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Exact and Perturbed Friedmann-Lemaıtre Cosmologies

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Exact and Perturbed Friedmann-Lemaıtre Cosmologies Exact and Perturbed Friedmann-Lemaˆıtre Cosmologies by Paul Ullrich A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Mathematics in Applied Mathematics Waterloo, Ontario, Canada, 2007 °c Paul Ullrich 2007 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis we first apply the 1+3 covariant description of general relativity to analyze n-fluid Friedmann- Lemaˆıtre (FL) cosmologies; that is, homogeneous and isotropic cosmologies whose matter-energy content consists of n non-interacting fluids. We are motivated to study FL models of this type as observations sug- gest the physical universe is closely described by a FL model with a matter content consisting of radiation, dust and a cosmological constant. Secondly, we use the 1 + 3 covariant description to analyse scalar, vec- tor and tensor perturbations of FL cosmologies containing a perfect fluid and a cosmological constant. In particular, we provide a thorough discussion of the behaviour of perturbations in the physically interesting cases of a dust or radiation background. iii Acknowledgements First and foremost, I would like to thank Dr. John Wainwright for his patience, guidance and support throughout the preparation of this thesis. I also would like to thank Dr. Achim Kempf and Dr. C. G. Hewitt for their time and comments. iv Contents 1 Introduction 1 1.1 Cosmological Models . 1 1.2 Geometrical Background . 3 1.2.1 Projection and Differentiation . 4 1.2.2 Kinematic Quantities . 5 1.2.3 Decomposition of the Energy-Momentum Tensor . 7 1.2.4 The Weyl Curvature Tensor . 7 1.2.5 Spatial Curvature . 8 1.2.6 Evolution and Constraint Equations . 10 1.3 Overview of the Thesis . 11 2 The Friedmann-Lemaˆıtre Cosmologies 13 2.1 Properties of the RW Metric . 13 2.2 Characterizations of the FL Cosmologies . 17 2.2.1 The Kinematic Characterization . 17 2.2.2 The Weyl Characterization . 18 3 Dynamics of n-Fluid FL Cosmologies 20 3.1 Behaviour and Classification of n-Fluid FL Cosmologies . 20 3.1.1 Basic Evolution Equations . 21 3.1.2 n-Fluid FL Cosmologies . 23 v 3.1.3 Hubble-Normalized Scalars . 26 3.1.4 A Compact State Space for n-fluid FL Cosmologies . 27 3.1.5 Invariant Sets and Equilibrium Points in the Compact State Space . 32 3.1.6 Classification of n-Fluid FL Cosmologies . 33 3.1.7 Asymptotic Behaviour of Solutions . 35 3.2 Parameterization of FL Models . 43 3.2.1 The Intrinsic Parameters . 43 3.2.2 The Observational Parameters . 45 3.2.3 Conserved Quantities . 47 3.3 2-Fluid FL Cosmologies - Qualitative Analysis . 49 3.3.1 General Features . 49 3.3.2 FL Cosmologies with Radiation and Dust (RDC-universes) . 51 3.3.3 FL Cosmologies with Dust and Cosmological Constant . 53 3.4 3-Fluid FL Cosmologies - Qualitative Analysis . 56 3.4.1 General Features . 56 3.4.2 A Model of the Physical Universe . 59 3.5 Explicit FL Cosmologies . 63 3.5.1 Single Fluid FL Cosmologies . 64 3.5.2 FL Vacuum Cosmologies . 66 3.5.3 Flat FL with a Single Fluid and Λ .......................... 67 3.5.4 FL Cosmologies with Radiation and Dust (RDC-Universes) . 68 3.5.5 FL Cosmologies with Radiation and Λ (RCΛ-Universes) . 69 3.5.6 Explicit Solutions in Terms of Observational Parameters . 71 3.6 Discussion . 72 4 Perturbations of FL Models 75 4.1 Historical Development . 75 4.1.1 Metric Approach . 75 4.1.2 Geometrical Approach . 77 4.2 The Linearized Einstein Field Equations . 78 4.2.1 The Linearization Process . 79 4.2.2 The Linearized Evolution and Constraint Equations . 80 vi 4.3 Harmonic Decomposition . 83 4.4 Scalar Perturbations . 84 4.4.1 The Governing DE . 84 4.4.2 Kinematic Quantities and Weyl Curvature . 86 4.5 Vector Perturbations . 87 4.5.1 The Governing DE . 87 4.5.2 Kinematic Quantities and Weyl Curvature . 88 4.6 Tensor Perturbations . 89 4.6.1 The Governing DE . 89 4.6.2 Kinematic Quantities and Weyl Curvature . 90 4.7 Dynamics of the Linear Perturbations: General Features . 91 4.7.1 Conformal Time and the Particle Horizon . 91 4.7.2 Governing DEs Using Conformal Time . 93 4.7.3 General Solutions for Scalar and Tensor Perturbations with Λ = 0 . 96 4.7.4 Functional Dependence of First-Order Quantities . 98 4.8 Perturbations of FL Models with Pressure-Free Matter . 101 4.8.1 The General Solution in Integral Form for Scalar Perturbations . 101 4.8.2 Scalar Perturbations with Λ = 0 . 103 4.8.3 Scalar Perturbations with Λ > 0, K = 0 . 105 4.8.4 Tensor Perturbations for Λ = 0 ............................106 4.9 Perturbations of Radiation-Filled FL Models . 108 4.10 Perturbations in Asymptotic Epochs of FL Models . 112 4.10.1 The Flat FL Asymptotic Epoch . 114 4.10.2 The Milne Asymptotic Epoch . 117 4.10.3 The de Sitter Asymptotic Epoch . 119 4.11 Discussion . 122 A Evolution and Constraint Equations 124 B Spatial Curvature 128 B.1 The Generalized Gauss Equation . 128 B.2 The Spatial Gradient of the 3-Ricci Scalar . 129 vii C Solutions of the Friedmann Equation 131 C.1 Solutions with a Stiff Fluid . 132 C.2 Existence-Uniqueness for the Friedmann Equation . 134 D Conformal Time 135 E Covariant Differential Identities 138 F Covariant Harmonics 141 F.1 Scalar Harmonics . 142 F.2 Vector Harmonics . 143 F.3 Tensor Harmonics . 145 G The 3-Cotton-York Tensor 147 G.1 Evolution Equation for the 3-Cotton-York Tensor . 148 G.2 Tensor Perturbations . 149 H Bessel and Associated Legendre Functions 150 H.1 Normal Forms of Bessel and Associated Legendre DEs . 150 H.2 Special Solutions of the Governing DEs . 151 H.3 Resonant Solutions of the Legendre DE . 153 H.4 Asymptotic Behaviour . 154 Bibliography 155 viii List of Figures 3.1 The 2-fluid FL state space for RDC-universes . 52 3.2 The 2-fluid FL state space for DCΛ-universes . 54 3.3 Thumbnail plots depicting behaviour of the length scale in FL cosmologies . 55 3.4 The 3-fluid FL state space for radiation, dust and cosmological constant. 60 3.5 3-fluid FL state space cross-section along the expanding flat FL manifold . 60 3.6 Past and future evolution of the RDCΛ model of the physical universe . 62 4.1 Conformal time and the particle horizon. 93 4.2 Scalar perturbation modes for dust and spatial curvature. 105 4.3 Scalar perturbation modes for dust and cosmological constant. 107 4.4 Tensor modes of the Hubble-normalized electric Weyl tensor in open FL with dust . 109 4.5 Tensor modes of the Hubble-normalized electric Weyl tensor in open FL with radiation . 111 ix List of Tables 3.1 The past and future attractors of generic n-fluid FL cosmologies. 42 3.2 Asymptotic behaviour of Hubble-normalized quantities in generic n-fluid FL cosmologies. 42 4.1 Functional dependence of first-order quantities . 100 4.2 Perturbations of flat FL with kη ¿ 1 .............................115 4.3 Perturbations of flat FL, with kη À 1 ............................116 4.4 Perturbations of Milne . 118 4.5 Perturbations of de Sitter (kη˜ ¿ 1)..............................120 4.6 Perturbations of de Sitter (kη˜ À 1)..............................121 x CHAPTER 1 INTRODUCTION 1.1 Cosmological Models The goal of cosmology is to describe the large-scale structure,.
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