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Institute of Cosmology and Gravitation University of Portsmouth Institute of Cosmology and Gravitation University of Portsmouth Cosmological Tests of Gravity Ph.D. Thesis Supervisors Candidate Prof. Kazuya Koyama Benjamin Bose Prof. Gong-Bo Zhao arXiv:1802.00989v1 [astro-ph.CO] 3 Feb 2018 THIS THESIS IS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY BY THE UNIVERSITY OF PORTSMOUTH February 6, 2018 1 Declaration and Dissemination Whilst registered as a candidate for the above degree, I have not been registered for any other research award. The results and conclusions embodied in this thesis are the work of the named candidate and have not been submitted for any other academic award. This thesis is based on the published works: 1. B. Bose and K. Koyama. A Perturbative Approach to the Redshift Space Power Spectrum: Beyond the Standard Model. JCAP, 1608, 032 (2016), arXiv:1606.02520 [astro-ph.CO]. 2. B. Bose and K. Koyama. A Perturbative Approach to the Redshift Space Correla- tion Function: Beyond the Standard Model. JCAP, 1608, 032 (2017), arXiv:1705.09181 [astro-ph.CO]. 3. B. Bose, K. Koyama, W. A. Hellwing, G.-B. Zhao and H. A. Winther. Theoretical Accuracy in Cosmological Growth Estimation. Phys.Rev. D96 no.2, 023519 (2017), arXiv:1702.02348 [astro-ph.CO]. Approximate Word Count : 31,644 c 2017 by Benjamin Bose. All rights reserved. The copyright of this thesis rests with the Author. Copies (by any means) either in full, or in extracts, may not be made without the prior written consent of the Author. 2 Abstract General relativity has proved itself to be an incredibly robust theory having passed many high precision solar system tests as well as being able to describe the evolution of the universe and its constituents. Its cosmological application necessarily requires large and mysterious energy components to match observations. In particular, the observation of cosmic acceleration forces general relativity to adopt a dominant dark energy component. This has prompted a flurry of investigation into alternatives to the concordance model of gravity which can offer self-acceleration. These alternatives must deal with strong priors coming from the local tests of gravity. This has lead to the development of theories that exhibit so called screening mechanisms which allow them to be observationally equivalent to general relativity at small scales. On the other hand, these theories generically predict distinguishing phenomena at larger scales such as an enhanced gravitational force. This makes galaxy clusters and the LSS of the universe a great testbed for modifications to gravity. In particular, the anisotropy of galaxy clustering in redshift space, which arises due to the peculiar velocities of galaxies, offers a promising means of testing gravity. These so called redshift space distortions involve the velocity components of galaxies which in turn strongly involves the gravitational force and the growth rate of structure. This links us up with the fundamental laws of nature at large scales. As astronomical surveys become more and more precise, our measurements of the growth of structure become ever more powerful to detect departures from general relativity. This is true only if our theoretical modelling is up to the challenge these surveys propose. Specifically, with upcoming large volume, spectroscopic surveys such as Euclid and DESI, statistical er- rors will become tiny leaving room for various theoretical biases to enter the game. One such bias is that from using general relativity as a standard in data comparisons. Put in other words, the next era of astronomy will put us in a position to move beyond consistency tests of general relativity. 3 In this thesis we develop a c++ code that numerically and consistently constructs LSS observables, accounting for the redshift space distortion phenomenon. By con- sistently we mean that this can be done for a large class of alternative theories of gravity and dark energy models. This is done using perturbation theory which treats over-densities and velocities as small perturbations upon a homogeneous and isotropic expanding background spacetime. The construction provides the first order contribu- tion in non-linear dynamics which gives a more accurate description of the observables, an ever growing necessity when looking to extract the most information in data com- parisons. We focus on the redshift space power spectrum and correlation function, two commonly used statistics that are measured from galaxy surveys. Specifically, we adopt the Taruya, Nishimichi and Saito redshift space power spectrum model and Gaussian streaming model redshift space correlation function, which both employ beyond-linear treatments of the redshift space distortions. The perturbative approach makes the pipeline ideal for statistical inference analyses that require very quick model computa- tions. We test this framework against suites of numerical simulations that by and large de- scribe the full dynamics of structure growth. Specifically, we make use of simulations within three models of gravity : general relativity, the 5-dimensional DGP brane world model and an f(R) model. The two latter models both exhibit screening which makes them viable contenders to the former. Comparing the perturbative approach to these simulations gives us a handle on its accuracy in modelling the dynamics of the pertur- bations. We also validate the pipelines' numerical predictions against well established analytic forms for DGP and general relativity. We find very good agreement in all comparisons and a significant improvement in accuracy over the linear treatment. This is followed by a comparison of different approaches to the redshift space cor- relation function, specifically the Gaussian streaming model and the Fourier transform of the Taruya, Nishimichi and Saito spectrum. We find that these two approaches are consistent to within a few percent at scales around the baryon acoustic oscillation. 4 Finally, we use dark matter simulation data to perform a test for bias introduced by incorrectly modelling gravity. This is done by using DGP simulation data in a Markov Chain Monte Carlo analysis for two different templates for the redshift space power spectrum. The first template employs general relativity to model the perturba- tions while the second models them within the DGP model. We fit next generation like survey error bars on the data to put the analysis in that context. It is then found that for upcoming surveys, using general relativity as a benchmark model may only safely be used in consistency tests of general relativity but not to infer constraints on alternative models. Contents 1 Introduction 24 1.1 General Relativity . 29 1.2 Standard Model of Cosmology . 32 1.2.1 The Metric, Hubble Parameter, Redshift and Distance . 33 1.2.2 Matter Content and Dynamics . 37 2 Perturbation Theory 44 2.1 The Evolution of Perturbations . 45 2.1.1 Conservation: The Vlasov Equation . 45 2.1.2 Evolution: The Continuity and Euler Equations . 47 2.2 Separability and the EdS Approximation . 51 2.3 Correlations in the Density and Velocity Fields . 55 2.4 Successes, Failures and Improvements . 62 2.4.1 Regularised Perturbation Theory . 66 3 Modelling Gravity 69 3.1 Modified Gravity . 72 3.2 Screening Mechanisms . 77 3.2.1 Self Interaction Screening . 79 3.2.2 Vainshtein Screening . 81 5 6 CONTENTS 3.3 Generalised Cosmological Perturbations . 83 4 Redshift Space Distortions 97 4.1 Fourier Space . 99 4.2 Configuration Space . 105 4.2.1 FT of TNS Spectrum . 106 4.2.2 Streaming Models . 106 4.2.3 Mean Infall Velocity u12(r) . 110 2 2 4.2.4 Velocity Dispersion σ12(r; µ ) . 111 5 Beyond the Standard Model: The Redshift Space Power Spectrum 115 5.1 Comparing to Analytic Forms . 116 5.2 Scale Dependent Perturbations in f(R) . 122 6 Beyond the Standard Model: The Redshift Space Correlation Func- tion 125 6.1 Real Space . 127 6.2 Redshift Space: TNS vs GSM . 129 7 Validating Theoretical Templates for Stage IV Surveys 135 7.1 Theory vs Simulations . 139 7.1.1 Real Space Comparisons: N-body . 139 7.1.2 Redshift Space Comparisons : MG-PICOLA . 145 7.2 Template Performance . 149 7.3 An Ideal Survey: SPT Mock Data . 157 8 Discussion and Conclusions 163 8.1 Summary of Results Presented . 164 8.1.1 Numerical Results for Power Spectrum in MG . 164 7 CONTENTS 8.1.2 Comparison of Correlation Function Modelling . 166 8.1.3 Relevance of MG-Modelling for Stage IV surveys . 168 8.2 The Path Underfoot and Ahead . 170 8.2.1 Tracer Bias . 171 8.2.2 Modifying the Dark Sector . 173 8.2.3 Extending Framework to Smaller Scales . 174 8.2.4 Higher Order Statistics . 177 A Numerical Algorithm 180 B Horndeski A,B, and C Terms 185 C Statistical Inference and MCMC 187 D N-Body Simulations and COLA 191 E Fourier Space Comparisons: nDGP 195 F Validating use of MG-PICOLA 197 G Covariance Between Spectra Multipoles 202 List of Tables 1 Physical Constants . 22 2 Abbreviations . 23 1.1 Common EOS Parameter Values . 39 1.2 Present Cosmological Parameters . 43 3.1 Dark Sector Evidence . 96 7.1 Summary of template performances at z = 0:5 for nDGP simulations where fiducial f = 0:783. 156 7.2 Summary of template performances for nDGP simulations at z = 1 where fiducial f = 0:909. 157 7.3 Summary of template performances for GR simulations at z = 1 where fiducial f = 0:859. 157 8.1 Summary of Chapter 5 Results. 165 8.2 Summary of Chapter 6 Results. 168 8 List of Figures 1.1 Hubble's original plot (left) of the recessional velocities (redshift of spec- tra measurements) of 24 nebulae and their physical distances inferred from Leavitt's cepheid period-luminosity relation, a recent version of which is shown on the right [1].
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