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Cosmology in Modified Theories of Gravity nikolaos th. chatzarakis COSMOLOGY IN MODIFIED THEORIES OF GRAVITY Master of Science “Computational Physics” Department of Physics Faculty of Sciences Aristoteleion University of Thessaloniki COSMOLOGY IN MODIFIED THEORIES OF GRAVITY A search for viable cosmological models in f (R) and generalised Gauss-Bonnet theories nikolaos th. chatzarakis Master of Science “Computational Physics” Department of Physics Faculty of Sciences Aristoteleion University of Thessaloniki June 2019 – Nikolaos Th. Chatzarakis: Cosmology in Modified Theories of Gravity, A search for viable cosmological models in f (R) and generalised Gauss-Bonnet theories, © June 2019 “Cosmologists are often in error, but never in doubt.” Lev Davidovich Landau “We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time. Through the unknown, remembered gate When the last of earth left to discover Is that which was the beginning; At the source of the longest river The voice of the hidden waterfall And the children in the apple-tree Not known, because not looked for But heard, half-heard, in the stillness Between two waves of the sea.” T.S. Eliot, The Four Quartets “Space may be the final frontier but it’s made in a Hollywood basement And Cobain can you hear the spheres singing songs off Station To Station? And Alderaan’s not far away, it’s Californication.” Red Hot Chilly Peppers, Californication ABSTRACT The purpose of this dissertation is to present the most typical modified theories of gravity and discuss their validity and viability in constructing a cosmological model. Current issues in cosmology, related to the Standard Cosmological model, i. e.the “Big Bang model” and the L-CDM FLRW model, and including the infla- tionary scenario, the post-inflation reheating and the late-time accelerating expan- sion leading to “dark energy”, are considered obstacles for the traditional theory of gravity, i. e.General Theory of Relativity, since they can be explained only by means of exogenous fields a posteriori introduced. The goal of many modified theories of gravity is to extend General Relativity in such a way that quantum or sting-theory corrections ate taken into account, but also in order for many such cosmological -or astrophysical- problems to be resolved endogenously, without any additional hypotheses. We shall briefly mention the key modified theories of gravity, namely the f (R) theory, the f (R, G) theory and the mimetic Einstein- Gauss-Bonnet theory; then, we will proceed by testing their viability in specific cosmological problems, such as the early-time and late-time dynamics. Index terms— f (R) theory of gravity, f (G) theory of gravity, Loop Quantum Cosmology, Inflation, Dark Energy, Reconstruction techniques, Dynamical sys- tems vii PUBLICATIONS Part of this dissertation has appeared previously in the following publications: Shin’ichi Nojiri, S.D. Odintsov, V.K. Oikonomou, N. Chatzarakis, and Tan- moy Paul. “Viable Inflationary Models in a Ghost-free Gauss-Bonnet The- ory of Gravity.” In: European Physical Journal C (under revision) (2019). arXiv: 1907.00403 [gr-qc]. V. K. Oikonomou and N. Chatzarakis. “The Phase Space of k-Essence f (R) Gravity Theory.” In: (2019). arXiv: 1905.01904 [gr-qc]. ix ACKNOWLEDGMENTS The elaboration of a Master Dissertation is usually a hard work compressed in a schedule of some months, perhaps a year, giving the M.Sc. student little time to settle himself/herself into the deep waters of the actual work done. Consequently, much of it is not due to personal work, but rather to enlightenment and encour- agement provided by the supervisor and the other professors. Personally, I would like to thank Prof. C.G. Tsagas for the initial spark in cosmology over three years ago, Prof. N. Stergioulas for any comments, as well as introducing me to Dr. V.K. Oikonomou, and last but not least, dr. V.K. Oikonomou himself for supervising the work and assisting in all parts that demanded so. Through the latter, I would also like to thank Profs. S. Nojiri and S.D. Odintsov for the collaboration. Furthermore, the time and labour expended during the completion of such a Dissertation is often more than originally calculated and demands extreme moral and material support, as well as understanding. In my case, all demands were more that equally met by my parents, my sister and my friends, being not only a continuous resort, but also unnaturally patient. Through this, though not in its absolute measure, I would like to express my graduate to all these people who stood by me, even in times I proved myself not as worthy. xi CONTENTS i introduction1 1 introduction3 1.1 The Standard Model and the Cosmological Eras . 10 1.2 Cosmic Inflation and Scalar Fields . 18 1.3 Accelerating Expansion and Cosmological Constant . 26 1.3.1 Canonical Dark Energy . 27 1.3.2 Phantom Dark Energy . 29 1.3.3 Quintessence . 30 1.4 Past and Future Finite-time Singularities . 31 1.5 A Notice on Notions and Notation . 33 ii a roadmap to modified theories of gracity 35 2 the f (R) theory of gravity 37 2.1 General properties . 37 2.1.1 The metric tensor . 37 2.1.2 The Affine Connection: torsion-free Levi-Civita case . 39 2.1.3 Curvature . 41 2.2 Field equations and Conservation laws . 46 2.3 Scalar-tensor description and Brans-Dicke equivalence . 50 2.4 Viable f (R) theories . 53 2.5 Viable f (R, f, X) theories . 55 2.5.1 Canonical scalar field description . 55 2.5.2 Non-canonical scalar field inflation . 58 2.5.3 Inflation with f (R, f) theories of gravity . 59 2.6 A cosmological model . 61 3 the gauss-bonnet theories of gravity 65 3.1 General Properties . 65 3.2 Field equations and Conservation Laws . 67 3.2.1 Minimal coupling . 69 3.2.2 Non-minimal coupling . 70 3.3 “Mimetic” ghost-free theory . 72 3.3.1 The Flat FRW Vacuum without ghosts . 76 3.4 A Cosmological Model . 78 a b 3.4.1 Minimal coupling: f (R, G) = R + f1R + f2G ........ 79 a b 3.4.2 Non-minimal coupling: f (R, G) = f0R G .......... 80 3.4.3 The finite-time singularities: f (R, G) = R + p(t)G + q(t) .. 81 iii recosnstruction of cosmological models 85 4 the f (R) theory under testing 87 xiii xiv contents 4.1 The reconstruction technique . 87 4.1.1 Inflationary dynamics of f (R) gravity: Formalism . 88 4.2 The deSitter and the quasi-deSitter expansion . 91 4.3 The radiation-dominated and the matter-dominated eras . 93 4.4 An exponential early-time expansion . 95 4.5 A hyperbolic tangent late-time evolution . 100 4.6 Conclusions . 102 5 the ghost-free einstein-gauss-bonnet theory 107 5.1 Inflationary Dynamics . 107 5.2 The deSitter expansion . 110 5.2.1 An exponential coupling function, h(c) = aebc ........ 111 5.2.2 A power-law coupling function, h(c) = acb ......... 114 5.3 The quasi-deSitter expansion . 117 5.3.1 An exponential coupling function, h(c) = aebc ........ 118 5.3.2 A power-law coupling function, h(c) = acb ......... 121 5.3.3 The coupling-free inflation of a quasi-DeSitter background 123 5.4 Conclusions . 124 iv phase space analysis of alternative cosmologies 127 6 the interacting fluids description of dark matter and dark energy 129 6.1 The two-fluids description . 130 6.2 The Classical case (1): setting up the model . 131 6.3 The Classical case (2): working out the model . 133 6.3.1 The generalised dark energy . 134 6.3.2 The superfluid dark matter and generalised dark energy . 135 6.3.3 The superfluid dark matter and generalised Chaplygin dark energy . 139 6.4 The Loop Quantum Cosmology case (1): setting up the model . 143 6.5 The Loop Quantum Cosmology case (2): working out the model . 146 6.5.1 The superfluid dark matter . 146 6.5.2 The superfluid dark matter and generalised Chaplygin dark energy . 148 6.6 Conclusions . 149 7 the phase space of the k-essence theory 153 7.1 Introduction . 153 7.2 Setting up the model . 154 7.2.1 Equations of Motion of the k-Essence f (R) Gravity Theory 154 7.2.2 The choice of the dynamical variables and their evolution . 155 7.2.3 Friedmann Constraint and the Effective Equation of State . 159 7.2.4 Integrability of the Differential Equations for x3 and x4 ... 161 7.2.5 The three free parameters . 163 7.3 Working out the model (1): main analytical and numerical results . 164 contents xv 7.3.1 Model Ia (1): the case of canonical dark energy (c1 = −1) . 164 7.3.2 Model Ia (2): the case of phantom fields (c1 = 1) . 171 7.3.3 Model Ib: the case of bounce (c1 = 0) . 176 7.4 Working out the model (2): additional characteristics . 179 7.4.1 Model Ia: a possible 2 − d attractor . 179 7.4.2 Model Ib: a possible 1 − d attractor . 181 7.4.3 Infinities encountered in the models . 183 7.5 Conclusions . 184 v conclusion 187 8 conclusion 189 bibliography 191 LISTOFFIGURES Figure 1 The phase space of eqs. (20) and (21) for different values of w, corresponding to different types of fluid. 16 Figure 2 The solution to the Horizon problem. 21 Figure 3 A typical potential for the scalar field, f........... 24 Figure 4 The spectral index and the scalar-to-tensor ratio for “nega- tive times” with respect to C0 for N = 50 and C1 = C2 = 1.
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