Cosmological Observables in the Quasi-Spherical Szekeres Model

Citation preview

COSMOLOGICAL OBSERVABLES IN THE QUASI-SPHERICAL SZEKERES MODEL

APPROVED BY SUPERVISING COMMITTEE: Eric Schlegel, Ph.D., Chair Rafael Lopez-Mobilia, Ph.D. Chris Packham, Ph.D. Marcelo Marucho, Ph.D. Mustapha Ishak-Boushaki, Ph.D. Accepted: Dean, Graduate School

Copyright 2014 Robert G. Buckley All rights reserved.

COSMOLOGICAL OBSERVABLES IN THE QUASI-SPHERICAL SZEKERES MODEL

by

ROBERT G. BUCKLEY, B.S.

DISSERTATION Presented to the Graduate Faculty of The University of Texas at San Antonio In Partial Fulfillment Of the Requirements For the Degree of DOCTOR OF PHILOSOPHY IN PHYSICS

THE UNIVERSITY OF TEXAS AT SAN ANTONIO College of Sciences Department of Physics and Astronomy December 2014

ACKNOWLEDGEMENTS

I would like to thank the UTSA physics department for showing me the world of physics for so many years, through both my undergraduate and graduate career. In particular, I would like to thank Dr. Liao Chen for advising me with my first big research project, Dr. Patrick Nash for introducing me to the Mathematica software, and Dr. Eric Schlegel for advising me through this work. It’s been a long and winding road. I would also like to thank Dr. Mustapha Ishak-Boushaki for some useful discussions. This work was made possible in part by financial support from the Texas Space Grant Consortium, the UTSA Department of Physics and Astronomy, and the UTSA Graduate School. This work made extensive use of the Wolfram Mathematica software. I would also like to thank my parents for their love and support, and my best friend Billie Harvey for keeping me company and helping me stay (mostly) sane throughout this process. This Masters Thesis/Recital Document or Doctoral Dissertation was produced in accordance with guidelines which permit the inclusion as part of the Masters Thesis/Recital Document or Doctoral Dissertation the text of an original paper, or papers, submitted for publication. The Masters Thesis/Recital Document or Doctoral Dissertation must still conform to all other requirements explained in the Guide for the Preparation of a Masters Thesis/Recital Document or Doctoral Dissertation at The University of Texas at San Antonio. It must include a comprehensive abstract, a full introduction and literature review, and a final overall conclusion. Additional material (procedural and design data as well as descriptions of equipment) must be provided in sufficient detail to allow a clear and precise judgment to be made of the importance and originality of the research reported. It is acceptable for this Masters Thesis/Recital Document or Doctoral Dissertation to include as chapters authentic copies of papers already published, provided these meet type size, margin, and legibility requirements. In such cases, connecting texts, which provide logical bridges between different manuscripts, are mandatory. Where the student is not the sole author of a manuscript, the student is required to make an explicit statement in the introductory material to that manuscript describing the students contribution to the work and acknowledging the contribution of the other author(s). The signatures of the Supervising Committee which precede all other material in the Masters Thesis/Recital Document or Doctoral Dissertation attest to the accuracy of this statement.

December 2014 ii

COSMOLOGICAL OBSERVABLES IN THE QUASI-SPHERICAL SZEKERES MODEL

Robert G. Buckley, Ph.D. The University of Texas at San Antonio, 2014 Supervising Professor: Eric Schlegel, Ph.D., Chair

The standard model of cosmology presents a homogeneous universe, and we interpret cosmological data through this framework. However, structure growth creates nonlinear inhomogeneities that may affect observations, and even larger structures may be hidden by our limited vantage point and small number of independent observations. As we determine the universe’s parameters with increasing precision, the accuracy is contingent on our understanding of the effects of such structures. For instance, giant void models can explain some observations without dark energy. Because perturbation theory cannot adequately describe nonlinear inhomogeneities, exact solutions to the equations of general relativity are important for these questions. The most general known solution capable of describing inhomogeneous matter distributions is the Szekeres class of models. In this work, we study the quasi-spherical subclass of these models, using numerical simulations to calculate the inhomogeneities’ effects on observations. We calculate the large-angle CMB in giant void models and compare with simpler, symmetric void models that have previously been found inadequate to match observations. We extend this by considering models with early-time inhomogeneities as well. Then, we study distance observations, including selection effects, in models which are homogeneous on scales around 100 Mpc—consistent with standard cosmology—but inhomogeneous on smaller scales. Finally, we consider photon polarizations, and show that they are not directly affected by inhomogeneities. Overall, we find that while Szekeres models have some advantages over simpler models, they are still seriously limited in their ability to alter our parameter estimation while remaining within the bounds of current observations.

iii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Chapter 1: Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

The FLRW metric and the concordance model . . . . . . . . . . . . . . . . . . . .

5

1.2

Inhomogeneous models as alternatives . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

The Lemaître-Tolman-Bondi model . . . . . . . . . . . . . . . . . . . . .

8

1.2.2

Past studies using the LTB model . . . . . . . . . . . . . . . . . . . . . . 12

1.2.3

Fundamental weaknesses of the LTB model . . . . . . . . . . . . . . . . . 14

1.3

1.4

The Szekeres model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1

Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2

Effects of the Szekeres functions . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.3

Geodesic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Goal of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 2: CMB dipoles in simple Szekeres models . . . . . . . . . . . . . . . . . . . . 23 2.1

Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2

Test models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3

2.2.1

Base LTB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2

Szekeres functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1

Spherical harmonics formalism and the observed CMB . . . . . . . . . . . 28 iv

2.3.2

Calculating the dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3

Higher order multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4

Origin of the dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6

2.5.1

Fitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.2

Size of “allowed” region . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.3

Higher order multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 3: Effects of varying bang time on the CMB . . . . . . . . . . . . . . . . . . . . 43 3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2

Model definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

3.4

3.5

3.6

3.2.1

Base LTB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2

Szekeres functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Direct effects of varying bang time . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1

Density distribution and evolution . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2

Expansion rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Results (primary models) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.1

CMB Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2

CMB Quadrupoles and Octupoles . . . . . . . . . . . . . . . . . . . . . . 51

3.4.3

Comparison with Doppler shifts . . . . . . . . . . . . . . . . . . . . . . . 54

3.4.4

Luminosity distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.5

Kinetic Sunyaev-Zel’dovich effect . . . . . . . . . . . . . . . . . . . . . . 60

Results (alternate models) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5.1

Varying M (r) instead of k(r) . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.2

Models with Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

v

Chapter 4: Assessing luminosity distance bias from large scale structure . . . . . . . . . 69 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2

Calculating distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3

The Dyer-Roeder approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6

4.7

4.5.1

Large coarse-grained structure model . . . . . . . . . . . . . . . . . . . . 77

4.5.2

Swiss Cheese model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6.1

Large coarse-grained model . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6.2

Swiss Cheese model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 5: Polarization with anisotropic expansion . . . . . . . . . . . . . . . . . . . . 94 5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2

Setup: electrodynamics in anisotropic spacetime . . . . . . . . . . . . . . . . . . . 95

5.3

The wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4

Solving the wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5

Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 6: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.1

Outlook for inhomogeneous models . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.1.1

Can they fit within current observational constraints? . . . . . . . . . . . . 105

6.1.2

Can they explain observed anomalies? . . . . . . . . . . . . . . . . . . . . 105

6.1.3

Can they affect parameter estimations? . . . . . . . . . . . . . . . . . . . 106

6.2

Observational tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3

Future lines of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 vi

6.4

Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Appendix A: Mathematical conventions and notations . . . . . . . . . . . . . . . . . . . 111 Appendix B: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Vita

vii

LIST OF TABLES

Table 2.1

Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Table 2.2

“Allowed” region sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 3.1

Dipole parameters in tB models . . . . . . . . . . . . . . . . . . . . . . . 52

Table 3.2

Ranges of quadrupoles and octupoles in the low-dipole region . . . . . . . 53

viii

LIST OF FIGURES

Figure 1.1

Illustration comparing Szekeres and LTB models . . . . . . . . . . . . . . 17

Figure 1.2

Shell crossing in a Szekeres model . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.1

Density plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Figure 2.2

Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 2.3

Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 2.4

Dipole parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 2.5

Dipole magnitudes in three dimensions . . . . . . . . . . . . . . . . . . . . 37

Figure 2.6

Large-angle CMB maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 2.7

Quadrupole magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 2.8

Quadrupole and octupole magnitudes . . . . . . . . . . . . . . . . . . . . 41

Figure 3.1

Curvature with varying bang-time . . . . . . . . . . . . . . . . . . . . . . 48

Figure 3.2

Density with varying bang-time . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 3.3

Expansion rates with varying bang-time . . . . . . . . . . . . . . . . . . . 50

Figure 3.4

CMB Quadrupoles and octupoles with varying bang-time . . . . . . . . . . 52

Figure 3.5

Doppler Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 3.6

Residual distance moduli along axes . . . . . . . . . . . . . . . . . . . . . 59

Figure 3.7

Kinetic Sunyaev-Zel’dovich velocities . . . . . . . . . . . . . . . . . . . . 61

Figure 3.8

Varied M (r) profile comparison . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 3.9

CMB multipoles with varied M (r) . . . . . . . . . . . . . . . . . . . . . . 63

Figure 3.10 Kinetic Sunyaev-Zel’dovich velocities with varied M (r) . . . . . . . . . . 64 Figure 3.11 Results for models with Λ . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.12 Kinetic Sunyaev-Zel’dovich velocities with Λ . . . . . . . . . . . . . . . . 66 Figure 4.1

Coarse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

ix

Figure 4.2

Swiss Cheese model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Figure 4.3

Coarse-grained model test 1 . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 4.4

Demagnification demonstration . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 4.5

Luminosity distances across the whole sky in the coarse model . . . . . . . 86

Figure 4.6

Distances through isolated structures . . . . . . . . . . . . . . . . . . . . . 87

Figure 4.7

Luminosity distances in the Swiss Cheese model . . . . . . . . . . . . . . 89

Figure 4.8

Luminosity distances in the Swiss Cheese model with larger holes . . . . . 90

x

Chapter 1: INTRODUCTION AND BACKGROUND The field of cosmology has made remarkable progress over the past century. Before Hubble studied the redshifts and distances of Cepheid variables, many (including Einstein) believed in a static universe. Now we know it to be expanding. Before the discovery of the cosmic microwave background (CMB) in 1964, there was great uncertainty about the origin of the universe, with many holding to the steady state theory. Now we know that the universe began in a hot, dense state following the Big Bang. And before 1998, it was commonly assumed that the universe’s expansion was decelerating due to gravity. This assumed model led to puzzling discrepancies, such as the missing density problem—the fact that the universe should be flat due to inflation, yet only about 30% of the critical density could be accounted for. This problem seemed so daunting that cosmologists Sean Carroll and Brian Schmidt made a bet that the density of the universe would not be known to within 0.3 by 2011 [1]. Then, the discovery of the dimming of distant supernovae radically changed the accepted picture of the universe. Most cosmologists now believe that the universe is accelerating, due to a negative-pressure component which accounts for the missing 70% of the total energy density. The current concordance model is dominated by two mysterious components: a cosmological constant Λ and cold dark matter (CDM). The ΛCDM model, as it is thus called, has proven successful at fitting a multitude of observations, and it has paved the way to more precise connections between theory and observation; the error bars on the matter density parameter are now ±0.017 [2], far below the ±0.3 predicted by Carroll and Schmidt. And yet, this model is still based on assumptions. It is founded on the elegant but simplistic Friedmann-Lemaître-Robertson-Walker (FLRW) metric, a solution to Einstein’s equations of general relativity for a spacetime that is homogeneous and isotropic—the same everywhere and in all directions. Cosmological observations are all interpreted using this metric as a starting point, and so any conclusions we draw are only valid if this metric is a good description of our universe. The notion of a homogeneous and isotropic universe is deeply entrenched in modern cosmology. It is such a fundamental part of our understanding that it has been dubbed the “cosmological 1

principle”. The idea of an isotropic universe arises from the fact that the sky looks essentially the same in all directions, as long as you pay no attention to fine details and ignore the galactic foreground. Combined with the insight that we should expect we do not occupy a “special” location in the universe (called the “Copernican principle,” as it grew from natural extensions of Copernicus’s heliocentric model), this also leads to homogeneity. Clearly, this is not an exact truth of our universe, as we see dense structures such as galaxy clusters and superclusters, as well as empty voids. However, it is widely believed that statistical homogeneity holds true if we average over sufficiently large scales, and that anything smaller than these scales can be ignored or treated only as small perturbations when considering cosmological observations. The simplicity of the concordance model has done much to facilitate progress, allowing for straightforward comparisons between theory and observations. It has been successful in explaining a variety of observations, but some observations lead to puzzles when interpreted in this paradigm. Foremost among these puzzles is dark energy. In 1998, researchers discovered that high-redshift supernovae appear dimmer than expected [3, 4]. Using an FLRW model, this implies that the expansion of the universe is accelerating. If Einstein’s equations hold true, this acceleration can only be explained by an unseen energy component with negative pressure, accounting for nearly three fourths of the total energy content of the universe. The nature of this dark energy is a complete mystery. Attempts to explain it from a particle physics perspective predict a value 54 orders of magnitude higher than what is observed [5].1 Furthermore, as the universe expands, the dark energy density is comparable to the matter density for only a brief period, and it seems to be quite a coincidence that we happen to find ourselves living in this short time [6]. There are other puzzles as well, some of which challenge the very notion of isotropy. These include the quadrupole/octupole alignment [7], the CMB cold spot [8], parity asymmetry and hemispherical asymmetry in the CMB [9, 10], bulk flows [11, 12], some anomalously large early structures [13], and others. Taken individually, none of these anomalies are large enough to pose 1

The discrepancy is often quoted as 120 orders of magnitude in the literature, but this is an overestimation—see discussion in [5] around eqs 516 and 548. Nevertheless, 54 orders of magnitude is still a very large mismatch.

2

a serious threat to the concordance model. However, several of them suggest a special axis of the universe, and these “preferred” directions from different observations align with each other to a surprising degree, pointing to the same region of the sky [14, 15]. The possibility remains that these are all simply due to systematic errors from instrumentation or local effects of the galactic plane, but they may also find a more compelling explanation in alternative universe models. In light of these anomalies, we must ask ourselves how confident we are in our initial premise. How sure are we that the universe really is homogeneous and isotropic at large scales? We do not have the luxury of multiple vantage points from which to confirm our observations. The vast majority of our data comes strictly from our single past light cone. The Copernican principle states that observers in other parts of the universe should see the same basic picture that we do, but while this thought is sensible and appealing, it is a philosophical assumption, not a scientific fact. We have but one universe to observe and one Earth to view it from, so we cannot rely on such assumptions without rigorous testing. As we attempt to push forward towards a more precise ΛCDM model, we should also check other paths to ensure that we are not on the wrong road entirely. With this in mind, a number of researchers have recently turned their attention to inhomogeneous universe models. Some have suggested that even if the universe is statistically homogeneous, the random inhomogeneities may be enough to influence the overall evolution [16, 17]. This idea, called “backreaction,” is possible because the equations of general relativity governing the evolution are nonlinear, so the averaged behavior of an inhomogeneous universe is not the same as the behavior of a universe that is first “smoothed over” into an averaged homogeneous universe. Others have suggested that there may be much larger inhomogeneities, at or near the scale of the Hubble radius, which are not clearly visible due to our limited perspective. These could give the appearance of acceleration when none actually exists. Because most of our observations are limited to our past light cone, space and time are intrinsically linked in our view of the universe. Perhaps, then, we are simply confusing temporal change in the expansion rate with spatial variation. To study this hypothesis, researchers have used toy models built from exact solutions to Einstein’s 3

equations. While the only tractable models are simplistic, not completely realistic representations of our universe, they provide a testing ground to connect theory with observation. The simplest models are based on the Lemaître-Tolman-Bondi (LTB) metric [18–20], which describes a spherically symmetric dust universe. This metric is well-suited to describing a giant void in which the lower density near the center results in a higher expansion rate. If we are near the center, this would account for the curve in our observations that we interpret as acceleration. This is not the only way for LTB models to reproduce the distance-redshift curve [21], but it is the most straightforward, and therefore most-studied. Although these models are capable of fitting the distance-redshift curve, they have difficulty fitting other observations at the same time. There is reason to suspect that some of these shortcomings are due to the symmetry of the models. Some researchers have thus begun to study a less simplified class of models based on another exact solution to Einstein’s equations, the Szekeres metric 2 [22], in particular the quasi-spherical subclass. This metric is a generalization of the LTB solution, but it has no symmetries, except in special cases. It is therefore capable of describing more complex arrangements of matter. This class of models forms the basis for this dissertation. The rest of this chapter will delve deeper into this progression of models, from the standard FLRW to Szekeres. It is structured as follows. Section 1.1 will provide a mathematical description of the FLRW metric, and a further discussion of the foundations of the ΛCDM model and its strengths and weaknesses. In section 1.2 we will review some key past studies of inhomogeneous models, particularly ones based on the LTB metric. We will again give a mathematical description, and explain its capabilities and limitations, and the criticisms against it. Section 1.3 will describe the quasi-spherical Szekeres metric. Finally, section 1.4 will lay out the goals of this work and give an overview of the remaining chapters. 2

pronounced “SEH-keh-desh”

4

1.1

The FLRW metric and the concordance model

A spacetime of an homogeneous and isotropic universe is described by the FLRW metric:

ds2 = −dt2 +

a(t)2 dr2 + r2 a(t)2 dΩ2 , 1 − kr2

(1.1)

where dΩ2 ≡ dθ2 + sin2 θ dφ2 . It is defined by only one curvature parameter k and one function of time, the scale factor a(t). The curvature parameter can be positive, negative, or 0, corresponding with spherical, hyperbolic, and flat geometries, respectively. The scale factor gives the overall size of the universe; as it increases, distances between objects increase in direct proportion, and so the universe expands. Its evolution, as described by the Einstein equations, reduces to only two simple equations, called the Friedmann equations [23]:

H2 =

8πG kc2 Λc2 a˙ 2 = ρ − + 2 a2 3 a 3 4πG a ¨ 3p Λc2 =− , ρ+ 2 + a 3 c 3

(1.2) (1.3)

where the dot denotes a time derivative, and we have introduced the energy density ρ, pressure p, and cosmological constant Λ; G and c hold their usual meanings. H is the Hubble parameter; its value at the present time is the Hubble constant H0 . Through these equations, the behavior of a homogeneous and isotropic universe under different conditions is well-understood. Without this simplification, it would have been far more difficult for theoretical cosmology to get on its feet. We can see from 1.3 that the scale factor’s expansion decelerates from the gravitational pull of the energy density, unless there is either a negative pressure or a cosmological constant to counteract it. From the role of Λ in the two equations, we can see that it is in fact equivalent to a fluid with positive energy density and negative pressure of equal magnitude. A perfect fluid obeys a law

ρ = wp

5

(1.4)

where w is a constant, so the cosmological constant is a kind of dark energy with w = −1. It is convenient to express the different energy components of the universe in terms of a density parameter relative to the critical density—the total density required for a flat universe, given the Hubble value. For a component x, then,

Ωx =

8πGρx ρx = . ρc 3H 2

(1.5)

The current concordance model is the ΛCDM model. The universe in this model is flat, as indicated by the CMB [24–26], so the sum of the density parameters must equal 1. As suggested by the first letter of the name, the model includes a cosmological constant dark energy with ΩΛ about 0.7. The remainder of the required density is in the form of non-relativistic (“cold”) pressureless matter, with Ωm roughly 0.3. About 3/4 of this matter is dark (not interacting electromagnetically). This cold dark matter, or CDM, is the other main component of the model. Radiation played a role in the early development of the universe, as seen in the CMB, but is too weak today to contribute significantly.

1.2

Inhomogeneous models as alternatives

A number of authors have proposed several different types of inhomogeneous universe models. Some, such as Thomas Buchert [16] and David Wiltshire [17], argue that even a statistically homogeneous universe cannot be smoothed out without losing important cosmological information. The key problem is that, in an expanding universe, averaging and time evolution are non-commutative operations. That is, for any quantity, the rate of change of the average is different than the average of the rate of change: d hΨi − dt



dΨ dt

 = hΨθi − hθi hΨi ,

6

(1.6)

where θ is the expansion rate. The right-hand-side terms constitute the “backreaction”, which alters the Friedmann equations in a way that can in principle mimic dark energy. This can be understood intuitively by the following example: consider a universe with a large static region and a small region of steadily expanding space. The average expansion rate is small at first, but eventually the expanding region will grow to a size comparable to the static region. At this point the average expansion rate will be considerable, larger than it was in the beginning. Thus we get apparent acceleration even though no region was individually accelerated, and there was no “dark energy”. There is significant disagreement over whether there could be enough variance in the expansion rate in our universe to produce a significant amount of backreaction [27, 28]. There is also some uncertainty regarding how exactly it would be reflected in our observations, since we cannot measure the average expansion rate of the universe directly. It is not obvious how to translate these quantities into ones relevant to observations. Fortunately, there are more direct, mathematically exact ways to deal with inhomogeneities, which reveal other ways inhomogeneities could affect observations, possibly to the point of making dark energy unnecessary. Using exact solutions to the Einstein equations, we can construct toy models in which we can study precisely how different kinds of inhomogeneities would affect observations. This is a key motivation for this dissertation. As mentioned previously, some have suggested that there may be larger-scale inhomogeneities that could directly and systematically skew our observations. In many cases, this requires us to be in a special location, in violation of the Copernican principle. However, this principle is an assumption held mainly for its philosophical appeal, not because of any solid observational proof, so it is worthwhile to see what happens when we relax this assumption. Suppose, then, that we are, by chance or otherwise, in a privileged location in the universe, such as near the center of a large spherical inhomogeneity. From this vantage point, we might see that distant galaxies are not receding as quickly as nearby ones (relative to their distance). This resembles our present state of affairs, so we would likely conclude that the universe’s expansion is accelerating, but in this hypothetical scenario this is a faulty conclusion derived from mistaken assumptions. The 7

truth would be that the expansion rate decreases with distance, rather than increases with time. Because nearly all of our observations fall on our past light cone, time and radial distance are tightly intertwined in our data. We only see a slice of the full 4-D spacetime. It is therefore very difficult to distinguish between spatial variation and temporal variation, until we find reliable observations that probe off the light cone. In this scenario, then, dark energy is merely an illusion resulting from the inhomogeneous distribution of matter. This inhomogeneous distribution would not be clearly visible, again due to our limited perspective. This kind of scenario can be well described by the LTB class of models [18–20]. This is not the class of models that will serve as a basis for our work, but the model we use will be an extension of the LTB model. It is therefore important to have a firm grasp of what the LTB model is and how it works, as well as the limitations that prompted us to use a more complex model. Understanding the LTB model will make it much easier to picture and comprehend later models. 1.2.1

The Lemaître-Tolman-Bondi model

This is perhaps the simplest kind of inhomogeneous model whose evolution can be solved exactly in general relativity. It is spherically symmetric, but the matter density and curvature can vary in the radial direction. The only energy content is a comoving, irrotational, pressureless dust, though a cosmological constant can be added with a simple modification. It is described by the metric R0 (t, r)2 2 dr + R(t, r)2 dΩ2 . ds = −dt + 1 − k(r) 2

Here,

0

2

(1.7)

denotes a partial derivative with respect to r (and, as before, a dot will denote partial

derivation with respect to t). The radial coordinate r can be thought of as a label marking concentric spherical shells. The function k(r) is a curvature function; note that curvature can vary radially, even switching signs if so defined. The function R(t, r) is a generalization of the scale factor a(t) from the FLRW model. It tracks the sizes (areal radii) of the shells, which do not in general expand

8

(or contract) in tandem. Its evolution is dictated by the equation: 1 ˙ 1 2M (r) − k(r) + ΛR(t, r)2 . R(t, r)2 = 2 c R(t, r) 3

(1.8)

This mirrors the first Friedmann equation. Note that we have introduced a new function, M (r), which appears as a constant of integration. It can be considered to represent the total effective gravitational mass inside the shell at r. Specifically, it is related to physical quantities by

4π

G M 0 (r) ρ(t, r) = . c2 R(t, r)2 R0 (t, r)

(1.9)

With these two equations, the evolution of any shell can be determined. We see that each shell expands much like a pressureless FLRW universe. In fact, the LTB model includes the FLRW model as a special case—when k(r) ∝ r2 , M (r) ∝ r3 , and R(t, r) = a(t)r, all of the equations reduce to the Friedmann equations, and we have a homogeneous isotropic model again. There is, however, one further detail—the time at which different shells begin their evolution. Whereas the FLRW model has all of space emerging from the big bang singularity simultaneously, the LTB model is, mathematically at least, free to have different shells emerge at different times. This gives rise to another functional degree of freedom called the “bang-time function” tB (r). Imagine the big bang singularity as a point at which all of the shells are collapsed, with R equal to 0 for all r. Consider two shells in particular, r1 and r2 , with r2 > r1 . At time tB (r2 ), shell r2 begins to expand, but shell r1 remains collapsed—that is, for tB (r2 ) < t < tB (r1 ), R(t, r2 ) > 0 while R(t, r1 ) = 0. Later, at time tB (r1 ), shell r1 emerges from the singularity and begins to expand. While this picture may seem ridiculous, be aware that it is only a mathematical description, and does not necessarily reflect the physical reality represented by such models. Remember that the LTB model is only valid in regimes where the energy content is well-approximated as pressureless dust. If we look back too close to the big bang, we reach the point of matter-radiation equality, at which point this approximation fails. The LTB model, then, can only realistically describe what happens later. Because the big bang itself is before the time of the LTB model’s validity, the 9

bang-time function should not be taken to indicate a literally non-simultaneous big bang. Rather, it encodes velocity and density variations near the beginning of the time when the LTB model applies. It can be used to simulate decaying modes of perturbations [29]. Because most models of the early universe are more homogeneous at earlier times, we choose a constant bang-time function in chapters 2 and 4, precluding decaying modes. However, in chapter 3, we consider the possibility of a varying bang-time function and further examine its implications. If, for some time t and radial coordinate r, the value of R(t, r) is known, the value of the bang-time function at that r is found by integrating eq (1.8), obtaining Z t − tB (r) = 0

R

dΦ p . 2M (r)/Φ − k(r) + ΛΦ2 /3

(1.10)

We would appear to have three free functions of r: M , k, and tB . However, since r is merely a label, it carries with it a gauge freedom; all of the above equations are invariant under transformations of the form r˜ = f (r) (as long as f (r) is monotonic). This means that we can impose a definition of our choosing on one function of r, granting some welcome convenience. For instance, in some situations, it is advantageous to define M (r) = r3 , or even M (r) = r. For my purposes, I will require R(t0 , r) = r in units of Mpc, where t0 is the present time. With this requirement, tB (r) can be determined from eq (1.10) once M (r) and k(r) are chosen. Whatever gauge choice we make, we are left with two free functions to define the model. Note a potential danger in eq (1.9). If R0 ever vanishes (while M 0 does not), the density becomes infinite. This is known as a shell crossing singularity, since it occurs as a result of one shell trying to expand through another (and thus compressing all the ones in between). When this happens, the model becomes unphysical, because in reality the matter would react to such compression with heat and pressure, preventing the singularity from forming. Because the LTB model is based in part on the assumption of a pressureless dust matter, it loses validity in this circumstance. Shell crossing singularities are therefore to be avoided. This is a non-trivial task, since the shells generally evolve essentially independently of one another. Many of the most useful

10

models for explaining apparent acceleration do so by having inner shells expand more quickly than the outer ones, making shell crossings even more likely. Methods for preventing shell crossing in construction differ with different construction methods (i.e. different choices of starting points for defining the functions), but we can always perform a simple check on R0 once our model is defined. Once our model is defined, we can begin to calculate what observations an observer in such a universe would make. It will be particularly useful to be able to determine the paths of light rays. We can calculate such paths numerically by solving the (affinely parameterized) null geodesic equations:

α β k;β k = 0,

(1.11)

where k α ≡ dxα /dλ is the tangent vector, λ is the affine parameter along the geodesic, and a semicolon indicates a covariant derivative. Integrating these equations backwards in time, starting at the observer, allows us to connect distant sources to observations in the sky. The redshift z at any point along the geodesic is defined by (kα uα )s , 1+z = (kα uα )o

(1.12)

where subscripts s and o denote source and observer respectively, and uα is the four-velocity of the source or observer, in this case (1, 0, 0, 0) because the matter is comoving. We can normalize the null geodesic tangent vector at the observer so that kot = −1, so we are left with simply 1 + z = −kst .

(1.13)

In this way, we can calculate how the model’s inhomogeneities change the CMB temperature at any point in the sky. Similarly, we can calculate angular diameter distances DA along the geodesic by using the Sachs optical equations [30], as we will see in chapter 4. One of the key advantages to using an ex11

act model to describe inhomogeneities is that such observations can be calculated straightforwardly and unambiguously. 1.2.2

Past studies using the LTB model

Much work has already gone into studying the LTB model and its observational properties. It was first formulated by Lemaître and Tolman in 1933–1934 [18, 19], and further developed by Bondi in 1947 [20]. It did not receive a great deal of attention initially, but interest gradually increased as more theoretical, observational, and computational tools became available. Interest started ramping up more quickly around the turn of the century, as researchers began to see the potential of the idea. Around 2006, researchers first started seriously considering LTB models as explanations for cosmological observations, and early findings were promising (e.g. [31]). In 2008, Garcia-Bellido and Haugbølle found an LTB model that fit the SNIa distance-redshift data, the baryon acoustic oscillation (BAO) data, and the scale of the first peak of the cosmic microwave background (CMB) simultaneously [32]. This model featured a huge 2.5 Gpc void around us, but it was able to fit the data at the time without the need to resort to dark energy. More recently, though, Garcia-Bellido and others returned to this model and found that new BAO data is in severe tension with the SNIa data, making the model no longer a truly viable alternative [33]. They did point out that isocurvature perturbation modes would help, but these are limited by CMB observations. Other authors considered a different kind of model based on the LTB metric—Swiss Cheese models, e.g. [34]. These are made of not one spherical inhomogeneity, but many, embedded in an FLRW background. Typically, one constructs a single LTB model to represent all of the “holes”, which matches to the FLRW background “cheese” at a certain radius. Matching the expansion rate requires that the total effective mass within the hole must be equal to the mass of the same region in the FLRW model—so if the holes have underdense voids in the center, they must be surrounded by overdense walls. This was intended to replicate the kind of wall/void structure we see in reality. The effects of these inhomogeneities, however, were typically mild, and could not explain the apparent acceleration without dark energy [35]. 12

As interest in LTB models grew, several authors built up theoretical frameworks to help others deal with certain aspects more rigorously. Perturbation theory has long been used in FLRW models, but it does not translate straightforwardly to the LTB metric, particularly because the radial inhomogeneity causes scalar, vector, and tensor modes to interact. Nevertheless, in 2008 Zibin laid out a working method ignoring only scalar-tensor coupling [36], and later Clarkson gave a complete solution to the problem [37]. Krasi´nski and Hellaby showed how to define a model based on arbitrary initial and final density profiles [38], and Ellis, Hellaby, and several others showed how to construct an LTB model directly from various sets of observations, the so-called “inverse problem” [21, 39–42]. Over time, however, criticisms of the LTB model also grew. Moss and Zibin found that models which fit both the CMB and SNIa observations clash with observations of the local Hubble rate (lower than observations) and the age of the universe (older than observations). They also find tension with the radial BAO scale in the outer regions, the Compton y distortion (defined in appendix B), and the local amplitude of matter fluctuations [43]. Foreman argued that LTB void models replace the temporal fine-tuning problem of dark energy with its own spatial fine-tuning problem: it requires the observer to be very close to the center of the void [44]. If we are too far off-center, we would see a much larger CMB dipole than what we actually observe, since photons passing through the center of the void experience more of the higher expansion rate inside the void, and are also subject to a large-scale Rees-Sciama effect [45]. Alnes first estimated the relationship between the observer’s position and the CMB dipole [46], and later calculated that this constrains us to a position within 15 Mpc of the void center for a 1500 Mpc-radius void [47]. Foreman later calculated the constraint at 80 Mpc using a different model and somewhat different methods, still a small fraction of the total void radius [44]. This goes against the Copernican principle. Even if the observer is lucky enough to be in this small low-dipole region, the high dipoles seen by hypothetical observers farther from the center poses a problem due to the kinetic SunyaevZel’dovich (kSZ) effect. Free electrons scatter CMB photons towards the observer, and if those 13

electrons see a large dipole along the line of sight, this will affect the observed spectrum. This creates an additional contribution to the CMB power spectrum at small angular scales, tracing the anisotropy of the projected free electron surface density [45,48–50]. This effect was first studied in relation to clusters in LTB void models by Garcia-Bellido and Haugbølle [45], then estimated for intra-cluster gas [49], and finally shown by Moss and Zibin to rule out most LTB models without dark energy [48]. Furthermore, due to the symmetry, CMB anisotropies in the next several multipoles beyond the dipole receive very little contribution from the inhomogeneity for observers near the center. This is disappointing, because one of the most significant anomalies seen in the WMAP CMB data is the improbable alignment of the quadrupole and octupole, first pointed out by Tegmark [7]. The preferred axes of these two multipoles lie within 1 ◦ , due to no specific feature, and there is currently no model to explain this [51]. (Initial studies of the Planck data, however, find a misalignment between 9 ◦ and 13 ◦ —still a significant alignment, but not necessarily anomalous [52].) One might imagine that a very-large-scale inhomogeneity of the sort proposed to explain the distance-redshift curve could explain these large-scale anomalies, but Alnes found that, for observers in the region allowed by the dipole, such a void produces only a very small quadrupole and octupole, insignificant compared to what is found in the WMAP [47]. 1.2.3

Fundamental weaknesses of the LTB model

While the LTB model has more freedom than the FLRW model, it is still limited by its spherical symmetry, and there are reasons to suspect this property is to blame for its failures. For Swiss Cheese models, the requirement that the holes match to FLRW at the boundary necessitates a compensating overdense wall, and the symmetry makes it impossible for photons to pass through the void without also passing through the wall. This makes the change to z average out to only a small adjustment [34]. The Copernican objections are also somewhat tied to the LTB geometry. Having the entire universe symmetric about one point makes that a special point indeed, but this is not the only 14

picture inhomogeneous models can make. A less symmetric model would not have such a unique central location. The symmetry of LTB also limits its ability to explain any kind of anisotropic anomaly, and it cannot describe any more detailed structure than spherical shells. Further shortcomings will be detailed in later sections. Célérier and others have frequently asserted that others have misapplied the LTB model. They argue that it should not be considered to be an exact model of the universe, but rather the result of smoothing out the structure in two dimensions instead of three, as FLRW does [53]. It is not surprising, then, that we should seem to be at the center of a spherically symmetric universe— this is merely an unavoidable consequence of the smoothing. Likewise, we should not expect that distant gas in an LTB model should cause a large kSZ effect, since observers in these locations would also see themselves as the center of the universe after performing this smoothing procedure, and would therefore not see a large CMB dipole. From this viewpoint, though, the question of what we should expect to see becomes somewhat ambiguous. It would be interesting, then, to consider a model that is similar to LTB but one step more general—one that does not completely smooth out the angular coordinates. The Szekeres model satisfies these needs, and it has seen a comparatively small (though significant) amount of study.

1.3

The Szekeres model

The Szekeres model is a generalization of the LTB model, introduced in 1975 [22]. It, too, contains only a comoving, irrotational, pressureless dust. The Szekeres model, however, in general has no symmetry; there are no Killing vectors, except in special cases [54]. There are three subclasses of Szekeres models, each with different geometry: quasi-hyperbolic, quasi-planar, and quasi-spherical. The quasi-spherical Szekeres model will be the focus of this study, as it is simplest to understand, includes LTB as a specific case (allowing for direct comparisons), and is best suited to describing localized structures. (From here on, when we refer to the 15

Szekeres model, we will mean the quasi-spherical subclass.) It is described by the metric 0

(R0 − R EE )2 2 R2 ds = −dt + dr + 2 (dp2 + dq 2 ). 1−k E 2

2

(1.14)

This is remarkably similar to the LTB metric. Like LTB, the Szekeres model consists of a series of spherical shells labeled by the coordinate r, and the functions R and k play the same roles as before. The coordinates on the shell, p and q, relate to the more familiar θ and φ by a stereographic projection, as we will explain in the next subsection; the shells are still perfectly spherical. The only real difference comes from the function E = E(r, p, q), which describes the departure from LTB—unlike LTB, the shells are not concentric, nor is matter distributed evenly across a given shell. The function E(r, p, q) is defined in terms of three arbitrary functions of r as

E(r, p, q) =

[p − P (r)]2 + [q − Q(r)]2 + S(r)2 . 2S(r)

(1.15)

The functions P (r), Q(r), and S(r) are hereafter referred to as the “Szekeres functions”. 1.3.1

Spherical coordinates

We can bring the coordinates to a more familiar form with a simple transformation:   θ p − P = S cot cos φ, 2

(1.16a)

  θ sin φ. q − Q = S cot 2

(1.16b)

In these coordinates, the metric is significantly more complicated and no longer diagonal, but for some applications they provide greater clarity. For instance, we can write E0 S 0 cos θ + (P 0 cos φ + Q0 sin φ) sin θ =− . E S

16

(1.17)

Figure 1.1: A simple illustration of an LTB model (left) and a Szekeres model (right).

This makes it clear that P defines anisotropy in the direction (θ, φ) = (π/2, 0), Q in the direction (π/2, π/2), and S in the direction θ = 0—what we would call the “x”, “y”, and “z” directions in pseudo-Cartesian coordinates. Note that on the positive “z” axis the Szekeres p and q coordinates diverge; we will have to steer clear of this region to avoid problems with our numerical calculations. There is nothing physically special about this region (except in the case of P 0 = Q0 = 0 and S 0 6= 0, but even then a simple coordinate transformation can switch the positive and negative “z” directions), so systematically avoiding this region should not significantly affect our analyses. We should take a moment to gain some intuition about the shape of E 0 /E, since this combination dictates many of the physical properties of the model, as we will soon see. On a 2-sphere of constant r, E 0 /E forms a dipole distribution, ranging from a maximum value of 

E0 E

 max

p (P 0 )2 + (Q02 ) + (S 0 )2 = S

(1.18)

to the negative of this same value at the antipodal point. On the great circle between the maximum and the minimum, E 0 /E = 0. Therefore, each 2-sphere is symmetric about an axis. The direction of this axis can change from one 2-sphere to the next, though, so the model as a whole is generally not symmetric.

17

1.3.2

Effects of the Szekeres functions

The Szekeres functions have three effects on the model: • They displace the centers of the shell r+δr relative to the shell r by δrRP 0 /S in the direction (θ, φ) = (π/2, 0), by δr RQ0 /S in the direction (π/2, π/2), and by δr RS 0 /S in the direction (0, 0); • They rotate the shells by δr P 0 /S about the axis (π/2, −π/2) and by δr Q0 /S about the axis (π/2, 0); • They redistribute the matter on each shell in the shape of a dipole along the direction of shifting. If P 0 , Q0 , and S 0 all vanish, the model reduces to LTB. The Einstein equations still reduce to only two equations. The first is exactly the same as eq (1.8), meaning the shells expand just as they would without the shifting from the Szekeres functions. The second, describing the density, is a modification to eq (1.9): 0

(r,p,q) M 0 (r) − 3M (r) EE(r,p,q) G h i. 4π 2 ρ(t, r, p, q) = 0 (r,p,q) c R(t, r)2 R0 (t, r) − R(t, r) EE(r,p,q)

(1.19)

We can see that the sign of the change in density from LTB depends on whether the LTB density on the shell is greater than or less than the average effective density inside the shell; i.e., a void model will see higher density where the Szekeres push shells together on the edge of the void, and a model with a central overdensity will see the opposite. The density of an FLRW model written in LTB form is unaffected by the Szekeres functions. Because the shells’ evolution is the same as in the LTB model, the bang-time function Eq (1.10) also holds in the same form. We noted previously that LTB models face a danger of forming singularities when one shell crosses another. In Szekeres models, this danger is amplified. Even if an inner shell remains 18

Figure 1.2: A simple illustration of shell crossing in a Szekeres model.

smaller than an outer shell, the Szekeres functions can shift them enough that they bump against each other, and even intersect, as illustrated in figure 1.2, resulting in a point or ring-shaped singularity. Specifically, whenever R0 − R E 0 /E = 0, the density diverges. We can usually avoid this situation by being careful in our choices for the Szekeres functions. We can easily calculate the extrema of E 0 /E using eq 1.18, so if we are using a gauge in which R(t0 , r) = r (as we do for the rest of this work), we can prevent shell collisions at the present time simply by ensuring (E 0 /E)max < 1/r. If our model begins in a near-homogeneous state at early times and grows more inhomogeneous as structures form (which is the case when tB (r) = const), this condition is usually sufficient. It does, however, place limits on the amount of anisotropic structure that is allowed. In Szekeres models, as in LTB models, there are two different spatial expansion rates at any point: the longitudinal expansion rate Hk and the transverse expansion rate H⊥ . The transverse expansion rate is in the two directions along the surface of the shell. Since the shells are simply 2-spheres in both LTB and Szekeres models, the expressions for H⊥ are the same in both, but Hk , which acts in the direction normal to the shell, is modified by the displacement from the Szekeres functions. 0 R˙ 0 − R˙ EE Hk = 0 0 R − R EE

H⊥ =

19

R˙ R

(1.20)

These variations over a shell give Szekeres models greater freedom than the simpler LTB models. They are capable of describing a wider variety of structures, and can show qualitatively different effects. For instance, Szekeres models can exhibit faster nonlinear structure growth than spherically symmetric collapse models [55]. Of course, this wider landscape of possibilities also means the model has more degrees of freedom, with the addition of the Szekeres functions. In total, after making our gauge choice, we need five free functions of r to define the model: M , k (or tB ), S, P , and Q. 1.3.3

Geodesic equations

While the geodesic equations are fairly lengthy when written out explicitly, Nwankwo and Ishak have shown that a simplification is possible through the definitions R , E 0 (R0 − R EE )2 . H= 1−k F =

(1.21) (1.22)

With these compactified functions, the geodesic equations become [56] dk t 1 1 + H,t (k r )2 + dλ 2 2 dk r dH r 1 1 + k − H,r (k r )2 − H dλ dλ 2 2 p 2 dk 1 d(F ) 1 F2 − H,p (k r )2 + kp − dλ 2 dλ 2 q 2 dk 1 d(F ) q 1 F2 − H,q (k r )2 + k − dλ 2 dλ 2

1.4

F2


 p 2  q 2 (k ) + (k ) = 0, ,t

(1.23a)

F2


 p 2  q 2 (k ) + (k ) = 0, ,r

(1.23b)

F2


 p 2  (k ) + (k q )2 = 0, ,p

(1.23c)

F2


 p 2  q 2 (k ) + (k ) = 0. ,q

(1.23d)

Goal of this work

This work aims to investigate possible cosmological applications of quasi-spherical Szekeres models. We will develop observational tests using numerical simulations, with which we will assess the capabilities of Szekeres models. We will look at how well they can fit existing observations, how 20

much they could affect our interpretation of cosmological parameters such as the dark energy density, whether they could explain the observed anomalies, and how we could potentially distinguish such models from the standard FLRW. The work is divided into four projects. The first, presented in chapter 2 deals primarily with the CMB dipole. It will test a variety of Szekeres models based on a giant void model, to find whether this simple observation places as strong a constraint on the observer’s location as it does in LTB models. We will demonstrate that this constraint is quantitatively similar, but qualitatively less objectionable, as the location to which the observer is constrained is not as geometrically “special” as in the LTB case. Furthermore, we will show that, unlike in LTB models, observers in this low-dipole region can still see a significant quadrupole, opening the way for Szekeres models to potentially explain the large-angle CMB anomalies. The second project, detailed in chapter 3, is an extension of the first project, in which we consider more extreme models. In particular, we include variations in the bang-time function (eq 1.10), as well as stronger anisotropies through the Szekeres functions. The goal is to more thoroughly test the limits of the effects Szekeres models can have on the CMB in order to place observational constraints on the parameter space, as well as to seek a model which can generate both a quadrupole and octupole comparable to our observations. Such a model would be a possible explanation for the anomalies we see in the low CMB multipoles. Chapter 4 explains the third project, which considers Szekeres models which do not feature a Hubble-scale inhomogeneity, instead containing only relatively small inhomogeneities in concordance with statistical homogeneity. The objective is to test the hypothesis that (relatively) small inhomogeneities can systematically bias observations through selection effects, as it is easier to see through less dense regions. This hypothesis is similar to one made by Zel’dovich [57] that led to the “empty beam” approximation and later the Dyer-Roeder approximation [58]; however, these approximations have been shown to be inadequate. We believe that using exact inhomogeneous models will help clarify the issue. We will test both a large, complex Szekeres model with many structures, and a Swiss Cheese model built out of smaller, simpler Szekeres models in an FLRW 21

background. For a range of models and observer locations, we will assess the magnitude of the effect and determine whether it could impact our interpretation of cosmological parameters such as dark energy. The fourth project is presented in chapter 5, and involves polarization of photons passing through inhomogeneous structures. This is a more purely theoretical work, in which we derive equations describing the evolution of the components of an electromagnetic wave in a Szekeres or LTB model. We work through the electrodynamics equations in general relativity to show that, despite the anisotropic expansion rates present in such models, the polarization is not directly affected by the inhomogeneities. That is, unpolarized light passing through a region of anisotropic expansion, such as a gravitational lens, will remain unpolarized, unless affected by more conventional effects from the matter in the region. Chapter 6 will summarize the results from these three projects and the conclusions that can be drawn from them. We will consider the implications for this line of research and speculate on how it could develop in the future. We will show how our results provide new insight into the level of significance of inhomogeneities for building an accurate understanding of the universe, and we will lay out what future observational tests could further constrain or support inhomogeneous models.

22

Chapter 2: CMB DIPOLES IN SIMPLE SZEKERES MODELS

2.1

Introduction and motivation

This project deals with the question of how the CMB dipole behaves in the presence of Szekeres inhomogeneities. Some analysis is also done on the quadrupole and octupole as well. This chapter is based on a paper published in Phys. Rev. D 1 . I am the primary author of this paper, and performed the calculations therein, with advice and guidance from Dr. Eric Schlegel. As mentioned in section 1.2.2, the CMB dipole imposes a strong constraint on the observer’s location in many LTB models, particularly giant void type models. Also recall that the high CMB dipoles seen by observers far from the center would result in a large kSZ effect, contrary to observations. Within the “allowed region”, observers also see very little quadrupole and octupole in the CMB, so LTB void models are unable to explain the related anomalies we see. The LTB models have many shortcomings related to the large-angle CMB, but remember that they are still a simplification of a possible reality. It is therefore worthwhile to strip away a layer of simplification by testing Szekeres models by the same criteria. Ishak et al. and others have argued that the lack of a unique center gives these models an advantage over LTB models with regards to the Copernican principle [59, 60]. If there is no single unique center, our position may not be so special after all. Nevertheless, we must still satisfy the requirement that the CMB dipole seen at the observer is not unacceptably large compared to observations. We must then ask, in what region of a Szekeres void model would an observer see a suitably small dipole? How does the volume of this region compare to that of the corresponding LTB model? If the region is still small, it would seem that even in the Szekeres model we must reside in a special location—the place where the observed dipole is small—even if it is not the “center”. This provides a more quantitative test of the model’s compliance with the Copernican principle. Once we have located the low-dipole region, we can also investigate other properties this region 1

R. G. Buckley and E. M. Schlegel, Phys. Rev. D 87, 023524 (2013), “CMB dipoles and other low-order multipoles in the quasi-spherical Szekeres model” DOI: http://dx.doi.org/10.1103/PhysRevD.87.023524

23

can have, such as the CMB quadrupole and octupole. This may show further advantages over LTB—in LTB, the low-dipole region has a inhomogeneity-induced quadrupole and octupole too small to explain the anomalous alignment seen in the real CMB [47], but we should not expect Szekeres to be so limited. Studying the dipoles across the void will also provide hints on whether Szekeres models suffer the same constraints from the kSZ effect as LTB models. In summary, the LTB void model has four shortcomings related to its symmetry, the Copernican principle, and the CMB, against which we wish to test the Szekeres model: • Quantitatively, there is only a small region in an LTB universe model in which an observer would see a CMB dipole consistent with observations. This fine-tuning requirement violates the Copernican principle, which implies that any location should be equally valid. • Qualitatively, the LTB model further violates the Copernican principle because this “allowed” region is geometrically special. • Due to the symmetry, CMB anisotropies in the next several multipoles beyond the dipole receive very little contribution from the inhomogeneity, so the LTB model offers no explanation for the observed anomalies in the quadrupole and octupole. • The kSZ effect at l ' 2000–3000 is too strong to be reconciled with observations [48].

2.2

Test models

We will construct a set of Szekeres test models by starting with one base LTB model and adding several different Szekeres functions to it, each one resulting in a different Szekeres model. 2.2.1

Base LTB model

For a base LTB model, we use a constrained Garcia-Bellido Haugbølle (GBH) model [32]. This model describes a large void with a homogeneous Big Bang (tB (r) = const), which asymptotically approaches a flat FLRW background at large r. It is defined in terms of two related functions of r: 24

the radially dependent matter density parameter Ωm (r) and the present-day transverse expansion rate H0 (r). 

 1 − tanh [(r − r0 )/2∆r] Ωm (r) = Ωout + (Ωin − Ωout ) , 1 + tanh [r0 /2∆r] s " # 1 Ωm (r) 1 − −1 , sinh−1 H0 (r) = H0 1 − Ωm (r) [1 − Ωm (r)]3/2 Ωm (r)

(2.1) (2.2)

which are in turn related to the functions M (r) and k(r) by 1 M (r) = H0 (r)2 Ωm (r)r3 , 2

(2.3)

k(r) = H0 (r)2 (Ωm (r) − 1) r2 .

(2.4)

Our choices for the parameters, Ωin (the matter density at the center of the void), r0 (the characteristic size of the void), ∆r (the sharpness of the void wall), t0 (the present age of the universe), and H0 (related to the local Hubble constant at the center of the void) are given in Table 2.1. This is similar to the best-fit model found in [32], which was selected by combining SNIa distanceredshift data, baryon acoustic oscillation (BAO) data, and the scale of the first peak in the CMB power spectrum, without dark energy. We define our r coordinate so that R(t0 , r) = r in units of Mpc. 2.2.2

Szekeres functions

For our test models, we desired something simple enough to be readily analyzable, yet without symmetries which could hide more general effects. We chose to set S(r) = 1 and Q(r) = 0. This leaves only one function to work with, P (r), yet does not result in axial symmetry (though there is a discrete bilateral symmetry). We constructed our P (r) functions to start at P (0) = 0, and increase smoothly between two radii ri and rf , after which it is again constant (meaning no Szekeres effects above rf ). The rate of increase is proportional to another parameter, labeled C.

25

Table 2.1: The parameters used to define our test models. The same base LTB model parameters apply to all 6 models. Quantities in Mpc refer to area distances of shells at the present time.

Ωin

0.13

Base LTB Model Parameters r0 ∆r t0 H0 km Mpc Mpc Gyr s Mpc

2300

620

15.3

Model 1 2 3 4 5 6

64

Szekeres Parameters C ri rf Mpc Mpc 0. 630 0 575 0. 315 575 2875 0. 945 100 300 0. 945 0 575 0. 630 0 1150 0. 945 1900 2100

The precise form, based on the form used in [60], is given by integrating

P 0 (r) =

    0    

if r ≤ ri −99/100 −0.0003r (r−ri )(2rf −ri −r) e (rf −ri )2

(1 + r)       0

if ri < r < rf .

(2.5)

if r ≥ rf

In this manner, we construct six test models, with parameters given in Table 2.1. In model 1, the P function is moderately strong and extends from the origin to one fourth the void radius. In model 2, the P function is weaker, but covers a broader range, and does not begin until one fourth the void radius. This allows us to separate the radial dependence of the dipole from local effects of the Szekeres function. The third model’s P function has only a relatively narrow spike, allowing us to examine the effects of an isolated segment of Szekeres anisotropy from locations in the interior, exterior, and middle of the anisotropic shells. These three models will be the focus of our investigation, but we will also examine three more: model 4, which is like 1 but with a stronger P function, model 5, again like 1 but with broader range, and model 6, like 3 but with the spike at a much higher r value, to compare the effects of distant anisotropies and nearby ones. Figure 2.1 shows two-dimensional cross-sections of the density distributions of each of the first three models. These are not intended to be realistic models. Their purpose is to provide insight 26

(a)

(b)

(c)

Figure 2.1: Density plots of (a) model 1, (b) model 2, and (c) model 3, covering a two-dimensional cross-section corresponding with the symmetry plane. The plotting range for each model is chosen to cover the Szekeres anisotropies, and the color scale is adjusted for each plot to maximize the visual contrast. Densities on the scale are written as fractions of ρF LRW , the density of the background FLRW model, which the test models approach asymptotically at very high r. Black circles show shells of constant r. The green triangles on each marked shell show the direction of shell shifting, and yellow dots show the geometric centers of the shells.

into the observational effects that can arise from a Szekeres-type anisotropy and to establish a baseline from which we can extrapolate to more general cases. A more realistic model would require that an observer in the region allowed by the dipole would also see a luminosity distanceredshift curve with directional variation within the constraints set by supernova observations, as well as consistency with baryon acoustic oscillations, galaxy age data, and other such observations (all now direction-dependent due to the lack of perfect isotropy), all while containing structures in some manner consistent with the shape and statistics of observed large-scale structure.

27

2.3

Methods

2.3.1

Spherical harmonics formalism and the observed CMB

The cosmic microwave background consists of photons last scattered when matter and radiation decoupled in the early universe. This last scattering surface (LSS) occurred at a redshift of about 1090, when the universe was less than 400,000 years old [61]. The photons are distributed in a near blackbody spectrum, but the temperature varies with direction, due to fluctuations at the LSS, the effects of intervening structures in the photon’s path, and our motion relative to the LSS rest frame. In this work, we are investigating the latter two effects, so we will ignore the intrinsic anisotropies from LSS fluctuations. The mean CMB temperature, Tˆ, is 2.725 K [61]. Temperatures across the sky are typically expressed relative to this mean, T (θ, φ) = Tˆ + ∆T (θ, φ). This temperature field can be decomposed into spherical harmonic functions,2 l ∞ ∆T (θ, φ) X X alm Ylm (θ, φ). = Tˆ

(2.6)

l=1 m=−l

This groups the anisotropies into fluctuations on different scales, beginning with the dipole l = 1, which measures variation across the whole sky. As l increases, the scale of the fluctuations decreases. The power spectrum collects terms of each l as l X 1 Cl = |alm |2 . 2l + 1 m=−l

(2.7)

The dipole term (l = 1) is by far the largest, with a maximum ∆T of 3.355 mK. This is assumed to be due to our solar system’s velocity relative to the LSS. To match the observed dipole, this velocity must be 369 km s−1 [61]. We will use this as our limit—if we calculate a larger dipole, that observation point is not consistent with our observations. The observed quadrupole and octupole 2

The l = 0 term is simply the monopole, which is simply the mean Tˆ, which we have separated out.

28

Figure 2.2: The six geodesics used to calculate the dipole at a particular point in model 5. Each geodesic is shown in a different color, and each dot represents one step in the numerical integration. The black line indicates where p and q diverge to ±∞.

(l = 2 and l = 3) have power

l C /(4π) l+1 l

approximately 300 µK2 and 1000 µK2 , respectively

(though the error bars are large), corresponding to fluctuations on the order of 10−5 . [62] 2.3.2

Calculating the dipole

We can calculate the observed CMB temperature at any point in the sky by generating a null geodesic from the observer backwards in time to the LSS3 , with the initial tangent vector at an angle corresponding to the point in the sky in question. If the CMB anisotropies are dominated by the dipole term (as we will check), we can calculate the dipole with a small number of geodesics. The method we will use is an extension of that used in [44]. We will generate three spatially orthogonal pairs of null geodesics backwards in time from the observer, with the geodesics in each pair propagating in opposite spatial directions. A basic illustration is shown in Fig. 2.2. 3

In practice, we only need to integrate out to where the model is sufficiently close to FLRW. From there, we can use standard FLRW equations to calculate the redshift at the LSS.

29

Assuming the temperature of the LSS is uniform, the CMB temperature measured at any point in the sky is found from the redshift of the geodesic in that direction by

T =

T∗ , 1 + z∗

(2.8)

where asterisks mark quantities at the LSS. We further assume that the intersections of the geodesics with the LSS occurs at an equal time t∗ in the synchronous gauge, regardless of direction of propagation.4 The temperature difference in each pair of geodesics can be treated as a component of a vector. The magnitude of this vector gives the total dipole. By dividing by the mean temperature, we get the apparent dipole velocity,

v=

q (T1 − T4 )2 + (T2 − T5 )2 + (T3 − T6 )2 1 3

(T1 + T2 + T3 + T4 + T5 + T6 )

,

(2.9)

where geodesics 4, 5, and 6 are in the directions opposite of geodesics 1, 2, and 3, respectively. q v. The dipole magnitude is given by D = 4π 3 In practice, we do not need to integrate the geodesics all the way back to the LSS. At large radii, our models asymptotically approach FLRW. If we integrate to a sufficiently early time t1 , all of the geodesics will be far enough outside of the inhomogeneity that from then on the redshifts evolve nearly exactly as in FLRW. Then we can write

T =

T∗ a(t1 ) × , 1 + z(t1 ) a(t∗ )

where a(t) is the scale factor of the FLRW background. The factors

(2.10) a(t1 ) a(t∗ )

in the numerator and

denominator of (2.9) cancel out, and can therefore be ignored. Likewise, we do not need to assume any particular value for T∗ , since it does not affect the final result in (2.9). 4

This is not strictly accurate, as the void in our test models approaches FLRW only asymptotically, without a compensating overdensity; since geodesics going in different directions from a non-central observer reach different radial distances, the LSS may occur at slightly different times for each. However, in our calculations, all of the geodesics reach distances where the density approaches FLRW closely enough that such differences are insignificant, as further verified in section 2.4.

30

To check that the result is the true dipole, and has not been overly contaminated by higher multipoles, we can repeat the process with a different set of orthogonal geodesics, and compare the results. Using this method, we estimate that the relative error in our data due to this effect is on the level of 10−3 or less. 2.3.3

Higher order multipoles

We have seen how to calculate the CMB dipoles generated by the inhomogeneities, but this is not the only effect the inhomogeneities have on the CMB. The inhomogeneities leave higher-order multipole imprints on the CMB as well. To analyze the extent of the inhomogeneity-induced spherical harmonics, we adopt a procedure similar to that used to calculate the dipoles, but with many more geodesics, propagating in evenly spaced directions across the entire sky. We will use a spacing of 4 degrees, for a total of 2534 data points for each location we test. To obtain the strength of a given multipole, we calculate the alm coefficients by numerical integration (limited by the resolution of the data)5 :

alm ≈

2534 X

∗ T (θn , φn )Ylm (θn , φn ) sin θn δθ δφ

(2.11)

n=1

We pixelise the sphere in a rectangular manner, with rows of points of constant θ and a uniform spacing between rows of δθ = 4 ◦ . Within each row, δφ varies to fill the circle6 :

δφ =

180 ◦ . b180 sin θ/4c

5

(2.12)

In practice, we remove each multipole (starting with the monopole) from the data before calculating the next, to avoid spurious results from integration error. 6 This simplistic pixelisation scheme is prone to certain systematic errors in the calculations, but tests suggest that in the present work these errors are on the order of 1 µK or less, small enough to be ignored. We used this scheme because it was simple to implement in Mathematica, but for the rest of work we use the HEALPix scheme, which is ultimately more reliable.

31

2.4

Origin of the dipole

In the LTB model, one can understand the CMB dipole in terms of the Rees-Sciama effect [45]. Since the void is in the nonlinear regime, its density contrast grows faster than the scale of the universe, causing the gravitational potential well to deepen over time. CMB photons passing through the void lose energy because the well they climb out of is deeper than the well they fell into. An off-center observer will therefore see that photons which pass through the center of the void are redshifted more than those coming from the opposite direction. Another way to understand the dipole is by directly looking at two null geodesics, one passing through the center of the void (ingoing) and the other extending radially in the opposite direction (outgoing). If the observer is near the center, the ingoing geodesic crosses the center and returns to the original shell without picking up much redshift. The two geodesics then both propagate outwards (and backwards in time), but since the ingoing geodesic took a nonzero amount of time to cross the center (linearly proportional to the initial shell radius), they cross each shell at slightly different times throughout the journey. We identify three potential ways in which the Szekeres functions can influence photon redshifts, and therefore the dipole: • By directly affecting the longitudinal expansion rate Hk (see eq. 1.20). In a void model, this means that the expansion rate is slower where shells are pressed together, and faster where they are stretched apart. This induces greater photon redshift on the stretched side and lesser redshift on the compressed side. • By altering the times at which photons pass through shells. Looking backwards in time from the observer, photons traveling along the direction of shell shifting must travel a greater distance to reach the outer shells than for an observer at the same coordinates in the corresponding LTB model, thus reaching them at an earlier time; conversely, traveling in the opposite direction appears faster. Even when the Szekeres functions do not extend to high radii, this effect causes the observer to see the outer shells as though looking from a shifted 32

position. • By influencing the total distance from the void the photons reach when they hit the surface of last scattering. This is related to the second case, but only applies when the model does not approach FLRW sufficiently quickly. The third contribution is undesirable, since it explicitly violates the assumption of a statistically uniform surface of last scattering. We have confirmed that it does not play a significant role in our model by comparing the difference in the change in redshifts for a typical observer’s geodesics between the times t0 /600 and t0 /20000. We find that they differ by less than 0.1%. The dipoles found using these two times as ending times also differ by less than 0.1%. We can therefore be confident that the dipole is not greatly influenced by effects near the surface of last scattering.

2.5

Results and discussion

For each model, we choose several r values, and for each of these we calculate the CMB dipoles seen by observers at evenly spaced locations covering the sphere. 2.5.1

Fitting function

We have found that the dipoles on each shell of constant r can be well approximated by a simple function of three parameters: h i D(r, θ, φ) = a(r)ˆ r(θ, φ) + b(r) cos θ0 (r)kˆ − sin θ0 (r)ˆi ,

(2.13)

where rˆ(θ, φ) is the radial unit vector (normal to the shell), and ˆi and kˆ are unit vectors in the directions (π/2, 0) and (0, 0), respectively. This describes the sum of two vectors, one of magnitude a and radial direction, and one of magnitude b and constant direction (θ0 , π). The magnitude of the former corresponds very closely to that of the dipole seen in the corresponding LTB model (i.e. a model with S(r), P (r), and Q(r) set to constant values, but otherwise unchanged). The other vector can be thought of as a “Szekeres dipole,” as it is the result of the Szekeres S(r), P (r), and 33

(a)

(b)

(c)

(d)

Figure 2.3: Magnitudes and directions of dipoles in a two-dimensional cross-section corresponding to the symmetry plane. Solid black arrows represent the total dipoles from numerical calculations, green dashed arrows are the LTB dipole, and blue dot-dashed arrows are the Szekeres component of the dipole. Smaller red dotted arrows are the fitting errors—the data minus the fit— magnified by a factor of 2000. (a): model 1, r = 100 Mpc; (b): model 1, r = 200 Mpc; (c): model 2, r = 150 Mpc; (d): model 3, r = 200 Mpc. Q(r) functions. A few examples are shown in Fig. 2.3. This simple function is able to fit the data to within 0.1 mK in all cases. Though the Szekeres dipole appears to be nearly constant on a given shell, its magnitude and direction do change as we move between shells. Figure 2.4 shows the radial dependence for each of the six models. A few key features are immediately apparent. In models 2 and 6, we see that the Szekeres dipole has very little radial dependence in the range tested. In model 3, shells outside the Szekeres anisotropy spike see virtually no Szekeres dipole at all, while interior shells see a significant amount. The shell in the middle of the spike sees a Szekeres dipole roughly (but not exactly) half the magnitude (and half the θ0 deviation from π/2) of the interior shells. In general, as we traverse through shells which have Szekeres anisotropies, the magnitude of the Szekeres 34

(a)

DT 3.355 mK 10 8 6 4 2 0

ô

ô

ô

ô

ô

(b)

Θ0 -А2 HradiansL 0.6

æ 1

ô ô à 2 ô ò ò ô ò ò ì 3 ò ò æ æ æ æ ò 4 ò ææææ à à ààææ æ à àò à ô 5 æ ò æ ç 6 ì ì ì LTB ì ç ç ç ç ç ç ì ì r HMpcL 50 100 150 200 250 300

0.5

ò æ

0.4 0.3

ì

æ

ô æ

ì

0.2 0.1 0.0

à

à

æ

ò ô

ò

æææ æ ææ

ì àà

ç ç ç ç ç ç 50 100 150

ò ô ò ô ò æ

æ

ô

ò

ò

æ

ì à

à

200

ì 250

ôô

ô

ò ì r HMpcL 300 à

Figure 2.4: (a): magnitudes of the Szekeres dipoles at various r values in each of the three models, and the LTB dipoles in brown, all as a factor of the actual observed dipole. (b): θ0 for the same dipoles.

dipole decreases and the angle decreases towards π/2. Furthermore, models in which the Szekeres anisotropies occur at higher r (e.g. model 5 compared to 4, or 6 compared to 3) appear to generate Szekeres dipoles with angles deviating less from π/2, compared to the differences in the Szekeres dipole magnitudes. The picture appears to be that the behavior of the Szekeres functions at r values lower than that of the observer has much less effect on the dipole than the behavior at higher r values. As we move outwards from r1 to r2 , the portion of the Szekeres functions between r1 and r2 loses its impact. At least for observers reasonably close to the origin (on the order of a few hundred Mpc or less), the effects of the Szekeres functions in the interior are virtually nonexistent. As found by [47], the LTB dipole component increases approximately linearly with r near the origin. However, the Szekeres modifications shift the region of interest to higher r values. We find that a cubic fit matches the data to within 3 × 10−4 mK for r ≤ 400. 2.5.2

Size of “allowed” region

For the model to be consistent with observations, one requirement is that the CMB dipole does not greatly exceed the actual observed dipole. A number of authors have found that in LTB models large enough to explain the observed acceleration, this is only true within a very small region near the center; everywhere else, the dipole is much larger [44–48, 50, 63, 64]. It would therefore

35

seem highly improbable that we would find ourselves in such a specific region where the dipole is relatively small. We wish to repeat this calculation in our Szekeres test models, to see the size and shape of this “allowed” region and determine whether there is any measurable advantage over LTB. We will use our fits for the magnitudes and directions of the Szekeres and LTB dipole components to find the region where the total dipole is less than the 3.35 mK dipole observed by COBE [65]. (A more complete calculation would incorporate an additional stochastic dipole component arising from peculiar velocities, but a rigorous calculation of this sort would require knowledge of the evolution of perturbations in a Szekeres model, so we will leave this to future work.7 ) The dipole is only low where the LTB dipole and Szekeres dipole nearly cancel; it must therefore be centered around a point on the shell where the Szekeres dipole magnitude lines intersect the LTB dipole line in Fig. 2.4. Once we have calculated the boundaries of this “allowed” region, we will numerically integrate over it to find the mass and volume contained within it. Our results are summarized in Table 2.2. Figure 2.5 shows this region visually for each of the three models. We see that they are still small, roughly spherical regions (even when they reside in a region of significant shell shifting and twisting), though they are displaced away from the coordinate origin. They are often larger than in the base LTB model, but still small compared to the size of the void, by a factor on the order of 10−6 . The mass is the more relevant quantity, since it determines the number of “allowed” galaxies. And we should expect that the “allowed” mass is in general larger in Szekeres models than in LTB, because the Szekeres anisotropy shifts the “allowed” region away from the center of the void. This means it is in a higher density region, with more galaxies where we may find ourselves located. It seems that removing the spherical symmetry of LTB does tend to somewhat alleviate the need for fine-tuning of the observer’s location, but not by nearly enough to fix the problem entirely. 7

Since the region of interest is not necessarily near the coordinate origin, the shear may be significant, so we cannot assume that perturbations evolve the same way as in FLRW, as done for LTB in [44].

36

(a)

(b)

(c)

Figure 2.5: The total dipole magnitudes across models 1 (a), 2 (b), and 3 (c). The region where the dipole is less than the actual observed dipole is shown as a green sphere. Its range is also marked in green on the axes. Shells of constant r are also shown, in increments of 33 Mpc, colored according to the magnitude of the total dipole, with lighter being larger.

Table 2.2: Volumes and masses of the “allowed” region in each of the six models, compared to that of the base LTB model. Model 1 2 3 4 5 6

Vi /VLTB 1.14 1.14 0.93 1.18 1.30 1.04

37

mi /mLTB 1.26 1.27 0.99 1.35 1.58 1.05

2.5.3

Higher order multipoles

Because a complete CMB map is far more computationally intensive, we have fewer data points for the higher order multipoles at this time, so our analysis is limited. We leave a more thorough analysis for future work, and present our preliminary results here. We performed the calculation for an observer at the point of zero total dipole in each of the six models. We found that model 2 has a significant quadrupole at this location—about 5 × 10−6 , compared to the real observed anisotropies of the order 10−5 [47]—and a very small octupole, on the order of 10−7 . Figure 2.6 shows the cmb map at this point, as well as at a random point near the edge of the “allowed” region. In the other five models, the quadrupole and octupole at the null-dipole point are below the level of the random noise from numerical errors, and are therefore not measurable. It is clearly not a fluke that the quadrupole vanishes at the null-dipole, since it happened in five very different models, but it does not appear to be a general rule for Szekeres models either, as seen in model 2. The distinguishing feature of model 2 is that the Szekeres anisotropies cover a broad range, reaching a very high r value. We may hypothesize that this high r Szekeres anisotropy is the reason for the difference in behavior—why the quadrupole is nonzero at the point where the dipole vanishes. Model 6 also has anisotropy at high r, but Fig. 2.4 shows that its total impact on the dipole is relatively small, and it stands to reason that its effect on the quadrupole might be small as well; because the Szekeres functions are so weak overall, they cannot push the quadrupole significantly away from zero at the null-dipole point. To test the hypothesis that high-r Szekeres anisotropies are what affects the quadrupole separately from the dipole, we need a model with stronger Szekeres functions, so we created a seventh model with C = 0.945, ri = 1500, and rf = 2500. This is similar to model 6 in that the Szekeres functions only act in the outer regions of the void, but the broader range gives the Szekeres dipoles greater strength. In fact, the magnitude of the Szekeres dipole seen in the inner regions (r < 300 Mpc) is within 3% of what is seen in model 2, with the direction the same to within

38

(a)

(b)

(c)

(d)

Figure 2.6: The full CMB sky induced by the Szekeres void of model 2. Maps are oriented such that the z axis (the top of the map) points in the model’s radial direction, and the center of the map points in the model’s θˆ direction. (a): raw CMB sky map for an observer near the center of the low-dipole region, with only the uniform 2.725 K monopole removed; (b): same, but with dipole and quadrupole removed, showing that no higher moments are visible above the noise. (c): raw map for an observer at a random point near the edge of the low-dipole region; (d): same, but with dipole removed.

39

a20 6. ´ 10-5 4. ´ 10-5 2. ´ 10-5

-100

0

100

-2. ´ 10-5

200

r HMpcL

Figure 2.7: The primary quadrupole coefficient a20 at several points along the radial line containing the null-dipole point in model 2, and a quadratic fitting curve. Negative r values simply refer to points on the opposite side of the origin. The larger red dot indicates the null-dipole point.

0.01 radians. The quadrupole at the null-dipole point in model 7, however, is double what it is in model 2—a full 10−5 , comparable to observations. The octupole is still only on the order of 10−7 , though. Comparing models 2 and 7 seems to confirm that, given equal Szekeres dipole strength, the model with Szekeres functions weighted at higher r values will have a larger CMB quadrupole at the null-dipole point. To better understand the more general behavior of the quadrupole and octupole, we gathered data at a number of different points in model 2 (with only 6 degree resolution for faster computations). Along the radial line passing through the null-dipole point, we found that the quadrupole is dominated by a20 , which follows a simple quadratic curve, as shown in Fig. 2.7. This parabola is centered neither at the origin nor at the null-dipole point, and its minimum dips significantly into the negative. The total quadrupole magnitude thus hits zero at two points on this line, with a hill in between (where the null-dipole point falls). Off of this line, the quadrupole displays more complex behavior, which we do not yet have enough data points to fully describe or explain. Figure 2.8 summarizes both the quadrupole and octupole data. The quadrupoles seem to roughly follow a quadratic trend, consistent with what Alnes found for LTB models [47], but it is clearly not an exact fit. For the octupoles, it is even less clear that a cubic fit is accurate. Finally, a test of the CMB at r = 300 in model 3, compared with a similar test in the corresponding LTB model, revealed that Szekeres behavior on shells interior to the observer’s shell has 40

(a)

DT HmKL

(b)

DT HmKL 0.025

0.15

0.020 0.015

0.10

0.010 0.05

50

100

150

200

250

Dquad HMpcL

0.005 50

100

150

200

250

Doct HMpcL

Figure 2.8: (a) The quadrupoles at all tested points in model 2, in terms of ∆T , as a function of the distance from the center of the fit shown in Fig. 2.7. The blue curve is a simple extrapolation of the fit from Fig. 2.7. (b) The octupoles at all tested points, as a function of the distance from the center of a cubic fit on the line containing the null-dipole point. Both quadratic (green) and cubic (blue) fitting curves are shown for comparison.

negligible effects on the entire CMB, not just the dipole. The differences between the two maps are on the level of 1 µK (a tenth the strength of even the octupole), and appear to follow a random noise pattern across the entire sky; we can thus attribute these small differences to numerical error.

2.6

Conclusions

In this chapter, we have studied the CMB dipole seen by observers in a Szekeres model. We have established a procedure for calculating dipoles at general locations, and we have shown that they follow a simple, consistent pattern. While the models tested show little quantitative advantage over LTB in terms of the size of the region allowed by dipole observations, Szekeres models do offer greater freedom in where this region is located. We are no longer required to be at the center of the void, where the density is low and anisotropies are only significant at the dipole level. We have found that the CMB quadrupole seen by observers in the low-dipole region is not always as small as in the corresponding LTB model, and significant compared to the quadrupole seen in the WMAP data. The octupole was still small in this region in all the models tested, but it is possible that a more extreme Szekeres model would amplify that mode as well. There is then some hope that a Szekeres model may offer a possible explanation for the WMAP quadrupole and octupole anomalies.

41

Of the four shortcomings of LTB listed in section 2.1, it appears that Szekeres models offer improvements on one and a half. The region allowed by the dipole requirements is still small, so there is still a need for fine-tuning of the observer’s location, but this region is not necessarily “special” in other ways, as it is in LTB void models. That is, LTB void models constrain the observer to a small region that sees a small CMB dipole, and also happens to see a very small quadrupole and octupole, lie near the unique symmetry center of the entire model, and typically be the region of minimum density, whereas a Szekeres void model constrains the observer to a region that is only special in the first of these ways. The strength of the quadrupole and octupole in this region show significant improvement over LTB for some models, but not for others, and it is still unclear whether they can truly match the anomalies seen by WMAP. The kSZ effect, though not calculated here, is expected to still be a problem for Szekeres void models, since the total dipoles still follow a roughly linear trend similar to the LTB model. It is worth noting that the test models considered here used a homogeneous bang time function, meaning no decaying modes are present. While this is consistent with the standard view of inflation and the early universe, it has been suggested that even slight variations in the bang time could significantly reduce the kSZ effect and allow for very different void profiles. We will study such models in the next chapter.

42

Chapter 3: EFFECTS OF VARYING BANG TIME ON THE CMB

3.1

Introduction

In the previous chapter, we restricted our tests to relatively simple models. The LTB density profile was a smooth curve shared among all the models, and the Szekeres functions were designed to create anisotropies of fairly low magnitude and a single general direction. There was no dark energy of any sort. Perhaps the most significant simplification was the assumption of a constant bang time function. A full functional degree of freedom was left unexplored. There are good reasons for adopting this simplification. A varying bang time would result in large inhomogeneities at early times. Not only is this at odds with the standard models of inflation, which result in a nearly homogeneous universe with only small quantum fluctuations to serve as the seeds of structure growth, but it also violates the assumptions of standard CMB analysis [66]. Nevertheless, it is worthwhile to test what happens when we loosen our standard assumptions about the early universe. Our understanding of inflation is not solid enough to discount other possibilities. This chapter will explore models that push the boundaries of what is allowed by observations, in order to determine the limits of the possible effects of inhomogeneities. We will also perform a deeper analysis of some of the results in order to gain a more complete understanding of the connections between a model’s physical characteristics and the observables seen within that model. The rest of this chapter is organized as follows. In section 3.2, we describe the basic model definitions we will use. We define a set of 16 models we will use in our standard analysis, with different combinations of Szekeres functions and bang-time variations. Section 3.3 describes the direct physical effects of the varying bang-time on the models, and section 3.4 presents the results from out numerical calculations of the observables within the models, beginning with the CMB dipoles, followed the quadrupoles and octupoles, then luminosity distances in select models, and finally a brief look at the kinetic Sunyaev-Zel’dovich effect. Section 3.5 tests a few alternate

43

models: ones with a different way of incorporating the varying bang-time, and then ones with a non-zero cosmological constant paired with a shallower void. Finally, we discuss our conclusions in section 3.6.

3.2

Model definitions

Guided by our experiences from the previous chapter, we will define a new model parameterization that will expedite efficient and precise calculations. We will choose our LTB profile such that it transitions to FLRW exactly at a certain r, rather than asymptotically, so there will be no uncertainty as to whether our calculations reach “close enough” to FLRW. Our Szekeres functions will be chosen to test a range of degrees of anisotropy, from moderate to extreme, in order to probe the limits of the possible effects. We will limit our tests to axially symmetric models, since we can take advantage of the symmetry to greatly accelerate our calculations. 3.2.1

Base LTB model

We will use a present-day matter density profile based on the empirical formula used by Hamaus, et al. to describe a general void [67]:  ρ(r) = ρ0F LRW

1 − (r/rs )α 1 + δc 1 + (r/rv )β

 .

(3.1)

This describes a void with an outer compensating overdensity, with a density contrast of δc at the center, and an overall radius rv . The density far from the void approaches the present density of the FLRW background, ρ0F LRW . The remaining parameters determine the size and shape of the compensating overdensity. This form, however, does not give a model that matches to FLRW exactly. A perfect match requires that, at some maximum r value rm , both the density and total effective mass equal their FLRW values: ρ(rm ) = ρ0F LRW and M (rm ) =

4π 3 r 3 m

(where we have used eq 1.9, recalling that

we are still using a gauge where R(t0 , r) = r). We will therefore add additional terms to achieve 44

this matching without drastically altering the shape of the void. 1 − (r/rs )α 1 − (rm /rs )α − δ c 1 + (r/rv )β 1 + (rm /rv )β   Z rm 2 (r − rm /2)2 − rm /4 2 δρ (r1 )r1 dr1 . ρ(r) = ρ0F LRW 1 + δρ (r) − R rm 2 /4) r 2 dr ((r2 − rm /2)2 − rm 2 0 2 0

δρ (r) = δc

(3.2) (3.3)

The mass function, then, is

M (r) =

 R    r 4πρ(r1 )r12 dr1 0

if r < rm

   4π ρ0F LRW r3

if r ≥ rm

3

(3.4)

We choose values for the parameters as δc = −0.75, rv = 2000 Mpc, rs = 0.91 rv , α = 2.18, and β = 9.48. We initially match this to a flat background FLRW model with Λ = 0 and H0 = 57 km s−1 Mpc−1 . This is a low value for the Hubble constant, as giant void models typically require a low H0 to fit the CMB data [32,66]. 1 The local expansion rates inside the void will be somewhat higher, depending on the location; with a constant bang-time, the expansion rate at the origin is 74.7 km s−1 Mpc−1 . (Recall that we are not trying find a model to fit all of the data at once, but rather to probe the effects of varying bang time in a “typical” scenario and place rough constraints on the parameters.) This Hubble value determines the background density ρ0F LRW through the Friedmann equation 1.2, as well as the present age of the universe t0 . The bang-time function must also match exactly to the background FLRW model at rm . We will use a Gaussian curve, similarly modified to allow exact matching.

tB (r) =

   δ

tB



2

e−(2r/rm ) −

8e−4 4 4 r rm

+

20e−4 2 r 2 rm

  0

 − 13e−4 /(1 − 13e−4 )

if r < rm (3.5) if r ≥ rm

1

A good fit to the CMB actually requires an even higher H0 than used here, but this causes tension with local H0 measurements.

45

This ensures not only that tB (r) is continuous, but also its first two derivatives. It remains a monotonic function. We will test four cases, with δtB of −248.5, 0, 124.25, and 248.5 Myr. The only remaining LTB function is the curvature function k(r), which is determined by numerically solving eq 1.10. 3.2.2

Szekeres functions

In contrast to the models of chapter 2, the models used here will be axially symmetric. We can take advantage of this symmetry to greatly reduce the number of calculations we will need. For an observer on the axis, the symmetry effectively reduces the sky to variations in one dimension. To reliably determine the first few CMB multipoles, we need geodesics covering only half the unit circle rather than the entire unit sphere. Of course, this argument only applies as long as the observer is exactly on the symmetry axis. To test observation points off the axis, we would still need to perform the full set of calculations. We will therefore focus on this axis. This does not give as complete a picture spatially, but this is a necessary sacrifice if we are to explore a broader range of parameter space, with the addition of a varying bang time function. For simplicity, we will only use S(r) of the three Szekeres functions—that is, P (r) = Q(r) = 0. The functional form for S(r) will differ somewhat from the P (r) functions used in chapter 2. The first difference is necessary and straightforward. Because the S(r) function (in the absence of the other functions) acts on the metric and density through S 0 (r)/S(r), as opposed to the P (r) function which acts through P 0 (r)/S(r) (where we previously set S(r) = 1), our current S(r) functions need an exponential wrapped around them to have the same effect. In other words, S(r) = ef (r) results in the same amount of shell shifting and matter anisotropy as P (r) = f (r). Secondly, we will remove the exponential damping we used in chapter 2. We find that this factor is an unnecessary complication. Because we are defining all of our functions to transition smoothly and exactly to FLRW, we do not require an extra factor ensuring the Szekeres anisotropies approach 0 at large distances. Without this factor, it is easier to directly infer the amount of shell shifting from the function’s parameters. 46

In addition, we have found that the functional form we previously used is limiting in terms of the maximum amount of total anisotropy allowed. Using a quartic curve to smoothly transition from 0, up to the maximum value, and then back down to 0 results in a function that is only near the maximum value for a fraction of the total range. Because the ban against shell crossing puts an upper limit on how much anisotropy is possible at any r, the quartic curve form artificially forces the anisotropy to be well below this limit for most of the range. To allow more freedom in our choice of anisotropies, we will instead break this quartic curve into parts: a smooth rise, a flat plateau, and a smooth decline. More explicitly, we define our function as follows:     0        (r−r1 )(2r2 −r1 −r)   (r2 −r1 )2    1 d(r) = × 1 r       (r4 −r)(r−2r3 +r4 )   (r4 −r3 )2       0 Z r eCd(r) dr. S(r) =

if r ≤ r1 if r1 < r < r2 if r2 ≤ r ≤ r3 ,

(3.6)

if r3 < r < r4 if r ≥ r4 (3.7)

0

The constant C determines the overall magnitude of the anisotropy (with shell crossing occuring at roughly 1), while the ri constants allow us to control the shape of the curve—how wide the anisotropy plateau is and how gradual the transitions. We will test four versions of the Szekeres function. The first we call “far”, because it has asymmetry only far from the origin: ri = (1000, 1500, 2500, 3000) and C = 0.3. A “wide” Szekeres function, on the other hand, covers nearly the entire hole, with ri = (1, 100, 3900, 4000) and C = 0.1. For both of these types, we will also test a “strong” version with double the C. These four Szekeres functions, combined with the four cases for δtB , give us our 16 models.

47

0.

kHrL•r 2

-2. ´ 10-8

-4. ´ 10-8

-6. ´ 10-8 0

1000

2000

r HMpcL

3000

4000

Figure 3.1: The curvature function k(r) with constant bang-time (solid, blue) and varying bangtime (dashed, red) with δtB = 248.5 Myr.

3.3

Direct effects of varying bang time

As mentioned in section 1.2.1, variations in the bang-time function correspond to decaying perturbation modes. This means that if we introduce variations in the bang-time function while keeping the present-day density profile the same, we will generate larger inhomogeneities at early times, which then quickly dissipate out. This also means that the velocity profile changes, as does the curvature function. For illustrative purposes, let us consider a model with “strong, far” Szekeres functions. In one case, the bang-time function is constant, while in the other, δtB = 248.5 Myr. 3.3.1

Density distribution and evolution

As long as we keep M (r) fixed, the present day density profile is unchanged. Beware, however, that the curvature function does change, so the physical geometric structure of the model is subtly altered. Figure 3.1 shows a comparison of the curvature function with and without a varying bang time. At earlier times, the density profile is altered. As we look back in time, the void grows shallower. In the model with δtB = 248.5 Myr, this occurs more rapidly, causing the void to disappear at a time when the constant-tB model still has a considerable underdensity. At very early times, there is an overdensity where the void is now, which diverges as we approach t = 248.5 Myr, when 48

2 r•rFLRW

r•rFLRW

2

1

0

-2000

1

0 0

2000

4000

Mpc

-2000

0

2000

4000

Mpc

Figure 3.2: The evolution of the density profile along the axis with constant bang-time (left) and δtB = 248.5 Myr (right). Profiles at times t0 , t0 /2, t0 /3 . . . t0 /20 are plotted, with lighter lines representing earlier times. The x-axis shows distance from the coordinate origin. The scale of the x-axis applies to the present-time profile only; other curves are stretched to fit the same total size.

the inner shells were still in a singularity.2 This is illustrated in figure 3.2. We also notice that the effect of the Szekeres functions on the density reverses as well, creating an underdensity at early times where there is an overdensity today. 3.3.2

Expansion rates

By our gauge choice, each shell must reach a size of r Mpc by time t0 . Varying the bang-time function changes the amount of time they have to grow, so naturally the expansion rates must change, both in the longitudinal and transverse directions. Figure 3.3 shows a set of comparative illustrations. We see that both expansion rates are higher in the void at the present time, while only the longitudinal expansion rate decreases where the Szekeres functions create an overdensity. With a constant bang-time, the expansion rates flatten out as we go back in time, like the density. If the bang-time varies, though, the expansion rates turn around, increasing where the central overdensity appears. This is how the early-time overdensity dissipates. 2

Of course, as mentioned before, the model loses validity before reaching this point due to the pressureless dust approximation breaking down.

49

2

HÞ •HFLRW

HÞ •HFLRW

2

1

1

0

0 -2000

0

2000

-2000

4000

0

Mpc 2

4000

2000

4000

HÈÈ •HFLRW

HÈÈ •HFLRW

2

1

0

2000 Mpc

1

0 -2000

0

2000

-2000

4000

Mpc

0 Mpc

Figure 3.3: Temporal and spatial dependence of transverse expansion rates (top row) and longitudinal expansion rates (bottom row) for models with constant bang-time (left) and δtB = 248.5 Myr (right). Other details are the same as in figure 3.2.

50

3.4

Results (primary models)

The physical effects described above are not directly observable. We use our simulated observational tests to compute how these models would look to an observer within them. 3.4.1

CMB Dipoles

Testing a variety of models with and without varying bang-time, we find that the CMB dipole is shifted a moderate amount by variations in tB (r), but maintains a similar basic pattern. Because it increases roughly linearly with distance in the region where the dipole is low, we can summarize the results with two parameters per model: the r value at which the dipole vanishes, r0 ; and the slope of the dipoles with respect to r, a010 . The latter (when adjusted to compensate for shell shifting) determines the physical volume of the “allowed” region, assuming it is still roughly spherical. Table 3.1 lists these parameters. Changing the amount of variation in the bang-time function alters the location of the null dipole point in a roughly linear matter. The slope of the dipoles through this point increases linearly as δtB increases. A greater dipole slope means a smaller region in which the magnitude of the dipole is small, so models with positive δtB have a smaller “allowed” region. Stronger and wider Szekeres functions, on the other hand, result in a larger “allowed” region, consistent with the results of chapter 2. 3.4.2

CMB Quadrupoles and Octupoles

In contrast to the dipole, the CMB quadrupoles and octupoles undergo drastic changes when the bang-time function is altered. Figure 3.4 shows a comparative example. The quadrupole still follows an approximately quadratic curve in this region, and the octupole is still cubic, but the minimum values, displacements, and curvatures are all influenced by the bang-time variations. In chapter 2, we saw that without a varying bang-time, it is difficult to generate significant quadrupoles where the dipoles are low, and even more difficult to generate octupoles. We see now

51

Table 3.1: Parameters of the CMB dipoles along the axis in the models tested. See section 3.2.2 for definitions of the Szekeres functions. V refers to the volume of the region allowed by the dipole, and VLT B is this volume for an LTB model with a constant bang-time. δtB (Myr) -248.5 0 124.25 248.5

Szekeres function far far far far

r0 150.8 141.3 136.8 132.4

a010 1.13 × 10−4 1.20 × 10−4 1.24 × 10−4 1.28 × 10−4

V /VLT B 1.26 1.03 0.94 0.85

-248.5 0 124.25 248.5

wide wide wide wide

153.5 149.2 147.1 145.1

1.24 × 10−4 1.33 × 10−4 1.37 × 10−4 1.42 × 10−4

1.26 1.03 0.93 0.85

-248.5 0 124.25 248.5

strong far strong far strong far strong far

305.4 286.1 277.0 268.1

1.08 × 10−4 1.16 × 10−4 1.20 × 10−4 1.24 × 10−4

1.43 1.16 1.05 0.95

-248.5 0 124.25 248.5

strong wide strong wide strong wide strong wide

282.9 275.0 271.3 267.6

1.33 × 10−4 1.41 × 10−4 1.46 × 10−4 1.51 × 10−4

1.36 1.13 1.03 0.93

4. ´ 10-3 6. ´ 10-4 4. ´ 10-4 a30

a20

2. ´ 10-3 2. ´ 10-4 0. 0. -2. ´ 10-4 -4. ´ 10-4 -2. ´ 10-3

-200

0

200

400

600

-200

800

r

0

200

400

600

800

r

Figure 3.4: CMB quadrupoles (left) and octupoles (right) along the axis in models with “far” Szekeres function and different bang-time functions. Blue, solid: δtB = 0; red, dashed: δtB = 248.5 Myr; green, dotted: δtB = 124.25 Myr; purple, dot-dashed: δtB = −248.5 Myr. Insets show the δtB = 0 curves at a smaller scale.

52

Table 3.2: Ranges of the quadrupoles and octupoles in the region in which the dipole is less than the observed dipole of ∆T = 3.355 mK. δtB (Myr) -248.5 0 124.25 248.5

Szekeres function far far far far

|a20 | range 2.65–4.51×10−4 2.88–3.88×10−5 1.06–1.96×10−4 2.52–4.33×10−4

|a30 | range 1.01–3.67×10−5 0–2.34×10−6 1.18–1.81×10−5 2.14–3.91×10−5

-248.5 0 124.25 248.5

wide wide wide wide

0.34–1.21×10−4 0–8.87×10−6 2.24–6.16×10−5 0.47–1.26×10−4

0–5.07×10−6 0–1.45×10−6 0.98–2.50×10−6 2.22–5.10×10−6

-248.5 0 124.25 248.5

strong far strong far strong far strong far

1.21–1.59×10−3 1.45–1.56×10−4 4.58–6.41×10−4 1.11–1.48×10−3

1.33–2.35×10−4 0–8.48×10−6 0.92–1.20×10−4 1.88–2.61×10−4

-248.5 0 124.25 248.5

strong wide strong wide strong wide strong wide

2.08–3.83×10−4 1.92–3.28×10−5 0.89–1.65×10−4 2.07–3.68×10−4

0.64–2.91×10−5 0-3.67×10−6 1.11–1.61×10−5 1.85–3.32×10−5

that a varying bang-time makes it easier to find models with significant quadrupoles and octupoles. On the other hand, too much variation in the bang-time results in a quadrupole that is far too large everywhere inside the void. This puts a limit on how much bang-time variation is allowed. Table 3.2 shows the ranges of the quadrupoles and octupoles that are possible along the axis inside the “allowed” region for all 16 models. The ranges have a (very) roughly linear relationship to the magnitudes of the bang-time variations, but a nonlinear relationship to the Szekeres functions— doubling the strength of the shell shifting more than doubles the quadrupoles and octupoles. To explain the observed anomalous alignment, the magnitudes of the quadrupole and octupole must both simultaneously be on the order of the real anisotropies, 10−5 . Because the octupole is cubic, it tends to increase more slowly than the quadrupole in the region of interest, but the fact that the curves are displaced unequally opens up the possibility of finding a model and a location that is near the nadir of the quadrupole curve, yet some distance away from the minimum of

53

the quadrupole curve. Our tests have found, however, that the displacements are similar enough that the octupoles in the “allowed” region still tend to be an order of magnitude lower than the quadrupoles, so the prospects for finding such a model are dubious. In all models except those with the “wide” Szekeres function, any variation in the bang-time function on the levels we have tested pushes the quadrupoles well above the acceptable levels for the entire range. The most promising model is the one with “wide” Szekeres function and δtB = −248.5 Myr, but even then only a fraction of the range could be consistent with observations. (In fact, the low end of the quadrupole range and the high end of the octupole range are on opposite ends of the “allowed” region.) The quadrupoles and octupoles put limits on both the Szekeres functions and the bang-time variation, as too much of either results in values in contradiction with observations. In many cases, the “allowed” region is further restricted by requiring consistency with the observed quadrupole and octupole in addition to the dipole. If we wish for the inhomogeneities to be able to explain the observed quadrupole/octupole anomalies as well, we will require fine-tuning of the Szekeres functions, the bang-time function, and the observer’s location. 3.4.3

Comparison with Doppler shifts

Studies of low CMB multipoles done in LTB models have found that the first few multipoles can be understood from a Newtonian perspective as a simple Doppler shift [47]. Because the inner shells expand at a different rate than the background, comoving observers have a non-zero velocity β relative to the CMB frame of reference, equal to the distance from the origin times the difference in the expansion rate. The leading contribution to al0 is then proportional to β l . In a Szekeres model, the velocity of comoving observers relative to the homogeneous background is not so simple. Not only does each shell have a different expansion rate, but the shells shift as they evolve as well. Even the origin is not fixed relative to the background! The equations quickly become cumbersome, but we can simplify them if we focus only on axially symmetric models. (For convenience, we will assume P (r) = Q(r) = 0.) We also assume that the density 54

and curvature are constant for r > rm , and S 0 (r) vanishes beyond rm as well. To compute the comoving velocity relative to the background, we need to imagine overlaying the model with a homogeneous FLRW model. At large distances, the two should match exactly, since the FLRW model corresponds to the background comoving with the CMB frame. A point comoving with the Szekeres model is not comoving with the FLRW model; this discrepancy is the velocity we are looking for. The natural choice for the origin of the background FLRW model is not the origin of the Szekeres model, but rather the center of the outermost shells. The total displacement of a shell at r1 from this background origin is Z

rm

R(t, r) r1

S 0 (r) dr. S(r)

(3.8)

Ignoring variations in curvature, then, the FLRW coordinates and Szekeres coordinates can be related by  Z {x, y, z} = R(t, r) sin θ cos φ, R(t, r) sin θ sin φ, R(t, r) cos θ +

rm

r1

 S 0 (r) dr /a(t). R(t, r) S(r) (3.9)

The velocity, then, is simply the time derivative of this equation times the scale factor. Looking at our data, we can immediately rule out this Doppler shift as the sole source of the large-angle CMB anisotropies. We have seen that including a varying bang time function can greatly affect the quadrupole while only making small changes to the dipole. If there ever was a β l correspondence, this clearly breaks it. Even without a varying bang time, we have seen that some models can have significant quadrupoles in regions where the dipole is tiny. Clearly there are other factors at play. Nevertheless, it is useful to know to what extent the multipoles do correspond to Doppler shifts. To that end, we have compared the dipole, quadrupole, and octupole we calculated directly in certain models with the values we would obtain from pure Newtonian Doppler shifts, shown in figure 3.5. We tested both LTB models and models with “strong, far” Szekeres functions, with both 55

6. ´ 10-2

6. ´ 10-2 4. ´ 10-2 2. ´ 10-2 a10

a10

4. ´ 10-2

0 -2

-2. ´ 10 2. ´ 10-2

-4. ´ 10-2 -6. ´ 10-2 -8. ´ 10-2

0 0

100

200

300

400

500

-200

0

r

200

400

600

800

400

600

800

400

600

800

r

1. ´ 10-3 1.5 ´ 10-3 -4

8. ´ 10

1. ´ 10-3 a20

a20

6. ´ 10-4 4. ´ 10-4

5. ´ 10-4

2. ´ 10-4 0 0 0

100

200

300

400

-200

500

0

200 r

r 8. ´ 10-5

1.5 ´ 10-4 1. ´ 10-4

6. ´ 10-5

a30

a30

5. ´ 10-5 -5

4. ´ 10

0 -5. ´ 10-5

2. ´ 10-5

-1. ´ 10-4 -1.5 ´ 10-4

0 0

100

200

300

400

500

-200

0

200 r

r

Figure 3.5: The dipole (top), quadrupole (middle), and octupole (bottom) from Newtonian Doppler shifts (black) and the directly calculated values along the axis in (left column) an LTB model, and (right column) a model with a “strong, far” Szekeres function. Constant bang-time models are shown in solid blue for direct calculations and dashed black for Doppler approximations, and δtB = 124.25 Myr models are shown in the dot-dashed red lines (direct calculations) and dotted black line (Doppler approximations).

56

constant bang-time and δtB = 124.25 Myr. We see that in all cases, the dipole matches the Doppler prediction fairly well, while the quadrupole and octupole are increasingly different. The varying bang-time has a much greater effect on the total quadrupole and octupole than on the Doppler approximations, and it makes the dipole fit slightly worse as well. The octupoles in particular are greatly underestimated by the Doppler approximation. This indicates that the anisotropic matter distribution affects the low multipoles via a ReesSciama effect or a similar effect, to a degree that dominates the quadrupole and octupole while remaining below the Doppler contribution to the dipole. One consequence of this is that we can quickly and easily calculate approximate dipoles anywhere in the model simply by finding the Doppler velocity. This allows us to estimate the magnitude of the kinetic Sunyaev-Zel’dovich effect that would be seen in our models. However, the quadrupoles and octupoles still require the full calculation. 3.4.4

Luminosity distances

Our observations of the Hubble diagram are nearly isotropic across the sky. If the distribution of matter around us is anisotropic, as in the present toy models, we should not necessarily expect this to be the case. While an observer in a particular region might see only a small dipole in the CMB, this is only a single-redshift observation, measuring the total cumulative effect of the inhomogeneities from the observer to the CMB in different directions. The Hubble diagram, on the other hand, probes a range of redshifts. In order to be viable, the model must fit the observations over the whole range. While our observations of distances through type Ia supernovae are consistent with isotropy, they are not yet numerous enough to provide tight constraints with respect to different directions. We therefore do not require that our test models show a perfectly symmetric Hubble diagram, but rather only that the directional variation is not so large that it would be obvious in our observations. In fact, some researchers have found hints of a dipolar modulation in distances across our actual sky [68–71]. This is typically attributed to some sort of dark energy dipole, or else some systematic 57

measurement error. The significance of the detection is not very strong, largely due to the small number of data points, but the direction is closely aligned with several other anomalies, including a bulk flow seen in a dipole in the kSZ effect out to redshift z ∼ 0.2 [14]. We can use our models to test whether an anisotropic inhomogeneity might produce similar effects. The process for calculating luminosity distances is explained in greater depth in the next chapter, where it will be the primary focus of our tests. For now, it suffices to say that we numerically integrate the Sachs optical equations to track the size of a light bundle surrounding the geodesic [30]. These equations give the evolution of the bundle’s expansion rate and shear in terms of the Ricci and Weyl curvatures, which then determine the angular diameter distance through the size of the bundle. d ˆ ln DA = θ. dλ

(3.10)

The luminosity distance is directly related to the angular diameter distance through the cosmic distance duality relation

DA =

DL . (1 + z)2

(3.11)

Using these equations, we calculate the distance as a function of redshift along both axial directions. Figure 3.6 shows the results for a few example models. With or without a varying bang-time, we see that there is a noticeable difference in the distances in different directions at low redshifts. This asymmetry is greater when the Szekeres functions are stronger, unsurprisingly, as stronger Szekeres functions make the void more asymmetric. Sharp Szekeres functions appear to have a greater effect than smooth ones—even though the “far” and “wide” cases resulted in similar CMB dipoles (see table 3.1), the “wide” case (which covers a broader range with lower C) produces notably less asymmetry. Because a large asymmetry would be plainly visible in the supernova observations, this puts a limit on how strong and sharp we can make our Szekeres functions while remaining consistent with the data. 58

0 0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

mtB - m0

Dm

Dm

-0.01

-0.02

-0.8 0

0.2

0.4

0.6

0.8

1.

1.2

1.4

-0.03 0

0.2

0.4

0.6

z

0.8

1.

1.2

1.4

0

0.2

0.4

0.6

z

0.8

1.

1.2

1.4

1.

1.2

1.4

1.

1.2

1.4

z

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 0

0.2

0.4

0.6

0.8

1.

1.2

1.4

-0.02 mtB - m0

0.2

Dm

Dm

0 0.2

-0.04

-0.06 0

0.2

0.4

0.6

z

0.8

1.

1.2

1.4

0

0.2

0.4

0.6

z

0.8 z

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 0

0.2

0.4

0.6

0.8

1.

1.2

1.4

-0.02 mtB - m0

0.2

Dm

Dm

0 0.2

-0.04

-0.06 0

0.2

0.4

0.6

z

0.8 z

1.

1.2

1.4

0

0.2

0.4

0.6

0.8 z

Figure 3.6: Residual distance moduli ∆µ = µ−µM ilne viewed along both axial directions from an observer at the null dipole point, compared to a ΛCDM model (black, dashed) and an Einstein-de Sitter model (gray) 3 . The Milne, ΛCDM, and EdS models all have H0 = 74.3 km s−1 Mpc−1 . The models have (left column) constant bang-time and (middle column) δtB = 248.5 Myr (except for the top-middle, which has only half). The right column shows the differences in the distance up down moduli between these two cases, i.e. µup − µdown (dashed orange). 0 tB − µ0 (solid blue) and µtB (The jaggedness at low redshifts is simply interpolation error.) The Szekeres functions are “far” (top row), “strong far” (middle row), and “wide” (bottom row). (See section 3.2.2 for definitions). The red and blue lines are for geodesics directed away from the density peak, and the purple and orange lines are for geodesics going through the peak.

59

With a varying bang-time, there is hardly any discernable difference compared to the constant bang-time case. The separation between the curves is virtually identical, only slightly greater at low redshifts and slightly lower at high redshifts in the varying bang-time case (on the order of 10%). The only major difference is that both curves are shifted downwards slightly at low-intermediate redshifts, due to the higher expansion rate within the void. Directional distance data would thus be of little use in distinguishing models with different bang-time functions. In practice, we would likely compensate for this low-redshift downward shift by lowering the background H0 so that the local H0 within the void is roughly the same as in the constant bangtime case. Note that this would effectively shift the entire red and orange curves upwards, and that this would give them a better fit to the ΛCDM distance at high redshifts. Indeed, others have found that matching the CMB spectra (effectively the distance at very high redshifts) in LTB void models requires a very low H0 —lower than we have used in our models—which results in a local H0 lower than what is measured, despite the higher expansion rate inside the void [66]. A varying bang-time allows the expansion rate inside the void to be even higher, reducing this tension. Let us return to the asymmetry in the distances. In the supernova data, the asymmetry is currently most noticeable in the low redshifts z ≤ 0.2, but this may simply be because there is more and better data in this range [14]. In our models, the asymmetry extends to z ∼ 0.5, when the geodesics exit the void, so a smaller void may be favored. Mariano finds the magnitude of the asymmetry to be on the order of ∆µ/µ ∼ 10−3 , which is the same level as the peak asymmetries in our test. We therefore conclude that this level of Szekeres anisotropy cannot be ruled out on the basis of the presently available supernova distance data, and may even be supported by the data. 3.4.5

Kinetic Sunyaev-Zel’dovich effect

We have seen that the dipoles increase quickly away from the null-dipole point, which suggests a large kinetic Sunyaev-Zel’dovich effect. To truly see the effect, though, we need to know the dipoles along the observer’s past light-cone, not on a constant time slice. We will use Doppler velocities as an approximation of the true dipoles in order to gain a rough idea of the magnitude of 60

15

bHzL H103 km s-1 L

20

bHzL H103 km s-1 L

20

bHzL H103 km s-1 L

20

15

10

15

10

5

10

5

0 0.

0.5

1.

1.5

5

0 0.

0.5

z

1.

1.5

0 0.

z

0.5

1.

1.5

z

Figure 3.7: Velocities relative to the CMB frame along the geodesics in both axial directions, originating from the null-dipole point. The Szekeres functions are: far (left), strong far (middle), and wide (right). Solid blue and purple lines are for the constant bang-time case, dashed red and orange lines are for the positive varying bang-time cases (δtB = 124.25 Myr left, 248.5 Myr middle and right), and dot-dashed green and cyan lines are for the negative varying bang-time cases (δtB = −248.5 Myr).

the effect in these models. We will also assume the dipole dominates the anisotropy in the reflecting clusters’ skies; if higher multipoles are comparable, this dipole approximation will overestimate the magnitude of the effect [72]. The total kSZ temperature anisotropy in direction nµ is [48, 72] ∆T (nµ ) = T

Z

β(nµ , z) [1 + δ(nµ , z)] dτ ,

(3.12)

where β is the velocity along the line of sight direction, δ is the comoving free electron density perturbation, and τ is the optical depth. We will not attempt to model optical depth or the matter power spectrum in our models, so we will instead calculate only the cluster velocity β(nµ , z). Observations of individual clusters put upper limits of β(z) . 2000 km s−1 in the range 0.18 ≤ z ≤ 0.55 [45], and observations from ACT and SPT restrict the kSZ power spectrum at l = 3000 to less than or equal to 8µK2 and 13µK2 , respectively, corresponding to peculiar velocities around 400 km s−1 [72]. We will use the same models and observer locations as in section 3.4.4. Our results are shown in figure 3.7. We see that the velocities are several times higher than the observational limits. Due to the asymmetry, one direction has a higher peak while the other extends to a higher redshift, but both are well above the observed velocities. These models are in strong contradiction with 61

kSZ observations, as the LTB models were before them [48]. Positive variations in the bang-time increase the velocities along the light-cone by a more noticeable degree than they did the constanttime-slice dipoles. Negative bang-time variations decrease the velocities, but reducing them to acceptable levels would require very large bang-time variations, beyond the limits placed by the CMB quadrupole.

3.5

Results (alternate models)

3.5.1

Varying M (r) instead of k(r)

When we change the bang-time function, we must also change M (r) or k(r) in order to maintain consistency with our gauge condition R(t0 , r) = r. In all previous models, we have kept M (r) constant and changed k(r), so that the present-day density profile is unchanged. Here we explore the other option—keeping the curvature function constant while varying the mass function—by repeating two previous tests. In both cases, we begin with the same mass function as before, find the corresponding curvature function when tB (r) = 0, and then apply a new tB (r) and use eq 1.10 to numerically find the new M (r). Our first test uses the “far” Szekeres function (see section 3.2.2 for definitions) with δtB = 124.25 Myr. The second uses the “strong, far” Szekeres function with δtB = 248.5 Myr. Figure 3.8 shows how the density profile and expansion rates are affected in the latter case, compared to the previous test in which M (r) was fixed. The differences are slight. The void is presently slightly shallower, and the early-time overdensity is slightly higher, and the central expansion rate is slightly lower. The effect of this change on the CMB quadrupole and octupole is illustrated in figure 3.9. It is clear that the effect is quite similar to the effect when keeping M (r) fixed. The octupoles deviate slightly more from the constant bang-time case, but only away from the null-dipole point. As for the dipoles, compared to the data in table 3.1, the downwards shift in r0 is about 80−90% as much as with M (r) fixed, but the slope of the dipole a010 is virtually identical to the constant 62

2.5 1.5

r•rFLRW

2.

HÈÈ •H0

HÞ •H0

1.5

1.

1.

1.5 1.

0.5

0.5

0.5 0

0 -2000

0

2000

-2000

4000

0 0

2000

-2000

4000

0

Mpc

Mpc

2000

4000

Mpc

Figure 3.8: A comparison between the density profile and expansion rates for a model with strong, far S(r) and δtB = 248.5 Myr, with M (r) fixed (solid, colored) and with k(r) fixed (dashed, black), shown at both t0 (darker) and t0 /20 (lighter).

4. ´ 10-3 6. ´ 10-4 3. ´ 10-3 4. ´ 10-4 a30

a20

2. ´ 10-3 1. ´ 10-3

2. ´ 10-4 0

0 -2. ´ 10-4 -1. ´ 10-3 -4. ´ 10-4 -2. ´ 10-3

-200

0

200

400

600

-200

800

0

r

200

400

600

800

400

600

800

r

8. ´ 10-3 1. ´ 10-3 6. ´ 10-3

a30

a20

4. ´ 10-3

5. ´ 10-4

2. ´ 10-3 0 0 -2. ´ 10-3

-200

0

200

400

600

-5. ´ 10-4

800

r

-200

0

200 r

Figure 3.9: The quadrupoles (left) and octupoles (right) for models with far Szekeres function (top) and strong, far Szekeres function (bottom). Solid blue lines: δtB = 0, dotted green lines: δtB = 124.25 Myr (M (r) fixed), dashed red lines: δtB = 248.5 Myr (M (r) fixed), large-dashed black lines: same δtB with k(r) fixed.

63

bHzL H103 km s-1 L

20

15

10

5

0 0.

0.5

1.

1.5

z

Figure 3.10: Velocities relative to the CMB frame along the geodesics in both axial directions, originating from the null-dipole point with the “strong far” Szekeres function. Solid blue and purple lines are for the constant bang-time case, dashed red and orange lines are with δtB = 248.5 Myr and fixed M (r), and large-dashed black and grey lines are for the same δtB with k(r) fixed instead.

bang-time case. Thus, varying the bang-time while keeping k(r) fixed allows one to shift the null-dipole point without shrinking the “allowed” region. The luminosity distances are also quite similar, only shifted upwards slightly (by ∆µ ∼ 0.03) at low redshifts due to the lower expansion rate inside the void—75.5 km s−1 Mpc−1 , compared to 76.7 km s−1 Mpc−1 when M (r) is fixed and 74.7 km s−1 Mpc−1 when the bang-time is constant. This can be compensated by simply choosing a different value for H0 in the model definitions. The kSZ effect is adjusted in a similar way whether M (r) or k(r) is fixed, but slightly less with k(r) fixed and M (r) varied, as seen in figure 3.10. In most ways, then, the effects we have seen from varying the bang-time function come primarily from tB (r) itself, not from the choice of whether to keep M (r) or k(r) fixed, though this choice is not entirely inconsequential. 3.5.2

Models with Λ

It seems that any attempts to replace dark energy with a giant void are subject to fine-tuning problems and unrealistically large kSZ effects. Perhaps we should instead be searching for a compromise—a model in which inhomogeneity does not replace dark energy entirely, but rather 64

1.5 ´ 10-3 4. ´ 10-2 1. ´ 10-3

0.

a20

a10

2. ´ 10-2

-2

5. ´ 10-4

-2. ´ 10

-4. ´ 10-2 0.

-6. ´ 10-2 -200

0

200

400

-200

600

0

200

r

400

600

r

Dm

2. ´ 10-4

Dm

a30

1. ´ 10-4

0.

-200

0

200

400

600

0.2 0.1 0 -0.1 -0.2 0.2 0.1 0 -0.1 -0.2 0

0.2

0.4

0.6

0.8

1.

1.2

1.4

z

r

Figure 3.11: The CMB dipoles (upper-left), CMB quadrupoles (upper-right), CMB octupoles (lower-left), and luminosity distances (lower-right) for models with Λ and a shallower void. Solid blue lines: constant bang-time; dashed red lines (and the lower luminosity distance plot): δtB = 248.5 Mpc.

only alters its apparent value. Such a model would not solve the fine-tuning issues associated with dark energy in the standard model, but may provide a similar or superior fit to observations. We will briefly consider a model with ΩΛ = 0.5, H0 = 67 km s−1 Mpc−1 , and a void half as deep as the other models in this chapter (δc = −0.375). We will use the strong, wide version of S(r), and test two cases: constant bang-time and δtB = 248.5 Myr. Figure 3.11 shows the CMB dipoles, quadrupoles, and octupoles, as well as the luminosity distances for both of these cases. The luminosity distances are close to the ΛCDM values over the entire redshift range 0 < z < 1.5, a significantly better fit than the previous models. It would be difficult to distinguish this model from the ΛCDM model from supernova data alone. We also see that the separation between the distance curves in different directions is less, with a peak ∆µ/µ roughly half as high as the previous tests (section 3.4.4). This means that stronger Szekeres anisotropies are possible within the current bounds of the directional distance 65

8

bHzL H103 km s-1 L

6

4

2

0 0.

0.5

1.

1.5

z

Figure 3.12: Velocities relative to the CMB frame along the geodesics in both axial directions, originating from the null-dipole point with the “strong wide” Szekeres function. Solid blue and purple lines are for the constant bang-time case, dashed red and orange lines are with δtB = 248.5 Myr (keeping M (r) fixed).

data, though the actual matter anisotropy is lessened due to the shallower void. The null dipole points are nearly the same as in the previous tests (at r = 289.0 and 277.2), but the slopes of the dipoles are only half as strong (a010 = 5.14 × 10−5 and 6.04 × 10−5 ), meaning the observer can be in a much larger region while keeping the dipoles low. In the constant bangtime case, the quadrupole is 1.44 × 10−5 at one edge of the “allowed” region, where the octupole is simultaneously 3.17 × 10−6 —still small compared to the real octupole, but significant. In the varying bang-time case, the octupole can be as high as 2.31×10−5 , but the minimum quadrupole is still rather large (6.92 × 10−5 ). It is clear that even with a cosmological constant, void models have limitations from the low CMB multipoles, but they are somewhat less stringent than in pure-void models. The constraints from the kSZ effect are less stringent as well. As shown in figure 3.12, the kSZ velocities in this model are a fraction as large as in the models without Λ. The relative effects of the bang-time function on the kSZ are more significant, which makes it easier for a negative bang-time variation to push the velocities on the light-cone to acceptably low values. Fitting the kSZ constraints seems a more attainable goal in void models with Λ than in those without.

66

3.6

Conclusions

We have studied the observational effects of varying bang-time functions in Szekeres models. Adding another degree of freedom broadens the search area, making it more difficult to find the best fit, but potentially allowing us to find solutions that would otherwise remain hidden. We found that changing the bang-time function alters the position and size of the region consistent with the observed CMB dipole in a roughly linear manner. This allows the observer’s location relative to the void to have greater variety, but the total volume of the “allowed” region is still quite small. It is therefore not able to alleviate the Copernican principle violation inherent in giant void models—at least, not by itself. The effect on the CMB quadrupole and octupole is more dramatic. Small changes in the bangtime variation cause large changes in the magnitude and shape of the curves, to a degree that cannot be attributed to Newtonian Doppler shifts (though they are still approximately quadratic and cubic with distance, respectively). This makes it easier to find models in which the inhomogeneities contribute significantly to these terms while the dipole is low, but it also creates situations where the quadrupole everywhere inside the void is far higher than what is observed. The octupole is usually an order of magnitude too low when the quadrupole is near the observed value. Finding a model in which the curves are displaced relative to each other just enough to have the dipole, quadrupole, and octupole simultaneously comparable to observations has proven difficult. Even if such a model is possible, it would require fine-tuning of the model parameters. This makes these models unattractive as a potential explanation for the anomalous quadrupole-octupole alignment. The bang-time function also adjusts luminosity distance observations, primarily by changing the expansion rate within the void. Due to the models’ asymmetry, the luminosity distance in different directions differs as well, mostly at low redshifts (up to the edge of the void). This effect cannot be ruled out on the basis of supernova observations—in fact, a weak asymmetry has been detected [68–71], somewhat similar to what we see in these models, but there is not enough data yet to confirm this detection. We found that this asymmetry is insensitive to variations in the

67

bang-time function. Although we vary the curvature function along with the bang-time function for consistency, all of these effects come primarily from the bang-time function itself, as we found when we varied the mass function instead while keeping the curvature function fixed. One of the stronger criticisms against LTB void models is that they result in a very strong kinetic Sunyaev-Zel’dovich effect, beyond the limits placed by observations. We have found that Szekeres models still face this same problem, with high velocities relative to the CMB frame along the past light-cone. Bang-time variations can alter these velocities, but large variations would be needed to erase the effect entirely. All of these observations put limits on the model functions. Too much asymmetry from the Szekeres functions causes a luminosity distance dipole that would be readily apparent, and too much bang-time variation combined with too strong Szekeres functions results in a CMB quadrupole that cannot be reconciled with observations. We can therefore generally rule out models with much higher parameters than those tested here. Finally, we tested models combining a shallower void with a cosmological constant. The limitations on the model parameters are somewhat less restrictive in this case. The region “allowed” by the CMB dipole is larger, and the distance dipole is smaller than in the pure void model with the same S(r). The CMB quadrupoles still limit the bang-time function, but not as severely. (A match to the observed quadrupole and octupole still seems unlikely.) We therefore have less fine-tuning of the void parameters, but at the cost of re-introducing the fine-tuning problems associated with Λ in the standard model. A void model with Λ is therefore a plausible alternative that could alter our interpretation of cosmological data, but is disfavored by Occam’s razor. If future observations strengthen the case for anisotropic anomalies such as the distance dipole, though, such models could gain support.

68

Chapter 4: ASSESSING LUMINOSITY DISTANCE BIAS FROM LARGE SCALE STRUCTURE

4.1

Introduction

We have seen how a giant, non-Copernican structure could distort our view of the universe, but even within the bounds of a statistically homogeneous universe there is a potential for inhomogeneities to influence our observations. When we interpret supernova observations, we typically use an FLRW framework, treating the universe as though matter were smoothly distributed. In actuality, matter is lumped into dense structures on a range of scales—galaxies, clusters, and superclusters— with relatively empty voids in between. Even if fluctuations average out on larger scales, these structures influence the light beams we observe. Might our observations in the real, lumpy universe differ from observations in an idealized smooth one? There are reasons to suspect there could be a systematic effect on our observations from these structures. First, the gravity of structures affects the surrounding space. Because of this, the expansion rate of space is not uniform, but varies with the density—voids expand more quickly than the average Hubble rate, while space around dense structures expands more slowly. This expansion affects both the redshift and the luminosity distance of light beams. Secondly, the light beams which constitute our observations do not necessarily traverse underdense and overdense regions with equal probability. They likely travel preferentially through voids. There are several reasons for this [73]. Light beams that pass through foreground galaxies may be absorbed or scattered by dust, or simply lost amidst the foreground galaxy’s emissions. Dense dark matter halos can strongly lens an observation, making the resulting data point appear as an outlier. Such distorted or contaminated data is often discarded, and surveys tend to target regions with empty foregrounds for just this reason [74]. Thus, selection effects bias observations in favor of beams that travel through emptier regions of space. It is unknown how large or how important this bias is, but it introduces a systematic error 69

into distance observations. As previously mentioned, dense structures can strongly lens a light beam. The converse of this is that underdense regions add a negative lensing effect compared to what we would see in a true FLRW universe. If we also consider the growth of structure—how the “lumpiness” of the universe increases with time—we begin to see how this effect could contribute to the apparent acceleration. This idea is the basis of the Dyer-Roeder (DR) approximation. This simple method accounts for the bias by inserting a “smoothness parameter” into the FLRW equation for distance in terms of redshift, multiplying the density the light “sees”. A smoothness parameter of 1 is the standard smooth FLRW scenario, whereas a value of 0 corresponds to a universe in which all of the matter is condensed into lumps, and the light beams see none of it—the “empty beam” approximation. However, attempts to apply this technique have failed to yield convincing results [75–77]. Several authors argue that the DR approximation does not capture the full effect of the density variations, and have suggested modifications, but it is not clear which, if any, is truly accurate. The present project seeks to remove some of this ambiguity by analyzing the effects of the bias in exact inhomogeneous models. These are only toy models, and do not reproduce the full hierarchy of structures at all scales, but because they satisfy the full Einstein equations, not just approximations, we can be sure they capture the full effect of the inhomogeneities. With Szekeres models, we can simulate complex arrangements of structures on the scale of superclusters. The rest of this chapter is organized as follows. Section 4.2 reviews how to calculate distances in general relativity through the Sachs optical equations. Section 4.3 then gives further background on the Dyer-Roeder approximation, and explains its shortcomings. Section 4.4 explains the methods we will use to collect and analyze data, and section 4.5 describes the models we will use. Section 4.6 presents our analysis of the results, and section 4.7 gives our conclusions.

4.2

Calculating distances

In order to fix the parameters of a given model and test its predictions against observations, we need to be able to calculate distances to compare with observed luminosity distances from type Ia 70

supernovae. Regardless of the model used, the Sachs optical equations can be used as a starting point [30]. These equations describe the evolution of a bundle of light rays surrounding the primary geodesic. These neighboring geodesics begin their journey at the same point, but traveling in slightly different angles, so that the bundle spreads over time. The size of the cross-section of the bundle on any time slice gives us the angular diameter distance, which is related to the luminosity distance by the distance duality relation,

DA =

DL . (1 + z)2

(4.1)

The rate of increase of this cross-sectional area corresponds to the expansion of the null bundle, ˆ labeled θ: d ˆ ln DA = θ. dλ

(4.2)

(The hat indicates that the quantity refers to the expansion of the null bundle rather than expansion of spacetime itself, two different quantities that are easily confused.) The expansion is one of the three components of the evolution of the null bundle. The other two are the shear σ ˆ , which describes how the bundle changes shape as it travels, changing from circular to elliptical due to Weyl focusing; and the rotation ω ˆ . They are defined directly in terms of the tangent vector as follows: 1 α , θˆ = k;α 2 r 1 1 |ˆ σ| = k(α;β) k α;β − θˆ2 , 2 2 r 1 ω ˆ= k[α;β] k α;β . 2

(4.3) (4.4) (4.5)

The shear is a complex variable, whose phase indicates the direction of shearing relative to an arbitrary direction. The rotation vanishes for our purposes, because the Szekeres models are irro-

71

tational. Rotation terms will be omitted from the following equations. The first Sachs equation describes the evolution of the null bundle expansion: dθˆ ˆ2 1 + θ + |ˆ σ |2 = − Rαβ k α k β , ds 2

(4.6)

where Rαβ is the Ricci tensor. The null bundle shear evolves according to the second Sachs equation: dˆ σ ˆσ = Cαβµν ∗α k β ∗µ k ν . + 2θˆ ds

(4.7)

Here, Cαβµν is the Weyl tensor and α is a complex spatial three-vector with unit norm whose real and imaginary parts are both tangent to the wave front (orthogonal to k α ) and orthogonal to each other. The real and imaginary parts of α thus constitute a set of orthonormal basis vectors for the screen space of the light bundle. These vectors are orthogonal to both the comoving vector uµ (in our models equal to (1, 0, 0, 0) and the geodesic tangent vector k µ , so together the four vectors form a basis for the whole spacetime. The screen space basis vectors must be parallel transported along the geodesic to maintain consistency. The initial conditions for θˆ and σ ˆ are based on the null bundle’s behavior as it reaches the observer. At this point, the expansion diverges, because tangent vectors pointing in different directions all converge on a single point. In practice, then, we must set the initial conditions a small distance away from the observer. To first order, the angular diameter distance near the observer is simply equal to the affine parameter (because we begin with |∂t/∂λ| = 1), so by eq 4.2, θˆ0 ≈ 1/λ0 . The shear is initially zero to first order. Using these as initial conditions, we can numerically integrate the Sachs equations to obtain the complete distance-redshift curve.

72

4.3

The Dyer-Roeder approximation

The equations of the previous section are sufficient if the metric contains all of the relevant information about the matter distribution, but in the real universe light is absorbed, scattered, lensed, and distorted by clumps of matter. These objects are outside the purview of even a perturbed FLRW model, which can only describe small, smooth density variations. But these objects can have significant selection effects on our data. By absorption or scattering they can block our view of objects behind them—it is hard to see a galaxy when another galaxy is in the foreground. Even if we can see light from something behind a galaxy cluster, it is often contaminated to the point we have to discard the data [74]. These problems were first addressed by Zel’dovich, leading to the “empty-beam” approximation [57]. His reasoning was that because so much of the matter is clumped into small structures, most of the light we see from distant sources has traveled through a vacuum, not a fluid matching the cosmic average density. This means that, for narrow beam surveys such as for SNIa, most of the beams are slightly demagnified (compared to the average beam predicted by the idealized FLRW model), and they are only compensated by highly focused beams passing through the very dense centers of dark mater halos [78], which we either do not see, ignore due to contamination, or treat separately as rare anomalies. Therefore, in eq (4.6), density terms appearing in the Ricci tensor should be replaced with 0. Further contributions by Dyer and Roeder led to what is now known as the Dyer-Roeder approximation [58]. In its present form, it does not replace the effective density with 0, but rather multiplies it by a smoothness parameter α between 0 and 1, which represents the fraction of matter not in clumps, and evolves with redshift to reflect the growth of structure. The Dyer-Roeder approach applies the above reasoning to an FLRW universe. The shear then vanishes, and after substituting in the Ricci tensor and inserting the smoothness parameter, we get d2 DA + dz 2



1 dH 2 + H dz 1+z



dDA 3αΩm H02 + (1 + z)DA = 0. dz 2H 2

(4.8)

In practice, present data only weakly constrain the smoothness parameter, putting a lower limit 73

of α ≥ 0.42 to α ≥ 0.66 depending on the data sets used [76]. The underlying cosmological parameters, such as the dark energy and matter density parameters ΩΛ and Ωm and the dark energy equation of state w can be altered significantly, but not drastically—on the order of a few to several percent. A lower α tends to require a larger amount of dark energy to fit observations. [75, 77] Several authors have argued that the Dyer-Roeder (D-R) approximation does not sufficiently describe the effects of inhomogeneities on observed distances, and have suggested modifications. Mattsson has pointed out that, by inspecting the power series expansion of DL (z), we see that α actually seems to increase the apparent deceleration of the expansion, contrary to physical reasoning that emptier space should decelerate less [74]. He argues that the D-R approximation ignores two important effects: inhomogeneities in the expansion rate and the growth of nonlinear structures. He proposes a second parameter, β(z) 1 , which should multiply H(z) to reflect the higher expansion rate of voids. Bretòn and Montiel also found problems with the Dyer-Roeder approximation when they tried to use it in an analysis combining GRB observations with SNIa data [75]. Contrary to all physical expectations, they found a best fit for the smoothness parameter to be lower for z > 1.5 (α = 0.587) than for z < 1.5 (α = 0.856). If the approximation is to be trusted, it would seem to suggest that matter was clumpier in the past than it is now. Clarkson et al. have cast further doubt on the Dyer-Roeder approximation by formulating a modified version, which seems equally valid but gives drastically different results [78]. The modified D-R approximation substitutes for H(z) using eq (1.2), thus obtaining more density terms before applying the smoothness factor. There are several other similar ways to modify the approximation, and it is not clear which is “correct,” if any. But they are certainly not equivalent— the modified approximation they tested showed much a greater to the distance modulus at low redshifts (z . 1) with less adjustment to α(z), allowing it to come closer to mimicking dark energy. They conclude that the Dyer-Roeder approximation is an ambiguous and incomplete way of accounting for inhomogeneities. 1

Not to be confused with the cluster velocities along the light-cone used to calculate the kSZ effect in section 3.4.5

74

All of these problems indicate that the Dyer-Roeder approximation by itself is inadequate to relate the real lumpy universe to the ideal FLRW model. Perhaps we could bridge the gap by constructing an exact model that reproduces some of the key features of realistic structure, and apply the reasoning of Zel’dovich, Dyer, and Roeder from there.

4.4

Methods

To calculate observations in our test model simulations, we will again need to integrate the null geodesic equations (1.23) backwards in time from the observer, as we did when calculating CMB temperatures in the previous chapter. This time, however, we are interested in luminosity distances along the light beam, so we will need to simultaneously integrate the Sachs optical equations, eqs 4.7 and 4.6, as well as eq 4.2. To calculate the selection effects we are looking for, we need to also track a weight factor along each geodesic. As the geodesic passes through overdense regions, the probability of it encountering a galaxy increases, and so the probability that we will actually observe the geodesic’s signal decreases. When geodesics are generated across the whole sky, their relative weight factors will tell us how likely we would be to observe a signal in each direction at each redshift. Geodesics we are more likely to see will weigh more heavily on the inferred average distance-redshift relation. While overdense regions with an abundance of galaxies may be difficult to see through, there is more to see in such regions. Because our primary observations for measuring distances are from supernovae within galaxies, where there are more galaxies, we will make more observations. We therefore need two weight factors. One is a cumulative “extinction” factor, which we will label ξ, accounting for all the ways structures can block lines of sight or influence our surveys. It should decrease along the geodesic at a rate roughly proportional to the galaxy density. The other is an “emission” factor, γ, to account for the fact that denser structures also imply a greater number of possible sources to observe. Its value at any point along the geodesic should be roughly proportional to the galaxy density at that point, independent of the geodesic’s history. The total weight is the product of these two factors: w(z) = γ(z)ξ(z). 75

For both, we will use as an initial guess a form proportional to the density, but decreasing as we go backwards in time to reflect the growth of structures (in a form based on [79]). Specifically, ρ d ln ξ = 10−5 aF LRW (t)5/4 , dλ ρ0F LRW γ = ρ aF LRW (t)5/4 .

(4.9a) (4.9b)

There is enough uncertainty in this whole line of reasoning, particularly where the selection effects are concerned, that it is unfeasible to find a “true” form for these factors a priori. A trial-and-error approach is more practical. Each geodesic, then, will give us a distance-redshift curve, DL (z) along the corresponding line of sight, along with a weight factor, which is also a function of redshift, w(z). A weighted average distance-redshift function is then constructed by Pn w (z)DLi (z) Pn i , DL (z) = i=1 j=1 wj (z)

(4.10)

where n is the total number of geodesics. With this averaged function, we will use a fitting procedure to find the ΩΛ one would naively infer by interpreting this function in an FLRW framework. If the structures have no significant net effect on distance observations after averaging, as is commonly assumed, these inferred parameters should correspond closely to the “true” parameters input into the model, within the bounds of current observational precision. If, however, there is a significant discrepancy, we would argue that it serves as evidence that inhomogeneities must be taken into account when interpreting cosmological data.

4.5

Models

This work will test our approach on two types of Szekeres models. The first is a large coarsegrained model with many structures, similar to the one suggested in [60], but without a giant 76

void of any sort. Although the geometry of the model prohibits a consistent homogeneity scale everywhere, density variations are limited to a 100 Mpc scale as much as possible. The other type of model is a Szekeres Swiss Cheese model, which can maintain homogeneity below a certain scale everywhere, and is therefore useful for investigating the effects of a series of smaller structures. Both will be matched to a flat FLRW background with ΩΛ = 0.7. The large coarse-grained model has a background Hubble rate H0 = 74.3 km s−1 Mpc−1 , and the Swiss Cheese model has H0 = 70 km s−1 Mpc−1 . 4.5.1

Large coarse-grained structure model

Construction of the large coarse-grained model begins with a base LTB model. We desire a density profile that averages out to the background FLRW value over scales on the order of 100 Mpc. This rules out giant void models. However, we cannot simply use a homogeneous LTB density profile, because as remarked in section 1.3, the density contrast induced by the Szekeres functions depends on how the LTB density on the shell compares with the average density within the shell. Therefore, a homogeneous LTB model will remain homogeneous when we apply the Szekeres functions (as can be confirmed by careful consideration of eq 1.19). To create wall structures and voids, then, we need a density profile that oscillates around the base FLRW value:     πr M 0 (r) −r/1000 Mpc = ρF LRW 1 − 0.5 cos e . 4πr2 100 Mpc

(4.11)

The bang-time function is set to 0 everywhere, and k(r) is chosen for consistency with Φ(t0 , r) = r by numerically solving eq 1.10. This sets up a model consisting of alternating underdense and overdense spherical shells, roughly averaging out radially over scales of 100 Mpc. Although the larger shells necessarily exceed this homogeneity scale in the angular directions, the variations are damped down at high r. For the Szekeres functions, we want to create a series of structures within each radial section, but each in a random direction. We first need a standardized form for the Szekeres functions in

77

each section, keeping in mind that we need the first derivatives to be continuous in order to allow the calculation of geodesics. We will use a function

d(r, r1 , r2 ) = (1 + r)

−99/100



(r − r1 )(r2 − r) (r1 /2 − r2 /2)2

2 (4.12)

For the section r1 < r < r2 , this function peaks at just below 1/r when r = (rf /2 − ri /2)—that is, at the midpoint of the radial section—and it goes to 0 at the boundaries (as well as its first derivative 2 ). By setting S 0 /S = c d(r, r1 , r2 ) in this region, while keeping P and Q at 0, we create a band of shell shifting, with the associated density anisotropy. The parameter c allows us to tune the magnitude of the anisotropy, and as long as c < 1, we do not cause the shells to cross at the present time. (Recall that shell r + δr is shifted by δr R S 0 /S relative to shell r, and R(t0 , r) = r, so if S 0 /S < 1/r, the shells do not cross.) Because we have so far only used the S function, this only creates an axially symmetric anisotropy in a single direction. To change it to an arbitrary direction, we employ a modified Haantjes transformation. Normally, a Haantjes transformation is simply a coordinate transformation which globally rotates a Szekeres model. Here, we will instead apply a separate transformation to each radial section, so that the direction of the anisotropy in each section will be physically different. In a quasi-spherical Szekeres model, a Haantjes transformation modifies the p and q coordinates as follows [80]: p + D1 (p2 + q 2 ) , T q + D2 (p2 + q 2 ) q = q0 + , T

p = p0 +

T = 1 + 2D1 p + 2D2 q + (D12 + D22 )(p2 + q 2 ),

(4.13a) (4.13b) (4.13c)

Technically, we only need S 0 , P 0 , and Q0 to be continuous in order to have a continuous density distribution and coordinate system, but we have found that an additional level of continuity improves the stability of our numerical computations. 2

78

where p0 , q0 , D1 , and D2 are arbitrary constants. We want the metric to keep the same form after this transformation—that is, we want E,r (r, p, q)/E(r, p, q) = E ,r (r, p, q)/E(r, p, q) and (dp2 + dq 2 )/E(r, p, q)2 = (dp2 + dq 2 )/E(r, p, q)2 . This requires that the S, P , and Q functions are transformed as well. Because our initial model contains only an S function, the transformations are somewhat simplified.

P (r) = Q(r) = S(r) =

D1 S

2

1 + (D12 + D22 )S D2 S

2,

(4.14a)

2,

(4.14b)

2.

(4.14c)

2

1 + (D12 + D22 )S S 1 + (D12 + D22 )S

Using this transformation, we are able to turn an axially symmetric model that uses only S(r) into one that uses all three functions, and thus has shell shifting in a different direction, while maintaining the magnitude of the shell shifting and axial symmetry. Effectively, all we have done is rotate the model. The parameters D1 and D2 can be related to the angles of rotation by 1 − cos θ cos φ e−S(r) , sin θ 1 − cos θ D2 = sin φ e−S(r) , sin θ D1 =

(4.15) (4.16)

where (θ, φ) is the direction of the shell shifting in the rotated model at r. Using this prescription, we build S(r), P (r), and Q(r) as piecewise functions, with a different shifting direction in each section (chosen at random uniformly over the unit sphere). We include constant factors in each section which ensure continuity without affecting the shifting magnitudes. This gives us a complete model with randomly distributed, totally anisotropic structures. We divide the model into sections by finding the r values of the local minima in the density of the base LTB model. These r values serve as the boundaries of the sections. In each section, we choose a random angle on the unit sphere, and transform the Szekeres functions in that range

79

Figure 4.1: A two-dimensional analogue of the coarse model used. The actual model has anisotropies randomly distributed in all directions of the unit sphere.

accordingly, following the above procedure. Figure 4.1 shows a two-dimensional analogue of the result. Notice the scattered dense arcs, and the voids between them. 4.5.2

Swiss Cheese model

Swiss Cheese models were first envisioned by Einstein and Straus as a way to model compact objects embedded in the expanding universe. In their original construction, all the matter in a spherical region in an FLRW model is collapsed into a point mass at the center of the sphere, creating a “hole”. The spacetime inside the hole is described by a Schwarzschild or Kottler metric, which matches perfectly to the FLRW metric at the boundary, so that the space outside the hole (the “cheese”) is unaffected and remains homogeneous [81]. The process can be repeated to create any number of holes, as long as none overlap. This simple model was investigated with respect to the Dyer-Roeder approximation by Fleury [82]. Using holes with radii on the order of 1 Mpc, with the central point masses representing galaxies opaque out to radii of 10 kpc, he found that this Swiss Cheese model agreed with the Dyer-Roeder approximation quite well. However, this kind of model cannot account for the kinds of structures we see on larger scales. To describe these structures, we need to fill the holes with a

80

different kind of metric. The holes can use any metric that can match to FLRW on a spherical surface, such as LTB or even Szekeres metrics. This allows for a wider range of structures within the holes. In particular, one can model the holes as voids surrounded by compensating overdense walls, roughly resembling the kinds of structures we see at 100 Mpc scales. This gives a model that is more realistic than the perfectly smooth FLRW model, while still maintaining strict homogeneity on larger scales. This model is still an exact solution to Einstein’s equations, as long as junction conditions are satisfied at the boundaries between the holes and the cheese. These junction conditions require that the model-defining functions take on their FLRW values at the boundary—that is, if rH is the edge of the hole, M (rH ) =

4π ρ r3 , 3 0F LRW H

2 , and tB (r) = 0. The holes and cheese must k(rH ) = kF LRW rH

also share the same value of Λ. As described in section 1.2.3, using the LTB metric in this way results in a symmetric wall surrounding the void, which the photons cannot avoid. This limits the amount of selection effects we would be able to infer. Realistic wall structures are not so symmetric, so we should not expect this limitation in reality. Using a Szekeres metric, on the other hand, we can make the wall lopsided. Each hole would have a very dense arc on one side, representing a galaxy cluster or supercluster, while the other side would have little overdensity. Photons trying to pass through the dense side would tend to be blocked, while photons passing through the other side could enter the void unimpeded, and experience a greater expansion rate along the way. Szekeres Swiss Cheese models have already been studied by Peel [83] and Bolejko [84, 85]. Bolejko found that axially symmetric Szekeres Swiss Cheese models fare no better in matching the supernova than LTB models with the same hole size. In both cases, holes of at least 500 Mpc are needed to produce apparent acceleration comparable to what is observed. He also found that as long as the structures are compensated, the Rees-Sciama effect does not significantly alter the CMB fluctuations. Peel, on the other hand, found that a model with a hole radius of 30 Mpc can produce a small but potentially significant bias in the distance modulus, comparable to the expected scatter from the Dark Energy Survey (DES) 10-field survey for redshifts larger than 0.6 [86]. 81

The holes should resemble realistic voids. We therefore choose an LTB density profile similar to that used by Hamaus, et al. [67] They found that voids of all sizes conform to a similar density profile at all redshifts, determined by only two free parameters. We used their parameterization previously in chapter 3. The basic functional form was given in eq 3.1:  ρ(r) = ρ0F LRW

1 − (r/rs )α 1 + δc 1 + (r/rv )β

 .

(4.17)

As before, we modify this form to match exactly to FLRW (see eqs 3.2 and 3.3). Hamaus et al. found that this parameterization was overdetermined, so they were able to eliminate the α and β parameters through linear relations to rs /rv . For our purposes, we will use holes that match to FLRW at r = 30 Mpc, and we will choose values of rv = 20 Mpc, rs = 0.91rv , and δc = −0.88, which then gives α = 2.18 and β = 9.5. To create an anisotropy in the outer wall, we will use Szekeres functions similar in form to those used in chapter 3 (i.e. eq 3.6), combined with a Haantjes transformation as described in section 4.5.1. With our model for the holes now chosen, we can populate our SC model with numerous instances of the Szekeres hole. We need only to choose an arrangement. We begin by placing the holes in a cubic lattice, with adjacent holes touching at the boundaries. We can assign sets of three integer indices to these holes, with an arbitrarily chosen “central hole” at {0, 0, 0}. Our tests will consist of geodesics beginning in the cheese, in a plane just below the central hole. These geodesics will propagate upwards through a series of holes, initially parallel to each other. This is not meant to represent an entire path from source to observer, but rather a segment of the total light paths in a tight beam in the observer’s sky. To be realistic, the arrangement of holes must be randomized in some way. First, we will randomize the orientation of the holes, so that the dense sides are not aligned. For each hole in the test geodesics’ paths, we choose a random orientation angle, chosen uniformly over the unit

82

Figure 4.2: A two-dimensional analogue of the Swiss Cheese model used. The actual model has anisotropies randomly distributed in all directions of the unit sphere.

sphere.

3

Then, we randomize the lateral positions of the holes so that the voids are not aligned

in the geodesics’ direction of travel. For each “layer” of holes above the central hole’s layer, we displace the entire layer by random values in the Cartesian x and y directions. For upwardstraveling geodesics, this makes the impact parameter as it enters each hole independent of its impact parameter through other holes. A 2-dimensional analogue of this setup is shown in figure 4.2.

4.6 4.6.1

Results Large coarse-grained model

For our first test, we chose an observer location at a saddle point in the density at r = 300 Mpc. As we follow a single geodesic, shown in figure 4.3, we can clearly see the effects of the density variations on the distance-redshift curve. It oscillates around the FLRW distance-redshift curve, going higher when encountering overdense walls and lower when passing through voids. This 3

In practice, we keep the same model for each hole, but when a geodesic enters a hole, we rotate it around the outer surface according to the hole’s orientation, then reverse this rotation as it exits back into the cheese.

83

4 4 2 DL - DLCDM L

DL - DL LCDM HMpcL

0.5

0

0.

-2

dr•rLCDM

2

0 -2

-0.5

-4

-4 0.

0.1

0.2

0.3

0.4

0.5

0.0

z

0.2

0.4

0.6

0.8

1.0

z

Figure 4.3: Left: The distance-redshift curve for a single geodesic in the large coarse-grained model. The blue line shows the luminosity distances minus the values for the background ΛCDM model (in units of Mpc), and the dashed red line shows the density contrast. Right: The distanceredshift curves for all of the geodesics, covering the sky, originating from a saddle point in the density. The bright green line is the weighted average. The dashed cyan line is the weighted average with a stronger weighting function.

behavior is primarily due to the gravitational potential at the point of emission. Photons originating in walls, must climb out of a potential well, resulting in a demagnified image—i.e., larger apparent distance—whereas photons originating in voids gain energy as they exit. Objects behind a void or wall are demagnified or magnified, respectively, as we would expect. A demonstration of this for the case of a void is shown in figure 4.4. For the particular geodesic shown in the left of figure 4.3, the cumulative effect seems to be a slight magnification as redshift increases. But not all lines of sight will necessarily give the same results, so we collect a sample by generating a total of 48 geodesics from the same location, distributed across the sky in a HEALPix grid. This number is not large enough to constitute a truly representative distribution, but the calculations are computationally expensive, and we are limited in computing power and time. For now, this set will suffice to give us a rough idea of the behavior. Looking at the distance curves of this full set of geodesics, shown in the right of figure 4.3, we see that after a short distance they appear to mix randomly, with little correlation on short scales. On longer scales, the bulk appear to share a slight downwards trend at higher redshifts, with only a handful trending upwards to counterbalance. However, when we take the weighted average, the result shows little to no deviation from the background ΛCDM case. In fact, the weighted average 84

0.015

m - mLCDM

0.010 0.005 0.000 -0.005 -0.010 -0.015 0.0

0.2

0.4

0.6

0.8

1.0

z

Figure 4.4: The residual distance modulus vs redshift (relative to the background ΛCDM model) for a geodesic passing through a single uncompensated 200 Mpc void.

is nearly the same as the unweighted average. For this reason, we also tested a stronger weight function, in which ξ is proportional to ρ2 , and with an overall factor 100 times higher. The result is shown in the dashed cyan line. There is a small difference, with a slight bias towards magnification, but it dampens out at higher redshifts. We repeated this test for several other observer locations. To test for possible effects from local structures, we chose some special locations, including in an underdensity and in the middle of a wall. These two cases are shown in figure 4.5. In all of these tests, the weighted average distance curve still matches the ΛCDM curve closely once the geodesics exit the local structure. We apply a fitting procedure comparing the distances to ΛCDM distances with different ΩΛ , and in all cases we find the best fit is within 1% of ΩΛ = 0.7. Even the stronger weighting factor is not enough to bias the average a noticeable amount. Could the particular form of our weighting factor be to blame for this lack of effect? We have tried applying stronger weight factors, which respond nonlinearly to density and thus favor only the very emptiest beams. Regardless of the weighting function, though, the weighted average must lie within the envelope formed by grouping all of the individual curves. This envelope is simply too narrow to allow for a significant adjustment to the apparent cosmological parameters. Some of the geodesics are magnified and some are demagnified, and the distributions are different for different observer locations, but at larger distances the magnification effects taper off. By redshift 85

6

0.010

4 m - mLCDM

DL - DLCDM L

0.005 2 0

0.000

-2 -0.005 -4 -0.010 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

z

0.6

0.8

1.0

0.6

0.8

1.0

z 0.02

5

m - mLCDM

DL - DLCDM L

0.01 0

-5

0.00

-0.01

-0.02 0.0

0.2

0.4

0.6

0.8

1.0

0.0

z

0.2

0.4 z

Figure 4.5: The distance-redshift curve for geodesics across the whole sky in the large coarsegrained model, in terms of luminosity distance (left) and distance modulus (right). The thick green line is the weighted average. The dashed cyan line is the weighted average with stronger ξ. Top: observer in a wall, bottom: observer in an underdensity.

86

10

0

20

20

10 DL - DLCDM L

DL - DLCDM L

DL - DLCDM L

20

30

10

0

-10

-10

-20

-10 0.0

0.2

0.4

0.6

0.8

1.0

0

0.0

0.1

0.2

z

0.3

0.4

0.5

z

0.0

0.2

0.4

0.6

z

Figure 4.6: The distance-redshift curves for geodesics passing through isolated structures. The model consists of an uncompensated overdense spherical shell with a thickness of 240 Mpc and a radius of 200 Mpc at the peak, made asymmetric by shell shifting through Szekeres functions. Left: observer inside the overdense shell. The purple line is through the denser, narrower side, and the blue is through the wider, shallower side. Middle: the observer is positioned outside the shell at r = 500 Mpc, looking across it. The purple line starts on the denser side, and the blue on the less-dense side. Right: observer at r = 1000 Mpc.

z = 1.0, none of the geodesics deviate from the background DLΛCDM by more than 0.1%. There are two possible conclusions to draw from this. One is simply that structure on these scales does not affect light beams enough to significantly bias our observations. The other is that the model chosen is too unrealistic to correspond to real observations. Perhaps a Szekeres model’s effects on light beams are not the same as the effects of real structures, due to the particular mathematical construction. We can get a better sense of how the individual components of our complex model affect the distance-redshift relation by testing simplified models containing only single isolated components. Figure 4.6 shows a few examples. When the observer is inside the shell, the luminosity distance curve in one direction approaches a horizontal asymptote—the relative magnification at large distances is very small. In the other direction, there is a slight slope, but it is small compared to the background luminosity distance. For an observer far outside the shell (right), we see a considerably greater slope, meaning greater magnification. An observer closer in (middle) sees less magnification, but the difference between the two viewing directions is more clear, confirming that Szekeres structure does in fact affect luminosity distances in this situation. In the current model, with its structures arranged in an onion-like series of shells, nearly all of the structures reside in shells surrounding the observer. Furthermore, because the wall structures 87

are stretched across a portion of the total surface area of the shells, larger shells stretch the wall structures out across a larger area, resulting in smaller density contrasts. This is reflected in the reduced magnification at higher redshifts visible in figure 4.5. Perhaps, then, this kind of model simply cannot give a sufficiently realistic description of real structures. We need a model with selfcontained structures scattered around the observer. This is where the Swiss Cheese model comes in. 4.6.2

Swiss Cheese model

Swiss Cheese models are better able to describe smaller, independent, contained structures. Due to computational limitations associated with how the model is constructed, we did not sample a full sky, but a full sky sample is not necessary. Because the individual structures do not span wide distances in any direction, any patch of the sky should statistically look similar to any other patch. We therefore need only test a small patch at a time - enough geodesics to cover an even distribution of impact parameters with the first hole. Aside from the differences in the model and geodesic distribution, the procedure is essentially the same as before. Each geodesic gives a distance-redshift curve, along with a weight function. The results are presented in figure 4.7. Compared to the previous model, there is a more consistent periodicity, due to the way the holes are layered. Also, the spread of distances appears more even—just as many geodesics are demagnified by the inhomogeneities as are magnified. The weighted average does show a slight bias towards smaller distances at higher redshifts compared to the homogeneous ΛCDM case, but the difference is far too small to be measurable in the real world, on the order of a 10−4 drop in the distance modulus at a redshift of 0.6. This is well below the expected scatter from the DES, which is on the order of 10−2 [86]. A shift this small may be simply due to numerical error in the calculations. Even the most magnified or demagnified geodesics show only a ∆µ on the order of 10−3 . This difference is too small to significantly impact parameter estimations of the background cosmology. Because the structures are nonlinear perturbations to the background, the effects may scale 88

0.004 0.002

0

m - mLCDM

DL - DLCDM L

2

-2

0.000 -0.002

-4 -0.004 0.0

0.2

0.4

0.6

0.0

z

0.2

0.4

0.6

z

Figure 4.7: The distance-redshift curve for a set of geodesics in the Swiss Cheese model with rv = 20 Mpc. The thick green line is the weighted average. The dashed red line is the weighted average with stronger ξ (color changed from fig 4.5 for greater visibility in this case).

nonlinearly with the size of the structures, so we also tested a model with holes scaled up by a factor of 3. The void diameters are now 120 Mpc, which is quite large. We also generated a larger number of geodesics (117), with the hole orientations and positions randomized for each geodesic, in order to get a more complete random sample (though an even larger sample will be needed to draw more reliable statistics in future work). The results are shown in figure 4.8. We do in fact find that the dispersion in the distances is increased by a factor larger than 3—there is nearly a tenfold increase. However, this is still a small dispersion compared to the real supernova data, for which both the intrinsic dispersion and the lensing dispersion are of order 0.1 magnitudes at redshifts of 1.2 [87, 88]. The distribution is more uneven, as can be clearly seen in the histograms in figure 4.8. The mean magnification is close to zero, as it should be due to photon conservation, but the median is shifted in the magnification direction, with a small tail of high demagnification. This is the reverse of the skew in the distribution found by weak lensing analyses using N-body simulations and ray-tracing in a ΛCDM universe, which has a demagnified median and a tail of high magnification [89,90]. This is a significant difference in the effects of Szekeres Swiss Cheese structures compared to the structures found in N-body simulations. A key difference from the results in the previous tests is that here, the strongly-weighted average shows a large departure from the simple mean and the background distance. As we initially hypothesized, underdense paths contribute more strongly than overdense paths at higher redshifts, 89

0.02

0.01 m - mLCDM

DL - DLCDM L

40

20

0

0.00

-0.01

-20

-0.02 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

z

0.6

0.8

1.0

0.015

0.020

z

40

30

runs in bin H117 totalL

runs in bin H117 totalL

25 30

20

20

15 10

10

5 0 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010

0

m-mLCDM Hz = 0.5L

-0.005

0.000

0.005

0.010

m-mLCDM Hz = 1.0L

Figure 4.8: Top: The distance-redshift curve for a set of geodesics in the Swiss Cheese model with rv = 60 Mpc. The thick green line is the weighted average. The dashed red line is the weighted average with stronger ξ (color changed from fig 4.5 for greater visibility in this case). Bottom: distributions of the shift in the distance modulus from the background ΛCDM value at redshifts z = 0.5 (left) and z = 1.0 (right) for 117 geodesics, each through a randomized sequence of holes. Solid green lines mark the mean, dashed red lines the strongly-weighted average, and dot-dashed blue lines the median. The standard deviations are σ∆µ = 0.0020 at z = 0.5 and σ∆µ = 0.0048 at z = 1.0.

90

biasing the average towards demagnification. The effect is still relatively small, but significant, nearly a 0.5% increase in the luminosity distance at z = 1.0, and an increase of 0.015 in the distance modulus, above the expected scatter from the DES. Using this strongly-weighted average distance curve, the best-fit ΩΛ is 0.708, about a 1% increase over the “true” background value.

4.7

Conclusions

In this chapter, we have studied the effects of inhomogeneities on distance observations in models with large-scale homogeneity, consistent with the ΛCDM concordance model. Unlike previous studies using Swiss Cheese models, we have included selection effects favoring light paths that travel through emptier regions, following the spirit of the Dyer-Roeder method. We have also tested a wider range of models—not only Swiss Cheese models with anisotropic holes, but also a model consisting of many structures in a single Szekeres metric. Most of our tests failed to produce a sufficiently large dispersion in the distances at high redshifts to match a realistic distribution, or to affect the interpretation of cosmological parameters. Furthermore, even within the small range, the weighted averages experienced only tiny deviations from the mean due to selection effects. The only exception was with the Swiss Cheese model with larger holes. Here, the dispersion was considerably larger, though still below the observed dispersion from gravitational lensing. The distribution was also the wrong shape, skewed towards magnification with a small number of more strongly demagnified light paths, the opposite of distributions found from more conventional studies. This implies that the Swiss Cheese model differs from a realistic matter distribution in an way that has important effects on observations. Within the context of this test, we did find that the selection effects can bias the distance measurements to a significant degree. The few strongly demagnified geodesics dominate the weighted average at high redshifts, as they pass through the most underdense regions. An observer in this model, if considering the non-representative set of observations corresponding to this weighted average to be the “true” distance-redshift relation, would incorrectly calculate a slightly larger ΩΛ 91

than the actual background value. However, this result was only possible with a rather extreme weighting function, in which most lines of sight became effectively invisible. Considering also the large hole size that strains against homogeneity limits, as well as the unrealistic lensing distribution, the prospects of connecting this to a real observable effect are dubious at best. Although the Szekeres model is the most general known inhomogeneous dust solution to Einstein’s equations, it is far from a perfect representation of real-world structures. Its mathematical construction limits the possible arrangements of matter it can simulate. Our construction of the coarse model, for instance, required periodic radial density fluctuations in order to allow the Szekeres functions to create dense walls, a constriction real structures are not bound by. In addition, the amplitude of the Szekeres inhomogeneities is limited as we move to high r values, if we are to maintain a semblance of large-scale homogeneity. In Swiss Cheese models, the voids are required to be compensated by overdensities in order to facilitate the FLRW embedding, but in the real universe uncompensated voids are easily possible. Furthermore, Szekeres models are ill-equipped to simulate a hierarchy of structures on a range of scales. Even when passing through a supercluster, a light beam might travel through mostly empty space between galaxies. No known exact solution to Einstein’s equations can fully account for the full range of scales in which we see structures. A perturbation theory for Szekeres models would allow some improvement, but the lack of symmetry in general makes this a very difficult problem. Overall, our simulations give some support to the idea that we can ignore selection effects from inhomogeneities, and that we can accurately ascertain the cosmological parameters using a simple homogeneous framework as an approximation, plus weak lensing effects seen in the dispersion in the distances. Due to the unrealistic properties of the models used, though, this is not a conclusive refutation of systematic biases from inhomogeneities. A more powerful result would require something more than simple Szekeres models. Combining Szekeres structures on large scales with statistical lensing predictions from smaller-scale structures (facilitated by a perturbation 92

theory) would be a significant improvement, and may give results more closely resembling our observations of the real universe.

93

Chapter 5: POLARIZATION WITH ANISOTROPIC EXPANSION

5.1

Introduction

Besides intensity and redshift, light beams can carry information in their polarization. Photons can be polarized by Thomson scattering or gravitational waves from inflation. While these polarization signals can be difficult to see and require special instrumentation, we are reaching sensitivity levels high enough to see interesting results, such as the recent (provisional) discovery of gravitational wave signatures in the CMB by the BICEP-2 team [91]. Studies have also found a curious verylarge-scale alignment of polarization vectors from quasars [92]. As such, interest in cosmological polarization is increasing. It has been suggested by Ciarcelluti that regions with anisotropic expansion could induce a polarization on light passing through it [93]. Such regions should naturally occur wherever structures decouple from the Hubble flow. As gravitational forces slow down the expansion rate within a galaxy cluster, the region connecting the different expansion rates experiences anisotropic expansion. This is a very relevant issue for LTB and Szekeres models, since they include anisotropic expansion as a built-in feature. However, careful study of the equations of electrodynamics in general relativity shows that the polarization state of electromagnetic waves is not altered by anisotropic expansion. Conceptually, we can intuit that this should be the case by the symmetry of the electric and magnetic fields. If we consider a photon traveling in the z direction, polarized with its electric field pointing along the x direction, and another with its electric field along the y direction, their magnetic fields will point in the y and x directions respectively. If the expansion rate is higher in the x direction, we might suppose that the electric field of the first photon is damped more than that of the second. If we only look at the electric field, this looks like polarization in the y direction. But the magnetic field of the second photon would be damped in the same way. When we see the photons, they are both ordinary photons—neither can ever have more energy in its magnetic field than its electric field,

94

or vice versa. We must therefore conclude that both photons are damped or amplified by the same amount, regardless of anisotropic expansion, and so there is no change in polarization. While this argument makes intuitive sense, it is dangerous to be overly reliant on intuition, especially in the realm of general relativity. A more rigorous mathematical derivation follows.

5.2

Setup: electrodynamics in anisotropic spacetime

We begin by following the same methods as Ciarcelluti [93]. Rather than deal with the full metric of the entire spacetime, we focus our attention to the immediate vicinity of the photon at a given time, where the metric ds2 = gµν dxµ dxν can be expressed locally as ds2 = −dt2 + a1 (t)2 (dx1 )2 + a2 (t)2 (dx2 )2 + a3 (t)2 (dx3 )2 .

(5.1)

This is a Bianchi Type I metric, which describes a homogeneous anisotropic spacetime [94]. Even though we are ultimately interested in inhomogeneous models, we can use this metric as a local description of the photon’s environment at any given time. To calculate the total effects along a geodesic, we would switch between the global inhomogeneous metric and a different local metric at each step. This metric is similar to the familiar flat FLRW metric, but with three different scale factors, owing to the difference in the expansion rates in the three spatial directions. Accordingly, we need three Hubble rates to describe the local expansion, defined as

Hi =

a˙ i . ai

(5.2)

We choose our coordinates such that t = 0 at the beginning of the time step, and ai (0) = 1. In a general spacetime, the electromagnetic field tensor Fµν and its dual ∗ Fµν can be expressed

95

in terms of the electric and magnetic field four-vectors as

Fµν = uµ Eν − uν Eµ + µναβ B α uβ , ∗

(5.3a)

1 F αβ = αβµν Fµν 2 = αβµν uµ Eν + uα B β − B α uβ ,

(5.3b)

where uµ is the fundamental observer’s four-velocity, which in our case is simply uα = (1, 0, 0, 0), since we are working in comoving coordinates, and µναβ is the totally antisymmetric Levi-Civita tensor, √ −gAµναβ ,

(5.4a)

1 µναβ = √ Aµναβ . −g

(5.4b)

µναβ =

Here, g is the determinant of the metric tensor, and A is the totally antisymmetric symbol equal to +1 or −1 when its indices are even or odd permutations of (0, 1, 2, 3), respectively, and 0 otherwise. (Note that A0123 = 1, while A0123 = −1.) Because the EM tensor is antisymmetric, its diagonal elements vanish. In our metric, its other elements are

F 0i = −F i0 = E i , a3 3 B , a1 a2 a1 1 = B , a1 a3 a2 2 = B . a2 a3

(5.5a)

F 12 = −F 21 =

(5.5b)

F 23 = −F 32

(5.5c)

F 31 = −F 13

(5.5d)

We use Roman indices to denote spatial coordinates only. The fields are governed by the Maxwell

96

equations, written in tensorial form as

∗

F;νµν = 4πJ µ ,

(5.6a)

F;νµν = 0.

(5.6b)

Since we are concerned with photons propagating through empty space, not interacting with matter, the four-current density J µ = 0. From these, we derive the electrodynamic equations in a vacuum: ∂E i ∂B i = = 0, ∂xi ∂xi ∂B m ∂(a3 E i ) = ilm a2m , ∂t ∂xl ∂(a3 B i ) ∂E m = −ilm a2m . ∂t ∂xl

(5.7a) (5.7b) (5.7c)

Here, a is the geometric average scale factor, so a3 is simply shorthand for a1 a2 a3 . The symbol ilm is the fully antisymmetric symbol in three dimensions, with 123 = 1. Since we are no longer dealing with strictly tensorial quantities, we have broken from the standard index summation convention (note the three appearances of m on the right-hand side). However, we still have implied summation over all repeated indices that appear only on one side of the equation.

5.3

The wave equations

Up to this point, we have followed Ciarcelluti’s derivation. His next step is to convert to Fermi normal coordinates, given by

ri ≡ ai (t)xi , i

E ≡ ai (t)E i , i

B ≡ ai (t)B i .

97

(5.8a) (5.8b) (5.8c)

This is the natural reference frame for observers, and the E and B vector fields are what comoving observers would actually measure. However, because the spatial coordinate transformation is timedependent, it changes the meaning of partial derivatives with respect to time. Holding xi constant is not the same as holding ri constant, so

∂ | ∂t xi

6=

∂ | . ∂t ri

We will therefore keep our old coordinates

while finding a solution, and save the transformation to Fermi normal coordinates for a later step. We will, however, use a different expression for the electric and magnetic fields, for convenience:

e i ≡ a3 E i , E

(5.9a)

e i ≡ a3 B i . B

(5.9b)

To determine the evolution of an electromagnetic wave in our anisotropic spacetime, we must derive wave equations from eqs 5.7. This begins in the usual way—by taking another time derivative to obtain a second-order differential equation. For the moment, let us focus on the equation for the time derivative of the electric field, eq 5.7b.   em ei em a2m ∂ ∂ B ∂ a2m ∂ B ∂ 2E = ilm 3 + ilm . ∂t2 ∂t a3 ∂xl a ∂xl ∂t

(5.10)

In the first term, we can substitute in eq 5.7c. Counting all the terms from the two multiplied ilm symbols, and remembering that the divergence of the electric field is 0 (5.7a), we obtain a term similar to a Laplacian, which we will denote as 2 e2 ≡ 1 ∂ . ∇ a2i (∂xi )2

(5.11)

We can see the result will be very similar to the ordinary wave equation, but there is still the other term to deal with. The derivatives on the ai factors give the respective Hubble rates, so our wave equation has a mixing term: ei em ∂ 2E a2m ∂ B 2 ei e − ∇ E = ilm (Hm − Hi − Hl ) 3 . ∂t2 a ∂xl 98

(5.12)

We can slightly simplify this result by noticing that i is not summed over, since it appears on the left hand side of the equation. The Hi term therefore is just an overall factor added to the original equation 5.7b. The full set of equations now reads 2 em ei ei ∂ 2E e 2E e i = −Hi ∂ E + ilm (Hm − Hl ) am ∂ B . − ∇ ∂t2 ∂t a3 ∂xl 2 ei em ei ∂ 2B e 2B e i = −Hi ∂ B − ilm (Hm − Hl ) am ∂ E . − ∇ ∂t2 ∂t a3 ∂xl

(5.13a) (5.13b)

This is a system of six coupled second-order differential equations. An exact solution would be quite difficult, but we can find an approximate solution to first order in t. Since the metric we are using is only a local description anyway, we will need to examine the effects of the anisotropic expansion in a step-by-step manner along the wave’s path, with each step having a new local metric. This means that we can assume t remains small within each step.

5.4

Solving the wave equations

At the beginning of each step, we construct our local metric such that t = 0, and ai (0) = 1. We posit a form for the electric and magnetic fields as

ei = E e i exp (iωS + γi t) , E 0

(5.14)

and likewise for B. The frequency of the wave is denoted ω, which we assume to be much larger than the Hubble rates. S is a real function that satisfies ∂S/∂xµ = kµ , and γi is a yet unknown damping factor. The covariant tangent vector kµ evolves according to the geodesic equation, dkµ = kµ,ν k ν = Γαµν kα k ν . dλ

(5.15)

Working this out in the current metric, we find that the spatial components are constant: dki /dλ = 0. It will therefore keep the exact same form throughout the step. The time component k0 is not

99

constant due to the redshift caused by the expansion, but we will arrange our system such that it is (locally) constant in space, varying only in time. The initial electric and magnetic fields must be orthogonal to the tangent vector as well as to e0i ki = B e i ki = E ei B ei each other: E 0 0 0 = 0. (Recall that at t = 0, the local spatial metric is simple Euclidean space.) We therefore parameterize our three vectors as

ki = (sin θ cos φ, sin θ sin φ, cos θ),

(5.16a)

e i = (− sin φ cos ψ + cos θ cos φ sin ψ, cos φ cos ψ + cos θ sin φ sin ψ, − sin θ sin ψ), E 0

(5.16b)

e0i = (cos θ cos φ cos ψ + sin φ sin ψ, cos θ sin φ cos ψ − cos φ sin ψ, − sin θ cos ψ). B

(5.16c)

The angle ψ determines the initial polarization of the wave. We now plug our ansatz for the fields’ evolution, eq. 5.14, into the wave equations 5.13 in e 2 only order to determine γi , keeping only terms to the lowest order in t. On the left-hand side, ∇ acts on the iωS term, bringing out a factor −ω 2 ki2 /a2i . When both t derivatives also act on the iωS term, they bring out a factor −ω 2 k02 . Combined, these two terms vanish because kµ is a null vector. These terms represent the normal wave oscillation. The damping factor then relies on the e 1 ; the other components can remaining terms. We will focus on finding an explicit solution for E then be found in terms of permutations of the indices. On the left-hand side, the t derivative also results in terms involving γ1 , and on the right-hand e 1 , we side we have all of the terms involving the expansion rates. After dividing both sides by E are left with

γ12 + 2iωk0 γ1 + iωk0,t = −H1 (iωk0 + γ1 ) + (H3 − H2 )

e3 e2 B a2 B a3 + iωk2 iωk3 e 1 a1 a3 e1 a1 a2 E E

! . (5.17)

If the frequency is sufficiently high, we expect the damping factor to be independent of frequency. We can therefore assume that it is on the scale of the Hubble rates. We can therefore drop the γ12

100

and H1 γ1 terms, as they will be very small compared to terms containing ω. For the other terms, we are only interested in the zeroth order in t, since γi already multiplies t in eq 5.14. This means e i and E e i become B e i and E ei . we can replace the scale factors by their value at t = 0, i.e. 1, and B 0 0 Finally, dividing both sides by 2iωk0 gives us a simple solution for γ1 . e02 e03 + k3 B k2 B e01 k0 E

1 1 γ1 = − H1 + (H3 − H2 ) 2 2 1 1 = − H1 + (H3 − H2 ) 2 2



! −

1 k0,t 2 k0

cos 2θ sin φ cos ψ − cos θ cos φ sin ψ sin φ cos ψ − cos θ cos φ sin ψ

 −

1 k0,t . 2 k0

(5.18)

Similar expressions are easily obtainable for γ2 and γ3 . The last term, which relates to the redshift from the expansion, is the same for every component of the electric and magnetic fields. Since the polarization depends on the relative differences in the evolution of the different components, this term has no effect on the final results. We will therefore omit it in the following equations. We are finally ready to convert to Fermi normal coordinates. In eq 5.8, we can approximate ai (t) as exp(Hi t). We also need to undo the transformation in eq 5.9. After performing these transformations, the observed electric component of the photon will appear as "

1 1 E = exp iωS − H1 t − H2 t − H3 t + (H3 − H2 ) 2 2 " 2 e02 exp iωS − 1 H2 t − H1 t − H3 t + 1 (H1 − H3 ) E =E 2 2 " 3 e03 exp iωS − 1 H3 t − H1 t − H2 t + 1 (H2 − H1 ) E =E 2 2 1

5.5

e1 E 0

e2 e 3 + k3 B k2 B 0 0 1 e0 k0 E

! #

e3 e 1 + k1 B k3 B 0 0 2 e k0 E0

! #

e 2 + k2 B e1 k1 B 0 0 3 e k0 E0

! #

t , t , t .

(5.19a) (5.19b) (5.19c)

Polarization

Our solution gives the evolution of a photon polarized at angle ψ. Unpolarized light contains a mixture of photons polarized in all different directions. If the anisotropic expansion induces a polarization, that means photons polarized at some angles are amplified while photons at other polarizations are suppressed. Since we have left ψ arbitrary, we can determine the induced polar101

ization by looking at how the magnitude of the fields depends on ψ.r The magnitude of the electric field measured by the observer is

2 2 2 2 3 1 E + E + E . The

absolute value signs remove the iωS terms, and we can replace exp(ct) with 1 + ct, to first order e i has unit magnitude, we can condense common factors, leaving us with, we are left in t. Since E with # ( " 2 2 2  2 e 3 + k3 B e2 k B 1 2 3 2 0 0 e1 E H1 + (H3 − H2 ) E + E + E = 1 + t 0 e01 k0 E # "  2 e 1 + k1 B e3 k B 3 0 0 e2 + E H2 + (H1 − H3 ) 0 e02 k0 E " # )  2 e1 e 2 + k2 B k B 1 0 0 e03 + E H3 + (H2 − H1 ) − 2 (H1 + H2 + H3 ) . e03 k0 E (5.20) Substituting in our parameterization of the initial spatial vector components 5.16, this eventually reduces to 2 2 2
1 2 3 E + E + E = 1 + cos2 θ + sin2 θ sin2 φ − sin2 θ cos2 φ − 2 H1 t
+ cos2 θ + sin2 θ cos2 φ − sin2 θ sin2 φ − 2 H2 t − (cos 2θ + 2) H3 t. (5.21) Note that this is independent of ψ. Therefore, the magnitude of the measured electric field is independent of the photon’s polarization. As a mixture of photons passes through this region of anisotropic expansion, we now see clearly that none of them will be amplified or suppressed relative to the others, so the outgoing light will still be unpolarized.

5.6

Conclusion

One might wonder if this result is only due to our approximation only considering the first order of t. Essentially, our result only tells us that the first time derivative of the polarization is zero. Still, 102

due to the linearity of Maxwell’s equations, we argue that there should not be any “acceleration” term in the effects on the polarization, so the total effect is exactly zero. The only case when our approximations may fail is when ω is very small, on the same order as the Hubble rates, but this is a very unrealistic scenario. We therefore conclude that we should not expect to see a polarization signal arising from distortions in the expansion of spacetime. Any polarization seen around voids or lensing galaxies must be attributed to other sources. This is consistent with past work that has found no polarization rotation resulting from gravitational lenses [95, 96]. Gravitational distortions can affect the shape of polarization signals in the CMB or other sources, but only by bending already-polarized light beams. Unpolarized light passing through inhomogeneous regions, such as those that might be modeled by LTB or Szekeres models, will pass through without becoming polarized—at least, not due to the spacetime itself. Interactions with matter can still induce polarization.

103

Chapter 6: CONCLUSIONS We have examined several observables in a range of models. We have found that Szekeres models do offer some real advantages over the simpler LTB models. They allow the observer to be somewhere other than the center while remaining consistent with the observed CMB dipole, reducing the violation of the Copernican principle while also permitting different local environments for the observer. They allow more freedom in the inhomogeneity-induced CMB quadrupole and octupole relative to the dipole, allowing for partial explanation of the associated observed anomalies. And they generally allow description of more complex structures, with different shapes and structure formation rates, which may better resemble some real structures. These structures can in principle allow for selection effects to bias distance observations. However, they are still subject to many of the same limitations as LTB models. The dipole still constrains the observer to a small region. Because of its size, we would not expect to find ourselves there by chance. The octupole still tends to be an order of magnitude below the quadrupole, so a complete explanation of the anomalies is impossible without a great deal of fine-tuning. And when considering distance observations in a universe with homogeneity on large scales, despite the more diverse structures, it takes inhomogeneities straining the homogeneity scale to produce a sizeable dispersion in the distances, and even then the distribution is unrealistically skewed. What, then, can we say about this line of research?

6.1

Outlook for inhomogeneous models

The ΛCDM model is the standard model of cosmology for good reasons. It has successfully explained many observations, and made predictions that have borne fruit. Yet it is still important to explore alternative models, or else we risk being blinded by our preconceptions. Until we can thoroughly rule out these alternative models, we cannot be sure of our standard model. Inhomogeneous universe models are therefore an important area of investigation. They are difficult to rule out entirely. Our observations are limited, and the parameter space associated with 104

such models is vast. With two free functions, LTB models can be found to fit any two conceivable sets of isotropic observations. Szekeres functions add even more freedom with three more free functions. Special constructions such as Swiss Cheese models open up still more possibilities. Nevertheless, researchers have succeeded in ruling out wide swaths of the parameter space, tightening the regions in which inhomogeneities might hide. It has reached the point where even the former proponents of LTB giant void models are conceding that they simply cannot work without dark energy [66]. Our research extends the search into Szekeres models, and so far suggests that they are likely unable to address all of the issues of LTB models. 6.1.1

Can they fit within current observational constraints?

It is known that giant void models can fit the supernova data. The problems start when one adds in constraints from the CMB and local H0 measurements. The CMB forces the Hubble constant to be unrealistically small, so that even at the center of the void, it is too low to fit measurements [66]. A varying bang-time function can increase the local H0 to acceptable values, but this can create other problems. Bull found that the required variations create a kSZ signal at some redshift [72], which we would see by now if it existed. Our tests found that models with a variety of Szekeres and bang-time functions experience a large kSZ effect. We also found that varying bang-times lead to large CMB quadrupoles, which also contradict observations. While the Szekeres functions have the potential to explain a variety of anisotropic anomalies, they are also limited by the small amount of large-angle anisotropy we see in general. We have seen how they split the Hubble diagram in different directions. There is still room there to give more freedom than LTB models in terms of the CMB dipole and other observations, but there are constraints. 6.1.2

Can they explain observed anomalies?

The quadrupole-octupole alignment appears to be beyond the reach of Szekeres models. While the asymmetry of Szekeres models allows some freedom in the relative magnitudes of the different 105

multipoles, the octupoles are still generally too low to contribute to observations in regions where the dipole and quadrupole are not too large. Getting all three at the right magnitudes simultaneously to explain the observations would require a highly contrived model with a great deal of fine-tuning, which would pose more questions than it answered. Szekeres models would be able to explain the dark energy dipole some researchers have detected, though it is not widely agreed that this is a real signal. We do not yet have enough data points in our supernova catalogs to establish a directional dependence with statistical significance. As we collect more data, if they strengthen the case for a dipole in the apparent acceleration, a large-scale anisotropic matter distribution would be a plausible explanation. Due to the other problems with giant void models, a more subtle inhomogeneity with dark energy still included would be a preferable candidate. A Szekeres model could serve as a useful first approximation. Polarization signals cannot be explained by the gravitational effects of inhomogeneous models. 6.1.3

Can they affect parameter estimations?

Within the bounds of the cosmological principle, in which the size of structures is limited by the homogeneity scale, our tests indicate that inhomogeneities typically have only small effects on distance observations. If the inhomogeneities are somewhat larger than those commonly observed, on the scale of 100 Mpc or more, they can create a dispersion due to lensing effects, and if there are strong selection effects favoring observations of light paths through underdense regions, the observed average distances in our tests can be biased sufficiently far from the true average to cause a small but significant error in the measured ΩΛ . However, the shape of the distributions of distances is skewed in the opposite direction from more standard lensing analyses, indicating that the structures in our model are not sufficiently realistic to represent the real universe. The overall implications of our tests are that inhomogeneities on the largest observed scales likely do not systematically bias average distance observations a great deal, but there is a chance that analyses ignoring these effects could arrive at slightly inaccurate results. A more conclusive result would require additional tools to combine effects of structures on different scales. 106

If we allow some violation of the Copernican principle, parameters could still be tweaked in models containing large-scale, but low-amplitude inhomogeneities, with a cosmological constant. This approach is essentially a compromise between the standard model and typical giant void models; rather than replacing Λ, the inhomogeneity merely augments it at a level low enough to evade observational contradictions. This kind of model is more difficult to disprove, but Occam’s razor would favor the simpler standard model, and eliminating dark energy was one of the primary motivations of void models in the first place. Nevertheless, we cannot discount the possibility, and it may turn out that such models fit observations better than the standard model.

6.2

Observational tests

In order to determine whether or not (and to what extent) the universe we are living in is inhomogeneous, we need observational tests that can distinguish between homogeneous and inhomogeneous models. Some work has already been done in this direction. Galaxy surveys such as the WiggleZ survey have confirmed that the distribution of matter is consistent with a homogeneity scale on the order of 100 Mpc [97]. There are, however, other studies that have found signs of a large local underdensity extending as far as 300 Mpc, which may explain the tension between local H0 measurements and the H0 inferred from the CMB [98–101]. In any case, most of these studies only probe our past light-cone. Their consistency with homogeneity is a necessary condition for the standard model, but not a sufficient proof. Extrapolating these observations to draw conclusions about the entire spacetime relies on the Copernican principle, which we cannot take for granted if we wish to rigorously rule out or confirm alternative models. True tests of the Copernican principle will rely on off-light-cone observations. One such observation is the kinetic Sunyaev-Zel’dovich effect, which, as mentioned previously, already puts tight constraints on void models. The SZ effect comes from the scattering of CMB photons off of gas in distant clusters, allowing them to effectively act as mirrors, showing us information about the CMB on their past light cones, which lie inside our own. The thermal version of the effect tells us the monopole seen by the cluster, while the kinetic version tells us the dipole. 107

The SZ effect also applies to CMB polarization, providing additional information, including the cluster’s observed CMB quadrupole and octupole [102]. Thus, any detection of a large SZ signal beyond the levels explainable by perturbations would indicate a departure from homogeneity. Our studies in chapters 2 and 3 can be used as guides to connect inhomogeneities with signals; we have seen that the low multipoles can all grow large in the presence of a large inhomogeneity. More precise measurements of CMB spectra and polarizations are therefore valuable for establishing or refuting homogeneity. Further information can be gained from long-term observations of galaxies. In Szekeres models, distant sources would appear to drift across the sky as the light beams are bent by the nonsymmetric expansion [103]. Redshifts also change over time in a model-dependent way [104–106]. Both of these effects compare observations on two different past light-cones, giving us an extra dimension in our observations, however thin. Observations over periods of ten years could allow us to distinguish between models. Close study of distant galaxies gives us off-light-cone information as well. As we look back along our past light-cone, we see galaxies in earlier stages of evolution. If the universe is truly homogeneous, different regions of space should host galaxies in the same stages of development (statistically speaking) at the same time. If we are able to model star formation rates and stellar evolution over time with enough consistency, we can use galaxy “fossil records,” combined with lookback times calculated from the radial BAO scale, to compare galaxies at different points inside our past light-cone [107]. Simply collecting more data in more traditional areas will also be helpful. The more supernovae and other standard candles we observe, the tighter constraints we can cast on the Hubble diagram and anisotropies within it. The more galaxies we catalog, the stronger we can support or challenge the statistical homogeneity scale. In the end, the only way to truly understand the universe is to keep looking.

108

6.3

Future lines of research

There are still more observations to test. We did not examine the CMB power spectrum, in which some have reported hemispherical asymmetries [9,10]. Asymmetric inhomogeneous models could possibly be useful in explaining this. We also have not examined time drifts in our models. Our models could also be measured against galaxy age data and H(z) data, but one must be careful to avoid model dependence in the interpretation of these observations. There are more models to test as well. Some are radical departures from standard cosmology, such as the “giant hump” models proposed by Célérier [21], which matches distance data without a giant void by allowing a much greater variation in the bang-time than we have considered. Others are more subtle, keeping the cosmological constant from the concordance model and only using inhomogeneities to modify the details of the standard picture. Considering the difficulties giant void models have faced, the latter should be a greater focus, as any Hubble-scale inhomogeneity of sufficient magnitude to mimic dark energy would have effects on a multitude of other observables as well. A perturbation theory for Szekeres models would open up further tests. For instance, knowing how structure formation on comparatively smaller scales is affected by the larger-scale inhomogeneities would allow us to compare galaxy surveys with Szekeres models. Being able to calculate matter power spectrums within Szekeres models would also allow for deeper study of the kSZ power spectrum on smaller scales [48]. This is no easy task, though—even LTB perturbation theory is very complex due to the mixing of scalar, vector, and tensor modes [37]. The lack of symmetry in Szekeres models means that all four dimensions must be treated as independent, There may be more practical applications for Szekeres models on smaller scales. Lensing models typically work by approximations, which become insufficient as our data become more precise. Szekeres models would allow the simulation of light beams passing through lensing structures with the full effects of general relativity. Fanizza has already done this using LTB models [108], but Szekeres models would allow simulations of more diverse, non-spherical structures.

109

6.4

Closing remarks

Even if inhomogeneous models ultimately fail to measure up to the cosmological data, this was still a necessary path to explore. The only way to be sure of our standard model is to rule out the alternatives. In the process, we also gain insight into the inner workings of general relativity—the interplay between energy, spacetime curvature, and observations. This leaves us better equipped to face future challenges, whether on universe scales or the scale of a lensing cluster. The universe is a vast, strange place, and it has surprised us before. It may still have a surprise or two in store for us yet.

110

Appendix A: MATHEMATICAL CONVENTIONS AND NOTATIONS f˙ ≡ ∂f /∂t

f 0 ≡ ∂f /∂r

We use a metric signature of (−1, 1, 1, 1). Greek indices (α, β, µ, ν, etc.) cover all four spacetime coordinates, while Latin indices (i, j, k, etc.) represent only spatial coordinates. We follow the standard index summation convention, e.g. vα v α ≡

P3

α=0

vα v α , with some

exceptions explained in chapter 5. Commas followed by indices or coordinates denote partial derivatives, i.e. f,α ≡ ∂f /∂xα , or f,r ≡ ∂f /∂r. Semicolons denote covariant derivatives in a similar manner. Parentheses surrounding pairs of indices denote symmetric commutation: T(αβ) ≡ 21 (Tαβ + Tβα ). In a similar manner, square brackets denote anti-symmetric commutation: T[αβ] ≡ 21 (Tαβ − Tβα ).

111

Appendix B: DEFINITIONS Affine parameter: a parameter along a geodesic defined so that parallel transport preserves the tangent vector.

Baryon acoustic oscillations (BAO): fluctuations in baryonic matter caused by pressure waves in the early universe that were “frozen in” when photons decoupled from baryons at recombination.

Compton y distortion: a form of spectral distortion resulting from CMB photons Compton scattering off hot electrons after reionization, shifting low-frequency photons to higher frequencies.

Distance modulus (µ): µ = 5 log10 DL + 25.

Einstein-de Sitter (EdS) universe: a flat FLRW model with matter as the only energy component.

Friedmann-Lemaître-Robertson-Walker (FLRW) model: a homogeneous and isotropic universe model.

Integrated Sachs-Wolfe (ISW) effect: the change in a photon’s energy due to time variation of gravitational potentials. The linear effect vanishes in EdS universes because perturbations grow at the same rate as cosmic expansion, but it does not vanish in ΛCDM models because the cosmological constant increases the growth rate of cosmic expansion.

Kinetic Sunyaev-Zeldovich (kSZ) effect: a spectral distortion in the CMB caused by dust scat-

112

tering the CMB dipole.

Lemaître-Tolman-Bondi (LTB) model: a spherically symmetric inhomogeneous dust model.

Milne model: an FLRW model with no energy or pressure. This means the curvature is maximally negative, and the scale factor evolves as simply a(t) = H0 t.

Rees-Sciama (RS) effect: the change in a photon’s energy as it crosses nonlinear structures due to the change in the gravitational potential during transit. Because it is a nonlinear effect, it can occur in EdS universes, in contrast to the linear ISW effect, but it is expected to be subdominant to linear ISW in ΛCDM models.

Ricci scalar (R): the trace of the Ricci tensor: R = Rµµ .

Ricci tensor (Rµν): a curvature tensor describing the growth rates of volumes in a manifold. Deσ fined as the trace of the Riemann tensor: Rµν = Rµσν .

α Riemann tensor (Rβµν ): a tensor describing the complete spacetime curvature. Can be expressed α in terms of the Christoffel symbols as Rβµν = Γανβ,µ − Γαµβ,ν + Γαµσ Γσνβ − Γανσ Γσµβ .

Weyl tensor (Cαβµν ): a curvature tensor describing the changes to the shape of a body due to tidal forces. Defined in terms of the Riemann and Ricci tensors and Ricci scalar as

Cαβµν = Rαβµν −


2 2 gα[µ Rν]β − gβ[µ Rν]α − R gα[µ gν]β , n−2 (n − 1)(n − 2

where n is the number of dimensions.

113

Szekeres model: a nonsymmetric inhomogeneous dust model.

Szekeres functions: the functions P (r), Q(r), and S(r), which determine the shell shifting and density asymmetries.

114

BIBLIOGRAPHY

[1] T. M. Davis, Gen. Rel. Grav. 46, 1731 (2008) [arXiv:1404.7266]. [2] E. Komatsu et al., ApJS, 192, 18 (2011) [arXiv:1001.4538]. [3] A. G. Riess et al., Astron. J. 116, 1009 (1998) [arXiv:astro-ph/9805201]. [4] S. Perlmutter et al., Astron. J. 517, 565 (1999) [arXiv:astro-ph/9812133]. [5] J. Martin, Compets Rendus Physique 13, 566 (2012) [arXiv:1205.3365]. [6] I. Zlatev, L. Wang, and P. S. Steinhardt, Phys. Rev. Lett 82, 896 (1999). [7] M. Tegmark, A. de Oliveira-Costa, and A. J. S. Hamilton, Phys. Rev. D 68, 123523 (2003). [8] M. Cruz, M. Tucci, E. Martinez-Gonzales, and P. Vielva, MNRAS 369, 57 (2006). [9] P. K. Aluri and P. Jain, MNRAS 419, 3378 (2012) [arXiv:1108.5894]. [10] J. W. Moffat, JCAP 10, 012 (2005). [11] A. Kashlinsky, F. Atrio-Barandela, D. Kocevski, and H. Ebeling, ApJ 686, 49 (2008) [arXiv:0809.3734]. [12] F. Atrio-Barandela (2013) [arXiv:1303.6614]. [13] R. G. Clowes et al., MNRAS 429, 2910 (2012) [arXiv:1211.6256]. [14] L. Perivolaropoulos, Galaxies 2, 22 (2014) [arXiv:1401.5044]. [15] L. Perivolaropoulos (2011) [arXiv:1104.0539]. [16] T. Buchert, Gen. Rel. Grav. 40, 467 (2008) [arXiv:0707.2153]. [17] D. L. Wiltshire, New J. Phys. 9, 377 (2007) [arXiv:gr-qc/0702082]. [18] G. Lemaître, Ann. Soc. Sci. Bruxelles A 53, 51 (1933). 115

[19] R. C. Tolman, Proc, Nat. Acad. Sci. USA 20, 169 (1934). [20] H. Bondi, MNRAS 107, 410 (1947). [21] M.-N. Célérier, K. Bolejko, and A. Krasi´nski, Astron. Astrophys. 518, A21 (2010) [arXiv:0906.0905]. [22] P. Szekeres, Phys. Rev. D 12, 2941 (1975); Commun. Math. Phys. 41, 55 (1975). [23] A. Friedmann, Z. Phys. 10, 377 (1922). [24] P. de Bernardis et al., Nature, 404, 955 (2000) [arXiv:astro-ph/0004404]. [25] A. Melchiorri et al., ApJ Lett., 536, L63 (2000) [arXiv:astro-ph/9911445]. [26] S. Hanany et al., ApJ Lett., 545, L5 (2000) [arXiv:astro-ph/0005123]. [27] A. Ishibashi and R. M. Wald, Class. Quant. Grav. 23, 235 (2006) [arXiv:gr-qc/0509108]. [28] C. Clarkson and O. Umeh, Class. Quant. Grav. 28, 164010 (2006) [arXiv:1105.1886]. [29] A. Krasi´nski, Acta Physica Polonica B42, 2263 (2011) [arXiv:1110.1828]. [30] R. K. Sachs, Proc. Roy. Soc. London A 264, 309 (1961). [31] K. Enqvist and T. Mattsson, JCAP 04, 003 (2008) [arXiv:astro-ph/0609120]. [32] J. Garcia-Bellido and T. Haugbølle, JCAP 02, 019 (2007) [arXiv:0802.1523]. [33] M. Zumalacarregui, J. Garcia-Bellido, P. Ruiz-Lapuente, JCAP 10, 009 (2012) [arXiv:1201.2790]. [34] V. Marra, E. W. Kolb, S. Matarrese, and A. Riotto, Phys. Rev. D 76, 123004 (2007) [arXiv:0708.3622]. [35] S. J. Szybka, Phys. Rev. D 84, 044011 (2011) [arXiv:1012.5239].

116

[36] J. P. Zibin, Phys. Rev. D 78, 043504 (2008) [arXiv:0804.1787]. [37] C. Clarkson, T. Clifton, and S. February, JCAP 06, 025 (2009) [arXiv:0903.5040]. [38] A. Krasi´nski and C. Hellaby, Phys. Rev. D 65, 023501 (2002) [arXiv:gr-qc/0106096]. [39] R. W. Stoeger, G. F. R. Ellis, and S. D. Nel, Class. Quant. Grav. 9, 509 (1992). [40] N. Mustapha, C. Hellaby, and G. F. R. Ellis, MNRAS 292, 817 (1997) [arXiv:gr-qc/9808079]. [41] M. E. Araújo and W. R. Stoeger, Phys. Rev. D 82, 123513 (2010) [arXiv:1011.6110]. [42] K. Bolejko, C. Hellaby, and A. H. A. Alfedeel, JCAP 09, 011 (2011) [arXiv:1102.3370]. [43] A. Moss, J. P. Zibin, D. Scott, Phys. Rev. D 83, 103515 (2011) [arXiv:1007.3725]. [44] S. Foreman, A. Moss, J. P. Zibin, and D. Scott, Phys. Rev. D 82, 103532 (2010) [arXiv:1009.0273]. [45] J. Garcia-Bellido and T. Haugbølle, JCAP 09, 016 (2008) [arXiv:0807.1326]. [46] H. Alnes, M. Amarzguioui, and Ø. Grøn, Phys. Rev. D 73, 083519 (2006) [arXiv:astroph/0512006]. [47] H. Alnes and M. Amarzguioui, Phys. Rev. D 74, 103520 (2006) [arXiv:astro-ph/0607334]. [48] J. P. Zibin and A. Moss, Class. Quant. Grav. 28, 164005 (2011) [arXiv:1105.0909]. [49] P. Zhang and A. Stebbins, Phys. Rev. Lett. 107, 041301 (2011) [arXiv:1009.3967]. [50] C.-M. Yoo, K.-i. Nakao, and M. Sasaki, JCAP 10, 011 (2010) [arXiv:1008.0469]. [51] C. L. Bennett et al., ApJS 192, 17 (2011). [arXiv:1001.4758]. [52] Planck Collaboration: P. A. R. Ade et al. (2013) [arXiv:1303.5083]. [53] M.-N. Célérier, Astronom. Astrophys. 543, A71 (2012) [arXiv:1108.1373]. 117

[54] W. B. Bonnor, A. H. Sulaiman, and N. Tomimura, Gen. Relativ. Gravit. 8, 549 (1977). [55] M. Ishak and A. Peel, Phys. Rev. D 85, 083502 (2012) [arXiv:1104.2590]. [56] A. Nwankwo, M. Ishak, and J. Thompson, JCAP 05, 028 (2011) [arXiv:1005.2989]. [57] Ya. B. Zel’dovich, Soviet Astr. AJ 8, 13 (1964). [58] C. C. Dyer and R. C. Roeder, ApJ 174, L115 (1972). [59] M. Ishak et al., Phys. Rev. D 78, 123531 (2008) [arXiv:0708.2948]. [60] K. Bolejko and R. A. Sussman, Phys. Lett. B 4, 265 (2010) [arXiv:1008.3420]. [61] G. Hinshaw et al., Astrophys. J. Suppl. 180, 225 (2009). [62] Planck Collaboration: P. A. R. Ade et al. (2013) [arXiv:1303.5076]. [63] T. Biswas, A. Notari, and W. Valkenburg, JCAP 11, 030 (2010) [arXiv:1007.3065]. [64] J. Grande and L. Perivolaropoulos, Phys. Rev. D 84, 023514 (2011) [arXiv:1103.4143]. [65] C. L. Bennett et al., ApJ 464, L1 (1996) [arXiv:astro-ph/9601067]. [66] M. Redlich et al., AA 570, A63 (2014) [arXiv:1408.1872]. [67] N. Hamaus, P. M. Sutter, and D. Wandelt, Phys. Rev. Lett. 112, 251302 (2014) [arXiv:1403.5499]. [68] S. Gupta, T. D. Saini, and T. Laskar, Mon. Not. R. Astron. Soc. 388, 242 (2008). [69] A. Mariano and L. Perivolaropoulos, Phys. Rev. D 86, 083517 (2012) [arXiv:1206.4055]. [70] R.-G. Cai and Z.-L. Tuo (2011) [arXiv:1109.0941]. [71] R.-G. Cai et al., Phys. Rev. D 87, 123522 (2013) [arXiv:1303.0961]. [72] P. Bull, T. Clifton, and P. G. Ferreira, Phys. Rev. D 85, 024002 (2012) [arXiv:1108.2222]. 118

[73] K. Iwata, C.-M. Yoo (2014) [arXiv:1409.3297]. [74] T. Mattsson, Gen. Rel. Grav. 42, 567 (2010) [arXiv:0711.4264]. [75] N. Bretón and A. Montiel, Phys. Rev. D 87, 063527 (2013) [arXiv:1303.1574]. [76] V. C. Busti, J. A. S. Lima, MNRAS: Letters 426, L41 (2012) [arXiv:1204.1083]. [77] J. A. S. Lima, V. C. Busti, and R. C. Santos (2013) [arXiv:1301.5360]. [78] C. Clarkson et al., MNRAS 426, 1121 (2012) [arXiv:1109.2484]. [79] K. Bolejko, MNRAS 412, 1937 (2011) [arXiv:1011.3876]. [80] K. Bolejko, A. Krasi´nski, C. Hellaby, and M.-N. Célérier, Structures in the Universe by Exact Methods: Formation, Evolution, Interactions, Cambridge U P (2009). [81] A. Einstein and E. G. Straus, Reviews of Modern Physics 17, 120 (1945). [82] P. Fleury, JCAP 06, 054 (2014) [arXiv:1402.3123]. [83] A. Peel, M. A. Troxel, and M. Ishak (2014) [arXiv:1408.4390]. [84] K. Bolejko and M.-N. Célérier, Phys. Rev. D 82, 103510 (2010) [arXiv:1005.2584]. [85] K. Bolejko, Gen. Rel. Grav. 41, 1737 (2009) [arXiv:0804.1846]. [86] J. P. Bernstein et al., ApJ 753, 152 (2012). [87] D. E. Holz and E. V. Linder, ApJ 631, 678 (2005). [88] D. Sarkar, A. Amblard, D. E. Holz, and A. Cooray, ApJ 678, 1 (2008) [arXiv:0710.4143]. [89] Wambsganss, et al., ApJ 475 L81 (1997) [arXiv:astro-ph/9607084]. [90] R. Takahashi, M. Oguri, M. Sato, and T. Hamana, ApJ 742, 15 (2011) [arXiv:1106.3823]. [91] P. A. R. Ade et al., Phys. Rev. Lett. 112, 241101 (2014) [arXiv:1403.3985]. 119

[92] V. Pelgrims and J.-R. Cudell, MNRAS 442, 1239 (2014) [arXiv:1402.4313]. [93] P. Ciarcelluti, Mod. Phys. Lett. A 27, 1250221 (2012) [arXiv:1201.6096]. [94] A. K. Yadav, Rom. J. Phys. 56, 609 (2011) [arXiv:1005.0537]. [95] C. C. Dyer and E. G. Shaver, ApJ 390, L5 (1992). [96] V. Faraoni, Astron. Astrophys. 272, 385 (1993) [arXiv:astro-ph/9211012]. [97] M. Scrimgeour et al., MNRAS 425, 116 (2012) [arXiv:1205.6812]. [98] R. C. Keenan et al., Astrophys. J. Suppl. 186, 94 (2010) [arXiv:0912.3090]. [99] R. C. Keenan et al., ApJ 754, 131 (2012) [arXiv:1207.1588]. [100] R. C. Keenan, A. J. Barger, and L. L. Cowie, ApJ 775, 62 (2013) [arXiv:1304.2884]. [101] J. R. Whitbourn and T. Shanks (2013) [arXiv:1307.4405]. [102] R. Maartens, Phil. Trans. R. Soc. A 369, 5115 (2011) [arXiv:1104.1300]. [103] A. Krasi´nski and K. Bolejko, (2013) [arXiv:1212.4697]. [104] P. Mishra, M.-N. Célérier, and T. P. Singh, Phys. Rev. D 86, 083520 (2012) [arXiv:1206.6026]. [105] C.-M. Yoo, T. Kai, K.-i. Nakao, Phys. Rev. D 83, 043527 (2011) [arXiv:1010.0091]. [106] J.-P. Uzan, C. Clarkson, and G. F. R. Ellis, Phys. Rev. Lett. 100, 191303 (2008) [arXiv:0801.0068]. [107] A. F. Heavens, R. Jiminez, and R. Maartens, JCAP 09, 035 (2011) [arXiv:1107.5910]. [108] G. Fanizza and F. Nugier (2014) [arXiv:1408.1604]. [109] S. R. Green and R. M. Wald (2013) [arXiv:1304.2318]. 120

[110] D. Larson et al., Astrophys. J. Suppl. 192, 16 (2011) [arXiv:1001.4635]. [111] W. Percival et al., MNRAS 401, 2148 (2010). [112] R. Durrer, Phil. Trans. R. Soc. A 369, 1 (2011) [arXiv:1103.5331]. [113] M. P. Mood, J. T. Firouzjaee, and R. Mansouri (2013) [arXiv:1304.5062]. [114] P. Peter (2013) [arXiv:1303.2509]. [115] A. Mariano and L. Perivolaropoulos, Phys. Rev. D 87, 043511 (2013) [arXiv:1211.5915]. [116] P. K. Rath, T. Mudholkar, P. Jain, P. K. Aluri, and S. Panda (2013) [arXiv:1302.2706]. [117] D. Araujo et al., ApJ 760, 145 (2012) [arXiv:1207.5034]. [118] H. Martel and P. Premadi, ApJ 673, 657 (2008) [arXiv:0710.5452]. [119] M. Smith, et al., ApJ 780, 24 (2014) [arXiv:1307.2566]. [120] S. Weinberg, Cosmology (Oxford University Press, New York, 2008).

121

VITA

Robert Buckley was born in Fort Lewis, Washington on January 22, 1985. He graduated as valedictorian from Sandra Day O’Connor High School in San Antonio, Texas in 2003. From there, he enrolled at the University of Texas at San Antonio as a physics major, and received his Bachelor’s degree with honors in 2007. Partly due to physical disabilities, he remained in San Antonio to pursue his graduate studies, still at the University of Texas at San Antonio. He has dabbled in a few different areas of physics—his undergraduate Honors thesis was on the Brownian dynamics of a rigid rod, a problem relevant to molecular biophysics—but he was ultimately drawn to astrophysics and cosmology, which had interested him from an early age.