RevModPhys.84.671 - REVIEWS OF MODERN PHYSICS VOLUME 84 APRILJUNE 2012 Theoretical aspects of massive gravity Kurt Hinterbichler Center for Particle

Theoretical aspects of massive gravity Kurt Hinterbichler * Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA (published 7 May 2012) Massive gravity has seen a resurgence of interest due to recent progress which has overcome its traditional problems, yielding an avenue for addressing important open questions such as the cosmological constant naturalness problem. The possibility of a massive graviton has been studied on and off for the past 70 years. During this time, curiosities such as the van Dam, Veltman, and Zakharov (vDVZ) discontinuity and the Boulware-Deser ghost were uncovered. These results are rederived in a pedagogical manner and the Stu ¨ckelberg formalism to discuss them from the modern effective field theory viewpoint is developed. Recent progress of the last decade is reviewed, including the dissolution of the vDVZ discontinuity via the Vainshtein screening mechanism, the existence of a consistent effective field theory with a stable hierarchy between the graviton mass and the cutoff, and the existence of particular interactions which raise the maximal effective field theory cutoff and remove the ghosts. In addition, some peculiarities of massive gravitons on curved space, novel theories in three dimensions, and examples of the emergence of a massive graviton from extra dimensions and brane worlds are reviewed. DOI: 10.1103/RevModPhys.84.671 PACS numbers: 04.50.Kd, 11.10. ± z CONTENTS I. Introduction 671 A. General relativity is massless spin 2 671 B. Modifying general relativity 673 C. History and outline 674 II. The Free Fierz-Pauli Action 675 A. Hamiltonian and degree of freedom count 675 B. Free solutions and graviton mode functions 676 C. Propagator 678 III. Linear Response to Sources 679 A. General solution to the sourced equations 680 B. Solution for a point source 680 C. The vDVZ discontinuity 681 IV. The St €u ckelberg Trick 682 A. Vector example 682 B. Graviton Stu ¨ckelberg and origin of the vDVZ discontinuity 683 V. Nonlinear Interactions 685 A. Massive general relativity 685 B. Spherical solutions and the Vainshtein radius 687 C. Nonlinear Hamiltonian and the Boulware-Deser mode 688 VI. The Nonlinear St €u ckelberg Formalism 690 A. Stu ¨ckelberg for gravity and the restoration of diffeomorphism invariance 690 B. Another way to Stu ¨ckelberg 692 VII. St €u ckelberg Analysis of Interacting Massive Gravity 693 A. Decoupling limit and breakdown of linearity 693 B. Ghosts 694 C. Resolution of the vDVZ discontinuity and the Vainshtein mechanism 694 D. Quantum corrections and the effective theory 695 VIII. The ± 3 Theory 697 A. Tuning interactions to raise the cutoff 697 B. The appearance of Galileons and the absence of ghosts 698 C. The ± 3 Vainshtein radius 699 D. The Vainshtein mechanism in the ± 3 theory 700 E. Quantum corrections in the ± 3 theory 700 IX. Brane worlds and the Resonance Graviton 701 A. The DGP action 701 B. Linear expansion 703 C. Resonance gravitons 705 X. Conclusions and Future Directions 706 Appendix: Total Derivative Combinations 707 I. INTRODUCTION