Late-Time Acceleration: Interacting Dark Energy and Modified Gravity by Timothy Clemson This thesis is submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of the University of Portsmouth. September 23, 2013 Abstract In 1998 astronomical observations of distant stars exploding at the ends of their lives led to the discovery that the expansion of the Universe is acceler- ating. This is likely to be caused by an intrinsic part of Einstein’s General Theory of Relativity known as the cosmological constant, but naturalness issues and the need to improve observational tests have motivated the study of alternative models of the Universe. The research in this thesis is part of ongoing efforts to pin down the cause of late-time acceleration by better understanding these alternatives and their signatures in cosmological obser- vations. One such alternative is known as interacting dark energy and would be caused by additional matter in the Universe, as yet unknown to particle physics. This would interact with another unknown particle called dark matter that has been part of the standard model of cosmology since the 1970’s. The first part of this thesis contains a review of works on interacting dark energy and investigates a particular version of the model which had not been studied in detail before, placing recent observational constraints on its parameters. Another alternative to the cosmological constant is known as modified gravity, where General Relativity is extended by the addition of new degrees of freedom. Theories of modified gravity are mathematically related to some modelsofinteractingdarkenergyandcanappearverysimilarincosmological observations. The second part of this thesis investigates the extent to which the two can be distinguished using current observational data. i Table of Contents Abstract i Declaration ix Acknowledgements x Dissemination xi Abbreviations xii Definitions and Notation xiii 1 Perspectives on ΛCDM 1 1.1 The Cosmological Constant . . . . . . . . . . . . . . . . . . . 2 1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 The place of Λ in GR . . . . . . . . . . . . . . . . . . . 4 1.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 CMB and BAO . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 SNIa and H . . . . . . . . . . . . . . . . . . . . . . . 8 0 1.2.3 Other probes . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Alternatives to ΛCDM . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Dark energy . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Modified gravity . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Other possibilities . . . . . . . . . . . . . . . . . . . . . 14 ii 2 Cosmological Perturbations 15 2.1 Perturbing the Metric . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Gauge-Invariant Variables . . . . . . . . . . . . . . . . . . . . 21 2.4 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Dynamics of the Metric . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Specific Gauge Choices . . . . . . . . . . . . . . . . . . . . . . 26 2.6.1 Synchronous . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.2 Newtonian . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 Dynamics of the fluid . . . . . . . . . . . . . . . . . . . . . . . 29 3 IDE in the literature 32 3.1 An Overview of IDE . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Historical Context . . . . . . . . . . . . . . . . . . . . . . . . 34 ˙ 3.3 Interactions Proportional to ψ . . . . . . . . . . . . . . . . . . 36 3.3.1 Early works . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.2 Later works . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3 Particle theory . . . . . . . . . . . . . . . . . . . . . . 42 3.3.4 Massive neutrinos . . . . . . . . . . . . . . . . . . . . . 44 3.3.5 Halo collapse . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Interactions Proportional to . . . . . . . . . . . . . . . . . . 49 H 3.4.1 Constant coupling . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Variable coupling . . . . . . . . . . . . . . . . . . . . . 51 3.4.3 Observations . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.4 Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4.6 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Interactions Proportional to Γ . . . . . . . . . . . . . . . . . . 58 3.5.1 Introducing the Γ model . . . . . . . . . . . . . . . . . 58 3.5.2 First constraints . . . . . . . . . . . . . . . . . . . . . 59 3.5.3 Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 60 iii 3.5.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.5 Classification . . . . . . . . . . . . . . . . . . . . . . . 62 3.5.6 Quadratic couplings . . . . . . . . . . . . . . . . . . . 63 3.5.7 Improved constraints . . . . . . . . . . . . . . . . . . . 64 3.6 Other Forms of Couplings . . . . . . . . . . . . . . . . . . . . 64 3.6.1 Motivated by theory . . . . . . . . . . . . . . . . . . . 64 3.6.2 Designed for a purpose . . . . . . . . . . . . . . . . . . 65 3.7 Parameterisations of Interactions . . . . . . . . . . . . . . . . 66 3.7.1 The ξ parameterisation . . . . . . . . . . . . . . . . . . 67 3.7.2 The ǫ parameterisation . . . . . . . . . . . . . . . . . . 68 3.7.3 Other forms of parameterisation . . . . . . . . . . . . . 69 3.8 Other Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.8.1 Reconstruction of the coupling . . . . . . . . . . . . . . 69 3.8.2 Particle theory . . . . . . . . . . . . . . . . . . . . . . 70 3.8.3 Vacuum decay . . . . . . . . . . . . . . . . . . . . . . . 71 3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Constraining a Model of Interacting Dark Energy 74 4.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.1 IDE in the background . . . . . . . . . . . . . . . . . . 75 4.1.2 IDE in the perturbations . . . . . . . . . . . . . . . . . 76 4.1.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . 82 4.2 Coding for the Model . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.1 Modifying CAMB . . . . . . . . . . . . . . . . . . . . . 83 4.2.2 Maple checks . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.3 CosmoMC . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.4 Using the SCIAMA supercomputer . . . . . . . . . . . 87 4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3.1 Effects on the CMB and matter power spectra . . . . . 88 4.3.2 Likelihood analysis . . . . . . . . . . . . . . . . . . . . 90 4.3.3 Analysis of the best-fit models . . . . . . . . . . . . . . 94 4.3.4 Growth of structure . . . . . . . . . . . . . . . . . . . . 98 iv 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 Mathematically Equivalent Models of Interacting Dark En- ergy and Modified Gravity 104 5.1 Conformal Equivalence . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Historical Context . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 The Dual MG/IDE Descriptions . . . . . . . . . . . . . . . . . 105 5.3.1 Field equations . . . . . . . . . . . . . . . . . . . . . . 107 5.3.2 Scalar field equations of motion . . . . . . . . . . . . . 109 5.3.3 Metric perturbations . . . . . . . . . . . . . . . . . . . 111 5.3.4 Coupling terms . . . . . . . . . . . . . . . . . . . . . . 115 5.3.5 Growth equations . . . . . . . . . . . . . . . . . . . . . 116 5.4 Direct Comparison . . . . . . . . . . . . . . . . . . . . . . . . 117 6 TheDistinguishability ofInteracting DarkEnergyfromMod- ified Gravity 120 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.1.1 Distinguishing DE and MG . . . . . . . . . . . . . . . 121 6.1.2 Distinguishing IDE and MG . . . . . . . . . . . . . . . 121 6.1.3 The IDE and MG models . . . . . . . . . . . . . . . . 122 6.2 IDE/DGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.3 Distinguishability . . . . . . . . . . . . . . . . . . . . . 127 6.3 IDE/STT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.3.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.3.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . 132 6.3.3 Distinguishability . . . . . . . . . . . . . . . . . . . . . 134 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 Conclusions 137 Bibliography 140 v List of Tables 3.1 Observational constraints on the direction of energy transfer from a selection of IDE works . . . . . . . . . . . . . . . . . . 72 4.1 Cosmological parameters of the median and best-fit samples from CosmoMC for Q Γρ IDE models with w 1 and x ∝ ≥ − Γ 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 ≥ 4.2 Cosmological parameters of the median and best-fit samples from CosmoMC for w = 1 and when the entire parameter − space is considered for the Q Γρ IDE model . . . . . . . . 93 x ∝ vi List of Figures 4.1 Comparison of δ from CAMB and Maple for a Q Γρ IDE c x ∝ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Comparison of ρ from CAMB and Maple for a Q Γρ IDE x x ∝ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 CMB power spectra for an illustrative selection of Q Γρ x ∝ IDE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Total matter power spectra for an illustrative selection of Q ∝ Γρ IDE models . . . . . . . . . . . . . . . . . . . . . . . . . . 90 x 4.5 Marginalised probability distributions for Q Γρ IDE mod- x ∝ els with w > 1 and Γ 0 . . . . . . . . . . . . . . . . . . . . 91 − ≥ 4.6 Marginalised probability distributions for Q Γρ IDE mod- x ∝ els over the entire w Γ parameter space . . . . . . . . . . . . 92 − 4.7 Marginalised probability distributions for Q Γρ IDE mod- x ∝ els over the entire w Ω parameter space . . . . . . . . . . . 92 m − 4.8 Distributions of accepted steps in the MCMC chains for Q ∝ Γρ IDE models . . . . . . . . . . . . . . . . . . . . . . . . . . 94 x 4.9 CMB power spectra for best-fit Q Γρ IDE models . . . . . 96 x ∝ 4.10 Total matter power spectra for best-fit Q Γρ IDE models . 97 x ∝ 4.11 Normalised growth rates for ΛCDM and best-fit Q Γρ IDE x ∝ models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.12 Effective dark energy equation of state for best-fit Q Γρ x ∝ IDE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 vii 4.13 Effective Hubble parameter and effective Newton constant for best-fit Q Γρ IDE models . . . . . . . . . . . . . . . . . . . 101 x ∝ 5.1 Evolution of quantities in equivalent IDE and MG models . . 119 6.1 Evolution of the density perturbation and the density param- eters for matched DGP/IDE models . . . . . . . . . . . . . . . 126 6.2 Solutions of Ω′ in the matched IDE/DGP setup . . . . . . . . 127 6.3 Evolution of the sum of the metric potentials and the E pa- G rameter for matched DGP/IDE models . . . . . . . . . . . . . 128 6.4 Results in the µ Σ plane for the matched DGP/IDE models 129 − 6.5 Solutions of ψ′2 and Ω′ in the IDE/STT setup . . . . . . . . . 131 6.6 ψ′ and Ω′ limits on Ω as a function of ω for the IDE/STT i system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.7 Evolution of the density perturbation and the density param- eters for matched STT/IDE models . . . . . . . . . . . . . . . 133 6.8 Evolution of the sum of the metric potentials and the E pa- G rameter for matched STT/IDE models . . . . . . . . . . . . . 134 6.9 Results in the µ Σ plane for the matched STT/IDE models . 135 − viii Declaration Whilst registered as a candidate for the above degree, I have not been regis- tered 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. Word count: Approximately 32,000 words. ix