a background scalar field. In the case of f (R)-gravity, one gets  tαλ

=

f0 k λ kα

  ρσ    ˙h h˙ ρσ − 1 f0 δαλ kρ kσ h¨ ρσ 2 . 2

(33)

One is interested in calculating physical and observable quantities. Two examples are the rate of energy and of the linear momentum carried away from any system as consequence of the emission of GWs: E˙ = −

 ∞

t 0i d 2 Si ,

P˙ i = −

 ∞

t ij d 2 Sj ,

(34)

where t αi is the (α, i) component of the GW stress-energy tensor. These observables can be simplified through the shortwave approximation, which assumes the characteristic wavelength of radiation λc is much shorter than the observer’s distance to the center of mass r, i.e., the observer is in the so-called

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1563

far-away wave zone so that r  λc . When this is true, the propagating fields can be expanded as [69] λc φL = f1,L (τ ) + r



λc r



2 f2,L (τ ) + O

λc r

3 ,

(35)

where τ = t − r/v is retarded time and v is the speed of propagation of the field. Moreover, the space-time (partial) derivative of the field then satisfies 1 ∂α φL = − kα ∂τ φL + O v



λ2c r2

 ,

(36)

where k α is a unit normal 4-vector normal to the r = const surface (kα ≡ (−1, Ni )) with Ni = xi /r and x i the spatial coordinates on the 2-sphere. With this at hand, one can now simplify the observable quantities E˙ and P˙ i . In GR, the rate at which energy and linear momentum are lost by the system is r2 E˙ GR = − 32π G i r2 = − 32π P˙GR G

 i hγ δ d 2 S , h˙ TT ∂ i TT γδ   j γδ 2 ∂ i hTT γ δ ∂ hTT d Sj , 

(37) (38)

where h˙ αβ is the partial derivative of hαβ with respect to coordinate time t and d 2 Si = R 2 Ni dΩ. Incorporating the shortwave approximation into Eq. (37) gives r2 E˙ GR = − 32π G r2 i P˙GR =− 32π G

 

 γδ dΩ , ∂ h ∂τ hTT τ TT γδ

(39a)

 

 γδ N i ∂τ hTT γ δ ∂τ hTT dΩ ,

(39b)

Applying the same procedure to the scalar-tensor theories, the rate of energy and linear momentum carried away by all propagating degrees of freedom is    6 + 4ω TT γδ ∂τ ϕ ∂τ ϕ dΩ , ∂τ θγ δ ∂τ θTT + φ02   2  r φ 6 + 4ω 0 i i TT γ δ P˙ST = − ∂τ ϕ ∂τ ϕ dΩ . N ∂τ θγ δ ∂τ θTT + 32π G φ02

φ0 r 2 E˙ ST = − 32π G

(40a)

(40b)

In the GR limit, rates of energy and momentum loss are identical to that of above. Finally, following again the same procedure, in f (R)-gravity, one obtains [29]: .... ....  G    ... ij ...  ij f0 Q Qij − f0 Q Q ij . E˙ f (R) = 60

(41)

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M. De Laurentis and I. De Martino

Here, the quadrupole formulation has been used (see for more details [29]). An important remark is necessary at this point. Equation (41) can be written as E˙ f (R) =

  Gf0  ... ij ...  1 ....ij ....  Q Qij − 2 Q Q ij . 60 m

(42)

where the massive mode contribution is evident. This means that this further term affects both the total energy release and the waveform. To proceed further, one would have to specify a specific physical system, to solve the field equations for the metric perturbation, and then insert the solution in the above equations to integrate them. From here on, the results obtained above will be developed only for f (R).

Application to Binary Systems and Observational Constraints Binary systems of compact objects are optimal laboratories to test gravity in the strong field limit [27]. Such systems could be composed of two compact objects such as neutron stars, white dwarfs, or black holes. During their gradual evolution, inspiraling binaries lose energy and angular momentum, causing a shrinking of the orbit [46, 76, 87]. Nevertheless, the system also loses linear momentum as first suggested by [10] which computed the emitted flux of linear momentum in linearized gravity showing that it is a quadrupole-octupole effect. The flux of linear momentum carried out by the GWs emission is of lower order than that of the energy and angular momentum.

Time Variation of the Orbital Period in General Relativity To simplify the presentation, let us assume that the orbits of both stars are Keplerian and they move on the (x, y)-plane. Let us denote by mp the mass of the pulsar, mc mp the reduced mass of the system. In the mc of its companion, and μ = mc + mp center-of-mass frame, the relative coordinates are x(t) = r(t) cos ψ ,

y(t) = r(t) sin ψ ,

z(t) = 0 ,

(43)

and the radial distance between the two bodies evolves as r(t) =

a(1 −  2 ) , 1 +  cos(ψ(t))

(44)

where ψ is the eccentric anomaly. Both magnitudes depend on time. The only nonzero components of the mass momentum are in the plane of the motion and can be written as

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

 Mij = μr

2

cos2 ψ sin ψ cos ψ sin ψ cos ψ sin2 ψ

1565

 ,

(45)

where i, j = (x, y) are the indexes in the orbital plane. The third derivative of the quadrupole must be computed in order to achieve the radiated power. The calculation is simplified by using the following relation [61]: ψ˙ =



Gmc a3

1  − 3 2 2 1 − 2 (1 +  cos ψ)2 ,

(46)

where a is the semi-major-axis and  the eccentricity of the orbit. Defining the quantity H1 =

(2π )5/3 G2/3 mc mp 5/2 , 5/3  1 − 2 Pb

(47)

the only third derivatives of the quadrupole different from zero are ... M 11 = H1 sin 2ψ( cos ψ + 1)2 (3 cos ψ + 4) , ... M 22 = −H1 (3 cos 2ψ + 5) + 8 cos ψ) sin ψ( cos ψ + 1)2 , ... M 12 = −H1 ( cos ψ + 1)2 (5 cos ψ + 3 cos 3ψ + 8 cos 2ψ) .

(48) (49) (50)

The average radiated power on an orbital period is dE 1 = dt Pb and since



Pb 0

1 dE(ψ) = dt dt Pb

 0

2π

dψ dE(ψ) , dt ψ˙

(51)

P˙b 3 E˙ , the time variation of the orbital period becomes =− Pb 2E 3 P˙b = − 15



Pb 2π

− 5 3

5 2  μG 3 (mc + mp ) 3  4 2 37 + 292 + 96 . 7 c5 (1 −  2 ) 2

(52)

The time derivative of the orbital period is the best determined post-Keplerian parameter. Consequently, this magnitude provides the best constraints on theories of gravitation [27].

Periastron Advance in General Relativity The geodesic equations in a space-time can be derived from the Lagrangian via the Euler-Lagrange’s equations [25, 84]:

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M. De Laurentis and I. De Martino

2L = gμν

dxμ dxν , dλ dλ

(53)

where λ is the affine parameter along the geodesic of the metric gμν . Considering the Schwarzschild metric (here the equations are in units c = G = 1) 

Rs ds = 1 − r



2

  Rs −1 2 dr − r 2 dΩ , dt − 1 − r 2

(54)

where Rs ≡ 2GM = 2M is the Schwarzschild radius, the Lagrangian associated c2 with the metric elements of Eq. (54) is     2M 2 2M −1 2 ˙ 2L = 1 − r˙ − r 2 θ˙ 2 − r 2 sin2 θ φ˙ 2 . t − 1− r r

(55)

Here, the dot denotes the differentiation with respect to the affine parameter. Then, the canonical momenta are   ∂L 2M pt ≡ = 1− t˙ , ∂ t˙ r   2M −1 ∂L = 1− r˙ , pr ≡ − ∂ r˙ r

(56) (57)

pθ ≡ −

∂L = r 2 θ˙ , ∂ θ˙

(58)

pφ ≡ −

∂L = r 2 sin2 θ φ˙ . ∂ φ˙

(59)

Next, the time component of the Euler-Lagrange equations reads d ds

 1−

2M r

 = 0.

(60)

  2M pt ≡ 1 − ≡E. r

(61)

The latter implies there is a conserved quantity:

Then, the φ component of the Euler-Lagrange equation reads d ∂L ∂L = 0, = ˙ ds ∂ φ ∂φ

(62)

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1567

and also leads to define a conserved quantity: pφ ≡ r 2 sin2 θ φ˙ ≡ L ,

(63)

where L is the angular momentum per unit mass of the two bodies. Furthermore, the equation for the θ component, which is not a conserved quantity, leads to d 2 (r θ˙ ) = r 2 φ˙ 2 sin θ cos θ. ds

(64)

Since θ˙ = 0 when the values π/2 and zero are assigned, and θ will remain constant at the assigned value, and the geodesic is described in invariant plane. Using the constants of motion defined in the above equations, one can obtain the following identity [25]:     2M −1 L2 2M −1 2 E 1− − 2 − r˙ 1 − = {+1; 0} . r r r 2

(65)

depending on whether time-like or null geodesics are considered. For time-like geodesics, there are two integrals of motion: 

dr ds

2

   L2 2M 1 + 2 = E2 , + 1− r r

(66)

and L2 dφ = 2 . ds r

(67)

By considering r as a function of φ instead of s, the equation of motion reads 

dr dφ

2 = (E 2 − 1)

r4 2M + 2 r 3 − r 2 + 2Mr . 2 L L

(68)

Next, introducing the change of variable u ≡ 1/r, the previous equation can be recast as 

du dφ

2 =−

(1 − E 2 ) 2M + 2 u − u2 + 2Mu3 , L2 L

(69)

that can be solved once the equations for the proper and coordinate time are considered: dτ 1 = , dφ Lu2

(70)

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M. De Laurentis and I. De Martino

and E dt = . dφ Lu2 (1 − 2Mu)

(71)

An example of the numerical solution of the above equations is shown in Fig. 3, which also shows the periastron advance of the orbital plane. The solution of Eq. (69) depends on whether one is looking for bound orbits (E 2 < 1) or unbound orbits (E 2 > 1). For a comprehensive discussion on all possible solutions, the reader may refer to [25]. For bound orbits, one can make the following ansatz: u≡

1 1 = (1 + e cos χ ) , r p

(72)

where χ is the relativistic anomaly, p is the semi-latus rectum, and e is the eccentricity. Applying the following conditions

Fig. 3 Numerical solution of the geodesic equations illustrating the periastron advance in GR. The coordinates x and y in the equatorial plane are given in units of GM/c2 . The black dot point indicates the central object

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1569

χ = π −→ u =

1−e p

(73)

χ = 0 −→ u =

1+e , p

(74)

one finds an equation for the evolution of the relativistic anomaly: 

dχ dφ

 = −(1 − 6μ + 2μe) − 4μecos 2 (χ /2) .

(75)

Focusing on the first-order corrections to the Keplerian orbits, one must expand the equation for the relativistic anomaly to first order in μ and consider that after one complete revolution, the φ changes by 2π . Therefore, the advance of the perihelion per revolution of a point-like particle orbiting around a massive spherical object reads ΔφGR =

6π GM . c2 a(1 − e2 )

(76)

Binary systems composed by double pulsars or by a pulsar and a companion star provide an excellent laboratory to study both GR and alternative theories of gravity. Pulsars act as very precise clocks. Therefore, the monitoring of a pulsar allows to measure the time of arrival (TOA) of pulses at the telescopes and to obtain the pulse profile. In case the pulsar is part of a binary system, the pulse profile shows a periodic variation related to the orbital motion around the center of mass of the binary system, and it needs to be modeled. Starting from Eq. (76), a straightforward calculation of the precession (for more details, see [77]) gives the rate of precession in GR  ω˙ GR =

2π Pb

5/3

G2/3 (mp + mc )2/3   . c2 1 − e2

(77)

Other Post-Keplerian Parameters and Observational Constraints on General Relativity Post-Keplerian parameters reflect the departure from the Keplerian motion due to relativistic effects in binary motion [77]. In addition to the orbital decay in Eq. (52), and the orbital precession in Eq. (77), there is the parameter γ that encodes the combination of gravitational redshift and transverse Doppler shift:

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M. De Laurentis and I. De Martino



Pb γ =e 2π

1/3 

GM c3

2/3

mc (mp + 2mc )(mp + mc )−4/3 .

(78)

Two additional parameters are r=

GM mc , c3

s ≡ sin i =

    a sin i Pb −2/3 GM −1/3 (mp + mc )2/3 , cmc 2π c3

(79) (80)

which encode the Shapiro delay due to the curvature of the space-time surrounding the companion star. Three decades of timing measurements of PSR B1913+16 (the first known binary pulsar) led to strong evidences that gravitational waves existed and were emitted as predicted by Einstein’s theory. Looking at the left panel of Fig. 4, the orbital decay due to the loss of energy by GWs fits incredibly well the observations showing a strong agreement with GR. Moreover, the right panel of Fig. 4 shows the constraints on the masses of the pulsar and the companion star obtained using the five postKeplerian parameters, mp = 1.438 ± 0.001M , and mc = 1.390 ± 0.001M , respectively. Using the full information led to P˙bobs /P˙bGR = 0.9983 ± 0.0016 [85], which strongly confirms the prediction of GR.

Fig. 4 Left Panel: the plot depicts the orbital decay from GR (solid line), and compare it with data. Figure is taken from [86]. Right Panel: The plot shows the capability of the five post-Keplerian parameters to constrain the masses of the stars of the binary system. The colored bands are the 1σ errores. (Figure is taken from [85])

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1571

First-Time Derivative of the Orbital Period in f (R)-Gravity To make concrete prediction on f (R)-gravity, one may follow the GR case in section “Time Variation of the Orbital Period in General Relativity” [30, 31]. The power radiated in the form of GWs is given by Eq. (41). Therefore, using the quadrupole matrix, given by Eq. (45), the radius of the orbit (Eq. 44), and its eccentric anomaly (Eq. 46), the only nonzero time derivatives of degrees three and four in Eq. (41) are ... Q11 = H1 sin 2ψ( cos ψ + 1)2 (3 cos ψ + 4), ... Q22 = −H1 (8 cos ψ + (3 cos 2ψ + 5)) × sin ψ( cos ψ + 1)2 , ... Q12 = −H1 ( cos ψ + 1)2 × (5 cos ψ + 3 cos 3ψ + 8 cos 2ψ),

(81) (82) (83)

and   ....

Q 11 =H2 15 2 cos 4ψ + 50 cos 3ψ + 12 2 + 32 cos 2ψ+  6 cos ψ − 3 2 , (84)   ....

Q 22 = − H2 15 2 cos 4ψ + 50 cos 3ψ + 24 2 + 32 cos 2ψ+  14 cos ψ − 7 2 , (85)     .... Q 12 =2H2 sin ψ 15 2 cos 3ψ + 50 cos 2ψ + 33 2 + 32 cos ψ + 30 , (86) where H1 is defined as in Eq. (47) and H2 as H2 =

22/3 π 8/3 G2/3 mc mp ( cos ψ + 1)3 . 4 √ 8/3  2  − 1 3 mc + mp Pb

(87)

Averaging the radiated power over an orbital period, and using Eq. (41) and, again, P˙b 3 E˙ , the time variation of the orbital period is recast as [30] the relation =− Pb 2E 2 5   5    3 Pb − 3 μG 3 (mc + mp ) 3  4 2 ˙ Pb = − 37 × f + 292 + 96 0 7 20 2π c5 (1 −  2 ) 2

 f0 π 2 Pb−1  8 6 4 2 891 + 28016 + 82736 + 43520 + 3072 . − 3 2(1 +  2 )

(88)

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M. De Laurentis and I. De Martino

Before concluding, let us remark that if one sets f0 = 0, (i.e., f (R) = R), then one recovers the GR result of Eq. (52).

Observational Constraints on f (R)-Gravity from Binary Systems A sample of binary system whose variation of the orbital period was measured and has been used to constrain f (R)-gravity by comparing theoretical prediction with observations [30, 31]. All the observed properties together with the GR prediction of the time derivative of the period (P˙GR ) are given in Table 2. The galactic or Shklovskii effect (There exist several kinematic effects that need to be considered: first, the Shklovskii effect, related to the proper motion of the binary stars [57, 73]: 2

μ d . P˙bShk = Pb c

(89)

Second, the difference of the Galactic accelerations between the Sun and the system gives an apparent period increment ac P˙bAcc = Pb , c

(90)

where ac is the acceleration of the binary system around the galactic center.) has been subtracted where available. For the first five systems, the masses have been determined reliably, while for the remaining sample, masses were estimated taking mp = 1.4M and assuming either i = 60◦ or i = 90◦ . The last two columns show the constraints on f0 and on the error on f0 . Those results show that for most binary systems, the value of f0 is very different of zero (by many order of magnitudes). For those systems, one can only conclude that external factors like mass transfer, uncertainties on orbital parameters, or the mass of the stars are large and need to be taken into account. Additionally, higher-order terms in the Lagrangian or the induced scalar fields give rise to massive gravitons in a natural way [15], adding extra polarization modes that could potentially be detected on CMB. The binary pulsar PSR J 0348 + 0432, which is composed of a pulsar spinning at 39 ms with mass mp = 2.01 ± 0.04M and a WD companion with mass mc = 0.172 ± 0.003M , has been used to provide an upper limit on the mass of the graviton [7]. Using the range of the allowed values of f0 , and following [43], the upper limit on the graviton mass is [31] 24 F (e) m ≤ 5 2



2π h¯ c2 Pb

2 ˙ Pbobs − P˙bf (R) < 5.95 × 10−20 eV/c2 , P˙bf (R)

(91)

where F (e) =

37 4 2 1 + 73 24 e + 96 e , (1 − e2 )3

(92)

a We

(lsec) 2.518 2.342 1.415 1.859 3.729 0.343 0.397 0.040 0.253 0.045 1.899 20.044 0.089 10.164 3.367 174.257 38.768 64.809

(degrees) – 54.12◦ 88.69◦ 73◦ 78.4◦ 32.6◦ 65.8◦ 60◦ 84.7◦ 30◦ 86.4◦ < 47◦ 65◦ 137.58◦ 44◦ 63◦ 40◦ 0.681 0.617 0.088 0.171 0.274 3.4E-7 7.1E-7 0 0 0 1.30E-7 0.249 0 1.93E-5 1.92E-5 0.808 1.11E-4 0.025

refer to Ref. [30] for more details on how data were collected

J2129+1210C J1915+1606 J0737-3039A J1141-6545 J1537+1155 J1738+0333 J0751+1807 J0024-7204J J1701-3006B J2051-0827 J1909-3744 J1518+4904 J1959+2048 J2145-0750 J0437-4715 J0045-7319 J2019+2425 J1623-2631

(days) 0.335 0.323 0.102 0.198 0.420 0.355 0.263 0.121 0.144 0.099 1.533 8.634 0.382 6.839 5.741 51.169 76.511 191.443

(10−12 ) −3.96 −2.423 −1.252 −0.403 −0.138 −0.017 −0.031 −0.55 −5.12 −15.5 −0.55 0.24 14.7 0.4 0.159 −3.03E+5 −30.0 400.0

(10−12 ) −3.94 −2.403 −1.248 −0.387 −0.192 −0.0277 −0.017 −0.03 −0.09 −0.03 -0.003 −0.001 −0.003 −0.0005 −0.0004 −0.02242 −0.000006 −0.000003

(10−12 ) 0.05 0.001 0.017 0.025 0.0001 0.0031 0.009 0.13 0.062 0.8 0.03 0.22 0.8 0.3 0.283 9E+3 60.0 600.0 (M ) 1.358 1.4398 1.3381 1.27 1.3332 1.46 1.7 1.4 1.4 1.4 1.57 1.56 1.4 1.4 1.76 1.4 1.33 1.3

(M ) 1.354 1.3886 1.2489 1.02 1.3452 0.181 0.67 0.024 0.14 0.027 0.212 1.05 0.022 0.5 0.254 8.8 0.35 0.8

0.04 0.10 0.23 4.25 −37.90 −14.7 −157.0 1670 5710 3.24E+4 2.09E+05 −7.05E+5 −1.38E+6 −4.00E+6 −1.57E+6 2.74E+9 2.71E+11 −1.98E+13

0

0.08 0.05 0.09 6.44 0.07 29.2 10.02 415 70.4 1.68E+3 1.14E+04 6.43E+3 7.51E+4 2.99E+6 2.73E+6 8.13E+7 5.41E+11 2.97E+13

0

Table 2 Observational data on binary pulsars. The columns, from left to write, are name, orbital period Tb (in days), projected semi-major axis a(sin i) (in light seconds), eccentricity , time variation of the period P˙Obs , GR prediction P˙GR , observational uncertainty ±δ of P˙Obs , and mass of the components mp and mc (in solar masses). Then, there included our results: the difference between P˙bObs and T˙GR , the correction P˙bf (R) , f0 , and, finally, the error on f0 P˙bObs P˙GR Namea Tb a i  ±δ mp mc f  σf 

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories 1573

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which is comparable to the constraint derived from PSR B1913 + 16 and similar to the one obtained by [7] in Brans-Dicke theory.

Periastron Advance in f (R)-Gravity The starting point to periastron advance in f (R)-gravity is the line element of the post-Newtonian limit [32]: ds 2 = [1 + ΦYuk (r)] dt 2 − [1 − ΦYuk (r)] dr 2 − r 2 dΩ ,

(93)

where the gravitational potential is (for a comprehensive review, see [19, 20]) ΦYuk (r) = −

 r GM  1 + δe− λ , (1 + δ)r

(94)

 . where it holds the definitions 1 + δ = f0 , and λ = −6f0 /f0 . To compute the geodesic equations, one can use the Euler-Lagrange equations: d ∂L ∂L − μ = 0, μ ds ∂ x˙ ∂x

(95)

that are equivalent to the geodesic equations [84] and obtain [32] r¨ =

      r r 1 RS r˙ 2 − t˙2 δ(λ + r) + e λ λ + e λ λ(1 + δ)r 3 θ˙ 2 + sin2 θ φ˙ 2 , Δ (96)

θ¨ = cos θ sin θ φ˙ 2 −

2˙r θ˙ , r

 2φ˙

r˙ + cot θ r θ˙ , r   r    t¨ =Δ−1 2RS e λ + δ λ + δr r˙ t˙ ,

φ¨ = −

(97)

(98)

(99)

where, for the sake of convenience, it is defined   r Δ ≡ λr 2RS δ + e λ (2RS − (1 + δ)r) .

(100)

The above equations can be integrated numerically to obtain the orbital motion and precession of a two-body system. See, for instance, Fig. 5 which compares the GR

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1575

Fig. 5 Numerical solution of the geodesic equations in f (R)-gravity. The orbit was obtained using the same system of Fig. 3. Gray line depicts the GR solution, while the red line represents the solution in f (R)-gravity for δ = 0.3 and λ = 20000 AU

and f (R)-gravity solutions showing the difference in the periastron advance [32, 33]. Following the same approach described in section “Periastron Advance in General Relativity”, one gets an explicit equation for r˙ 2 [32]: r˙ 2 =

L2 [ΦYuk (r) + 1] − E 2 r 2 . r 2 [ΦYuk (r) − 1] [ΦYuk (r) + 1]

(101)

Next, doing the change of variable u = 1/r, imposing (du/dφ)2 = 0, and using Eqs. (72), (73), and (74), one finally obtains an equation for the relativistic anomaly [32]: 

dχ dφ

2

 2μ2 (e cos χ + 1)    (Υ + 1) + = 1 − e2 + 3 μ + 2μ(e cos χ + 1)2 δ+1   + e2 − 1 (1 − 4μ)μ2 − μ2 (e cos χ + 1)2 , (102)

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where we have defined the auxiliary variable  1 1 +1 . Υ =δ − λμ(e cos χ + 1) 2λ2 μ2 (e cos χ + 1)2 

(103)

To get an analytical solution, one may expand in Taylor series the exponential e up to the second order. (Expanding in Taylor’s series restricts the use of the results to the cases in which the semi-major axis of the orbit is much lower than the Yukawa scale length.) The integration of Eq. (102) leads to the expression for the periastron advance (which reduces to Eq. (76) for δ = 0) [32]: 1 − λu

Δφ =

 2δG2 M 2 2π δG2 M 2 ΔφGR 3δGM   −   1+ −  2 4 2 2 (δ + 1) ac 1 − e λ ac 1 − e2 3a 2 c4 1 − e2  δG2 M 2 δGM − 4 . (104) + 6c (δ + 1)λ2 3λc2

The above equation can be recast for binary systems as ω˙ =

 4/3 4/3  ω˙ GR 2π 2δ G 1+  (mp + mc )4/3 2 4 2 (δ + 1) P c b 1−e  2/3 5/3 2π G 2δ  − (mp + mc )2/3 c4 1 − e2 λ Pb  2/3 2/3 2π G 2δ  − (mp + mc )2/3 2 Pb c2 1−e  δ G2 δG 2 − 2 4 (mp + mc ) + (mp + mc ) , λ c2 2λ c

(105)

Constraining Alternative Theories of Gravity Using GW150914 and GW151226 The recently reported gravitational wave events GW150914 and GW151226 caused by the mergers of binary black holes [1–3] provide a formidable way to set constraints on alternative metric theories of gravity in the strong field regime. In particular, the signal from GW150914 exhibits the typical behavior predicted by the coalescence of compact systems where inspiral, merger, and ringdown phases are traversed [4]. The LIGO-VIRGO collaboration has analyzed the three regimes adopting a parametrized analytical family of inspiral-merger-ringdown waveforms [8, 13, 14, 60, 64, 91]. The signal is divided in terms of frequency: the early to late inspiral regime from ∼20 to ∼55 Hz; the intermediate region from ∼55 to

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

1577

∼130 Hz; and the merger-ringdown region from ∼130 Hz until the end of the waveform. The simplest and fastest parameterized waveform model that is currently available [47] sets bounds on the physical effects based on the inspiral phase only, where a calibrated post-Newtonian (PN) treatment is sufficient. For the later phases, phenomenological coefficients fitted to numerical relativity (NR) waveforms are used. In this part, we will show an approach with the possibility to set constraints on extended theories of gravity via the modified inspiral phase. It is worth noting that the existence of GWs confirms metric theories of gravity, among them general relativity (GR), but there is ample room for other possibilities (see [1, 35] for a detailed discussion). Any extended theory of gravity can be parameterized by means of a suitable post-Newtonian parametrization where the governing parameter is the effective gravitational coupling constant Geff and the effective gravitational potential Φ(r). Both these quantities are functions of the radial coordinate that influence the phase of the GW signal. In other words, the GW waveform could, in principle, single out the range of possible gravitational metric theories that are in agreement with the data. In particular, one can assume that under certain conditions, the potential in Eq. (94) becomes ΦYuk (r) = −

 r Geff M GM  1 + δe− λ − , (1 + δ)r r

(106)

where Geff GN (1 + α) ,

(107)

it is possible to constrain Geff and Φ(r) by the GW parameters reported for the events GW150914 and GW151226. Before this, let us review the post-Newtonian approximation required to perform this kind of analysis. Specifically, let us compute the 3.5PN approximation relevant for our analysis [61]. In particular, PN waveform models at the 3.5PN order are developed, e.g., in [17]. To compare the theoretical waveforms with experimental sensitivities, it is useful to write the Fourier transform of the two GW strains h+ , h× as h+ = Ae

iφ+ (f ) c



r

h× = Aeiφ× (f )

c r



Geff M c3 Geff M c3

5 6



1 7

f6 5 6

1 7

1 + cos2 i 2

cos i ,

 ,

(108)

(109)

f6

where i is the inclination angle of the line of sight and the constant A has the value A=

1 π

2 3



5 24

1 2

.

(110)

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The phase φ+ is given as  − 5  3 3 Geff M r π − ϕc − + φ+ (f ) = 2πf tc + 8πf , 3 c 4 4 c

(111)

where ϕc and tc are the value of the phase and the time at coalescence, respectively. Furthermore, the phases of the two strains are directly related, φ× = φ+ + π2 . An accurate computation of the phase going well beyond the Newtonian approximation is crucial for discriminating the signal of a coalescing binary from the noise. Therefore, one has to give the PN correction to the phase (111). In order to exploit the signal present in the detector, and thus detect sources at further distance, an accurate theoretical prediction on the time evolution of the waveform is required. In order to calculate the PN corrections, one writes the equation of motion in a more general form:     dv i xi 1 Geff M i (1 + A ) + Bv + O , =− 2 dt r r c8

(112)

such that it has a term proportional to the relative separation x i and a term proportional to the relative velocity v i in the center of mass frame. Here, the effective gravitational constant is not given by the standard Newton constant, but by Geff = GN (1 + α). Explicit expressions for the functions A and B are extremely long and are given in Ref. [63]. Here, it is important to address the issue of how to obtain constraints from GW150914 and GW151226 data in the framework of the frequency-domain waveform model [4]. One can proceed by considering the following relation for the frequency-domain phase:  5 7 i   3 Mf Geff − 3  π Mf Geff 3 π φ = 2πf tc −ϕc − + ϕ (Θ) π , i 4 128η c3 c3

(113)

i=0

where ϕi (Θ) are the PN expansion coefficients that are functions of the intrinsic binary parameters. The information on the spin χi (with i = 1, 2) is incorporated via the following relations: χs =

(χ1 + χ2 ) , 2

χa =

(χ1 − χ2 ) , 2

(114)

that appear in the functions ϕi (Θ). The 3.5PN expansion coefficients are ϕ0 = 1,

(115)

ϕ1 = 0,

(116)

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

3715 55η + , 756 9   113δχa 113 76η χs , ϕ3 = −16π + + − 3 3 3

ϕ2 =

1579

(117) (118)

  405 15293365 27145η 3085η2 + + + − + 200η χa2 508032 504 72 8   405 405 5η δχa χs + − + χs2 , − (119) 4 8 2       38645π 65π η 732985 140η Geff Mf − +δ − − χa ϕ5 = 1 + log π 756 9 2268 9 c3    732985 24260η 340η2 + + χs , + − 2268 81 9 (120)   2 2 640π 15737765635 2255π 11583231236531 6848γE − − + − + η ϕ6 = 4694215680 21 3 3048192 12   127825η3 6848 Geff Mf 76055η2 − − log 64π + 1728 1296 63 c3   2270 2270π π δχa + − 520π η χs , + 3 3 (121) ϕ4 =

378515π η 74045π η2 77096675π + − 254016 1512 756   25150083775 26804935η 1985η2 + − χa +δ − 3048192 6048 48   5345η3 25150083775 10566655595η 1042165η2 + − + χs . + − 3048192 762048 3024 36

ϕ7 =

(122) where ϕ0 , . . . , ϕ7 indicate the 0, . . . , 3.5PN approximation, respectively, and γE = 0.577 is the Euler-Mascheroni constant [91] and where the common definitions are used: δ=

(m1 −m2 ) , M

η=

(m1 m2 ) M

(123)

where m1 and m2 are the masses of the two compact objects. In Figs. 6 and 3 are plotted the frequency domain phase representations of GW150914 and GW151226

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and show the effect of varying the δϕi parameters as provided by the single parameter analysis of [2–4]. Note that have been followed their naming convention in introducing the quantities ϕ5l and ϕ6l that contain the logarithmic dependence with frequency. In addition, the variation with the leading order deviation α = ±10−2 is shown. The single parameter analysis of [2] was performed by setting all but the considered δϕi to 0. In contrast, the multiple parameter analysis was done by allowing all δϕi to vary freely. The latter leads to an error that is almost one order of magnitude larger due to the additional degrees of freedom. The GW150914 and GW151226 data for the multiple parameter analysis performed in [2, 4] due to the large error bars in the δϕi parameters were not considered. For the masses, the values given in [3] with m1 = 36.2M , m2 = 29.1M for GW150914 have been used, while m1 = 14.2M , m2 = 7.5M for GW151226. The initial spins were only constrained to be less than χ1 < 0.7, χ2 < 0.8 [3], and for our analysis, values of χ1 = 0.7, χ2 = 0.8 have been taken. Additionally, we adopted the values tc = 0.43s for GW150914 and tc = 1s for GW151226 [1, 2, 4] and set ϕc = 0 for both. Furthermore, the sensitivity of the results by varying the initial masses m1 , m2 and initial spins χ1 , χ2 within the errors and ranges obtained by [3] is studied. We found that the resulting changes were mostly quantitative, such as altering the slopes of the curves, and that the qualitative behaviors of the curves, such as the width of the constraints given by different α, remained unchanged. Thus, the results plotted in Figs. 6 and 3 are representative of the possible physical parameters reported in [3]. In Fig. 6, the frequency-domain phase representation for GW150914 and as shaded blue area the constraints due to the δϕi of the combined events, as provided by [3] and given in Table 3, is shown. The GR evolution is shown as a black solid line, while the extended theory is marked in the range of α ∈ [−10−2 , +10−2 ] as red curves. It has been reported on the early inspiral range f ∈ [20, 90]Hz and zoomed in on the range f ∈ [80, 90]Hz in the inset. As on can observe, at all the parameter orders, the single parameter analysis does not rule out |α| < 10−2 . The data of the second event, GW151226, is shown in Fig. 3; due to the lower mass involved in the merger, the frequency is much higher. Consequently, the inspiral regime lasts until 450Hz, and the phase in the range f ∈ [40, 200]Hz is shown. Inspecting Equations (120) and (121), we see that the variation with α is more pronounced for objects with lower total mass M. Hence, in principle, this event could yield tighter constraints on the value of the allowed α (Fig. 7). In order to investigate the required tolerances, in Fig. 8, we plot two PN orders, 0 and 4 where, for GW151226, we have decreased the variations by factors of 2, 5 and 10. For the PN terms of order 0, an increase by an order of magnitude would be sufficient to set constraints on α. More promising still are the higher-order terms which, even with a factor of 5 improvement, would be able to set constraints on |α| < 10−3 . We stress that these results are obtained from the single parameter analysis, while a more correct treatment would have to adopt the uncertainties of the multiple parameter analysis. Furthermore, we expect that as more GWs are detected, the

220

220

200

200

180

180

ψ

ψ

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

160 140

GR

20

30

40

50

60

f [Hz]

70

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160

100

30

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60

f [Hz]

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30

40

50

60

f [Hz]

70

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90

90

GR

20

30

40

50

60

f [Hz]

70

80

90

α = −0.01 α =0 .01 δϕ 5l = −0.13 δϕ 5l =0 .53 GR

20

30

40

50

60

f [Hz]

70

80

90

160 α = −0.01 α =0 .01 δϕ 7 = −0.25 δϕ 7 =3 .07

120

GR

20

80

α = −0.01 α =0 .01 δϕ 3 = −0.07 δϕ 3 =0 .17

140

α = −0.01 α =0 .01 δϕ 6l = −0.87 δϕ 6l =5 .99

120

f [Hz]

70

160

220

140

60

120

GR

20

50

140

α = −0.01 α =0 .01 δϕ 4 = −1.42 δϕ 4 =0 .21

120

40

160

220

140

30

120

GR

20

GR

20

140

α = −0.01 α =0 .01 δϕ 2 = −0.26 δϕ 2 =0 .07

120

ψ

100

90

220

140

ψ

80

α = −0.01 α =0 .01 δϕ 1 = −0.08 δϕ 1 =0 .49

120

ψ

ψ

100

160 140

α = −0.01 α =0 .01 δϕ 0 = −0.15 δϕ 0 =0 .03

120

1581

100

GR

20

30

40

50

60

f [Hz]

70

80

90

Fig. 6 Frequency-domain phase representation for GW150914 with masses m1 = 36.2M , m2 = 29.1M and initial spins χ1 < 0.7, χ2 < 0.8 [3] . The solid black curve is GR prediction where α = 0 and δϕi = 0. The red lines are α = ±0.01. The shaded blue area is the range allowed for δϕi parameter in accordance with Table 1 of [4]. From left to right: the first on the left column shows the phase at 0PN order, and the right one is for the 0.5PN order. The second on the left column is 1PN, while on the right, there is the 1.5PN. The third on the left column represents the phase at 2PN. On the right column, the phase at 2.5PN is shown. Finally, the fourth on left column shows the 3PN and on the right 3.5PN. Note that the error on the δϕ7 is so large that it falls outside the scale. The inset frequency ranges from 80 Hz to 90 Hz illustrating the curves at these frequencies

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Table 3 We report the frequency dependence of each parameter of Figure 6 in [3], median, and 90% credible regions. For each parameter, we report the corresponding quantities for the combined signals of GW150914 and GW151226 analyses as in [3, 4] Waveform regime

Parameter

f −Dependence

Median GW150914+GW151226

Early-inspiral regime

δϕ0

f −5/3

−0.05+0.08 −0.1

δϕ1

f −4/3

δϕ2

f −1

δϕ3

f −2/3

δϕ4

f −1/3

δϕ5l

log(f )

δϕ6

f 1/3

δϕ6l

f 1/3 log(f )

δϕ7

f 2/3

0.18+0.31 −0.26

−0.05+0.12 −0.21 0.11+0.06 −0.18

−0.6+0.81 −0.82

0.27+0.26 −0.4

−0.38+0.49 −0.72

2.66+3.33 −3.53 1.48+1.59 −1.73

statistics on the combined posterior density distributions [3] will improve, and we will be able to set stronger constraints on these types of alternative theories of gravity in the near future.

Constraints from the Shapiro Delay In this section are obtained constraints from the Shapiro delay using the relative time difference between observations at multiple frequencies. This allows one to infer violations of the equivalence principle using the observed time delay from astrophysical particle messengers like photons, gravitons, or neutrinos [53, 83, 90]. To date, the strongest constraints on the frequency dependence of the PPN-γ parameter are obtained by observations of fast radio bursts yielding Δγ (f ) ∼ 10−9 . In the case of fast radio bursts, the largest uncertainty is the signal dispersion due to the poorly understood line-of-sight free electron population [83]. The fact that this uncertainty is completely avoided for gravitational waves makes them an appealing messenger to test equivalence principle violations. The Shapiro gravitational time delay is caused by the slowing passage of light as it moves through a gravitational potential (106) as Δtgrav = −

1+γ c3



ro

Φ(r)dr ,

(124)

re

where γ is the (theory dependent) PPN parameter and ro and re are the positions of the observer and the source of emission. Let us conservatively assume a short burst of emission, that is, all wave frequencies are emitted at the same instant. Now, given the observed signal duration for GW150914 of ∼ 0.2s, we can obtain an estimate for

1200

1200

1000

1000

ψ

ψ

38 Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories

800 α = −0.01 α = 0.01 δϕ0 = −0.15 δϕ0 = 0.03

600

400 80

100

120

f [Hz]

140

160

1000

1000

800 α = −0.01 α = 0.01 δϕ2 = −0.26 δϕ2 = 0.07

80

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120

f [Hz]

140

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200

1000

1000

800 α = −0.01 α = 0.01 δϕ4 = −1.42 δϕ4 = 0.21

60

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f [Hz]

140

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ψ

1000

α = −0.01 α = 0.01 δϕ6l = −0.87 δϕ6l = 5.99

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800 α = −0.01 α = 0.01 δϕ7 = −0.25 δϕ7 = 3.07

400 180

180

GR

60

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GR

40

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α = −0.01 α = 0.01 δϕ5l = −0.13 δϕ5l = 0.53

40

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400 180

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f [Hz]

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40

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α = −0.01 α = 0.01 δϕ3 = −0.07 δϕ3 = 0.17

40

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400 180

ψ

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60

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GR

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α = −0.01 α = 0.01 δϕ1 = −0.08 δϕ1 = 0.49

400 180

ψ

ψ

60

800

600

GR

40

1583

GR

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f [Hz]

140

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180

200

Fig. 7 Frequency-domain phase representation for GW151226 with masses m1 =14.2M , m2 = 7.5M and initial spins χ1 < 0.7, χ2 < 0.8 [2, 3] . The solid black curve is GR prediction where α = 0 and δϕi = 0. The red lines are α = ±0.01. The shaded blue area is the range allowed for δϕi parameter in accordance with Table 1 of [4]. From left to right: the first on the left column shows the phase at 0PN order, and the right one is for the 0.5PN order. The second on the left column is 1PN, while on the right, there is the 1.5PN. The third on the left column represents the phase at 2PN. On the right column, the phase at 2.5PN is shown. Finally, the fourth on the left column shows the 3PN and on the right 3.5PN. The inset shows the frequency from 180 to 190 Hz

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430

430

|α| = 0.01 |α| = 0.001 δϕ0 = −0.075 δϕ0 = 0.015

428 426

|α| = 0.01 |α| = 0.001 δϕ4 = −0.071 δϕ4 = 0.0105

428 426

GR

ψ

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ψ

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36

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44

|α| = 0.001 δϕ0 = −0.03 δϕ0 = 0.006

426

38

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430

|α| = 0.01

428

36

|α| = 0.01 |α| = 0.001 δϕ4 = −0.0284 δϕ4 = 0.0042

428 426

GR

GR

ψ

424

ψ

424

44

422

422

420

420

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418

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42

430

44

416

|α| = 0.001 δϕ0 = −0.015 δϕ0 = 0.003

426

38

40

f [Hz]

42

430

|α| = 0.01

428

36

|α| = 0.01 |α| = 0.001 δϕ4 = −0.0142 δϕ4 = 0.0021

428 426

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GR

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424

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Fig. 8 Two PN orders for GW151226, 0 (left) and 4 (right), with errors in δϕi improved (from top to bottom) by factors of 2, 5, and 10. The scales have increased in order to more clearly show the curves

the frequency dependence of γ , respectively, α. In the absence of other dispersive propagation effects, e.g., due to Lorentz invariance violation (see also [35, 90]), an upper limit for Δα/Δf is obtained. For example, in scalar tensor theories, the γ PPN parameter is expressed in terms of non-minimal coupling function of a scalar field, equivalently, in terms of the α parameter, that is (for more details see [22]),

γ −1 = −

(f  (ϕ))2 α2 = −2 , f (ϕ) + 2[f  (ϕ)]2 1 + α2

(125)

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In this case, the delay (124) takes the form Δtgrav

2α 2 = −(2 − )/c3 1 + α2



ro

ΦN (r)(1 + αe−r/r1 )dr ,

(126)

re

of which the most important contribution comes from the term linear in α:  Δtgrav −2α/c3

ro

ΦN (r)e−r/r1 dr .

(127)

re

It is evident that for r1  re , ro , the value for Δα/Δf is just half the constraint that can be set on Δγ /Δf using the usual Shapiro delay. Thus, with the same assumptions for the potential encountered by the gravitons as [54] (corresponding to a Shapiro delay of 1800 days), we can set the limit |α(250Hz) − α(35Hz)| < 1.3 × 10−9 . It is important to note that this seemingly tight limit relates to the frequency dependence only, e.g., if we have α = α0 + α(f ), the constant term α0 is entirely unconstrained by this experiment. However, knowledge of Δα/Δf can be used to extrapolate measurements of the absolute value of α across the spectrum and thus extend their range of validity.

Conclusions The study of the generation, propagation, and detection of gravitational waves in the weak-field limit of a given relativistic theory of gravity is an important part of astrophysics. In this chapter is shown one of the methods to calculate the GW stress-energy tensor in higher-order theories of gravity, from which is calculated the rate of energy loss. The study has been developed using the “linearized theory” which consists in expanding the field equations around the flat Minkowski metric. The post-Minkowskian limit of ETGs exhibits new gravitational field modes which can easily be interpreted as massive gravitons. The field equations then reduce to linear wave equations from which radiation can be calculated. GR predicts radiation that, at the lowest order, is proportional to the third derivative of the quadrupole momentum of the mass-energy distribution. It is a consequence of conservation equations that the first derivative of the monopole momentum and the second derivative of the dipole momentum are zero. This means that the gravitational radiation is first seen at the quadrupole term. The dipole effects depend on the difference of the self-gravitational binding energy per unit mass for two bodies, and it is thus dependent also on the internal structures of the objects. When the objects are in circular orbits, the time variation of the scalar field at each object, due to the motion of the other, is zero, and the dipole contributions consequently drop out. Under these circumstances, the dominant surviving terms are of quadrupole order. In higher-order theory of gravity, as has been shown, the situation can differ due to the presence of further degrees of freedom of the gravitational field. However, GR has to

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be fully recovered. This “compatibility” with GR could be a test-bed for these ETGs. In particular, an example about the energy loss from binary systems was given and has been shown that when the nature of the binary systems can exclude energy losses due to trade or loss of matter, then, it is possible to explain the gap between the firsttime derivative of the observed orbital period and the theoretical one predicted by GR, using a higher-order theory of gravity. Of course, to improve the estimation of the higher-order theory coefficients, one needs to consider the hydrodynamic effects due to the transfer of the matter in the binary system, to analyze different systems from double NS, and to improve the mass estimations of the stars in the binary systems without prior pulsar mass and orbital inclination and so on. It was also shown that when the weak-field limit of ETGs is performed, it shows new aspects of gravitation which are not present in GR. The Newtonian and postNewtonian limits give weak-field potentials which are not of the standard Newtonian form. The corrections, in general, are Yukawa-like terms which could explain in a very natural fashion several astrophysical and also cosmological observations. In this chapter has been investigated the impact of the Yukawa-like gravitational potential on the periastron shift of an orbiting body. First have been computed and have been solved numerically the geodesic equations to visually show the presence of stable orbits and the orbital precession of a test particle moving around a massive body. Then, the expression of the periastron advance for a binary system composed of two neutron stars or pulsars with comparable masses has been generalized. Finally, the results shown above will represent a fundamental tool to be used with forthcoming observations of pulsars near the galactic center. Furthermore, the inspiral data for GW150914 and GW151226 have been analyzed. We would like to underline that corrections coming from alternative gravity to the standard relativistic equations and waveforms describing binary black hole systems are negligible up to the 2.5PN order. However, since, as we have demonstrated, extended theories of gravity give rise to an effective gravitational coupling constant Geff , the post-Newtonian dynamics of any metric formalism can be obtained straightforwardly for the lowest-order deviation parameter α = const.. The recently detected gravitational waveforms of GW150914 and GW151226 can thus give constraints on the theory. Using the fact that the gravitational wave frequencies are modulated through Geff , we have shown that this modulation will change the phase of the detected gravitational signal. Our conclusions are in agreement with [26] who found that GW150914 and GW151226 do not place strong constraints on the theory of gravity, since the parameters of the merging black holes are not measured with high enough precision. However, improved statistics on the deviations δφi could remedy this shortcoming in the future. Moreover, the Shapiro delay of GW150914 to set an upper limit |α(250H z) − α(35H z)| < 1.3 × 10−9 was used. Although this result was obtained for scalar tensor theories, this applies for all theories where the PPN-γ is at least quadratic in α. The constraints provided by GW150914 and GW151226 on GR and, in general, metric theories of gravity are unprecedented due to the nature of the sources and

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the strong field regime. However, they have not reached high enough precision to definitively discriminate among concurring theories. Furthermore, in order to extract new physical effects, one would need a wide range of GW waveforms beyond the standard forms adopted for GR and allow for polarizations beyond the standard × and + modes [15]. Binary pulsar observations and GW observations could then be used to stringently constrain these theories in a new independent way from previous constraints.

Cross-References  Binary Neutron Stars  Post-Newtonian Templates for Gravitational Waves from Compact Binary

Inspirals Acknowledgments IDM acknowledges support from MICINN (Spain) under de project IJCI2018-036198-I. IDM is also supported by Junta de Castilla y León (SA096P20), and Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER (PGC2018-096038-B-I00). MDL acknowledges INFN Sez. di Napoli (Iniziative Specifica TEONGRAV and QGSKY).

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Zack Carson and Kent Yagi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameterized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications to Specific Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inspiral-Merger-Ringdown Consistency Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications to Specific Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gravitational-Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graviton Mass and GW Propagation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generic GW Propagation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amplitude Birefringence in Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Gravitational-wave sources offer us unique testbeds for probing strong-field, dynamical, and nonlinear aspects of gravity. In this chapter, we give a brief overview of the current status and future prospects of testing general relativity with gravitational waves. In particular, we focus on three theory-agnostic

Z. Carson · K. Yagi () Department of Physics, University of Virginia, Charlottesville, VA, USA e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_41

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tests (parameterized tests, inspiral-merger-ringdown consistency tests, and gravitational-wave propagation tests) and explain how one can apply such tests to example modified theories of gravity. We conclude by giving some open questions that need to be resolved to carry out more accurate tests of gravity with gravitational waves. Keywords

Gravitational waves · Tests of general relativity · Black holes · Neutron stars

Introduction The famous observation of gravitational waves (GWs) radiated outward from the merger of two black holes (BHs) 1.3 billion lightyears away by the LIGO and Virgo Collaborations (LVC) has ushered in the birth of an entirely new era of astrophysics. For the first time ever, this discovery has allowed us to probe the extreme gravity regime where spacetime is extremely strong, nonlinear, and dynamical. GWs carry with them a multitude of fascinating information, including the astrophysical properties of the source BHs, the underlying theory of gravity driving the collision and radiation process, and many more. To date, this first GW event and the following ∼10 have failed to present any evidence of deviation from Einstein’s famous theory of gravity, general relativity (GR). This prevailing theory of gravity has remained as its post as the accepted model for the last century since its prescription by Einstein and has passed every test thrown at it since. However, a very limited number of tests have been studied in the extreme gravity regimes of spacetimes – such as those surrounding binary BH mergers as detected by the LVC. Thus, it remains vitally important that we continue to test GR – especially in the unexplored regions of the phase space. While the current LVC infrastructure is certainly a marvel of modern engineering, it may still not be enough to uncover the elusive traces of a modified theory of gravity, still hiding deep within the current levels of relatively large noise. On the other hand, the next generation of GW detectors promise entirely new sensitivity in the mHz regime, as well as incredible improvements on the order of 100 times the sensitivity of current detectors. As the famous Popper once said, we can never truly prove scientific theories (such as GR); however, we can rule out and constrain alternatives with observations [1]. Will these new and improved GW detectors allow us to constrain alternative theories to the point where we begin to see deviations from Einstein’s GR? Gravitational-wave sources have unique features compared to other systems that have been used to probe gravity as they are strong-field, dynamical, and nonlinear sources [2]. To illustrate this point, we present in Fig. 1 the gravitational potential and curvature of such systems. Notice that the two GW sources have a distinct feature that they lie on the top-right corner (strong-field). Moreover, they are shown by lines, which means that they swept through a wide range of potential and

39 Testing General Relativity with Gravitational Waves

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Φ=M/L √ Fig. 1 Gravitational potential Φ and (square root of) curvature R for systems with mass M and size L that have been used to probe GR. The two GW sources (GW150914 and GW151226) lie on the top-right corner, meaning that they are strong-field sources. Moreover, these are represented by lines instead of points, which indicates that these sources are also dynamical. (This figure is taken and edited from [2])

curvature during the observational period (∼0.1–1 s), which indicates that they are also dynamical sources (unlike others which are mostly indicated by points). These GW observations can be used to probe various fundamental aspects of GR. These fundamental pillars include the equivalence principle, Lorentz invariance, parity invariance, four-dimensional spacetime, massless gravitons, and coordinate commutativity [2–5]. Another interesting aspect one can probe are GW polarizations. In GR, there are only two tensor polarization modes, which are typically decomposed as plus and cross modes (see Fig. 2). In theories beyond GR, there can be also two scalar polarization modes (breathing and longitudinal) and two vector modes. An example of the breathing mode is shown in Fig. 2. The LVC has used GW170817 and carried out a model selection analysis among three different models: (i) tensor only, (ii) vector only, and (iii) scalar only models. They found that the first model was significantly preferred from the data over the other two models, and hence GR is consistent with such a test [5]. KAGRA has recently joined the collaboration and started its operation. This additional detector will help to further test additional polarizations. For example, one will be able to compare the tensor-only model against tensor + scalar model which may be more realistic.

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breathing mode time Fig. 2 Various GW polarizations. This figure shows how a ring of particles move under each GW mode propagating orthogonal to it. GR only contains the plus and cross modes, while in theories beyond GR, there can be additional polarization modes, such as the breathing mode presented here. The first two are tensor modes, while the last one is the scalar mode

In this article, we focus on reviewing three theory-agnostic tests of GR with GWs. The first one is the parameterized test where we add parameterized deviations from GR in the waveform. In certain modified theories of gravity, there are known mappings between such parameters and theoretical constants, and thus the parameterized tests can easily be applied to specific theories. The second test is the inspiral-merger-ringdown (IMR) consistency test. The idea here is to estimate the final mass and spin of the remnant BH from the inspiral and merger-ringdown parts of the waveform independently assuming GR and check the consistency between the two estimates. Although such a test was originally designed to perform a consistency test of GR, one can apply it to test specific modified theories of gravity as well. The third test is the GW propagation test, where we consider various nonGR effects related to GW propagation in a model-independent way, such as the modified dispersion relation of the graviton. Below, we will look at each of these tests in turn. We will use the geometric units of c = G = 1 throughout.

Parameterized Tests One simple way of probing gravity with GWs in a theory-agnostic way is to introduce general, arbitrary deviations to the GR waveform template in the frequency domain, with the latter given by h˜ GR (f ) = AGR (f )eiψGR (f ) ,

(1)

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where AGR and ψGR are the GR amplitude and phase and f is the GW frequency. In such a framework, the GW community can simply probe arbitrary deviations from the predictions of GR with future GW observations without assuming an alternative theory a priori. Once such constraints have been obtained, the results can finally be mapped to the large set of existing modified theories of gravity.

Formalism Parameterized Waveforms One method for performing model-independent tests of GR with GWs was proposed in [6]. The inspiral portion of the waveform is given in terms of post-Newtonian (PN) expansion, where the relative velocities of the binary constituents are assumed to be much smaller than the speed of light. When each body is non-spinning, each PN term is given in terms of the masses m1 and m2 . The authors proposed to measure each PN term independently and check the consistency between each other in the m1 –m2 plane. This idea is similar to tests of GR with binary pulsar observations. One downside of the above test is that one can only probe non-GR effect entering at PN orders where the GR terms are present, but there are many modified theories of gravity that predict the leading non-GR correction to enter at, e.g., negative-PN orders due to scalar radiation or the variation of the gravitational constant G that are absent in the GR waveform. Another downside is that it may be difficult to perform similar tests with spinning compact binaries. To overcome these issues, the so-called parameterized post-Einsteinian (ppE) formalism was proposed by Yunes and Pretorius [7], which allows one to characterize such arbitrary deviations to the GR amplitude and phase. In particular, we take corrections of the form A → AGR (1 + αua ),

ψ → ψGR + βub

(2)

for effective relative velocity between our binary compact objects u = (π M f )1/3 (m1 m2 )3/5 with a chirp mass of M ≡ (m 1/5 . From here on out, we shall refer to the 1 +m2 ) new parameters (α, a) and (β, b) as the amplitude and phase ppE parameters, respectively. α and β correspond to the size of the modifications in question, which we will focus on constraining. a and b correspond to the power of velocity u relative to GR at which a given correction alters the waveform. These parameters define the PN order at which an effect alters the waveform a = 2n,

b = 2n − 5,

(3)

where an n-PN correction is proportional to (u/c)2n relative to the leading-order GR term. The ppE modified waveform reduces to the GR one in the limit (α, β) → (0, 0). Figure 3 shows a sample comparison of the GR and ppE waveform.

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time (arbitrary units) Fig. 3 Example waveforms of a compact binary inspiral in GR and ppE. Observe that they differ both in the amplitude and phase

LVC used a slightly different formalism called generalized inspiral-mergerringdown (gIMR) formalism [3]. LVC first took the IMR phenomenological (IMRPhenom) GR waveforms that have various phenomenological parameters determined by fitting them with numerical relativity waveforms. LVC then allowed each of these parameters to vary from the GR one to account for generic non-GR effects. In the inspiral portion, there is a one-to-one mapping between PPE and gIMR formalisms [2]. The latter also naturally includes corrections in the mergerringdown portion. The original PPE formalism also has some proposals for the modified waveforms in the merger-ringdown portion [7].

Parameter Estimation Now that we have built in arbitrary corrections to the GR waveform, we must work on constraining the ppE parameters with actual GW observations. In practice, this is most commonly done by using a full statistical Bayesian analysis via, e.g., Markov-chain Monte Carlo method to estimate posterior probability distributions on parameters, such as α and β for various a and b. Working from the other end, one can derive the expressions for (α, a) and (β, b) corresponding to various modified theories of gravity, which can then be used to map the constraints to theory-specific coupling parameters. Such expressions can be found by first deriving corrections to the orbital separation and frequency evolution of a compact binary inspiral. These originate from corrections to the binary binding energy and GW luminosity due to, e.g., modifications to the gravitational potential and radiation from additional channels like scalar and vector fields in the modified theory of gravity one wants to consider. From here, one can calculate corrections to the GW amplitude and phase by following, e.g., [8], where the resulting expressions for (α, a) and (β, b) are displayed for a large list of modified theories of gravity

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that violate certain fundamental pillars of GR, including the equivalence principle, Lorentz/parity invariance, and massless gravitons. With this done, constraints on α and β can be estimated across the entire PN spectrum for each present and future GW event, improving the constraints onward. The resulting ruled-out regions of the parameter space can then be used to constrain the magnitude of various coupling parameters. As GW detection technology improves in the near future with upgraded ground-based and spacebased interferometers, the space of allowable modified theories of gravity may shrink until the possibility of detecting beyond-GR behavior finally presents itself. Another commonly used method to predict posterior probability distributions on α and β is known as the Fisher analysis method [9]. This analysis is a reliable approximation to the Bayesian analysis used by, e.g., the LVC, for large signalto-noise ratio (SNR) systems. In particular, the authors of Ref. [2] found that for events with SNRs of ∼25 (corresponding to that of GW150914), the Fisher analysis technique reliably reproduces the results of the Bayesian analysis done by the LVC for bounds on non-GR parameters. Furthermore, as future GW interferometers become increasingly sensitive, GW events will become louder and louder, closing the gap between the Bayesian and Fisher analyses, with the latter being significantly less computationally expensive. The Fisher analysis technique relies on the assumption that the inherent noise n within the GW detector is distributed Gaussian, so 

 1 p(n) ∝ exp − (n|n) , 2

(4)

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˜ ) are the corresponding function in the frequency domain and Here, a(f ˜ ) and b(f * refers to complex conjugate. We have also assumed that the noise is stationary. The upper and lower limiting frequencies fhigh and flow depend on the specific detector configuration used and are described thoroughly in, e.g., [10]. The ultimate goal of the Fisher analysis is to find best fit parameters θˆ a (including ppE parameters α, β) whose template waveform h maximally agrees with an observed signal s. By substituting in s = n+h(θ i ) into Eq. (4), we find the posterior probability distribution on θ a to be   1 (0) p(θ a |s) ∝ pθ a exp − Γij (θ i − θˆ i )(θ j − θˆ j ) . 2

(6)

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In the above expression, pθ a are the prior on parameters θ i and Γij is the Fisher information matrix, defined to be  Γij ≡

  ∂h  ∂h . ∂θ i  ∂θ j

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Equation (6) corresponds to a multivariate Gaussian probability distribution, and the resulting root-mean-square errors on parameters θ a are given simply as Δθi =

 Γii−1 .

(8)

The prior distribution is arbitrary, though in practice, the outcome results are kept simple if one uses a Gaussian prior. For example, if one desires to include Gaussian (0) prior distributions with a standard deviation σθ i on template parameters or to combine the results of multiple observations on detectors A = 1 . . . N , the Fisher information matrix simply becomes  1 Γij → Γ˜ij = ΓijA ,

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where ΓijA is the Fisher matrix of the Ath detector.

Current Status Now that we have built up the formalism of parameterized tests of GR, let us discuss their present considerations from current GW observations. In this section, we will mainly focus on constraints formed from the first two binary BH GW observations of GW150914 and GW151226 [11]. The former event was located at a distance of 430+150 −170 Mpc away with a large SNR of 25.1, with constituent masses of +4.8 +0.12 35.6−3.0 M and 30.6+3.0 −4.4 M and dimensionless effective spin of −0.01−0.13 (here, the dimensionless effective spin parameter is the mass-weighted average of compact objects’ angular momentum component normal to the orbital plane divided by their masses squared). The latter event was located 440+180 −190 Mpc away, was detected with +2.2 a smaller SNR of 13, and was comprised of 13.7+8.8 −3.2 M and 7.7−2.6 M BHs with an effective dimensionless spin of 0.18+0.2 −0.12 . For the remainder of this section, we focus only on constraints on the ppE phase parameter β. As found by Tahura et al. in [12], the inclusion of the ppE amplitude parameter only alters constraints on β by ∼10%. Thus, we and many other authors consider constraints on the phase correction only. We first turn our attention to the work done by the LVC in [13] for a catalog of binary BHs in [11] and for the binary neutron star (NS) merger event GW170817

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in [5]. In these works, the collaboration considered several tests of GR using a Bayesian analysis framework. For all cataloged events thus far, none of them deviate from GR to a statistically significant degree. This does not indicate that no deviations exist – they may simply be hidden in the relatively large detector noise which will only improve as time continues. Figure 4 of [13] and Fig. 2 of [5] present constraints on the arbitrary phase deviation parameter for each event considered across the spectrum of PN orders. The ones for GW150914 are shown as green crosses in Fig. 4. Figure 4 also displays the 90% upper-limit constraints on β as a function of PN order from both GW150914 and GW151226 [2] via a Fisher analysis, showing not only bounds on the positive-PN orders but also on negative-PN ones. Additionally shown are the results from the previous strongest solar system constraints and constraints from the double pulsar system PSR J0737-3039 [14]. We observe several interesting details from the listed figure: 1. The Bayesian and Fisher analyses on GW150914 agree with each other very well, confirming that the latter is valid as an order of magnitude estimate. 2. GW150914 and GW151226 produce similar results for positive-PN orders, while GW151226 places stronger bounds than GW150914 for negative-PN orders by up to two orders of magnitude.

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3. Solar system bounds outperform GW observations for 1PN order only. However, care must be taken in this comparison since the former only includes conservative effects (such as corrections to the Keplerian motion), while the latter includes both conservative and dissipative (GW emission) effects. 4. The binary pulsar observation from PSR J0737-3039 produces significantly stronger results at negative-PN orders than the GW observations, and vice versa for positive-PN orders. This binary pulsar bound only takes into account the dissipative effects. However, we note that while final two observations from the above list produce stronger constraints at certain PN orders than GW observations, they originate from weak and/or static field environments; thus, the GW constraints remain unique. Moreover, certain modified theories of gravity induce large corrections only to BHs, and hence it is important to derive bounds from various sources.

Future Prospects Now let us consider predictions we can make on the future of parameterized tests of GR. Typically, this is accomplished by choosing future-generation GW detector sensitivity predictions (see Fig. 5) and using a Fisher or a Bayesian analysis to predict constraints on the theory-agnostic ppE phase and/or amplitude parameters α and β. The top panel of Fig. 6 summarizes how the bound on β improves with future GW interferometers, assuming that they detect GW signals from a GW150914like event. Observe that LISA has a significant improvement, especially when corrections enter at negative-PN orders (similar for TianQin). This is because the relative velocity of BHs is much smaller for LISA than ground-based detectors, and thus (relative) negative-PN effects become enhanced (if they exist). Observe also that DECIGO outperforms all the interferometers due to its high detector sensitivity and a large number of GW cycles during its observation. One particularly promising line of inquiry is into the potential multi-band observations [17] of GWs from GW150914-like stellar-mass BH binaries between both space-based detectors (such as LISA, TianQin, (B-)DECIGO, and many others) and ground-based detectors (such as the Cosmic Explorer (CE), Einstein Telescope (ET), and many others). Figure 5 shows the GW spectra for GW150914, which can be detected both ground-based and space-based interferometers. Such an observation would take place during both the low-frequency bands (∼10−4 – 1 Hz) when the (stellar-mass) BHs are far apart and moving slowly, as well as the high-frequency bands (∼1–104 Hz) when the BHs are moving rapidly, making contact, and ringing down to their final state. By combining together the entire observation, we can expect stronger bounds to be placed on α and β along all PN orders and corresponding constraints on most modified theories of gravity. For example, the prospects of probing scalar dipole radiation (absent in GR) with multiband observations are discussed in [18].

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The bottom panel of Fig. 6 shows the improvement on the bound on β using multi-band observations with respect to the single-band case. Observe that LISA+CE may have an improvement over LISA-alone or CE-alone by up to a factor of 40. On the other hand, DECIGO does not have much improvement with the multi-band observations because its sensitivity is already good enough on its own. This predictive work is extremely valuable as they can strengthen the case for the development of future detectors and can prove to be informative to decisions based on detector design.

Applications to Specific Theories The above bounds on the arbitrary ppE phase parameter can further be mapped to desired modified theories of gravity by considering the mappings displayed in, e.g., [8]. The results are summarized in Table 1 for the following theories: • Einstein-dilaton Gauss-Bonnet (EdGB) gravity: A scalar field (dilaton) is coupled to a Gauss-Bonnet combination of the Riemann curvature squared in the action with a coupling constant αEdGB , motivated by string theory. • Scalar-tensor theories: Generic theories with a scalar field coupled either minimally or non-minimally to the metric. BHs can acquire a scalar charge when ˙ the scalar field is time evolving with a rate φ.

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• Dynamical Chern-Simons (dCS) gravity: A (pseudo) scalar field is coupled to the Pontryagin density (odd-parity curvature squared scalar) in the action with a coupling constant αdCS , motivated by string theory, loop quantum gravity, and effective field theory of inflation. • Noncommutative gravity: Quantizing spacetime by promoting the coordinates x μ to operators xˆ μ . For example, [tˆ, xˆ i ] = iθ 0i where θ 0i θ 0 i = Λ2 lp2 tp2 with Λ being the noncommutative parameter and lp and tp representing the Planck length and time, respectively. • Time-varying mass theories: Theories in which BH masses may change over time with a rate M˙ due to, e.g., enhanced Hawking radiation in some extra dimension models. Time variation for BH masses is also relevant in astrophysical/cosmological context in terms of accretion of gas and dark energy. ˙ • Time-varying G theories: Theories in which G varies over time with a rate G, such as scalar-tensor theories.

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• Massive graviton theories: Theories in which the graviton has a nonvanishing mass mg . The graviton mass affects the GWs both during generation and propagation. Notice that GW bounds are typically much weaker than other bounds, though the former have a meaning that they are the bounds obtained in the strong and dynamical field regime. Having said this, there are some theories, like noncommutative gravity, on which GW events placed stringent bounds. We now move onto describing future prospects on constraining example theories via parameterized tests. We first take a look at the work done by Chamberlain and Yunes in [19]. Here, the authors estimated constraints on several modified theories of gravity (including dipole radiation, extra dimensions, time-varying gravitational constant, Einstein-Æther, khronometric, and massive graviton theories of gravity) obtained with a Fisher analysis for a variety of astrophysical systems and groundand space-based detectors. In particular, they also considered several alternative design sensitivities for the space-based detector LISA to compare the resulting constraints. Table 1 of the same paper neatly displays the best constraints one can place on each theory of gravity considered from ground-based or space-based interferometers, including the current constraints, as well as the best astrophysical systems for placing constraints with both ground-based and space-based detectors. Table 1 Summary of constraints on sample modified theories of gravity with GW150914 and GW151226, together with other bounds. (1st column) sample theories; (2nd column) violation of GR fundamental pillars, such as the Strong Equivalence Principle (SEP), four-dimensional spacetime (4D), and massless gravitons (mg = 0); (3rd column) the PN order at which the leading correction enters; (4th column) representative theoretical parameters; (5th column) bounds on these representative parameters with GW150914; (6th column) same as the 5th column but for GW151226; (7th column) bounds from other observations. The top (bottom) row within massive graviton corresponds to corrections to the dynamical (propagation/conservative) sector. (This table is taken and edited from [2, 15]) Theory EdGB

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Figure 7 presents the future predicted bounds on various theoretical parameters in theories mentioned earlier with GW150914-like events with CE and space-based detectors (LISA, TianQin, B-DECIGO, and DECIGO) computed by ourselves [10]. Observe that space-based detectors place stronger bounds than CE for theories in which the leading correction enters at negative-PN orders (EdGB, scalar-tensor, varying-G and varying-mass theories). This is because such corrections (relative to GR) become larger at lower frequencies when the relative velocity is smaller. On the other hand, CE places stronger bounds than the space-based detectors for theories with positive-PN corrections (dCS and noncommutative theories). Here we can also see several cases where ground-based and/or space-based observations can provide constraints stronger than the current ones found in the literature. Figure 7 also presents the results for multi-band observations. We see that in every scenario, the multi-band observations produce stronger bounds than both the space-based and ground-based ones individually. Finally, we observe that in dCS gravity, both of the ground-based and space-based detectors (except for DECIGO) fail to provide constraints consistent with the theories’ small-coupling approximation (needed to

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treat the theory as a valid effective theory), and are thus not reliable. However, only when the multi-band observations are made is this approximation satisfied, and new constraints can be placed that are stronger than the current ones by seven orders of magnitude. We end this section by mentioning future prospects for testing GR with mixed NS/BH binaries. Several candidates for such mixed binaries were reported during the O3 run. These binaries are especially useful for probing theories with scalar fields. This is because such theories generically allow the presence of the scalar dipole radiation, which is proportional to the square of the difference in the scalar charges of the binary constituents. Thus, the amount of scalar radiation becomes larger for binaries consisting of different types of compact objects. To put the discussion into context, let us here focus on EdGB gravity as an example, which we can probe within the ppE framework by looking at the 1PN correction bound. In [20], we first studied the prospects of probing this theory with the O3 NS/BH candidates. We found that if the BH mass is smaller than ∼16.5 M , it is likely that one can place a bound that is stronger than the current bound. Figure 8 presents the projected future bounds on the EdGB coupling constant with NS/BH binaries using various ground-based and space-based GW detectors (including multi-band observations). We show the cases for both single events and multiple events. Observe that the bound can become stronger than the current one by up to four orders of magnitude. This, in turn, means that if the true theory of gravity is EdGB and the true coupling constant lies somewhere between √ 10−4 km < αEdGB < 2 km, future NS/BH GW observations have the potential to detect the EdGB correction encoded in the waveform.

Inspiral-Merger-Ringdown Consistency Tests Now let us consider another model-independent test of GR, this time by comparing the consistency between the inspiral and merger-ringdown portions of a given GW signal. This test is most commonly known as the IMR consistency test and has been widely used in the community to generically test arbitrary GW signals for any signs of deviation from GR. Somewhat unlike the parameterized tests considered in the previous section, the IMR consistency tests allow us to detect any arbitrary deviations residing within the entire GW signal, while the former allows one to constrain deviations at pre-specified PN orders in the inspiral portion (LVC used the gIMRPhenom formalism to probe parameterized deviations in the merger-ringdown as well.). In the IMR consistency tests, we will begin by checking the consistency between the inspiral GW signal when the compact objects are largely separated and moving relatively slow and the merger-ringdown signal when the objects first make contact and form a remnant BH which settles down to its final state via the radiation of quasi-normal modes (QNMs). The separation between the two signals in question is commonly defined to be when the compact objects reach the location of the innermost stable circular orbit (ISCO) (e.g., when the GW frequency reaches fISCO ), at which point they enter a plunging orbit and finally make contact with each

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other. If the two signals agree with each other to a statistically significant level within the detector noise using GR template waveforms, we will say that GR is consistent and no deviations from it can be found. On the other hand, if deviations between the two signals are found, one can claim a general deviation from GR present somewhere within the observed signal. Further follow-up studies can then be performed to determine the precise nature of the deviation and if it can be mapped to any known alternative theories of gravity. The consistency between the inspiral and merger-ringdown signals is performed by simply estimating the remnant BH’s final mass Mf and (dimensionless) spin χf given the observation of only the former or latter signal, without the presence of the other. This is estimated by using the fit for Mf and χf in terms of the initial masses and spins obtained from numerical relativity simulations, as we will discuss in more detail later. We can then draw the two-dimensional posterior probability distribution functions between the remnant BH’s mass and spin from both of the inspiral and merger-ringdown portions of the signal. The contours are then compared with each other to determine the consistency between the two portions of the GW signal. This is commonly done by transforming to the new coordinates ∝ (ΔMf , Δχf ), where ΔMf and Δχf describe the differences between the inspiral and merger-ringdown

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predictions of the final mass and spin. Thus, finally, the compatibility of the resulting contour with the “perfect-match” value of (0, 0) gives us a reliable test of GR which allows us to detect even minor deviations from GR, depending on the sensitivity of the GW detector in use.

Formalism Let us now discuss the various technicalities of the IMR consistency test [21] as is commonly used by the LVC and others. As discussed above, given an observed (or simulated) GW signal with sufficient SNR in both of the inspiral (I) and merger-ringdown (MR) portions (This corresponds to an SNR of at least 9 in both portions, as was achieved in the first GW observation GW150914 [22], which was measured with SNRs of 19.5 and 16, respectively.), which are separated by the ISCO frequency fISCO = (63/2 π M)−1 for total binary mass M ≡ m1 + m2 , we separate the signal into two distinct regions: “I” (f < fISCO ) and “MR” (f > fISCO ). If GR is correct, the remnant BH’s mass and spin can be uniquely predicted entirely from the intrinsic masses and spins of the initial BHs by the no-hair theorem, namely, Mf = Mf (m1 , m2 , χ1 , χ2 ) and χf = χf (m1 , m2 , χ1 , χ2 ). Such expressions are quite complicated and have been determined through various numerical relativity simulations of binary BH mergers, e.g., in [23] where the authors provided fits for Mf and χf within the IMRPhenomD gravitational waveform from a wide variety of binary BH simulations. Using such expressions, we can begin our process by first estimating posterior probability distributions on the initial parameters mi,1 MR , mi,2 MR , χ1i,MR , and χ2i,MR from the inspiral and merger-ringdown portions of the signal, individually. Such distributions are most reliably reconstructed using a Bayesian analysis as is done by the LVC. However, as we discuss later, this can also be approximated by the Fisher analysis techniques described in section “Parameter Estimation”. While the latter method is an approximation to the former, it becomes especially useful for the forecasting of future IMR consistency tests of GR from signals which have yet to be detected, for which the SNRs are expected to be high and the validity of the approximation increases. With the above probability distributions on the initial masses and spins in hand, we can then transform them into posterior probability distributions on the remnant BH’s mass and spin Mfi,MR and χfi,MR from the inspiral and merger-ringdown signals using, e.g., the numerical relativity fits of [23], or from any other chosen gravitational waveform model. Typically, to more qualitatively judge the comparison between the inspiral and merger-ringdown, we transform the two contours into a new shared coordinate system. Namely, we choose the new coordinates:

≡

ΔMf Mf

,

σ ≡

Δχf . χf

(10)

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Here ΔMf ≡ Mfi −MfMR and Δχf ≡ χfi −χfMR describe the differences between the inspiral and merger-ringdown mass and spin predictions, and M f ≡ 12 (Mfi + MfMR ) and χ f ≡ 12 (χfi + χfMR ) describe their averages. In this new coordinate system, we are allowed a simple way to directly compare the two probability distributions with only a single contour. In fact, notice how the condition for a perfect match between the two obeys the point (, σ ) = (0, 0), with the GR being the true theory of gravity found in nature. To transform the inspiral (Pi ) and merger-ringdown (PMR ) probability distributions in the (Mf , χf ) plane to the new (, σ ) coordinate system, we simply apply the following transformation found in the Appendix of Ref. [21]: 1 P (, σ ) =

∞ d χ¯ f

0

d M¯ f Pi

0

× PMR



1−



1+

 ¯ σ

Mf , 1 + χ¯ f 2 2

 ¯ σ ¯ Mf , 1 − χ¯ f Mf χ¯ f . 2 2

(11)

The compatibility of the new probability distribution with the value of (0, 0) determines the success or failure of the IMR consistency test, at various statistical confidence intervals. Finally, let us consider how one would utilize the IMR consistency testing framework to forecast results from future GW observations with improved observatories. First laid out in [10] by the same authors, we discuss a simple method via a Fisher analysis discussed in section “Parameterized Tests”. To do so, one must first estimate the statistical uncertainties (manifesting from, e.g., statistical detector noise) on Mf and χf from the inspiral and merger-ringdown signals independently from the variance-covariance matrices Σ i,MR . Next, we can estimate the systematic errors Δth Mf and Δth χf potentially present in such an observation by following the work of Ref. [24]. Such errors can manifest themselves from, e.g., waveform mismodeling uncertainties by assuming a GR waveform template and observing a non-GR signal. In general, the systematic, or “theoretical,” errors on parameters θ a from such a scenario can be given as 

 Δth θ a ≈ Σ ab [ΔA + iAGR ΔΨ ]eiΨGR ∂b h˜ GR ,

(12)

where Σ ab = (Γ −1 )ab is the covariance matrix found from the Fisher analysis, a summation over b is implied, and ΔA ≡ AGR − Anon−GR and ΔΨ ≡ ΨGR − Ψnon−GR are the differences between the amplitude and phase in GR and a given non-GR theory of gravity. Combining both types of uncertainties found above, we can find the final probability distributions in the Mf − χf plane to be Gaussian Pi,MR ≡



1

2π |Σ i,MR |

 exp −

T 1 X − X GR i,MR − Δth X i,MR 2

39 Testing General Relativity with Gravitational Waves

  GR ×Σ −1 − Δ X X − X , th i, MR i,MR i,MR

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(13)

where X ≡ (Mf , χf ) contains the final state variables; XGR i,MR contains their GR predictions from the inspiral and merger-ringdown portions, respectively; and Σ i,MR represents the covariance matrices of each portion. By assuming a specific theory of gravity, one can simply compute increasing sizes of systematic uncertainties present in the above distributions corresponding to increasing magnitudes of nonGR effects. While the statistical uncertainties manifest themselves as the size of probability distributions, the systematic uncertainties will manifest as shifts in the (Mf , χf ) plane, ultimately increasing in magnitude until the inspiral and mergerringdown signals no longer agree with one another. Finally, once again, the inspiral and merger-ringdown probability distributions can be transformed to the (, σ ) coordinate system as discussed previously, allowing us to predict the magnitude of non-GR effects required to make such a distribution disagree with the GR value of (0, 0). Once this point is reached, we can take the given size of non-GR effects to be a new constraint on that given theory. The method described here can then be applied to alternative theories of gravity found in the literature, allowing us to forecast the power of the IMR consistency test in future GW observations for several theories of gravity.

Current Status Now let us discuss the current status of the IMR consistency tests in the GW community. Figure 9 shows the posterior distribution of Mf and χf from inspiral alone, merger-ringdown alone, and the entire inspiral-merger-ringdown waveform for the event GW150914. We display the results found by the LVC in [3] using a Bayesian analysis on the actual observed signal. We see that, in the case of GW150914, the inspiral and merger-ringdown signals overlap significantly – showing very good agreement between the two portions of the signal and the GR assumption consistent with the data. In Ref. [13], the LVC applied the IMR consistency test for the events GW150914, GW170104, GW170729, GW170809, GW170814, GW170818, and GW170823 as found in the GWTC-1 catalog [11]. The resulting posterior probability distributions from the IMR consistency test are displayed in Fig. 2 of [13]. All of the observed GW events considered in this test agree strongly with GR, indicating that if deviations from GR were to exist, we would need more sensitivity to discover them. Further, the combined posterior probability distribution from all seven events similarly observes GR behavior. Table 3 of [13] displays important quantities for each event considered, including the cutoff frequency between the inspiral and merger-ringdown signals, the SNR’s within each region of the signal, and finally the GR quantile. The latter quantity denotes the fraction of the posterior contained by an isoprobability contour passing directly through the GR value of (0, 0) in the -σ plane defined in Eq. (10). Thus,

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1

0.8

χf

0.6

0.4 Inspiral Merger-Ringdown Inspiral-Merger-Ringdown

0.2

40

60

80 Mf [MO. ]

100

Fig. 9 90% credible region contours of the inspiral, merger-ringdown, and complete inspiralmerger-ringdown waveform posterior distributions in the Mf − χf plane, for GW150914. (The data was taken from [3])

smaller GR quantile fractions indicate better consistency between the observed signal and GR. Such values occur between 7.8% and 80.4% for each of the events considered, with the largest value corresponding to GW170823 and the smallest from GW170814. While the GR quantiles can vary by SNR (i.e., small-SNR signals will correspond to broad posteriors and smaller GR quantiles), we still see strong evidence in favor of GR as the true theory of gravity.

Future Prospects Now we will consider the future of IMR consistency tests with improved GW interferometers. In Ref. [25], the authors present the IMR consistency tests on “golden” BH binary coalescences. Here, golden events occur as large-SNR events with total masses of 50 M –200 M that can be observed by ground-based GW observatories in all of the inspiral, merger, and ringdown portions of the signal. The authors of [25] begin their analysis by simulating ∼100 GR signals as observed by the advanced LIGO with its design sensitivity with SNRs of 25. This number of signals is potentially observable within 1 year of advanced LIGO operation according to several population synthesis models. Figure 2 of the same paper displays the corresponding posterior probability distributions in the (ΔMf /M f ), Δχf /χ f ) plane as computed via a Bayesian analysis. All 100 posteriors agreed strongly with the GR value of (0, 0), and the combined posterior region could reach as low as a few percent. This indicates the high degree of

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precision one can gain to test GR upon the successful combination of several GW observations. Following this, the authors simulated general modified GR waveforms to predict the detectability of non-GR effects present within the signal with future observations. To do this, the authors modified the GW flux at 2PN order by a constant factor of αmodGR . The authors then presented the IMR consistency test with αmodGR = 400 and found the IMR consistency test to be strongly inconsistent, as shown in Fig. 1 of [25]. In particular, the inspiral and merger-ringdown posteriors clearly disagree with each other, and the transformed (ΔMf /M f , Δχf /χ f ) posterior clearly lies outside of the GR value of (0, 0). In fact, the authors found that with such a signal, GR can be ruled out with a confidence of 99%, indicating the strong possibility with which one can detect deviations from GR with future observations. Now let us switch our attention to future prospects of the IMR consistency test with third-generation ground-based detectors and space-based detectors. We show results using a predictive Fisher analysis as described in sections “Parameter Estimation” and “Formalism”. While this method is less robust than the ones discussed previously with Bayesian analyses, it offers one a quick method to obtain order-of-magnitude estimates on the future ability of the test in question. By first estimating the intrinsic source parameters m1 , m2 , χ1 , and χ2 using a Fisher analysis, the resulting Gaussian probability distributions are transformed into yet another two-dimensional Gaussian probability distribution in the (Mf , χf ) plane as shown in Fig. 10 for GW150914-like events with various future detectors. We also show the posterior distribution with the O2 noise curve obtained via a Fisher analysis, together with that obtained by LVC with a Bayesian analysis on the observed signal. Here we see that the former predict reasonably well those computed with the latter – both in the direction of correlation and size of the contour. Figure 10 also presents the results for multi-band observations. In particular, we considered the combination of future third-generation detector CE with space-based detectors LISA, TianQin, and (B-)DECIGO. While the former detector can observe the merger-ringdown portion of the signal quite well, the latter detectors are apt at observing the early-inspiral regime of the signal. The combination of the two was shown to decrease the area of the (ΔMf /M f ), Δχf /χ f ) posterior by up to four orders of magnitude as compared to that achievable by the first LIGO observing run shown in Fig. 9 (one order of magnitude improvement from the ground-based-only detection by CE). This considerable decrease in the area of the posterior points to the significant improvement in resolution one might gain with future observations.

Applications to Specific Theories Although the IMR tests were originally designed to check the consistency of the GR assumption, it can be used to constrain specific theories (or spacetimes) as in the parameterized tests. We here review Refs. [26,27] by the same authors that made a first attempt in this direction. In these investigations, we presented simple recipes one can use to introduce corrections to the inspiral, ringdown, and remnant BH

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0.01

0.4

σ = Δχf / χf

0

0.2 -0.01 -0.01

0

0.01

0 O1 (Fisher) O1 (Bayesian) CE CE+TianQin CE+LISA CE+B-DECIGO CE+DECIGO

-0.2

-0.4

-0.2

0 0.2 ε = ΔMf / Mf

0.4

Fig. 10 90% credible region contours of the transformed probability distributions in the –σ plane (see Eqs. (10)–(11)), describing the difference in the remnant mass and spin predictions between the inspiral and merger-ringdown estimate for GW150914-like event using the GR templates. Here we display the results for LIGO O1 (Fisher and Bayesian [13] for comparison), CE, and the multiband observations of CE and LISA, TianQin, B-DECIGO, and DECIGO. (This figure is taken from Ref. [10])

mass and spin portions of the GW signal for various modified theories of gravity or spacetime metrics. In this way, one can choose a specific theory of gravity or spacetime metric (and corresponding stress-energy tensor) and perform a theoryspecific IMR consistency test of gravity with full-waveform corrections to obtain order-of-magnitude constraints on beyond-GR and beyond-Kerr parameters. In particular, after applying the full-waveform corrections corresponding to the desired theory of gravity, one can estimate posterior probability distributions on the remnant BH mass and spin parameters using the Fisher analysis method. Following this, one can slowly increase the magnitude of beyond-GR or beyondKerr corrections present in the hypothetical signal, calculating the corresponding systematic mismodeling uncertainties from Eq. (12). Finally, one can compute the final posterior probability distribution via Eq. (13). Once the magnitude of beyond-GR effects is large enough to fail the IMR consistency test, we can take the corresponding value of the beyond-GR or beyond-Kerr parameter as a new constraint on that parameter if the observed signal is consistent with GR. In [26, 27], the above technique was applied to both the EdGB gravity, and parameterized beyond-Kerr spacetime metrics proposed, e.g., by Johannsen and Psaltis (JP) [28]. In the former case, constraints on the EdGB coupling parameter were shown to improve upon current constraints by up to an order of magnitude with

39 Testing General Relativity with Gravitational Waves

0.01

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0 km

CE + LISA

0 Δχf / χf

GR

-0.01

0.2 km

-0.02 0.3 km

-0.03 -0.04

-0.015

-0.01

-0.005 ΔMf / Mf

0

0.005

Fig. 11 Future prospects for the IMR consistency to test EdGB gravity with multi-band observations. We assumed that both CE and LISA detect signals from a GW150914-like event. We show the 90% credible posterior distribution for three difference choices of the EdGB coupling constant √ √ αEdGB . If the signal is consistent with GR, this test can place an upper bound of αEdGB  0.2 km, which is about 10 times stronger than the current bounds from low-mass X-ray binaries and GW observations. (This figure is taken from [26])

future multi-band observations between ground-based and space-based detectors, CE and LISA (see Fig. 11). In the latter case, constraints on the JP deviation parameter via the space-based observation of extreme-mass-ratio inspirals with LISA were found to be stronger than current bounds by up to three orders of magnitude. The theory-specific constraints using the IMR consistency test were also found to be comparable with those found using the parameterized tests as discussed in section “Parameterized Tests”.

Gravitational-Wave Propagation Up until now, we have been focusing on probing non-GR effects entering at the level of GW generation. Now, we switch gears to review probing non-GR effects during the GW propagation.

Modified Dispersion Relation The GW propagation acquires non-GR corrections if the propagation speed of GWs is different from the speed of light. This is indeed the case in, e.g., massive gravity theories in which the graviton has a nonvanishing mass [29]. A theory-agnostic way

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of probing such effects to the waveform was proposed by Mirshekhari et al. [30], in which the generic modified dispersion relation of the graviton with energy E and momentum p has been proposed as E 2 = (pc)2 + A(pc)α¯ .

(14)

Here A is the overall magnitude of the correction to the dispersion relation, while the index α¯ represents the p dependence of the correction. Such a correction to the dispersion relation changes the gravitational waveform from the GR one. The correction can be mapped to the ppE formalism in section “Parameterized Tests” as [30] β=

M 1−α¯ π 2−α¯ Dα¯ , 2− α ¯ 1 − α¯ λA (1 + z)1−α¯

b = 3(α¯ − 1),

(15)

¯ where λA ≡ hA1/(α−2) is similar to a Compton wavelength, while the distance Dα¯ is given by

(1 + z)1−α¯ Dα¯ = H0

 0

z



¯ (1 + z )α−2

Ωm (1 + z )3 + ΩΛ

dz ,

(16)

where z is the source redshift, H0 is the current Hubble constant, and Ωm and ΩΛ are the energy density of matter and dark energy, respectively. The generic modification to the dispersion relation of the graviton in Eq. (14) can capture modifications in specific theories, including massive gravity, doubly special relativity, extra dimension theories, Hoˇrava-Lifschitz gravity, multifractional spacetime theory, and gravitational Standard Model Extension (see [2] for details on the mapping). Figure 12 presents the upper bound on A from GW150914 as a function of α¯ [2]. This bound was originally derived via a Fisher analysis, which was later confirmed by the LVC with a Bayesian analysis on the observed signal [13]. We also present bounds from cosmic rays due to the absence of Cherenkov radiation, which can place stringent bounds on the subluminal propagation of GWs. Notice that the GW observations place a unique bound on the positive A parameter space. Figure 12 also shows bound from GW150914 obtained by comparing the arrival time difference between the LIGO detectors at Hanford and Livingston [31]. Although such a bound is much weaker than that from the phase difference within the ppE framework, it can place a unique bound at α¯ = 2 (corresponding to the correction to the GW propagation speed being independent of the GW frequency), where the phase correction degenerates with the time of coalescence. Future prospects on constraining A from GW150914-like events with upgraded detectors and multi-band observations are discussed in [10]. Since the corrections that we are considering enter at positive-PN orders, ground-based detectors seem already good enough for probing such GW propagation effect with GW150914-like events, and an addition of space-based detectors does not help much in this case.

39 Testing General Relativity with Gravitational Waves

10

1

2.5

n PN 4

5.5

7

2

3

4

GW150914, phase diff. GW150914, time delay cosmic rays

20

+A [eV

2-α

]

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10 10 10

-20

-40

-A [eV

2-α

]

10

20

10 10 10 10

0

0

-20

-40

-60

0

1

α Fig. 12 Constraints on A in Eq. (14) as a function of the index α¯ from GW150914. The top axis shows the corresponding PN order the correction enters to the phase. We show the one from the difference in the phase relative to GR (red solid) [2] and that from the time delay between Hanford and Livingston detectors (magenta dotted-dashed) [31]. We also present the bound from cosmic rays (via the absence of gravitational Cherenkov radiation). Observe that GW150914 places unique bounds on the positive A parameter space. (This figure is taken and edited from [2])

Graviton Mass and GW Propagation Speed When the graviton has a nonvanishing mass, there is a constant shift in the dispersion relation with a correction corresponding to α¯ = 0. The bound on A from GW150914 at α¯ = 0 can be mapped to that on the graviton mass mg as mg < 10−22 eV (see also Table 1), which is consistent with what the LVC found [3]. The updated bound with all 10 binary BH events during the O1 and O2 runs combined is given by mg < 5 × 10−23 eV [13], which is slightly weaker than the updated solar system bound [32]. We next discuss future prospects. The GW bound on mg is expected to improve significantly with GWs from supermassive BH binaries using LISA [19,33]. Future prospects for constraining mg with GW150914-like events including multi-band observations are studied in [10]. Multi-band observations can make the bound stronger than the single-band case by one order of magnitude, though such bounds are not as stringent as those from supermassive BH binaries with LISA.

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The propagation speed of GWs, cGW , has also been constrained strongly from the binary NS event GW170817. Comparing the arrival time difference between GW and electromagnetic wave signals, the LVC placed the bound [4] − 3 × 10−15 ≤

cGW − c ≤ 7 × 10−16 . c

(17)

Such a stringent bound effectively ruled out many modified theories of gravity, including various scalar-tensor and vector-tensor theories (see, e.g., [34]). We note that the comparison of GW and electromagnetic signals can also probe Shapiro time delays to constrain (or rule out) theories like relativistic MOdified Newtonian Dynamics (MOND) (see, e.g., [35]).

Generic GW Propagation Tests A more generic way of testing GW propagation has been proposed by Nishizawa [36]. The author begins by taking a generic tensor perturbation hij in cosmological background spacetime as [37] 2 2 h ij + (2 + ν)H h ij + (cGW k + a 2 m2g )hij = a 2 Γ γij .

(18)

Here k is the wave number, a is the scale factor, H = a /a is the Hubble parameter in a conformal time, a prime represents a derivative with respect to the conformal time, ν is the Planck mass run rate, and Γ γij is the source term due to anisotropic stress. The above equation reduces to GR when c = 1 and (ν, mg , Γ ) = (0, 0, 0). The generic modified dispersion relation considered in section “Modified Dispersion Relation” corresponds to cGW = cGW (k) and (ν, Γ ) = 0. Corrections to the waveform from the non-GR parameters in Eq. (18) can be mapped to the ppE (and gIMR) formalism [36]. Future prospects of probing the generic GW propagation with second-generation ground-based detectors are discussed in [36]. Let us now focus on the case where ν = 0. The parameter ν acts as a friction term that changes the amplitude, and if it is independent of k, the corresponding ppE parameter is given by α=−

1 2

 0

z

ν dz , 1 + z

a = 0.

(19)

This effect can be absorbed into the luminosity distance dL . This means that the luminosity distance dLGW measured with GWs would be different from dLGW . Such a difference can generically be parameterized with two parameters Ξ0 and n as [38, 39] Ξ (z) ≡

dLGW (z) 1 − Ξ0 = Ξ0 + , GW dL (z) (1 + z)n

(20)

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which is related to ν as ν(z) = −

d ln Ξ (z) n(1 − Ξ0 ) = . d ln(1 + z) 1 − Ξ0 + Ξ0 (1 + z)n

(21)

The mapping between (Ξ0 , n) and parameters in various example theories is presented in Table 1 of [39]. Future prospects for measuring and constraining Ξ0 with GW standard sirens using third-generation ground-based detectors and LISA are discussed in [38] and [39], respectively.

Amplitude Birefringence in Parity Violation One way to probe the gravitational parity violation is through the amplitude birefringence [40, 41]. When GWs propagate, the amplitudes of right-handed and left-handed circular polarizations behave differently. Let us take Chern-Simons (CS) gravity as an example. We begin by considering a metric perturbation under the Friedmann-Robertson-Walker spacetime given by

ds 2 = a 2 (η) −dη2 + (δij + h¯ ij )dχ i dχ j ,

(22)

where χ i is the comoving spatial coordinates, while h¯ ij (η, χ i ) is the comoving metric perturbation. We decompose this metric perturbation into right-handed (h¯ R ) and left-handed (h¯ L ) circular polarizations, √ which are related to the √ plus and cross mode polarizations as h¯ R = (h¯ + −i h¯ × )/ 2 and h¯ L = (h¯ + +i h¯ × )/ 2, respectively. We further decompose h¯ R,L into the amplitude AR,L and phase φ(η) as h¯ R,L = AR,L e−i



φ(η)−κnk χ k



(23)

,

where κ is the conformal wave number, while nk represents the unit vector pointing toward the direction of GW propagation. Plugging these into the modified field equations, one finds an equation for the phase (dispersion relation) given by

2

iφ + (φ ) − κ = −2i 2

SR,L

SR,L



φ ,

SR,L ≡ a 1 − λR,L

κϑ , a2

(24)

with λR = +1, λL = −1, and ϑ representing the scalar field in CS gravity. /S We next impose the following assumptions: (φ )2  φ , and κ  SR,L R,L . The second condition is satisfied when κ  H and imposing the weak CS approximation given by κ|ϑ |  a 2 ,

κ|ϑ |  2a 2 H .

(25)

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Under these assumptions, one can solve the dispersion relation in Eq. (24). Evaluating the solution at the current conformal time of η = 1, one finds ˙ φR,L (1) = ±κ(1 − ηs ) + iλR,L πf Θ.

(26)

Here ηs is the conformal time at which GWs are emitted while Θ˙ ≡ ϑ˙ 0 − (1 + z)ϑ˙ s , where the subscripts 0 and s represent a quantity being evaluated at η = 1 and η = ηs , respectively. An over-dot refers to a derivative with respect to the physical time t. Notice that the CS correction in Eq. (26) is purely imaginary, which means that such a correction actually affects the amplitude. Moreover, it is proportional to λR,L , which leads to the conclusion that if one of the circular polarization is amplified, the other one is suppressed. This effect can be summarized as   ˙ h¯ R,L = h¯ GR R,L 1 + λR,L πf Θ ,

(27)

where h¯ GR R,L is the comoving metric perturbation for circular polarizations in GR. This can be mapped to the PPE framework, and the amplitude correction enters at 1.5PN order relative to GR. Let us now discuss the current bounds and future prospects of probing parity violation from GW amplitude birefringence. We begin with isolated GW sources. Unfortunately, current observation of GW150914 cannot place any meaningful bounds because they do not satisfy the weak CS bound in Eq. (25) [41]. On the other hand, if aLIGO with its design sensitivity or CE detects signals from a GW150914-like event, one should be able to place bounds on parity violation from birefringence. Another interesting possibility of using multi-messenger observations to probe amplitude birefringence is discussed in [42]. We next look at using stochastic GW background (GWB) from binaries of stellarmass BHs, which we assume to be stationary, Gaussian, and isotropic. The quadratic expectation values for each polarization are given by     1 h˜ R (f, n) h˜ ∗R (f , n ) I (f ) + V (f ) 2 δ(f − f )δ (n, n ) × = , h˜ L (f, n) h˜ ∗L (f , n ) I (f ) − V (f ) 2

(28)

where the angular brackets denote ensemble averaging, while I (f ) and V (f ) are the Stokes parameters corresponding to the total intensity and parity violation, respectively. We will focus on the latter, which is absent in the standard GR spectrum of GWs so that its detection will hint at the presence of parity violation. The fractional energy density spectrum of the V-mode is given by (V ) (f ) ≡ ΩGW

4π 2 f 3 V (f ), ρc

where ρc is the critical energy density of the Universe.

(29)

39 Testing General Relativity with Gravitational Waves

10

(V)

ΩGW

10

10

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-8

-9

150MO.

-10

. .. (ϑ0, ϑ0/H0) = (10km, 0) . .. (ϑ0, ϑ0/H0) = (70km, 0) . .. (ϑ0, ϑ0/H0) = (0, 10km) . .. (ϑ0, ϑ0/H0) = (0, 70km) aLIGO (HLV) aLIGO (HLVK) Voyager (HLV) 3rd-gen (HV1V2V3)

28MO. 10

10

-11

-12

10

-13

10

1

10

2

f [Hz] Fig. 13 Astrophysical GW background from binary BH with two different average chirp masses for the Stokes’ V-mode (parity-violating mode) in dCS gravity, together with the power-law integrated sensitivity curves for various combinations of detectors at Hanford (H), Livingston (L) Virgo (V), and KAGRA (K) sites. For the spectrum, we consider different combinations of ϑ˙ 0 and ϑ¨ 0 /H0 . If the spectrum goes above the sensitivity curve at any frequency, the signal-to-noise ratio is beyond unity. Parity violation in the V-mode GW background may be detected with thirdgeneration detectors. (This figure is taken from [41])

Figure 13 presents the spectrums of the V-mode for various choices of the evolution of the scalar field and the average chirp mass of binary BHs. We also show the power-law integrated sensitivity curves (with an SNR of 1) for networks of second- and third-generation ground-based detectors. Observe that such spectrums may be detectable with the third-generation detectors. Amplitude birefringence for a single tensor perturbation can be extended to chiral mixing between two tensor perturbations. Such a chiral mixing is one example of GW oscillations (similar to neutrino oscillations) considered in [43] which describes the mixed flavor of two tensor perturbations. Other examples include friction, (1) velocity, and mass mixing. The flavor mixing between the tensor perturbations hij (2)

and hij can be written in a unified framework as 

d2 ˆ 2 + Nk ˆ + Mˆ + νˆ + Ck dη2

  (1)  h = 0. h(2)

(30)

Here, νˆ is the friction matrix, Cˆ is the velocity matrix, Nˆ is the chirality matrix, Mˆ is the mass matrix, and k is the wave number. It would be interesting to study

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how accurately one can probe each of these GW oscillation effects with current and future GW observations.

Open Questions We end this chapter by presenting several open questions that need to be addressed to further improve the ability of testing GR with GWs: 1. Higher PN corrections in the inspiral: One of the current major issues is that methods for extracting physical implications from theory-agnostic tests are limited due to the lack of complete inspiral-merger-ringdown waveforms in theories beyond GR. In the inspiral part, most studies focused on finding the leading PN corrections to the waveform. Higher PN corrections that are usually neglected are necessary to construct complete waveforms in non-GR theories, especially since the PN approximations become inaccurate close to merger. Toward constructing complete waveforms, one needs to consider generic binary systems, including spin precession and large eccentricity. 2. Corrections in the merger phase: Numerical relativity simulations of compact binary mergers have been carried out in a few non-GR theories, such as scalartensor theories and EdGB/dCS gravity [44,45], though more simulations need to be performed with a broader class of theories, with parameters (mass ratio, spins, etc.) varied systematically. When carrying out numerical relativity simulations, one also needs to check the well-posedness of the theories, though one way to overcome this is to treat the theories as effective theories and solve order by order in theoretical parameters that are assumed to be small, as done within EdGB/dCS simulations [44, 45]. 3. Corrections in the ringdown phase: From numerical relativity simulations, one can also find corrections in the ringdown phase. Another approach is to find corrections to the QNMs through the BH perturbation. However, most studies focused on nonrotating BHs, and we still lack a systematic analysis of including spins. Perhaps the first step is to carry out calculations for slowly rotating BHs. Ideally, one would want to extend the analysis for arbitrary spins, though Kerrlike solutions are still lacking in many modified theories of gravity. 4. Phenomenological waveforms: Once the above inspiral, merger, and ringdown corrections to the waveform are computed, one can then attempt to construct phenomenological IMR waveforms in specific non-GR theories. Such waveforms will be useful for, e.g., extracting physics from merger-ringdown gIMR parameters. It would be also interesting to investigate whether one can construct phenomenological waveforms for scalar and vector polarization modes that are typically present in non-GR theories. 5. GW memory: There are nonlinear effects in the waveform that are subdominant and have not been studied much in theories beyond GR. One example is the GW memory that may be measured with future ground-based and space-based detectors. It would be important to extend the Bondi-Sachs formalism and

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compute the memory waveforms in modified theories of gravity, including scalar and vector memories. Such nonlinear effects may give new insights to beyondGR theories. 6. Cosmological screening: Another nonlinear effect that may arise in non-GR theories motivated to explain cosmological problems is various screening mechanisms [46]. Some studies exist on how such mechanisms affect the GW generation [47] and propagation [48], though more analysis needs to be done along this direction. 7. Astrophysical systematics: In order to test GR, one needs to have systematic errors under control. One possible source of systematics is through astrophysical effects [49], such as accretion disks [50], tidal resonances [51], and dark matter halos. More work needs to be done on how such systematics may limit the ability of testing GR with GWs. 8. Electromagnetic counterparts: GW170817 opened a new window for multimessenger astronomy, and we expect to find similar events in the future. It would be interesting to perform merger simulations of binary NSs in modified theories of gravity and predict how the electromagnetic counterpart signals may get modified from GR. One can then compare the GW and electromagnetic signals to perform multi-messenger tests of GR, which is beyond what has already been done by, e.g., comparing the arrival time difference between gravitons and photons to constrain the propagation speed of GWs and the graviton mass [4].

Cross-References  Emission of Gravitational Radiation in Scalar-Tensor and f(R)-Theories  Testing the Nature of Dark Compact Objects with Gravitational Waves Acknowledgments Z.C. and K.Y. acknowledge support from the Owens Family Foundation. K.Y. also acknowledges support from NSF Award PHY-1806776, NASA Grant 80NSSC20K0523, and a Sloan Foundation Research Fellowship.

References 1. Popper K (1934) The logic of scientific discovery. Mohr Siebeck 2. Yunes N, Yagi K, Pretorius F (2016) Theoretical physics implications of the binary black-hole mergers gw150914 and gw151226. Phys Rev D 94:084002 3. Abbott BP et al (2016) Tests of general relativity with GW150914. Phys Rev Lett 116(22):221101. [Erratum: (2018) Phys Rev Lett 121(12):129902] 4. Abbott BP et al (2017) Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys J 848(2):L13 5. Abbott BP et al (2019) Tests of general relativity with GW170817. Phys Rev Lett 123(1):011102 6. Arun KG, Iyer BR, Qusailah MSS, Sathyaprakash BS (2006) Testing post-Newtonian theory with gravitational wave observations. Class Quantum Gravity 23:L37–L43

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7. Yunes N, Pretorius F (2009) Fundamental theoretical bias in gravitational wave astrophysics and the parameterized post-Einsteinian framework. Phys Rev D80:122003 8. Tahura S, Yagi K (2018) Parameterized post-Einsteinian gravitational waveforms in various modified theories of gravity. Phys Rev D98(8):084042 9. Cutler C, Flanagan EE (1994) Gravitational waves from merging compact binaries: how accurately can one extract the binary’s parameters from the inspiral waveform? Phys Rev D 49:2658–2697 10. Carson Z, Yagi K (2020) Parametrized and inspiral-merger-ringdown consistency tests of gravity with multiband gravitational wave observations. Phys Rev D 101:044047 11. Abbott BP et al (2019) GWTC-1: a gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. Phys Rev X9(3):031040 12. Tahura S, Yagi K, Carson Z (2019) Testing gravity with gravitational waves from binary black hole mergers: contributions from amplitude corrections. Phys Rev D100(10):104001 13. Abbott BP et al (2019) Tests of general relativity with the binary black hole signals from the LIGO-Virgo catalog GWTC-1. Phys Rev D100(10):104036 14. Yunes N, Hughes SA (2010) Binary pulsar constraints on the parameterized post-Einsteinian framework. Phys Rev D82:082002 15. Carson Z, Yagi K (2019) Parameterized and consistency tests of gravity with gravitational waves: current and future. In: Proceedings, recent progress in relativistic astrophysics: Shanghai, 6–8 May 2019 16. Carson Z, Yagi K (2020) Multi-band gravitational wave tests of general relativity. Class Quantum Gravity 37(2):02LT01 17. Sesana A (2016) Prospects for multiband gravitational-wave astronomy after GW150914. Phys Rev Lett 116(23):231102 18. Barausse E, Yunes N, Chamberlain K (2016) Theory-agnostic constraints on black-hole dipole radiation with multiband gravitational-wave astrophysics. Phys Rev Lett 116(24):241104 19. Chamberlain K, Yunes N (2017) Theoretical physics implications of gravitational wave observation with future detectors. Phys Rev D96(8):084039 20. Carson Z, Seymour BC, Yagi K (2020) Future prospects for probing scalar–tensor theories with gravitational waves from mixed binaries. Class Quantum Gravity 37:065008 21. Ghosh A, Johnson-McDaniel NK, Ghosh A, Mishra CK, Ajith P, Pozzo WD, Berry CPL, Nielsen AB, London L (2017) Testing general relativity using gravitational wave signals from the inspiral, merger and ringdown of binary black holes. Class Quantum Gravity 35: 014002 22. Abbott BP et al (2016) Properties of the binary black hole merger GW150914. Phys Rev Lett 116(24):241102 23. Husa S, Khan S, Hannam M, Pürrer M, Ohme F, Forteza XJ, Bohé A (2016) Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal. Phys Rev D 93:044006 24. Cutler C, Vallisneri M (2007) LISA detections of massive black hole inspirals: parameter extraction errors due to inaccurate template waveforms. Phys Rev D76:104018 25. Ghosh A et al (2016) Testing general relativity using golden black-hole binaries. Phys Rev D94(2):021101 26. Carson Z, Yagi K (2020) Probing string-inspired gravity with the inspiral-merger-ringdown consistency tests of gravitational waves. Class Quantum Gravity 37:215007 27. Carson Z, Yagi K (2020) Probing beyond-Kerr spacetimes with inspiral-ringdown corrections to gravitational waves. Phys Rev D 101:084050 28. Johannsen T, Psaltis D (2011) A metric for rapidly spinning black holes suitable for strong-field tests of the no-hair theorem. Phys Rev D83:124015 29. Will CM (1998) Bounding the mass of the graviton using gravitational wave observations of inspiralling compact binaries. Phys Rev D57:2061–2068 30. Mirshekari S, Yunes N, Will CM (2012) Constraining generic Lorentz violation and the speed of the graviton with gravitational waves. Phys Rev D85:024041

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31. Blas D, Ivanov MM, Sawicki I, Sibiryakov S (2016) On constraining the speed of gravitational waves following GW150914. JETP Lett 103(10):624–626. [(2016) Pisma Zh Eksp Teor Fiz 103(10):708] 32. Will CM (2018) Solar system versus gravitational-wave bounds on the graviton mass. Class Quantum Gravity 35(17):17LT01 33. Berti E, Buonanno A, Will CM (2005) Estimating spinning binary parameters and testing alternative theories of gravity with LISA. Phys Rev D71:084025 34. Baker T, Bellini E, Ferreira PG, Lagos M, Noller J, Sawicki I (2017) Strong constraints on cosmological gravity from GW170817 and GRB 170817A. Phys Rev Lett 119(25):251301 35. Boran S, Desai S, Kahya EO, Woodard RP (2018) GW170817 falsifies dark matter emulators. Phys Rev D97(4):041501 36. Nishizawa A (2018) Generalized framework for testing gravity with gravitational-wave propagation. I. Formulation. Phys Rev D97(10):104037 37. Saltas ID, Sawicki I, Amendola L, Kunz M (2014) Anisotropic stress as a signature of nonstandard propagation of gravitational waves. Phys Rev Lett 113(19):191101 38. Belgacem E, Dirian Y, Foffa S, Maggiore M (2018) Modified gravitational-wave propagation and standard sirens. Phys Rev D98(2):023510 39. Belgacem E et al (2019) Testing modified gravity at cosmological distances with LISA standard sirens. JCAP 1907:024 40. Alexander S, Finn LS, Yunes N (2008) A gravitational-wave probe of effective quantum gravity. Phys Rev D78:066005 41. Yagi K, Yang H (2018) Probing gravitational parity violation with gravitational waves from Stellar-mass black hole binaries. Phys Rev D97(10):104018 42. Yunes N, O’Shaughnessy R, Owen BJ, Alexander S (2010) Testing gravitational parity violation with coincident gravitational waves and short gamma-ray bursts. Phys Rev D82:064017 43. Jimenez JB, Ezquiaga JM, Heisenberg L (2020) Probing cosmological fields with gravitational wave oscillations. JCAP 2004:027 44. Witek H, Gualtieri L, Pani P, Sotiriou TP (2019) Black holes and binary mergers in scalar Gauss-Bonnet gravity: scalar field dynamics. Phys Rev D99(6):064035 45. Okounkova M, Stein LC, Scheel MA, Teukolsky SA (2019) Numerical binary black hole collisions in dynamical Chern-Simons gravity. Phys Rev D100(10):104026 46. Jain B, Khoury J (2010) Cosmological tests of gravity. Ann Phys 325:1479–1516 47. de Rham C, Tolley AJ, Wesley DH (2013) Vainshtein mechanism in binary pulsars. Phys Rev D87(4):044025 48. Perkins S, Yunes N (2019) Probing screening and the graviton mass with gravitational waves. Class Quantum Gravity 36(5):055013 49. Barausse E, Cardoso V, Pani P (2014) Can environmental effects spoil precision gravitationalwave astrophysics? Phys Rev D89(10):104059 50. Kocsis B, Yunes N, Loeb A (2011) Observable signatures of EMRI black hole binaries embedded in thin accretion disks. Phys Rev D84:024032 51. Bonga B, Yang H, Hughes SA (2019) Tidal resonance in extreme mass-ratio inspirals. Phys Rev Lett 123(10):101103

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Georgios Lukes-Gerakopoulos and Vojtˇech Witzany

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief Introduction to Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous and Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hamiltonian Systems and Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré Surfaces of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of Orbits in Maps and Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KAM, Poincaré-Birkhoff Theorem, and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tools to Study Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inspirals Through Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Generic Inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-resonant Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Near-Resonant Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbital Motion in Kerr Spacetimes and Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deviating Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spinning Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact of Non-integrability on Extreme-Mass-Ratio Systems . . . . . . . . . . . . . . . . . . . . . . . . Resonance Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prolonged Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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G. Lukes-Gerakopoulos () Astronomical Institute of the Czech Academy of Sciences, Prague, Czech Republic V. Witzany School of Mathematics and Statistics, University College Dublin, Dublin, Ireland e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_42

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Abstract

The largest part of any gravitational-wave inspiral of a compact binary can be understood as a slow, adiabatic drift between the trajectories of a certain referential conservative system. In many contexts, the phase space of this conservative system is smooth, and there are no “topological transitions” in the phase space, meaning that there are no sudden qualitative changes in the character of the orbital motion during the inspiral. However, in this chapter, we discuss the cases where this assumption fails and nonlinear and/or non-smooth transitions come into play. In integrable conservative systems under perturbation, topological transitions suddenly appear at resonances, and we sketch how to implement the passage through such regions in an inspiral model. Even though many of the developments of this chapter apply to general inspirals, we focus on a particular scenario known as the extreme mass ratio inspiral (EMRI). An EMRI consists of a compact stellar-mass object inspiraling into a supermassive black hole. At leading order, the referential conservative system is simply geodesic motion in the field of the supermassive black hole, and the rate of the drift is given by radiation reaction. In Einstein gravity, the supermassive black hole field is the Kerr spacetime in which the geodesic motion is integrable. However, the equations of motion can be perturbed in various ways so that prolonged resonances and chaos appear in phase space as well as the inspiral, which we demonstrate in simple physically motivated examples. Keywords

Black holes · LISA · EMRI · Celestial mechanics · Chaos · Dynamical systems

Introduction An Extreme Mass Ratio Inspiral (EMRI) is an event that is expected to occur once a stellar-mass compact object (the secondary) is captured on a sufficiently tight orbit by a supermassive black hole (the primary) in a center of a galaxy. As in the case of any compact-object binary, the motion of the two bodies in this specific binary creates gravitational waves that carry energy and angular momentum to infinity. Consequently, the orbit decays in a spiralling motion, and the stellar-mass compact object eventually plunges into the supermassive black hole. The gravitational waves carrying the imprint of this motion peak in the mHz frequency band and are expected to be observed by future gravitational-wave observatories such as the Laser Interferometer Space Antenna (LISA) [1] and Taiji [29]. However, in contrast to stellar-mass compact binaries observed by the ground-based detectors LIGO and Virgo, a sizable fraction of EMRIs should enter the sensitivity band in an eccentric, orbitally precessing state. As a result, we have to consider EMRI scenarios from a richer and more complex landscape of dynamics.

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A common approximate framework for the description of this special class of gravitational-wave inspirals is that the secondary is replaced by a point particle in the field of the much more massive primary with equations of motion ordered by the mass ratio q = μ/M, where μ is the mass of the secondary and M the mass of the primary. The correspondence between the position of the “particle” and the real body is established through matched asymptotic expansions [53]; see also  Chap. 36, “Black Hole Perturbation Theory and Gravitational Self-Force” in this handbook. Roughly speaking, the position of the particle corresponds to the center of mass of the body. At zeroth order, the equations of motion are those of a free test particle, or a geodesic, in the field of the primary black hole. First- and second-order corrections are then “self-force terms” obtained from black-hole perturbation theory as well as effects due to the finite size of the body [6, 70]. The corrections to the zeroth-order equations then cause small local deviations from geodesic motion, but the most important global effect is the gradual decay of the orbit. Geodesic motion in the field of an isolated spinning black hole in Einstein gravity, the Kerr spacetime, is integrable [14]. Consequently, if we only model an EMRI as motion adiabatically drifting from geodesic to geodesic in Kerr spacetime, we obtain a reasonably simple waveform with slowly drifting fundamental frequencies and harmonics. Will this simplicity hold once we refine the inspiral model? Unfortunately no, at least generically. Once we include immediate, “conservative” effects to the equations of motion or introduce even slight modifications to Kerr spacetime, the integrability is mostly broken. Discussing the precise properties of the resulting near integrable of weakly chaotic systems and the consequences for the inspiral is the subject of this chapter. In an EMRI system, chaos itself is not expected to be a prominent effect, even if an EMRI can pass through a chaotic layer for an extremely brief period of time. On the other hand, passage through resonant regions in an EMRI poses the biggest challenge for waveform models [9, 12, 24]. At this point, we should define what do we even mean by the word resonance. The broadest definition of a resonance is when two or more characteristic frequencies of a system match in integer ratios. This means that there is a relative phase of the motions that is “frozen by kinematic coincidence,” and the resonant orbit stops sampling the available phase space (see Fig. 1). This has more than one consequence, which has led to some confusion in the literature. An obvious issue is that an orbit sampling only a part of the dynamically available phase space may not have the symmetries of the equations of motion. Hence, even though the equations of motion may possess rotational symmetry, the resonant orbit may not. In the context of gravitational radiation, this can mean that it radiates anisotropically, which results in a resonantly enhanced kick to the binary in question [see 67]. In many gravitational-wave applications, one is interested only in the time-averaged flux of gravitational waves over the orbit, and for non-resonant orbits, the time average is interchangeable with the more convenient phase-space average. However, the resonant orbits evolve through a smaller subset of the phase space, which means the averaging formulas have to be modified for them in such computations [30, 31, 64].

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Fig. 1 Three orbits with varying ratios of the radial oscillation frequency Ωr and azimuthal rotation frequency Ωφ plotted for ten azimuthal cycles. The leftmost, almost generic orbit very quickly samples all the possible phase shifts between the r and φ motion and densely covers the available space. However, as the ratio Ωr /Ωφ approaches close to 1/2, the orbit is averaging over the available space at a slower and slower rate, which can even be slower than the inspiral itself

Nevertheless, the subject of this chapter is different. We are concerned with the fact that an integrable, conservative dynamical system under perturbation often develops a special dynamical phenomenon in a region of nonzero phase-space volume around the original kinematic resonances, which we will call a prolonged resonance [17, 37, 73]. A prolonged or perhaps “inflated” resonance is a region in phase space where trajectories are observed to oscillate around a finite subset of the original resonant trajectories. The qualitative transition between generic motion and the prolonged resonance has to be treated with special care in an inspiral computation. In the EMRI literature, one can encounter the terms transient and sustained resonances during an inspiral [23, 66]. What is the relation of these terms to our “prolonged resonance?” The self-force on the inspiraling body in the EMRI can be formally decomposed into parts that cause the secular decay of the orbit and those which do not (conventionally called “averaged dissipative self-force” and “oscillating dissipative and conservative self-force,” respectively). Taking only the second part into account, we obtain a virtual conservative dynamical system that has the character of some sort of perturbed geodesic motion in Kerr spacetime. Even though it has been posited that even this perturbed system could stay integrable [23], we generically expect that it will not be and that it will contain prolonged resonances. Depending on the character of both the prolonged resonances and the secular part of the self-force, the encounter of the real inspiral with these structures can be transient or, under special conditions, sustained for a time comparable with the inspiral time. That is, transient and sustained resonances as referred to in the EMRI literature can be understood as different modes of interaction with the topological structure of the prolonged resonance. In this chapter, we first establish the general mathematical theory of dynamical systems, focusing on perturbed Hamiltonian systems. The center stage is occupied by prolonged resonances, and we also sketch how to treat the inspiral through it.

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After establishing the general theory and tools, we pass to specific cases of nearintegrable systems and provide some numerical examples. Throughout the chapter, we use the G = c = 1 geometric units, the (−, +, +, +) signature of the metric, and bold characters denote vectors (n-tuples). From section “Orbital Motion in Kerr Spacetimes and Perturbations” on the Einstein summation convention is employed, and the Greek indices μ, ν, κ, λ, . . . run from 0 to 3.

Brief Introduction to Dynamical Systems Continuous and Discrete Dynamical Systems A system evolved forward in time by a set of equations is a dynamical system. Depending on whether the time evolution is taking place in continuous or discrete time steps, the dynamical systems split into continuous systems and mappings, respectively. Formally, a set of first-order differential equations dx = f(x, t) dt

(1)

defines a continuous dynamical system. In Eq. (1) x is a vector belonging to the phase space S , f is a vector function on the phase space, and t is a continuous evolution variable, which usually has the meaning of time. For the purposes of our discussion, we always assume that f is sufficiently smooth. We say that the phase space has dimension D if the phase-space vector x has dimension D. Equation (1) is called the equation of motion, and its solution with respect to specific initial conditions is called a trajectory in phase space. On the other hand, a set of difference equations xn+1 = F(xn , n)

(2)

defines a discrete mapping. In Eq. (2), xn is again a vector in phase space, F is a vector function, and n ∈ N is a label for the discrete time steps. In the framework of general relativity, the dynamical systems in question are continuous. Even though continuous systems can be reduced to mappings, as we will discuss later on, for now, we are going to focus only on the former. In continuous dynamical systems, the equations of motion define a flow in phase space Ft : S → S along which an initial condition x0 evolves to x in time t, i.e., x(t) = Ft (x0 ). If there is a volume element on S such that the size of any volume of initial conditions along the flow does not change, then the system is called conservative. If f in Eq. (1) does not depend explicitly on time, then the system is autonomous. An example of a phase-space flow of an autonomous conservative system with a phase space of dimension 2 is given in Fig. 2.

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2

px

1

0

-1

-2

-3

-2

-1

0

1

2

3

x Fig. 2 The phase-space portrait of a nonlinear pendulum with the Hamiltonian H = px2 /2 − cos(x). The arrows denote the direction of the flow, and the black line is the separatrix H = 0 separating the topologically distinct oscillations around x = 0 (also known as librations) from the rotations (in the sense that the motion continuously rotates through the periodic x ∈ (0, 2π ]). Both the types of motion can be understood as bound and the phase-space trajectory a topological circle (T1 = S 1 ), but they belong to a different homotopy class on the phase-space manifold

Hamiltonian Systems and Integrability Consider a Hamiltonian system with N degrees of freedom, that is, a phase space consisting of N-dimensional positional vector q = (q 1 , . . . , q n ) and a corresponding set of N conjugate momenta p = (p1 , . . . , pN ). Hamilton’s equations then read dq ∂H = , dt ∂p

dp ∂H =− . dt ∂q

(3)

These can be put in the form   dx ∂H 0 IN = = f(x), · −IN 0 dt ∂x

(4)

where IN is the N-dimensional identity matrix and x = (q, p). In other words, a Hamiltonian system with N degrees of freedom is a dynamical system with a phase space of dimension 2N. Hamilton’s equations can be shown to conserve the volume

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form dN p dN x, which means Hamiltonian systems are conservative [52]. An even stronger statement can be proven, and the equations can be shown to conserve the so-called symplectic differential form dp ∧ dq from which the conservation of the volume form trivially follows. The Hamiltonian system is autonomous when ∂H /∂t = 0, which also implies dH /dt = 0 along the trajectories. In other words, in autonomous systems, H is a constant of motion, and its value depends only on the initial condition x0 . We say that any differentiable function I (x) is a constant of motion (or an integral of motion) iff d I /dt = 0 for any trajectory in S . An autonomous Hamiltonian system is called integrable iff it has at least as many constants of motion as degrees of freedom, and this set of constants of motion is functionally independent and in involution. Two functions on phase-space I1 (x), I2 (x) are said to be in involution if {I1 , I2 } =

  ∂I1 ∂I2 0 IN · = 0, · −IN 0 ∂x ∂x

(5)

where { } is the Poisson bracket. In that case, the trajectories stay on the hypersurface defined by Ii = const., i = 1, . . . , N . Such a set of conditions defines an Ndimensional manifold M , which is smooth and invariant under the action of the flow Ft in S . If the manifold M is compact and connected, then according to the Liouville-Arnold theorem, M is diffeomorphic to the N-dimensional torus TN . This case corresponds to bound integrable motion. Consequently, one can transform to a convenient set of canonical variables known as action-angle coordinates (θθ , J) such that H (x, p) = H (J) and the equations of motion reduce to ∂H ∂H ˙ , J=− = 0. θ˙ ≡ ω (J) = ∂J ∂θθ

(6)

The θ variables correspond to the angles on the torus TN , while their conjugate momenta, “the actions,” J correspond to the integrals of motion. If we specify all of the J, then both the torus on which the motion takes place and the fundamental frequencies of motion ω (J) are also specified. We can vary J, to explore the foliation of the phase space S by the tori. On the other hand, if the set of frequencies ω is to uniquely specify the torus on which we are moving, the following non-degeneracy condition must also hold globally:  det

ω ∂ω ∂J

 = 0 .

(7)

In particular, this condition is violated for harmonic motion and near non-degenerate equilibrium points where ω is locally a constant vector. In other words, we cannot tell the values of actions (which are proportional to oscillation amplitudes) from the frequencies of near-equilibrium oscillations alone.

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The character of the motion on the torus depends on whether any of the fundamental frequencies match in an integer ratio. More generally, it depends on whether there exist linearly independent integer vectors k such that k·ω ≡

N 

ki ωi = 0, where ki ∈ Z and |k| ≡

i=1

N 

|ki | = 0 .

(8)

i=1

Equation (8) is called a resonance condition. The number of linearly independent k for which this holds true is called the number of resonant conditions fulfilled by the motion. The motion on a torus TN is quasiperiodic if no resonant condition is fulfilled. A quasiperiodic orbit will densely cover the torus in infinite time, and it will not return to the initial condition from where it started in finite time. On the other hand, the motion is also ergodic, which means that an infinite-time average of a phase-space function along the motion can be replaced by a phase-space average over the torus. If there are m < N − 1 independent resonant conditions, then the quasiperiodic orbit will cover densely a TN −m torus which is a submanifold of the respective TN torus. If there are m = N − 1 resonant conditions, then the motion is periodic. By solving the respective system of resonant conditions in the fully periodic case, we can pick one of the frequencies, e.g., ω1 , and express all the other frequencies as ωi = ri ω1 , where ri ∈ Q.

Poincaré Surfaces of Section Let us now discuss the visualization of the foliation of tori in autonomous Hamiltonian systems of 2 degrees of freedom. Such dynamical systems have fourdimensional phase spaces and T2 tori foliating them, depending on two integrals of motion (one of the integrals can be chosen to be the Hamiltonian H ). If we restrict ourselves to a hypersurface where one of the integrals of motion is kept fixed, we are reduced to a three-dimensional space filled with two-dimensional tori, which is still hard to visualize. Thus, we have to make a well-chosen section through this space that transversely (non-tangentially) cuts through the tori and allows us to examine the foliation (see Fig. 3). This is a so-called Poincaré surface of section, which actually corresponds to a discrete mapping, called Poincaré mapping, as defined in Eq. (2), because every consecutive point is uniquely determined by the previous one. The term Poincaré surface of section is most often reduced to just Poincaré section or surface of section in the bibliography. Following this tradition, we use these reduced terms in the article interchangeably. To construct the surface of section numerically, one has to integrate the equations of motion and identify the constant of motion to be held fixed (often the Hamiltonian) and a good section condition Φ(x, p) = 0. For instance, if we know that all orbits of interest oscillate about a certain equilibrium point, it is good to put the section into that point. Once the trajectory passes through Φ(x, p) = 0, the only two

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Fig. 3 An illustration of sections obtained from 40 intersections of various phase-space trajectories (red) through a Poincaré surface of section (cyan). The trajectories stay on the torus defined by the integrals of motion (green). The top-left trajectory is a 3:2 resonant trajectory (the frequency of motion over the small circle of the torus is two thirds of that over the large circle), and it repeatedly intersects only a set of three points on the section. The rest of the trajectories are non-resonant, and they asymptotically trace out a cross-section of the torus

remaining phase-space coordinates are recorded and plotted, and this is repeated until many points from a single trajectory are gathered. Since in the integrable case a quasi-periodic trajectory densely fills the torus, the set of points from it gradually circle out a single closed curve on the plot, and it is called an invariant curve, since on a Poincaré section, it maps itself on itself. As this is repeated with a number of independent trajectories, the foliation is revealed as a set of nested non-intersecting closed curves (see Fig. 5). On the other hand, trajectories fulfilling a resonant condition only fill a subspace of the torus and appear as a finite set of periodically repeating points on the section. For each resonance, the number of these sets is infinite. Each set, however, consists of a finite number of points equal to the periodicity of the resonance, i.e., to the number of mappings needed for a periodic orbit to return to its initial condition on the Poincaré section. The periodicity of the resonance is also called the multiplicity of the resonance. The applicability of Poincaré surfaces of section goes well beyond bound integrable systems. On a surface of section, one can observe the breaking of integrability, mainly by studying the neighborhood of periodic points as explained in section “Stability of Orbits in Maps and Continuous Systems”. The case of two degrees of freedom is the lowest number of degrees of freedom in which non-integrability can occur, and it is thus also the best studied case. On the other hand, most physical systems of interest have more degrees of freedom, so it would seem the Poincaré surface of section is not useful for them. In that case, one has to carefully consider the symmetries of the problem and see whether the essential dynamics can be observed only in a subsector of the system, i.e., in a reduced system. For example, motion of a particle in a three-dimensional

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axisymmetric potential has 3 degrees of freedom, but the azimuthal symmetry angle is redundant, and the azimuthal angular momentum Lz is constant along orbits. We can then understand the motion of all particles with the same Lz as a family of dynamical systems of two degrees of freedom where Lz plays the role of a parameter of the system.

Stability of Orbits in Maps and Continuous Systems Periodic and fixed points can be considered as the skeleton of a dynamical system, since they are tied to “topological transitions” in the flow or, in physical terms, to qualitative changes of the motion in phase space. Their stability can be found by applying linear perturbation theory. This can be done in the sense of a discrete dynamical system on a Poincaré surface of section or for the continuous flow in the full phase space, both of which will now be discussed.

Fixed Points in Discrete Dynamical Systems Let xf be a fixed point of the mapping F(xn ), i.e., xf = F(xf ). For every periodic point xp with periodicity j , a new mapping H = Fj (xn ) can be defined so that xp = H(xp ). Thus, we can implicitly treat both periodic and fixed points by only discussing fixed points. The stability of a fixed point can be examined by a linear perturbation xn = xf + δxn around the fixed point. The resulting variational equations read δxn+1 = Aδxn ≡

 ∂F  δxn , ∂xn xn =xf

(9)

where the Jacobian matrix calculated at the fixed point and A is known as the monodromy matrix. The solution of this equation is found by projecting the initial conditions into the eigenbasis of A, and the individual components then evolve as ∝ λn , where λ are the respective eigenvalues. In other words, the stability of an eigendirection depends on whether |λ| is smaller or larger than one. For Poincaré maps generated by Hamiltonian systems, it holds that det(A) = 1. In the case of two-dimensional maps, this implies that the eigenvalues will appear in pairs of the type λ1 , λ2 = 1/λ1 . The explicit formula for the two eigenvalues reads λ1,2 =

Tr(A) ±



Tr(A)2 − 4 , 2

(10)

where Tr(A) is the trace of the monodromy matrix. • If |Tr(A)| < 2, then λ1,2 ∈ C, and the eigenvalues can then be rewritten as λ = exp±iϑ , where ϑ = cos−1 Tr(A) indicates the angular velocity with which 2 the nearby points are rotating around the fixed point. Due to this rotation, the

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point is sometimes called elliptic, but physically, it corresponds to a stable point and the phase-space rotations around it correspond to small oscillations in the configuration space. • If |Tr(A)| = 2, then λ1,2 = 1 or λ1,2 = −1. The point in this case is indifferently stable. Often this means the point is “fixed by kinematic coincidence.” In other cases, the appearance of an indifferently stable point means that variations of the system parameters will induce a topological transition in the phase-space flow. • If |Tr(A)| > 2, then λ1,2 ∈ R. Each eigenvalue corresponds to an eigenvector defining an eigendirection. If |λ1 | > 1, then λ1 corresponds to an unstable eigendirection and λ2 = 1/λ1 to a stable one. These eigendirections define a hyperbolic flow around the fixed point; therefore, the point is called hyperbolic. However, physically, the point corresponds to an unstable equilibrium; hence, it is called an unstable point; the presence of both the diverging and approaching directions typically corresponds to the same process just with flipped directions of time. In the case of higher-dimensional Hamiltonian maps, the eigenvalues have to come in quartets of the form λ, 1/λ, λ∗ , 1/λ∗ .

Stability of Periodic Trajectories and Fixed Points in Continuous Systems Let us consider a linear perturbation y = x+ on any point x of the phase-space S of a Hamiltonian dynamical system. This perturbation lies in the tangent space Tx S to the phase space at the point x. The deviation vector ξ can be evolved along the flow Ft by applying a linear operator from the tangent space at one point along the trajectory to a tangent space at a later point Dt : Tx S → TFt (x) S . The action of this operator takes the deviation vector ξ (t0 ) at time t0 and evolves it to ξ (t) = Dt ξ (to ) at time t. The respective evolution equations are given by a variation of Eq. (4) and read  dξξ ∂f  = ξ. dt ∂x x(t)

(11)

The vector ξ is then interpreted as the deviation between two infinitesimally close solutions of the system. In return, this deviation can show as how sensitive is a part of a dynamical system to its initial conditions. Let us first discuss the stability of a strictly fixed point xf such that dx/dt = f(xf ) = 0 where the evolution is simplified to  dξξ ∂f  ξ ξ. = Bξ ≡ dt ∂x xf

(12)

The fact that the matrix B is constant implies that the linearly independent solutions for ξ are proportional to eκt , where κ are eigenvalues of B. Since B is generated

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from a Hamiltonian system (4), its eigenvalues come in “loxodromic quartets” κ, κ ∗ , −κ, −κ ∗ (degenerate doublets are also possible if either κ ∈ R, or κ ∈ I). If the real part of κ is positive or negative, the respective eigenvector corresponds to a stable or unstable direction, respectively. Consider a fixed point of a Hamiltonian with a single degree of freedom, that is, a two-dimensional phase space. Now we have only three options similar to the fixed point in discrete systems: • κ1,2 ∈ I, κ1 = κ2 ∗ = −κ2 . We define ω1,2 = κ1,2 /i so that the solutions are rotating at angular velocity ω about the fixed point. Again, this case corresponds to a stable equilibrium. • κ1 = κ2 = 0. This is again an indifferently stable fixed point. • κ1,2 ∈ R, κ1 = −κ2 . The eigendirection corresponding to the positive κ is unstable, and the one to the negative stable, the flow is hyperbolic about xf , and it corresponds to an unstable equilibrium. An example of a stable fixed point is x = 0, px = 0 in Fig. 2. In the same figure, an unstable fixed point can then be found at x = π, px = 0. Let us now turn to the stability of periodic orbits in continuous systems. There we can define a discrete map of the form (2) as the evolution of the whole system by the period of the orbit T , F(x) = FT (x). The periodic orbit then reduces to a fixed point on this discrete map, and we can easily see that its monodromy matrix A is equal to DT . Hence, up to a few cosmetic changes, the stability theory of periodic orbits is identical to that of fixed points of discrete maps as given in section “Fixed Points in Discrete Dynamical Systems”. Additionally, it often turns out that in separable systems, periodic orbits can be examined in separable subspaces where they appear as fixed points. An important example of this would be circular geodesics in black hole spacetimes, in which the circular orbits appear as fixed points in the radial sector. The stability theory of periodic orbits in these separable subspaces is then identical to that of continuous-system fixed points discussed above. An example of such a reduction in Schwarzschild spacetime is given in Fig. 4.

Stable and Unstable Manifolds We have discussed the linear stability of fixed points and periodic orbits in discrete and continuous dynamical systems. However, how does that relate to a more global, nonlinear picture? According to the Hartman-Grobman theorem, the qualitative picture of the motion we obtain from the linearization around fixed points and periodic orbits is always correct in some small neighborhood of the point [26, 28]. Furthermore, the stable manifold theorem guarantees that one can prolong the stable and unstable directions into the so-called stable and unstable asymptotic manifolds [50]. These are formally defined as follows: • The stable asymptotic manifold M s is the set of points that asymptotically approach the unstable fixed point as t → ∞.

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Fig. 4 Left: The Schwarzschild effective potential for a free test particle at specific angular momentum L = 3.7M (blue) and the corresponding separatrix specific energy Esep . The potential is defined using specific energy E as Veff = E(pr = pz = 0, L = 3.7M) (see Eq. (55)). Right: The corresponding motion in the radial r − pr phase space. The eternally bound motion is separated from the plunging motion by the homoclinic separatrix (black), which originates at the unstable circular orbit at r ≈ 4.4M. The stable circular orbit is at r ≈ 9.4M, and it manifests as a stable fixed point in the diagram. Another separatrix between motion escaping and non-escaping to infinity would appear at E = 1 outside of the plot range

• The unstable asymptotic manifold M u is the set of points that approach the unstable fixed point as t → −∞. It can be shown that the stable asymptotic manifold of one fixed point cannot cross itself or the stable asymptotic manifold of another fixed point. The same holds for the unstable asymptotic manifolds. However, stable and unstable asymptotic manifolds can cross each other. Crossings of the same fixed point are then called homoclinic points, and crossings of stable and unstable manifolds belonging to different fixed points are called heteroclinic points. The character of the crossing is crucial. If M u and M s have a tangential intersection (their tangent manifolds coincide at the crossing), then they are necessarily just a part of a single smooth homoclinic or heteroclinic manifold, and each of their points is part both of M u and M s . These cases are known also as separatrices. However, if they cross transversely (at least parts of their tangent manifolds are independent at the crossing), then they both cannot form the same smooth manifold, and there has to be an infinite number of such crossings. The result is the infamous homoclinic (or heteroclinic) tangle where M u and M s are folded in an infinitely intricate manner into each other [63] giving rise to chaotic orbits. This is discussed in detail in the section “Chaotic Layers”.

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Stability of Generic Trajectories In some sense, it is also possible to measure the stability of any orbit in phase space. If the norm of the deviation vector ξ introduced in the last section grows linearly with time, then the orbit is characterized as regular (mildly sensitive to perturbations), while if it grows exponentially, it is either chaotic (highly sensitive to perturbations). A chaotic orbit is an orbit that is not periodic while being highly sensitive to perturbation; otherwise, it is simply an unstable periodic orbit. An indicator of chaos based on the orbit stability is the maximal Lyapunov Characteristic Exponent 1 |ξξ (t)| log , ξ (t0 ) t→∞ t |ξξ (t0 )|

mLCE = max lim

(13)

where |ξξ | is some norm of the deviation vector (the result is independent of the choice of the norm). It is straightforward to see that mLCE → 0 for regular orbits, while for the unstable ones, mLCE will converge to a constant value equal to the exponent of the exponential growth of ξ . In practice, mLCE for non-periodic orbits has to be evaluated numerically, and then the limit is approximated only by a finite-time integration. As a result, we can only detect characteristic exponents  1/tint , where tint is the integration time. In other words, we can often only notice chaos if it is sufficiently strong, and we may be unable to numerically distinguish between very mild instability and regularility. Also, even though the mLCE is defined as the maximum over the initial ξ , a generic initial condition will very often have a nonzero projection into the unstable part of the deviation subspace, which then always dominates the late-time growth. It is thus sufficient to compute the limit in (13) just for a handful of linearly independent vectors to determine the mLCE.

KAM, Poincaré-Birkhoff Theorem, and Chaos Chaos arises around resonances or unstable equilibria if an initially integrable system is perturbed. The basic features of the transition from integrability to nonintegrability are dominated by two theorems: the Kolmogorov-Arnold-Moser (KAM) theorem [3, 35, 47] and the Poincaré-Birkhoff theorem [11, 51].

KAM Theory and Birkhoff Chains Let us take an autonomous integrable Hamiltonian system H0 (J) with N degrees of freedom expressed in action-angle variables J, θ fulfilling the non-degeneracy conditions (7). Now let us consider a close smooth Hamiltonian system of the form H (θθ , J) = H0 (J) + εH1 (θθ , J) ,

(14)

where ε  1. The KAM theorem tells √ us that if the perturbation ε is sufficiently small, then there exists a K(ε)  O( ε) and a d > N − 1 such that the set of tori

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satisfying the Diophantine condition N    K(ε)   ki ω i  > ,    |k|d

(15)

i=1

will survive the perturbation with only small deformations. These tori are called KAM tori and their depiction on a Poincaré section KAM curves. In the KAM theorem, the Diophantine condition ensures that the surviving tori are sufficiently √ far away from a resonance. However, the condition is fulfilled in a ∼1 − O( ε) fraction of the volume of the phase space. Thus, the qualitative character of the motion in the system is mostly conserved under √ small perturbations. Nevertheless, generally, there is also an O( ε) volume around resonances where the character of the motion changes qualitatively, and this change is described by the Poincaré-Birkhoff theorem [11,51]. From the infinite number of orbits on a resonant torus, only 2n, n ∈ N stay periodic, n of which are stable and n of which unstable. In a subspace orthogonal to the periodic orbits, the phase space structure is reminiscent of a nonlinear pendulum (see section “Tools to Study Resonances”). This can be easily observed on Poincaré surfaces of section of systems of 2 degrees of freedom. Specifically, all unperturbed orbits on a resonance with ω1 : ω2 = r : s will have a periodicity j equal to either r or s on the section, depending on its construction. After the perturbation, only an even number 2nj of periodic points will survive on the section; half of them will be stable and the other half unstable. This chain of stable and unstable points in a resonance is called a Birkhoff chain. Remember that every periodic orbit corresponds to j points on the section. For instance, in the case n = 1, all the stable and unstable points correspond to a single stable or unstable phase space trajectory, respectively.

Chaotic Layers As discussed in section “Stability of Orbits in Maps and Continuous Systems”, unstable points are anchoring points of stable and unstable asymptotic manifolds. This is true both for unstable trajectories in resonances and unstable periodic trajectories and fixed points in integrable systems before their perturbation. These manifolds are the birthplace of chaos for non-integrable systems. When and if the stable and unstable manifolds intersect transversely at one location in phase space, they necessarily have to do so an infinite number of times, which causes an infinitely folded fractal-like structure called a homoclinic (or heteroclinic) tangle. A homoclinic tangle is non-integrable and implies chaos. For example, black hole spacetimes naturally contain unstable circular orbits. Once these orbits are slightly pushed in the radial direction while keeping their energy and angular-momentum constant, they become either asymptotically approaching or diverging zoom-whirl orbits. For unstable circular orbits with specific energy below one, the orbits that are pushed radially outward will at first spiral out to a finite distance from the original orbit but eventually return and start spiraling back in to the unstable circular orbits, which makes them homoclinic.

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These families of zoom-whirl orbits in phase space define homoclinic manifolds, and it is here where we most often find chaos in perturbed black hole fields [e.g., 54, 59, 71]. Nevertheless, for small perturbations, homoclinic chaos occurs only in a small layer around the √ asymptotic manifolds. In other words, the constants of motion J stay within O( ε) bounds as compared to the non-integrable system, and we mostly lose phase information (i.e., error in θ quickly becomes O(1)). However, as the perturbation further grows, KAM tori can dissolve into cantor sets called cantori [21]. The gaps in the cantori allow unstable and stable asymptotic manifolds of different resonances or unstable periodic orbits to cross each other, and hence heteroclinic chaos appears. Heteroclinic chaos means that the chaotic trajectory can √ drift through the non-integrable layers to values of J that are  O( ε) far from the original values, which makes the unpredictability of the system more dire. It is also generally observed that the absolute value of Lyapunov exponents grows once heteroclinic chaos is established. There is no unanimously accepted definition of chaos. For instance, in the discussion above, we have informally used the definition of a chaotic orbit as one that is not periodic and has a positive Lyapunov characteristic exponent. On the other hand, we have also worked with the assumption that broken integrability in a volume of phase space of a Hamiltonian system implies chaoticity of all trajectories in that volume. Fortunately, the link between the two can be numerically established as follows: 1. We observe the breaking of integrability as the trajectories not being bound to an N-dimensional torus in phase space. Instead, they densely fill a finite connected sub-manifold X of dimension larger than N. Such trajectories are known as transitive trajectories in X , and their existence implies that Ft is transitive in X [62]. This means that for every two non-empty regions U , V ∈ X , there exists a T such that FT (U ) ∩ V = 0. 2. If we further assume that periodic orbits are dense in X (while being of zero measure), it follows that Ft has sensitive dependence on initial conditions in X [5]. Here, sensitivity means that there is a distance δ such that in an arbitrarily close neighborhood of any x ∈ X , there always exists a point y and a constant T such that |FT (x) − FT (y)| > δ, where | · | is some metric distance on X . Of course, the second assumption that the non-integrable region is densely filled with periodic orbits is nontrivial. A dense set of periodic points can only be proven to exist along the asymptotic manifolds when a homoclinic tangle occurs due to a transversal intersection of the stable and unstable manifolds [63]. It is plausible that this structure is promoted to the rest of the non-integrable volume, but not rigorously proven. However, there is ample numerical evidence that non-integrability always implies sensitive dependence on initial conditions [16].

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Tools to Study Resonances Since the resonances are the places where chaotic motion arises, it would be useful to have a tool to spot them. A natural approach would be to survey the phase space by identifying fundamental frequencies of motion in the Fourier transform of the trajectories. However, the interpretation of the numerical results of this procedure can be tricky if the system is not in action-angle variables [37]. In systems of two degrees of freedom, one can instead use a Poincaré surface of section to evaluate the so-called rotation number νϑ . The rotation number νϑ is defined as the ratio between the two fundamental frequencies of the system. According to this method, one first identifies the center of a main island of stability, i.e., the fixed point xc on the Poincaré section around which the majority of the invariant curves are nested, and then finds the rotation angles ϑi := ang [(xi+1 − xc ) , (xi − xc )] between successive intersections xi of the trajectory with the section with respect to xc . The angles can be defined with respect to any reasonable polar coordinates centered on xc . The rotation number is then calculated as the average of the rotation angles N 1  ϑi . N →∞ 2π N

νϑ = lim

(16)

i=1

For a non-degenerate foliation of an integrable system, the rotation number changes strictly monotonically as one moves away from xc . The dependence of νϑ on the distance from the center of the island is also known as the rotation curve (see left set of panels in Fig. 5). Under perturbation, the curve stays qualitatively similar to the unperturbed system up to resonances, near which new features appear. Namely, chaotic layers in the resonance appear as a random fluctuations of the rotation curve, while the islands of stability of a Birkhoff chain will create plateaus of constant rotation number values. The width of the plateau in weakly √chaotic regions, when measured properly, corresponds quite accurately to the O( ε) width of the resonance. The width of a resonance is a very useful quantity, since in a perturbed Hamiltonian system (14), the width of the resonances relates to the perturbation parameter ε. In the following passage, we sketch the basic steps to reach this relation; for further details, the interested reader is referred to [2, 46]. Let us consider a 2 degrees of freedom Hamiltonian system; then, the system (14) reduces to

H = H0 (J1 , J2 ) + εH1 J1 , J2 , θ 1 , θ 2 . (17) Assuming that at the action values of the unperturbed Hamiltonian system H0 J1 = J1r , J2 = J2r lies a resonance k1 ω1 + k2 ω2 = 0; we will rotate the action-angle variables as

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Fig. 5 The top panels show Poincaré sections, while the bottom panels show rotation curves calculated along the ρ˙ = 0 line. All panels are calculated for a set of MSM parameters M = 1, a = 0.999 M and orbital parameters Lz = 3 M, E = 0.97. The left set of panels are plotted for b = 0, while the right for b = 2.1 M. The Poincaré section of the left case is dominated by KAM curves (top panel); thus, when one starts from the center of the foliation (ρ ≈ 15 M, ρ˙ = 0) and moves away along ρ˙ = 0, the respective rotation curve appears to be strictly monotonic (bottom panel). The right case focuses on a resonance. The top panel is dominated by an island of stability of the 6/7 resonance, which is reflected on the rotation curve in the bottom panel by the characteristic plateau

J1 J2 + , θ˜ 1 = k1 θ 1 + k2 θ 2 , J˜1 = 2k1 2k2

(18)

J1 J2 − , θ˜ 2 = k1 θ 1 − k2 θ 2 . J˜2 = 2k1 2k2

(19)

One can easily see that these new variables are canonical by evaluating the symplectic form dJ˜1 ∧dθ˜ 1 +dJ˜2 ∧dθ˜ 2 . It should also be noted that the θ coordinates ˜ which then means that have to be wound more than once to reach periodicity in θ, ˜θ 1 and θ˜ 2 turn out to be 2(k 2 + k 2 )π and 4|k1 k2 |π -periodic, respectively. 1 2

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In the new variables, the resonance condition has been reduced to ω˜ 1 = 0. This implies that at the resonance for the perturbed system holds θ˙˜1 = O(ε). By applying an “averaging” near-identity transform J˜2 → J˜2 + O(ε), θ˜ 2 → θ˜ 2 + O(ε), we can eliminate the phase θ˜ 2 (see [2] and section “Inspirals Through Resonances”). This averaging will render our system (17)

to be approximated by the integrable system 1 ˜ ˜ ˜ ˜ ˜ H = H0 J1 , J2 +εH1 J1 , J2 , θ , in which the action J˜2 is a constant of motion. The next step is to expand this Hamiltonian function in terms of the deviation of the action ΔJ˜1 = J˜1 − J˜1r from the resonance value to the leading order, which

2

results in H = β2 ΔJ˜1 + εF θ˜ 1 , where β, F depend on the constants J˜2 , J˜1r . To arrive to the final form of the Hamiltonian, F is expanded in a Fourier series from which we keep only the leading harmonic and introduce a phase shift to θ˜ 1 to obtain H =

β ˜ 2 ΔJ1 + εα cos n θ˜ 1 , 2

(20)

where n corresponds to the one shown when discussing the Poincaré-Birkhoff theorem in section “KAM Theory and Birkhoff Chains”. Without loss of generality, we can choose conventions such that α, β, ε are positive parameters and the function (20) is then essentially the Hamiltonian of a nonlinear pendulum. Hence, what we have shown that the phase portrait of a resonance can be approximately mapped to that of a pendulum.

For ΔJ˜1 = 0 the minima and the maxima of cos nθ˜ 1 correspond to stable and unstable fixed points, respectively. From the unstable fixed points stem separatrices, separating the near-resonant Birkhoff chain from the rest of the KAM tori. The location of the separatrices is approximately given as the level set H = εα of the Hamiltonian (20), which yields

2

2εα ΔJ˜1 |sep = 1 − cos n θ˜ 1 β

(21)

The width of the resonance is defined as the difference between

maximum and the the minimum value of ΔJ˜1 on the separatrix (i.e., when cos n θ˜ 1 = −1) 



α√ width := max ΔJ˜1 |sep − min ΔJ˜1 |sep = 4 ε. β

(22)

Another way to find the width of the resonance is from the opening angle dΔJ˜1 |sep /dϑ˜ 1 between the separatrices at an unstable point. For example, for small deviations from the unstable point at θ˜1 = 0, Eq. (21) reduces to

2

2

2 width n θ 1 εα 1 ΔJ˜1 |sep,θ˜ 1 →0 = n θ˜ . = β 4

(23)

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Hence, by having the coordinates ΔJ˜1 , θ 1 of the separatrix near the unstable point, we can estimate the width of the resonance. Let us also establish a direct correspondence with the discussion of Birkhoff chains in section “KAM Theory and Birkhoff Chains”. If we go back to the initial system (17) and choose a Poincaré section on the plane θ 2 = 0, then from the transformation (18), we get θ˜ 1 = k1 θ 1 everywhere on the section. This implies that on the surface of section and at the k1 ω1 + k2 ω2 = 0 resonance, there will appear a total of nk1 islands of stability and nk1 unstable points. In the following, labels 1, 2 can be swapped to obtain a different section.

Inspirals Through Resonances Here we briefly sketch the necessary theory and methods that are needed to efficiently integrate a set of dissipative equations through a resonance in a nearintegrable system. At certain points, we fast-forward to the discussion of the meaning of this theory in the context of gravitational-wave inspirals; the reader is welcome to skip these portions of the text and come to them later.

A Generic Inspiral Consider an unperturbed (not necessarily Hamiltonian) dynamical system of N degrees of freedom in action-angle coordinates j, θ . For clarity of certain complicated expressions in this section, we switch to components ja , θ b ; a, b = 1 . . . N, θ b ∈ (0, 2π ], and we will use the Einstein summation convention. Now we subject this system to a generic non-Hamiltonian smooth perturbation of order ε  1 such that the equations of motion become j˙a = εfa(1) (jb , θ c ) + ε2 fa(2) (jb , θ c ) + . . . ,

(24)

b b θ˙ b = ωb (ja ) + εg(1) (ja , θ c ) + ε2 g(2) (ja , θ c ) + . . .

(25)

In the language of the gravitational self-force, ε corresponds to the mass ratio, f (n) (j, θ ), and then involves both the averaged and oscillating dissipative self-force and some of the conservative self-force of order n. On the other hand, g(n) (j, θ ) involves only the oscillating dissipative and conservative parts of the self-force of order n. Now we are interested in the approximate evolution of this system over a long time tinsp that can be characterized as tinsp ∼ 1/(εω). Specifically, we would like the error of the final phase θ b (tinsp ) to go to zero in the ε → 0 limit. In the completely generic case, one needs to switch between the treatment for non-resonant (weakly resonant) and strongly resonant parts of phase space. Recall that generic resonances are hypersurfaces in the action space characterized by Eq. (8), which in the current discussion reads

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ka ωa (jb ) = 0, ka ∈ ZN . Such hypersurfaces create a dense net in the phase space, and in practice one needs to identify a finite set of “strong” resonances for a separate treatment. This is done b into trigonometric polynomials (a Fourier by expanding the functions fa(n) , g(n) coefficient expansion) over θ b . The magnitude of the terms in this expansion will quickly fall off due to the smoothness of the functions f, g, and one can thus split the expansion into a finite number of dominant O(1) terms and sub-dominant terms (n) b of relative order ε. This can be viewed as a smoothing of the functions fa , g(n) on the torus, where oscillations in the function values of relative magnitude  ε are pushed to higher order. Even though this discarding scheme is dependent on the absolute value of ε, it is the only way to isolate a finite number of resonances and build an effective ε → 0 limit of the perturbed system. In fact, the understanding of this “moving target” character of the perturbation theory as ε → 0 is one of the essential points of the proof of the famous Kolmogorov-Arnol’d-Moser theorem. A strong resonance at O(εn ) is then a resonance with a wavenumber kc which has a trigonometric polynomial of order kc in the dominant part of fa(n) (resonances b cause minor resonances in our inspiral-type scenario). caused by g(n)

Non-resonant Motion Away from strong resonances, we can apply a near-identity transform Ja = ja + (1) (2) b (j , θ a ) + ε 2 ηb (j , θ a ) such that εξa (jb , θ c ) + ε2 ξa (jb , θ c ), Θ b = θ b + εη(1) c (2) c the equations of motion attain the form [2] J˙a = εFa(1) (Jb ) + ε2 Fa(2) (Jb ) + O(ε3 ) , ˙b

Θ =Ω

b

(Ja ) + εGb(1) (Ja ) + ε2 Gb(2) (Ja ) + O(ε3 ) ,

(26) (27)

where F, G are given as sums of averages of f, g over the angles θ b and various transformation terms. Ω b can be chosen to be functionally identical with ωb apart from the fact that Jb was inserted instead of jb as the argument. The transformation vectors ξ, η can then be computed from the requirement that the transformation leads to equations of motion of forms (26) and (27), which yields θ  ξa(1) = fa(1) , θ  ∂ωa a a η(1) = g(1) + b ξb(1) , ∂j  θ ∂fa(1) b ∂fa(1) (1) ∂ξa(1) (1) ∂ξa(1) b (2) (2) ξa = fa + η + ξ − F − G , ∂θ b (1) ∂jb b ∂jb b ∂θ b (1)

(28) (29)

(30)

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G. Lukes-Gerakopoulos and V. Witzany

 a η(1)

=

a g(2)

a a ∂ωa (2) 1 ∂ 2 ωa (1) (1) ∂η(1) (1) ∂η(1) b + ξb + ξb ξc − Fb − G ∂jb 2 ∂jb ∂jc ∂jb ∂θ b (1)

θ , (31)

 θ  hk (j a ) b b eikb θ , h(ja , θ b ) ≡ b a ikb ω (j )

(32)

kb =0

where hkb are the Fourier coefficients of h over θ b . The transformation is determined uniquely only up to “integration constants,” and one can add arbitrary bounded smooth functions of ja (but not of θ b ) to any of the ξ, η. The advantage of the transformed system (26) and (27) is the possibility to integrate the new actions Ja separately from the angles Θ b . The actions evolve on the inspiral time-scale ∼J0 /(εF (1) ) ∼ 1/(εΩ), and their decay is usually stiff. On the other hand, the phases evolve on a time-scale 1/Ω, and their evolution has the character of a steady increase with a slowly changing slope. It is then obvious that the J -Θ split allows for an efficient choice of time steps and integration methods for each of the sub-problems. (1) (2) In fact, once the functional forms of Fa , Fa , Gb(1) are explicitly known, we can drop the Gb(2) terms and integrate the equations, and the resulting solutions Ja (t), Θ b (t) will be globally ε-close to the exact solution ja (t), θ b (t) over a time interval of order 1/ε if no strong resonances are encountered [34]. Even more, (2) b since none of the needed quantities depend on ξa , η(2) , the second-order part of the transform does not need to be known explicitly for such a solution. In the language of gravitational self-force, this is equivalent to the statement that for accurate inspirals not passing through resonances, one requires the full phase dependence of the first-order dissipative and conservative self-force, but only the average dissipative piece of the second-order self-force.

Near-Resonant Motion It can be easily checked that the transformation functions ξ (n) , η(n) contain poten(1) tially singular terms. The leading-order singularity for ξ (n) is ∼εn fk /(kb ωb )2n−1 (1) and for η(n) even ∼εn fk /(kb ωb )2n . When we approach sufficiently close to a strong resonance such that ka ωa → 0 for some ka , the near-identity transform √ becomes ill-convergent and even completely meaningless when ka ωa ∼ ε. One thus needs to switch to a different description at some well-chosen point before the breakdown. To “hand over” the original variables j, θ with sufficient accuracy, one should compute as many terms of the second-order transform ξ (2) , η(2) as possible without the knowledge of g(2) (j, θ ), f (2) (j, θ ), since the known terms are also the most singular near resonance. Then, if we choose a cutoff index

40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

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β ∈ (0, 1/2) such that we cut off the evolution at ka ωa ∼ εβ , we will hand over the phases θ with error terms of order ∼ε3−6β and the actions with errors of order ∼ε3−5β . The optimal choice of β will be discussed in the section “Error Budget”. Let us define the resonant phase γ ≡ ka θ a and a set of non-resonant phases a ˜ θ˜ , a˜ = 1 . . . N − 1 obtained through various linear combinations of the original phases θ a such that θ a → γ , θ˜ a˜ is an invertible coordinate transform. We can then ˜ ˜ carry out a near-identity transform ja , θ˜ b , γ → J˜a , Θ˜ b , Γ eliminating the phases θ˜ a˜ (or Θ˜ a˜ ) analogous to the one for the full set θ a above [2]. However, in this case, we will have no convergence issues near the resonance, since the second-order transformation terms have only denominators of the type k˜a˜ ω˜ a˜ , k˜a˜ ∈ ZN −1 where ω˜ a˜ are the non-resonant frequencies. We then obtain evolution equations of the form J˙˜a = εF˜a(1) (J˜b , Γ ) + ε2 F˜a(2) (J˜b , Γ ) + O(ε3 ) ,

(33)

˜ ˜ ˜ b˜ (J˜a , Γ ) + ε2 G ˜ b˜ (J˜a , Γ ) + O(ε3 ) , Θ˙˜ b = Ω b (J˜a ) + εG (1) (2)

(34)

Γ˙ = ka ωa (J˜a ) + εχ(1) (J˜a , Γ ) + ε2 χ(2) (J˜a , Γ ) + O(ε3 ) ,

(35)

˜ Ω˜ is analogous to F, G, Ω in the previous where the meaning of the functions F˜ , G, paragraphs. Additionally, we see that the “frequency” of Γ -evolution is ka ωa  √ O( ε) in the resonant region and that χ(1,2) collects transformed terms of order ε, ε2 , respectively. This leads to Γ often being called a “semi-fast” variable, since it evolves much slower than the regular phases Θ˜ in the resonant region but generally faster than J˜a . The main advantage of this modified coordinate transform is that even though we increase the number of variables that need to be solved in the first step by one to the set Γ, J˜a , at least the N − 1 phases Θ˜ a˜ can be still solved later in a separate step. Hence, we will now only focus on the solution for Γ, J˜a . In the context of the gravitational self-force, we may not be able to easily evaluate terms such as F˜ (2) (J˜, Γ˜ ) everywhere, so we need to analyze the costs of omitting a part of them over the period of integrating through the non-resonant region. Specifically, it will be possible to evaluate F˜ (2) (J˜, Γ˜ ) exactly when ka ωa = 0, since then it amounts only to an infinite-time average over individual resonant trajectories, but we assume it will not be possible to evaluate this term anywhere else. This assumption is not set in stone; it is in principle possible to evaluate derivatives of F˜ (2) (J˜, Γ˜ ) by computing black hole perturbations based on a “blurred” stationary ˙˜ Γ˙ , the same way F (2) (J ) has to be computed on a blurred trajectory with nonzero J, stationary trajectory with a nonzero J˙ [44].

Error Budget ˜ integrate through the Let us assume that we switch to the coordinates J˜, Γ, Θ, resonance for a period Δtpass and then switch back to the coordinates J, Θ. Considering that the switch happens whenever ka ωa ∼ εβ and the frequencies drift

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G. Lukes-Gerakopoulos and V. Witzany

with a rate ∼εΩ,J F , we can estimate that Δtpass ∼ εβ−1 . Let us now estimate the errors by Taylor-expanding the solutions for J˜(t), Γ (t) around the instant tr at which ka ωa = 0 exactly. For the actual evolution, it is more practical to numerically integrate the equations of motion, but the Taylor expansion provides a good way to estimate the overall error budget and reads: J˜a (tr + Δtpass ) =

∞ ˜(l)  J (tr ) l=0

Γa (tr + Δtpass ) =

l!

(Δtpass )l ∼

∞  Γ (l) (tr ) l=0

l!

∞ 

J˜(l) (tr )εl(β−1) ,

(36)

l=0

(Δtpass )l ∼

∞ 

Γ (l) (tr )εl(β−1) .

(37)

l=0

We can then evaluate all the time-derivatives J˜(l) , Γ (l) from the equations of motion to a certain accuracy given that we are able to evaluate F˜ (1) (J˜, Γ ), χ(1) (J, Γ ) and all its derivatives. As mentioned above, we can also assume that we can evaluate F (2) (J, Γ )|tr but none of its derivatives, which leads to a phase error term that is bounded by ∼ε3β and an action error term bounded by ε1+2β . Another source of ˜ (2) (J˜, Γ ), which leads to a phase error error comes from the inability to evaluate G 1+β of order ε and action error again of order ε1+2β . Additionally, one must also consider that the phase Γ and actions J˜ were handed over with ∼ε3−5β , ε3−6β errors, respectively; this induces errors of order ∼ε2−4β in both phases and actions after the resonant evolution. What is then the optimal value for β given that we are interested in the overall inspiral faithfulness? A smaller β means that we integrate in the near-resonant coordinates for a longer time while possibly not increasing the accuracy of the total computation any more. On the other hand, a larger β means a larger error is accumulated as one approaches and drifts away from the resonant region in the fully averaged coordinates. We thus need to find a value of β such that the handover happens exactly at the point when further near-resonant integration would be redundant. However, the final answer also depends on whether we care more about the error in the phase or in the actions. Let us now assume that the inspiral encounters only a single strong resonance at a generic point, that is, a point such that there is still a ∼1/ε time left after resonance exit. Then any error in the action upon resonance exit translates into the final inspiral phase with a factor ∼1/ε while the resonant phase error is, at leading order, simply added to final phase error. It is then easy to see that the error budget is dominated by the actions, and it is optimized exactly when β = 1/4. This is because at that point, the error of the near-resonant integration and the handover in the actions are both ∼ε3/2 . In summary, assuming one has the complete first-order pieces of the perturbation and time-averaged dissipative pieces of the second-order perturbation, it is possible to squeeze the total inspiral phase error of the passage √ through a single resonance to ∼ ε in a well-defined √ procedure. However, ignoring the resonant terms entirely would lead to a ∼1/ ε error in the inspiral phase.

40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

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An informed reader will notice that this scenario corresponds to something which is known √ as a transient resonance, that is, the resonant behavior is dominant √ for a time ∼1/ ε. However, there may occur cases such that ka ωa remains  O( ε) for a longer time. The time derivative of the resonant condition is d ∂ωa fb (jc , θ ) + O(ε2 ) . (ka ωa (jb )) = ka ω˙ a = εka dt ∂jb

(38)

The assumption of our analysis then is that the average of ka ω˙ a over the nonresonant phases is never below order ε everywhere where the resonant condition ka ωa = 0 is met. If, however, a resonance has ka ω˙ a = O(ε2 ) anywhere, then it is in principle possible to obtain a sustained resonance such that the dominant behavior is dominant for a time of the order of the entire inspiral ∼1/ε [67]. We expect these to be nongeneric cases that are negligible in realistic systems. However, if some symmetry makes this degeneracy of the equations of motion generic, the most efficient system for the integration over the time of the entire inspiral is simply the “near-resonant” system (33)–(35).

Additional Perturbations Let us now consider additional perturbations of magnitude κ  1 such that the equations of motion become j˙a = εfa(1) (jb , θ c ) + ε2 fa(2) (jb , θ c ) + κh(jb , θ c ) + . . . ,

(39)

b b θ˙ b = ωb (ja ) + εg(1) (ja , θ c ) + ε2 g(2) (ja , θ c ) + κl(ja , θ c ) + . . .

(40)

We will classify the additional perturbation as conservative when h(jb , θ c ) has a zero average over θ c and thus the κ-terms do not contribute to the long-term decay of actions jb , and as dissipative otherwise. Depending on the context, one can also have various ∼κε cross-terms as well. For instance, for inspirals in stationary axisymmetric spacetimes that are some deformations of Kerr spacetime (see the section “Deviating Spacetimes”), the perturbation is conservative, and one can generally have κ  ε, and κε cross-terms would correspond to the fact that the self-force deviates from the Kerr self-force in the modified spacetime. On the other hand, for modifications of gravity that only alter the radiation-reaction dynamics, one would generally consider κ  ε. It is obvious that away from resonances, one can perform consecutive near-identity transforms that gradually eliminate the phase dependence of all the κ and ε terms from the equations of motion and put them into an “averaged” form. Once again, however, separate treatment is required near resonances. Let us discuss only the effect of large conservative perturbations near resonances in more detail. A conservative perturbation with κ  ε requires the switch to a near resonant description of the equations of motion already when ka ωa ∼ κ β , β  ∈

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G. Lukes-Gerakopoulos and V. Witzany

(0, 1/2), and then the near-resonant integration has to be carried out over an interval  Δtpass ∼ κ β /ε. In other √ words, the near-resonant behavior dominates the motion for an order of at least ∼ κ/ε cycles. By using the techniques sketched above, it is then easy to show that the leading-order contribution of the κ-terms to the final inspiral  phase due to the resonance passage is of the order κ 1+β /ε2 . If the contribution of κ-terms at resonance are completely ignored, the system will have a phase error of order κ 3/2 /ε2 at the end of the inspiral. To which degree this error can be removed depends on a more delicate analysis of the relative magnitudes of the ε and κ terms and our ability to evaluate them. Restricting now to the discussion of EMRIs and the gravitational self-force in non-Kerr spacetimes, one of the main issues would be the fact that there would be various ∼κ n εl cross-terms due to the fact that one has to use different self-force than in the Kerr spacetime. The dominant source of irremovable error would probably be an unknown ∼κε term in j˙ correcting the radiation-reaction.

Orbital Motion in Kerr Spacetimes and Perturbations The current consensus is that the spacetime around a black hole is described by the Kerr solution [33]. This paradigm, broadly used in the fields of astrophysics and gravitational-wave theory, is known also as the "Kerr black hole hypothesis" [4]. The metric elements of the Kerr spacetime in Boyer-Lindquist coordinates (t, r, ϑ, ϕ) read gtt = −1 + gϕϕ =

2Mr 2aMr sin2 ϑ Σ , gtϕ = − , grr = , Σ Σ Δ

Λ sin2 ϑ , gϑϑ = Σ , Σ

(41)

where Σ = r 2 + a 2 cos2 ϑ , Δ =  2 − 2Mr ,  2 = r 2 + a 2 , Λ =  4 − a 2 Δ sin2 ϑ ,

(42)

M is the mass and a is the angular momentum per mass. The Kerr metric describes an asymptotically flat vacuum spacetime that is stationary, axisymmetric, and symmetric with respect to reflections about the equatorial plane (ϑ = π/2). The Kerr metric describes the field of an isolated rotating black hole as long as a horizon is covering the ring singularity, which holds for a < M. For a > M the Kerr spacetime corresponds to a naked singularity. For a = 0 the Schwarzschild solution is recovered, and the Boyer-Lindquist coordinates are reduced to Schwarzschild coordinates. A Hamiltonian function giving the geodesic motion of a massive test particle with respect to the proper time τ in curved spacetime reads

40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

H =

1 νκ 1 g pν pκ = − μ2 , 2 2

1651

(43)

where μ is the mass of the test particle. In the framework of general relativity, the system has four degrees of freedom. In the case of Kerr spacetime, the system is integrable, since there are four independent integrals of motion in involution. Namely, the stationarity and the axisymmetry imply that energy E = −pt = and angular momentum Lz = pϕ along the symmetry axis z are constants of motion. Additionally, the Hamiltonian itself expresses the conservation of the test particle’s mass, and the fourth constant  Lz 2 + a 2 μ2 cos2 ϑ K = pϑ + aE sin ϑ − sin ϑ  = 2  2 E − a Lz pr − (Δpr 2 + μ2 r 2 ) 

2

(44)

discovered by Carter [14] reflects a hidden symmetry. In the non-spinning limit of Kerr spacetime, i.e., the spherically symmetric Schwarzschild spacetime, the Carter constant reduces to the total angular momentum, which is constant as well. The existence of the four integral of motion suggests the system should be separable, i.e., we should be able to evolve each degree of freedom independently, but in order to achieve this, the Carter-Mino time has to be employed [14, 45].

Deviating Spacetimes The Carter constant is a unique feature of the Kerr spacetime, since it appears that there is no other stationary, axisymmetric, and asymptotically flat spacetime in general relativity that possesses similar “hidden” symmetry [15, 25]. Attempts to construct solutions possessing Carter constant by perturbing the Kerr metric led to solutions obeying alternative theories of gravity, but not the Einstein’s field equations [32, 69]. Even on the level of a Newtonian and electromagnetic analogue of Kerr, the Carter constant appears to be a unique feature of the Kerrlike Newtonian and electromagnetic fields [22, 39, 42]. This implies that in the framework of general relativity, any deviation from Kerr spacetime destroys the integrability of the geodesic motion.

Bumpy Black Holes One way to parametrize a solution of Einstein’s field equations that deviates continuously from the Kerr one is to introduce one or more parameters changing the Geroch-Hansen multipole moments of the Kerr field [27] Mn + iSn = M(ia)n , n ∈ N ,

(45)

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G. Lukes-Gerakopoulos and V. Witzany

where Mn and Sn are the mass and the current-mass multipole moments, respectively. For example, the Manko-Novikov solution [40] introduces an extra parameter for each mass multipole moment. The fact that the solution is characterized by another parameter that can be seen in its structure outside the horizon necessarily implies that such solutions have at least broken horizons; otherwise, the no-hair theorem would be violated. In the case of the Manko-Novikov solution, this manifests as a ring singularity on the horizon, which disappears as the extra multipole moments are switched off and the Kerr solution is recovered. Such solutions are often called bumpy or non-Kerr black holes and reflect the possibility of having compact objects in the general relativity framework that challenges the Kerr black hole hypothesis, even though it is currently unclear how such objects should form. As an example of a bumpy black hole, we are going to use a reduced version of an exact solution known as the Manko, Sanabria-Gómez, Manko (MSM) solution [41]. The original MSM spacetime depends on five real parameters: the mass M, the spin a (per unit mass M), the charge, the magnetic dipole moment, and the mass-quadrupole moment Q. However, in the reduced version, the charge and the magnetic dipole are set to zero. This allows the mass-quadrupole moment  Q = −M

(M 2 − (a − b)2 )2 + 4 M 2 b2 − a b + a2 4(M 2 − (a − b)2 )

 (46)

to deviate from the Kerr one QKerr = −aM 2 by one free parameter b. The Kerr mass-quadrupole is retrieved from Eq. (46) for b2 = a 2 − M 2 . Note that since for black holes M > a, this implies that b is imaginary, which is not an issue for the MSM spacetime [41]. If the quadrupole deviation parameter is defined as δQ := Q − QKerr =

2 M M 2 + b2 − a 2 , 4(a 2 − 2a b + b2 − M 2 )

(47)

then for δQ > 0, the MSM describes a more prolate bumpy black hole than the Kerr one and for δQ < 0 a more oblate one. For b = 0, the Tomimatsu Sato δ = 2 M solution is retrieved [41], and δQ = − (M 2 − a 2 ). 4 For presenting the spacetimes deviating from Kerr, it is useful to introduce the Weyl set of coordinates, which relates to the Boyer-Lindquist coordinates as follows: ρ=

√

Δ sin ϑ, z = (r − M) cos ϑ

(48)

√ 2 2 Note that in the Weyl set of coordinates, the event horizon H = M + M − a for   r√ √ Kerr lays at ρ = 0 and stretches along z in the interval − M 2 − a 2 , M 2 − a 2 , i.e., the horizon is reduced to a line segment along the z-axis. Obviously, this set of coordinates does not cover the spacetime inside the event horizon. Even though it is

40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

1653

possible to use imaginary values of the coordinates to reach the black hole interior; see Ref. [8]. The line element in Weyl coordinates reads   ds 2 = −e2ν (dt − γ dϕ)2 + e−2ν e2ψ (dρ 2 + dz2 ) + ρ 2 dϕ 2 .

(49)

For the reduced MSM spacetime, which is used for the numerical examples in the article, the metric functions read e2ν =

E , (E + R P + (v 2 − 1)S T )

e2ψ =

E , 16κ 8 (u2 − v 2 )4

γ =

(50)

(v 2 − 1)(R T − κ 2 (u2 − 1)S P ) , E

E = R 2 + κ 2 (u2 − 1)(v 2 − 1)S 2 , S = −4(a − b)[κ 2 (u2 − v 2 ) + 2δv 2 ] + v 2 M 2 b , P = 2{κMu[(2κu + M)2 − 2v 2 (2δ + ab − b2 ) − a 2 + b2 ] − 2v 2 (4δd − M 2 b2 )}, R = 4[κ 2 (u2 − 1) + δ(1 − v 2 )]2 + (a − b)[(a − b)(d − δ) − M 2 b](1 − v 2 )2 , T = 4(2κMbu + 2M 2 b)[κ 2 (u2 − 1) + δ(1 − v 2 )] + (1 − v 2 ){(a − b)(M 2 b2 − 4δd) − (4κMu + 2M 2 )[(a − b)(d − δ) − M 2 b]},

(51)

where κ=

√

d +δ, δ = −

M2

M 2 b2 1 , d = [M 2 − (a − b)2 ] . 4 − (a − b)2

(52)

Note that all the metric functions are expressed in generalized spheroidal coordinates u, v, while the line element (49) is written in the Weyl coordinates ρ, z. The transformation between them reads  ρ = κ (u2 − 1)(1 − v 2 ) , z = κuv .

(53)

√ For b2 = a 2 − M 2 , it can be shown that κ = M 2 − a 2 , v = cos ϑ, and u = (r − M)/κ. A useful concept to study the geodesic motion in a spacetime (49) is that of the effective potential. If we take the Hamiltonian function (43) and divide it with the square of the test particle, we arrive to

− gρρ ρ˙ 2 + gzz z˙ 2 = Veff = (1 + g tt E 2 − 2g tϕ E Lz + g ϕϕ L2z ) ,

(54)

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G. Lukes-Gerakopoulos and V. Witzany

Fig. 6 Left panel: The effective potential when z = 0 for the Kerr spacetime with a = 0.999M and the specific orbital parameters set to E = 0.97 and Lz = 3M (red dotted curve) and the respective effective potential for MSM spacetime when b = 2.1M (black curve). Right panel: The curve of zero velocity Veff = 0 for the cases shown in the left panel

dx μ is the four-velocity, while E = −pt /μ and Lz = pϕ /μ are the where x˙ μ = dτ specific energy and angular momentum, respectively, i.e., the energy and angular momentum per unit mass μ. For simplicity, we keep the same symbols E, Lz as for the respective quantities not divided by particle rest mass. Which type is used each time is either stated explicitly or implied by the units. The lhs of Eq. (54) is a strictly non-positive quantity, since gρρ = gzz > 0 outside the horizon, implying that geodesic motion is allowed only if Veff ≤ 0 (Fig. 6). The Veff = 0 limit defines the curve of zero velocity, i.e., the limit along which a geodesic has zero radial and polar velocity. On the equatorial plane z = 0 bounded orbits are defined by two radii ρ1 < ρ2 , for which Veff (ρ1 ) = Veff (ρ2 ) = 0 (Fig. 6). For a circular orbit, it holds that Veff = ∂V∂ρeff = 0, while the stability of circular orbits is defined by the 2

sign of ∂ ∂ρV2eff : positive for the stable ones and negative for the negative. Note that the definition of the effective potential is not unique. For example, alternatively, one can define the following expression as effective potential:      g tϕ 2 g ϕϕ g tϕ 1  Veff (ρ, z; Lz ) = E(ρ˙ = z˙ = 0, Lz ) = tt Lz + − tt L2z − tt tt g g g g (55) as was done for Fig. 4. Despite the different definitions, the dynamics defined by any of the effective potentials are obviously the same.

40 Nonlinear Effects in EMRI Dynamics and Their Imprints on Gravitational Waves

1655

External Matter Deformation Another way to introduce deformations to the black hole field to cause chaos is by surrounding it with additional matter [61]. The gravitating matter can correspond to accretion disks or rings, clouds of dust, or matter halos. Exact solutions for spinning black holes surrounded by matter are rare [10,48]; however, solutions valid to linear order in the matter perturbation can be constructed using the Teukolsky equation and the so-called Chrzanowski & Kegeles formalism [see the example and references in 57]. On the other hand, in the case of a Schwarzschild black hole, many idealized exact solutions can be found [36, 58, 60]. The effect of the self-gravitating matter on the geodesic motion around a Schwarzschild black hole is that the total angular momentum ceases to be a constant of motion, since the spherical symmetry is lost and reduced usually to axisymmetry. The loss of the fourth integral of motion gives rise to non-integrability and chaos [61]. Let us take a look at how such a superposition of a Schwarzschild black hole with additional matter source looks like. For a static and axially symmetric spacetime, γ = 0 and ν, ψ are functions only of ρ and z in the line element (49). The vacuum Einstein’s field equations then reduce to a Laplace equation  2  ∂ ν ∂ν ∂ 2ν ρ =0 (56) + 2 + 2 ∂ρ ∂ρ ∂z and a line-integral equation ∂ψ = −ρ ∂ρ



∂ν ∂z

2

 −

∂ν ∂ρ

2  ,

∂ν ∂ν ∂ψ = 2ρ . ∂z ∂z ∂ρ

(57) (58)

The Laplace equation (56) implies that ν “potentials” can be added linearly like in Newtonian theory, which, however, does not hold for the function ψ. Namely, if we have source described by ν1 , ψ1 and another source described by ν2 , ψ2 , then the superposition of these sources is given by the potential ν = ν1 + ν2 and the function ψ = ψ1 + ψ2 + ψint . The interaction term ψint can be obtained by integrating   ∂ν1 ∂ν2 ∂ν1 ∂ν2 ∂ψint = −2ρ − , ∂ρ ∂z ∂z ∂ρ ∂ρ   ∂ν1 ∂ν2 ∂ν1 ∂ν2 ∂ψint = 2ρ + . ∂z ∂ρ ∂z ∂z ∂ρ

(59)

In particular, a Schwarzschild black hole is described by the functions   1 2M 1 d+ + d− − 2 M νS = ln , = ln 1 − 2 d+ + d− + 2 M 2 r

(60)

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G. Lukes-Gerakopoulos and V. Witzany

ψS =

r(r − 2M) 1 (d+ + d− )2 − 4 M 2 1 ln , = ln 2 4 d+ d− 2 (r − M)2 − M 2 cos2 (ϑ)

(61)

where d± =

 ρ 2 + (z ± M)2 = r − M ± M cos ϑ .

(62)

For the respective functions describing the surrounding matter, the interested reader is referred to the literature; see, e.g., [36, 58, 60, 61]. Let us now discuss a simple case of a black hole immersed in the field of an axisymmetric faraway halo or ring of matter of characteristic distance R from the black hole and mass M . In the interior of this halo or ring, the leading-order effect will be the quadrupolar tidal. The metric functions for this tidal field read [20]

1 νQ = − Q ρ 2 − 2 z2 , 4   2 2 ρ ρ 2 − z Q2 , ψQ = 2 8

(63)

where Q ∼ M /R 3 is the quadrupolar parameter, and it is easy to check that the metric functions satisfy Eqs. (56) and (57). Obviously, this spacetime is physically valid only at distances much smaller than R from the center, and, as a consequence, it is not even asymptotically flat. However, for the purposes of our didactic example, this metric is sufficient. The interaction function ψint for the above sources is derived from integrating Eq. (59) and reads 1 ψint = − ((z + M)d− + (M − z)d+ )) Q . 2

(64)

Even if the functions of the superposition νSQ = νS + νQ superpose linearly, the metric functions are an infinite series in Q. However, we are able to write the Hamiltonian of the geodesic motion in this spacetime as HSQ =HS + QhQ + O(Q2 ) ,   

pϕ2 1 2 2 −2νS 2 2νS −2ψS 2 2 ρ − 2z hQ = − pt + e (pρ + pz ) + 2 e e 4 ρ

(65)

(66)

1 − ((z + M)d− + (M − z)d+ )) e2νS −2ψS (pρ2 + pz2 ) . 2 In other words, we can treat the external tidal field as a perturbation to the Hamiltonian, which is subject to the theory discussed in the previous sections of this paper. One can then use analytical perturbation methods such as the so-called

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Melnikov integral to see whether this perturbation will cause non-integrability along the homoclinic orbits in Schwarzschild spacetime [see 54].

Spinning Particle Perturbing the background spacetime is not the only way the motion can become non-integrable. Another way non-integrability can arise is when we take the internal multipole structure of the test body into account [65]. The Mathisson-PapetrouDixon (MPD) equations [19,43,49] describe the motion of an extended test body on a curved spacetime. If the multipole expansion of the body is truncated to the poledipole approximation, then the body is effectively reduced to a spinning particle, and the MPD equations read D Pμ 1 = − R μ νρσ U ν S ρσ , dλ 2 μν DS = P μU ν − P ν U μ , dλ

(67) (68)

where S μν is the spin-tensor, P μ is the four-momentum, U μ = dx μ /dλ is a tangent D vector, and the denotes a covariant derivative with respect to an affine parameter dλ λ. The MPD equations are underdetermined; one has to add four constraints in order to evolve the system. One constraint comes from defining the affine parameter, while the other three come from a spin supplementary condition (SSC) Vμ S μν = 0

(69)

fixing the center mass of the body, where V μ is a future-oriented time-like vector. If the affine parameter is the proper time (U μ Uμ = −1), then the MPD equations can be recovered from a Hamiltonian for certain SSCs, and its constant value expresses the conservation of the particle mass, similar to the case of time-like geodesics [72]. The spin of the particle introduces one additional degree of freedom as compared to the geodesic motion, even though there can be additional “gauge” degrees of freedom involved in the evolution in some cases [72]. The strength of the deviation of the spinning particle motion from geodesic motion and the coupling of the spin degree of freedom to the orbit are governed by the spin magnitude S defined by S2 =

1 μν S Sμν . 2

(70)

For each continuous symmetry of the spacetime background with a corresponding Killing vector ζ μ , this system admits an integral of motion

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1 C(ζ ) = Pσ ζ σ − ζρ;σ S ρσ . 2

(71)

For the Schwarzschild background, there are four Killing vectors corresponding to rotations and time translations: ∂ , ∂t ∂ , = ∂ϕ

ζ(t) =

(72)

ζ(z)

(73)

ζ(x) = − sin ϕ ζ(y) = cos ϕ

∂ ∂ − cos ϕ cot ϑ , ∂ϑ ∂ϕ

∂ ∂ − sin ϕ cot ϑ . ∂ϑ ∂ϕ

(74) (75)

These generate the following integrals of motion: the energy E := −C(ζ(t) ) and the three components of the total angular momentum Jx := C(ζ(x) ), Jy := C(ζ(y) ), Jy := C(ζ(z) ). From the three components of the angular momentum, we can obtain only at most two integrals in involution, since they have Poissoncommute with the commutation relations of the generators of rotations, e.g., {Jx , Jy } = Jz . We can choose the two integrals in involution, for instance, as J 2 , Jz . In the Kerr case, however, there are only the Killing vectors ζ(t) , ζ(z) and the respective integrals of motion E, Jz and no exact generalization of J 2 . From now on, we are going to constrain the discussion on the case when V μ = μ p , which is the Tulczyjew-Dixon (TD) SSC. In the case of the Schwarzschild background the extra degree of freedom comes without a respective integral of motion. Using E, J 2 , Jz , the motion of the spinning particle on the Schwarzschild background can be reduced to a two-degree system [73]. In the case of a Kerr background, the Carter constant is no longer constant for a spinning particle. Hence, by using the E and Jz , the motion of the spinning particle on a Kerr background can be reduced to a three-degree-of-freedom system [72]. In both the Kerr and Schwarzschild cases, the spin of the test particle leads to a weakly non-integrable system; however, in the Kerr case, one has to deal with more degrees of freedom leading to more complex dynamics [38]. From the MPD equations, the spin is naturally identified as the perturbation parameter. One might expect that once the perturbation is present, the system should become non-integrable. However, the picture can be more complicated. For example, in the linear-in-spin approximation of MPD equations on a Kerr background at least for the TD SSC, integrability can be approximately recovered up to O(S 2 ) [70], since two other integrals are conserved to linear order in spin [55, 56]. Namely, these two integrals are a Carter-like quantity and a quantity corresponding to the projection of the orbital momentum on the spin [70]. Hence, when an integrable system is perturbed, the system might hold some of its integrable features up to an order in its expansion with respect to the perturbation parameter.

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Let us see what is the expected contribution of a spin induced prolonged resonance crossing to an inspiral. The width of the spin-induced prolonged resonances S grows linearly with the value of the dimensionless spin σ = μM [73]. Taking into √ account the relation (22), this implies that the crossing will last σ 2 /q cycles (see section “Additional Perturbations”). Since the dimensionless spin σ of a secondary compact object, like a black hole or a neutron star, is of the same order as to the mass ratio q, hence, the spin-induced resonance will dominate at least over O(1) cycles during an inspiral.

Impact of Non-integrability on Extreme-Mass-Ratio Systems We would now like to demonstrate in a qualitative model what kind of effect can the non-integrability of geodesics have on an inspiral in the spacetime. Recall that in the simplest approximation an EMRI can be considered as an isolated binary system, in which the secondary body is regarded as a non-extended test particle, while the primary body is a Kerr black hole defining the background in which the secondary moves. In this picture the secondary drifts adiabatically from a geodesic to a geodesic trajectory due to gravitational radiation reaction, essentially tracing out the phase space of integrable Kerr geodesics. In the long run, the motion depends on the slowly dissipating action variables, which in physical terms translate to losses in energy, orbital angular momentum Lz , and Carter constant K . Seen from the lens of dynamical-systems theory, the EMRI in the above setup can be viewed as a regular dissipative dynamical system where the phase space is separable into the space of actions and the dependent phases, which do not feed back into the evolution in the action space at all. In a more accurate EMRI approximation, the secondary body is extended, and the self-force has also a conservative part affecting the immediate motion. The full description of self-force to the first order terms has been reached only recently for the Kerr black hole background [6, 68], and it is very difficult to achieve for other spacetimes like the bumpy black holes. Moreover, the issue is that the self-force computation in a given spacetime represents a huge investment, and it is not clear from astrophysical observations for which metric in particular should one carry this computation out. Therefore, to simulate the dissipative part, one can employ simple qualitative formulas such as the quadrupole formula or some appropriately modified kludge prescriptions to gain qualitative insight into the effect. As long as an inspiral crosses regular parts of the phase space, the adiabatic approximation holds, and, from a dynamical point of view, one cannot tell whether the system is globally integrable or not. Discrepancies will arise only when the inspiral reaches a resonance.√Each strong resonance should introduce to the inspiral a phase shift of the order of ε/q, where q is the mass ratio and ε the perturbation parameter. Therefore, before including the dissipation into a perturbed system deviating from the simplest approximation, it is important to estimate how the perturbation parameter ε leads to non-integrability, i.e., how resonances and, hence,

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chaos grow with respect to ε. Analytical estimates of the resonant growth are quite difficult, since they require handling the geodesic motion in action-angle coordinates. Thus, we will now present how one can investigate this question numerically.

Resonance Growth Assume that we have enough integrals to reduce a non-integrable system to two degrees of freedom. In such a system, we can use Poincaré sections and rotation curves to detect a resonance and to follow its width modification as the perturbation parameter ε changes. This can be achieved either by looking for the unstable points of the resonance in order to measure the angle between the asymptotic manifold branches as done in [73] or by looking for stable points in order to measure the width of the plateaus on the rotation curves as done to obtain Fig. 7. In particular, for each of the b parameter values shown in Table 1, the width w of the plateau was measured. Since the perturbation parameter ε is expected to be in a power law relation with the parameter of the system deviating it from integrability, plotting the width w with respect to this parameter should give a straight line in a logarithmic plot. The left panel of Fig. 7 shows that b is not that parameter, since the points on the plot do not correspond to a line. On the other hand, the right panel indicates that the quadrupole deviation parameter ΔQ = ε, since the points fit well on a line with an inclination ≈0.5, and there is Eq. (22) relating the width of the resonance with the perturbation parameter.

Prolonged Resonances Now that we have established the relation between the quadrupole deviation parameter and the perturbation parameter, let’s see what happens to the inspiral during a resonance crossing. For that we follow the recipe given in [37]. Namely, we have used modified kludge formulae as in [37] to calculate the energy dE/dt|g and angular momentum dLz /dt|g fluxes of the secondary on a geodesic trajectory. Then we subtract these fluxes from the energy E|g and angular momentum Lz |g dLz in a linear approximation, i.e., E(t) = E|g − dE dt |g t and Lz (t) = Lz |g − dt |g t, to introduce dissipation into the system. Note that we do not attempt to introduce dissipation to the other components of angular momentum or some generalization of the Carter constant in the MSM spacetime. We have employed the above-described procedure on the 6/7 resonance shown in the right panel of Fig. 5 and on the 1/3 resonance for b = 10−4 M (one of the cases of Fig. 7). Starting on a given surface of section with a resonance, it is not easy to guess where to place an initial condition on the section so that it crosses the resonance. This is because the section is actually constructed at constant E, Lz and the

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Fig. 7 Both panels show how the resonance 1/3 grows, the left panel with respect to the parameter b and the right panel with respect to the quadrupole deviation from Kerr ΔQ . Spin is set to a = 0.999M and the orbital parameters to E = 0.97 and Lz = 2M. On Poincaré sections similar to the right panel of Fig. 5, rotation curves have been created along the ρ˙ = 0 with a step Δρ = 2 10−5 M. By measuring the plateaus on the rotation curve, the width of the resonances has been found with accuracy 2 Δρ. By applying a linear fit on the points shown in the right panel, the inclination has been √ found to be 0.5069 ± 0.018, which indicates that the width of the resonance is proportional to ΔQ Table 1 The values of b used to produce Fig. 7 and the obtained respective widths of the 1/3 resonance b (104 /M) width (103 /M)

−4 9.06

−3.5 8.68

−3 8.36

−2.5 8.08

−2 7.82

−1.5 7.58

−1 7.36

−0.5 7.18

−0.1 7.02

0 6.98

dissipation thus makes us drift between various sections. This can also be stated in orbital parameter terms, and it is not clear whether during the evolution of the inspiral, a resonance lying at higher inclinations and eccentricities (at fixed E, Lz ) in the phase space than the inspiraling body will catch up with the inspiral or whether the inspiraling body starting from higher inclinations and eccentricities (at fixed E, Lz ) will catch up with the resonance and cross it. In our numerical investigations and in [37], the resonance crossings takes place only when the initial conditions are set between the resonance and the main island of stability. Another unclear aspect of prolonged resonance crossings is the time that the inspiral will spend in the resonance. For example, the 1/3 resonance crossing examined here and the 2/3 resonance crossing discussed in [37] indicate that the inspiral will enter and leave the resonance in finite time. The amount of time spent in the resonance varies from initial condition to initial condition, and the only way to tell why would be to carry out an analysis as sketched in section “Inspirals Through Resonances”. A more puzzling case is that there appears to be cases that the inspiral gets trapped in the resonance. In particular, when we examined the crossings of the

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Fig. 8 An inspiral with mass ratio q = 10−4 going through 6/7 resonance on a MSM spacetime with M = 1, a = 0.999 M, b = 2.1 M. The inspiral starts from the equatorial plane with ρ = 9.0790461M, ρ˙ = 0 with energy E|g = 0.97μ and angular momentun Lz |g = 3μM. This initial condition is not unique, and similar inspiraling behavior can be found in the interval 9.07898 M  ρ  9.07902 M on the equatorial plane. Left panel: The inspiral depicted on a section lying on the equatorial plane. Right panel: Detail from the stroboscopic depiction of left panel panel; the full picture is shown as an inset

6/7 resonance, we have found that there are initial conditions producing cases that the inspiral enters the resonance, but does not seem to be able to leave it (see Fig. 8). Let us examine more carefully the trapping case. The left panel of Fig. 8 shows the intersections of the inspiraling orbit and the equatorial plane on the ρ, ρ˙ surface, where ρ˙ = pρ /μ is the ρ component of the four-velocity. Note that such a plot would be a Poincaré section if not for the dissipation. Having this in mind, we borrow the terminology for the description of the plot from the conservative counterpart. The plot shows an inspiral with mass ratio q = 10−4 starting from a KAM torus (a continuous contour) before it enters the resonant island of stability (7 distinct regions in the plot). As the inspiral progresses, the islands shift to lower ρ coming closer to the central object located at ρ = 0, and at the same time, the eccentricity of the trajectory lowers. The depiction of the trapping is easier to see, when a stroboscopic depiction is employed on the section (right plot of Fig. 8), i.e., from the time series of the left plot, only every seventh consequent point is kept. The inset in the right panel of Fig. 8 shows the whole stroboscopic evolution of the inspiral on the section, while the main right panel focuses on the trapping in the islands of stability leaving out most of the “KAM phase” of the evolution. Let us take another look at the trajectories “trapped” in resonance. The initial condition giving the behavior seen in Fig. 8 is not unique; there is a range of them. For all of them, the trapping lasts at least 104 sections, which is the number of sections we allowed the inspiral to evolve to. Recall that this is the number of times the inspiral crosses the equatorial plane in a certain polar direction, so we can infer that 104 is also roughly the total number of orbital cycles made by the particle.

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Fig. 9 An inspiral with mass ratio q = 10−5 crossing a 1/3 resonance on a MSM spacetime with a = 0.999 M, b = −4 10−4 M. Left panel: A stroboscopic depiction of the inspiral on a section lying on the equatorial plane with energy E|g = 0.97 and angular momentun Lz |g = 2M. The inspiral starts from the gray point ρ = 2.73 M, ρ˙ = 0, enters to the resonance at a position indicated by a red point, and exits from the resonance at a point indicated by the green point. The arrows indicate the direction that the inspiral evolves. Right panel: The rotation curve corresponding to the trajectory shown in the left panel as a function of the coordinate time. The dashed line indicates the borders of the 1/3 resonance

Since an entire EMRI takes ∼1/q orbital cycles, we then see that the particle has been trapped in the resonance for a time comparable with the entire inspiral time! The quadrupole deviation of the spacetime, on which we have evolved the inspiral, is ΔQ ≈ 23M 3 . This number is extremely large, and we expect that if there are any quadrupole deviations from Kerr, realistically, they would be estimated to be of the order of O(10−4 M 3 ) [7]. Even if the initial conditions giving such a prolonged resonance are several, they appear to be of zero measure in comparison with the initial conditions leading to the usual non-trapping resonances. There might be an issue also with the approximation we have employed to obtain this result. Namely, the linear approximation in energy and angular momentum makes sense as long as the trajectory of the inspiral does not get too far from the initial geodesic, on which we have calculated the fluxes. This is not the case in our example. We have tried to address this issue by giving by hand fluxes deviating from the calculated ones up to at least the second significant digit in order to see if the trapping was a flux fine tuning effect. It turns out that this trapping of the inspiral does not appear to ˙ L˙ z . A depend heavily on the specific values of the dissipative fluxes and the ratio E/ more meticulous way to approximate the radiation reaction would be to recalculate the values of the fluxes every few cycles during the resonance passing as done in [18]. However, it may be that the trapping occurs due to the fact that we are applying dissipation only to E, LZ and not to other components of angular momentum. Either way, this example would correspond to a sustained resonance as discussed in the section “Inspirals Through Resonances”.

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A more standard crossing corresponding to the transient-resonance scenario presented in section “Inspirals Through Resonances” can be seen in our 1/3 resonance crossing example. This has been calculated on a background with ΔQ ≈ 2 10−4 M 3 and with mass ratio set to q = 10−5 . For presenting this example, we use only the stroboscopic depiction of a section (left panel of Fig. 9) showing how the trajectory enters (red point) and leaves the resonance (green point). In this stroboscopic depiction, the points belonging to the trajectory follow a clockwise direction until they reach the resonance; they follow the resonance as long as they are trapped in it, which is in this case is just a small part of an island of stability as the most cases during a crossing appear to be; once they leave the resonance, they follow a counterclockwise rotation. The crossing of the resonance lasted approximately for 650 sections, which as explained previously corresponds to the number of cycles around the primary. √ When we compare it to the number of cycles expected from a transient resonance ΔQ/q ≈ 1400, we see that there are of the same order of magnitude. The number of cycles spent in this resonant represents a ∼1% of the total expected cycles of a EMRI with q = 10−5 . The actual information about how long the inspiral stayed in the resonance can be obtained by a rotation curve. The right panel of Fig. 9 shows the rotation curve corresponding to the left panel, in which the horizontal axis is the coordinate time instead of initial condition shown in Fig. 5. There are two ways of producing this curve. The first is to apply Fourier analysis on the trajectory’s time series to find the frequencies as done and described in detail in [37]. However, this approach is cumbersome, and the produced rotation curve at the resonance might give oscillations around the plateau as in [37] instead of a clear plateau. An easier way, providing also a clearer plateau (right panel of Fig. 9), is to take the points producing the curve shown in the left panel as initial conditions and evolve those points as geodesics for enough sections in order to produce the respective rotation curve (right panel of Fig. 9). Note that by imposing the dissipation in our scheme has a result the non-conservation of the contraction of the test body’s four-velocity. The initial value (vμ v μ = −1) increases very slowly tending to zero; since the change is extremely slow, the four-velocity remains time-like throughout the calculation. However, the actual values of the contraction have to be taken into account when reproducing the geodesics for the rotation curve.

Discussion One of the other phenomena associated with resonances is the emergence of chaos. However, at the moment, the impact of chaos on EMRIs and the corresponding gravitational waves is largely unexplored. Even if we do not expect chaos to have a significant impact on such systems, since the chaotic layers in phase space are very small under realistic perturbations, there might still be surprises around the corner. Some work has been done on the prolonged resonances [e.g. 13], but carrying out

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realistic computations of crossing such resonances was impossible until recently. The issue was that the self-force, as well as many idealized perturbations, only creates axisymmetric perturbations to the motion, so the ϕ coordinate stays redundant. As a result, resonances only arise in the r, ϑ sector, that is, for generically inclined and eccentric motion. However, both the dissipative fluxes and the self-force itself have been computed for the generic orbital case only quite recently [68]. It is thus the next order of business to use this realistic self-force to study resonances in inspirals beyond the level of toy models such as the one presented in the past sections. There has not yet been a consistent calculation or a systematic and exhaustive investigation of the crossings of resonances, even though these crossings, as shown in our examples, can last for a non-negligible number of cycles. The crossings will have a definite imprint on the emitted gravitational waves and the detection using matched filters [18], since the ratios of the frequencies of these waves will be rational for a larger number of cycles, thus modifying the frequency domain shape of the waveform template. In fact, since the impact of the resonance on √ the phase is 1/ q, resonant effects have a higher priority for inclusion than any other post-adiabatic effect. On the other hand, as discussed in section “Inspirals Through Resonances”, we need the resonant phase Γ at the beginning of the resonant crossing with sufficient accuracy to estimate the passage well, so the passage through non-resonant parts of the phase space still needs to be computed at accuracy beyond adiabatic order. Alternatively, the resonant phase Γ with which the inspiral enters the resonance can be understood as a free parameter of the waveform, even though that would reduce the predictivity and usefulness of the model. One of the strongest causes for studying resonances is the non-integrability the gravitational self-force itself can introduce into an inspiral. However, the prolonged resonances induced by the spin of the secondary [73] may provide another good case to study resonant effects. The numerical tools needed to calculate the fluxes of gravitational waves from the spinning particle such as numerical Teukolsky equations solvers will be available in the near future, if they are not already available. Taking into account only the first-order dissipative part of the self-force might not provide the whole picture of a prolonged-resonance crossing, but the main challenge at this stage is just to understand the phase-space features of a resonance leading to the different behaviors of the inspiral during the crossing. Ultimately, the adiabatic approximation of the inspiral will have to be replaced by a full self-force and self-torque computation for an accurate picture. The analytical perturbative treatment sketched in section “Inspirals Through Resonances” should provide a more robust systematic framework to tackle the issue of passing through a resonance in semi-analytical inspiral models. However, this formalism currently only represents an order of magnitude estimate of the potential efficiency of a certain computation scheme. Nevertheless, it is yet to be shown whether and how such a scheme can work in practice and, in particular, what the “factors of order one” are in front of the leading-order terms in the estimates in the ε → 0 limit. We leave this question for future work.

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Cross-References  Black Hole Perturbation Theory and Gravitational Self-Force Acknowledgments GL-G has been supported by the fellowship Lumina Quaeruntur No. LQ100032102 of the Czech Academy of Sciences. VW was supported by European Union’s Horizon 2020 research and innovation program under grant agreement No 894881. The authors would like to thank Lukáš Polcar for allowing them to use the didactic example in section “External Matter Deformation”.

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48. Neugebauer G, Meinel R (1993) The Einsteinian gravitational field of the rigidly rotating disk of dust. Astrophys J Lett 414:L97 49. Papapetrou A (1951) Spinning test particles in general relativity. 1. Proc R Soc Lond A209:248–258 50. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32(4):55–114 51. Poincaré H (1912) Sur un théorème de géométrie. Rendiconti del Circolo Matematico di Palermo 33:375–407 52. Poincaré H (1993) New methods of celestial mechanics. American Institute of Physics, Woodbury 53. Poisson E, Pound A, Vega I (2011) The motion of point particles in curved spacetime. Living Rev Relativ 14(1):7 54. Polcar L, Semerák O (2019) Free motion around black holes with discs or rings: Between integrability and chaos. VI. the Melnikov method. Phys Rev D 100(10):103013 55. Rüdiger R (1981) Conserved quantities of spinning test particles in general relativity. I. Proc R Soc Lond A 375(1761):185–193 56. Rüdiger R (1983) Conserved quantities of spinning test particles in general relativity. II. Proc R Soc Lond A 385(1788):229–239 57. Sano Y, Tagoshi H (2014) Gravitational perturbation induced by a rotating ring around a Kerr black hole. arXiv preprint arXiv:1412.8607 58. Semerák O (2003) Gravitating discs around a Schwarzschild black hole: III. Class Quan Grav 20(9):1613–1634 59. Semerák O, Suková P (2010) Free motion around black holes with discs or rings: between integrability and chaos–I. Mon Not R Astron Soc 404(2):545–574 ˇ 60. Semerák O, Cížek P (2020) Rotating disc around a Schwarzschild black hole. Universe 6(2):27 61. Semerák O, Suková P (2015) On geodesic dynamics in deformed black-hole fields. Fund Theor Phys 179:561–586 62. Silverman S (1992) On maps with dense orbits and the definition of chaos. Rocky Mt J Math 22(1):353–375 63. Smale S (1965) Diffeomorphisms with many periodic points. In: Cairns SS (ed) Differential and combinatorial topology: a symposium in honor of Marston Morse. Princeton University Press 64. Speri L, Gair JR (2021) Assessing the impact of transient orbital resonances. arXiv e-prints, page arXiv:2103.06306 65. Suzuki S, Maeda K-I (1997) Chaos in Schwarzschild spacetime: the motion of a spinning particle. Phys Rev D 55(8):4848–4859 66. van de Meent M (2014) Conditions for sustained orbital resonances in extreme mass ratio inspirals. Phys Rev D 89(8):084033 67. van de Meent M (2014) Resonantly enhanced kicks from equatorial small mass-ratio inspirals. Phys Rev D 90(4):044027 68. Van De Meent M (2018) Gravitational self-force on generic bound geodesics in Kerr spacetime. Phys Rev D 97(10):104033 69. Vigeland S, Yunes N, Stein LC (2011) Bumpy black holes in alternative theories of gravity. Phys Rev D 83(10):104027 70. Witzany V (2019) Hamilton-Jacobi equation for spinning particles near black holes. Phys Rev D 100(10):104030 71. Witzany V, Semerák O, Suková P (2015) Free motion around black holes with discs or rings: between integrability and chaos–IV. Mon Not R Astron Soc 451(2):1770–1794 72. Witzany V, Steinhoff J, Lukes-Gerakopoulos G (2019) Hamiltonians and canonical coordinates for spinning particles in curved space-time. Class Quan Grav 36(7):075003 73. Zelenka O, Lukes-Gerakopoulos G, Witzany V, Kopáˇcek O (2020) Growth of resonances and chaos for a spinning test particle in the Schwarzschild background. Phys Rev D 101(2):024037

Part V Data Analysis Techniques

Principles of Gravitational-Wave Data Analysis

41

Andrzej Królak

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Theory of Signal Detection and Parameter Estimation . . . . . . . . . . . . . . . . . . . . . Random Variables and Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Matched Filter in Gaussian Noise: Known Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood Ratio Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of the Likelihood Ratio Statistic to Detection of Gravitational-Wave Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-Noise Ratio and Fisher Information Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . False Alarm and Detection Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suboptimal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deterministic Gravitational-Wave Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal from a Supernova Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Colored Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Basic mathematical concepts of gravitational-wave data analysis are introduced. In particular statistical principles of detection of signals in noise and estimation of their parameters are presented. Applications to the main signals of gravitational waves that can be modeled as functions of time dependent on several unknown parameters are given.

A. Królak () Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2022 C. Bambi et al. (eds.), Handbook of Gravitational Wave Astronomy, https://doi.org/10.1007/978-981-16-4306-4_43

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Keywords

Gravitational-wave data analysis · Signal detection · Parameter estimation · Likelihood ratio statistic · Matched filtering

Introduction In this chapter, we introduce basic mathematical tools needed for the extraction of gravitational-wave signals from the detector noise. We shall restrict ourselves to the sources of gravitational radiation for which we can model the signal as a certain function of time. Such models are available for the main sources of gravitational waves in terms of a certain set of unknown parameters. As gravitational-wave signals are buried in the noise of the detector, we need statistical methods to detect the signal and estimate its parameters. The signal will always be detected with a certain probability, and estimators of the parameters will always have some statistical errors. In this chapter, we shall summarize basic notions of statistics and random processes. We shall present various approaches to the detection of signals and estimation of their parameters. The main tool that is useful for the detection of a gravitational-wave signal in noise and at the same time for estimation of its parameters is the likelihood ratio statistic. We shall present application of the likelihood ratio statistic to the main expected gravitational-wave signals. Many concepts presented in the chapter are discussed in more detail in the review article [1] and the monograph [2].

Statistical Theory of Signal Detection and Parameter Estimation In this section, we shall define the basic notions of statistics and of random processes. We shall present several alternative approaches to the problem of detection of a signal in noise. We shall highlight the likelihood ratio statistic as the basic tool for extraction of a gravitational-wave signal from the noise of the detector and estimation of signal parameters.

Random Variables and Random Processes The basis for analysis of data from gravitational-wave detector is the theory of probability theory. In the probability theory, a random experiment is modelled by some probability space, which is formally defined as a triplet (X , O, P ), where the set X is called an observation set (or a sample space), O are subsets of X called observation events, and P is a probability distribution on O. Observation set is a set of values that our data can assume. Usually X is n-dimensional Euclidean space Rn or a discrete set Γ = {γ1 , γ2 , . . .}. Observation events O are subsets of X to which we can assign consistent probabilities.

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Probability distribution P in space X is a function assigning number P (A) ≥ 0 to any set A ∈ O such that P (X ) = 1 and such that for any countable family of disjoint sets Ak ∈ O (k = 1, 2, . . .), we have P

∞ 

∞   Ak = P (Ak ).

k=1

(1)

k=1

For probability spaces, usually, there exists a probability density function p : Rn → [0, ∞) such that the probability P (A) of the event A ∈ O is given by  P (A) =

p(x) dx,

(2)

A

where x are some coordinates on Rn . Example 1 (Probability space for a coin toss). The sample space X is the set of all outcomes of a coin toss, X ={heads, tail}. The observation space O is the set of all subsets A of X (there are 22 of them). The probability measure P can be defined as P (A) := |A|/2, where |A| is the number of elements in the event A (called cardinality of the set A). For example, the event of getting tails is A ={tails}, its cardinality |A| = 1, and the probability P (A) of this event is 1/2. A random variable X is a real-valued function X : X → R, whose domain is the observation set X , such that {ξ ∈ X : X(ξ ) ≤ x} ∈ O

for all x ∈ R.

The cumulative distribution function (often also called just distribution function) PX of the random variable X is defined as PX (x) := P (X ≤ x),

(3)

where P (X ≤ x) denotes the probability of the event {ξ ∈ X : X(ξ ) ≤ x}. The cumulative distribution function PX is a nondecreasing function on the real axis R with the properties that limx→−∞ PX (x) = 0 and limx→+∞ PX (x) = 1. The derivative of distribution function is called the probability density function (pdf) pX of the random variable X: pX (x) =

dPX (x) . dx

(4)

As any PX is nondecreasing, pX (x) ≥ 0 everywhere. The pdf pX is also normalized: 

∞ −∞

pX (x) dx = 1.

(5)

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The expectation value of the continuous random variable X possessing the pdf pX is defined as  ∞ x pX (x) dx. (6) E[X] := −∞

The variance of the random variable X is Var[X] := E[(X − E[X])2 ].

(7)

Let us consider two random variables X and Y defined on the same probability space (X , O, P ). The covariance between random variables X and Y is defined as   Cov(X, Y ) := E (X − E[X])(Y − E[Y ]) .

(8)

Random variables X, Y for which E[XY ] = E[X]E[Y ] are called uncorrelated. If E[XY ] = 0, then the random variables X, Y are called orthogonal. Let the random variables X and Y be continuous and let pX and pY be the pdf of, respectively, X and Y ; and let pX,Y be the joint pdf of the variables X, Y . Then the conditional pdf pX|Y of X given that Y = y is equal to pX|Y (x|Y = y) =

pX,Y (x, y) . pY (y)

(9)

Random variables X, Y are said to be independent provided pX,Y (x, y) = pX (x) pY (y). For independent random variables, pX|Y (x|Y = y) = pX (x). The most frequently encountered distribution in data analysis is the normal (or Gaussian) distribution N(μ, σ 2 ) with probability density function p(x) = √

1 2π σ

exp

(x − μ)2 , − 2σ 2

−∞ < x, μ < ∞,

σ > 0,

(10)

where μ is the mean of the distribution and σ 2 is its variance. Under very general conditions, which are made precise by central limit theorems, this is the approximate distribution of the sum of a large number of independent random variables when the relative contribution of each term to the sum is small. Each datum registered from a detector is a random variable. A sequence of data constitutes a random (or stochastic ) process which is formally defined as follows. Let T be a subset of real numbers, T ⊂ R. A random process x(t), t ∈ T , is a family of random variables x(t), labelled by the numbers t ∈ T , all defined on the same probability space (X , O, P ). For each finite subset {t1 , . . . , tn }⊂ T , the random variables x(t1 ), . . . , x(tn ) have a joint n-dimensional cumulative distribution function F defined by  Ft1 ,...,tn (x1 , . . . , xn ) = P x(t1 ) ≤ x1 , . . . , x(tn ) ≤ xn .

(11)

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When T is a set of discrete points, T = {t1 , t2 , . . .}, the random process is called a random sequence, and its values x(tk ) are denoted by xk (k = 1, 2, . . .). Very often, the variable t is just time, and then the random sequence is called the time series. For example, data from a gravitational-wave detector (like a laser interferometer) are time series. The random process x(t), t ∈ T , is Gaussian if the cumulative distribution function Ft1 ,...,tn is Gaussian for any n and any t1 , . . . , tn ∈ T . The stochastic process x(t), t ∈ T , is called stationary (sometimes referred to as strongly stationary, strictly stationary, or completely stationary) if all the finite-dimensional cumulative distribution functions (11) defining x(t) remain the same if the set of points {t1 , . . . , tn } ⊂ T is shifted by an arbitrary constant amount τ , i.e., if Ft1 +τ,...,tn +τ (x1 , . . . , xn ) = Ft1 ,...,tn (x1 , . . . , xn )

(12)

for any n, t1 , . . . , tn , and τ . In other words, the probabilistic structure of a stationary process is invariant under a shift of the parameter t. The mean value of the stochastic process x(t) (denoted by m(t)) is defined as m(t) := E[x(t)].

(13)

The autocorrelation function (denoted by K(t, s)) is given by K(t, s) := E[x(t)x(s)].

(14)

In addition to the autocorrelation function K, it is useful to define the autocovariance function C of the stochastic process x(t)   C(t, s) := E[ x(t) − m(t) x(s) − m(s) ] = K(t, s) − m(t) m(s).

(15)

For the stationary stochastic process x(t), the mean value m is constant, and the autocorrelation function K(t, s) depends only on the difference t − s, i.e., there exists a function R of one variable such that E[x(t)x(s)] = R(t − s).

(16)

A very important property of a stationary random process is that it possesses a well-defined spectrum. This is the content of the Wiener-Khinchin theorem which can be stated as follows. Theorem 1 (The Wiener-Khinchin theorem). A necessary and sufficient condition for ρ(τ ) to be the autocorrelation function of some stochastically continuous stationary process x(t) is that there exists a function F (f ), having the properties of a cumulative distribution function on (−∞, ∞) [i.e., F (−∞) = 0, F (∞) = 1, and F is nondecreasing], such that, for all τ , ρ(τ ) may be expressed in the form

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 ρ(τ ) =

∞ −∞

ei2πf τ dF (f ).

(17)

In the case when F is differentiable everywhere, we have a purely continuous spectrum, and we can introduce a function S(f ) such that dF (f ) = 2π S(f ) df.

(18)

The function S(f ) is defined for frequency f over the interval (−∞, ∞), and it is called the two-sided spectral density of the random process. Ergodic theorem. The practical value of stationary stochastic processes is to a considerable extent due to the fact that its mean value and its autocorrelation function can usually be calculated by using just one realization of the stochastic process. The possibility of calculating these characteristics of stationary processes from a single realization is a consequence of the fact that ergodic theorem (or law of large numbers) is applicable. According to the ergodic theorem, the mathematical expectation obtained by averaging the corresponding quantities over the whole space of experimental outcomes can be replaced by time averages of the same quantities. More precisely, if xt is a stationary time series satisfying certain very general conditions (which are almost always met in practice), then with a suitable definition of the limit of a sequence of random variables, the following limiting relations for the mean m and the autocorrelation function R are valid: N 1  xt , N →∞ N

m = E[xt ] = lim

(19a)

t=1

N 1  xt xt+τ . N →∞ N

R(τ ) = E[xt xt+τ ] = lim

(19b)

t=1

Hypothesis Testing The presence of the signal in the noisy data x changes the statistical characteristics of the data, in particular its probability distribution. We need to find a way to detect these changes. When the signal is absent, the data have probability density function p0 (x), and when the signal is present, the pdf of the data is p1 (x). The problem of detecting the signal in noise can be posed as a statistical hypothesis testing problem. The null hypothesis H0 is that the signal is absent in the data, and the alternative hypothesis H1 is that the signal is present. A hypothesis test (or a decision rule) δ is a partition of the observation set X into two subsets R and its complement

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R := X − R. If data are in R, we accept the null hypothesis; otherwise, we reject it. The problem is to find a test that is in some way optimal. We shall present two approaches to find such a test: Bayesian approach and Neyman-Pearson approach. The theory of hypothesis testing has begun with the work of Reverend Thomas Bayes [3] on “inverse” of conditional probability. Tests that minimize the chance of error were proposed by Neyman and Pearson [4]. The notion of cost and risk was introduced by Wald [5].

Bayesian Approach In Bayesian approach, we assign costs to our decisions; in particular, we introduce nonnegative numbers Cij (i, j = 0, 1), where Cij is the cost incurred by choosing hypothesis Hi when hypothesis Hj is true. We shall call matrix C the cost matrix. We define the conditional risk Rj (j = 0, 1) of a decision rule δ for each hypothesis as Rj (δ) := C0j Pj (R) + C1j Pj (R ),

j = 0, 1,

(20)

where Pj is probability distribution of the data when hypothesis Hj is true. Next we assign probabilities π0 and π1 = 1 − π0 to the occurrences of hypothesis H0 and H1 , respectively. These probabilities are called a priori probabilities or priors. We define the Bayes risk as the overall average cost incurred by decision rule δ: r(δ) := π0 R0 (δ) + π1 R1 (δ).

(21)

Finally we define the Bayes rule as the rule that minimizes, over all decision rules δ, the Bayes risk r(δ). Combining Eqs. (20) and (21), the Bayes risk can be written as r(δ) = π0 C00 + π1 C01  +

R

  π0 (C10 − C00 ) p0 (x) + π1 (C11 − C01 ) p1 (x) dx.

(22)

Then it is not difficult to see that the Bayes rule is to accept the hypothesis H1 if the ratio Λ(x) :=

p1 (x) p0 (x)

(23)

is greater than the threshold value λ given by λ :=

π0 C10 − C00 . π1 C01 − C11

(24)

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The ratio Λ is called the likelihood ratio, and as we shall see, it plays a key role in signal detection theory. Let us introduce the probabilities  PF :=

R

p0 (x) dx,

(25a)

p1 (x) dx.

(25b)

 PD :=

R

Probabilities PF and PD are the probabilities that the data has the values in the region R when, respectively, hypotheses H0 and H1 are true. In theory of signal detection, the probabilities PF and PD are the probabilities of false alarm and detection, respectively. Example 2. Consider the following two hypotheses concerning a real-valued observation X: H0 : X = n,

(26a)

H1 : X = n + μ,

(26b)

where μ > 0. Let us assume that n has normal distribution N(0, σ 2 ) then n + μ has also normal distribution N(μ, σ 2 ). It is useful to introduce the quantity ρ := μ/σ that measures relative strength of the “signal” μ and noise. We shall call the quantity ρ the signal-to-noise ratio. The likelihood ratio is then given by Λ(x) = exp

1 x ρ − ρ2 . σ 2

(27)

As the likelihood ratio (27) is an increasing function of the observation x, the Bayes rule is equivalent to comparing the data x with the threshold x0 = σ

ln λ ρ + ρ 2

.

(28)

When the data x exceeds the threshold value x0 , we announce the presence of the signal μ in the data.

Neyman-Pearson Approach The Neyman-Pearson approach involves a trade-off between the two types of errors that one can make in choosing a particular hypothesis. Type I error is choosing hypothesis H1 when H0 is true, and type II error is choosing H0 when H1 is true. In signal detection theory, probability of type I error is called false alarm probability, whereas probability of type II error is called false dismissal probability.

41 Principles of Gravitational-Wave Data Analysis

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The probability of detection of the signal is 1−(false dismissal probability). In hypothesis testing, probability of type I error is called significance of the test, whereas 1−(probability of type II error) is called power of the test. The NeymanPearson design criterion is to maximize the power of the test (probability of detection) subject to a chosen significance of the test (false alarm probability). We are testing the hypothesis H0 that the probability distribution of the data X is P0 (no signal in the data) against the hypothesis H1 that the data probability distribution is P1 (signal present in the data). The fundamental lemma of Neyman-Pearson states that the most powerful test consists of comparing the likelihood ratio, i.e., the ratio of two pdfs p1 and p0 (see Eq. (23)), to a threshold. In order to assess the performance of the likelihood ratio test based on NeymanPearson approach, one usually examines power of the test (probability of detection) as a function of significance level (false alarm probability). For different signalto-noise ratios ρ, one obtains different curves. The parameterized plot (with ρ being the parameter) of functions PD (PF ; ρ) is called the receiver operating characteristic (ROC). By differentiating probabilities of detection and false alarm, one immediately gets dPD p1 = = λ. dPF p0

(29)

Thus, the slope of the ROC curve for a given false alarm probability α is equal to the threshold for Neyman-Pearson test at the level α.

Likelihood Ratio Test It is remarkable that the two very different approaches in hypothesis testing, Bayesian and Neyman-Pearson, lead to the same test called the likelihood ratio test. The likelihood ratio Λ is the ratio of the pdf when the signal is present to the pdf when it is absent (see Eq. (23)). We accept the hypothesis H1 if Λ > λ, where λ is the threshold that is calculated from the costs Cij , priors πi , or the significance of the test depending on what approach is being used.

The Matched Filter in Gaussian Noise: Known Signal Cameron-Martin Formula Let s be a known signal we are looking for and let n be the detector’s noise. For convenience, we assume that the signal s is a continuous function of time t and that the noise n is a continuous random process. Assuming that the noise n is additive, the data x collected by the detector can be written as x(t) = n(t) + s(t).

(30)

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Suppose that the noise n is a zero-mean and Gaussian random process with autocorrelation function K(t, t ) := E[n(t)n(t )].

(31)

Let the observational interval be [0, To ]. Then it can be shown that the logarithm of the likelihood function is given by 

To

ln Λ[x] = 0



1 q(t)x(t) dt − 2

To

q(t)s(t) dt,

(32)

0

where the function q is the solution of the integral equation 

To

s(t) =

K(t, t )q(t ) dt .

(33)

0

The equation (32) for the likelihood ratio in the Gaussian noise case is called the Cameron-Martin formula. From the expression (32), we see that the likelihood ratio test is equivalent to comparing the following quantity, which is the linear correlation of the data  G :=

To

q(t)x(t) dt,

(34)

0

to a threshold G0 . One easily finds that the expectation values of G when the signal s is absent or present in the data read  E0 {[G]} = 0,

E1 {[G]} =

To

q(t)s(t) dt,

(35)

0

and the variance of G is the same independently of whether signal is present or absent  Var[G] =

To



q(t1 )q(t2 )E[n(t1 )n(t2 )]dt1 dt2 0

 =

0 To



To

q(t1 )q(t2 )K(t1 , t2 ) dt1 dt2 0

 =

0 To

q(t)s(t) dt. 0

The quantity ρ given by

To

(36)

41 Principles of Gravitational-Wave Data Analysis



1681

To

ρ2 =

q(t)s(t) dt

(37)

0

is called the signal-to-noise ratio. Since data x are Gaussian and the integral G is linear in x, it has a normal pdf; therefore, pdfs p0 and p1 of G when respective signal is absent or present in the data are given by p0 (G) =

 1 G2  , exp − 2 ρ2 2πρ 2

(38a)

p1 (G) =

 1 (G − ρ 2 )2  . exp − 2 ρ2 2πρ 2

(38b)

1

1

Probability of false alarm PF and of detection PD is readily expressed in terms of error functions G  0 , PF = erfc ρ

(39a)

 G 0 −ρ , PD = erfc ρ

(39b)

where the complementary error function erfc is defined as 1 erfc(x) := √ 2π



∞ x

 1  exp − t 2 dt. 2

(40)

We see that in the Gaussian case a single parameter – signal-to-noise ratio ρ – determines both probabilities of false alarm and detection and consequently the receiver operating characteristic. For a given false alarm probability, the greater the signal-to-noise ratio, the greater the probability of detection of the signal.

Stationary Noise When the noise n is a stationary random process its autocorrelation function K depends only on the difference of times at which it is evaluated, i.e. there exists an even function R of one variable such that E[n(t) n(t )] = R(t − t ).

(41)

Moreover, assuming an idealized situation of observation over the infinite interval (−∞, ∞), the integral equation (33) takes the form  s(t) =

∞

−∞

R(t − t ) q(t ) dt .

(42)

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This integral equation can easily be solved by taking the Fourier transform of its both sides. One then obtains that the Fourier transform q˜ (the tilde indicates here and below taking the Fourier transform) of the function q is given by s˜ (f ) , S(f )

q(f ˜ )=

(43)

where the function S(f ) is the two-sided spectral density of the detector’s noise. By Wiener-Khinchin theorem (Theorem 1), S(f ) is related to the Fourier transform of the noise autocorrelation function:  ∞ S(f ) = R(τ ) exp(−2π if τ ) dτ. (44) −∞

The two-sided spectral density S can also be determined through the equation for the correlation between the Fourier components of the noise E[n(f ˜ ) n(f ˜ )∗ ] = S(f ) δ(f − f ),

−∞ < f, f < +∞.

(45)

For real data, we have that S(−f ) = S(f ). Consequently, it is useful to introduce a one-sided spectral density Sh (f ) defined as Sh (f ) = 2S(f ).

(46)

Making use of the result (43), one can show that for the Gaussian and stationary noise, the log likelihood function (32) can be expressed as 1 ln Λ[x] = (x|s) − (s|s), 2

(47)

where the scalar product ( · | · ) is defined by (here  denotes the real part of a complex expression) 

∞

(x|y) := 4 0

x(f ˜ )y˜ ∗ (f ) df. Sh (f )

(48)

Using the scalar product defined above, the signal-to-noise ratio ρ introduced in Eq. (37) can be written as 

∞

ρ 2 = (s|s) = 4 0

|˜s (f )|2 df. Sh (f )

(49)

Matched Filtering A linear filter is a map which relates its output function O = O(t) to its input function I = I (t) by means of equation

41 Principles of Gravitational-Wave Data Analysis

 O(t) = (H ∗ I )(t) :=

+∞

−∞

1683

H (τ )I (t − τ ) dτ,

(50)

where ∗ denotes the convolution. The function H is called the impulse response of the filter. The Neyman-Pearson test for the zero-mean Gaussian noise consists of passing the data through a linear filter whose impulse response H is  H (τ ) :=

q(To − τ ), 0 ≤ τ ≤ To , 0,

(51)

τ < 0 or τ > To .

It is easy to show that for the response function H given above, it holds (H ∗ x)(To ) = G,

(52)

where G is the detection statistic from Eq. (34). Thus the output of the filter (51) at the time t = To is equal to G and can be compared to a decision level G0 to decide whether a signal is present or absent. Such a filter is called the matched filter. In the case of white noise, its autocorrelation function is proportional to the Dirac delta function R(τ ) =

1 S0 δ(τ ), 2

(53)

where the constant S0 is the one-sided spectral density of noise. In this case, the statistic G introduced in Eq. (34) is given by G=

2 S0



∞

−∞

s(t)x(t) dt.

(54)

Thus, for the case of white noise, the likelihood ratio test consists of correlating the data with a filter matched to the signal we are searching for in the noise. Let us consider another interpretation of the matched filter. Suppose that we pass data x = n + s through linear filter F . The output signal-to-noise ratio d is defined by d2 =

(s|F )2 . E[(n|F )(n|F )]

(55)

By Cauchy-Schwarz inequality, we have (s|F )2 ≤ (s|s)(F |F ), where equality holds when F (t) = A0 s(t) with A0 being some constant. Because E[(n|F )(n|F )] = (F |F ),

(56)

we have d 2 ≤ (s|s). Thus, the signal-to-noise ratio d is maximum and equal to ρ (cf. Eq. (49)) when the filter F is proportional to the signal s. Consequently, the

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matched filter is a filter that maximizes the signal-to-noise ratio over all linear filters. This property is independent of the probability distribution of the noise in the data.

Estimation of Parameters In the case of gravitational-wave signals, we know the form of the signal that we are searching for in the data in terms of a finite number of unknown parameters. We would like to find optimal procedures of estimating these parameters. An estimator of a parameter θ is a function θˆ (x) that assigns to each data x the “best” guess of ˆ the true value of θ . Note that as θ(x) depends on the random data, an estimator is always a random variable.  In this section, we consider a statistical model X , O, {Pθ , θ ∈ Θ} , where {Pθ , θ ∈ Θ} is a family of probability distributions parameterized by parameters θ = (θ1 , . . . , θK ) which take their values from the set Θ ⊂ RK . Functions T : X → Rk are called (k-dimensional) statistics.

Fisher Information Let p(x; θ ) be a pdf of a random variable X that depends on the parameters θ = (θ1 , θ2 , . . . , θK ). The square K × K matrix Γ (θ) with components   ∂ ln p(x; θ ) ∂ ln p(x; θ ) , Γij (θ ) := E ∂θi ∂θj

i, j = 1, . . . , K,

(57)

is called the Fisher information matrix about parameters θ based on the random variable X. The random vector with coordinates ∂ ln p(x; θ )/∂θi (i = 1, . . . , K) is called the score function. We have the following results:   ∂ ln p(x; θ ) = 0, E ∂θi

i = 1, . . . , K,

 2  ∂ ln p(x; θ ) E = −Γij (θ), ∂θi ∂θj

i, j = 1, . . . , K.

(58a)

(58b)

Example 3. For the normal distribution N(μ, σ 2 ), the Fisher matrix is given by ⎛

⎞ 1 0 ⎜ 2 ⎟ Γ (θ) = ⎝ σ 2 ⎠ . 0 σ2

(59)

Bayesian Estimation We assign a cost function C(θ , θ ) of estimating the true value of θ as θ . We then associate with an estimator θˆ a conditional risk or cost averaged over all realizations of data x for each value of the parameter θ :

41 Principles of Gravitational-Wave Data Analysis

ˆ = Eθ {C(θ, ˆ θ)} = Rθ (θ)



1685

 C θˆ (x), θ pθ (x) dx,

(60)

X

where X is the set of observations and pθ (x) is the joint probability distribution of data x and parameter θ . We further assume that there is a certain a priori probability distribution π(θ ) of the parameter θ . We then define the Bayes estimator as the estimator that minimizes the average risk defined as ˆ = E[Rθ (θ)] ˆ = r(θ)

  X

 C θˆ (x), θ pθ (x) π(θ ) dθ dx,

(61)

Θ

where E is the expectation value with respect to a priori distribution π and Θ is the set of observations of the parameter θ . It is not difficult to show that for a commonly used cost function C(θ , θ ) = (θ − θ )2 ,

(62)

the Bayesian estimator is the conditional mean of the parameter θ given data x, i.e., ˆ θ(x) = E[θ |x] =

 θ p(θ |x) dθ,

(63)

Θ

where p(θ |x) is the conditional probability density of parameter θ given the data x.

Maximum Likelihood Estimation In this approach, we introduce the likelihood function Λ(θ, x) which is simply the probability density p(x, θ ) treated as a function of parameter θ . The maximum likelihood estimator θˆ of the parameter θ is the value of θ that maximizes the likelihood function Λ(θ, x). Thus maximum likelihood estimators are obtained by solving the equation ∂ ln Λ(θ, x) = 0. ∂θ

(64)

A very important property of maximum likelihood estimators is that asymptotically (i.e., for signal-to-noise ratio tending to infinity), they are (i) unbiased, and (ii) they have a Gaussian distribution with the covariance matrix equal to the inverse of the Fisher information matrix.

Lower Bounds on the Variance of Estimators Let us first consider the case of one parameter θ . Assuming a number of mild conditions on the pdf p(x; θ ), we have the following theorem [6, 7]. Theorem 2 (Cramèr-Rao lower bound). Let Γ (θ ) be the Fisher information. Suppose that Γ (θ ) > 0, for all θ . Let φ(X) be a one-dimensional statistic with

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E[φ(X)] < ∞ for all θ . Then Var[φ(X)] ≥

d E[φ(X)] dθ Γ (θ )

2 (65)

.

If the variance of an unbiased estimator of the parameter θ attains the lower Cramèr-Rao bound, the estimator is called efficient. In the multiparameter case where the vector θ = (θ1 , . . . , θK ) collects all the K parameters, we have the following lower bound. Theorem 3 (Multiparameter Cramèr-Rao lower bound). Let Γ (θ ) be the Fisher information matrix. Suppose that Γ (θ) is positive definite for all θ. Let φ(X) be a statistic with E[φ(X)] < ∞ for all θ , and let γ (θ ) :=

∂E[φ(X)] ∂E[φ(X)] ,..., ∂θ1 ∂θK

T ,

(66)

where T denotes the matrix transposition. Then Var[φ(X)] ≥ γ (θ)T · Γ (θ )−1 · γ (θ ).

(67)

For an unbiased estimator φ(X) of the parameter θi , the diagonal (i, i) component of the inverse of the Fisher matrix is the smallest possible variance. Jeffreys’ Prior. Fisher information plays a role in the method for choosing a prior density π with the property that when the parameter θ is transformed by a function g (which is one-to-one and differentiable), then also prior density is transformed in the same way. In turns out that a prior probability density π(θ ) with this property is given by

π(θ ) = π0 ΓX (θ ),

(68)

where ΓX (θ ) is the Fisher information about θ based on random variable (i.e., data) X and π0 is a constant so that the integral of π(θ ) is 1.

Likelihood Ratio Statistic In Section “Hypothesis Testing,” we have shown that the optimal test to detect a known signal in noise is the likelihood ratio test. In the case of gravitational-wave signal, we often have a very good model of the signal in terms of a number of unknown parameters. In Section “Estimation of Parameters,” we have considered the problem of parameter estimation. In this case, there are several approaches. The simplest one is the maximum likelihood estimation. A convenient tool for testing the

41 Principles of Gravitational-Wave Data Analysis

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hypothesis of the presence of a signal s(t, θ ) with unknown parameters θ in noisy data x(t) is the likelihood ratio statistic which is defined as 

 Λ(θˆ ; x) L = 2 log . Λ(θˆr ; x)

(69)

Λ(θˆ ; x) is the likelihood function where unknown parameters were replaced by their maximum likelihood estimators, and Λ(θˆr ; x) is the likelihood function where r parameters are assigned a fixed value. This statistic is convenient when we can find explicit analytic expression for some of the maximum likelihood estimators. Asymptotically (i.e., for sample size n approaching to ∞), from Wilks’ theorem [8], the likelihood ratio statistic is χ 2 -distributed with r degrees of freedom independently of the distribution of the data x(t). In the next section, we shall present how the likelihood ratio statistic can be applied to all the basic types of deterministic gravitational-wave signals.

Application of the Likelihood Ratio Statistic to Detection of Gravitational-Wave Signals The expected gravitational-wave sources may be of two types – deterministic sources where signal can be modelled by a certain function of time depending on a number of parameters and stochastic sources which can be characterized by a certain spectral density depending on some parameters. The deterministic sources consist of two categories – transient sources that last for a short time and periodic sources that last for a long time, much longer than the length of the available data. The main transient sources are gravitational-wave impulses expected from supernova explosions and gravitational-wave chirps originating from binary star inspiral. The main periodic sources are rotating neutron stars or pulsars. The stochastic sources arise from superposition of a very large number of unresolved gravitational-wave sources. Such background may occur in the early universe, for example, from the inflation. It may also arise from populations of sources like supernovae, coalescing binaries, or pulsars. In this section, we shall limit ourselves to the case of deterministic signals. In Section “Random Variables and Random Processes,” for mathematical convenience, we considered the data x(t) as a continuous random process. In practice, the data from a gravitational-wave detector is a discrete random process, and we shall limit ourselves to such processes in this section. Thus, we assume that the noise n(t), t = 1, . . . , n in the detector is Gaussian and uncorrelated with the same variance σ 2 for each sample t and mean μ = 0. In this case, the one-sided spectral density of noise Sh (f ) is a constant independent of frequency and related to variance by the relation Sh = 2σ 2 dt,

(70)

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where dt is the sampling time. We also assume that the signal s(t) present in the data x(t), t = 1, . . . , n is additive, i.e., x(t) = n(t) + s(t).

(71)

Finally, we assume that the signal s(t; θ ) depends on a set of unknown parameters θ . We shall see that for all gravitational-wave signals, we can assume that the signal s(t; θ ), t = 1, . . . , n can be expressed as a linear combination of functions hl , l = 1, . . . , L:

s(t; θ ) =

L 

Al hl (t; η),

(72)

l=1

where the set of unknown parameters θ consists of L unknown amplitude parameters Al and m unknown parameters η. The amplitude parameters are called extrinsic parameters, whereas η parameters are called intrinsic. In this case, the probability density distribution p(x) of the data is given by p(x; A , η) =

1

√ 2π σ 2

n

   L n 2 t=1 (x(t) − l=1 Al hl (t; η)) exp − , 2σ 2

(73)

where A = (A1 , . . . , AL ) and η = (η1 , . . . , ηm ). The likelihood function Λ is given by Λ(A , η; x) = p(x; A , η),

(74)

i.e., the likelihood function is just the pdf treated as a function of the parameters of the pdf. The maximum likelihood estimators (see Section “Maximum Likelihood Estimation”) of amplitudes Al are the values of the parameters that maximize Λ, and they are obtained by solving the following set of equations: ∂Λ = 0, for l = 1, . . . , L. ∂Al

(75)

From Eqs. (2) and (74), we have ∂Λ = Λ(x) ∂Al

n

t=1 x(t)hl (t, η) −

L

n

l =1 Al σ2

t=1

n

t =1 hl (t

, η)h

From the above equations, the maximum likelihood estimators Aˆl read

l (t, η)

. (76)

41 Principles of Gravitational-Wave Data Analysis

Aˆl =

L  l =1

1689

Ml−1 l Nl ,

(77)

where the operator · is defined as n

t=1 x(t)y(t) σ2

x y :=

(78)

and we have introduced a vector N and a matrix M with components Nl = x hl ,

and

Mll = hl hl ,

(79)

where l, l = (1, . . . , L). We define the likelihood ratio statistic as 

 Λ(Aˆl , η; ˆ x) L = 2 log , Λ(Al = 0; x)

(80)

where we assign fixed values equal to 0 to the L amplitude parameters Al . Using the expression for maximum likelihood estimators obtained above (Eq. (77)), we explicitly have L = 2L ,

(81)

where L is the L -statistic given by 1  Nl Ml−1 l Nl . 2 L

L =

L

(82)

l=1 l =1

The statistic L depends on the maximum likelihood estimators ηˆ of the m intrinsic parameters η. When the explicit analytic form of the estimators ηˆ is unknown, we maximize numerically the statistic L with respect to intrinsic parameters η. If the value of L is statistically significant, we announce the detection of the signal. The values of the parameters η that maximize the statistic are maximum likelihood estimators of η. In this way, with the likelihood ratio statistic, we can detect the signal and estimate the parameters at the same time.

Signal-to-Noise Ratio and Fisher Information Matrix We first introduce the following compact notation. Let us introduce two L − dimensional column vectors.

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⎞ A1 ⎜ ⎟ A := ⎝ ... ⎠ , ⎛

⎞ h1 (t; η) ⎟ ⎜ .. h(t; η) := ⎝ ⎠. . ⎛

AL

(83)

hL (t; η)

With this notation, the signal s(t) can compactly be written in the following form: s(t) = A T · h(t; η),

(84)

where T stands for the matrix transposition and · denotes matrix multiplication. The the L + m signal parameters consist of L extrinsic amplitude parameters A and m intrinsic parameters η. The signal-to-noise ratio ρ for the signal (84) is then given by ρ=

< s2 > =

AT·M·A,

(85)

where the components of the matrix M are given by Eq. (79). The Fisher matrix Γ for the general signal s(t) can be written in terms of block matrices as  Γ (A , η) =

ΓA A (η)

ΓA η (A , η)

ΓA η (A , η)T Γηη (A , η)

 ,

(86)

where ΓA A is an L × L matrix with components ∂s/∂Ai ∂s/∂Aj (i, j = 1, . . . , L), ΓA η is an L × m matrix with components ∂s/∂Ai ∂s/∂ηk (i = 1, . . . , L, k = 1, . . . , m), and Γηη is an m × m matrix with components ∂s/∂ηk ∂s/∂ηl (k, l = 1, . . . , m). The explicit form of these matrices is ΓA A (η) = M(η),

(87a)

 ΓA η (A , η) = F(1) (η) · A · · · F(m) (η) · A ,

(87b)

⎞ A T · S(11) (η) · A · · · A T · S(1m) (η) · A ⎟ ⎜ ................... Γηη (A , η) = ⎝ ⎠. T T A · S(m1) (η) · A · · · A · S(mm) (η) · A ⎛

(87c)

The components of m matrices F(k) (note here the index within parentheses labels the matrices) and the components of m2 matrices S(kl) read   ∂hj (t; η) , F(k)ij (η) := hi (t; η) ∂ηk

k = 1, . . . , m,

i, j = 1, . . . , L,

(88a)

41 Principles of Gravitational-Wave Data Analysis

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 ∂hi (t; η) ∂hj (t; η) S(kl)ij (η) := , ∂ηk ∂ηl

k, l = 1, . . . , m,

i, j = 1, . . . , L, (88b)

respectively.

False Alarm and Detection Probabilities False Alarm Probability We shall first obtain the false alarm and detection probability density functions in the case when the intrinsic parameters η of the signal are known. In this case, the L statistic is a quadratic function of random variables that are linear correlations of the data with filters; see Eq. (82). Thus, in the case of the Gaussian noise that we assume here, L -statistic is a quadratic form of Gaussian random variables. Consequently, the L -statistic has a distribution related to the χ 2 distribution. One can show that 2L has a χ 2 distribution with L degrees of freedom (where L is the number of the extrinsic parameters A ) when the signal is absent and noncentral χ 2 distribution with L degrees of freedom and noncentrality parameter equal to square of the signalto-noise ratio ρ when the signal is present. As a result, the pdf