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Modified Gravity and Cosmology on the Largest Scales Modified gravity and cosmology on the largest scales Alexandra Terrana A Dissertation submitted to the Faculty of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate program in Physics and Astronomy York University Toronto, Ontario May 2017 © Alexandra Terrana 2017 Abstract This dissertation presents a series of my contributions to research in theoretical cosmology, focusing on aspects of the very large scale universe, particularly dark energy, cosmic acceleration, modified gravity, and cosmic variance. Following an overview of the current understanding of the standard cosmological model in chapter 1, three pertinent topics are discussed in detail. A common theme among all chapters is the desire to explain the properties of the universe on the largest scales. One of the biggest mysteries on large scales is the need for dark energy to explain the observed accelerated expansion of the late universe. The unsatisfying explanation offered by the standard cosmological model and the associated enormous fine tuning problem have driven considerable interest in infrared (long-distance) modifications of general relativity. In this work, we consider a particularly well motivated modified theory, massive gravity, in which the modification is to simply assume that the particle mediating the gravitational force has a non-zero mass. For a mass on the order of the Hubble constant, this theory offers an alternative explanation of the accelerated cosmic expansion. Chapter 2 lays the theoretical groundwork for massive gravity, summarizing its history and formalism. A fundamental challenge for any modified gravity theory is sequestering the modification to large enough distance scales, so that the predictions match general relativity on solar system scales where it has been tested to high precision. Chapter 3 provides a detailed analysis of massive gravity's ability to screen its extra degrees of freedom, allowing for continuity with general relativity on short distance scales. Further, in chapter 4, we explore the cosmological production and propagation of gravitational waves in an extension of massive gravity, bigravity, determining whether there may be any testable deviations from general relativity. Understanding these predictions is crucial, as there is now a vigorous observational program to probe possible deviations from our standard model. As rapid progress in observational cosmology unfolds, not only is it paramount to construct viable modified gravity theories to test against general relativity, it is necessary to explore which observational methods will be the most powerful for constraining them. This dissertation contains progress on both of these fronts: analyzing potential modified gravity theories, and analyzing potential novel observational probes of the large scale universe. Chapter 5 provides the theoretical framework for one such novel probe, the large scale kinetic Sunyaev-Zeldovich effect. This effect is particularly intriguing because of its ability to overcome cosmic variance, and thus help us unlock the secrets of the universe on the largest scales. ii Acknowledgments I am most thankful for the guidance and support provided by my supervisor, Matthew Johnson. I have learned so much from him. He has been a mentor and role model with respect to how to be a good researcher, but even more so, how to be a good person. I sincerely appreciate all of his advice concerning research and careers, and all he has done to make my experience here stimulating and enjoyable. It was an honor to be his first Ph.D. student! It has been a privilege to discuss and work with many outstanding physics faculty and students at York University and Perimeter Institute. I am truly grateful for productive collaborations with Richard Brito, Vitor Cardoso , and most recently, Anne-Sylvie Deutsch and Moritz Munchmeyer. I must also thank my wonderful family and friends for continual encouragement throughout this pursuit. I am especially thankful to Fil for emotional support, help with all computer related issues, and inspirational work ethic. I gratefully acknowledge the funding sources that made my doctoral work possible. I was supported by the Ontario Graduate Scholarship for the first year and the Vanier Canada Graduate Scholarship for the last three years. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Min- istry of Research, Innovation and Science. Some results presented here were obtained us- ing the Healpix package (https://sourceforge.net/projects/healpix/), the Cosmicpy package (http://cosmicpy.github.io/), and the LAMBDA web-based CAMB interface (https://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm). iii Table of Contents Abstract . ii Acknowledgments . iii Table of Contents . iv List of Tables . viii List of Figures . ix Citations to Published Work . xii 1 Our universe 1 1.1 Previewing the Cosmological Model . .2 1.2 General Relativity . .4 1.3 Standard Cosmology . .8 1.3.1 The expanding universe with FRW . .8 1.3.2 Inflation . 16 1.3.3 Cosmological perturbation theory . 22 1.4 The dark side of the universe . 25 1.4.1 Dark matter . 25 1.4.2 Dark energy . 26 1.5 Modified Gravity . 31 1.6 Cosmic Microwave Background . 34 2 Massive gravity history and formalism 42 iv 2.1 From linear to non-linear massive gravity . 43 2.2 Development of dRGT Massive Gravity . 45 2.2.1 The dRGT Action . 46 2.2.2 The Decoupling Limit . 47 2.2.3 Galileons . 50 2.3 The DGP Model . 51 2.4 Bigravity . 54 2.4.1 Bigravity cosmology . 56 3 Vainshtein screening in massive gravity 59 3.1 Introduction . 60 3.2 Vainshtein mechanism . 62 3.3 Screening in the Decoupling Limit of DGP . 64 3.3.1 DGP equations of motion . 64 3.3.2 Dynamical stability analysis . 66 3.3.3 Quasinormal modes and tails of screening solutions . 70 3.4 Cauchy Breakdown . 75 3.4.1 Coordinate invariance of Cauchy breakdown . 79 3.5 Collapsing and Exploding Sources . 80 3.6 Asymmetric screening solutions . 83 3.7 Screening in the Decoupling Limit of dRGT . 86 3.7.1 dRGT equations of motion . 86 3.7.2 Dynamical stability analysis . 89 3.7.3 Collapsing and Exploding Sources . 91 3.7.4 Asymmetric screening solutions . 92 3.8 Conclusions . 93 v 4 Gravitational waves in massive bigravity 96 4.1 Introduction . 97 4.2 Bigravity Tensor Perturbations . 98 4.3 CMB Tensor Power Spectrum . 103 4.4 Present Day Stochastic Gravitational Wave Background . 104 4.5 Initial Conditions . 106 4.5.1 An inflationary model in bigravity . 110 4.6 Conclusion . 111 5 Reducing cosmic variance with the large-scale kSZ effect 114 5.1 Introduction . 115 5.2 The large-scale kSZ effect . 117 5.2.1 Fourier kernel for the effective velocity . 119 5.2.2 Angular decomposition of the effective velocity . 121 5.3 Simulations . 122 5.4 kSZ Tomography . 124 5.5 Cosmic Variance Limited Noise . 127 5.6 Cosmic Variance Limited Signal to Noise . 132 5.6.1 RMS Estimate . 133 5.6.2 Comparing with the signal from simulations . 134 5.7 Mode counting . 136 5.8 Detectability . 140 5.9 Discussion and Conclusions . 145 6 Conclusions 147 References 151 vi Appendices 172 A Convergence properties 172 B Cauchy problem and stability 173 C The effective velocity in Fourier space 176 C.1 Sachs-Wolfe . 176 C.2 Doppler . 177 C.3 Integrated Sachs-Wolfe . 178 C.4 Effective velocity . 178 D A pure gradient is pure gauge 180 E Cancellation of the kernel contributions as k ! 0 for ΛCDM without radia- tion 182 F Random Gaussian fields 184 vii List of Tables 5.1 Large-scale kSZ mode counting . 139 5.2 Experimental requirements for the lower resolution scenarios of the kSZ effect . 144 viii List of Figures 1.1 The evolution of the density parameter for matter, radiation, and a cosmological constant as a function of scale factor . 27 1.2 The CMB temperature map . 35 1.3 The CMB temperature power spectrum from Planck . 37 2.1 The evolution of the ratio of scale factors for the expanding branch and bouncing branch of bigravity . 58 3.1 Static spherically symmetric screening solutions in DGP . 65 3.2 Sound speed profiles of fluctuations on top of the DGP screening solutions . 67 3.3 The evolution of the DGP screening solution with a gaussian perturbation . 68 3.4 The evolution of the DGP screening solution with vacuum initial conditions . 69 3.5 Waveform for the time evolution of gaussian wavepacket . 70 3.6 Effective potential of the screening solutions in DGP . 72 3.7 Fundamental quasinormal modes of the scalar field in the positive DGP branch 73 3.8 The evolution of Ztt to Cauchy breakdown in DGP . 76 3.9 Regions of parameter space that lead to Cauchy breakdown in DGP . 77 3.10 The values of the ratio RV =R0 for which Cauchy breakdown occurs . 78 3.11 Evolution of a collapsing source toy model that leads to Cauchy breakdown . 82 3.12 Evolution of an exploding source toy model that evades Cauchy breakdown . 83 3.13 Representation of two localized bodies that are each axially symmetric . 84 3.14 Higher multipole moments of the positive branch DGP static screening solution 86 ix tt 3.15 The profile of the ZMG factor for dRGT massive gravity . 89 3.16 Fundamental quasinormal modes of the scalar field in massive gravity . 91 3.17 Higher multipole moments of the static screening solution in massive.
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