Phenomenological Tests of Modified Gravity Ana Aurelia Avilez-López Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy July 2015 For Ana Lucía and Jorge . DECLARATION I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. This dissertation contains fewer than 100,000 words including appendices, bibliography, footnotes, tables and equations. The thesis is based in the following research • Avilez, A. and Skordis, C., Cosmological Constraints on Brans-Dicke Theory, Phys.Rev.Lett.113(2014)011101, arXiv:1303.4330 [astro-ph.CO]. • Ana Avilez-Lopez, Antonio Padilla, Paul M. Saffin, Constantinos Skordis, The Parametrized Post-Newtonian-Vainshteinian Formalism, (Submitted to JCAP) arXiv:1501.01985 [gr-qc]. Paper 1 is described in chapters 3 and 4. The motivation and part of the calculations of chapter 6 are based paper 2. Ana Aurelia Avilez-López July 2015 ACKNOWLEDGEMENTS I would like to thank my supervisor Costas Skordis for his dedicated, patient and fruitful guidence and support during my doctoral course. I am deeply grateful to him for giving me his trust and support even in hard moments. Also, thank you Costas for the freedom you gave me to explore new ideas and your flexibility about times and forms that made possible to carry on with this research work. Also, I am grateful to the people in the department for supported me and encourage me to give my best effort. I am grateful to my collaborators Tony Padilla and Paul Saffin for their guidence, discussions and support. I would also like to thank my external collaborator Yong-Seon Song for his interesting questions and discussions. I am grateful with the national council of science and technology of Mexico CONACYT for the financial and institutional support. Many thanks to my husband Jorge for all his patience, love and support. Thanks to Ana Lucia for the time she donated to this research I promisse it will be worth. Thank you to my mother Margarita Elena López and Bertha Fernandez-Quiroz, without their support this work would have not been possible at all. Thank you: Cristina, Victoria, Gerardo, Janette, Fernando, Giovanna, Bruno, Paul Tognarelli, Mou Zong Gang and Ana Chies for your friendship and support during this period. ABSTRACT The main goal of this thesis is to test the viability of some modified theories of gravity suitable to describe gravitational phenomena at cosmological and astronomical scales. In the first part of the Thesis we study the viability of the Brans-Dicke theory (BDT) and the effective scalar-tensor theory according (gBDT) to cosmological observations. We assume that either BDT as gBDT are limiting cases on very large scales of more general scalar-tensor theories involving derivative self-interactions which have running Newton’s constant. In order to implement this assumption in a simple way we consider two types of models. The restricted models that correspond to the standard BDT with Newton constant today equal to measured Newton constant in solar-system experiments. The unrestricted models, correspond to the case where the Newton’s constant today is a free parameter, and the cosmological GN is allowed to be different than in the solar system as in more general theories. We first explore the relevant theoretical aspects of these models. Afterwards, by using different analysis techniques we fitted cosmological observations. Finally we forecast limits of BDT by considering estimated covariance matrices for measurements of the matter power spectrum in redshift space from Euclid. The effective scalar-tensor theory gBDT arises from a phenomenological setup of parametrization of the LSS growth equations, we found estimates of modifications of the growth by using the correspondance between the estimates for the gBDT parameters. In the second part of the Thesis we present an extension of the Parameterized Post- Newtonian (PPN) formalism that is able to handle Vainsteinian corrections. We argue that theories with a Vainshtein mechanism must be expanded using two small parameters. In this Parameterized Post-Newtonian-Vainshteinian (PPNV) expansion, the primary expansion parameter which controls the PPN order is as usual the velocity v. The secondary expansion parameter, a, controls the strength of the Vainshteinian correction and is a theory-specific combination of the Schwarzschild radius and the Vainshtein radius of the source that is independent of its mass. We present the general framework and apply it to the Cubic galileon theory both inside and outside the Vainshtein radius. The PPNV framework can be used to determine the compatibility of such theories with solar system and other strong-field data. TABLE OF CONTENTS List of figures xv 1 Introduction1 1.1 Historical Review Of General Relativity . .1 1.1.1 Solar System Precision Tests . .1 1.1.2 Origins Of The Standard Cosmology . .4 1.2 Challenges for the Standard Cosmology . 11 1.2.1 Dark Matter-Energy Mysteries . 11 1.2.2 Other Observations in Tension with the Standard Model . 19 1.3 Alternatives to The Standard Cosmology . 22 1.3.1 Lovelock’s Theorem . 23 1.3.2 Overlook to the Zoo of Modified Theories of Gravity . 23 1.3.3 The Need of Screening Mechanisms in Modified Theories of Gravity . 27 1.4 Thesis Goals and Outline . 28 2 Standard Cosmology Based On General Relativity 31 2.1 Introduction . 31 2.2 The Standard Homogeneous and Isotropic Universe . 32 2.3 The Birth of Baryonic Matter: Nucleosynthesis and Recombination . 37 2.3.1 The Boltzmann Equation . 37 2.3.2 The Epoch of Nucleosynthesis . 39 2.3.3 Recombination of Neutral Hydrogen . 45 2.3.4 Linear Cosmological Perturbations . 47 2.4 Cosmological Testable Observables from Perturbations . 63 2.4.1 The Cosmic Microwave Background . 63 2.4.2 The Large Scale Structure of the Universe . 74 xii Table of contents 2.5 Summary of the Chapter . 77 3 The Brans-Dicke Cosmology 79 3.1 Historical Review of the Brans-Dicke Theory and its Generalizations . 79 3.2 The Model . 82 3.3 Background Cosmology . 83 3.4 Conservative Bounds on the Horndeski’s Subclass . 88 3.5 Big Bang Nucleosynthesis . 90 3.6 Recombination History . 92 3.7 Linear Cosmological Perturbation Theory . 98 3.7.1 Equations of Motion for Perturbations . 98 3.7.2 Scalar Perturbations And GR As A Late-Time Attractor . 101 3.7.3 Matter Perturbations And Matter Power Spectrum . 102 3.7.4 Peculiar Velocities . 106 3.7.5 Anisotropies In The Cosmic Radiation Background . 111 3.8 Conclusion and Summary of the Chapter . 124 4 Cosmological Tests Of Brans-Dicke Theory 127 4.1 Introduction . 127 4.2 Theory v.s. Data: Some Tests And Methods . 129 4.2.1 Parametric And Phenomenological Approaches . 129 4.2.2 Tools For Comparing Predictions v.s. Data . 131 4.3 Cosmological Constraints On Brans-Dicke Gravity . 140 4.3.1 MCMC Constraints From The CMB Anisotropies . 140 4.3.2 MCMC Constraints from the CMB and LSS . 155 4.3.3 Constraints from WMAP7 and The Alcock-Paszynski Test . 162 4.4 Forecasts on BDT from Redshift Space Distortions . 167 4.4.1 Theoretical models and upper bound of the detectability . 167 4.4.2 Alcock-Paczynski Effect from RSD . 169 4.4.3 Forecasting Constraints From Future RSD Surveys . 170 4.5 Summary and Conclusions of this Chapter . 173 5 The Effective Scalar-Tensor Theory At Cosmological Scales 177 5.1 Introduction . 177 5.2 A Phenomenological Setup for Modified Gravity . 179 Table of contents xiii 5.3 Cosmological Implications of The Effective Scalar-Tensor Theory . 183 5.3.1 The Model . 183 5.3.2 Background Cosmology . 185 5.3.3 Linear Cosmological Perturbation Theory . 190 5.3.4 The Alcock-Paczynski Distortion in gBDT .......... 194 5.3.5 Anisotropies in the Cosmic Microwave Background . 195 5.4 Constraints On The Effective Scalar-Tensor Theory . 208 5.4.1 Methodology and Analysis . 208 5.4.2 Results and Discussion . 209 5.5 Conclusion . 214 6 The Parametrized Post Newtonian-Vainshteinian Formalism 215 6.1 Introduction . 215 6.2 The PPNV Expansion: Vainshteinian Correction to the PPN Formalism217 6.2.1 The Standard Post-Newtonian Approximation . 217 6.2.2 Vainshtein Mechanism for a Spherically Symmetric Source . 221 6.2.3 Paving the Way Towards PPNV: the Vainshtein Order . 223 6.2.4 The PPNV Formalism . 228 6.3 Case study: The Cubic Galileon Theory . 230 6.3.1 A Short Introduction to the Cubic Galileon Theory . 230 6.3.2 The PPNV Expansion Outside the Vainshtein Radius . 232 6.4 The Dual Cubic Galileon and The Strong Field Region . 243 6.4.1 Dualizing the Cubic Galileon . 243 6.5 The PPNV Expansion Inside the Vainshtein Region . 244 6.5.1 Leading Order Solutions: General Relativity . 244 6.5.2 Solving the Post-Newtonian Scalar Equation . 247 6.5.3 PPNV At Higher PPN Orders . 250 6.5.4 On the Need of Basis for Post-Post Newtonian Vainstenian Solutions . 252 6.5.5 The Post Post Newtonian Vainshteinian Metric . 253 6.6 Back to Spherical Symmetry . 254 6.7 Conclusion . 258 7 Discussion and Conclusions 259 xiv Table of contents References 267 Appendix A The Standard Post-Newtonian Tools 289 A.1 Post-Newtonian Potentials . 289 A.2 Renomalized Post-Newtonian Potentials . 291 A.2.1 Definitions For j(4;0) Solution .