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, 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 solutions 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.

Massive Gravity

Beyond the cosmological standard model

After a decade and a half of research motivated by the accelerating universe, theory and experiment have a reached a certain level of maturity. The development of theoretical models beyond \Lambda, or smooth dark energy, often called modified gravity, has led to broader insights into a path forward, and a host of observational and experimental tests have been developed. In this review we present the current state of the field and describe a framework for anticipating developments in the next decade. We identify the guiding principles for rigorous and consistent modifications of the standard model, and discuss the prospects for empirical tests. We begin by reviewing attempts to consistently modify Einstein gravity in the infrared, focusing on the notion that additional degrees of freedom introduced by the modification must screen themselves from local tests of gravity. We categorize screening mechanisms into three broad classes: mechanisms which become active in regions of high Newtonian potential, those in which first derivatives become important, and those for which second derivatives are important. Examples of the first class, such as f(R) gravity, employ the familiar chameleon or symmetron mechanisms, whereas examples of the last class are galileon and massive gravity theories, employing the Vainshtein mechanism. In each case, we describe the theories as effective theories. We describe experimental tests, summarizing laboratory and solar system tests and describing in some detail astrophysical and cosmological tests. We discuss future tests which will be sensitive to different signatures of new physics in the gravitational sector. Parts that are more relevant to theorists vs. observers/experimentalists are clearly indicated, in the hope that this will serve as a useful reference for both audiences, as well as helping those interested in bridging the gap between them.

Beyond the Cosmological Standard Model

Aiming at nonexperts, the key mechanisms of higher-spin extensions of ordinary gravities in four dimensions and higher are explained. An overview of various no-go theorems for low-energy scattering of massless particles in flat spacetime is given. In doing so, a connection between the $S$-matrix and the Lagrangian approaches is made, exhibiting their relative advantages and weaknesses, after which potential loopholes for nontrivial massless dynamics are highlighted. Positive yes-go results for non-Abelian cubic higher-derivative vertices in constantly curved backgrounds are reviewed. Finally, how higher-spin symmetry can be reconciled with the equivalence principle in the presence of a cosmological constant leading to the Fradkin-Vasiliev vertices and Vasiliev's higher-spin gravity with its double perturbative expansion (in terms of numbers of fields and derivatives) is outlined.

How higher-spin gravity surpasses the spin-two barrier

Cubic interaction vertices for massive and massless higher spin fields

In this Letter we consider the problem of partial masslessness and unitarity in (A)dS using gauge invariant description of massive high spin particles. We show that for S = 2 and S = 3 cases such formalism allows one correctly reproduce all known results. Then we construct a gauge invariant formulation for massive particles of arbitrary integer spin s in arbitrary space-time dimension d. For d = 4 our results confirm the conjecture made recently by Deser and Waldron.

On Massive High Spin Particles in (A)dS

On massive spin 2 interactions

Frame-like gauge invariant formulation for massive high spin particles

In this paper we give explicit gauge invariant Lagrangian formulation for massive theories based on mixed symmetry tensors \Phi_{[\mu\nu],\alpha}, T_{[\mu\nu\alpha],\beta} and R_{[\mu\nu],[\alpha\beta]} both in Minkowski as well as in (Anti) de Sitter spaces. In particular, we study all possible massless and partially massless limits for such theories in (A)dS.

On Massive Mixed Symmetry Tensor Fields in Minkowski Space and (A)dS

Recently Boulanger and Leclercq have constructed a cubic four derivative 3 − 3 − 2 vertex for the interaction of spin 3 and spin 2 particles. This vertex is trivially invariant under the gauge transformations of the spin 2 field, so it seemed that it could be expressed in terms of the (linearized) Riemann tensor. And indeed in this paper we managed to reproduce this vertex in the form R∂Φ∂Φ, where R is the linearized Riemann tensor and Φ is the completely symmetric third rank tensor. Then we consider the deformation of this vertex to (A)dS space and show that such deformation produces a 'standard' gravitational interaction for spin 3 particles (in the linear approximation) in agreement with general construction of Fradkin and Vasiliev. Then we turn to the massive case and show that the same higher derivative terms allow one to extend the gauge invariant description of a massive spin 3 particle from constant curvature spaces to arbitrary gravitational backgrounds satisfying Rμν = 0.

https://iopscience.iop.org/article/10.1088/0264-9381/26/3/035022/pdf

Yu. M. Zinoviev

Papers

On Massive High Spin Particles in (A)dS

On massive spin 2 interactions

Frame-like gauge invariant formulation for massive high spin particles

On Massive Mixed Symmetry Tensor Fields in Minkowski Space and (A)dS

On spin 3 interacting with gravity