"Massive Gravity"
Claudia de Rham 
1 Introduction
2 Massive and Interacting Fields
2.1 Proca field
2.2 Spin-2 field
2.3 From linearized diffeomorphism to full diffeomorphism invariance
2.4 Non-linear Stückelberg decomposition
2.5 Boulware–Deser ghost
I Massive Gravity from Extra Dimensions
3 Higher-Dimensional Scenarios
4 The Dvali–Gabadadze–Porrati Model
4.1 Gravity induced on a brane
4.2 Brane-bending mode
4.3 Phenomenology of DGP
4.4 Self-acceleration branch
4.5 Degravitation
5 Deconstruction
5.1 Formalism
5.2 Ghost-free massive gravity
5.3 Multi-gravity
5.4 Bi-gravity
5.5 Coupling to matter
5.6 No new kinetic interactions
II Ghost-free Massive Gravity
6 Massive, Bi- and Multi-Gravity Formulation: A Summary
7 Evading the BD Ghost in Massive Gravity
7.1 ADM formulation
7.2 Absence of ghost in the Stückelberg language
7.3 Absence of ghost in the vielbein formulation
7.4 Absence of ghosts in multi-gravity
8 Decoupling Limits
8.1 Scaling versus decoupling
8.2 Massive gravity as a decoupling limit of bi-gravity
8.3 Decoupling limit of massive gravity
8.4 Λ3-decoupling limit of bi-gravity
9 Extensions of Ghost-free Massive Gravity
9.1 Mass-varying
9.2 Quasi-dilaton
9.3 Partially massless
10 Massive Gravity Field Theory
10.1 Vainshtein mechanism
10.2 Validity of the EFT
10.3 Non-renormalization
10.4 Quantum corrections beyond the decoupling limit
10.5 Strong coupling scale vs cutoff
10.6 Superluminalities and (a)causality
10.7 Galileon duality
III Phenomenological Aspects of Ghost-free Massive Gravity
11 Phenomenology
11.1 Gravitational waves
11.2 Solar system
11.3 Lensing
11.4 Pulsars
11.5 Black holes
12 Cosmology
12.1 Cosmology in the decoupling limit
12.2 FLRW solutions in the full theory
12.3 Inhomogenous/anisotropic cosmological solutions
12.4 Massive gravity on FLRW and bi-gravity
12.5 Other proposals for cosmological solutions
IV Other Theories of Massive Gravity
13 New Massive Gravity
13.1 Formulation
13.2 Absence of Boulware–Deser ghost
13.3 Decoupling limit of new massive gravity
13.4 Connection with bi-gravity
13.5 3D massive gravity extensions
13.6 Other 3D theories
13.7 Black holes and other exact solutions
13.8 New massive gravity holography
13.9 Zwei-dreibein gravity
14 Lorentz-Violating Massive Gravity
14.1 SO(3)-invariant mass terms
14.2 Phase m1 = 0
14.3 General massive gravity (m0 = 0)
15 Non-local massive gravity
16 Outlook

15 Non-local massive gravity

The ghost-free theory of massive gravity proposed in Part II as well as the Lorentz-violating theories of Section 14 require an auxiliary metric. New massive gravity, on the other hand, can be formulated in a way that requires no mention of an auxiliary metric. Note however that all of these theories do break one copy of diffeomorphism invariance, and this occurs in bi-gravity as well and in the zwei dreibein extension of new massive gravity.

One of the motivations of non-local theories of massive gravity is to formulate the theory without any reference metric.34 This is the main idea behind the non-local theory of massive gravity introduced in [328*].35

Starting with the linearized equation about flat space-time of the Fierz–Pauli theory

1 δG μν − -m2 (h μν − h ημν) = 8πGT μν, (15.1 ) 2
where δG μν = ˆℰαμβν hαβ is the linearized Einstein tensor, this modified Einstein equation can be ‘covariantized’ so as to be valid about for any background metric. The linearized Einstein tensor δG μν gets immediately covariantized to the full Einstein tensor G μν. The mass term, on the other hand, is more subtle and involves non-local operators. Its covariantization can take different forms, and the ones considered in the literature that do not involve a reference metric are
{ ( −1 )T 1(hμν − hη μν) −→ □(g G μν ) Ref. [328] , (15.2 ) 2 38 gμν□ −g1R T Refs. [395, 229, 228]
where □g is the covariant d’Alembertian μν □g = g ∇ μ∇ ν and −1 □ g represents the retarded propagator. One could also consider a linear combination of both possibilities. Furthermore, any of these terms could also be implemented by additional terms that vanish on flat space, but one should take great care in ensuring that they do not propagate additional degrees of freedom (and ghosts).

Following [328*] we use the notation where T designates the transverse part of a tensor. For any tensor S μν,

T Sμν = S μν + ∇ (μSν), (15.3 )
with ∇ μST = 0 μν. In flat space we can infer the relation [328]
2 1 STμν = Sμν − --∂(μ∂αS ν)α + --2∂μ ∂ν∂α∂βS αβ. (15.4 ) □ □

The theory propagates what looks like a ghost-like instability irrespectively of the exact formulation chosen in (15.2*). However, it was recently argued that the would-be ghost is not a radiative degree of freedom and therefore does not lead to any vacuum decay. It remains an open question of whether the would be ghost can be avoided in the full nonlinear theory.

The cosmology of this model was studied in [395, 228]. The new contribution (15.2*) in the Einstein equation can play the role of dark energy. Taking the second formulation of (15.2*) and setting the graviton mass to m ≃ 0.67H0, where H0 is the Hubble parameter today, reproduces the observed amount of dark energy. The mass term acts as a dark fluid with effective time-dependent equation of state ωeff(a) ≃ − 1.04 − 0.02(1 − a), where a is the scale factor, and is thus phantom-like.

Since this theory is formulated at the level of the equations of motion and not at the level of the action and since it includes non-local operators it ought to be thought as an effective classical theory. These equations of motion should not be used to get some insight on the quantum nature of the theory nor on its quantum stability. New physics would kick in when quantum corrections ought to be taken into account. It remains an open question at the moment of how to embed nonlocal massive gravity into a consistent quantum effective field theory.

Notice, however, that an action principle was proposed in Ref. [402*], (focusing on four dimensions),

∫ [ ] 4 √ --- M-2Pl ¯ M-2Pl- ( -□--) μν S = d x − g 2 R + λ + 2M 2R μνh − M 2 G , (15.5 )
where the function h is defined as
2 h(z ) = □-+-m--1-|p (z)|e12Γ (0,p2s(z))+ 12γE, (15.6 ) □ z s
where γE = 0.577216 is the Euler’s constant, Γ (b,z) = ∫ ∞ tb−1e− t z is the incomplete gamma function, s is a integer s > 3 and ps(z) is a real polynomial of rank s. Upon deriving the equations of motion we recover the non-local massive gravity Einstein equation presented above [402],
m2 M P2l G μν + ---G μν = ----T μν, (15.7 ) □ 2
up to order 2 R corrections. We point out however that in this action derivation principle the operator □ −1 likely correspond to a symmetrized Green’s function, while in (15.2*) causality requires □−1 to represent the retarded one.

We stress, however, that this theory should be considered as a classical theory uniquely and not be quantized. It is an interesting question of whether or not the ghost reappears when considering quantum fluctuations like the ones that seed any cosmological perturbations. We emphasize for instance that when dealing with any cosmological perturbations, these perturbations have a quantum origin and it is important to rely of a theory that can be quantized to describe them.

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