New viable theories of modified gravity : Minimal Theories Title and Quasidilaton( Dissertation_全文 ) Author(s) Michele, Oliosi Citation 京都大学 Issue Date 2019-09-24 URL https://doi.org/10.14989/doctor.k22028 Right Type Thesis or Dissertation Textversion ETD Kyoto University Doctoral thesis New viable theories of modified gravity: Minimal Theories and Quasidilaton Michele Oliosi Yukawa Institute for Theoretical Physics, Kyoto University July 23, 2019 Abstract In the last few decades, several alternative models of gravity have been developed in the hope to produce large-scale deviations from general relativity (GR). The principal aim of this program is to uncover what physics could be at works behind the present-day acceleration of the Universe, other than a fine-tuned cosmological constant. Among other motivations to explore new models of gravity are also the other puzzles of cosmology, such as dark matter, as well as the incompleteness of our description of gravity in the UV. However, satisfying the constraints from current observations while guaranteeing full theoretical consistency is still a challenging task for several such constructions; both do indeed put strong restrictions on the theory space of gravitational theories. Two notable examples are the instabilities that plague some of the cosmological realizations of massive gravity theories and its extensions, or the recent restrictive bound on the speed of tensor modes from the multi-messenger observation of a neutron star binary merger. In this thesis we show that there exist new classes of alternative theories of gravity that are obser- vationally and theoretically viable, and produce interesting phenomenology. In particular, we focus first on minimally modified gravity (MMG) theories, which propagate only the two tensor modes by means of violations of the Lorentz symmetry in the gravitational sector. In this context, we present a class of theories constructed on the basis of the existence of an Einstein frame, in which the gravitational Lagrangian is equivalent to GR. As observational constraints, we consider in particular the bound on the speed of tensor modes, as well as on the variation of the gravitational constant. We find that there subsists a wide class of interesting possibilities to modify GR. In addition to this construction, we review other theories that fall into the MMG class. As a second example of a new alternative theory of gravity, we construct and study the minimal theory of quasidilaton massive gravity (MQD). This theory is motivated by some difficulties to find viable Friedmann-Lema^ıtre-Robertson-Walker cosmologies in the context of quasidilaton massive gravity theories, but can also be effectively understood as an extension of a specific MMG theory, the minimal theory of massive gravity, by rendering dynamical part of the fiducial metric structure. We show that MQD is viable for a wide region of its parameter space, that it will be efficiently constrained by future cosmological surveys, and can sustain interesting phenomenology, in particular produce weak gravity while propagating the same number of degrees of freedom as usual scalar-tensor theories. In order to motivate the models we present, this thesis also includes two compact review chapters, which cover respectively the standard model of gravity and cosmology, and several alternatives to GR together with current and prospective constraints. In light of the large number of future observational efforts to constrain the cosmological dynamics as well as the behavior of gravity on these scales, model- building efforts come by as crucial tools to be able to interpret the future data most efficiently. 2 Acknowledgements My deepest gratitude goes to my advisor Shinji Mukohyama, who has always, with all his kindness, made a priority of the growth of his students. Both for research as in humanity, he has been a really excellent model and guide. I would also like to give my special thanks to my vice-advisor Antonio de Felice. His constant generosity, and understanding (and house-made delicacies) made many days and moments nicer (and sweeter). Both of them have always shown uncommon intuition and deep knowledge; I have learned a lot during all our weekly meetings. I hope I will be able to nurture enough all these lessons to keep them as central tools along my future path. This thesis would not have been possible without the special atmosphere that enveloped every day of work. I think Kyoto University and especially YITP are truly remarkable places, imprinted of a tranquil atmosphere that does much for the growth of humans. In particular the Institute, the Theoretical Astrophysics group at the Department of Physics, and the respective numerous visitors, have blessed me with incredible colleagues, mentors, and friends, and it would be hard to list them without forgetting at least as many excellent people. Hats off in particular to the administrative staff for all the amazing and supporting work. Thank you all, sincerely. I would like to give my best thanks to my collaborators, including, along with my supervisors, Katsuki Aoki, Fran¸coisLarrouturou, Chunshan Lin, Hayato Motohashi, and Yota Watanabe. Their help and discussions have made me grow a lot, and helped me push further outside of initially much narrower horizons. Friends and family, outside my academic grounds, have of course played an enormous role in sustaining a brittle human through a complex adventure, and I could not acknowledge them enough for that. I would like to say that I sincerely admire all those in Kyoto who offered me their friendships, as well as my friends from Switzerland and the world, who have been always present with me despite the distances. Also, thanks to my family for their love and patience. Finally, I would like to thank all the people who invested time in proofreading all or part of this thesis. Of course my supervisors, who helped me a lot to give a direction to the thesis, and supported me throughout all the stages of the writing. For his considerable help with reading this thesis with his customary sagacity and application I would like to thank Fran¸coisLarrouturou. For the very helpful last minute adjustments and encouragements, I would like to thank Paul Martens, Pierre-Olivier Guimond, and Ozg¨unMavuk.¨ Thanks also to Ryo and Masroor for their kind support and all the entertaining discussions and smiles. 3 4 Contents 1 Introduction 7 2 Theories of gravity 13 2.1 General relativity, behind the scenes . 13 2.1.1 Notation . 13 2.1.2 Equivalence principles . 13 2.1.3 Basic elements . 14 2.1.4 Symmetries . 15 2.1.5 Lovelock's theorem . 16 2.1.6 Minimal coupling . 16 2.1.7 General Relativity . 16 2.1.8 Constraint structure of GR . 17 2.1.9 Remark on the field content of gravity . 20 2.2 Cosmology . 20 2.2.1 The cosmological background . 21 2.2.2 The dark Universe . 24 2.2.3 Cosmological perturbation theory . 27 2.2.4 Phenomenology of perturbations . 29 3 Beyond GR 33 3.1 Theories beyond Lovelock's theorem . 33 3.1.1 Adding fields . 33 3.1.2 Symmetry breaking . 39 3.1.3 Massive gravity . 41 3.2 Constraining modified gravity . 48 3.2.1 Cosmological constraints . 48 3.2.2 Observations in the weak field regime . 50 3.2.3 Strong field regimes . 52 3.2.4 Constraints with GW . 54 4 Minimally modified gravity 55 4.1 Philosophy of minimal theories . 55 4.2 Unbroken symmetries . 56 4.3 Type-I and type-II MMGs . 57 4.4 Type-I theories . 58 4.4.1 Construction . 58 4.4.2 Cosmology . 60 4.4.3 Considering constraints . 61 4.5 Type-II theories . 65 4.5.1 Constrained vielbein formalism . 65 4.5.2 Precursor of MTMG . 67 4.5.3 MTMG . 69 4.6 Summary . 72 5 CONTENTS CONTENTS 5 Minimal quasidilaton 73 5.1 Quasidilaton theories . 73 5.1.1 Dilaton . 73 5.1.2 Original quasidilaton . 75 5.1.3 Cosmology with a quasidilaton . 75 5.1.4 Quasidilaton: extensions . 76 5.2 Minimal quasidilaton . 78 5.2.1 Properties of the minimal quasidilaton . 78 5.2.2 Action of the minimal quasidilaton . 78 5.2.3 Total Hamiltonian of the minimal quasidilaton . 80 5.3 Cosmology in MQD . 81 5.3.1 Cosmological attractor . 82 5.3.2 Perturbed attractor . 83 5.4 Predictions of MQD . 86 5.5 Summary . 88 6 Conclusion 89 A Tools for cosmology 93 A.1 Gauge transformations . 93 A.2 Integrating by parts the friction matrix . 93 A.3 Extracting the sound speeds . 94 A.3.1 Matter action for cosmology . 94 B Expressions for MTMG 97 B.1 Action and Hamiltonian . 97 C Cubic Horndeski 99 C.1 Lagrangian of H3 . 99 C.2 Primary constraints . 100 C.3 Primary Hamiltonian . 100 C.4 Secondary Hamiltonian . 102 C.5 Closedness of the algebra . 104 C.6 Total Hamiltonian . 104 D Expressions for MQD 105 D.1 Definition of MQD . 105 D.2 Subhorizon expressions . 106 D.2.1 Mass terms of scalar perturbations and Geff=GN ................... 107 D.2.2 GΦ=GN .......................................... 108.