Astronomy Unit School of Physics and Astronomy Queen Mary University of London Post-Newtonian Gravity in Cosmology Viraj A A Sanghai Submitted in partial fulfillment of the requirements of the Degree of Doctor of Philosophy 1 Declaration I, Viraj Alias Adhir Prabhas Sanghai, confirm that the research included within this thesis is my own work or that where it has been carried out in collaboration with, or supported by others, that this is duly acknowledged below and my contribution indicated. Previously published material is also acknowledged below. I attest that I have exercised reasonable care to ensure that the work is origi- nal, and does not to the best of my knowledge break any UK law, infringe any third party's copyright or other Intellectual Property Right, or contain any confidential material. I accept that the College has the right to use plagiarism detection software to check the electronic version of the thesis. I confirm that this thesis has not been previously submitted for the award of a degree by this or any other university. The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author. Details of collaboration and publications: Part of this work is done in collabo- ration with Timothy Clifton and Pierre Fleury. I have made a major contribution to the original research presented in this thesis. It is based on the following papers that have been published or submitted for publication: Ray tracing and Hubble diagrams in post-Newtonian cosmology • V. A. A. Sanghai, P. Fleury and T. Clifton, JCAP 07 (2017) 028, Arxiv: 1705.02328 [astro-ph.CO] Parameterized post-Newtonian cosmology • V. A. A. Sanghai and T. Clifton, Class. Quantum. Grav. 34 (2017) 065003, Arxiv: 1610.08039 [gr-qc] 2 3 Cosmological backreaction in the presence of radiation and a cosmological constant • V. A. A. Sanghai and T. Clifton, Phys. Rev. D 94, 023505 (2016), Arxiv: 1604.06345 [gr-qc] Post-Newtonian cosmological modelling • V. A. A. Sanghai and T. Clifton, Phys. Rev. D 91, 103532 (2015), Erratum: Phys. Rev. D 93, 089903 (2016), Arxiv: 1503.08747 [gr-qc] Post-Newtonian cosmological models • V. A. A. Sanghai, Contribution to 14th Marcel Grossmann proceedings (2015), to be published in World Scientific proceedings, Arxiv: 1512.04824 [gr-qc] Signature: Viraj A A Sanghai Date: 12/08/2017 Abstract The post-Newtonian (PN) perturbative framework has been successful in understand- ing the slow-motion, weak field limit of Einstein's theory of gravity on solar system scales, and for isolated astrophysical systems. The parameterized post-Newtonian (PPN) formalism extended the PN framework to put very tight constraints on deviations from Einstein's theory on the aforementioned scales and systems. In this work, we extended and applied the post-Newtonian formalism to cosmological scales. We first used it to construct a cosmological model to understand the effect of regularly arranged point sources on the background expansion. Here we found that at higher orders we obtained a small radiation-like correction to the standard Friedmann-Lema^ıtre-Robertson-Walker (FLRW) equations, for a matter-dominated universe. This radiation-like correction was purely due to the inhomogeneity of our model, and the non-linearity of Einstein's field equations. We also extended the post-Newtonian formalism to include other forms of matter that are cosmologically relevant, such as radiation and a cosmological constant, and studied the non-linear effects they might have on the background expansion. Then we constructed an extension of the parameterized post-Newtonian formalism (PPN) to cosmological scales. We used it to parameterize the background expansion of the universe as well as first-order perturbations in cosmology, using four functions of time. In the future, this could allow us to put constraints on deviations from Einstein's theory of gravity on cosmological scales. We gave examples of how our parameterization would work for dark energy models and scalar-tensor and vector-tensor theories of gravity. In the final part of this work, we studied how light propagation behaves in an inhomogeneous post-Newtonian cosmology with matter and a cosmological constant. We used it to understand the effect that inhomogeneities would have on observables such as angular diameter distances as compared to those that are expected from a homogeneous and isotropic FLRW universe. 4 Acknowledgements I would like to begin by thanking my supervisor, Timothy Clifton, for his constant encouragement and support throughout my PhD. I would also like to thank my grandfather, Santosh Maskara, who sadly passed away during my PhD, but not before inspiring me to always pursue what I am passionate about. I also thank my parents, Prabhas and Aneetha Sanghai, my brother, Devesh, and my family in the UK and in India, without whom I wouldn't be where I am today. I thank Karim Malik, Pierre Fleury, David Mulryne, Richard Nelson, Juan Valiente Kroon, Phil Bull and Chris Clarkson for their guidance and support at various times during my PhD. I would especially like to thank Pierre Fleury for a fruitful collaboration. Finally I also thank my friends, especially Pedro Carrilho, John Ronayne, Shailee Imrith, Sophia Goldberg, Charalambos Pittordis, Zac Kenton, Mike Cole, Andries Waelkens, Sara Ricc`o, Gerben Oling, Dominic Dold, Patrick Mangat, Matt Mutter, Sanson Poon, Domenico Trotta, Clark Baker, Jorches Fuentes, Kit Gallagher, Louis Coates and Jessie Durk for many enjoyable work and non-work related conversations. This work was supported by the Science and Technology Facilities Council (STFC), grant number ST/K50225X/1. 5 6 To the loving memory of my grandfather, Santosh Maskara Contents Abstract4 Acknowledgements5 List of Figures 10 List of Tables 13 1. Introduction 14 1.0.1. Notations and Conventions . 18 2. Background 19 2.1. General Relativity and its Alternatives . 19 2.1.1. Test of the Weak Equivalence Principle . 20 2.1.2. Test of Local Position Invariance . 21 2.1.3. Test of Local Lorentz Invariance . 22 2.1.4. General Relativity . 22 2.1.5. Lovelock's Theorem . 23 2.1.6. Scalar-Tensor Theories of Gravity . 24 2.1.7. Vector-Tensor Theories of Gravity . 26 2.1.8. Birkhoff's Theorem . 28 2.1.9. Post-Newtonian Formalism . 29 2.1.10. Parameterized Post-Newtonian Formalism . 36 2.2. Standard Approaches in Cosmology . 41 2.2.1. Friedmann-Lema^ıtre-Robertson-Walker Background . 41 2.2.2. Cosmological Perturbations . 44 2.2.3. Newtonian N-body Simulations . 49 2.3. Inhomogeneous Cosmology . 50 2.3.1. The Cosmological Principle . 50 2.3.2. Backreaction Problem in Cosmology . 50 2.3.3. Inhomogeneous Models of Cosmology . 53 2.3.4. Optics in Inhomogeneous Cosmologies . 56 3. Post-Newtonian Cosmological Modeling 62 3.1. Introduction . 62 3.2. Applying the Post-Newtonian Formalism to Cosmology . 63 3.3. Building a Cosmology Using Junction Conditions . 65 3.4. Comparison with other Approaches . 70 7 Contents 8 3.5. Governing Equations . 71 3.5.1. Einstein's Field Equations . 72 3.5.2. Extrinsic Curvature Equations . 74 3.6. Cosmological Expansion . 76 3.6.1. Newtonian Order . 77 3.6.2. Post-Newtonian Order . 79 3.7. Post-Newtonian Cosmological Solutions . 84 3.7.1. The General Solution: An Application of the Green's Function Formalism............................. 85 3.7.2. A Special Case: Point Sources . 88 3.8. Relationship with FLRW models . 94 3.8.1. The Proper Length of Cell Edges . 94 3.8.2. Transformation to a Time-dependent Background . 96 3.9. Discussion . 100 4. Post-Newtonian Cosmological Model with Radiation and Λ 102 4.1. Introduction . 102 4.2. The Geometry of a Lattice Cell . 103 4.2.1. Post-Newtonian Expansion . 104 4.2.2. Matter Content . 105 4.2.3. Einstein's Field Equations . 106 4.3. Cosmological Expansion . 109 4.3.1. Newtonian Accuracy . 110 4.3.2. Post-Newtonian Accuracy . 111 4.4. Point Sources with Radiation, Spatial Curvature and Λ . 114 4.4.1. Solutions.............................. 115 4.4.2. Results............................... 117 4.5. Discussion . 125 5. Parameterized Post-Newtonian Cosmology 127 5.1. Introduction . 127 5.2. From Weak Fields to Cosmology . 129 5.2.1. Expanding and Non-Expanding Coordinate Systems . 129 5.2.2. Junction Conditions . 130 5.2.3. Emergent Cosmological Expansion . 132 5.3. An Extended PPN formalism . 134 5.3.1. Review of the Standard PPN formalism . 134 5.3.2. Additional Matter Content . 136 5.3.3. Additional Potentials . 137 5.3.4. Additional Time Dependence . 138 5.4. A Parameterized Approach to Cosmology . 139 5.4.1. Review of the Conservation Equations . 139 5.4.2. Background Expansion . 140 5.4.3. First-order Perturbations . 142 5.5. Worked Examples . 143 5.5.1. Dark Energy Models . 144 Contents 9 5.5.2. Scalar-Tensor Theories of Gravity . 146 5.5.3. Vector-Tensor Theories of Gravity . 150 5.6. Discussion . 157 6. Ray Tracing and Hubble Diagrams in Post-Newtonian Cosmology159 6.1. Introduction . 159 6.2. Method .................................. 160 6.2.1. A Universe in a Cell . 161 6.2.2. Initial Conditions . 164 6.2.3. Reflection of Light at the Cell's Boundary . 165 6.2.4. The Ray-Tracing Code . 167 6.3. Results . 169 6.3.1. Redshift . 170 6.3.2. Angular Distance . 172 6.3.3. HubbleDiagram ......................... 174 6.4. Comparisons . 176 6.4.1. Weinberg-Kibble-Lieu Theorem . 177 6.4.2. Comparison with Stochastic Lensing . 178 6.4.3. Comparison with Numerical Relativity . 181 6.5. Discussion . 184 7. Conclusions and Future Work 185 7.1. FutureWork................................ 187 A.