SISSA - International School for Advanced Studies The Threefold Way to Cosmological Tests of Gravity Thesis Submitted for the Degree of \Doctor Philosophiæ” Supervisors Candidate Prof. Carlo Baccigalupi Marco Raveri Prof. Alessandra Silvestri June 2016 ii Contents Abstract v Collaborations vii Notation ix 1 Introduction 1 I Modern Cosmology 5 2 Cosmological Background 7 2.1 FLRW models . .9 2.2 Matter Components . 10 2.3 Distances and Horizons . 11 2.4 Cosmic Acceleration . 13 3 Cosmological Perturbations 15 3.1 Perturbing the universe . 15 3.2 Jeans analysis - Newtonian perturbation theory . 16 3.3 Linear Equations . 17 3.4 Initial conditions . 21 4 Cosmological Observables 25 4.1 Random fields . 25 4.2 Growth of Structure: the Matter Power Spectrum . 26 4.3 The Cosmic Microwave Background . 30 4.4 Cosmological Data Sets . 33 II Null Tests of Cosmological Models 37 5 Testing the Concordance of Evidences 39 i ii CONTENTS 5.1 Bayesian Statistical Tools . 40 5.2 The Data Concordance Test . 41 5.3 DCT Properties . 41 5.4 Data Sets and Model . 45 5.5 Results . 46 5.6 Chapter Summary . 48 III Parametrized Test of DE/MG Models 51 6 The Effective Field Theory Approach to DE/MG Models 53 6.1 The EFT action in the Unitary Gauge . 54 6.2 The St¨uckelberg field . 55 6.3 Background dynamics . 56 6.4 Perturbations dynamics . 57 6.5 Stability of perturbations in the dark sector . 58 7 A Background Dynamical Analysis 61 7.1 Dynamical system and cosmological viability . 62 7.2 Stability Analysis . 64 7.3 Zeroth order analysis . 65 7.3.1 Reconstructing quintessence models . 67 7.4 First order analysis . 69 7.4.1 Numerical investigation of different transitions . 73 7.5 Second order analysis . 76 7.6 Nth order analysis: exploiting the recursive nature of the system . 80 7.7 Chapter Summary . 84 8 Implementation in CAMB 87 8.1 Code implementation of the background cosmology . 88 8.2 Numerically Evolving Perturbations . 89 8.3 Cosmological Observables . 90 8.4 Some Numerical Results . 92 8.5 Chapter Summary . 94 9 Observational Results 97 9.1 EFTCosmoMC: sampling of the parameter space under stability conditions . 97 9.2 Linear EFT Observational Results . 98 9.2.1 Results Without Massive Neutrinos . 99 CONTENTS iii 9.2.2 Results With Massive Neutrinos . 101 9.3 Investigating Tensions with Parametrizations . 103 9.3.1 Simulating Tensions in Planck Data . 105 9.3.2 Results . 106 9.4 Observational Signatures of Modified Gravitational Waves . 109 9.4.1 A Worked Example . 112 9.5 Chapter Summary . 114 IV Test of Specific DE/MG Models 119 10 Designer f(R) Gravity 121 10.1 Introduction to Designer f(R) gravity . 121 10.2 Mapping f(R) to the EFT framework . 122 10.3 Comparison with MGCAMB on Cosmological Predictions . 123 10.4 Designer f(R) models on non-ΛCDM background . 128 10.5 Cosmological Constraints Without Massive Neutrinos . 131 10.6 Cosmological Constraints With Massive Neutrinos . 134 10.7 Investigating Tensions with Designer f(R)............. 139 10.8 Chapter Summary . 141 11 Hu-Sawicki f(R) Gravity 145 11.1 Introduction to Hu-Sawicki f(R).................. 146 11.2 HS f(R) Background Cosmology . 147 11.3 Scale Dependence of the Growth of Structures . 149 11.4 Cosmological constraints . 152 11.5 Chapter Summary . 154 12 Hoˇrava Gravity 159 12.1 Introduction to Hoˇrava Gravity . 160 12.2 Mapping Hoˇrava Gravity into EFT . 162 12.3 Degrees of freedom: dynamics and stability . 165 12.4 Hoˇrava Background Cosmology . 168 12.5 Cosmological Perturbations in Hoˇrava Gravity . 171 12.6 Cosmological constraints . 177 12.6.1 H3 case: results . 178 12.6.2 H2 case: results . 180 12.7 Chapter Summary . 181 iv CONTENTS V Conclusions and Outlook 185 VI Appendix 191 .1 Dynamical Analysis Continued . 193 .1.1 Second Order . 193 .1.2 Nth order analysis continued . 194 .2 EFTCAMB Implementation Details . 197 .2.1 EFT Action from Second Order Operators . 197 .2.2 Cosmological Perturbations from Second Order Operators 198 .2.3 EFTCAMB Notation . 200 .2.4 Structure and Evolution of the EFTCAMB code . 201 .2.5 EFTCAMB Background Equations: Code Notation . 205 .2.6 EFTCAMB Perturbations Equations: Code Notation . 208 .2.7 Viability Conditions . 214 .2.8 Initial Conditions . 216 .2.9 Pure EFT Models . 216 .2.10 Designer Mapping EFT Models . 218 .2.11 Alternative Parametrizations . 222 .2.12 Full EFT Mapping . 224 .3 The Hoˇrava L4 and L6 Lagrangians . 228 .4 Hoˇrava Cosmological Parameters . 232 Acknowledgments 235 Abstract Explaining the physical origin of cosmic acceleration still poses a challenge to modern cosmology. On one hand, observational evidence corroborating this phenomenon is compelling and continuously becoming stronger and stronger. On the other hand a physical explanation for it is still missing. Cosmic accel- eration might be explained by a cosmological constant having the same effect of vacuum energy. Indeed cosmological observations point in this direction and the cosmological constant is a cornerstone of the standard cosmological model. This explanation, however, suffers from several naturalness problems that re- flect the fact that, while a cosmological constant is allowed in the gravitational sector by symmetry arguments, there is no theory for the gravitational effect of quantum vacuum. To address this issue, or at least to have an intuition of the phenomenology related to the solution of this problem, one might want to add other dark fluids to the cosmic budget or modify the laws of gravity on large scales to drive the accelerated expansion of the universe. In this thesis we develop and exploit a threefold approach to the study of the phenomenological aspects of cosmic acceleration with the aim of systematizing the investigation of models beyond the standard one. The first path that we shall follow is that of quantifying the level of agreement of cosmological observations, within the standard cosmological model; this will allow us to determine whether there is already some indication that this model might be inappropriate in describing present day observations. Then we shall move along the second path to study parametrized approaches to the phenomenology of Dark Energy and Modified Gravity theories. We de- velop the relevant tools to exploit an Effective Field Theory description for this phenomenon and we investigate some of its observational consequences. At last we shall move along the third path that consists in testing specific non-standard models. Exploiting the unifying power of the Effective Field The- ory approach, we study the cosmological implications and corresponding data constraints on two f(R) models and on Hoˇrava gravity. Overall we find that, already at present, cosmological observations are precise enough to substantially improve our knowledge about the space of Dark Energy and Modified Gravity models. While doing so we developed the relevant tools to perform massive and systematic studies of non-standard cosmologies aiming at explaining the physical origin of Cosmic Acceleration with present data and the next generation of cosmological surveys. v vi ABSTRACT Collaborations This thesis is the result of the research done during my four years of Ph.D. and of collaborations with C. Baccigalupi, N. Frusciante, B. Hu, M. Rizzato, A. Silvestri, D. Vernieri, S.Y. Zhou. The bulk of the present work is based on the following papers published in refereed Journals: 1. Bin Hu, Marco Raveri, Matteo Rizzato, Alessandra Silvestri, \Testing Hu-Sawicki f(R) gravity with the Effective Field Theory ap- proach", Monthly Notices of the Royal Astronomical Society (in press) [arXiv:1601.07536 [astro-ph.CO]] [1]. 2. Marco Raveri, \Are cosmological data sets consistent with each other within the Λ cold dark matter model?" PRD 93 (2016) 043522 [arXiv:1510.00688 [astro-ph.CO]] [2]. 3. Noemi Frusciante, Marco Raveri, Daniele Vernieri, Bin Hu, Alessandra Silvestri, \Horava Gravity in the Effective Field Theory formalism: from cosmology to observational constraints", Phys. Dark Univ. 13, 7 (2016) [arXiv:1508.01787 [astro-ph.CO]] [3]. 4. Bin Hu, Marco Raveri, \Can modified gravity models reconcile the tension between CMB anisotropy and lensing maps in Planck-like observations?", PRD 91, (2015) 123515 [arXiv:1502.06599 [astro-ph.CO]] [4]. 5. Bin Hu, Marco Raveri, Alessandra Silvestri, Noemi Frusciante, \EFT- CAMB/EFTCosmoMC: massive neutrinos in dark cosmologies", PRD 91, (2015) 063524 [arXiv:1410.5807 [astro-ph.CO]] [5]. 6. Marco Raveri, Carlo Baccigalupi, Alessandra Silvestri, Shuang-Yong Zhou, \Measuring the speed of cosmological gravitational waves", PRD 91, (2015) 061501 [arXiv:1405.7974 [astro-ph.CO]] [6]. 7. Marco Raveri, Bin Hu, Noemi Frusciante, Alessandra Silvestri, “Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data", PRD 90 (2014) 043513 [arXiv:1405.1022 [astro-ph.CO]] [7]. vii viii COLLABORATIONS 8. Bin Hu, Marco Raveri, Noemi Frusciante, Alessandra Silvestri, “Effective Field Theory of Cosmic Acceleration: an implementation in CAMB", PRD 89 (2014) 103530 [arXiv:1312.5742 [astro-ph.CO]] [8]. 9. Noemi Frusciante, Marco Raveri, Alessandra Silvestri, “Effective Field Theory of Dark Energy: a Dynamical Analysis", JCAP 1402, 026 (2014) [arXiv:1310.6026 [astro-ph.CO]] [9]. Supplementary material includes: • EFTCAMB/EFTCosmoMC codes available at http://www.lorentz.leidenuniv. nl/~hu/codes/ • Bin Hu, Marco Raveri, Noemi Frusciante, Alessandra Silvestri \EFTCAMB/EFTCosmoMC: Numerical Notes v2.0", arXiv:1405.3590 [astro-ph.IM], (2014) [10]. Notation Here, we provide a brief guide to the notation and a list of acronyms. c = 1 the speed of light is set to be equal to one; G Newtonian Gravitational Constant; m0 Planck mass; (−; +; +; +) metric signature; i; j; k::: 3D spatial indices in vectors and tensors; µ, ν; γ::: 4D indices in vectors and tensors; gµν metric tensor; g determinant of the metric tensor gµν τ conformal time; t cosmic time; H Hubble parameter in conformal time; H Hubble parameter in cosmic time H = H=a; Rµν , R Ricci tensor and its trace; Tµν , T Stress energy tensor and its trace; Gµν Einstein Tensor (Gµν = Rµν − 1=2gµν R); φ Scalar field; χm Matter fields; Sm Matter action of all matter fields, χm; rµ Covariant derivative; r¯ Spatial covariant derivative; Λ Cosmological Constant; δ Perturbative density field (δ = ρ/ρ − 1).