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 models of interacting dark energy and can appear very similar in cosmological 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 TheCosmologicalConstant . 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 CMBandBAO ...................... 7 1.2.2 SNIa and H0 ....................... 8 1.2.3 Otherprobes ....................... 9 1.3 AlternativestoΛCDM ...................... 11 1.3.1 Darkenergy........................ 11 1.3.2 Modified gravity . 13 1.3.3 Other possibilities . 14 ii 2 Cosmological Perturbations 15 2.1 PerturbingtheMetric ...................... 15 2.2 GaugeTransformation . 19 2.3 Gauge-Invariant Variables . 21 2.4 Observables ............................ 23 2.5 DynamicsoftheMetric. .. .. 24 2.6 SpecificGaugeChoices . .. .. 26 2.6.1 Synchronous........................ 26 2.6.2 Newtonian . 27 2.7 Dynamicsofthefluid....................... 29 3 IDE in the literature 32 3.1 AnOverviewofIDE ....................... 32 3.2 HistoricalContext ........................ 34 3.3 Interactions Proportional to ψ˙ .................. 36 3.3.1 Early works . 37 3.3.2 Laterworks ........................ 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 InteractionsProportionaltoΓ . 58 3.5.1 IntroducingtheΓmodel . 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 Improvedconstraints . 64 3.6 OtherFormsofCouplings . 64 3.6.1 Motivatedbytheory . 64 3.6.2 Designedforapurpose . 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 OtherWorks............................ 69 3.8.1 Reconstruction of the coupling . 69 3.8.2 Particle theory . 70 3.8.3 Vacuumdecay....................... 71 3.9 Discussion............................. 71 4 Constraining a Model of Interacting Dark Energy 74 4.1 ModelDevelopment........................ 74 4.1.1 IDEinthebackground . 75 4.1.2 IDE in the perturbations . 76 4.1.3 Initial conditions . 82 4.2 CodingfortheModel....................... 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 Growthofstructure. 98 iv 4.4 Conclusions ............................ 101 5 Mathematically Equivalent Models of Interacting Dark En- ergy and Modified Gravity 104 5.1 ConformalEquivalence . 104 5.2 HistoricalContext ........................ 105 5.3 TheDualMG/IDEDescriptions. 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 Growthequations.. .. 116 5.4 DirectComparison ........................ 117 6 The Distinguishability of Interacting Dark Energy from Mod- 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 fromaselectionofIDEworks . 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 IDEmodel ........ 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 IDEmodels ............................ 88 4.4 Total matter power spectra for an illustrative selection of Q ∝ Γρx IDEmodels.......................... 90 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 Γparameterspace. 92 − 4.7 Marginalised probability distributions for Q Γρ IDE mod- ∝ x els over the entire w Ω parameterspace. 92 − m 4.8 Distributions of accepted steps in the MCMC chains for Q ∝ Γρx IDEmodels.......................... 94 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 IDEmodels ............................ 99 vii 4.13 Effective Hubble parameter and effective Newton constant for best-fit Q Γρ IDEmodels................... 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- etersformatchedDGP/IDEmodels. 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 EG pa- rameterformatchedDGP/IDEmodels . 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 Ωi as a function of ω for the IDE/STT system............................... 132 6.7 Evolution of the density perturbation and the density param- etersformatchedSTT/IDEmodels . 133 6.8 Evolution of the sum of the metric potentials and the EG pa- 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 Acknowledgements My thanks go to Kazuya and Gong-Bo for giving their time during my PhD, Roy for his insight into the model and to Rachael and my parents for sup- porting me during my write-up. x Dissemination Interacting Dark Energy – constraints and degeneracies Timothy Clemson, Kazuya Koyama, Gong-Bo Zhao, Roy Maartens and Jussi Valiviita. Physical Review D, volume 85, (2012), 043007. arXiv:1109.6234 [astro-ph.CO] The Distinguishability of Interacting Dark Energy from Modified Gravity Timothy Clemson and Kazuya Koyama. Journal of Cosmology and Astroparticle Physics, volume 1301, (2013), 010. arXiv:1209.2618 [astro-ph.CO] Presentations September 2011 Interacting Dark Energy, UKCosmo, ICG. July 2012 Interacting Dark Energy and Modified Gravity, Thirteenth Marcel Grossmann Meeting, Stockholm. xi Abbreviations General Relativity GR Dark Energy DE Interacting Dark Energy IDE Dark Matter DM Cold Dark Matter CDM Modified Gravity MG Scalar-Tensor Theory STT Dvali-Gabadadze-Porrati