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Living Rev. Relativity, 17, (2014), 7 ,)6).' 2%6)%73 http://www.livingreviews.org/lrr-2014-7 doi:10.12942/lrr-2014-7 INRELATIVITY Massive Gravity Claudia de Rham CERCA & Physics Department Case Western Reserve University 10900 Euclid Ave, Cleveland, OH 44106, USA email: [email protected] http://www.phys.cwru.edu/~claudia/ Accepted: 18 July 2014 Published: 25 August 2014 Abstract We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali{Gabadadze{Porrati model (DGP), cascading gravity, and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware{Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solu- tions in these models. Finally, we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity. Keywords: General relativity, Gravity, Alternative theories of gravity, Modified theories of gravity, Massive gravity This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License. http://creativecommons.org/licenses/by-nc/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License: http://creativecommons.org/licenses/by-nc/3.0/de/. Figures that have been pre- viously published elsewhere may not be reproduced without consent of the original copyright holders. Because a Living Reviews article can evolve over time, we recommend to cite the article as follows: Claudia de Rham, \Massive Gravity", Living Rev. Relativity, 17, (2014), 7. URL (accessed <date>): http://www.livingreviews.org/lrr-2014-7 The date given as <date> then uniquely identifies the version of the article you are referring to. Article Revisions Living Reviews supports two ways of keeping its articles up-to-date: Fast-track revision. A fast-track revision provides the author with the opportunity to add short notices of current research results, trends and developments, or important publications to the article. A fast-track revision is refereed by the responsible subject editor. If an article has undergone a fast-track revision, a summary of changes will be listed here. Major update. A major update will include substantial changes and additions and is subject to full external refereeing. It is published with a new publication number. For detailed documentation of an article's evolution, please refer to the history document of the article's online version at http://www.livingreviews.org/lrr-2014-7. Contents 1 Introduction 7 2 Massive and Interacting Fields 10 2.1 Proca field......................................... 10 2.1.1 Maxwell kinetic term ............................... 10 2.1.2 Proca mass term ................................. 12 2.1.3 Abelian Higgs mechanism for electromagnetism . 13 2.1.4 Interacting spin-1 fields ............................. 14 2.2 Spin-2 field ........................................ 14 2.2.1 Einstein{Hilbert kinetic term .......................... 15 2.2.2 Fierz{Pauli mass term .............................. 16 2.2.3 Van Dam{Veltman{Zakharov discontinuity . 18 2.3 From linearized diffeomorphism to full diffeomorphism invariance . 20 2.4 Non-linear St¨uckelberg decomposition ......................... 22 2.5 Boulware{Deser ghost .................................. 24 I Massive Gravity from Extra Dimensions 26 3 Higher-Dimensional Scenarios 26 4 The Dvali{Gabadadze{Porrati Model 27 4.1 Gravity induced on a brane ............................... 28 4.1.1 Perturbations about flat spacetime ....................... 28 4.1.2 Spectral representation ............................. 30 4.2 Brane-bending mode ................................... 30 4.3 Phenomenology of DGP ................................. 33 4.3.1 Friedmann equation in de Sitter ........................ 33 4.3.2 General Friedmann equation .......................... 34 4.3.3 Observational viability of DGP ......................... 35 4.4 Self-acceleration branch ................................. 35 4.5 Degravitation ....................................... 37 4.5.1 Cascading gravity ................................ 38 5 Deconstruction 42 5.1 Formalism ......................................... 42 5.1.1 Metric versus Einstein{Cartan formulation of GR . 42 5.1.2 Gauge-fixing ................................... 43 5.1.3 Discretization in the vielbein .......................... 44 5.2 Ghost-free massive gravity ................................ 45 5.2.1 Simplest discretization .............................. 45 5.2.2 Generalized mass term .............................. 46 5.3 Multi-gravity ....................................... 47 5.4 Bi-gravity ......................................... 48 5.5 Coupling to matter .................................... 50 5.6 No new kinetic interactions ............................... 51 II Ghost-free Massive Gravity 53 6 Massive, Bi- and Multi-Gravity Formulation: A Summary 53 7 Evading the BD Ghost in Massive Gravity 56 7.1 ADM formulation ..................................... 57 7.1.1 ADM formalism for GR ............................. 57 7.1.2 ADM counting in massive gravity ........................ 58 7.1.3 Eliminating the BD ghost ............................ 58 7.2 Absence of ghost in the St¨uckelberg language ..................... 63 7.2.1 Physical degrees of freedom ........................... 63 7.2.2 Two-dimensional case .............................. 64 7.2.3 Full proof ..................................... 65 7.2.4 St¨uckelberg method on arbitrary backgrounds . 66 7.3 Absence of ghost in the vielbein formulation ...................... 68 7.4 Absence of ghosts in multi-gravity ........................... 69 8 Decoupling Limits 71 8.1 Scaling versus decoupling ................................ 71 8.2 Massive gravity as a decoupling limit of bi-gravity . 73 8.2.1 Minkowski reference metric ........................... 73 8.2.2 (A)dS reference metric .............................. 74 8.2.3 Arbitrary reference metric ............................ 74 8.3 Decoupling limit of massive gravity ........................... 76 8.3.1 Interaction scales ................................. 76 8.3.2 Operators below the scale Λ3 .......................... 76 8.3.3 Λ3-decoupling limit ................................ 77 8.3.4 Vector interactions in the Λ3-decoupling limit . 80 8.3.5 Beyond the decoupling limit ........................... 81 8.3.6 Decoupling limit on (Anti) de Sitter ...................... 82 8.4 Λ3-decoupling limit of bi-gravity ............................ 86 9 Extensions of Ghost-free Massive Gravity 89 9.1 Mass-varying ....................................... 90 9.2 Quasi-dilaton ....................................... 91 9.2.1 Theory ...................................... 91 9.2.2 Extended quasi-dilaton ............................. 93 9.3 Partially massless ..................................... 94 9.3.1 Motivations behind PM gravity ......................... 94 9.3.2 The search for a PM theory of gravity ..................... 95 10 Massive Gravity Field Theory 97 10.1 Vainshtein mechanism .................................. 97 10.1.1 Effective coupling to matter ........................... 97 10.1.2 Static and spherically symmetric configurations in Galileons . 99 10.1.3 Static and spherically symmetric configurations in massive gravity . 101 10.2 Validity of the EFT ................................... 102 10.3 Non-renormalization ................................... 103 10.4 Quantum corrections beyond the decoupling limit . 105 10.4.1 Matter loops ................................... 105 10.4.2 Graviton loops .................................. 105 10.5 Strong coupling scale vs cutoff ............................. 107 10.6 Superluminalities and (a)causality . 108 10.6.1 Superluminalities in Galileons . 109 10.6.2 Superluminalities in massive gravity . 111 10.6.3 Superluminalities vs Boulware{Deser ghost vs Vainshtein . 115 10.7 Galileon duality ...................................... 116 III Phenomenological Aspects of Ghost-free Massive Gravity 118 11 Phenomenology 118 11.1 Gravitational waves ................................... 118 11.1.1 Speed of propagation ............................... 118 11.1.2 Additional polarizations ............................. 119 11.2 Solar system ....................................... 120 11.3 Lensing .......................................... 122 11.4 Pulsars ........................................... 123 11.5 Black holes ........................................ 126 12 Cosmology 128 12.1 Cosmology in the decoupling limit . 128 12.2 FLRW solutions in the full theory . 131 12.2.1 Absence of flat/closed FLRW solutions . 131 12.2.2 Open FLRW solutions .............................. 131 12.3 Inhomogenous/anisotropic cosmological solutions . 133 12.3.1 Special isotropic and inhomogeneous solutions . 133 12.3.2 General anisotropic and inhomogeneous solutions . 136 12.4 Massive gravity on FLRW and bi-gravity . 136 12.4.1 FLRW reference metric ............................. 136 12.4.2 Bi-gravity ..................................... 138 12.5 Other proposals for cosmological solutions . 139 IV Other Theories of Massive Gravity 141 13
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