Introduction to Particle Physics and Cosmology 9781032683539, 9781032657035, 9781032683546

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
PART I: Particle Physics and Cosmology Phenomenology
Chapter 1: Introduction to the Standard Model of Particle Physics
1.1. Particles of the Standard Model
1.1.1. Particle Data Book
1.1.2. Discrete Symmetries
1.2. Forces of the Standard Model
1.2.1. The Electromagnetic Force
1.2.2. The Strong Force
1.2.3. The Weak Force
1.3. Range of Forces
1.4. Relative Strength of Forces
1.5. Higgs Boson
1.6. Shortcomings of the Standard Model
1.6.1. Parameters in the Standard Model
1.6.2. Gravity
1.6.3. Dark Matter
1.6.4. Dark Energy
1.6.5. Neutrino Masses
1.6.6. Matter-Anti-Matter Asymmetry
1.6.7. Unexplained Experimental Results
1.6.8. Hierarchy Problem
1.6.9. Grand Unification
1.7. End of Chapter Problems
Chapter 2: The Standard Model of Cosmology
2.1. Cosmic Expansion
2.2. The Friedmann Equation
2.3. Ages
2.3.1. Hubble Time
2.3.2. Nuclear Cosmochronology
2.3.3. Stellar Ages
2.4. Cosmic Microwave Background
2.5. Light Element Abundances
2.6. Observed Light Element Abundances
2.6.1. Deuterium
2.6.2. 3He
2.6.3. 4He
2.6.4. 7Li
2.7. Evidence for Dark Matter
2.8. Evidence for Dark Energy
2.9. End of Chapter Problems
PART II: Introduction to Relativity and Cosmology
Chapter 3: Introduction to Special Relativity
3.1. Starting Points
3.1.1. The Inertial Frame
3.1.2. Spacetime Diagrams
3.2. Lorentz Transformations
3.2.1. Time Dilation
3.3. Differences between Newtonian and Relativistic Spacetime
3.4. Four-Vectors, Basis Vectors, and One-Forms
3.4.1. Differentiation
3.5. Four-Velocity and Four-Momentum
3.5.1. Four-Momentum
3.5.2. Relativistic Energy
3.5.3. Relativistic Doppler Effect
3.5.4. Velocity Addition
3.6. Relativistic Kinematics
3.7. Energy-Momentum Tensor Tμν
3.7.1. Electromagnetism
3.7.2. Electromagnetic Energy-Momentum Tensor
3.8. End of Chapter Problems
Chapter 4: Introduction to General Relativity
4.1. Introduction
4.1.1. Principle of Equivalence
4.1.2. Bending of Light and Gravitational Redshift
4.2. Derivative of a Vector
4.3. Christoffel Symbols
4.3.1. Covariant Derivative of One-Forms and Tensors
4.3.2. Covariant Derivative and the Metric
4.4. Parallel Transport
4.5. Geodesic Equation
4.5.1. Newtonian Limit
4.6. Riemann Tensor
4.7. Einstein’s Equations
4.8. Schwarzschild Metric
4.8.1. Falling toward a Black Hole
4.8.2. Conserved Quantities
4.9. What Happens at r = 2M?
4.9.1. Hawking Radiation and Black-Hole Entropy
4.9.2. Robertson-Walker Metric
4.10. End of Chapter Problems
Chapter 5: Cosmology Models
5.1. Introduction
5.1.1. Derivation of the Friedmann-Robertson-Walker Metric
5.2. Christoffel Symbols
5.3. The Einstein Tensor
5.3.1. Energy Conservation
5.3.2. Friedmann Equation
5.3.3. Horizons and Distances
5.4. End of Chapter Problems
PART III: Mathematical Foundation of Particle Physics
Chapter 6: Introduction to Classical Field Theory and the Klein-Gordon Equation
6.1. Overview
6.2. Lagrangian Mechanics and Lagrangian Field Theory
6.2.1. The Action
6.3. Lagrangian Field Theory
6.4. The Lagrangian
6.4.1. A Vector Field
6.4.2. Complex Scalar Field
6.4.3. Gauge Covariant Derivative for an Electromagnetic Field
6.5. Solution to the Klein-Gordon Equation
6.6. Quantizing the Klein-Gordon Equation
6.6.1. Second Quantization
6.6.2. Simple Harmonic Oscillator
6.7. End of Chapter Problems
Chapter 7: Introduction to the Dirac Equation
7.1. Background
7.1.1. Derivation of the Dirac Equation
7.1.2. Dirac γ Matrices
7.2. Solution to the Dirac Equation
7.3. Coupling the Dirac Equation with Electromagnetism
7.4. Quantizing the Dirac Field
7.5. End of Chapter Problems
Chapter 8: Introduction to Group Theory
8.1. Group Theory
8.1.1. What Is a Group?
8.2. Representation and Generators of a Group
8.2.1. Generalizing
8.2.2. Generators of the Standard Model
8.3. Gauge Covariant Derivatives and the Lagrangian
8.3.1. Quantum Electrodynamics
8.3.2. Weak Interactions and Electroweak Theory
8.3.3. Quantum Chromodynamics (QCD)
8.4. End of Chapter Problems
Chapter 9: Summarizing the Standard Model Lagrangian
9.1. Kinetic Terms
9.2. Mass Terms, Yukawa Couplings and the Higgs Mechanism
9.2.1. Yukawa Couplings
9.2.2. Particles of the Standard Model
9.2.3. Parameters
9.3. Beyond the Standard Model: Supersymmetry
9.3.1. Motivations for Supersymmetry
9.3.2. A Short Introduction to Supersymmetry
9.3.3. Supersymmetric Quantum Mechanics
9.3.4. Minimal Supersymmetric Model (MSSM)
9.4. End of Chapter Problems
Chapter 10: Feynman Diagrams, Cross Sections, and Decay Rates
10.1. Cross Sections and Decay Rates
10.1.1. From Lagrangian to Cross Section
10.2. Feynman Diagrams
10.2.1. Propagators
10.2.2. Evaluating Feynman Diagrams
10.3. End of Chapter Problems
PART IV: Big Bang Cosmology
Chapter 11: Big Bang Thermodynamics
11.1. Densities and Pressure
11.2. Vacuum Energy Density and Dark Energy
11.3. End of Chapter Problems
Chapter 12: Relic Abundances and Non-Equilibrium Thermodynamics
12.0.1. The Boltzmann Equation
12.0.2. Abundance of Weakly Interacting Dark Matter
12.0.3. Neutrino Decoupling
12.0.4. Big Bang Nucleosynthesis
12.0.5. Nuclear Statistical Equilibrium and the Deuterium Bottleneck
12.0.6. Photon Decoupling and the CMB
12.1. End of Chapter Problems
Chapter 13: Inflation, Perturbations, and Structure Formation
13.1. Introduction
13.1.1. Nature of Observed Temperature Fluctuations
13.1.2. Inflation Resolves Some Shortcomings of the Standard Big Bang
13.2. Inflation Basics
13.2.1. Inflation Effective Potentials
13.2.2. End of Inflation
13.2.3. Reheating
13.2.4. Warm Inflation
13.2.5. How Much Inflation Occurs
13.2.6. Evolution of Scales
13.2.7. Initial Conditions for Inflation
13.3. Vacuum Fluctuations of the Inflation Field and the Primordial Power Spectrum
13.3.1. Power Spectrum of Primordial Fluctuations
13.3.2. Gravitational Waves
13.3.3. Growth of Structure
13.3.4. Growth of Structure in the Linear Regime
13.3.5. CMB Anisotropies
13.4. Transfer Functions and Computation of the CMB Power Spectra
13.4.1. Formation of Large-Scale Structure
13.5. End of Chapter Problems
References
Index

Citation preview

Introduction to Particle Physics and Cosmology This textbook provides an accessible introduction to the basic concepts of relativistic cosmology and the standard big bang model of cosmology, along with an introduction to quantum field theory and the standard model of particle physics. Readers are guided through the key concepts associated with the standard model of cosmology and the standard model of particle physics, providing them with the basic foundation needed to understand current research and literature on the physics of the early universe and modern particle physics. It culminates with an introduction to the physics of the early universe and its imprint on the large-scale structure and the cosmic microwave background. It assumes a basic understanding of quantum mechanics, classical mechanics and electromagnetism. It is aimed at advanced undergraduates and first-year beginning graduate students studying particle physics and/or cosmology. Key Features: • Provides a summary of the state-of-the-art tools and developments in cosmology and features end of chapter problems, alongside the basic tools for studies of inflation theory and early-universe cosmology. • Provides and understandable introduction to special and general relativity. • Includes an understandable introduction to the standard model of particle physics including group theory, gauge theories, quantum field theory, the Higgs mechanism and the Electroweak Lagrangian. Grant Mathews is a Professor in the Department of Physics and Astronomy at the University of Notre Dame, USA. His research interests involve the origin and evolution of matter in the universe from the first instants of cosmic expansion in the big bang to the present complex interactions of stars and gas in galaxies. He has developed supernova models and simulations of binary neutron stars to explore effects of the nuclear equation of state at high density. He is also studying inflation models of the early universe and the formation of primordial black holes. He is a fellow of the American Physical Society. Guobao Tang is a graduate student in the Department of Physics and Astronomy at the University of Notre Dame, USA. Her research interests involve the origin and evolution of galaxies as well as models for the influence of dark matter and galaxy environments on properties of large-scale structures and galaxy systems.

Series in Particle Physics, Cosmology, and Gravitation Series Editors: Brian Foster, Oxford University, UK             Edward W Kolb, Fermi National Accelerator Laboratory, USA This series of books covers all aspects of theoretical and experimental high energy physics, cosmology and gravitation and the interface between them. In recent years, the fields of particle physics and astrophysics have become increasingly interdependent and the aim of this series is to provide a library of books to meet the needs of students and researchers in these fields. Other recent books in the series: Neutrino Physics, Third Edition Kai Zuber The Standard Model and Beyond, Second Edition Paul Langacker An Introduction to Beam Physics Martin Berz, Kyoko Makino, and Weishi Wan Neutrino Physics, Second Edition K Zuber Group Theory for the Standard Model of Particle Physics and Beyond Ken J Barnes The Standard Model and Beyond Paul Langacker Particle and Astroparticle Physics Utpal Sakar Joint Evolution of Black Holes and Galaxies M Colpi, V Gorini, F Haardt, and U Moschella (Eds) Gravitation: From the Hubble Length to the Planck Length I Ciufolini, E Coccia, V Gorini, R Peron, and N Vittorio (Eds) The Galactic Black Hole: Lectures on General Relativity and Astrophysics H Falcke, and F Hehl (Eds) The Mathematical Theory of Cosmic Strings: Cosmic Strings in the Wire Approximation M R Anderson Geometry and Physics of Branes U Bruzzo, V Gorini, and U Moschella (Eds) Modern Cosmology S Bonometto, V Gorini, and U Moschella (Eds)Gravitation and Gauge SymmetriesM Blagojevic Gravitational Waves I Ciufolini, V Gorini, U Moschella, and P Fré (Eds) Introduction to Particle Physics and Cosmology Grant Mathews and Guobao Tang

Introduction to Particle Physics and Cosmology

Grant Mathews and Guobao Tang

Designed cover image: © Shutterstock First edition published 2025 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2025 Grant Mathews and Guobao Tang Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Mathews, Grant, 1950- author. | Tang, Guobao, author. Title: Introduction to particle physics and cosmology / Grant Mathews and Guobao Tang. Description: First edition. | Boca Raton, FL : CRC Press, 2025. | Series: Series in particle physics, cosmology & gravitation | Includes bibliographical references and index. | Summary: “This textbook provides an accessible introduction to the basic concepts of relativistic cosmology and the standard big bang model of cosmology, along with an introduction to quantum field theory and the standard model of particle physics. Readers are guided through the key concepts associated with the standard model of cosmology and the standard model of particle physics, providing them with the basic foundation needed to understand current research and literature on the physics of the early universe and modern particle physics. It culminates with an introduction to the physics of the early universe and its imprint on the large-scale structure and the cosmic microwave background. It assumes a basic understanding of quantum mechanics, classical mechanics and electromagnetism. It is aimed at advanced undergraduates and first year beginning graduate students studying particle physics and/or cosmology”-- Provided by publisher. Identifiers: LCCN 2024036301 | ISBN 9781032683539 (hbk) | ISBN 9781032657035 (pbk) | ISBN 9781032683546 (ebk) Subjects: LCSH: Particles (Nuclear physics)--Textbooks. | Cosmology--Textbooks. | Quantum field theory--Textbooks. Classification: LCC QC793.24 .M38 2025 | DDC 539.7/2--dc23/eng/20241206 LC record available at https://lccn.loc.gov/2024036301 ISBN: 978-1-032-68353-9 (hbk) ISBN: 978-1-032-65703-5 (pbk) ISBN: 978-1-032-68354-6 (ebk) DOI: 10.1201/9781032683546 Typeset in Nimbus Roman font by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.

Contents Preface.......................................................................................................................xi

PART I Chapter 1

Particle Physics and Cosmology Phenomenology Introduction to the Standard Model of Particle Physics ..................3 1.1

1.2

1.3 1.4 1.5 1.6

1.7 Chapter 2

Particles of the Standard Model.............................................. 4 1.1.1 Particle Data Book .....................................................5 1.1.2 Discrete Symmetries .................................................. 6 Forces of the Standard Model.................................................7 1.2.1 The Electromagnetic Force ........................................7 1.2.2 The Strong Force........................................................8 1.2.3 The Weak Force ......................................................... 8 Range of Forces ......................................................................9 Relative Strength of Forces...................................................10 Higgs Boson..........................................................................10 Shortcomings of the Standard Model ...................................11 1.6.1 Parameters in the Standard Model ...........................12 1.6.2 Gravity ..................................................................... 12 1.6.3 Dark Matter..............................................................12 1.6.4 Dark Energy ............................................................. 12 1.6.5 Neutrino Masses....................................................... 13 1.6.6 Matter-Anti-Matter Asymmetry...............................13 1.6.7 Unexplained Experimental Results..........................13 1.6.8 Hierarchy Problem ................................................... 13 1.6.9 Grand Unification..................................................... 13 End of Chapter Problems......................................................14

The Standard Model of Cosmology ..............................................16 2.1 2.2 2.3

Cosmic Expansion ................................................................16 The Friedmann Equation ...................................................... 17 Ages ...................................................................................... 19 2.3.1 Hubble Time ............................................................19 2.3.2 Nuclear Cosmochronology ...................................... 20 2.3.3 Stellar Ages.............................................................. 20

v

Contents

vi

2.4 2.5 2.6

2.7 2.8 2.9

PART II Chapter 3

Introduction to Relativity and Cosmology Introduction to Special Relativity.................................................. 31 3.1

3.2 3.3 3.4 3.5

3.6 3.7

3.8 Chapter 4

Cosmic Microwave Background........................................... 21 Light Element Abundances...................................................21 Observed Light Element Abundances .................................. 22 2.6.1 Deuterium ................................................................22 2.6.2 3 He............................................................................ 23 2.6.3 4 He............................................................................ 23 2.6.4 7 Li ............................................................................ 23 Evidence for Dark Matter .....................................................24 Evidence for Dark Energy ....................................................24 End of Chapter Problems......................................................27

Starting Points.......................................................................31 3.1.1 The Inertial Frame....................................................31 3.1.2 Spacetime Diagrams ................................................ 31 Lorentz Transformations....................................................... 34 3.2.1 Time Dilation ........................................................... 36 Differences between Newtonian and Relativistic Spacetime.............................................................................. 37 Four-Vectors, Basis Vectors, and One-Forms ....................... 38 3.4.1 Differentiation..........................................................39 Four-Velocity and Four-Momentum .....................................40 3.5.1 Four-Momentum ......................................................41 3.5.2 Relativistic Energy...................................................41 3.5.3 Relativistic Doppler Effect.......................................42 3.5.4 Velocity Addition .....................................................43 Relativistic Kinematics .........................................................43 Energy-Momentum Tensor T µν ........................................... 46 3.7.1 Electromagnetism ....................................................48 3.7.2 Electromagnetic Energy-Momentum Tensor ...........49 End of Chapter Problems......................................................50

Introduction to General Relativity................................................. 51 4.1

4.2

Introduction........................................................................... 51 4.1.1 Principle of Equivalence ..........................................51 4.1.2 Bending of Light and Gravitational Redshift........... 52 Derivative of a Vector ...........................................................52

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vii

4.3

Christoffel Symbols .............................................................. 54 4.3.1 Covariant Derivative of One-Forms and Tensors.....55 4.3.2 Covariant Derivative and the Metric ........................ 55 4.4 Parallel Transport.................................................................. 56 4.5 Geodesic Equation ................................................................56 4.5.1 Newtonian Limit ......................................................56 4.6 Riemann Tensor ....................................................................57 4.7 Einstein’s Equations..............................................................58 4.8 Schwarzschild Metric ........................................................... 59 4.8.1 Falling toward a Black Hole .................................... 59 4.8.2 Conserved Quantities ...............................................61 4.9 What Happens at r = 2M?..................................................... 63 4.9.1 Hawking Radiation and Black-Hole Entropy ..........63 4.9.2 Robertson-Walker Metric.........................................66 4.10 End of Chapter Problems......................................................66 Chapter 5

Cosmology Models........................................................................ 68 5.1

5.2 5.3

5.4

Introduction........................................................................... 68 5.1.1 Derivation of the Friedmann-Robertson-Walker Metric.......................................................................68 Christoffel Symbols .............................................................. 71 The Einstein Tensor ..............................................................71 5.3.1 Energy Conservation................................................71 5.3.2 Friedmann Equation.................................................72 5.3.3 Horizons and Distances............................................75 End of Chapter Problems......................................................76

PART III

Mathematical Foundation of Particle Physics

Chapter 6

Introduction to Classical Field Theory and the Klein-Gordon Equation......................................................................................... 81 6.1 6.2 6.3 6.4

6.5

Overview...............................................................................81 Lagrangian Mechanics and Lagrangian Field Theory .......... 81 6.2.1 The Action ............................................................... 83 Lagrangian Field Theory ......................................................83 The Lagrangian.....................................................................85 6.4.1 A Vector Field ..........................................................86 6.4.2 Complex Scalar Field...............................................86 6.4.3 Gauge Covariant Derivative for an Electromagnetic Field .............................................. 87 Solution to the Klein-Gordon Equation................................87

Contents

viii

6.6

6.7 Chapter 7

Introduction to the Dirac Equation ................................................93 7.1

7.2 7.3 7.4 7.5 Chapter 8

8.2

8.3

8.4

Group Theory...................................................................... 101 8.1.1 What Is a Group? ................................................... 102 Representation and Generators of a Group.........................102 8.2.1 Generalizing........................................................... 103 8.2.2 Generators of the Standard Model ......................... 104 Gauge Covariant Derivatives and the Lagrangian .............. 105 8.3.1 Quantum Electrodynamics..................................... 105 8.3.2 Weak Interactions and Electroweak Theory .......... 106 8.3.3 Quantum Chromodynamics (QCD) ....................... 106 End of Chapter Problems....................................................108

Summarizing the Standard Model Lagrangian............................ 109 9.1 9.2

9.3

9.4 Chapter 10

Background........................................................................... 93 7.1.1 Derivation of the Dirac Equation ............................. 93 7.1.2 Dirac γ Matrices....................................................... 95 Solution to the Dirac Equation ............................................. 96 Coupling the Dirac Equation with Electromagnetism ..........98 Quantizing the Dirac Field.................................................... 99 End of Chapter Problems....................................................100

Introduction to Group Theory .....................................................101 8.1

Chapter 9

Quantizing the Klein-Gordon Equation................................88 6.6.1 Second Quantization ................................................ 89 6.6.2 Simple Harmonic Oscillator ....................................89 End of Chapter Problems......................................................92

Kinetic Terms...................................................................... 110 Mass Terms, Yukawa Couplings and the Higgs Mechanism.......................................................................... 110 9.2.1 Yukawa Couplings .................................................112 9.2.2 Particles of the Standard Model............................. 113 9.2.3 Parameters..............................................................113 Beyond the Standard Model: Supersymmetry ....................114 9.3.1 Motivations for Supersymmetry ............................114 9.3.2 A Short Introduction to Supersymmetry................ 115 9.3.3 Supersymmetric Quantum Mechanics ................... 116 9.3.4 Minimal Supersymmetric Model (MSSM)............117 End of Chapter Problems....................................................118

Feynman Diagrams, Cross Sections, and Decay Rates ...............119 10.1 Cross Sections and Decay Rates......................................... 119 10.1.1 From Lagrangian to Cross Section ........................120

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ix

10.2 Feynman Diagrams ............................................................. 121 10.2.1 Propagators ............................................................122 10.2.2 Evaluating Feynman Diagrams..............................122 10.3 End of Chapter Problems....................................................125

PART IV

Big Bang Cosmology

Chapter 11

Big Bang Thermodynamics.........................................................129 11.1 Densities and Pressure ........................................................129 11.2 Vacuum Energy Density and Dark Energy .........................134 11.3 End of Chapter Problems....................................................136

Chapter 12

Relic Abundances and Non-Equilibrium Thermodynamics........137 12.0.1 12.0.2 12.0.3 12.0.4 12.0.5

The Boltzmann Equation ....................................... 137 Abundance of Weakly Interacting Dark Matter .....138 Neutrino Decoupling..............................................141 Big Bang Nucleosynthesis ..................................... 143 Nuclear Statistical Equilibrium and the Deuterium Bottleneck ............................................ 143 12.0.6 Photon Decoupling and the CMB ..........................146 12.1 End of Chapter Problems....................................................147 Chapter 13

Inflation, Perturbations, and Structure Formation .......................149 13.1 Introduction......................................................................... 149 13.1.1 Nature of Observed Temperature Fluctuations ......149 13.1.2 Inflation Resolves Some Shortcomings of the Standard Big Bang ................................................. 150 13.2 Inflation Basics ...................................................................153 13.2.1 Inflation Effective Potentials..................................155 13.2.2 End of Inflation ......................................................156 13.2.3 Reheating ............................................................... 156 13.2.4 Warm Inflation .......................................................157 13.2.5 How Much Inflation Occurs...................................157 13.2.6 Evolution of Scales ................................................157 13.2.7 Initial Conditions for Inflation ...............................158 13.3 Vacuum Fluctuations of the Inflation Field and the Primordial Power Spectrum................................................ 159 13.3.1 Power Spectrum of Primordial Fluctuations.......... 160 13.3.2 Gravitational Waves ...............................................161 13.3.3 Growth of Structure ...............................................163 13.3.4 Growth of Structure in the Linear Regime.............165 13.3.5 CMB Anisotropies .................................................166

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13.4 Transfer Functions and Computation of the CMB Power Spectra ................................................................................167 13.4.1 Formation of Large-Scale Structure.......................169 13.5 End of Chapter Problems....................................................171 References ............................................................................................................. 173 Index...................................................................................................................... 177

Preface This book is based upon notes from a course on particle physics and cosmology. This course and these notes are intended for advanced undergraduate students and firstyear graduate students. I have found that the existing textbooks on these subjects are in need of updating and simplification for the level of this course. The course only assumes some basic understanding of quantum mechanics, classical mechanics, and electromagnetism. Beyond that, all topics are introduced in a simple compact way that should be understandable to anyone interested in these topics and familiar with basic undergraduate physics. Portions of the material have been extracted from the author’s Chapters 90 [38], 91 [36], and 93 [37] in the Handbook of Nuclear Physics and early reviews [35, 39] by GJM. Needless to say, today we have utilized material from a number of excellent sources. Most notable among them are the book on cosmology and particle physics by Bergstr¨om and Goobar [24], the classic work on the early universe by Kolb and Turner [23], the excellent cosmology texts by Dodelson [12], Liddle and Lythe [27], Peebles [45], and Peacock [44], along with the excellent relativity textbooks by Schutz [53] and Carroll [5], and the useful introduction to quantum field theory by McMahon [40]. The course begins with a schematic overview of the Standard Model of particle physics and the Standard Model of cosmology. This is followed by an introduction to the notation and concepts of special and general relativity, followed by a basic introduction to Friedmann-Robertson-Walker cosmology. After this, the basic concepts of particle physics are summarized, and the mathematical framework of the Standard Model of particle physics is presented. The final section deals with big bang thermodynamics and the underpinning of inflationary cosmology, the cosmic microwave background, and the evolution of largescale structure. The authors wish to acknowledge the careful reading and editing by the course teaching assistant, Wei Sun, along with the students in the 2024 Spring Semester of Physics 50602/60602. We wish to particularly acknowledge the valuable editing and feedback from: Dorthy Gan, Anousha Grieveldinger, David Klim, Ryan Klim, and Alexander Stirling, along with valuable corrections and feedback from: Henry Bloss, Alec Cannon, Ryan Hersey, Enosh Sholey, Abbey Karazewski, Gio Kharchilara, Michelle Kwok, McKenna Leichy, Dan McGuire, James MItchel, Sarah Nowak, Nolan Powers, Nick Reichert, Rob Snuggs, and Gabiji Ziemyte.

xi

Part I Particle Physics and Cosmology Phenomenology

1

Introduction to the Standard Model of Particle Physics

One of the great achievements of the past century and the early decades of the present century has been the accumulation of experimental and observational evidence for the development of two standard models. On the one hand, there is a Standard Model of particle physics that explains almost all experimental data with amazing precision. On the other hand, there is a standard model of cosmology that is the culmination of many decades of improving cosmological observations. It is the goal of the following chapters of this book to familiarize the reader with the basic mathematical underpinning of both standard models. In this chapter, however, we review the Standard Model of particle physics. The Standard Model of particle physics is a means to classify all known elementary particles and their interactions via three fundamental forces, i.e., the electromagnetic, the weak, and the strong interactions. In subsequent chapters, we will learn that this model is mathematically a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3)× SU(2) × U(1). However, the reader should not be concerned if this statement is unfamiliar. The meaning will become clear in subsequent chapters. Historically, the Standard Model was developed in a number of steps, culminating in its well-established formulation in the 1970s. This model has since been supported by experimental confirmation of the existence of quarks, including the identification of the top quark in 1995, the tau neutrino in 2000, and the Higgs boson in 2012. The Standard Model also accurately predicted the existence of the W, Z, and Higgs bosons. There are by now more than 200 known subatomic particles [62]. Among these, there are only six weakly interacting particles, called leptons. The rest are classified as strongly interacting particles called hadrons. Hadrons appear in two types. These are baryons and mesons. The simplest baryons are the proton and neutron. The simplest mesons are the pions, π 0 , π ± . The large number of known hadrons are describable in terms of relatively few building blocks consisting of six flavors of quarks. Baryons consist of 3 quarks, while mesons consist of a quark-antiquark pair. Each of the six “flavors” of quarks can also have three “colors.” The strong interactions between quarks are only attractive for color-less (color singlet) combinations. Color forces between quarks are mediated through the exchange of 8 gluons. Quarks can also transform from one to another through the exchange of W bosons via the weak interaction. Quarks also exchange photons via the electromagnetic interaction. The name “quark” was taken by Murray Gell-Mann from the book Finnegans Wake by James Joyce, where the line “Three quarks for Muster Mark...” appears.

DOI: 10.1201/9781032683546-1

3

Introduction to Particle Physics and Cosmology

4

Gell-Mann received the 1969 Nobel Prize for his work in classifying elementary particles. The mathematical underpinning of the quark model is based upon an SU(3) symmetry that will be explained in subsequent chapters.

1.1

PARTICLES OF THE STANDARD MODEL

The fundamental particles of the Standard Model, including the final Higgs boson, are shown in Figure 1.1. The Standard Model consists of 12 fermions (spin = 1/2 particles) and the 4-gauge bosons (spin = 0 or 1), which are associated with the three forces. In addition, there is the Higgs boson that introduces the masses of certain elementary particles. The measured masses are given at the top of each box [62]. The three columns of fermions correspond to three generations. Within each generation, there is an up-type quark, a down-type quark, a charged lepton, and a neutral lepton. On the left of each box there is the relevant charge and spin for each particle. Three Generations of Fermions

I

Leptons

Quarks

Mass Charge Spin Name

II

III

Bosons 0 0 1

g

2.16MeV/c 2 2/3 1/2

uup

1.27GeV/c 2 2/3 1/2

c

172.7GeV/c 2

4.67MeV/c 2 −1/3 1/2

93.4MeV/c 2 −1/3 1/2

4.18GeV/c 2 −1/3 1/2

0 0 1

105.7MeV/c 2

1.777GeV/c 2

muon

tau

τ

91.2GeV/c 2 0 1

< 0.19MeV/c 2

< 18.2MeV/c 2

muon neutrino

tau neutrino

80.4GeV/c 2 ±1 1

d

down

0.511MeV/c 2

−1 1/2

e

electron

< 2eV/c 2 0 1/2

νe

electron neutrino

charm

s

strange

−1 1/2

0 1/2

μ

νμ

2/3 1/2

t

top

b

bottom

−1 1/2

0 1/2

ντ

gluon

γ

photon

Z0

Z boson

W±

W boson 125.3GeV/c 2

0 0

H

Higgs boson

Figure 1.1 Summary of particles in the Standard Model. The first three columns are the three generations of fermions, the fourth column indicates the gauge bosons associated with the three forces of the Standard Model. Within each generation there are the up-type quarks, the down-type quarks, the charged leptons, and the neutral leptons.

In addition to charge and spin, a number of other discrete quantum numbers are conserved in the Standard Model. These include baryon number B, charmness C, strangeness S, bottomness B, topness T, isospin I, the third component of isospin Iz , and parity P. Table 1.1 summarizes the characteristic quantum numbers for the quarks.

Introduction to the Standard Model of Particle Physics

5

Table 1.1 Quark quantum numbers Property Up (u) Down (d) Charm (c) Strange (s) Top (t) Bottom (b) Q-Charge 2/3 -1/3 2/3 -1/3 2/3 -1/3 j-spin 1/2 1/2 1/2 1/2 1/2 1/2 I-isospin 1/2 1/2 0 0 0 1/2 Iz - z-component of isospin 1/2 -1/2 0 0 0 1/2 P-parity 1 1 1 1 1 1 B-Baryon number 1/3 1/3 1/3 1/3 1/3 1/3 C-charmness 0 0 +1 0 0 0 S-strangeness 0 0 0 -1 0 0 B-bottomness 0 0 0 0 0 -1 T-topness 0 0 0 0 +1 0

We shall see in subsequent chapters that each of these conserved quantum numbers corresponds to a symmetry in the Standard Model Lagrangian. However, with what we have already discussed, we can construct the phenomenology of the Standard Model of particle physics. Let’s start with the simplest case of the up-and-down quarks. These are the constituents of the protons and neutrons. The proton is a uud three quark state. The neutron consists of udd quarks. The π 0 is a linear combination of uu¯ and d d.¯ The charged pions are π + = ud¯ and π − = ud. ¯ Example: Find the quark combination for the Σ+ baryon. Solution: According to the Particle Data Book [62], the Σ+ has baryon number B = 1, I = 1, S = −1, and jP = 1/2+ . Referring to Table 1.1, one can see that to obtain S = −1 requires an s quark. Since an s quark has a charge of −1/3, then a charge of Q = +1 requires 2 charges of Q = +2/3. Hence, the quark combination for the Σ+ is uus. As a check, one can easily confirm that B = 1 and I = 1 are satisfied. One can also deduce which reactions can occur and which are suppressed by conservation laws. Example: Can the following reaction occur? t → s + b¯ Solution: On the left side there is Q = 2/3, j = 1/2, baryon number B = 1/3, T = +1, S = 0, B = 0. On the right side is Q = −1/3 + 1/3 = 0, j = 0 or 1, B = 1/3 − 1/3 = 0, T = 0, S = −1, B = +1 This reaction cannot occur. It does not conserve charge, angular momentum, baryon number, topness, strangeness, or bottomness. 1.1.1

PARTICLE DATA BOOK

The Particle Data Book [62] is a valuable reference for particle physics. It is available at https://pdg.lbl.gov. The pages of this summary include physical constants,

6

Introduction to Particle Physics and Cosmology

descriptions of the Standard Model, and a summary of known and speculated elementary particles. Figure 1.2 shows an example of the first few lines for the baryon, meson, lepton, and gauge boson summaries from Ref. [62]. In addition to the name and mass for each particle, the spin, parity, isospin, strangeness (s), charm (c), bottomness (b), and quark content are summarized. For example, the top entry for the proton lists isospin, spin, and parity as:   1 1+ P (1.1) I(J ) = 2 2 Note, that for mesons it is listed as 

1+ I (J ) = 1 (0 ) 2 G

−

P

−



(1.2)

Citation: R.L. Workman et al. (Particle Data Group), Prog.Theor.Exp.Phys. 2022, 083C01 (2022) and 2023 update

N BARYONS (S = 0, I = 1/2) p, N + = uud;

I (J P ) = 12 ( 21 + )

p

Citation:

n, N 0 = udd

Mass m = 1.007276466621 ± 0.000000000053 u 0.00000029 MeV [a] !Mass m = !938.27208816 ±−10 !m p − m p !/m p < 7 × 10 , CL = 90% [b] ! q pet !al. (Particle R.L. Workman Group), Prog.Theor.Exp.Phys. 2022, 083C01 (2022) and 2023 update ! !/( qp ) =Data 1.000000000003 ± 0.000000000016 mp mp ! ! !q + qp !/e < 7 × 10−10 , CL = 90% [b] ! p ! !qp + qe !/e < 1 × 10−21 [c]

LI GH T U NF LAVORED MESONS Magnetic moment µ = 2.7928473446 ± 0.0000000008 µN " C = B = 0) −6 (µ p + µ p(S ) µ= p = (0.002 ± 0.004) × 10

√ Electric × 10−232,e du; cm , (uu−dd)/ F or I =dipole 1 (π,moment b , ρ, a):d < ud 0.021 −4 fm3 α f=, (11.2 0.4) × d10d) for I Electric = 0 (η, polarizability η 0 , h , h 0 , ω, φ, f 0 ): c±1 (u u+ + c 2 (s s ) Magnetic polarizability β = (2.5 ± 0.4) × 10−4 fm3 (S = 1.2) Charge radius, µ p Lamb shift = 0.84087 ± 0.00039 fm [d] G (J P ) fm − (0− ) = 1[d] Charge radius = 0.8409 ±I 0.0004 π± Magnetic radius = 0.851 ± 0.026 fm [e] ] 1.8) Mass life m =τ 139.57039 ± 0.00018 (S[f= Mean > 9 × 1029 years, CLMeV = 90% (p → invisible mode) 31 to 10 [f ] 10−8 s (S = 1.2) Mean life life ττ = (2.6033 ± 33 0.0005) years × (mode dependent) Mean > 10 c τ = 7.8045 m D50, See the “Note on Nucleon Decay” in our 1994 edition (Phys. Rev. D50 1173) →a short ` ± ν γreview. form π ± for

factors [a]

The “partial life” limits tabulated here are the limits on τ /Bi , where F V mean = 0.0254 ± 0.0017 τ is the total mean life and Bi is the branching fraction for the mode in = 0.0119 ± 0.0001 F question. A For N decays, p and n indicate proton and neutron partial lifetimes.F V slope parameter a = 0.10 ± 0.06 0. 009 R = 0.059 + − 0. 008 Partial mean life p 30 years) − modes are charge conj ugates of p DECAY πMODES (10the modes below. Confidence level (MeV/c)

Figure 1.2 Illustration from [62] of the first few lines of particle properties in the Particle Data Book for baryons, mesons, gauge bosons, and leptons. For decay limits to particles which are+not established, see the section on Antilepton meson Antilepton + meson for Ax ions and Other Very L ight Bosons. N → e +Searches π > 5300 (n), > 16000 (p) 90% 459 N → µ+ π > 3500 (n), > 7700 (p) 90% 453 p Nπ +→DECAY ν π MODES > 1100 (n), > 390 (p) 90% 459 Fraction (Γ /Γ) Confidence level (MeV /c) i p → e+ η > 10000 90% 309 +ν + µ [b] (99. 98770± 0. 00004) % 30 p →µ µ η > 4700 90% 297 µ+ννηµ γ [c]> 158 ( 2. 00 ± 0. 25 ) × 10−4 90% 30 n→ 310 Ne +→ (p) ) × 10−4 90% 149 νe e + ρ [b]> 217 ( 1. (n), 230 >±720 0. 004 70 N → 113 e + νµe+γρ [c]> 228 ( 7. (n), 39 >±570 0. 05(p) ) × 10−7 90% 70 ( 1. 036 ± 0. 006 ) × 10−8 4 e + νe π 0 https://pdg.lbl.gov Page 1 Created: 5/31/2023 09:09 + + − e νe e e ( 3. 2 ± 0. 5 ) × 10−9 70 + µ νµ ν ν < 9 × 10−6 90% 30 e + νe ν ν < 1. 6 × 10−7 90% 70

1.1.2

DISCRETE SYMMETRIES

The Parity P of a particle is another example of a discrete symmetry. In nonrelativistic quantum mechanics, parity refers to the sign of the wave function under inversion of coordinates, i.e., Lepton F amily number (L F ) or Lepton number (L ) violating modes L [d] < 1. 5 × 10−3 90% µ+ ν e L F [d] < 8. 0 × 10−3 90% µ+ νe − + µ e e+ ν L F < 1. 6 × 10−6 90%

ψ(−x) = αψ(x), 30

30 30

(1.3)

− +) I G (J PC ) = in 1− (0quantum where α = ±1. However, field theory, parity P is an intrinsic property π0 Mass m = 134.9768 ± 0.0005 MeVParticles (S = 1.1) of elementary particles. can have either even parity (positive parity) or odd H TTP://PDG.LBL.GOV Page 1 Created: 7/10/2023 parity (negative parity). Bosons have15:48 the same intrinsic parity for both particles and antiparticles, whereas fermions and their corresponding antiparticles can have opposite parities. Parity is conserved in electromagnetic and strong interactions. However,

Introduction to the Standard Model of Particle Physics

7

parity is not conserved in the weak interaction. A spin-0 particle with positive parity is called a scalar; a spin-0 particle with negative parity (like the π meson) is called a pseudoscalar. Charge conjugation C, or C parity, is another discrete symmetry. The charge conjugation operator C converts a particle into its antiparticle, i.e., if |ψ⟩ is a particle ¯ is its antiparticle then, state and |ψ⟩ ¯ = C|ψ⟩ |ψ⟩

(1.4)

Since charge conjugation converts a particle into its antiparticle, it also reverses the sign of all quantum numbers. However, just as for parity, charge conjugation is conserved only in strong and electromagnetic interactions but is not conserved by weak interactions. Since parity and charge conjugation are not conserved by weak interaction, one might speculate that the product CP would be conserved. Indeed, that is almost the case. However, the neutral |K 0 ⟩ meson is actually a linear combination of states with its own antiparticle and is observed to spontaneously transition between its particle and antiparticle. Moreover, the |K 0 ⟩ can decay either into two or three π mesons, i.e., |K 0 ⟩ → 2π mesons C = +1, P = +1 ≡ CP = +1 |K 0 ⟩ → 3π mesons C = +1, P = −1 ≡ CP = −1.

(1.5)

This clearly violates CP conservation. In order to obtain a conserved discrete symmetry, one more symmetry is required. This is the CPT theorem. In this case, one can add a time reversal operator T that converts a state into one for which time flows in the negative direction. Under time reversal, both linear momentum and angular momentum change sign, while all other discrete symmetries remain the same. G parity, mentioned in Figure 1.2, is a generalization of C parity for multiplets of particles. It is a combination of charge conjugation and a 180-degree rotation in isospin space.

1.2

FORCES OF THE STANDARD MODEL

The first four entries in the last column of Figure 1.1 show what are called the gauge bosons of the Standard Model. They mediate three of the four forces of nature (except gravity). In the Standard Model, the forces arise from the exchange of virtual gauge bosons. This is represented by a Feynman diagram, as illustrated in Figure 1.3. Imagine time moving upward. In this illustration, two electrons approach each other. The electrons then exchange a photon, causing them to be repelled by the electromagnetic interaction. As we shall see in a subsequent chapter, such diagrams are a short-hand representation for a great deal of mathematics describing the interaction. 1.2.1

THE ELECTROMAGNETIC FORCE

The most familiar force is the electromagnetic interaction. It is based upon a unitary transformation symmetry called U(1). As described in later chapters, the “generator”

Introduction to Particle Physics and Cosmology

8

e−

e−

time

γ

e−

e−

Figure 1.3 Feynman diagram illustration of electron scattering.

of this group is a massless, chargeless, spin-1 photon γ with two polarization states. As shown in Figure 1.3, the exchange of photons is the source of the electromagnetic force. 1.2.2

THE STRONG FORCE

The strong interaction is based upon an SU(3) symmetry. As we shall see, there are 8 “generators” of this symmetry called gluons. The gluons are spin-1 massless particles that mediate the interaction between quarks. Gluons carry the “charge” of the strong interaction that is called color. Since the gluons themselves carry color charge, they interact among themselves. This makes it a non-Abelian gauge theory. The theory of the strong interaction is called quantum chromodynamics. 1.2.3

THE WEAK FORCE

The weak force in the Standard Model is based on a SU(2) symmetry which has three “generators.” The physical gauge bosons that are exchanged to mediate the weak force are the spin-1 W+ , W− and the Z0 particles. However, in the Standard Model, these particles are actually a superposition of the true “generators” of the gauge group. What distinguishes the weak force from the other two forces is that, in this case, the gauge bosons are massive.

Introduction to the Standard Model of Particle Physics

1.3

9

RANGE OF FORCES

In the Standard Model, the range of a force is determined by the mass of the gauge boson that mediates the force. This range can be estimated from the Heisenberg uncertainty principle. h¯ (1.6) ∆E∆t = 2 Now, insert ∆E = mc2 , where m is the mass of the gauge boson. The timescale for the appearance of the gauge boson is simply: ∆t =

h¯ 2mc2

(1.7)

Next, the range of the force ∆x can be given by how far a particle could move at the speed of light, i.e., ∆x = c∆t. This gives ∆x =

h¯ c h¯ = . 2mc 2mc2

(1.8)

It is convenient to remember that h¯ c = 197 MeV-fm. As an example, the mass of the W ± is 80 GeV/c2 . Inserting this value implies a range for the weak force of only ∆x = 1.2 × 10−3 fm = 1.2 × 10−18 m.

(1.9)

This is only about 10−3 times the size of a proton. This explains why the weak force is only apparent on the scale of the nuclei. In the case of the electromagnetic interaction, the gauge boson is massless. Then, the range of the force is infinite ∆x → ∞. This argument, however, does not apply to the massless gluons of the strong force. That is because the gluons self-interact. That is, the color force carried by gluons involves the concept of confinement. The lines of force collapse to a string between quarks. If one attempts to separate two quarks, the force becomes stronger. This is analogous to stretching a rubber band. This “string tension” limits the range of the strong force to about 10−15 m, i.e., the size of a nucleon. At very short distances, the force weakens, and the quarks behave like free particles. This is the notion of asymptotic freedom. If one attempts to separate two quarks, the force becomes strong enough to produce a quark + anti-quark pair. This replaces the “string” with two shorter strings connected to a member of the pair. The quark + anti-quark pair is a meson. Because of confinement, gluons are only indirectly responsible for the binding of neutrons and protons to nuclei. Rather, the force between nucleons is mediated by the exchange of mesons. The interior of a nucleus contains a cloud of such virtual mesons and nucleons. The nuclear potential is hence rather complicated. However, roughly speaking, the nucleon-nucleon potential can be approximated by a Yukawa expression: e−µr (1.10) VY (r) = −g2 r with g2 ≈ 10 MeV and µ ≈ 1 fm−1 .

Introduction to Particle Physics and Cosmology

10

1.4

RELATIVE STRENGTH OF FORCES

The relative strength of forces depends on distance or energy and the value of characteristic coupling constants. Experimentally, one measures the cross section for scattering, as in the Feynman diagram (Figure 1.3). As we shall see, roughly speaking, the cross section depends upon a coupling constant time the range (Compton wavelength). For example, the cross section for the Compton scattering of a photon by an electron is of the order:   h¯ 2 , (1.11) σ ≈ α□ me c where me is the electron mass and the dimensionless coupling constant α is α=

1 e2 . ≈ h¯ c 137

(1.12)

The strength of the forces in the Standard Model can therefore be described by dimensionless coupling constants given by the square of a charge divided by h¯ c. At low energy, the coupling strengths are: • • • •

Strong Force αs = (g2s /¯hc) ≈ 1/3 2 Electromagnetic Force = α = he¯ c ≈ 1/137 Weak Force = αW = e2 /(sin2 θW h¯ c) ≈ 1/30 Gravity = αG = (Gm2e /¯hc) ≈ 10−45

Here, θw is the Weinberg angle, which has a numerical value of sin2 θW = 0.23. It is worth noting that the intrinsic strength of the weak interaction is actually greater than that of the electromagnetic interaction. At low energies, it appears weak because of the small cross section caused by the small Compton wavelength of the massive gauge boson. The relative magnitudes of these dimensionless coupling constants change, however, as one goes to shorter distances and/or higher energies. Figure 1.4 illustrates schematically how the coupling constants change with energy scale. The intersections of these curves suggest the existence of a unified theory at higher energies. Indeed, the strong, weak, and electromagnetic forces appear to merge near an energy scale of ∼ 1015 GeV.

1.5

HIGGS BOSON

The last boson in Figure 1.1 is the spin-0, neutral Higgs boson. It is introduced in the Standard Model to give masses to the particles. That is, the Standard-Model particles are all massless. However, the Higgs represents a field that is added by hand. It pervades the universe. The coupling of different particles to this field impedes their motion. This is as though the particles have mass. The stronger the coupling, the larger the mass. The analogy is often made of moving through water. Motion is impeded by the resistance of the fluid.

Introduction to the Standard Model of Particle Physics

Ele

ctro

ma

gne

tic

11

forc

e

orce

kf Wea

ng

ro St

e rc

fo

Figure 1.4 Illustration of approximate relative strengths of forces vs. energy scale.

1.6

SHORTCOMINGS OF THE STANDARD MODEL

We shall see that the Standard Model can predict properties of electromagentic interactions with exceedingly high precision based upon the machinery of quantum field theory. Nevertheless, the Standard Model is incomplete. A list of shortcomings is as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

19 parameters with no explanation No gravity No dark matter Dark energy Neutrino masses unexplained Matter-Antimatter asymmetry Unexplained experimental results Hierarchy problem Strong CP problem Grand unification

Introduction to Particle Physics and Cosmology

12

1.6.1

PARAMETERS IN THE STANDARD MODEL

We shall see that at the heart of the Standard Model of particle physics is a rather complicated Lagrangian. The various terms have coupling constants that must be specified by hand. These constants presumably arise from a higher theory, such as string theory. 1.6.2

GRAVITY

We will see that it is difficult to describe gravity. One way to think of this difficulty is that since gravity is a result of space-time, then space-time itself needs to be quantized. In the quantum field theory machinery of the Standard Model, quantizing the graviton field leads to unrenormalizable infinities. This will be discussed later. 1.6.3

DARK MATTER

There is no natural candidate for dark matter or dark energy within the Standard Model in spite of the overwhelming evidence for its existence, as discussed in the next chapter. 1.6.4

DARK ENERGY

The problem with dark energy in the Standard Model is that there is way too much of it. We will analyze this properly in Part II, but a simple classical quantum mechanics argument illustrates this problem. It goes as follows: The Heisenberg uncertainty principle fluctuations in energy and time are related via: h¯ ∆E∆t = . 2

(1.13)

Hence, we could imagine that on a short enough time interval, fluctuations of energy could be large enough to create a particle and its antiparticle corresponding to ∆E = 2mc2 . The energy density ρ corresponding to such fluctuations would be: ρ=

64m4 c5 2mc2 = , 3 (c∆t) h¯ 3

(1.14)

where the final expression comes from taking the time interval from the uncertainty principle: ∆t = h¯ /(4mc2 ). It is left as an exercise to show that inserting a value of the most massive standardmodel particles of mc2 ≈ 100 GeV implies an energy density of ρ ≈ 1045 J m−3 . This is to be compared with the value inferred from observation of ρObs = 6 × 10−10 J m−3 . Clearly, this is a major discrepancy.

Introduction to the Standard Model of Particle Physics

1.6.5

13

NEUTRINO MASSES

In the Standard Model, the neutrinos are all massless. It is now known that neutrinos do not exist as flavor eigenstates. Rather, they mix to form mass eigenstates. This has been confirmed in multiple neutrino oscillation experiments. The masses of these neutrinos eigenstates are not predicted in the Standard Model. 1.6.6

MATTER-ANTI-MATTER ASYMMETRY

At some point in the early universe, matter and anti-matter particles of the Standard Model annihilated, leaving behind a net excess of matter particles. As we shall see, the standard model of particle physics under-predicts the excess of matter over antimatter by orders of magnitude. Hence, new physics is required. 1.6.7

UNEXPLAINED EXPERIMENTAL RESULTS

1. µ anomalous magnetic moment As we shall see in later chapters, The anomalous magnetic moment of a particle is an additional quantum contribution to the magnetic moment of a particle that can be expressed by Feynman diagrams with loops. For years, however, there has been a controversy in that the best prediction from Standard Model physics did not agree with the experimentally deduced value by more than 3 standard deviations. This may or may not require new physics. 2. B¯ meson decay channel There is an excess of the B¯ → D+ + τ − + ν¯ τ 3. The Koide Formula The Standard Model does not explain why the following relation holds: me + mµ + mτ 2 = Q≡ √ √ √ 2 ( me + mµ + mτ ) 3 1.6.8

(1.15)

HIERARCHY PROBLEM

One aspect of the Standard Model is that the Higgs mass has enormous quantum corrections. Hence, the Standard Model must be finely tuned to cancel these quantum corrections. 1.6.9

GRAND UNIFICATION

As noted above, the four forces of nature vary with distance. In particular, the strong force diminishes at closer distances, while the remaining forces increase. Moreover, the weak and electromagnetic forces become comparable on the scale of the range of the weak interaction. The range of forces can also be related to an energy scale via the uncertainty principle. This leads to the kind of diagram schematically drawn in Figure 1.4. The intersection of these strengths at various scales suggests a unification of the forces of nature. Indeed, the unification of the weak and electromagnetic interaction has already been established observationally, and the explanation of the

Introduction to Particle Physics and Cosmology

14

electro-weak theory within the Standard Model led to Wineberg and Salam receiving the 1979 Nobel Prize. Moreover, at a high enough energy scale (or temperature), the forces should become part of a grand unified theory (GUT) where the strong and electroweak strengths merge at a scale of 1015 GeV. It was initially thought that a minimal SU(5) symmetry might provide this. However, the prediction of this version was that the proton should decay on a timescale of 1030 years. This has not been observed, implying a more complicated version of the theory of grand unification. Ultimately, at a scale of 1019 GeV, gravity should also become comparable with the other forces. This should be embedded in a theory of everything (TOE). The hope is that string theory may provide the ultimate theory.

1.7

END OF CHAPTER PROBLEMS

1. Determine the quark (or anti-quark if needed) composition of the following particles in Table 1.2 based upon Table 1.1:

Table 1.2 Particle properties Particle

Q

S

C

B

T

p

Baryon number 1

+1

0

0

0

0

n

1

0

0

0

0

0

K−

0

-1

-1

0

0

0

Σ+

1

+1

-1

0

0

0

D+

0

+1

0

+1

0

0

Quark composition

2. Give the names of the leptons (not the antiparticles). 3. Explain why the following processes are not allowed for the particles listed in Table 1.1: a. K − → e− + γ b. Σ+ → n + e+ + γ c. n → p + e− + γ d. D+ → p + π 0 4. Find the eigenvalues of the charge conjugation and parity operators by operating twice with the C or P operators.

Introduction to the Standard Model of Particle Physics

5. Use the fact that the size of a nucleon is about 10−15 m to estimate the mass of the pion. 6. Verify that the existence of a Standard-Model particle of mc2 = 100 GeV implies a vacuum energy density of ρ ∼ 1045 J m−3 . 7. Verify the numerical values of the coupling constants for the electromagnetic, weak, and gravitational forces.

15

2

The Standard Model of Cosmology

The standard model of cosmology is the hot big bang. It is the culmination of investigations via a number of cosmological probes, including supernovae, observations of the large-scale distribution of galaxies and the intergalactic medium, analyses of the cosmic microwave background, observations of the earliest stars to form, and studies of the nucleosynthesis of the elements in the first few moments of cosmic expansion. In this chapter, we review some of the fundamental observational evidence for the standard cosmological model and its simplest formulation. There are at least 5 fundamental observations that led to the current cosmological standard model. These are the: 1. 2. 3. 4. 5.

2.1

Recession of galaxies Cosmic microwave background Cosmological ages Light element abundances Evidence for Dark Matter and Dark Energy

COSMIC EXPANSION

The first observational evidence for the standard hot big bang model of cosmology was the deduction, attributed to Edwin Hubble [20], of the cosmic expansion velocity versus distance relation. Hubble utilized redshifted lines to infer the recession velocities together with observations of Cepheid variable stars in what were called “spiral nebulae” to determine their distances. This placed them well outside the Milky Way. For illustration, Figure 2.1 shows the recession velocity vs. distance relation reproduced from his original paper. As an historical anecdote, Hubble was unaware that there was more than one type of Cepheid variable star, and he calibrated the distances with the wrong type. This caused an overestimate of the distances by about an order of magnitude. Cepheid variable stars exhibit regular periodic pulsations due to the formation of ionization zones within the stars. The changes in the stellar opacity cause them to expand and collapse. For each type, these stars exhibit a unique correlation between their pulsational period and their absolute luminosity. This relation was also used in the Hubble Key Project [15] that better identified the current value of the Hubble constant. Figure 2.1 is the basis for the Hubble law, v = H0 d

DOI: 10.1201/9781032683546-2

(2.1)

16

The Standard Model of Cosmology

17

Figure 2.1 Figure from the original publication by Edwin Hubble [20]. Reprinted by permission.

where H0 is usually expressed in units of km sec−1 Mpc−1 . So, for example, if H0 = 70 km sec−1 Mpc−1 and a galaxy is observed to be receding at 7000 km sec−1 , then the distance to the galaxy should be 100 Mpc.

2.2

THE FRIEDMANN EQUATION

The basic premise of modern cosmology is that the universe can be described as a nearly homogeneous and isotropic expanding space-time. This is obviously not true on the small scales of stars and galaxies. However, on the largest scales, the universe on average does indeed appear to be homogeneous and isotropic. The first formulation [16] of this was by the Russian physicist Alexander Alexandrovich Friedmann in 1922. Independently, Belgian priest Georges Lemaiˆtre found in 1927 [25] that the solutions of the general relativistic equations for an isotropic (the same in all directions) and homogeneous (constant density) universe require that the universe be either expanding or contracting. We will describe a proper relativistic derivation of the cosmic expansion in a later chapter. However, some insight can be gained from a semi-Newtonian derivation. To see this, consider Figure 2.2. This figure schematically depicts a uniform distribution of matter. In Newtonian mechanics, the net force on a particle of mass m in an infinite space would vanish because the force from any particle would be canceled by a particle in the opposite direction. However, in relativity a distribution of mass produces a curvature. The effect of this can be seen by choosing an arbitrary sphere of uniform density ρ and radius r next to the particle. The mass M within the sphere would be

18

Introduction to Particle Physics and Cosmology

m M

r

Figure 2.2 Schematic view of a Newtonian homogeneous distribution of matter in an infinite space.

M = (4/3)πr3 ρ. Now consider the energy of the particle relative to the sphere. A particle moving radially outward has a total energy: GmM 1 . (2.2) E = m˙r2 − 2 r where r˙ denotes the time derivative of r. Then, inserting the expression for M in terms of ρ and rearranging gives:  2 8 2E r˙ = πGρ + 2 (2.3) r 3 mr If we now choose coordinates to be uniformly expanding in time according to a scale factor, i.e., r(t) = a(t)r0 , (2.4) then Eq. (2.3) simplifies to  2 8 2E a˙ = πGρ + 2 2 . a 3 mro a

(2.5)

The Standard Model of Cosmology

19

If we replace 2E/(mr02 ) with a constant −k, this becomes the Friedmann equation that describes the Hubble parameter H in terms of densities ρ and curvature k. One can also add a cosmological constant Λ corresponding to a vacuum (dark) energy density. The Friedmann equation then becomes:  2 Λ 8 k a˙ ≡ H 2 = πGρ − 2 + , (2.6) a 3 a 3 As we shall see, in general relativity this constant k is a measure of the curvature of space and relates to what is called the Ricci scalar. In the limit that k = Λ = 0 at the present time, one can define a critical density ρc : ρc =

H02 = 1.88 × 10−27 h2 kg m3 , (8/3)πG

(2.7)

where h = H0 /100 is a dimensionless present value of the Hubble constant. So, for h = 0.68 the critical density is 8.7 × 10−28 kg m3 . If the universe were closed by hydrogen atoms with a mass of 1.67 × 10−27 kg, there would, on average, only be about 1 hydrogen atom within a 2 m3 volume. The universe is indeed sparce. The ratio of the present matter density to the critical density is called the closure parameter: ρ (2.8) Ωm = ρc One can then define the various other closure contributions from relativistic particles, non-relativistic baryons, curvature, and dark energy: Ωr

= 8πGρr /(3H02 ), Ωm = 8πGρm /(3H02 ), Ωk = −k/H02 , ΩΛ = Λ/(3H02 ).

(2.9) (2.10)

The Planck Collaboration [47] has determined a present value of H0 = 67.66 ± 0.42 km sec−1 Mpc−1 . However, there is presently a tension between this value and the value deduced from local standard candles [61]. Based upon the analysis from the Planck Collaboration [47] described in a subsequent chapter, the best current values of the closure parameters and other cosmological parameters are as given in Table 2.1. The quantity on the left hand side of the Friedmann equation is the square of the Hubble parameter. If this quantity were time independent then indeed one could write the Hubble law, r˙ = (a/a)r. ˙ However, this is only true nearby. A modern plot of the expansion vs. distance deviates from the Hubble law. As we shall see, it was the deviation from the Hubble law that led to the discovery of dark energy.

2.3 2.3.1

AGES HUBBLE TIME

The Hubble parameter has units of inverse time. If the universe were expanding at a constant rate (which it does not), then the age of the universe would be given by the

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20

Table 2.1 Cosmological parameters from the Planck Collaboration [47]. Ωr is based upon a current CMB temperature of T = 2.7255(6) [14]. Parameter ΩΛ Ωm Ωk Ωr H0 t0

Value 0.6889 0.3111 0.0007 4.43 × 10−5 67.66 km s−1 Mpc−1 13.787 Gyr

σ (68%) ± 0.0056 ± 0.0056 ± 0.0037 ± 0.01×10−5 ±0.42 ±0.020

inverse of the Hubble constant. It is straightforward to show that the Hubble age tH is approximately 1012 yrs, (2.11) tH ≈ H0 where H0 is in units of km s−1 Mpc−1 . So for a Hubble constant of 70 km/sec/Mpc, the age of the universe would be about 14 Gyr. This is close to the present age deduced from observations of the cosmic microwave background. That is a coincidence, however. It is the combined effects of normal expansion followed by cosmic acceleration that leads to an age so close to the Hubble age. For example, in a matter dominated cosmology, the actual age would be (2/3)tH . 2.3.2

NUCLEAR COSMOCHRONOLOGY

The ages of the oldest rocks in the solar system can be deduced from the amount of decay of long-lived radioactivity like 232 Th and 238 U. If one can deduce the initial abundance, the amount of residual abundance of the decaying isotope provides the age. The isotopes and their radioactive half-lives that can used for dating solar system material include: •

235 U

•

238 U

• • • •

- 1.02 Gyr - 6.45 Gyr 238 Th - 20.27 Gyr 87 Rb - 49 Gyr 187 Re - 45 Gyr 176 Lu - 37 Gyr

The oldest meteorites consistently date the age of the solar system to about 4.6 Gyr. 2.3.3

STELLAR AGES

Ages of stars can be deduced in several ways. One way is to look at the brightness vs. temperature for stars in the oldest globular clusters. Since stars in globular clusters

The Standard Model of Cosmology

21

appear to be mostly due to a single initial burst of star formation, then a fit of stellar evolution models for different masses and lifetimes can produce an age estimate. More recently, stellar ages can be deduced for stars that show absorption lines from thorium and/or uranium. These lead to stellar ages of the oldest stars consistent with the cosmological age. Independently, the ages of the oldest white dwarf stars can be determined from their color and temperature, leading to ages of about 10 Gyr. In all, there is no evidence for stars older than the cosmological age of about 14 Gyr.

2.4

COSMIC MICROWAVE BACKGROUND

The cosmic microwave background (CMB) has been measured to quite high precision. The current best value is T0 = 2.7255 ± 0.0006 K [14]. However, photons with an average thermal energy of kT have a wavelength λ = hc/kT . As the universe expands, the wavelength of a photon stretches and the temperature decreases. Conversely, as one goes back in time to a smaller scale factor, the temperature must increase as T0 , (2.12) T (t) = a(t) where convention is to set a(t0 ) = 1 at the present time t0 . As we shall see, the scale factor also relates to the cosmological redshift according to: 1 (2.13) z = − 1. a If one goes back in time to when the universe was 1/1100 its present size, the temperature was about 3,000 K. The early universe was then ionized and opaque. After this epoch, neutral hydrogen quickly formed and the universe became transparent to photons. Those photons from the last epoch of the ionized universe are what we see as the current cosmic microwave background. The observation of the CMB is probably the strongest evidence for the hot big bang. Arno Allan Penzias and Robert Woodrow Wilson were awarded the Nobel Prize in 1978 for the discovery of the CMB, while John Mather and George Smoot were also awarded the Nobel Prize in 2006 for their discovery of the pure blackbody form and anisotropy of the cosmic microwave background radiation via the Cosmic Background Explorer (COBE) satellite.

2.5

LIGHT ELEMENT ABUNDANCES

The epoch of nuclear reactions is referred to as big bang nucleosynthesis (BBN). BBN occurred when the universe was at a temperature of between 108 and 1010 K, i.e., kT ∼ 10 keV to 1 MeV. As we shall see, at a temperature of about 1 MeV, the weak reactions involving neutrons and protons fall out of equilibrium. This fixes the n/p abundance ratio to about 1/7 by the time nuclear reactions can begin. From T = 1 MeV to 100 keV, nuclei are in statistical equilibrium with photons. The equilibrium

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Table 2.2 Comparison [8] between observationally inferred and standard BBN predictions for light element abundances Isotope Yp D/H 3 He 7 Li/H

Observed 0.2449 ± 0.0040 2.53 ± 0.04 × 10−5 = 0.7 ± 0.5 × 10−5 = 1.6 ± 0.3 × 10−10

Predicted 0.2471 ± 0.0003 2.58 ± 0.13 × 10−5 = 1.004 ± 0.009 × 10−5 = 4.7 ± 0.2 × 10−10

favors free neutrons and protons until about T = 100 keV. Once nuclear reactions begin, almost all free neutrons are converted to 4 He. This implies a mass fraction of 4 He, designated as Y , given by: p Yp ≈

2(n/p) 2n = . n + p 1 + (n/p)

(2.14)

Inserting n/p ≈ 1/7 implies a primordial helium mass fraction of Yp ≈ 0.25, which is very close to the presently observed value. In addition to 4 He, a small amount of other light elements are produced. The present observationally inferred light element abundances compared with the predictions of the standard model of cosmology [8] are summarized in Table 2.2. It is a remarkable triumph of the standard BBN model that the observed light element abundances are so well reproduced over 9 orders of magnitude.

2.6

OBSERVED LIGHT ELEMENT ABUNDANCES

It is worthwhile to summarize how the observational abundances have been inferred, as each isotope is determined via a different means. The best abundance determinations are summarized here from Ref. [8] as follows: 2.6.1

DEUTERIUM

Deuterium is measured in the spectra of narrow-line Lyman-α absorption systems in the foreground of high-redshift quasi-stellar objects (QSOs). Unfortunately, only about a dozen of such systems have been found. Taken altogether, they exhibit an unexpectedly large dispersion. This suggests that there could be unaccounted systematic errors. This enhanced error can be approximately accounted for by constructing the weighted mean and standard deviation directly from the data points. Based upon this, a conservative range for the primordial deuterium abundance of −5 D/H = (2.87+0.22 −0.19 ) × 10 . This implies a 2σ (95% C.L.) concordance region of: −5 2.49 × 10 < D/H < 3.3 × 10−5 . However, if one restricts the data to the six well-resolved systems for which there are multiple Lyman-α lines, the deuterium

The Standard Model of Cosmology

23

constraint is slightly lower. This is the presently adopted value [8] of D/H = 2.53 ± 0.04 × 10−5 . 2.6.2

(2.15)

3 HE

The abundance of 3 He is best measured in Galactic HII (ionized atomic hydrogen) regions by the 8.665 GHz hyperfine transition of 3 He+ . A plateau with a relatively large dispersion with respect to metallicity has been found at a level of 3 He/H = (1.90 ± 0.6) × 10−5 . It is not yet understood, however, whether 3 He has increased or decreased through the course of stellar and galactic evolution. Whatever the case, the lack of observational evidence for the predicted galactic abundance gradient supports the notion that the cosmic average 3 He abundance has not diminished from that produced in BBN by more than a factor of 2 due to processing in stars. This is contrary to the case of deuterium for which the observations and theoretical predictions are consistent with a net decrease since the time of the BBN epoch. Moreover, there are results from 3D modeling of the region above the core convective zone for intermediate-mass giants which suggest that in net, 3 He is neither produced nor destroyed during stellar burning. Fortunately, one can avoid the ambiguity in galactic 3 He production by making use of the fact that the sum of (D +3 He)/H is largely unaffected by stellar processing. This leads to a best estimate of 3 He/H = (0.7 ± 0.5) × 10−5 which implies a reasonable 2σ upper limit of 3 He/H < 1.7 × 10−5 and a lower limit of zero. 2.6.3

4 HE

The primordial 4 He abundance, denoted Yp, is best determined from hightemperature regions characterized by ionized hydrogen (HII regions) in metal-poor irregular galaxies extrapolated to zero metallicity. However, the extraction of the final helium abundance has been fraught with uncertainties due to correlations among errors in the neutral hydrogen determination and the inferred helium abundance. A reasonable value for the primordial 4 He abundance is [8]: Yp = 0.2449 ± 0.0040,

(2.16)

which is in general agreement with the predicted value from standard BBN, as shown in Table 2.2. 2.6.4

7 LI

The primordial abundance of 7 Li is best determined from old metal-poor halo stars at temperatures corresponding to what is called the Spite plateau. That refers to a plot of lithium abundance vs. stellar temperature, which flattens in a certain temperature range. This range of constant lithium abundance suggests that these stars have not destroyed their surface primordial lithium. Moreover, these stars also exhibit a

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constant abundance as a function of metallicity for the most metal-poor stars. This uniform lithium abundance is taken to be the primordial lithium abundance. There is, however, an uncertainty in this determination due to the fact that the surface lithium in these stars may have experienced gradual depletion due to mixing with the higher temperature stellar interiors over the stellar lifetime. On the other hand, there are limits on the amount of such depletion that could occur since most lithium destruction mechanisms would imply a larger dispersion in abundances determined from stars of different masses, rotation rates, magnetic fields, etc., than that currently observed. In view of this uncertainty, a reasonable upper limit on the 7 Li abundance has been taken to be Li/H< 6.15 × 10−10 . This is based upon allowing for a possible depletion of up to a factor of ∼5 down to the present observationally determined value of: 7 Li/H = 1.6 ± 0.3 × 10−10 , (2.17) based upon an average [8] of halo stars with [Fe/H]≥ −3.

2.7

EVIDENCE FOR DARK MATTER

As we shall see, the CMB photons and light-element abundances from the big bang can be thought of as relics from the early moments of cosmic expansion. We will also consider other relics of the cosmic expansion in Part III. Besides neutrinos, there is the puzzling presence of dark matter. One of the earliest pieces of evidence for the existence of dark matter was the application of Newton’s laws to the motion of the Sun about the Galaxy. The Sun circles the Galactic center with a velocity of 220 km s−1 at a distance of 8.5 kpc. From the simple balance of centripetal force and gravitation, Newton’s circular velocity formula is: r GM , (2.18) vc = r It is left as an exercise to show that this rotation velocity at this distance implies a galactic mass interior to the location of the Sun of 1011 M⊙ . However, the mass in visible stars and remnants is only about 1010 M⊙ . This implies that most of the mass of the Galaxy is in the form of invisible dark matter.

2.8

EVIDENCE FOR DARK ENERGY

The first convincing evidence of the existence of dark energy came in the late 1990’s from observations of Type Ia Supernovae at high redshift. Type Ia supernovae are believed to arise from the thermonuclear explosion of a carbon-oxygen white dwarf that has exceeded its maximum mass (Chandrasekhar mass). This can be caused either by accretion from a companion star or from the merger of two white dwarfs. The thermonuclear explosion is very bright, and hence, visible at large redshift. Moreover, the luminosity of Type Ia supernovae is mainly due to the conversion of about 1.5 M⊙ of carbon and oxygen to iron and other elements. Because the nuclear binding

The Standard Model of Cosmology

25

energy is well known, as is the mass of the white dwarf, the absolute luminosity of Type Ia Supernovae can be determined. A summary Ref. [46] of the magnitude vs. distance relation from the combined data of the High-Z Supernova Search, and the Supernova Cosmology Project is shown in Figure 2.3. This shows the distance modulus versus the redshift z, where the distance modulus m − M relates to the distance d by: m − M = −5 + 5 log10 d,

(2.19)

while m − M is the difference between apparent magnitude and absolute magnitude, and d is the distance in pc. At the time, it was anticipated that the slowing of the expansion due to the gravitational pull of matter would cause the supernovae to appear closer and brighter at very high red-shift than that of a simple Hubble law. On this plot, that would be the lower dashed curve on Figure 2.3. Instead, it was observed that the Type Ia supernovae were in fact dimmer, as evidenced by the fit upper solid line. This suggested that the universe has begun accelerating.

Figure 2.3 Hubble diagram based upon the SNIa light curves indicating the need for dark energy from [46]. Used by permission.

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Figure 2.4 Constraints on the matter closure fraction (ΩM ) and dark energy fraction (ΩDE ) as derived from measurements of large-scale-structure baryon acoustic oscillations (BAO, green), the cosmic microwave background (CMB, orange), and supernovae (SNe, blue). Together they place very strong limits on both, as indicated by the gray region at their intersection. Credit: Supernova Cosmology Project, Suzuki et al. 2012, Astrophysical Journal. 746, 85.

The best evidence for the existence of dark energy, however, is not from a single observational method, but a cosmic concordance plot of constraints obtained by other means. This is illustrated in Figure 2.4. This shows an overlap of the probability contours on the closure content in dark energy (ΩDE ) and matter in the form of cold dark matter and baryons, Ωm . Type Ia supernovae only constrain a correlation between ΩDE and Ωm . However, observations of the cosmic microwave background mainly constrain the sum of Ωm + ΩDE ≈ 1. Indeed, the CMB data and the SNIa data are nearly orthogonal. The observation of the total matter content can be obtained from the observations of the total mass in galactic clusters or via baryon acoustic oscillations (BAO). We shall see in a later chapter that the BAO is a measure of the total matter content of the universe as the large-scale structures were formed. To understand the BAO, consider that baryons were ionized in the early universe. Radiation and baryons were then was tightly coupled mainly because photons scattered from electrons. As baryons collapsed into gravitational wells, the photon gas also collapsed. The photon pressure then halted the collapse and expanded the plasma. This

The Standard Model of Cosmology

27

repetition of collapse followed by expansion is referred to as the acoustic oscillations. The pattern of these oscillations became imprinted in the large-scale structure as the dark matter and baryons collapsed into stars, galaxies, and the cosmic web. This caused ripples to form in the spectrum of matter fluctuations (matter power spectrum). The amplitude of these ripples is a measure of the total matter density. This subject will be dealt with more thoroughly in Part III. The concordance of these three independent measurements places strong constraints on the contributions from both matter and dark energy.

2.9

END OF CHAPTER PROBLEMS

1. Evaluate the numerical factor for tH in Eq. (2.11). 2. Solve the Friedmann equation (2.2) for time t in a matter dominated (i.e., ρ = ρ0 /a3 ) cosmology with k = 0 and Λ = 0. Show that in this case the age of the universe is (2/3)tH . 3. If the Sun is at a distance of 8.5 kpc from the Galactic center and its rotational speed is 223 km s−1 , what is the total mass interior to the Sun in units of M⊙ based upon Newton’s circular velocity formula? If the mass in stars, interior to the Sun is 3 × 1010 M⊙ , what is the mass-to-light ratio? 4. Show that a flat rotation curve requires a mass density distribution ρ(r) ∝ r−2 . 5. Suppose the mass energy density of dark matter around the Sun is 0.3 GeV cm−3 , and that the dark matter consists of particles with a mass-energy of 100 GeV moving at 200 km s−1 . a. How many of these dark matter particles on average are inside the volume of your body? b. What is the flux (number of these particles per m2 s−1 ) passing through your body? 6. If the cosmic microwave background temperature when the CMB was emitted was 3300 K, and the present CMB temperature is 2.725 K, by what factor has the universe expanded? What is the red-shift at which the CMB was emitted? 7. Suppose the neutron to proton ratio at the beginning of nucleosynthesis was n/p = 1/6. What would be the mass fraction Yp of primordial 4 He? 8. How much 4 He could have been formed by stars over the history of the Galaxy based upon the following assumptions: a. The luminosity of the galaxy has remained at L(Gal) = 4 × 1036 J s−1 for the past 1010 yr. b. The mass of the galaxy in stars is M(Gal) = 4 × 1041 kg. c. The conversion of hydrogen to helium in stars produces 6×1014 J kg−1 .

Part II Introduction to Relativity and Cosmology

3 3.1

Introduction to Special Relativity

STARTING POINTS

The starting point for the mathematical foundation of particle physics and general relativity is Einstein’s special theory of relativity. In this chapter we review the foundations of special relativity and its consequences for spacetime, four-space velocity, and relativistic kinematics. In the process, we can introduce the Einstein notation that pervades modern physics. 3.1.1

THE INERTIAL FRAME

To begin a discussion of special relativity one must first clarify the meaning of an inertial frame. This is a frame at rest or in simple uniform motion. When referring to an observer in an inertial frame, one implies an idealized coordinate system such that clocks everywhere in this frame are synchronized. This will be important to unfold the mixing of space and time later in this chapter. Next, there are two postulates to special relativity. These can be stated as follows: 1. The principle of relativity: The laws of physics take the same form in any inertial frame. Similarly, there is no absolute velocity. One can only determine relative motion between two inertial frames. 2. Universality of the speed of light: The speed of light c in vacuum is universally the same for any observer in any inertial frame regardless of motion. 3.1.2

SPACETIME DIAGRAMS

The easiest way to see that these two postulates lead to a mixing of space and time is with a spacetime diagram. Figure 3.1 illustrates a two-dimensional spacetime diagram in one spatial dimension x and a time dimension t. Here, and henceforth, time is taken as a spatial coordinate ct. Moreover, adopting natural units c = 1, we can write a spacetime diagram in x and t coordinates. Next, imagine that a rocket ship passes by with a constant speed of v along the positive x axis. In this case, in an interval ∆t the ship would move a distance ∆x = v∆t in x and its motion on a spacetime diagram would appear as the tilted line depicted in Figure 3.1. Now consider the frame of an observer in the rocket. The observer in this inertial frame (x′ ,t ′ ) is at rest with respect to the rocket. Rather, this observer would say that the observer in the (x,t) coordinates is moving along the negative x′ axis a distance −v∆t ′ . This satisfies the relativity requirement that there is no absolute

DOI: 10.1201/9781032683546-3

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Introduction to Particle Physics and Cosmology

32

t v𝛥t

t’

𝜑

x Figure 3.1 Illustration of the spacetime diagram for a stationary observer observing a passing object with velocity v

velocity. However, the two observers will still discern differences as illustrated by the following example. 3.1.2.1

Simultaneous Events in One Frame Are Not Simultaneous in Another

Imagine that a flash bulb in the rocket is ignited at a time t ′ = −∆t ′ relative to the origin of x′ so that the light front moves in (x′ ,t ′ ) frame a distance x′ = ∆t ′ . This motion of the light wavefront would appear as lines moving outward at ±45o w.r.t the coordinate axes as depicted in Figure 3.2. The light then simultaneously reflects from two equally spaced mirrors and returns to the observer at time +∆t ′ Now, consider the viewpoint of a stationary terrestrial observer noting the experiment in the passing rocket. In this case, the t ′ coordinate is tilted as in Figure 3.3. However, the universality postulate requires that the speed of light be constant in this frame as well. Hence, the two outgoing light fronts propagate at ±45o and

Introduction to Special Relativity

33

Figure 3.2 Illustration of the spacetime diagram for a stationary observer igniting a flash at −∆t ′ that simultaneously strikes two mirrors at equal distances and then returns to the observer at +∆t ′ .

t

t’ 𝜑

x’ 𝜑

x

Figure 3.3 Illustration of a spacetime diagram for a stationary observer noting a passing object with velocity v for which a flash strikes two equidistant mirrors and returns to the moving observer.

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must reflect at 45o back to simultaneously intersect the timeline of the observer in the rocket. Thus, the light would appear the (x,t) spacetime diagram at as drawn on Figure 3.3. This shows that an event that appears simultaneous in one frame [the (x′ ,t ′ ) frame] is not necessarily simultaneous in another frame. That is, in the (x,t) frame the observer would determine that the light arrives at the left mirror first and arrives at the right mirror sometime later before the two wave fronts reflect and arrive together at the observer in the moving rocket. The line of simultaneous reflections for the observer in the rocket defines the x′ axis at t ′ = 0 as depicted in Figure 3.3. This implies a mixing of space and time coordinates as illustrated in Figure 3.4. That is, the motion of one inertial frame relative to another induces a scissor-like effect on the coordinates. This is the Lorentz transformation that explains how to describe the coordinates in one system relative to another.

t

t’ B

𝜑

A

𝜑

x’ x

Figure 3.4 Illustration of the mixing of space and time deduced by a stationary observer noting the coordinates of an observer in a passing rocket with velocity v.

3.2

LORENTZ TRANSFORMATIONS

As shown above, the time and distance as deduced by one inertial observer are not the same as that of a different inertial observer in uniform motion. Hence, unlike Newtonian physics, one cannot envision a universal set of spatial coordinates in which events unfold in a fixed time. Rather, one needs a new framework in which events in space and time can be determined unambiguously. This is achieved by introducing an invariant interval. Consider the interval between events A and B depicted in Figure 3.4. This interval remains invariant as one switches from the (x,t) to the (x′ ,t ′ ) coordinates and vice

Introduction to Special Relativity

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versa. In simple Cartesian coordinates, one can define an invariant interval ∆s with: ∆s2 = −∆t 2 + ∆x2 + ∆y2 + ∆z2 .

(3.1)

Note that in some contexts, one uses a notation with positive time and negative spatial coordinates. Here, we adopt the convention more often used in relativity: that the minus sign is attached to the time component. When considering particle physics, however, we will revert to the convention of a positive value for time-like intervals. 3.2.0.1

Index Notation

One can define the four-dimensional Cartesian displacement vector which points from one event to another: ∆⃗s ≡ ∆t⃗et + ∆x⃗ex + ∆y⃗ey + ∆z⃗ez .

(3.2)

Next, one can introduce coordinate notation: ∆sµ ≡ (∆t, ∆x, ∆y, ∆z) = (∆x0 , ∆x1 , ∆x2 , ∆x3 ),

(3.3)

where, henceforth, the time coordinate is labeled as x0 and the spatial x, y, z are labeled x1 , x2 , x3 . Also, it is common convention to use Greek indices α, β , ... to denote four-vectors, while Latin indices i, j, k... are reserved for spatial coordinates. So the displacement vector in 3-space is written ∆s j ≡ (∆x, ∆y, ∆z)

(3.4)

Next, the Einstein summation convention can be introduced. That is, the fourdimensional displacement vector can be written: ∆⃗s ≡

∑ µ=0,3

∆xµ⃗eµ ≡ ∆xµ⃗eµ ,

(3.5)

where a repeated index automatically implies a summation over all four indices. 3.2.0.2

The Metric

Having introduced the new notation, we can return to the interval given in Eq. (3.2). Although the four-dimensional displacement vector is simply a list of coordinates, the square of the displacement vector requires that the time coordinate have the opposite sign to that of the spatial coordinates. This can be expressed in matrix notation: ⃗∆s ·⃗∆s = ∆s2 = ηαβ xα xβ , where the quantity ηαβ is the metric for the spacetime.   −1 0 0 0  0 1 0 0   ηαβ =   0 0 1 0 . 0 0 0 1

(3.6)

(3.7)

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The metric is thus a machine that converts the product of two vectors into a number [e.g., Eq. (3.6)] in such a way that it remains invariant for different inertial coordinate systems. Indeed, as we shall see below, the metric is the machinery to take the product of any two vectors ⃗A and ⃗B, i.e., ⃗A · ⃗B = ηαβ Aα Bβ .

(3.8)

Note that while components of space and time are mixed by a Lorentz transformation, the vector product ⃗∆x ·⃗∆x or ⃗A · ⃗B is invariant. 3.2.1

TIME DILATION

Having defined the invariant interval, it is straightforward to deduce the existence of time dilation from Figure 3.1. Consider the interval ∆s2 = ∆x2 − ∆t ′2 experienced by the observer at rest in the frame of the rocket. In the terrestrial frame, there will be a displacement ∆x = v∆t during the time interval ∆t. The invariance of the intervals then requires that: −(∆t ′ )2 = −(∆t)2 + (∆x)2 . (3.9) Now since ∆x = v∆t, we have

which reduces to:

(∆t ′ )2 = (∆t)2 (1 − v2 ),

(3.10)

∆t = γ∆t ′ ,

(3.11)

where

1 . (3.12) γ≡√ 1 − v2 Thus, the time measured by the observer in the non-moving frame is always greater than the time measured by the observer in the moving frame by a factor of γ. By a similar, but slightly more tedious derivation, one could deduce that the length L of the rocket ship as measured by the terrestrial observer is always shorter than the length L′ measured by the observer in the moving rocket by a factor of γ, L=

L′ . γ

(3.13)

Now, one must take into account both the changes in length and time to transform between the coordinates depicted on Figure 3.4. This coordinate transformation can be specified by the Lorentz transformation: ′

′

xα = Λα β xβ . ′

(3.14)

The Lorentz transformation matrix Λα β converts the terrestrial coordinates to the rocket frame coordinates in which the earth appears to be moving in the −x′

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37

direction. The components of the transformation matrix are given by   γ −γv 0 0  −γv ′ γ 0 0  . Λα β =   0 0 1 0  0 0 0 1

(3.15)

The reverse transformation from the rocket frame to the terrestrial frame is given by ′

xα = Λαβ ′ xβ , with Λα β ′

(3.16)



 γ γv 0 0  +γv γ 0 0  . =  0 0 1 0  0 0 0 1

(3.17)

One should also be aware of the general Lorentz transformation, in which there is more than one component of velocity in the moving frame. In this case the general transformation matrix can be written as: ′

Λ0 0 = γ ′

′

′

′

Λ0 j = Λ j 0 = −γv j Λ j k = Λk j = (γ − 1)

3.3

v j vk + δ jk v2

(3.18)

DIFFERENCES BETWEEN NEWTONIAN AND RELATIVISTIC SPACETIME

It is worthwhile to examine the differences between the Newtonian view of space and time versus the special relativistic spacetime. In the Newtonian world, there is a three-dimensional space in which every observer would agree that events occur in a series of time slices that are the same to any observer. The difference in the special relativistic view can be seen in Figure 3.5. This figure shows a spacetime diagram for a three-dimensional (t, x, y) spacetime. In this case, the propagation of a light flash is represented by a cone spreading out at 45◦ . Two events can only be causally connected within this light cone. Inside the light cone is the real world in which all events are timelike. That is, the time coordinate is the largest part of the displacement vector between events. Events outside of the light cone are referred to as spacelike. In this case, the spatial component of the displacement vector exceeds the time component. Spacelike events cannot be causally connected until their light cones intersect. This is illustrated in Figure 3.6. The difference between special relativity and Newtonian spacetime is that in Newtonian spacetime every observer would agree on the future and past. In special relativity, however, the future and past light cone of each observer depends upon the

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Figure 3.5 Illustration of the spacetime diagram for a light cone in two spatial dimensions.

Figure 3.6 Illustration of the spacetime diagram for the light cones of two spacelike events.

inertial frame of the event and the observer. Although each observer will agree that there is a future and a past, events that are simultaneous in one frame may not be simultaneous in another.

3.4

FOUR-VECTORS, BASIS VECTORS, AND ONE-FORMS

Four-vectors are represented by their components Aµ = (A0 , A1 , A2 , A3 ).

(3.19)

They can be written in terms of basis vectors as: ⃗A = Aα⃗eα .

(3.20)

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The basis vectors are normalized vectors along each Cartesian coordinate that can be written: e0 = (1, 0, 0, 0), e1 = (0, 1, 0, 0), e2 = (0, 0, 1, 0), e3 = (0, 0, 0, 1).

(3.21)

In relativity, a vector is an entity that transforms under Lorentz transformations like the displacement four-vector ∆sα . That is, ′

Aβ = Λ

β′ α αA .

(3.22)

Also, the vector product is specified by the metric ⃗A · ⃗B = Aα ′ ηαβ Bβ = Aα Bα = Aβ ′ Bβ ′ .

(3.23)

That is, the vector product is a scalar and is invariant under Lorentz transformations. Note that in Eq. (3.23), a new kind of vector has been generated by multiplying by the metric: Bα = ηαβ Bβ . (3.24) This new entity is not a four-vector. It is designated by a subscript rather than a superscript as for a vector. This entity goes by the name one-form, dual vector, or covariant vector. In 3-space, there is no difference between a vector and a one-form, so indices are often written as superscripts or subscripts for convenience in Newtonian physics. In special relativity, however, a one-form differs from a vector by the sign of the time component. Moreover, in general relativity the one-form can be different in spatial components as well. Some familiar vectors in Newtonian mechanics, such as the gradient and the electromagnetic vector potential, are actually one-forms. Moreover, even the basis vectors within a vector are one-forms. A one-form transforms differently under a Lorentz transformation than a vector. That is, Aβ ′ = Λα β ′ Aα . (3.25) 3.4.1

DIFFERENTIATION

It is instructive to show that the gradient is a one-form. Consider some scalar field φ (t, x, y, z). The gradient is just the set of partial derivatives with respect to the spacetime coordinates:   ∂φ ∂φ ∂φ ∂φ ∂φ = , , , . (3.26) ∂ xµ ∂t ∂ x ∂ y ∂ z To see that this is a one-form, one can use the chain rule to transform the partial derivatives. That is, ∂ xµ ∂ φ ∂φ = . (3.27) ′ ′ ∂ xµ ∂ xµ ∂ xµ

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Introduction to Particle Physics and Cosmology

However, this is just the Lorentz transformation of a one-form: ∂φ ∂φ µ . ′ = Λ µ′ µ ∂ xµ ∂x

(3.28)

It is convenient to now introduce the subscript and coma notations for derivatives. That is, one can denote differentiation by ∂φ ≡ ∂µ φ ≡ φ,µ . ∂ xµ

3.5

(3.29)

FOUR-VELOCITY AND FOUR-MOMENTUM

A very special vector in relativity is the four-velocity. The Newtonian definition of velocity dvi /dxi will not suffice in relativity. This is because time and space depend upon the observer frame. A generalization to four-dimensional spacetime of the velocity requires replacing xi with its four-dimensional vector xα and replacing time with invariant proper time τ. The four-velocity U α is then Uα =

dxα . dτ

(3.30)

Where dτ 2 = −ds2 . To find the general components of the four-vector U α , consider a particle moving along a path in spacetime as depicted in Figure 3.7. Now imagine an observer momentarily moving along with the particle in a momentarily co-moving rest frame. The motion of this observer will be given by the tangent vector to the curve as drawn on Figure 3.7.

Figure 3.7 Spacetime diagram for a particle. Arrow indicates the momentarily co-moving reference frame.

Introduction to Special Relativity

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In that momentarily co-moving rest frame, the observer is at rest. Hence, the only nonzero component of motion is along the time coordinate. In other words, the fourvelocity of that comoving observer, is just the time basis vector: ⃗ MCRF =⃗e0U 0 = (1, 0, 0, 0). U

(3.31)

If we now apply the general Lorentz transformation of Eq. (3.18) to see what the four velocity looks like to an arbitrary inertial-frame observer, we obtain U α = (γ, γvx , γvy , γvz ).

(3.32)

Note that in the comoving reference frame, ⃗ ·U ⃗ = U α ηαβ U β = U α Uα = −1. U

(3.33)

Also, since the product of vectors is invariant under Lorentz transformations, this product remains as -1 for any observer in an arbitrary inertial frame. This is a powerful result. It implies that the magnitude motion through four-dimensional space-time is always at the speed of light, independent of motion in three space. Moreover, this is also true in the general theory of relativity in curved space. 3.5.1

FOUR-MOMENTUM

The generalization of Newtonian 3-momentum pi = mvi for a particle of mass m to four-momentum pα is simply to multiply the four-velocity by the rest-mass energy m. Hence, (3.34) pα = mU α = (γm, mγvx , mγvy , mγvz ), ⃗p ·⃗p = pα ηαβ pβ = pα pα = −m2 .

(3.35)

Since a photon has no mass, the square of the momentum of a photon simply obeys pµ pµ = 0 = ηµν pµ pν .

(3.36)

Although the square of the relativistic four-momentum for a photon vanishes in Eq. (3.36), a photon can have both energy E = hν and momentum whose magnitude is hν/c = hν (for c=1). Hence, the four-momentum for a photon moving along the +x axis is: ′ pν = (E, E, 0, 0). (3.37) 3.5.2

RELATIVISTIC ENERGY

It is left as an exercise to show that expanding Eq. (3.35) into time and spatial components can be used to introduce the concept of relativistic energy: ⃗p ·⃗p = −(γm)2 + (px )2 + (py )2 + (pz )2 = −m2 .

(3.38)

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If we now call the relativistic energy E = γm, and ifp we designate p as the magnitude of the spatial components of four-momentum p = (px )2 + (py )2 + (pz )2 , then we have p (3.39) E = ± m2 + p2 . Expanding the positive solution in the limit of v