Modified Theories of Relativistic Gravity: Theoretical Foundations, Phenomenology, and Applications in Physical Cosmology by c David Wenjie Tian A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Theoretical Physics (Interdisciplinary Program) Faculty of Science Date of graduation: October 2016 Memorial University of Newfoundland March 2016 St. John’s Newfoundland and Labrador Abstract This thesis studies the theories and phenomenology of modified gravity, along with their applications in cosmology, astrophysics, and effective dark energy. This thesis is organized as follows. Chapter 1 reviews the fundamentals of relativistic gravity and cosmology, and Chapter 2 provides the required Co-authorship 2 2 Statement for Chapters 3 ∼ 6. Chapter 3 develops the L = f (R; Rc; Rm; Lm) class of modified gravity 2 µν 2 µανβ that allows for nonminimal matter-curvature couplings (Rc B RµνR , Rm B RµανβR ), derives the “co- herence condition” f 2 = f 2 = − f 2 =4 for the smooth limit to f (R; G; ) generalized Gauss-Bonnet R Rm Rc Lm gravity, and examines stress-energy-momentum conservation in more generic f (R; R1;:::; Rn; Lm) grav- ity. Chapter 4 proposes a unified formulation to derive the Friedmann equations from (non)equilibrium (eff) thermodynamics for modified gravities Rµν − Rgµν=2 = 8πGeffTµν , and applies this formulation to the Friedman-Robertson-Walker Universe governed by f (R), generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, f (R; G) generalized Gauss-Bonnet and dynamical Chern-Simons gravities. Chapter 5 systemati- cally restudies the thermodynamics of the Universe in ΛCDM and modified gravities by requiring its com- patibility with the holographic-style gravitational equations, where possible solutions to the long-standing confusions regarding the temperature of the cosmological apparent horizon and the failure of the second law of thermodynamics are proposed. Chapter 6 proposes the Lovelock-Brans-Dicke theory of alternative 1 φ pa ∗ G − !L r φrαφ ∗ G gravity with LLBD = 16π R + −g RR + b φ α , where RR and respectively denote the topological Chern-Pontryagin and Gauss-Bonnet invariants; as a quick application, Chapter 7 looks into traversable wormholes and energy conditions in Lovelock-Brans-Dicke gravity, along with an extensive comparison to wormholes in Brans-Dicke gravity. Chapter 8, for a large class of scalar-tensor-like grav- R 4 p ity S = d x −g LHE + LG + LNC + Lφ + Sm whose action contains nonminimal couplings between a scalar field φ(xα) and generic curvature invariants fRg beyond the Ricci scalar, proves the local energy- momentum conservation and introduces the “Weyl/conformal dark energy”. Chapter 9 investigates the pri- 2−2β β −2 4 mordial nucleosynthesis in L = " R +16πmPl Lm gravity from the the semianalytical approach for He, and from the empirical approach for D, 4He, and 7Li; also, consistency with the gravitational baryogenesis is estimated. Within the same gravitational framework as in Chapter 9, Chapter 10 continues to study thermal relics as hot, warm, and cold dark matter, and revises the Lee-Weinberg bound for the mass of speculated heavy neutrinos. Key Words cosmic acceleration, dark energy, modified gravity, physical cosmology, early Universe ii Publications and Declaration Here is the list of publications (including manuscripts under review) during my PhD studies, where the first item is the fruit of my MSc studies, and the others belong to my PhD projects. [1] Ivan Booth, David Wenjie Tian. Some spacetimes containing nonrotating extremal isolated horizons. Classical and Quantum Gravity 30 (2013), 145008. [arXiv:1210.6889] 2 2 [2] David Wenjie Tian, Ivan Booth. Lessons from f (R; Rc; Rm; Lm) gravity: Smooth Gauss-Bonnet limit, energy-momentum conservation, and nonminimal coupling. Physical Review D 90 (2014), 024059. [arXiv:1404.7823] [3] David Wenjie Tian, Ivan Booth. Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity. Physical Review D 90 (2014), 104042. [arXiv: 1409.4278] [4] David Wenjie Tian, Ivan Booth. Apparent horizon and gravitational thermodynamics of the Universe: Solutions to the temperature and entropy confusions, and extensions to modified gravity. Physical Review D 92 (2015), 024001. [arXiv:1411.6547] [5] David Wenjie Tian, Ivan Booth. Lovelock-Brans-Dicke gravity. Classical and Quantum Gravity 33 (2016), 045001. [arXiv:1502.05695] [6] David Wenjie Tian. Local energy-momentum conservation in scalar-tensor-like gravity with generic curvature invariants. General Relativity and Gravitation 48 (2016): 110. [arXiv:1507.07448]. [7] David Wenjie Tian. Traversable wormholes and energy conditions in Lovelock-Brans-Dicke gravity. [arXiv:1508.02291]. [8] David Wenjie Tian. Big Bang nucleosynthesis in power-law f (R) gravity: Corrected constraints from the semianalytical approach. [arXiv:1511.03258] [9] David Wenjie Tian. Thermal relics as hot, warm and cold dark matter in power-law f (R) gravity. [arXiv:1512.09117] This thesis is written in the manuscript style following the university guidlines, and the papers [2]∼[9] above have been adopted to produce Chapter 3 to Chapter 10, where [2]∼[5] are coauthored with Dr. Ivan Booth. I hereby confirm that I am the main contributor to [2]∼[5], and I have been granted the permission by Dr. Ivan Booth to use [2]∼[5] in this thesis. iii Acknowledgement When looking back at my graduate studies, above all I am most grateful to Prof. Ivan Booth, who is the supervisor for both my MSc and PhD studies. I joined Dr. Booth’s research group in September 2010, with a poor background in relativistic gravity and physical cosmology at that time. Dr. Booth patiently taught me general relativity (GR) and differential geometry, from the introductory to the advanced levels; when I was not good enough at GR, he patiently answered all my naive (sometimes even funny or stupid) questions during our weekly meetings, suggested helpful reading materials, and guided me to move ahead towards correct directions; he funded my MSc and PhD studies, so that I was able to focus on research and academics without worrying about milk and bread; every year he took me to GR conferences, so that I can give presentations, learn from other relativists, and make friends with them; when I run into trouble in my life, he always stood behind me and helped me. Pains and gains, I gradually grew up in GR after years of effort; Dr. Booth witnessed every step of my progress, and did his best to help me establish my physics career. Nearly six years have passed: how time flies! Nowadays I have become capable to do independent research, to design and accomplish projects on my own, while from last year on, I found Dr. Booth began to get gray hairs. In my heart, he is not just my supervisor, but also like an uncle to me. Also, I would like to thank Prof. Hari Kunduri in our gravity group. Although he is not my supervisor, he helped a lot for the development of my physics career. Time and time again, he generously supported my scholarship applications and job applications. He is a very friendly person, and it is always very pleasant, peaceful and inspiring to discuss physics and talk with him. In addition, I want to thank Prof. John K.C. Lewis (Physics Department of our university), Prof. Valerio Faraoni (Bishop’s University), and Prof. Sanjeev Seahra (University of New Brunswick). They were very friendly to me, always encouraged me to bravely pursued my physics career, kindly had stimulating talks with me, and generously helped with my postdoctoral applications. Last but not least, the works comprising this thesis were financially supported by the Natural Sciences and Engineering Research Council of Canada, under the grant 261429-2013 held by Prof. Ivan Booth. I was also funded by the scholarship from the School of Graduate Studies in our university. iv Dedication To all unsung heroes in Earth history R v Table of Contents Abstract ii Publications and Declaration iii Acknowledgment iv Dedication v Table of Contents vii List of Symbols viii 1 Introduction and overview: Physical cosmology and relativistic gravity1 1.1 Standard cosmology.......................................1 1.1.1 Einstein’s equation...................................1 1.1.2 Friedmann and continuity equations..........................2 1.1.3 Multiple components in the Universe.........................3 1.1.4 Acceleration of the late-time Universe and dark energy................5 1.1.5 Cosmological constant and ΛCDM cosmology....................6 1.1.6 Observational data...................................9 1.1.7 Some implications of ΛCDM............................. 10 1.1.8 wCDM and more complicated examples of dark energy................ 12 1.1.9 Acceleration of the early Universe: Inflation...................... 13 1.2 Modified gravity........................................ 15 1.2.1 Lovelock theorem and modifications of GR...................... 15 1.2.2 f (R) gravity and construction of effective dark energy................ 18 1.2.3 Quadratic gravity.................................... 20 1.2.4 f (R; G) gravity..................................... 22 1.2.5 Generalized Brans-Dicke gravity with self-interaction potential........... 24 1.2.6 Equivalence between f (R) and nondynamical Brans-Dicke gravity.......... 26 1.2.7 Scalar-tensor-chameleon gravity............................ 27 1.2.8 A unified form of modified gravity........................... 29 (m) 1.2.9 More insights into