The Effective Field Theory Approach to Fluid Dynamics, Modified Gravity Theories, and Cosmology Junpu Wang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 c 2013 Junpu Wang All Rights Reserved ABSTRACT The Effective Field Theory Approach to Fluid Dynamics, Modified Gravity Theories, and Cosmology Junpu Wang The effective field theory approach is powerful in understanding the low energy phenom- ena without invoking the UV degrees of freedom. We construct a low energy Lagrangian for ordinary fluid systems (in constrast to superfluid), pure from symmetry considerations and EFT principles. The dynamical fields are the Goldstone excitations, associated with spontaneously broken spacetime translations. It is organized as derivatively coupled theory involving multiple scalar fields. This formalism enables us to study fluid’s quantum me- chanical properties and dissipative effects. Cosmological models can be built by naturally coupling the fluid EFT to gravity. From the EFT point of view, GR is the unique low energy theory for the spin-2 graviton field and any infrared modification corresponds to adding new degrees of freedom. We focus on two popular classes of modified gravity models, | the chameleon like theories and the Galileon theory, | and perform a few reliability checks for their qualifications as modified gravity theories. Furthermore, guiled by the EFT spirit, we develop a cosmological model where primor- dial inflation is driven by a `solid', defined, in a similar manner as the EFT of fluid The symmetry breaking pattern differs drastically from that of standard inflationary models: time translations are unbroken. This prevents our model from fitting into the standard EFT description of adiabatic perturbations, with crucial consequences for the dynamics of cosmological perturbations, and exhibits various unusual features. Table of Contents 1 Introduction 1 2 EFT Construction for Fluids 9 2.1 Effective Field Theory for Perfect Fluids . .9 2.1.1 Fluids with No Conserved Charge . .9 2.1.2 Fluids with Conserved Charge . 12 2.1.3 Hydrodynamics and Thermal Relations . 13 2.1.4 The Goldstone Field Action . 14 2.1.5 Duality Between Irrotational Ordinary Fluid and Superfluid . 15 2.2 EFT for Imperfect Fluids . 17 2.2.1 The general idea . 17 2.2.2 Low frequency, long distance behavior of correlators . 22 2.2.3 Dissipative couplings: Sint to linear order . 25 2.2.4 Rediscovering Kubo relations . 28 2.3 A Glance at the Quantum Mechanics of Perfect Fluids . 32 2.3.1 A Naive Deformation . 32 2.3.2 Scattering Amplitude of Some Simple Processes | the Vortex Strong Coupling . 34 2.3.3 The infrared situation . 38 2.4 Application of the Effective Fluid Model to Cosmology . 40 2.4.1 Perfect Fluids in a Cosmological Background . 40 2.4.2 The Conserved Curvature Perturbation to Non-linear Order . 43 i 2.4.3 Remarks on the Conservation of ζ ................... 46 3 EFT Approach to Gravity Theories: A Modification of Einstein Gravity 49 3.1 An Effective Field Theoretic Point of View of General Relativity . 49 3.2 A Modification of GR in the Infrared: Motivations and Examples . 55 3.3 No-Go Theorems for Generalized Chameleon Field Theories . 62 3.3.1 Set-Up . 64 3.3.2 Generalized Screening Condition . 65 3.3.3 Proof of Theorems . 66 3.4 Classical Stability of Galileon . 68 3.4.1 Conditions for Stability . 69 3.4.2 General Program . 71 3.4.3 Details of Analysis . 73 2 3.4.4 Conditions for intersection of the E = ρ/MPl and Z0 = 0 surfaces (c4 6=0).................................. 75 2 3.4.5 Conditions for intersection of the E = ρ/MPl and Z1 = 0 surfaces (c4 6=0).................................. 78 3.4.6 Stable Region: From Local stability to Absolute Stability . 79 4 An Inflation Model from EFT Construction: Solid Inflation 81 4.1 The Effective Theory for a Solid in Flat Spacetime . 81 4.2 Solids in a Cosmological Background: Inflation . 83 4.3 Cosmological perturbations . 90 4.4 Two-point functions . 93 4.4.1 Tensor perturbations . 94 4.4.2 Scalar Perturbations . 96 4.5 The three-point function . 99 4.5.1 Analytic Calculation of Integral . 101 4.6 The size and shape of non-gaussianities . 103 4.7 Why Is ζ Not Conserved? . 106 4.8 Reheating and Post-Inflationary Evolution . 109 ii 5 Conclusions 115 Bibliography 119 A Linear couplings of a U(1) Goldstone 125 B Phase Space for Phonons 129 µ C Expressions for Z ν in the Galileon Theory 131 D Further Details of the Galileon Stability Analysis 132 E Unitary Gauge vs. Spatially Flat Slicing Gauge for Solid Inflation 135 F Full trilinear action in SFSG for Solid Inflation 137 G Time Dependence of Background Quantities in Solid Inflation 139 iii List of Figures 2.1 Feynman diagram representation of Eqn. (2.39) . 20 2.2 Tree Diagrams for LL! LL Scattering. 36 2.3 Tree Diagrams for TT! TT Scattering. 37 4.1 The shape of non-gaussianities in solid inflation model . 105 iv Acknowledgments First and foremost, I want to thank my parents. Although having their only child tens of thousand miles away, they never complained. On the contrary, they constantly encourage and support me to pursue what I like. They also have influenced , to some extent, the way I am doing research today. Without them, I can hardly imagine that I could achieve a Ph.D degree some day. I can not overestimate my gratitude towards my thesis supervisor Professor Alberto Nicolis. He is my mentor; he is the person who leads into the palace of modern physics; he forges many of my viewpoints about quantum field theory and about gravity. But more often, he is like a close friend: I can walk in his office asking random questions and even making jokes. I would not expect any more from a professor than I get from Alberto. I want to always express my deepest gratitude towards my co-supervisor Professor Lam Hui. It is my honor to get guidance from two young, brilliant scientists during my doctoral study. Lam has an incredible ability to master knowledge from a vast of amount of fields. He is an example, stimulating me to dabble in other areas that I am not familiar with. I owe a debt of gratitude to Professor Justin Khoury, who has been a tremendous help to me in our collaboration and in the course of my applying for a postdoc job. I appreciate the encouragement and the advice about future plan that he gave. v Friends have always been a big part throughout my life; especially in the journey of pursuing my Ph.d degree, they are my listeners and cheerleaders. Shan Gao, my college roommate, colleague and best friend, thank you for all the conversations we have, about physics, about future and about all kinds of random things. Yaocheng Fei, Xiaoqiang Zhong and Shuyu Zhou, thank you buddies! Despite the long distance and infrequent communication, every time I talked to/saw you, I felt most relaxing and pleasantest. I remember 6 years ago when I came to Columbia, it was my first time to leave my home country and my first time to live alone it a foreign city. Fortunately, I met lots of new friends here. To Kehui Coffee Xiang, Yinsheng Guo, Dazhi Tan, Wenbo Li, Zhou Ye and our captain Tianxia Jia, you are making my life in this city delightful. Thank you for the basketball games we played, the basketball/soccer/American football/baseball games we watched and the dinners we had together. Nothing works better than these gathering events to dispel loneliness. I am grateful to my colleagues (also friends). I want to thank Solomon Endlich | the person who I have done most of my projects with and also the person who offered me enormous help in overcoming the culture gap. I would like to thank Andrej Ficnar, Xiao Xiao, Bart Horn, Riccardo Penco, Rachel Rosen for all the discussions we had, from which I benefited tremendously. To Professor Eduardo Ponton, thanks for letting me sit in your QFT and particle physics classes and for patiently answering my confusions in or outside the scope of classes. To Zhongjie Lin and Hantao Yin, thanks for all your assistance on programming, algorithms and computer hard/soft-wares. There are many more people I want to thank; at least allow me to list their names here: Chris Chan, Qi Liu, Yujiao Chen, Jianglei Yu, Feng Tian, Daiqian Zhang, Denial Hu (my two year roommate), Alex Chen, Junichi Okamoto, Hung The Dang, Dashun Wang, Lan Gong ... I am also grateful to my advisors in college Professor Zhifang Lin and Professor Rukeng Su. The former suggested me learn Mathematica, which now becomes my main toolkit to attack hard analytic problems and/or non-analytic problems. The latter kept reminding me the hardness in the theoretical physics journey, which makes me never too optimistic about what I have achieved. I would like to show my gratitude to the coordinators in the department of physics: Lalla vi Grimes (retired in 2009), Rasma Mednis (May she be at peace in heaven) and Randy Torres, who have kindly helped me deal with all sorts of paperwork, especially the documents for visa application every time when I travel. I would also thank our finance officer Yasmin Yabyabin and Lydia Argote for their taking care of my tedious reimbursement issues. I would like to mention the name of my favorite basketball player, Steve Nash | an not so physically gifted player in NBA who has won the Most Valuable Player award twice | whose experience reminds me how important hard-working and persistence could be in achieving success.