Astronomy Unit School of Physics and Astronomy Queen Mary University of London Two-parameter Perturbation Theory for Cosmologies with Non-linear Structure Sophia Rachel Goldberg Submitted in partial fulfillment of the requirements of the Degree of Doctor of Philosophy 1 Declaration I, Sophia Rachel Goldberg, 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 in- dicated. Previously published material is also acknowledged below. I attest that I have exercised reasonable care to ensure that the work is original, 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. Signature: Date: 19th November 2017 Details of collaboration and publications: the research in this thesis is based on the publications and collaborations listed below. `Cosmology on all scales: a two-parameter perturbation expansion' Sophia R. Goldberg, Timothy Clifton and Karim A. Malik Physical Review D 95, 043503 (2017) `Perturbation theory for cosmologies with nonlinear structure' Sophia R. Goldberg, Christopher S. Gallagher and Timothy Clifton Physical Review D 96, 103508 (2017) 2 Abstract We propose and construct a two-parameter expansion around a Friedmann-Lema^ıtre- Robertson-Walker geometry which uses both large-scale and small-scale perturba- tions analogous to cosmological perturbation theory and post-Newtonian gravity. We justify this observationally, derive a set of field equations valid on a fraction of the horizon size and perform a detailed investigation of the associated gauge problem. We find only the Newtonian gauge, out of the standard gauges used in cosmological perturbation theory, is applicable to post-Newtonian perturbations; we can identify a consistent set of perturbed quantities in the matter and gravity sec- tors and construct corresponding gauge-invariant quantities. The field equations, written in terms of these quantities, takes on a simpler form, and allows the effects of small-scale structure on the large-scale properties of the Universe to be clearly identified and discussed for different physical scenarios. With a definition of statis- tical homogeneity, we find that the cosmological constant and the average energy density, of radiation and dust, source the Friedmann equation, whereas only the inhomogeneous part of the Newtonian energy density sources the Newton-Poisson equation { even though both originate from the same equation. There exists field equations at new orders in our formalism, such as a frame-dragging field equation a hundred times larger than expected from using cosmological perturbation theory alone. Moreover, we find non-linear gravity, mode-mixing and a mixing-of-scales at orders one would not expect from intuition based on cosmological perturbation theory. By recasting the field equations as an effective fluid we observe that these non-linearities lead to, for example, a large-scale effective pressure and anisotropic stress. We expect our formalism to be useful for accurately modelling our Universe, and for investigating the effects of non-linear gravity in the era of ultra-large-scale surveys. 3 Acknowledgements First and foremost, I would like to thank my supervisors Karim Malik and Tim- othy Clifton: your guidance and support have helped me not only complete my thesis, but enjoy it too. I also thank my collaborator Christopher Gallagher: it has been a pleasure. When I broke the cosmos Pedro Carrilho would often be the first person I would turn to; thank you for being that person. Asmi and Serena, you have supported me in an uncountable number of ways, I am infinitely grateful. I am thankful for the support of all my colleagues and friends in the Physics and Mathematics departments; I will miss both Astronomy Unit lunches and room 502. I must express my very profound gratitude to my parents for providing me with unfailing support and continuous encouragement throughout my years of study. I also thank my sister, Gran, Annie, Viv, Phil, Bambie, Ginger, Hannah, Beth, Jessie, Jamie, Jeremie, Ian, Jessica, Imperial buddies, the Bennets and Bristolians, extended family and close friends: you are a smart and fun bunch who I am proud to be associated with. Thank you for reminding me of what is important. Finally, Andrew: I count myself very lucky to call you my partner in crime, thank you for keeping me grounded, making me laugh and bringing me joy literally every day. I dedicate this thesis to my sister, best friend and dæmon, Rebecca Goldberg. Your kindness, intelligence and love of life never ceases to comfort and inspire me. Forever enriching my life, you are never forgotten. This work was supported by the Science and Technology Facilities Council (STFC) grant number ST/K50225X/1. 4 Contents Abstract 3 Acknowledgements 4 List of Figures and Tables 8 1. Introduction 9 1.1. Notation . 18 1.2. Overview . 19 2. General Relativity 21 2.1. Introduction . 21 2.2. Differential geometry . 23 2.2.1. Metrics . 23 2.2.2. Geodesics and Christoffel symbols . 24 2.2.3. Covariant derivatives and curvature . 25 2.2.4. Lie derivatives . 26 2.3. Einstein's field equations . 26 2.4. Exact solutions . 27 2.4.1. Minkowski solution . 27 2.4.2. FLRW solution . 28 2.5. Introduction to perturbative solutions . 29 2.5.1. Limits of the field equations . 30 2.5.2. Defining perturbations . 31 3. Cosmological Perturbation Theory 33 3.1. Introduction . 33 3.2. The formalism . 33 3.2.1. An expansion in the wave-zone . 33 3.2.2. Defining perturbations . 34 3.2.3. Perturbed metric . 34 3.2.4. Perturbed matter sources . 36 3.2.5. Summary of book-keeping . 37 3.3. Gauges . 38 3.3.1. Gauge transformations . 39 3.3.2. Gauge invariant quantities . 41 3.3.3. Choice of gauge . 42 3.4. Dynamics . 43 3.4.1. The field equations . 43 5 Contents 6 3.4.2. Conservation equations . 44 4. Post-Newtonian gravity 46 4.1. Introduction . 46 4.2. Newtonian gravity . 47 4.2.1. Newton's laws . 47 4.2.2. Newtonian N-body simulations . 48 4.3. Post-Newtonian formalism . 49 4.3.1. A slow-motion expansion . 49 4.3.2. Defining perturbations . 50 4.3.3. Perturbed metric and matter sources . 51 4.3.4. Summary of book-keeping . 54 4.4. Post-Newtonian equations . 55 4.4.1. Field equations and standard post-Newtonian gauge . 55 4.4.2. Parameterised post-Newtonian gravity . 57 5. Two-parameter formalism 59 5.1. Introduction . 59 5.2. Formalism . 59 5.2.1. Defining perturbations . 59 5.2.2. Summary of book-keeping . 62 5.2.3. Matter perturbations . 65 5.2.4. Including radiation and a cosmological constant . 68 5.2.5. Derivatives . 72 5.2.6. Metric . 74 5.3. Observational justifications . 75 5.3.1. Post-Newtonian gravity . 76 5.3.2. Cosmological perturbation theory . 78 5.3.3. A realistic universe . 80 6. Einstein's field equations with a two-parameter expansion 82 6.1. Ricci and total energy-momentum tensors . 82 6.1.1. Ricci tensor . 82 6.1.2. Total energy-momentum tensor . 85 6.2. The field equations . 87 6.2.1. Background-order potentials . 88 6.2.2. Vector potentials . 90 6.2.3. Higher-order scalar potentials . 91 6.2.4. Tensor potentials . 93 7. Two-parameter gauge transformations 95 7.1. Two-parameter gauge transformations . 96 7.2. Transformation of the metric . 99 7.2.1. Transformation of metric components . 99 7.2.2. Transformation of irreducibly-decomposed potentials . 102 Contents 7 7.3. Transformation of matter sources . 107 7.3.1. Transformations of components . 108 7.3.2. Transformation of irreducibly-decomposed matter sources . 110 7.4. Allowed gauge choices . 112 7.5. Gauge invariant quantities . 115 7.5.1. Gauge-invariant metric perturbations . 116 7.5.2. Gauge invariant quantities from the matter sector . 120 8. Dynamics of gauge invariant quantities 123 8.1. Field equations . 123 8.1.1. Background-order potentials . 123 8.1.2. Vector potentials . 124 8.1.3. Higher-order scalar potentials . 124 8.1.4. Tensor potentials . 126 8.2. Discussion . 127 8.2.1. Large-scale limit: l ∼ η ...................... 127 8.2.2. Small-scale limit: l η ..................... 133 8.2.3. Other systems . 134 9. Effective fluid dynamics 136 9.1. Perturbations . 136 9.2. Effective field equations . 138 9.3. Discussion . 143 10.Conclusions and further work 146 Appendix A. Energy-momentum tensor for dust 150 Appendix B. The field equations for dust 152 B.1. Background-order potentials . 152 B.2. Vector potentials . 153 B.3. Higher-order scalar potentials . 153 B.4. Tensor potentials . 154 Appendix C. Two-parameter gauge transformations for dust 156 C.1. Transformation of the energy-momentum tensor for dust . 156 C.2. Transformation of irreducibly-decomposed sources for dust . 157 C.3. Gauge invariant quantities for dust . 159 Appendix D. Dynamics of gauge invariant quantities for dust 160 D.1. Background-order potentials . 160 D.2. Vector potentials . 161 D.3. Higher-order scalar potentials . 161 D.4. Tensor potentials . 162 Bibliography 163 List of Figures and Tables Figures 5.1.