Massive Gravity
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Abstract
We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higherdimensional theory of general relativity, leading to the DvaliGabadadzePorrati model (DGP), cascading gravity, and ghostfree massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of BoulwareDeser 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, Lorentzviolating massive gravity and nonlocal massive gravity.
Keywords
General relativity Gravity Alternative theories of gravity Modified theories of gravity Massive gravity1 Introduction
For almost a century, the theory of general relativity (GR) has been known to describe the force of gravity with impeccable agreement with observations. Despite all the successes of GR the search for alternatives has been an ongoing challenge since its formulation. Far from a purely academic exercise, the existence of consistent alternatives to describe the theory of gravitation is actually essential to test the theory of GR. Furthermore, the open questions that remain behind the puzzles at the interface between gravity/cosmology and particle physics such as the hierarchy problem, the old cosmological constant problem and the origin of the latetime acceleration of the Universe have pushed the search for alternatives to GR.
While it was not formulated in this language at the time, from a more modern particle physics perspective GR can be thought of as the unique theory of a massless spin2 particle [287, 483, 175, 225, 76], and so in order to find alternatives to GR one should break one of the underlying assumptions behind this uniqueness theorem. Breaking Lorentz invariance and the notion of spin along with it is probably the most straightforward since nonLorentz invariant theories include a great amount of additional freedom. This possibility has been explored at length in the literature; see for instance [398] for a review. Nevertheless, Lorentz invariance is observationally well constrained by both particle and astrophysics. Another possibility is to maintain Lorentz invariance and the notion of spin that goes with it but to consider gravity as being the representation of a higher spin. This idea has also been explored; see for instance [466, 52] for further details. In this review, we shall explore yet another alternative: Maintaining the notion that gravity is propagated by a spin2 particle but considering this particle to be massive. From the particle physics perspective, this extension seems most natural since we know that the particles carrier of the electroweak forces have to acquire a mass through the Higgs mechanism.
Giving a mass to a spin2 (and spin1) field is an old idea and in this review we shall summarize the approach of Fierz and Pauli, which dates back to 1939 [226]. While the theory of a massive spin2 field is in principle simple to derive, complications arise when we include interactions between this spin2 particle and other particles as should be the case if the spin2 field is to describe the graviton.
At the linear level, the theory of a massless spin2 field enjoys a linearized diffeomorphism (diff) symmetry, just as a photon enjoys a U (1) gauge symmetry. But unlike for a photon, coupling the spin2 field with external matter forces this symmetry to be realized in a different way nonlinearly. As a result, GR is a fully nonlinear theory, which enjoys nonlinear diffeomorphism invariance (also known as general covariance or coordinate invariance). Even though this symmetry is broken when dealing with a massive spin2 field, the nonlinearities are inherited by the field. So, unlike a single isolated massive spin2 field, a theory of massive gravity is always fully nonlinear (and as a consequence nonrenormalizable) just as for GR. The fully nonlinear equivalent to GR for massive gravity has been a much more challenging theory to obtain. In this review we will summarize a few different approaches to deriving consistent theories of massive gravity and will focus on recent progress. See Ref. [309] for an earlier review on massive gravity, as well as Refs. [134] and [336] for other reviews relating Galileons and massive gravity.
When dealing with a theory of massive gravity two elements have been known to be problematic since the seventies. First, a massive spin2 field propagates five degrees of freedom no matter how small its mass. At first this seems to suggest that even in the massless limit, a theory of massive gravity could never resemble GR, i.e., a theory of a massless spin2 field with only two propagating degrees of freedom. This subtlety is at the origin of the vDVZ discontinuity (van DamVeltmanZakharov [465, 497]). The resolution behind that puzzle was provided by Vainshtein two years later and lies in the fact that the extra degree of freedom responsible for the vDVZ discontinuity gets screened by its own interactions, which dominate over the linear terms in the massless limit. This process is now relatively well understood [463] (see also Ref. [35] for a recent review). The Vainshtein mechanism also comes hand in hand with its own set of peculiarities like strong coupling and superluminalities, which we will discuss in this review.
A second element of concern in dealing with a theory of massive gravity is the realization that most nonlinear extensions of FierzPauli massive gravity are plagued with a ghost, now known as the BoulwareDeser (BD) ghost [75]. The past decade has seen a revival of interest in massive gravity with the realization that this BD ghost could be avoided either in a model of soft massive gravity (not a single massive pole for the graviton but rather a resonance) as in the DGP (DvaliGabadadzePorrati) model or its extensions [208, 209, 207], or in a threedimensional model of massive gravity as in ‘new massive gravity’ (NMG) [66] or more recently in a specific ghostfree realization of massive gravity (also known as dRGT in the literature) [144].

First, one can now more rigorously test massive gravity as an alternative to GR. We will summarize the different phenomenologies of these models and their theoretical as well as observational bounds through this review. Except in specific cases, the graviton mass is typically bounded to be a few times the Hubble parameter today, that is m ≲ 10^{−30} − 10^{−33} eV depending on the exact models. In all of these models, if the graviton had a mass much smaller than 10^{−33} eV, its effect would be unseen in the observable Universe and such a mass would thus be irrelevant. Fortunately there is still to date an open window of opportunity for the graviton mass to be within an interesting range and providing potentially new observational signatures.

Second, these developments have opened up the door for theories of interacting metrics, a success long awaited. Massive gravity was first shown to be expressible on an arbitrary reference metric in [296]. It was then shown that the reference metric could have its own dynamics leading to the first consistent formulation of bigravity [293]. In bigravity two metrics are interacting and the mass spectrum is that of a massless spin2 field interacting with a massive spin2 field. It can, therefore, be seen as the theory of general relativity interacting (fully nonlinearly) with a massive spin2 field. This is a remarkable new development in both field theory and gravity.

The formulation of massive gravity and bigravity in the vielbein language were shown to be both analytic and much more natural and allowed for a general formulation of multigravity [314] where an arbitrary number of spin2 fields may interact together.

Finally, still within the theoretical progress front, all of these successes provided full and definite proof for the absence of BoulwareDeser ghosts in these types of theories; see [295], which has then been translated into a multitude of other languages. This also opens the door for new types of theories that can propagate fewer degrees than naively thought.
This review is organized as follows: We start by setting the formalism for massive and massless spin1 and 2 fields in Section 2 and emphasize the Stückelberg language both for the Proca and the FierzPauli fields. In Part I we then derive consistent theories using a higherdimensional framework, either using a braneworld scenario à la DGP in Section 4, or via a discretization^{1} (or KaluzaKlein reduction) of the extra dimension in Section 5. This second approach leads to the theory of ghostfree massive gravity (also known as dRGT) which we review in more depth in Part II. Its formulation is summarized in Section 6, before tackling other interesting aspects such as the fate of the BD ghost in Section 7, deriving its decoupling limit in Section 8, and various extensions in Section sec:Extensions. The Vainshtein mechanism and other related aspects are discussed in Section 10. The phenomenology of ghostfree massive gravity is then reviewed in Part III including a discussion of solarsystem tests, gravitational waves, weak lensing, pulsars, black holes and cosmology. We then conclude with other related theories of massive gravity in Part IV, including new massive gravity, Lorentz breaking theories of massive gravity and nonlocal versions.
Notations and conventions: Throughout this review, we work in units where the reduced Planck constant ℏ and the speed of light c are set to unity. The gravitational Newton constant is related to the Planck scale by \(8\pi {G_N} = M_{{\rm{P1}}}^{ 2}\). Unless specified otherwise, d represents the number of spacetime dimensions. We use the mainly + convention (−+ ⋯ +) and space indices are denoted by i, j, ⋯ = 1, ⋯, d − 1 while 0 represents the timelike direction, x^{0} = t.
We also use the symmetric convention: \((a,b) = {1 \over 2}(ab + ba)\) and \([a,b] = {1 \over 2}(ab  ba)\). Throughout this review, square brackets of a tensor indicates the trace of tensor, for instance \([{\mathbb X}] = {\mathbb X}_\mu ^\mu, [{{\mathbb X}^2}] = {\mathbb X}_v^\mu {\mathbb X}_\mu ^v\), etc. … We also use the notation Π_{ μν } = d_{ μ }d_{ ν } and \({\mathcal I} = \delta _{v \cdot}^\mu \,{\varepsilon _{\mu v\alpha \beta}}\) and ε_{ abcde } represent the LeviCevita symbol in respectively four and five dimensions, ε_{0123} = ε_{01234} = 1 = ε^{0123}.
2 Massive and Interacting Fields
2.1 Proca field
2.1.1 Maxwell kinetic term
We now proceed to establish the behavior of the different degrees of freedom (dofs) present in this theory. A priori, a Lorentz vector field A_{ μ } in four dimensions could have up to four dofs, which we can split as a transverse contribution \(A_\mu ^ \bot\) satisfying \({\partial ^\mu}A_\mu ^ \bot = 0\) bearing a priori three dofs and a longitudinal mode χ with \({\mathcal X}\)
2.1.1.1 Helicity0 mode
2.1.1.2 Helicity1 mode and gauge symmetry
2.1.2 Proca mass term
Before moving to the Abelian Higgs mechanism, which provides a dynamical way to give a mass to bosons, we first comment on the discontinuity in number of dofs between the massive and massless case. When considering the Proca action (2.16) with the properly normalized fields \(A_\mu ^ \bot\) and χ, one does not recover the massless Maxwell action (2.9) or (2.10) when sending the boson mass m → 0. A priori, this seems to signal the presence of a discontinuity which would allow us to distinguish between for instance a massless photon and a massive one no matter how tiny the mass. In practice, however, the difference is physically indistinguishable so long as the photon couples to external sources in a way which respects the U (1) symmetry. Note however that quantum anomalies remain sensitive to the mass of the field so the discontinuity is still present at this level, see Refs. [197, 204].
Notice that in the massive case no U (1) symmetry is present and the source needs not be conserved. However, the previous argument remains unchanged so long as ∂_{ μ }J^{ μ } goes to zero in the massless limit at least as quickly as the mass itself. If this condition is violated, then the helicity0 mode ought to be included in the exchange amplitude (2.21). In parallel, in the massless case the nonconserved source provides a new kinetic term for the longitudinal mode which then becomes dynamical.
2.1.3 Abelian Higgs mechanism for electromagnetism
Associated with the absence of an intrinsic discontinuity in the massless limit is the existence of a Higgs mechanism for the vector field whereby the vector field acquires a mass dynamically. As we shall see later, the situation is different for gravity where no equivalent dynamical Higgs mechanism has been discovered to date. Nevertheless, the tools used to describe the Abelian Higgs mechanism and in particular the introduction of a Stückelberg field will prove useful in the gravitational case as well.
2.1.4 Interacting spin1 fields
2.2 Spin2 field
As we have seen in the case of a vector field, as long as it is local and Lorentzinvariant, the kinetic term is uniquely fixed by the requirement that no ghost be present. Moving now to a spin2 field, the same argument applies exactly and the EinsteinHilbert term appears naturally as the unique kinetic term free of any ghostlike instability. This is possible thanks to a symmetry which projects out all unwanted dofs, namely diffeomorphism invariance (linear diffs at the linearized level, and nonlinear diffs/general covariance at the nonlinear level).
2.2.1 EinsteinHilbert kinetic term
In d ≥ 3 spacetime dimensions, gravitational waves have d (d +1)/2−2d = d (d −3)/2 independent polarizations. This means that in three dimensions there are no gravitational waves and in five dimensions they have five independent polarizations.
2.2.2 FierzPauli mass term
As seen in seen in Section 2.2.1, for a local and Lorentzinvariant theory, the linearized kinetic term is uniquely fixed by the requirement that longitudinal modes propagate no ghost, which in turn prevents that operator from exciting these modes altogether. Just as in the case of a massive spin1 field, we shall see in what follows that the longitudinal modes can nevertheless be excited when including a mass term. In what follows we restrict ourselves to linear considerations and spare any nonlinearity discussions for Parts I and II. See also [327] for an analysis of the linearized FierzPauli theory using Bardeen variables.
2.2.2.1 Propagating degrees of freedom
Terms on the first line represent the kinetic terms for the different fields while the second line represent the mass terms and mixing.
The degrees of freedom have not yet been split into their mass eigenstates but on doing so one can easily check that all the degrees of freedom have the same positive mass square m^{2}.
Most of the phenomenology and theoretical consistency of massive gravity is related to the dynamics of the helicity0 mode. The coupling to matter occurs via the coupling \({h_{\mu v}}{T^{\mu u}} = {\tilde h_{\mu v}}{T^{\mu v}} + \pi T\), where T is the trace of the external stressenergy tensor. We see that the helicity0 mode couples directly to conserved sources (unlike in the case of the Proca field) but the helicity1 mode does not. In most of what follows we will thus be able to ignore the helicity1 mode.
2.2.2.2 Higgs mechanism for gravity
As we shall see in Section 9.1, the graviton mass can also be promoted to a scalar function of one or many other fields (for instance of a different scalar field), m = m (ψ). We can thus wonder whether a dynamical Higgs mechanism for gravity can be considered where the field(s) ψ start in a phase for which the graviton mass vanishes, m (ψ) = 0 and dynamically evolves to acquire a nonvanishing vev for which m (ψ) ≠ 0. Following the same logic as the Abelian Higgs for electromagnetism, this strategy can only work if the number of dofs in the massless phase m = 0 is the same as that in the massive case m ≠ 0. Simply promoting the mass to a function of an external field is thus not sufficient since the graviton helicity0 and 1 modes would otherwise be infinitely strongly coupled as m → 0.
To date no candidate has been proposed for which the graviton mass could dynamically evolve from a vanishing value to a finite one without falling into such strong coupling issues. This does not imply that Higgs mechanism for gravity does not exist, but as yet has not been found. For instance on AdS, there could be a Higgs mechanism as proposed in [431], where the mass term comes from integrating out some conformal fields with slightly unusual (but not unphysical) ‘transparent’ boundary conditions. This mechanism is specific to AdS and to the existence of timelike boundary and would not apply on Minkowski or dS.
2.2.3 Van DamVeltmanZakharov discontinuity
As in the case of spin1, the massive spin2 field propagates more dofs than the massless one. Nevertheless, these new excitations bear no observational signatures for the spin1 field when considering an arbitrarily small mass, as seen in Section 2.1.2. The main reason for that is that the helicity0 polarization of the photon couple only to the divergence of external sources which vanishes for conserved sources. As a result no external sources directly excite the helicity0 mode of a massive spin1 field. For the spin2 field, on the other hand, the situation is different as the helicity0 mode can now couple to the trace of the stressenergy tensor and so generic sources will excite not only the 2 helicity2 polarization of the graviton but also a third helicity0 polarization, which could in principle have dramatic consequences. To see this more explicitly, let us compute the gravitational exchange amplitude between two sources T^{ μν } and T′^{ μν } in both the massive and massless gravitational cases.
In the massless case, the theory is diffeomorphism invariant. When considering coupling to external sources, of the form h_{ μν }T^{ μν }, we thus need to ensure that the symmetry be preserved, which implies that the stressenergy tensor T^{ μν } should be conserved ∂_{ μ }T^{ μν } = 0. When computing the gravitational exchange amplitude between two sources we thus restrict ourselves to conserved ones. In the massive case, there is a priori no reasons to restrict ourselves to conserved sources, so long as their divergences cancel in the massless limit m → 0.
2.2.3.1 Massive spin2 field
2.2.3.2 Massless spin2 field
Another ‘nonGR’ effect was also recently pointed out in Ref. [280] where a linear analysis showed that massive gravity predicts different spinorientations for spinning objects.
2.3 From linearized diffeomorphism to full diffeomorphism invariance
When considering the massless and noninteractive spin2 field in Section 2.2.1, the linear gauge invariance (2.38) is exact. However, if this field is to be probed and communicates with the rest of the world, the gauge symmetry is forced to include nonlinear terms which in turn forces the kinetic term to become fully nonlinear. The result is the wellknown fully covariant EinsteinHilbert term \(M_{{\rm{Pl}}}^2\sqrt { gR}\), where R is the scalar curvature associated with the metric g_{ μν }, = η_{ μν } + h_{ μν }/M_{pl}.
The situation is very different from that of a spin1 field as seen earlier, where coupling with other fields can be implemented at the linear order without affecting the U (1) gauge symmetry. The difference is that in the case of a U (1) symmetry, there is a unique nonlinear completion of that symmetry, i.e., the unique nonlinear completion of a U (1) is nothing else but a U (1). Thus any nonlinear Lagrangian which preserves the full U (1) symmetry will be a consistent interacting theory. On the other hand, for spin2 fields, there are two, and only two ways to nonlinearly complete linear diffs, one as linear diffs in the full theory and the other as full nonlinear diffs. While it is possible to write selfinteractions which preserve linear diffs, there are no interactions between matter and h_{ μν }. which preserve linear diffs. Thus any theory of gravity must exhibit full nonlinear diffs and is in this sense what leads us to GR.
2.4 Nonlinear Stückelberg decomposition
2.4.1 On the need for a reference metric
We have introduced the spin2 field h_{ μν } as the perturbation about flat spacetime. When considering the theory of a field of given spin it is only natural to work with Minkowski as our spacetime metric, since the notion of spin follows from that of Poincaré invariance. Now when extending the theory nonlinearly, we may also extend the theory about different reference metric. When dealing with a reference metric different than Minkowski, one loses the interpretation of the field as massive spin2, but one can still get a consistent theory. One could also wonder whether it is possible to write a theory of massive gravity without the use of a reference metric at all. This interesting question was investigated in [75], where it shown that the only consistent alternative is to consider a function of the metric determinant. However, as shown in [75], the consistent function of the determinant is the cosmological constant and does not provide a mass for the graviton.
2.4.2 Nonlinear Stückelberg
Full diffeomorphism invariance (or covariance) indicates that the theory should be built out of scalar objects constructed out of the metric g_{ μν } and other tensors. However, as explained previously a theory of massive gravity requires the notion of a reference metric^{6} f_{ μν } (which may be Minkowski f_{ μν } = η_{ μν }) and at the linearized level, the mass for gravity was not built out of the full metric g_{ μν }, but rather out of the fluctuation h_{ μν } about this reference metric which does not transform as a tensor under general coordinate transformations. As a result the mass term breaks covariance.
This Stückelberg trick for massive gravity dates already from Green and Thorn [267] and from Siegel [446], introduced then within the context of open string theory. In the same way as the massless graviton naturally emerges in the closed string sector, open strings also have spin2 excitations but whose lowest energy state is massive at tree level (they only become massless once quantum corrections are considered). Thus at the classical level, open strings contain a description of massive excitations of a spin2 field, where gauge invariance is restored thanks to same Stückelberg fields as introduced in this section. In open string theory, these Stückelberg fields naturally arise from the ghost coordinates. When constructing the nonlinear theory of massive gravity from extra dimension, we shall see that in that context the Stückelberg fields naturally arise at the shift from the extra dimension.
2.4.3 Alternative Stückelberg trick
2.4.4 Helicity decomposition
2.4.5 Nonlinear FierzPauli
2.5 BoulwareDeser ghost
The easiest way to see the appearance of a ghost at the nonlinear level is to follow the Stückelberg trick nonlinearly and observe the appearance of an Ostrogradsky instability [111, 173], although the original formulation was performed in unitary gauge in [75] in the ADM language (Arnowitt, Deser and Misner, see Ref. [29]). In this section we shall focus on the flat reference metric, ƒ_{ μν } = η_{ μν }
Alternatively the mass term was also generalized to include curvature invariants as in Ref. [69]. This theory was shown to be ghostfree at the linear level on FLRW but not yet nonlinearly.
2.5.1 Function of the FierzPauli mass term
Instead to prevent the presence of the BD ghost fully nonlinearly (or equivalently about any background), one should construct the mass term (or rather potential term) in such a way, that all the higher derivative operators involving the helicity0 mode (∂^{2}π)^{ n } are total derivatives. This is precisely what is achieved in the “ghostfree” model of massive gravity presented in Part II. In the next Part I we shall use higher dimensional GR to get some insight and intuition on how to construct a consistent theory of massive gravity.
3 Part I Massive Gravity from Extra Dimensions
4 HigherDimensional Scenarios

The fivedimensional theory is explicitly covariant.

A massless spin2 field in five dimensions has five degrees of freedom which corresponds to the correct number of dofs for a massive spin2 field in four dimensions without the pathological BD ghost.
Recently, another higher dimensional embedding of bigravity was proposed in Ref. [495]. Rather than performing a discretization of the extra dimension, the idea behind this model is to consider a twobrane DGP model, where the radion or separation between these branes is stabilized via a GoldbergerWise stabilization mechanism [255] where the brane and the bulk include a specific potential for the radion. At low energy the mass spectrum can be truncated to a massless mode and a massive mode, reproducing a bigravity theory. However, the stabilization mechanism involves a relatively low scale and the correspondence breaks down above it. Nevertheless, this provides a first proof of principle for how to embed such a model in a higherdimensional picture without discretization and could be useful to tackle some of the open questions of massive gravity.
In what follows we review how fivedimensional gravity is a useful starting point in order to generate consistent fourdimensional theories of massive gravity, either for softmassive gravity à la DGP and its extensions, or for hard massive gravity following a deconstruction framework.
The DGP model has played the role of a precursor for many developments in modified and massive gravity and it is beyond the scope of this review to summarize all of them. In this review we briefly summarize the DGP model and some key aspects of its phenomenology, and refer the reader to other reviews (see for instance [232, 390, 234]) for more details on the subject.
In this section, A, B, C ⋯ = 0, …, 4 represent fivedimensional spacetime indices and μ, ν, α ⋯ = 0, …, 3 label fourdimensional spacetime indices. y = x^{4} represents the fifth additional dimension, {x^{ A }} = {x^{ μ }, y}. The fivedimensional metric is given by ^{(5)}g_{ ab } (x, y) while the fourdimensional metric is given by g_{ μν } (x). The fivedimensional scalar curvature is ^{(5)}R [G ] while R = R [g ] is the fourdimensional scalarcurvature. We use the same notation for the Einstein tensor where ^{(5)}G_{ ab } is the fivedimensional one and G_{ μν } represents the fourdimensional one built out of g_{ μν }.
When working in the EinsteinCartan formalism of gravity, \({\mathbb A},{\mathbb B},{\mathbb C}\) label fivedimensional Lorentz indices and a,b,c = ⋯ label the fourdimensional ones.
5 The DvaliGabadadzePorrati Model
The idea behind the DGP model [209, 208, 207] is to start with a fourdimensional braneworld in an infinite sizeextra dimension. A priori gravity would then be fully fivedimensional, with respective Planck scale M_{5}, but the matter fields localized on the brane could lead to an induced curvature term on the brane with respective Planck scale M_{Pl}. See [22] for a potential embedding of this model within string theory.
At small distances the induced curvature dominates and gravity behaves as in four dimensions, while at large distances the leakage of gravity within the extra dimension weakens the force of gravity. The DGP model is thus a model of modified gravity in the infrared, and as we shall see, the graviton effectively acquires a soft mass, or resonance.
5.1 Gravity induced on a brane
5.1.1 Perturbations about flat spacetime
5.1.2 Spectral representation
We see that the spectral density is positive for any μ^{2} > 0, confirming the fact that about the normal (flat) branch of DGP there is no ghost.
5.2 Branebending mode
5.2.1 Fivedimensional gaugefixing
In Section 4.1.1 we have remained vague about the gaugefixing and the implications for the brane position. The branebending mode is actually important to keep track of in DGP and we shall do that properly in what follows by keeping all the modes.
After fixing the de Donder gauge (4.5), we can make the addition gauge transformation x^{ A } → x^{ A } + ξ^{ A }, and remain in de Donder gauge provided satisfies linearly □_{5}ξ^{ A } = 0. This residual gauge freedom can be used to further fix the gauge on the brane (see [389] for more details, we only summarize their derivation here).
5.2.2 Fourdimensional Gaugefixing
Keeping the brane at the fixed position y = 0 imposes = 0 since we need ξ^{ A } (y = 0) = 0 and should be bounded as y → ∞ (the situation is slightly different in the selfaccelerating branch and this mode can lead to a ghost, see Section 4.4 as well as [361, 98]).
5.2.3 Decoupling limit
We will be discussing the meaning of ‘decoupling limits’ in more depth in the context of multigravity and ghostfree massive gravity in Section 8. The main idea behind the decoupling limit is to separate the physics of the different modes. Here we are interested in following the interactions of the helicity0 mode without the complications from the standard helicity2 interactions that already arise in GR. For this purpose we can take the limit M_{Pl} → ∞ while simultaneously sending \({m_0} = M_5^3/M_{{\rm{Pl}}}^2 \to 0\) while keeping the scale \(\Lambda = {(m_0^2{M_{{\rm{Pl}}}})^{1/3}}\) fixed. This is the scale at which the first interactions arise in DGP.
5.3 Phenomenology of DGP
The phenomenology of DGP is extremely rich and has led to many developments. In what follows we review one of the most important implications of the DGP for cosmology which the existence of selfaccelerating solutions. The cosmology and phenomenology of DGP was first derived in [159, 163] (see also [388, 385, 387, 386]).
5.3.1 Friedmann equation in de Sitter
5.3.2 General Friedmann equation
This modified Friedmann equation has been derived assuming a constant H, which is only consistent if the energy density is constant (i.e., a cosmological constant). We can now derive the generalization of this Friedmann equation for nonconstant H. This amounts to account for Ḣ and other derivative corrections which might have been omitted in deriving this equation by assuming that was constant. But the Friedmann equation corresponds to the Hamiltonian constraint equation and higher derivatives (e.g., Ḣ ⊃ ä and higher derivatives of H) would imply that this equation is no longer a constraint and this loss of constraint would imply that the theory admits a new degree of freedom about generic backgrounds namely the BD ghost (see the discussion of Section 7).
However, in DGP we know that the BD ghost is absent (this is ensured by the fivedimensional nature of the theory, in five dimensions we start with five dofs, and there is thus no sixth BD mode). So the Friedmann equation cannot include any derivatives of H, and the Friedmann equation obtained assuming a constant H is actually exact in FLRW even if H is not constant. So the constraint (4.47) is the exact Friedmann equation in DGP for any energy density ρ on the brane.
The same trick can be used for massive gravity and bigravity and the Friedmann equations (12.51), (12.52) and (12.54) are indeed free of any derivatives of the Hubble parameter.
5.3.3 Observational viability of DGP
Independently of the ghost issue in the selfaccelerating branch of the model, there has been a vast amount of investigation on the observational viability of both the selfaccelerating branch and the normal (stable) branch of DGP. First because many of these observations can apply equally well to the stable branch of DGP (modulo a minus sign in some of the cases), and second and foremost because DGP represents an excellent archetype in which ideas of modified gravity can be tested.

Tests of the Friedmann equation. This test was performed mainly using Supernovae, but also using Baryonic Acoustic Oscillations and the CMB so as to fix the background history of the Universe [162, 217, 221, 286, 391, 23, 405, 481, 304, 382, 462]. Current observations seem to slightly disfavor the additional term in the Friedmann equation of DGP, even in the normal branch where the latetime acceleration of the Universe is due to a cosmological constant as in ΛCDM. These put bounds on the graviton mass in DGP to the order of m_{0} ≲ 10^{−1} H_{0}, where H_{0} is the Hubble parameter today (see Ref. [492] for the latest bounds at the time of writing, including data from Planck). Effectively this means that in order for DGP to be consistent with observations, the graviton mass can have no effect on the latetime acceleration of the Universe.

Tests of an extra fifth force, either within the solar system, or during structure formation (see for instance [362, 260, 452, 451, 222, 482] Refs. [453, 337, 442] for Nbody simulations as well as Ref. [17, 441] using weak lensing).
Evading fifth force experiments will be discussed in more detail within the context of the Vainshtein mechanism in Section 10.1 and thereafter, and we save the discussion to that section. See Refs. [388, 385, 387, 386, 444] for a fivedimensional study dedicated to DGP. The study of cosmological perturbations within the context of DGP was also performed in depth for instance in [367, 92].
5.4 Selfacceleration branch
This realization has opened a new field of study in its own right. It is beyond the scope of this review on massive gravity to summarize all the interesting developments that arose in the past decade and we simply focus on a few elements namely the presence of a ghost in this selfaccelerating branch as well as a few cosmological observations.
5.4.1 Ghost
The existence of a ghost on the selfaccelerating branch of DGP was first pointed out in the decoupling limit [389, 411], where the helicity0 mode of the graviton is shown to enter with the wrong sign kinetic in this branch of solutions. We emphasize that the issue of the ghost in the selfaccelerating branch of DGP is completely unrelated to the sixth BD ghost on some theories of massive gravity. In DGP there are five dofs one of which is a ghost. The analysis was then generalized in the fully fledged fivedimensional theory by K. Koyama in [360] (see also [263, 361] and [98]).

In [307] it was shown that when the effective mass is within the forbidden Higuchiregion, the helicity0 mode of graviton has the wrong sign kinetic term and is a ghost.

Outside this forbidden region, when \(m_{{\rm{eff}}}^2 > 2{H^2}\), the zeromode of the graviton is healthy but there exists a new normalizable branebending mode in the selfaccelerating branch^{8} which is a genuine degree of freedom. For \(m_{{\rm{eff}}}^2 > 2{H^2}\) the branebending mode was shown to be a ghost.

Finally, at the critical mass \(m_{{\rm{eff}}}^2 > 2{H^2}\) (which happens when no matter nor cosmological constant is present on the brane), the branebending mode takes the role of the helicity0 mode of the graviton, so that the theory graviton still has five degrees of freedom, and this mode was shown to be a ghost as well.
5.4.2 Evading the ghost?
Different ways to remove the ghosts were discussed for instance in [325] where a second brane was included. In this scenario it was then shown that the graviton could be made stable but at the cost of including a new spin0 mode (that appears as the mode describing the distance between the branes).
Alternatively it was pointed out in [233] that if the sign of the extrinsic curvature was flipped, the selfaccelerating solution on the brane would be stable.
We do not discuss this model any further in what follows since the graviton admits a zero (massless) mode. It is feasible that this model can be understood as a bigravity theory where the massive mode is a resonance. It would also be interesting to see how this model fits in with the Galileon theories [412] which admit stable selfaccelerating solution.
In what follows, we go back to the standard DGP model be it the selfaccelerating branch (ϵ = 1) or the normal branch (ϵ = − 1).
5.5 Degravitation
 1.
First, gravity must be weaker in the infrared and effectively massive [216] so that the effect of IR sources can be degravitated.
 2.
Second, there must exist some (nearly) static attractor solutions towards which the system can evolve at latetime for arbitrary value of the vacuum energy or cosmological constant.
5.5.1 Flat solution with a cosmological constant
The first requirement is present in DGP, but as was shown in [216] in DGP gravity is not ‘sufficiently weak’ in the IR to allow degravitation solutions. Nevertheless, it was shown in [164] that the normal branch of DGP satisfies the second requirement for any negative value of the cosmological constant. In these solutions the fivedimensional spacetime is not Lorentz invariant, but in a way which would not (at this background level) be observed when confined on the fourdimensional brane.
For positive values of the cosmological constant, DGP does not admit a (nearly) static solution. This can be understood at the level of the decoupling limit using the arguments of [216] and generalized for other mass operators.
5.5.2 Extensions
This realization has motivated the search for theories of massive gravity with 0 ≤ α < 1/2, and especially the extension of DGP to higher dimensions where the parameter can get as close to zero as required. This is the main motivation behind higher dimensional DGP [359, 240] and cascading gravity [135, 148, 132, 149] as we review in what follows. (In [433] it was also shown how a regularized version of higher dimensional DGP could be free of the strong coupling and ghost issues).
Furthermore, it was shown in [133] that it satisfies both requirements presented above to potentially help degravitating a cosmological constant. Unfortunately at higher orders this model is plagued with the BD ghost [291] unless the boundary conditions are chosen appropriately [59]. For this reason we will not review this model any further in what follows and focus instead on the ghostfree theory of massive gravity derived in [137, 144]
5.5.3 Cascading gravity
5.5.3.1 Deficit angle
This interesting feature has lead to many potential ways to tackle the cosmological constant by considering our Universe to live in a 3 + 1dimensional brane embedded in two or more large extra dimensions. (See Refs. [4, 3, 408, 414, 80, 470, 458, 459, 86, 82, 247, 333, 471, 81, 426, 409, 373, 85, 460, 155] for the supersymmetric largeextradimension scenario as an alternative way to tackle the cosmological constant problem). Extending the DGP to more than one extra dimension could thus provide a natural way to tackle the cosmological constant problem.
5.5.3.2 Spectral representation
Working back in terms of the spectral representation of the propagator as given in (4.19), this means that the propagator goes to 1/k in the IR as μ → 0 when n = 1 (as we know from DGP), while it goes to a constant for n > 1. So for more than one extra dimension, the theory tends towards that of a hard mass graviton in the far IR, which corresponds to α → 0 in the parametrization of (4.50). Following the arguments of [216] such a theory should thus be a good candidate to tackle the cosmological constant problem.
5.5.3.3 A brane on a brane
Both the spectral representation and the fact that codimensiontwo (and higher) branes can accommodate for a cosmological constant while remaining flat has made the field of highercodimension branes particularly interesting.
However, as shown in [240] and [135, 148, 132, 149], the straightforward extension of DGP to two large extra dimensions leads to ghost issues (sixth mode with the wrong sign kinetic term, see also [290, 70]) as well as divergences problems (see Refs. [256, 131, 130, 423, 422, 355, 83]).

Either smoothing out the brane [240, 148] (this means that one should really consider a sixdimensional curvature on both the smoothed 4 + 1 and on the 3 + 1dimensional branes, which is something one would naturally expect^{9}).

Or by including some tension on the 3 + 1 brane (which is also something natural since the setup is designed to degravitate a large cosmological constant on that brane). This was shown to be ghost free in the decoupling limit in [135] and in the full theory in [150].
As already mentioned in two large extra dimensional models there is to be a maximal value of the cosmological constant that can be considered which is related to the sixdimensional Planck scale. Since that scale is in turn related to the effective mass of the graviton and since observations set that scale to be relatively small, the model can only take care of a relatively small cosmological constant. Nevertheless, it still provides a proof of principle on how to evade Weinberg’s nogo theorem [484].
6 Deconstruction
As for DGP and its extensions, to get some insight on how to construct a fourdimensional theory of single massive graviton, we can start with fivedimensional general relativity. This time, we consider the extra dimension to be compactified and of finite size R, with periodic boundary conditions. It is then natural to perform a KaluzaKlein decomposition and to obtain a tower of KaluzaKlein graviton mode in four dimensions. The zero mode is then massless and the higher modes are all massive with mass separation m = 1/R. Since the graviton mass is constant in this formalism we omit the subscript 0 in the rest of this review.
Rather than starting directly with a KaluzaKlein decomposition (discretization in Fourier space), we perform instead a discretization in real space, known as “deconstruction” of fivedimensional gravity [24, 25, 170, 168, 28, 443, 340]. The deconstruction framework helps making the connection with massive gravity more explicit. However, we can also obtain multigravity out of it which is then completely equivalent to the KaluzaKlein decomposition (after a nonlinear field redefinition).
The idea behind deconstruction is simply to ‘replace’ the continuous fifth dimension y by a series of N sites y_{ j } separated by a distance ℓ = R/N. So that the fivedimensional metric is replaced by a set of interacting metrics depending only on x.
In what follows, we review the procedure derived in [152] to recover fourdimensional ghostfree massive gravity as well as bi and multigravity out of fivedimensional GR. The procedure works in any dimensions and we only focus to deconstructing fivedimensional GR for sake of concreteness.
6.1 Formalism
6.1.1 Metric versus EinsteinCartan formulation of GR
The counting of the degrees of freedom in both languages is of course equivalent and goes as follows: In dspacetime dimensions, the metric has d (d + 1)/2 independent components. Covariance removes 2d of them,^{10} which leads to \({{\mathcal N}_d} = d(d  3)/2\) independent degrees of freedom. In fourdimensions, we recover the usual \({{\mathcal N}_4} = 2\) independent polarizations for gravitational waves. In fivedimensions, this leads to \({{\mathcal N}_5} = 5\) degrees of freedom which is the same number of degrees of freedom as a massive spin2 field in four dimensions. This is as expect from the KaluzaKlein philosophy (massless bosons in d + 1 dimensions have the same number of degrees of freedom as massive bosons in d dimensions — this counting does not directly apply to fermions).
In the EinsteinCartan formulation, the counting goes as follows: The vielbein has d^{2} independent components. Covariance removes 2d of them, and the additional global Lorentz invariance removes an additional d (d − 1)/2, leading once again to a total of \({{\mathcal N}_d} = d(d  3)/2\) independent degrees of freedom.
In the bosonic sector, one can convert the covariant action of bosonic fields (e.g., of scalar, vector fields, etc.…) between the vielbein and the metric language without much confusion, however this is not possible for the covariant Dirac action, or other halfspin fields. For these types of matter fields, the EinsteinCartan Formulation of GR is more fundamental than its metric formulation. In doubt, one should always start with the vielbein formulation. This is especially important in the case of deconstruction when a discretization in the metric language is not equivalent to a discretization in the vielbein variables. The same holds for KaluzaKlein decomposition, a point which might have been underappreciated in the past.
6.1.2 Gaugefixing
6.1.3 Discretization in the vielbein
From the metric language, we thus see that the discretization procedure amounts to converting the extrinsic curvature to an interaction between neighboring sites through the building block \({\mathcal K}_v^\mu [{g_j},{g_{j + 1}}]\)
6.2 Ghostfree massive gravity
6.2.1 Simplest discretization
In this subsection we focus on deriving a consistent theory of massive gravity from the discretization procedure (5.19, 5.20). For this, we consider a discretization with only two sites j = 1, 2 and will only be considered in the fourdimensional action induced on one site (say site 1), rather than the sum of both sites. This picture is analogous in spirit to a braneworld picture where we induce the action at one point along the extra dimension. This picture gives the theory of a unique dynamical metric, expressed in terms of a reference metric which corresponds to the fixed metric on the other site. We emphasize that this picture corresponds to a trick to build a consistent theory of massive gravity, and would otherwise be more artificial than its multigravity extension. However, as we shall see later, massive gravity can be seen as a perfectly consistent limit of multi (or bi)gravity where the massless spin2 field (and other fields in the multicase) decouple and is thus perfectly acceptable.
6.2.2 Generalized mass term
This mass term is not the unique acceptable generalization of FierzPauli gravity and by considering more general discretization procedures we can generate the entire 2parameter family of acceptable potentials for gravity which will also be shown to be free of ghost in Section 7.
We see that in the vielbein language, the expression for the mass term is extremely natural and simple. In fact this form was guessed at already for special cases in Ref. [410] and even earlier in [502]. However, the crucial analysis on the absence of ghosts and the reason for these terms was incorrect in both of these presentations. Subsequently, after the development of the consistent metric formulation, the generic form of the mass terms was given in Refs. [95]^{13} and [314].
This procedure is easily generalizable to any number of dimensions, and massive gravity in d dimensions has (d − 2)free parameters which are related to the (d − 2) discretization parameters.
6.3 Multigravity
In Section 5.2, we showed how to obtain massive gravity from considering the fivedimensional EinsteinHilbert action on one site.^{14} Instead in this section, we integrate over the whole of the extra dimension, which corresponds to summing over all the sites after discretization. Following the procedure of [152], we consider N = 2M + 1 sites to start which leads to multigravity [314], and then focus on the twosite case leading to bigravity [293].
The counting of the degrees of freedom in multigravity goes as follows: the theory contains 2M massive spin2 fields with five degrees of freedom each and one massless spin2 field with two degrees of freedom, corresponding to a total of 10M + 2 degrees of freedom. In the continuum limit, we also need to account for the zero mode of the lapse and the shift which have been gauged fixed in five dimensions (see Ref. [443] for a nice discussion of this point). This leads to three additional degrees of freedom, summing up to a total of 5N degrees of freedom of the four coordinates x^{2}.
6.4 Bigravity
Let us end this section with the special case of bigravity. Bigravity can also be derived from the deconstruction paradigm, just as massive gravity and multigravity, but the idea has been investigated for many years (see for instance [436, 324]). Like massive gravity, bigravity was for a long time thought to host a BD ghost parasite, but a ghostfree realization was recently proposed by Hassan and Rosen [293] and bigravity is thus experiencing a revived amount of interested. This extensions is nothing other than the ghostfree massive gravity Lagrangian for a dynamical reference metric with the addition of an EinsteinHilbert term for the now dynamical reference metric.
6.4.1 Bigravity from deconstruction
Let us consider a twosite discretization with periodic boundary conditions, j = 1, 2, 3 with quantities at the site j = 3 being identified with that at the site j = 1. Similarly, as in Section 5.2 we denote by \({g_{\mu v}} = e_\mu ^ae_v^b{\eta _{ab}}\) and by \({f_{\mu v}} = f_\mu ^af_v^b{\eta _{ab}}\) the metrics and vielbeins at the respective locations y_{1} and y_{2}.
Notice that the most naive discretization procedure would lead to M_{ g } = M_{Pl} = M_{ f }, but these can be generalized either ‘by hand’ by changing the weight of each site during the discretization, or by considering a nontrivial configuration along the extra dimension (for instance warping along the extra dimension^{16}), or most simply by performing a conformal rescaling of the metric at each site.
Here, \({{\mathcal L}_0}[{\mathcal K}[g,f]]\) corresponds to a cosmological constant for the metric g_{ μν } and the special combination \(\sum\nolimits_{n = 0}^4 {{{( 1)}^n}C_4^n{{\mathcal L}_n}[{\mathcal K}[g,f]]}\), where the \(C_n^m\) are the binomial coefficients is the cosmological constant for the metric f_{ μν }, so only ℒ_{2,3,4} correspond to genuine interactions between the two metrics.
In the deconstruction framework, we naturally obtain α_{2} = 1 and no tadpole nor cosmological constant for either metrics.
6.4.2 Mass eigenstates
6.5 Coupling to matter
Notice, however, that the current full proofs for the absence of the BD ghost do not include such couplings between matter fields living on different metrics (or vielbeins), nor matter fields coupling directly to more than one metric (vielbein).
6.6 No new kinetic interactions
In more than four dimensions, the GR action can be supplemented by additional Lovelock invariants [383] which respect diffeomorphism invariance and are expressed in terms of higher powers of the Riemann curvature but lead to second order equations of motion. In four dimensions there is only one nontrivial additional Lovelock invariant corresponding the GaussBonnet term but it is topological and thus does not affect the theory, unless other degrees of freedom such as a scalar field is included.
So, when dealing with the theory of a single massless spin2 field in four dimensions the only allowed kinetic term is the wellknown EinsteinHilbert one. Now when it comes to the theory of a massive spin2 field, diffeomorphism invariance is broken and so in addition to the allowed potential terms described in (6.9)–(6.13), one could consider other kinetic terms which break diffeomorphism.
Now let us turn to a theory of gravity. In that case, we have seen that the coupling to matter forces linear diffeomorphisms to be extended to fully nonlinear diffeomorphism. So to be viable in a theory of massive gravity, the derivative interaction (5.57) should enjoy a ghostfree nonlinear completion (the absence of ghost nonlinearly can be checked for instance by restoring nonlinear diffeomorphism using the nonlinear Stückelberg decomposition (2.80) in terms of the helicity1 and 0 modes given in (2.46), or by performing an ADM analysis as will be performed for the mass term in Section 7.) It is easy to check that by itself \({\mathcal L}_3^{{\rm{(der)}}}\) has a ghost at quartic order and so other nonlinear interactions should be included for this term to have any chance of being ghostfree.
Within the deconstruction paradigm, the nonlinear completion of \({\mathcal L}_3^{{\rm{(der)}}}\) could have a natural interpretation as arising from the fivedimensional GaussBonnet term after discretization. Exploring the avenue would indeed lead to a new kinetic interaction of the form \(\sqrt { g} {{\mathcal K}_{\mu v}}{{\mathcal K}_{\alpha \beta}}^*{R^{\mu v\alpha \beta}}\), where *R is the dual Riemann tensor [339, 153]. However, a simple ADM analysis shows that such a term propagates more than five degrees of freedom and thus has an Ostrogradsky ghost (similarly as the BD ghost). As a result this new kinetic interaction (5.57) does not have a natural realization from a fivedimensional point of view (at least in its metric formulation, see Ref. [153] for more details.)
We can push the analysis even further and show that no matter what the higher order interactions are, as soon as \({\mathcal L}_3^{{\rm{(der)}}}\) is present it will always lead to a ghost and so such an interaction is never acceptable [153].
As a result, the EinsteinHilbert kinetic term is the only allowed kinetic term in Lorentzinvariant (massive) gravity.
This result shows how special and unique the EinsteinHilbert term is. Even without imposing diffeomorphism invariance, the stability of the theory fixes the kinetic term to be nothing else than the EinsteinHilbert term and thus forces diffeomorphism invariance at the level of the kinetic term. Even without requiring coordinate transformation invariance, the Riemann curvature remains the building block of the kinetic structure of the theory, just as in GR.
Before summarizing the derivation of massive gravity from higher dimensional deconstruction/KaluzaKlein decomposition, we briefly comment on other ‘apparent’ modifications of the kinetic structure like in f (R) — gravity (see for instance Refs. [89, 354, 46] for f (R) massive gravity and their implications to cosmology).
Such kinetic terms à la f (R) are also possible without a mass term for the graviton. In that case diffeomorphism invariance allows us to perform a change of frame. In the Einsteinframe f (R) gravity is seen to correspond to a theory of gravity with a scalar field, and the same result will hold in f (R) massive gravity (in that case the scalar field couples nontrivially to the Stückelberg fields). As a result f (R) is not a genuine modification of the kinetic term but rather a standard EinsteinHilbert term and the addition of a new scalar degree of freedom which not a degree of freedom of the graviton but rather an independent scalar degree of freedom which couples nonminimally to matter (see Ref. [128] for a review on f (R)gravity.)
7 Part II Ghostfree Massive Gravity
8 Massive, Bi and MultiGravity Formulation: A Summary
The previous ‘deconstruction’ framework gave a intuitive argument for the emergence of a potential of the form (6.3) (or (6.1) in the vielbein language) and its bi and multimetric generalizations. In deconstruction or KaluzaKlein decomposition a certain type of interaction arises naturally and we have seen that the whole spectrum of allowed potentials (or interactions) could be generated by extending the deconstruction procedure to a more general notion of derivative or by involving the mixing of more sites in the definition of the derivative along the extra dimensions. We here summarize the most general formulation for the theories of massive gravity about a generic reference metric, bigravity and multigravity and provide a dictionary between the different languages used in the literature.
Both massive gravity and bigravity break one copy of diff invariance and so the Stückelberg fields can be introduced in exactly the same way in both cases \({\mathcal U}[g,f] \to {\mathcal U}[g,\tilde f]\) where the Stückelbergized metric \({\tilde f_{\mu v}}\) was introduced in (2.75) (or alternatively \({\mathcal U}[g,f] \to {\mathcal U}[\tilde g,f]\). Thus bigravity is by no means an alternative to introducing the Stückelberg fields as is sometimes stated.
We have introduced the constant \({{\mathcal L}_0}\) (\({{\mathcal L}_0} = 4!\) and \(\sqrt { g{{\mathcal L}_0}}\) is nothing other than the cosmological constant) and the tadpole ℒ_{1} for completeness. Notice however that not all these five Lagrangians are independent and the tadpole can always be reexpressed in terms of a cosmological constant and the other potential terms.
8.1 Inverse argument
8.2 Alternative variables
8.3 Expansion about the reference metric
The quadratic expansion about a background different from the reference metric was derived in Ref. [278]. Notice however that even though the mass term may not appear as having an exact FierzPauli structure as shown in [278], it still has the correct structure to avoid any BD ghost, about any background [295, 294, 300, 297].
9 Evading the BD Ghost in Massive Gravity
The deconstruction framework gave an intuitive approach on how to construct a theory of massive gravity or multiple interacting ‘gravitons’. This lead to the ghostfree dRGT theory of massive gravity and its bi and multigravity extensions in a natural way. However, these developments were only possible a posteriori.
The deconstruction framework was proposed earlier (see Refs. [24, 25, 168, 28, 443, 168, 170]) directly in the metric language and despite starting from a perfectly healthy fivedimensional theory of GR, the discretization in the metric language leads to the standard BD issue (this also holds in a KK decomposition when truncating the KK tower at some finite energy scale). Knowing that massive gravity (or multigravity) can be naturally derived from a healthy fivedimensional theory of GR is thus not a sufficient argument for the absence of the BD ghost, and a great amount of effort was devoted to that proof, which is known by now a multitude of different forms and languages.
Within this review, one cannot make justice to all the independent proofs that have been formulated by now in the literature. We thus focus on a few of them — the Hamiltonian analysis in the ADM language — as well as the analysis in the Stückelberg language. One of the proofs in the vielbein formalism will be used in the multigravity case, and thus we do not emphasize that proof in the context of massive gravity, although it is perfectly applicable (and actually very elegant) in that case. Finally, after deriving the decoupling limit in Section 8.3, we also briefly review how it can be used to prove the absence of ghost more generically.
We note that even though the original argument on how the BD ghost could be circumvented in the full nonlinear theory was presented in [137] and [144], the absence of BD ghost in “ghostfree massive gravity” or dRGT has been the subject of many discussions [12, 13, 345, 342, 95, 341, 344, 96] (see also [350, 351, 349, 348, 352] for related discussions in bigravity). By now the confusion has been clarified, and see for instance [295, 294, 400, 346, 343, 297, 15, 259] for thorough proofs addressing all the issues raised in the previous literature. (See also [347] for the proof of the absence of ghosts in other closely related models).
9.1 ADM formulation
9.1.1 ADM formalism for GR
This result is fully generalizable to any number of dimensions, and in spacetime dimensions, gravitational waves carry d (d − 3)/2 polarizations. We now move to the case of massive gravity.
9.1.2 ADM counting in massive gravity
If \({\mathcal U}\) depends nonlinearly on the shift or the lapse then these are no longer directly Lagrange multipliers (if they are nonlinear, they still appear at the level of the equations of motion, and so they do not propagate a constraint for the metric but rather for themselves). As a result for an arbitrary potential one is left with (2 × 6) degrees of freedom in the threedimensional metric and its momentum conjugate and no constraint is present to reduce the phase space. This leads to 6 degrees of freedom in field space: the two usual transverse polarizations for the graviton (as we have in GR), in addition to two ‘vector’ polarizations and two ‘scalar’ polarizations.
These 6 polarizations correspond to the five healthy massive spin2 field degrees of freedom in addition to the sixth BD ghost, as explained in Section 2.5 (see also Section 7.2).
This counting is also generalizable to an arbitrary number of dimensions, in spacetime dimensions, a massive spin2 field should propagate the same number of degrees of freedom as a massless spin2 field in d + 1 dimensions, that is (d + 1)(d − 2)/2 polarizations. However, an arbitrary potential would allow for d (d − 1)/2 independent degrees of freedom, which is 1 too many excitations, always corresponding to one BD ghost degree of freedom in an arbitrary number of dimensions.
The only way this counting can be wrong is if the constraints for the shift and the lapse cannot be inverted for the shift and the lapse themselves, and thus at least one of the equations of motion from the shift or the lapse imposes a constraint on the threedimensional metric γ_{ ij }. This loophole was first presented in [138] and an example was provided in [137]. It was then used in [144] to explain how the ‘nogo’ on the presence of a ghost in massive gravity could be circumvented. Finally, this argument was then carried through fully nonlinearly in [295] (see also [342] for the analysis in 1 + 1 dimensions as presented in [144]).
9.1.3 Eliminating the BD ghost
9.1.3.1 Linear FierzPauli massive gravity
In Ref. [111], the most general potential was considered up to quartic order in the h_{ μν }, and it was shown that there is no choice of such potential (apart from a pure cosmological constant) which would prevent the lapse from entering nonlinearly. While this result is definitely correct, it does not however imply the absence of a constraint generated by the set of shift and lapse N^{ μ } = {N, N^{ i }}. Indeed there is no reason to believe that the lapse should necessarily be the quantity to generates the constraint necessary to remove the BD ghost. Rather it can be any combination of the lapse and the shift.
9.1.3.2 Example on how to evade the BD ghost nonlinearly
9.1.3.3 Condition to evade the ghost
To summarize, the condition to eliminate (at least half of) the BD ghost is that the det of the Hessian (7.13) L_{ μν } vanishes as explained in [144]. This was shown to be the case in the ghostfree theory of massive gravity (6.3) [(6.1)] exactly in some cases and up to quartic order, and then fully nonlinearly in [295]. We summarize the derivation in the general case in what follows.
9.1.3.4 Primary constraint
The field redefinition naturally involves a square root through the expression of the matrix D in (7.21), which should come as no surprise from the square root structure of the potential term. For the potential to be writable in the metric language, the square root in the definition of the tensor \({\mathcal K}_{\,\,\,v}^\mu\), should exist, which in turns imply that the square root in the definition of \(D_j^i\) in (7.21) must also exist. While complicated, the important point to notice is that this field redefinition remains linear in the lapse (and so does not spoil the standard constraints of GR).
9.1.3.5 Secondary constraint
The important point to notice is that the secondary constraint (7.33) only depends on the phase space variables γ_{ ij }, p^{ ij } and not on the lapse N. Thus it constraints the phase space variables rather than the lapse and provides a genuine secondary constraint in addition to the primary one (7.30) (indeed one can check that \({{\mathcal C}_2}{\vert_{{{\mathcal C}_{0 = 0}} \ne 0}}\).
To conclude, we have shown in this section that ghostfree (or dRGT) massive gravity is indeed free from the BD ghost and the theory propagates five physical dofs about generic backgrounds. We now present the proof in other languages, but stress that the proof developed in this section is sufficient to infer the absence of BD ghost.
9.1.3.6 Secondary constraints in bi and multigravity
In bi or multigravity where all the metrics are dynamical the Hamiltonian is pure constraint (every term is linear in the one of the lapses as can be seen explicitly already from (7.25) and (7.26)).
In this case, the evolution equation of the primary constraint can always be solved for their respective Lagrange multiplier (lapses) which can always be set to zero. Setting the lapses to zero would be unphysical in a theory of gravity and instead one should take a ‘bifurcation’ of the Dirac constraint analysis as explained in [48]. Rather than solving for the Lagrange multipliers we can choose to use the evolution equation of some of the primary constraints to provide additional secondary constraints instead of solving them for the lagrange multipliers.
Choosing this bifurcation leads to statements which are then continuous with the massive gravity case and one recovers the correct number of degrees of freedom. See Ref. [48] for an enlightening discussion.
9.2 Absence of ghost in the Stückelberg language
9.2.1 Physical degrees of freedom
Another way to see the absence of ghost in massive gravity is to work directly in the Stückelberg language for massive spin2 fields introduced in Section 2.4. If the four scalar fields ϕ^{ a } were dynamical, the theory would propagate six degrees of freedom (the two usual helicity2 which dynamics is encoded in the standard EinsteinHilbert term, and the four Stückelberg fields). To remove the sixth mode, corresponding to the BD ghost, one needs to check that not all four Stückelberg fields are dynamical but only three of them. See also [14] for a theory of two Stückelberg fields.
9.2.2 Twodimensional case
9.2.3 Full proof
The full proof in the minimal model (corresponding to α_{2} = 1 and α_{3} = −2/3 and α_{4} = 1/6 in (6.3) or β_{2} = β_{3} = 0 in the alternative formulation (6.23)), was derived in Ref. [297]. We briefly review the essence of the argument, although the full technical derivation is beyond the scope of this review and refer the reader to Refs. [297] and [15] for a fullyfledged derivation.
9.2.4 Stückelberg method on arbitrary backgrounds
When working about different nonMinkowski backgrounds, one can instead generalize the definition of the helicity0 mode as was performed in [400]. The essence of the argument is to perform a rotation in field space so that the fluctuations of the Stückelberg fields about a curved background form a vector field in the new basis, and one can then employ the standard treatment for a vector field. See also [10] for another study of the Stückelberg fields in an FLRW background.
Recently, a covariant Stückelberg analysis valid about any background was performed in Ref. [369] using the BRST formalism. Interestingly, this method also allows to derive the decoupling limit of massive gravity about any background.
In what follows, we review the approach derived in [400] which provides yet another independent argument for the absence of ghost in all generalities. The proofs presented in Sections 7.1 and 7.2 work to all orders about a trivial background while in [400], the proof is performed about a generic (curved) background, and the analysis can thus stop at quadratic order in the fluctuations. Both types of analysis are equivalent so long as the fields are analytic, which is the case if one wishes to remain within the regime of validity of the theory.
Consider a generic background metric, which in unitary gauge (i.e., in the coordinate system {x} where the Stückelberg background fields are given by \({\phi ^a}(x) = {x^\mu}\delta _\mu ^a\), the background metric is given by \(g_{\mu v}^{{\rm{bg}}} = e_\mu ^a(x)e_v^b{(x)_{{\eta _{ab}}}}\), and the background Stückelberg fields are given by \(\phi _{{\rm{bg}}}^a(x) = {x^a}  A_{{\rm{bg}}}^a(x)\).
9.2.4.1 Flat background metric
9.2.4.2 Nonsymmetric background Stückelberg
If the background configuration is not symmetric, then at every point one needs to perform first an internal Lorentz transformation Λ(x) in the Stückelberg field space, so as to align them with the coordinate basis and recover a symmetric configuration for the background Stückelberg fields. In this new Lorentz frame, the Stückelberg fluctuation is \({\tilde a^\mu} = \Lambda _v^\mu (x){a_v}\). As a result, to quadratic order in the Stückelberg fluctuation the part of the ghostfree potential which is independent of the metric fluctuation and its curvature goes symbolically as (7.60) with f replaced by \(f \to \tilde f + (\partial \Lambda){\Lambda ^{ 1}}\tilde a\), (with = \({\tilde f_{\mu v}} = {\partial _\mu}{\tilde a_v}  {\partial _v}{\tilde a_\mu}\)). Interestingly, the Lorentz boost (∂ Λ)Λ^{−1} now plays the role of a mass term for what looks like a gauge field ã. This mass term breaks the U (1) symmetry, but there is still no kinetic term for ã_{0}, very much as in a Proca theory. This part of the potential is thus manifestly ghostfree (in the sense that it provides a dynamics for only three of the four Stückelberg fields, independently of the background).
As for the second type of term in (7.61), since F = 0 on the background field \(A_\mu ^{{\rm{bg}}}\), the second type of terms is forced to be proportional to f_{ μν } and cannot involve any ∂_{0} a_{0} at all. As a result a_{0} is not dynamical, which ensures that the theory is free from the BD ghost.
This part of the argument generalizes easily for non symmetric background Stückelberg configurations, and the same replacement \(f \to \tilde f + (\partial \Lambda){\Lambda ^{ 1}}\tilde a\) still ensures that ã_{0} acquires no dynamics from (7.61).
9.2.4.3 Background curvature
9.3 Absence of ghost in the vielbein formulation
Finally, we can also prove the absence of ghost for dRGT in the Vielbein formalism, either directly at the level of the Lagrangian in some special cases as shown in [171] or in full generality in the Hamiltonian formalism, as shown in [314]. The later proof also works in all generality for a multigravity theory and will thus be presented in more depth in what follows, but we first focus on a special case presented in Ref. [171].
Let us start with massive gravity in the vielbein formalism (6.1). As was the case in Part II, we work with the symmetric vielbein condition, \(e_\mu ^af_v^b{\eta _{ab}} = e_v^af_\mu ^b{\eta _{ab}}\). For simplicity we specialize further to the case where \(f_\mu ^a = \delta _\mu ^a\), so that the symmetric vielbein condition imposes e^{ aμ } = e^{ μa }. Under this condition, the vielbein contains as many independent components as the metric. The symmetric veilbein condition ensures that one is able to reformulate the theory in a metric language. In spacetime dimensions, there is a priori d (d + 1)/2 independent components in the symmetric vielbein.
9.4 Absence of ghosts in multigravity
We now turn to the proof for the absence of ghost in multigravity and follow the vielbein formulation of Ref. [314]. In this subsection we use the notation that uppercase Latin indices represent ddimensional Lorentz indices, A, B, ⋯ = 0, ⋯, d − 1, while lowercase Latin indices represent the d − 1dimensional Lorentz indices along the space directions after ADM decomposition, a, b, ⋯ = 1, ⋯, d − 1. Greek indices represent ddimensional spacetime indices μ, ν, = 0, ⋯, d − 1, while the ‘middle’ of the Latin alphabet indices i, j ⋯ represent pure space indices i, j, ⋯ = 1, ⋯, d, − 1. Finally, capital indices label the metric and span over I, J, K, ⋯ = 1, ⋯, N.
10 Decoupling Limits
10.1 Scaling versus decoupling
Before moving to the decoupling of massive gravity and bigravity, let us make a brief interlude concerning the correct identification of degrees of freedom. The Stückelberg trick used previously to identify the correct degrees of freedom works in all generality, but care must be used when taking a “decoupling limit” (i.e., scaling limit) as will be done in Section 8.2.
This procedure is true in all generality: a decoupling limit is a special scaling limit where all the fields in the original theory are scaled with the highest possible power of the scale in such a way that the decoupling limit is finite.
A decoupling limit of a theory never changes the number of physical degrees of freedom of a theory. At best it ‘decouples’ some of them in such a way thai they are inaccessible from another sector.

Noninteracting Limit: The most natural question to ask is what happens in the limit where the interactions between the two fields are ‘switched off’, i.e., when sending the scale m ⊒ 0, (the limit m ⊒ 0 is studied more carefully in Sections 8.3 and 8.4). In that case if the two Planck scales M_{ g,f } remain fixed as m → 0, we then recover two massless noninteracting spin2 fields (carrying both 2 helicity2 modes), in addition to a decoupled sector containing a helicity0 mode and a helicity1 mode. In bigravity matter fields couple only to one metric, and this remains the case in the limit m → 0, so that the two massless spin2 fields live in two fully decoupled sectors even when matter in included.

Massive Gravity: Alternatively, we may look at the limit where one of the spin2 fields (say f_{ μν }) decouples. This can be studied by sending its respective Planck scale to infinity. The resulting limit corresponds to a massive spin2 field (carrying five dofs) and a decoupled massless spin2 field carrying 2 dofs. This is nothing other than the massive gravity limit of bigravity (which includes a fully decoupled massless sector).
If one considers matter coupling to the metric which scales in such a way that a nontrivial solution for f_{ μν } survives in the \({M_f} \to \infty \,\lim {\rm{it}}\,{f_{\mu v}} \to {\overset  f _{\mu v}}\), we then obtain a massive gravity sector on an arbitrary nondynamical reference metric \({\overset  f _{\mu v}}\). The dynamics of the massless spin2 field fully decouples from that of the massive sector.

Other Decoupling Limits Finally, one can look at combinations of the previous limits, and the resulting theory depends on how fast M_{ f }, M_{ g } → ∞ compared to how fast m → 0. For instance if one takes the limit M_{ f }, M_{ g } → ∞ and m → 0, while keeping both M_{ g }/M_{ f } and \(\Lambda _3^3 = {M_g}{m^2}\) fixed, then we obtain what is called the Λ_{3}decoupling limit of bigravity (derived in Section 8.4), where the dynamics of the two helicity2 modes (which are both massless in that limit), and that of the helicity1 and 0 modes can be followed without keeping track of the standard nonlinearities of GR.
If on top of this Λ_{3}decoupling limit one further takes M_{ f } → ∞, then one of the massless spin2 fields fully decoupled (no communication between that field and the helicity1 and 0 modes). If, on the other hand, we take the additional limit m → 0 on top of the Λ_{3}decoupling limit, then the helicity0 and 1 modes fully decouple from both helicity2 modes.
In all of these decoupling limits, the number of dofs remains the same as in the original theory, some fields are simply decoupled from the rest of the standard gravitational sector. These prevents any communication between these decoupled fields and the gravitational sector, and so from the gravitational sector view point it appears as if these decoupled fields did not exist.
It is worth stressing that all of these limits are perfectly sensible and lead to sensible theories, (from a theoretical view point). This is important since if one of these scaling limits lead to a pathological theory, it would have severe consequences for the parent bigravity theory itself.
Similar decoupling limit could be taken in multigravity and out of N interacting spin2 fields, we could obtain for instance N decoupled massless spin2 fields and 3(N − 1) decoupled dofs in the helicity0 and 1 modes.
In what follows we focus on massive gravity limit of bigravity when M_{ f } ⊒∞
10.2 Massive gravity as a decoupling limit of bigravity
10.2.1 Minkowski reference metric
At the level of the equations of motion, in the limit M_{ f } → ∞ we obtain the massive gravity modified Einstein equation for g_{ μν }, the free massless linearized Einstein equation for which fully decouples and the equation of motion for all the matter fields ψ_{ f } on flat spacetime, (see also Ref. [44]).
10.2.2 (A)dS reference metric
10.2.3 Arbitrary reference metric
As is already clear from the previous discussion, to recover massive gravity on a nontrivial reference metric as a limit of bigravity, one needs to scale the Matter Lagrangian that couples to what will become the reference metric (say the metric f for definiteness) in such a way that the Riemann curvature of f remains finite in that decoupling limit. For a macroscopic description of the matter living on this is in principle always possible. For instance one can consider a point source of mass M_{bh} living on the metric f. Then, taking the limit M_{ f }, M_{bh} → ∞ while keeping the ratio M_{BH}/M_{ f } fixed, leads to a theory of massive gravity on a Schwarzschild reference metric and a decoupled massless graviton. However, some care needs to be taken to see how this works when the dynamics of the matter sourcing is included.
As a result, massive gravity with an arbitrary reference metric can be seen as a consistent limit of bigravity in which the additional degrees of freedom in the metric and matter that sources the background decouple. Thus all solutions of massive gravity may be seen as M_{ f } → ∞ decoupling limits of solutions of bigravity. This will be discussed in more depth in Section 8.4. For an arbitrary reference metric which can be locally written as a small departures about Minkowski the decoupling limit is derived in Eq. (8.81).
Having derived massive gravity as a consistent decoupling limit of bigravity, we could of course do the same for any multimetric theory. For instance, out of Ninteracting fields, we could take a limit so as to decouple one of the metrics, we then obtain the theory of (N − 1)interacting fields, all of which being massive and one decoupled massless spin2 field.
10.3 Decoupling limit of massive gravity
We now turn to a different type of decoupling limit, whose aim is to disentangle the dofs present in massive gravity itself and analyze the ‘irrelevant interactions’ (in the usual EFT sense) that arise at the lowest possible scale. One could naively think that such interactions arise at the scale given by the graviton mass, but this is not so. In a generic theory of massive gravity with FierzPauli at the linear level, the first irrelevant interactions typically arise at the scale Λ_{5} = (m^{4}M_{Pl})^{1/5}. For the setups we have in mind, m ≪ Λ_{5} ≪ M_{Pl}. But we shall see that interactions arising at such a lowenergy scale are always pathological (reminiscent to the BD ghost [111, 173]), and in ghostfree massive gravity the first (irrelevant) interactions actually arise at the scale Λ_{3} = (m^{3}M_{Pl})^{1/3}.
We start by deriving the decoupling limit in the absence of vectors (helicity1 modes) and then include them in the following section 8.3.4. Since we are interested in the decoupling limit about flat spacetime, we look at the case where Minkowski is a vacuum solution to the equations of motion. This is the case in the absence of a cosmological constant and a tadpole and we thus focus on the case where α_{0} = α_{1} = 0 in (6.3).
10.3.1 Interaction scales
Clearly, the lowest interaction scale is Λ_{j=0,k= 0,ℓ =3} ≡ Λ_{5} = (M_{Pl}m^{4})^{1/5} which arises for an operator of the form (∂^{2}π)^{3}. If present such an interaction leads to an Ostrogradsky instability which is another manifestation of the BD ghost as identified in [173].
Any interaction with j > 0 or k > 0 automatically leads to a larger scale, so all the interactions arising at a scale between Λ_{5} (inclusive) and Λ_{3} are of the form (∂^{2}π)^{ ℓ } and carry an Ostrogradsky instability. For DGP we have already seen that there is no interactions at a scale below Λ_{3}. In what follows we show that same remains true for the ghostfree theory of massive gravity proposed in (6.3). To see this let us identify the interactions with j = k = 0 and arbitrary power ℓ for (∂^{2}π).
10.3.2 Operators below the scale Λ_{3}
As a result the potential term constructed proposed in Part II (and derived from the deconstruction framework) is free of any interactions of the form (∂^{2}π)^{ℓ}. This means that the BD ghost as identified in the Stückelberg language in [173] is absent in this theory. However, at this level, the BD ghost could still reappear through different operators at the scale Λ_{3} or higher.
10.3.3 Λ_{3}decoupling limit
Since there are no operators all the way up to the scale Λ_{3} (excluded), we can take the decoupling limit by sending M_{Pl} ⊒ ∞, m ⊒ 0 and maintaining the scale Λ_{3} fixed.
10.3.3.1 Decoupling limit
From the expression of these tensors in terms of the fully antisymmetric LeviCevita tensors, it is clear that the tensors are transverse and that the equations of motion of \({h^{\mu v}}\overset  {{X_{\mu v}}}\) with respect to both h and π never involve more than two derivatives. This decoupling limit is thus free of the Ostrogradsky instability which is the way the BD ghost would manifest itself in this language. This decoupling limit is actually free of any ghostlie instability and the whole theory is free of the BD even beyond the decoupling limit as we shall see in depth in Section 7.
Not only does the potential term proposed in (6.3) remove any potential interactions of the form (∂^{2}π)^{ ℓ } which could have arisen at an energy between Λ_{5} = (M_{Pl}m^{4})^{1/5} and Λ_{3}, but it also ensures that the interactions that arise at the scale Λ_{3} are healthy.
10.3.3.2 Unmixing and Galileons
10.3.3.3 X^{(3)}coupling
10.3.4 Vector interactions in the Λ_{3}decoupling limit
As can be seen from the relation (8.19), the scale associated with interactions mixing two helicity1 fields with an arbitrary number of fields π, (j = 0, k = 1 and arbitrary ℓ) is also Λ_{3}. So at that scale, there are actually an infinite number of interactions when including the mixing with between the helicity1 and 0 modes (however as mentioned previously, since the vector field always appears quadratically it is always consistent to set them to zero as was performed previously).
The full decoupling limit including these interactions has been derived in Ref. [419], (see also Ref. [238]) using the vielbein formulation of massive gravity as in (6.1) and we review the formalism and the results in what follows.
This decoupling limit includes nonlinear combinations of the secondderivative tensor Π_{ μν } and the first derivative Maxwell tensor F_{ μν }. Nevertheless, the structure of the interactions is gauge invariant for A_{ μ }, and there are no higher derivatives on in the equation of motion for A, so the equations of motions for both the helicity1 and 2 modes are manifestly second order and propagating the correct degrees of freedom. The situation is more subtle for the helicity0 mode. Taking the equation of motion for that field would lead to higher derivatives on π itself as well as on the helicity1 field. Since this theory has been proven to be ghostfree by different means (see Section 7), it must be that the higher derivatives in that equation are nothing else but the derivative of the equation of motion for the helicity1 mode similarly as what happens in Section 7.2.
When working beyond the decoupling limit, the even the equation of motion with respect to the helicity1 mode is no longer manifestly wellbehaved, but as we shall see below, the Stückelberg fields are no longer the correct representation of the physical degrees of freedom. As we shall see below, the proper number of degrees of freedom is nonetheless maintained when working beyond the decoupling limit.
10.3.5 Beyond the decoupling limit
10.3.5.1 Physical degrees of freedom
Recently, much progress has been made in deriving the decoupling limit about arbitrary backgrounds, see Ref. [369].
10.3.6 Decoupling limit on (Anti) de Sitter
10.3.6.1 Linearized theory and Higuchi bound
Before deriving the decoupling limit of massive gravity on (Anti) de Sitter, we first need to analyze the linearized theory so as to infer the proper canonical normalization of the propagating dofs and the proper scaling in the decoupling limit, similarly as what was performed for massive gravity with flat reference metric. For simplicity we focus on (3 + 1) dimensions here, and when relevant give the result in arbitrary dimensions. Linearized massive gravity on (A)dS was first derived in [307, 308]. Since we are concerned with the decoupling limit of ghostfree massive gravity, we follow in this section the procedure presented in [154]. We also focus on the dS case first before commenting on the extension to AdS.
While this observation is correct on AdS, in the dS one cannot take the massless limit without simultaneously sending H → 0 at least the same rate. As a result, it would be incorrect to deduce that the helicity0 mode decouples in the massless limit of massive gravity on dS.
To be more precise, the linearized action (8.62) is free from ghost and tachyons only if m ≡ 0 which corresponds to GR, or if m^{2} > 2H^{2}, which corresponds to the wellknow Higuchi bound [307, 190]. In d spacetime dimensions, the Higuchi bound is m^{2} > (d − 2)H^{2}. In other words, on dS there is a forbidden range for the graviton mass, a theory with 0 < m^{2} < 2H^{2} or with m^{2} < 0 always excites at least one ghost degree of freedom. Notice that this ghost, (which we shall refer to as the Higuchi ghost from now on) is distinct from the BD ghost which corresponded to an additional sixth degree of freedom. Here the theory propagates five dof (in four dimensions) and is thus free from the BD ghost (at least at this level), but at least one of the five dofs is a ghost. When 0 < m^{2} < 2H^{2}, the ghost is the helicity0 mode, while for m^{2} < 0, the ghost is he helicity1 mode (at quadratic order the helicity1 mode comes in as \( {{{m^2}} \over 4}F_{\mu v}^2\)). Furthermore, when m^{2} < 0, both the helicity2 and 0 are also tachyonic, although this is arguably not necessarily a severe problem, especially not if the graviton mass is of the order of the Hubble parameter today, as it would take an amount of time comparable to the age of the Universe to see the effect of this tachyonic behavior. Finally, the case m^{2} = 2H^{2} (or m^{2} = (d − 2)H^{2} in d spacetime dimensions), represents the partially massless case where the helicity0 mode disappears. As we shall see in Section 9.3, this is nothing other than a linear artefact and nonlinearly the helicity0 mode always reappears, so the PM case is infinitely strongly coupled and always pathological.

m^{2} < 0: Helicity1 modes are ghost, helicity2 and 0 are tachyonic, sick theory

m^{2} = 0: General Relativity: two healthy (helicity2) degrees of freedom, healthy theory,

0 < m^{2} < 2H^{2}: One “Higuchi ghost” (helicity0 mode) and four healthy degrees of freedom (helicity2 and 1 modes), sick theory,

m^{2} = 2H^{2}: Partially Massless Gravity: Four healthy degrees (helicity2 and 1 modes), and one infinitely strongly coupled dof (helicity0 mode), sick theory,

m^{2} > 2H^{2}: Massive Gravity on dS: Five healthy degrees of freedom, healthy theory.
10.3.6.2 Massless and decoupling limit

As one can see from Figure 4, in the case where H^{2} < 0 (corresponding to massive gravity on AdS), one can take the massless limit m ⊒ 0 while keeping the AdS length scale fixed in that limit. In that limit, the helicity0 mode decouples from external matter sources and there is no vDVZ discontinuity. Notice however that the helicity0 mode is nevertheless still strongly coupled at a low energy scale.
When considering the decoupling limit m ⊒ 0, M_{Pl} ⊒ ∞ of massive gravity on AdS, we have the choice on how we treat the scale H in that limit. Keeping the AdS length scale fixed in that limit could lead to an interesting phenomenology in its own right, but is yet to be explored in depth.

In the dS case, the Higuchi forbidden region prevents us from taking the massless limit while keeping the scale H fixed. As a result, the massless limit is only consistent if H ⊒ 0 simultaneously as m ⊒ 0 and we thus recover the vDVZ discontinuity at the linear level in that limit.
When considering the decoupling limit m ⊒ 0, M_{Pl} ⊒ ∞ of massive gravity on dS, we also have to send H ⊒ 0. If H/m ⊒ 0 in that limit, we then recover the same decoupling limit as for massive gravity on Minkowski, and all the results of Section 8.3 apply. The case of interest is thus when the ratio H/m remains fixed in the decoupling limit.
10.3.6.3 Decoupling limit
 First, as mentioned in Section 8.3.5, care needs to be applied to properly identify the helicity0 mode on a curved background. In the case of (A)dS, the formalism was provided in Ref. [154] by embedding a ddimensional de Sitter spacetime into a flat (d + 1)dimensional spacetime where the standard Stückelberg trick could be applied. As a result the ‘covariant’ fluctuation defined in (2.80) and used in (8.59) needs to be generalized to (see Ref. [154] for details)Any corrections in the third line vanish in the decoupling limit and can thus be ignored, but the corrections of order H^{2} in the second line lead to new nontrivial contributions.$$\begin{array}{*{20}c} {{1 \over {{M_{{\rm{Pl}}}}}}{H_{\mu \nu}} = {1 \over {{M_{{\rm{Pl}}}}}}{h_{\mu \nu}} + {2 \over {\Lambda _3^3}}{\Pi _{\mu \nu}}  {1 \over {\Lambda _3^6}}\Pi _{\mu \nu}^2\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over {\Lambda _3^3}}{{{H^2}} \over {{m^2}}}\left({{{(\partial \pi)}^2}({\gamma _{\mu \nu}}  {2 \over {\Lambda _3^3}}{\Pi _{\mu \nu}})  {1 \over {\Lambda _3^6}}{\Pi _{\mu \alpha}}{\Pi _{\nu \beta}}{\partial ^\alpha}\pi {\partial ^\beta}\pi} \right)} \\ {+ {H^2}{{{H^2}} \over {{m^2}}}{{{{(\partial \pi)}^4}} \over {\Lambda _3^3}} + \cdots \,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$(8.65)

Second, as already encountered at the linearized level, what were total derivatives in Minkowski (for instance the combination [Π^{2}] − [Π]^{2}), now lead to new contributions on de Sitter. After integration by parts, m^{−2}([Π^{2}] − [Π]^{2}) = m^{−2} = 12H^{2}/m^{2}(∂π)^{2}. This was the origin of the new kinetic structure for massive gravity on de Sitter and will have further effects in the decoupling limit when considering similar contributions from ℒ_{3,4}(Π), where ℒ_{3,4} are defined in (6.12, 6.13) or more explicitly in (6.17, 6.18).
10.4 Λ_{3}decoupling limit of bigravity
We now proceed to derive the Λ_{3}decoupling limit of bigravity, and we will see how to recover the decoupling limit about any reference metric (including Minkowski and de Sitter) as special cases. As already seen in Section 8.3.4, the full DL is better formulated in the vielbein language, even though in that case Stückelberg fields ought to be introduced for the broken diff and the broken Lorentz. Yet, this is a small price to pay, to keep the action in a much simpler form. We thus proceed in the rest of this section by deriving the Λ_{3}decoupling of bigravity and start in its vielbein formulation. We follow the derivation and formulation presented in [224]. As previously, we focus on (3 + 1)spacetime dimensions, although the whole formalism is trivially generalizable to arbitrary dimensions.
As already explained in Section 8.3.6, the first contribution ❶ arising from the mixing between the helicity0 and 1 modes is the same (in the decoupling limit) as what was obtained in Minkowski (and is independent of the coefficients β_{ n } or α_{ n }). This implies that the can be directly read of from the three last lines of (8.52). These contributions are the most complicated parts of the decoupling limit but remained unaffected by the dynamics of i.e., unaffected by the bigravity nature of the theory. This statement simply follows from scaling considerations. In the decoupling limit there cannot be any mixing between the helicity1 and neither of the two helicity2 modes. As a result, the helicity1 modes only mix with themselves and the helicity0 mode. Hence, in the scaling limit (8.74, 8.75) the helicity1 decouples from the massless spin2 field.
Furthermore, the first line of (8.52) which corresponds to the dynamics of \(h_\mu ^a\) and the helicity0 mode is also unaffected by the bigravity nature of the theory. Hence, the second contribution ❷ is the also the same as previously derived. As a result, the only new ingredient in bigravity is the mixing ❸ between the helicity0 mode and the second helicity2 mode \(\upsilon _\mu ^a\), given by a fixing of the form h^{ μν }X^{ μν }.
Notice as well the presence of a tadpole for υ if β_{1} ≠ 0. When this tadpole vanishes (as well as the one for h), one can further take the limit M_{ f } → ∞ keeping all the other β’s fixed as well as Λ_{3}, and recover straight away the decoupling limit of massive gravity on Minkowski found in (8.52), with a free and fully decoupled massless spin2 field.
In the presence of a cosmological constant for both metrics (and thus a tadpole in this framework), we can also take the limit M_{ f } → ∞ and recover straight away the decoupling limit of massive gravity on (A)dS, as obtained in (8.66).
11 Extensions of Ghostfree Massive Gravity

Reference metric f_{ ab },

Graviton mass m,

(d − 2) dimensionless parameters α_{ n } (or the β’s).
Another natural extension is to promote the graviton mass m, or any of the free parameters α_{ n } (or β_{ n }) to a function of a new dynamical variable, say of an additional scalar field In principle the mass and the parameters α’s can be thought as potentials for an arbitrary number of scalar fields m = m (ψ_{ j }), α_{ n } = α_{ n } (ψ_{ j }), and not necessarily the same fields for each one of them [320]. So long as these functions are pure potentials and hide no kinetic terms for any new degree of freedom, the constraint analysis performed in Section 7 will go relatively unaffected, and the theory remains free from the BD ghost. This was shown explicitly for the massvarying theory [319, 315] (where the mass is promoted to a scalar function of a new single scalar field, m = m (ϕ), while the parameters α remain constant^{23}), as well as a general massive scalartensor theory [320], and for quasidilaton which allow for different couplings between the spin2 and the scalar field, motivated by scale invariance. We review these models below in Sections 9.1 and 9.2.
Alternatively, rather than considering the parameters and as arbitrary, one may set them to special values of special interest depending on the reference metric f_{ μν }. Rather than an ‘extension’ per se this is more special cases in the parameter space. The first obvious one is m = 0 (for arbitrary reference metric and parameters), for which one recovers the theory of GR (so long as the spin2 field couples to matter in a covariant way to start with). Alternatively, one may also sit on the Higuchi bound, (see Section 8.3.6) with the parameters m^{2} = 2H^{2}, α_{3} = −1/3 and α_{4} = 1/12 in four dimensions. This corresponds to the partially massless theory of gravity, which at the moment is pathological in its simplest realization and will be reviewed below in Section 9.3.
The coupling massive gravity to a DBI Galileon [157] was considered in [237, 461, 261] leading to a generalized Galileon theory which maintains a Galileon symmetry on curved backgrounds. This theory was shown to be free of any Ostrogradsky ghost in [19] and the cosmology was recently studied in [315] and perturbations in [20].
Finally, as other extensions to massive gravity, one can also consider all the extensions applicable to GR. This includes the higher order Lovelock invariants in dimensions greater than four, as well as promoting the EinsteinHilbert kinetic term to a function f (R), which is equivalent to gravity with a scalar field. In the case of massive gravity this has been performed in [89] (see also [46, 354]), where the absence of BD ghost was proven via a constraint analysis, and the cosmology was explored (this was also discussed in Section 5.6 and see also Section 12.5). f (R) extensions to bigravity were also derived in [416, 415].
11.1 Massvarying
One of the important aspects of a massvarying theory of massive gravity is that it allows more flexibility for the graviton mass. In the past the mass could have been much larger and could have lead to potential interesting features, be it for inflation (see for instance Refs. [315, 378] and [282]), the HartleHawking noboundary proposal [498, 439, 499], or to avoid the Higuchi bound [307], and yet be compatible with current bounds on the graviton mass. If the graviton mass is an effective description from higher dimensions it is also quite natural to imagine that the graviton mass would depend on some moduli.
11.2 Quasidilaton
The Planck scale M_{pl}, or Newton constant explicitly breaks scale invariance, but one can easily extend the theory of GR to a scale invariant one M_{Pl} → M_{Pl}e^{λ(x)} by including a dilaton scalar field λ which naturally arises from string theory or from extra dimension compactification (see for instance [122] and see Refs. [429, 120, 248] for the role of a dilaton scalar field on cosmology).
Recently, the decoupling limit of the original quasidilaton model was derived in [239]. Interestingly, a new selfaccelerating solution was found in this model which admits no instability and all the modes are (sub)luminal for a given realistic set of parameters. The extension of this solution to the full theory (beyond the decoupling limit) should provide for a consistent selfaccelerating solution which is guaranteed to be stable (or with a harmless instability time scale of the order of the age of the Universe at least).
11.2.1 Theory
One of the relevance of this decoupling limit is that it makes the study of the Vainshtein mechanism more explicit. As we shall see in what follows (see Section 10.1), the Galileon interactions are crucial for the Vainshtein mechanism to work.
Note that in (9.13), the interactions with the quasidilaton come in the combination ((4 − n)α_{ n } −(n + 1)α_{n+1}), while in \({\mathcal L}_{{\Lambda _3}}^{(0)}\), the interactions between the helicity0 and 2 modes come in the combination ((4 − n)α_{ n } + (n + 1)α_{n+1}). This implies that in massive gravity, the interactions between the helicity2 and 0 mode disappear in the special case where α_{ n } = −(n + 1)/(4 − n)α_{n+1} (this corresponds to the minimal model), and the Vainshtein mechanism is no longer active for spherically symmetric sources (see Refs. [99, 56, 58, 57, 435]). In the case of QMG, the interactions with the quasidilaton survive in that specific case α_{3} = −4α_{4}, and a Vainshtein mechanism could still be feasible, although one might still need to consider nonasymptotically Minkowski configurations.
The cosmology of QMD was first discussed in [119] where the existence of selfaccelerating solutions was pointed out. This will be reviewed in the section on cosmology, see Section 12.5. We now turn to the extended version of QMG recently proposed in Ref. [127].
11.2.2 Extended quasidilaton
Keeping the same philosophy as the quasidilaton in mind, a simple but yet powerful extension was proposed in Ref. [127] and then further extended in [126], leading to interesting phenomenology and stable selfaccelerating solutions. The phenomenology of this model was then further explored in [45]. The stability of the extended quasidilaton theory of massive gravity was explored in [353] and was proven to be ghostfree in [406].
The key ingredient behind the extended quasidilaton theory of massive gravity (EMG) is to notice that two most important properties of QMG namely the absence of BD ghost and the existence of a global scaling symmetry are preserved if the covariantized reference metric is further generalized to include a disformal contribution of the form ∂_{ μ }σ ∂_{ ν }σ (such a contribution to the reference metric can arise naturally from the branebending mode in higher dimensional braneworld models, see for instance [157]).

Considering different coupling constants for the \(\tilde {\mathcal K}\)’s entering in \({{\mathcal L}_2}[\tilde {\mathcal K}]\), \({{\mathcal L}_3}[\tilde {\mathcal K}]\) and \({{\mathcal L}_4}[\tilde {\mathcal K}]\).

One can also introduce what would be a cosmological constant for the metric \(\bar f\), namely a new term of the for \(\sqrt { \bar f} {e^{4\sigma/{M_{{\rm{Pl}}}}}}\).

General shiftsymmetric Horndeski Lagrangians for the quasidilaton.
11.3 Partially massless
11.3.1 Motivations behind PM gravity
The multiple proofs for the absence of BD ghost presented in Section 7 ensures that the ghostfree theory of massive gravity, (or dRGT) does not propagate more than five physical degrees of freedom in the graviton. For a generic finite mass m the theory propagates exactly five degrees of freedom as can be shown from a linear analysis about a generic background. Yet, one can ask whether there exists special points in parameter space where some of degrees of freedom decouple. General relativity, for which m = 0 (and the other parameters α_{ n } are finite) is one such example. In the massless limit of massive gravity the two helicity1 modes and the helicity0 mode decouple from the helicity2 mode and we thus recover the theory of a massless spin2 field corresponding to GR, and three decoupled degrees of freedom. The decoupling of the helicity0 mode occurs via the Vainshtein mechanism^{24} as we shall see in Section 10.1.
As seen in Section 8.3.6, when considering massive gravity on de Sitter as a reference metric, if the graviton mass is precisely m^{2} = 2H^{2}, the helicity0 mode disappears linearly as can be seen from the linearized Lagrangian (8.62). The same occurs in any dimension when the graviton mass is tied to the de Sitter curvature by the relation m^{2} = (d − 2)H^{2}. This special case is another point in parameter space where the helicity0 mode could be decoupled, corresponding to a partially massless (PM) theory of gravity as first pointed out by Deser and Waldron [190, 189, 188], (see also [500] for partially massless higher spin, and [450] for related studies).

It would protect the structure of the potential.

In the PM limit of massive gravity, the helicity0 mode fully decouples from the helicity2 mode and hence from external matter. As a consequence, there is no Vainshtein mechanism that decouples the helicity0 mode in the PM limit of massive gravity unlike in the massless limit. Rather, the helicity0 mode simply decouples without invoking any strong coupling effects and the theoretical and observational luggage that goes with it.

Last but not least, in PM gravity the symmetry underlying the theory is not diffeomorphism invariance but rather the one pointed out in (9.16). This means that in PM gravity, an arbitrary cosmological constant does not satisfy the symmetry (unlike in GR). Rather, the value of the cosmological constant is fixed by the gauge symmetry and is proportional to the graviton mass. As we shall see in Section 10.3 the graviton mass does not receive large quantum corrections (it is technically natural to set to small values). So, if a PM theory of gravity existed it would have the potential to tackle the cosmological constant problem.
It is worth emphasizing that if a PM theory of gravity existed, it would be distinct from the minimal model of massive gravity where the nonlinear interactions between the helicity0 and 2 modes vanish in the decoupling limit but the helicity0 mode is still fully present. PM gravity is also distinct from some specific branches of solutions found in cosmology (see Section 12) on top of which the helicity0 mode disappears. If a PM theory of gravity exists the helicity0 mode would be fully absent of the whole theory and not only for some specific branches of solutions.
11.3.2 The search for a PM theory of gravity
11.3.2.1 A candidate for PM gravity:
The previous considerations represent some strong motivations for finding a fully fledged theory of PM gravity (i.e., beyond the linearized theory) and there has been many studies to find a nonlinear realization of the PM symmetry. So far all these studies have in common to keep the kinetic term for gravity unchanged (i.e., keeping the standard EinsteinHilbert action, with a potential generalization to the Lovelock invariants [298]).
Under this assumption, it was shown in [501, 330], that while the linear level theory admits a symmetry in any dimensions, at the cubic level the PM symmetry only exists in d = 4 spacetime dimensions, which could make the theory even more attractive. It was also pointed out in [191] that in four dimensions the theory is conformally invariant. Interestingly, the restriction to four dimensions can be lifted in bigravity by including the Lovelock invariants [298].
From the analysis in Section 8.3.6 (see Ref. [154]) one can see that the helicity0 mode entirely disappears from the decoupling limit of ghostfree massive gravity, if one ignores the vectors and sets the parameters of the theory to m^{2} = 2H^{2}, a._{3} = −1 and α._{4} = 1/4 in four dimensions. The ghostfree theory of massive gravity with these parameters is thus a natural candidate for the PM theory of gravity. Following this analysis, it was also shown that bigravity with the same parameters for the interactions between the two metrics satisfies similar properties [301]. Furthermore, it was also shown in [147] that the potential has to follow the same structure as that of ghostfree massive gravity to have a chance of being an acceptable candidate for PM gravity. In bigravity the same parameters as for massive gravity were considered as also being the natural candidate [301], in addition of course to other parameters that vanish in the massive gravity limit (to make a fair comparison once needs to take the massive gravity limit of bigravity with care as was shown in [301]).
11.3.2.2 Reappearance of the Helicity0 mode:
Unfortunately, when analyzing the interactions with the vector fields, it is clear from the decoupling limit (8.52) that the helicity0 mode reappears nonlinearly through their couplings with the vector fields. These never cancel, not even in four dimensions and for no parameters of theory. So rather than being free from the helicity0 mode, massive gravity with m^{2} = (d − 2)H^{2} has an infinitely strongly coupled helicity0 mode and is thus a sick theory. The absence of the helicity0 mode is simple artefact of the linear theory.
As a result we can thus deduce that there is no theory of PM gravity. This result is consistent with many independent studies performed in the literature (see Refs. [185, 147, 181, 194]).
11.3.2.3 Relaxing the assumptions:

One assumption behind this result is the form of the kinetic term for the helicity2 mode, which is kept to the be EinsteinHilbert term as in GR. A few studies have considered a generalization of that kinetic term to diffeomorphismbreaking ones [231, 310] however further analysis [339, 153] have shown that such interactions always lead to ghosts nonperturbatively. See Section 5.6 for further details.

Another potential way out is to consider the embedding of PM within bigravity or multigravity. Since bigravity is massive gravity and a decoupled massless spin2 field in some limit it is unclear how bigravity could evade the results obtained in massive gravity but this approach has been explored in [301, 298, 299, 184]. A perturbative relation between bigravity and conformal gravity was derived at the level of the equations of motion in Ref. [299] (unlike claimed in [184]).

The other assumptions are locality and Lorentzinvariance. It is well known that Lorentzbreaking theories of massive gravity can excite fewer than five degrees of freedom. This avenue is explored in Section 14.
To summarize there is to date no known nonlinear PM symmetry which could project out the helicity0 mode of the graviton while keeping the helicity2 mode massive in a local and Lorentz invariant way.
12 Massive Gravity Field Theory
12.1 Vainshtein mechanism
As seen earlier, in four dimensions a massless spin2 field has five degrees of freedom, and there is no special PM case of gravity where the helicity0 mode is unphysical while the graviton remains massive (or at least there is to date no known such theory). The helicity0 mode couples to matter already at the linear level and this additional coupling leads to a extra force which is at the origin of the vDVZ discontinuity see in Section 2.2.3. In this section, we shall see how the nonlinearities of the helicity0 mode is responsible for a Vainshtein mechanism that screens the effect of this field in the vicinity of matter.
Since the Vainshtein mechanism relies strongly on nonlinearities, this makes explicit solutions very hard to find. In most of the cases where the Vainshtein mechanism has been shown to work successfully, one assumes a static and spherically symmetric background source. Already in that case the existence of consistent solutions which extrapolate from a wellbehaved asymptotic behavior at infinity to a screened solution close to the source are difficult to obtain numerically [121] and were only recently unveiled [37, 39] in the case of nonlinear FierzPauli gravity.
This review on massive gravity cannot do justice to all the ongoing work dedicated to the study of the Vainshtein mechanism (also sometimes called ‘kinetic chameleon’ as it relies on the kinetic interactions for the helicity0 mode). In what follows, we will give the general idea behind the Vainshtein mechanism starting from the decoupling limit of massive gravity and then show explicit solutions in the decoupling limit for static and spherically symmetric sources. Such an analysis is relevant for observational tests in the solar system as well as for other astrophysical tests (such as binary pulsar timing), which we shall explore in Section 11. We refer to the following review on the Vainshtein mechanism for further details, [35] as well as to the following work [160, 38, 99, 332, 36, 244, 40, 338, 321, 440, 316, 53, 376, 366, 407]. Recently, it was also shown that the Vainshtein mechanism works for bigravity, see Ref. [34].
We focus the rest of this section to the case of four spacetime dimensions, although many of the results presented in what follows are well understood in arbitrary dimensions.
12.1.1 Effective coupling to matter
These types of interactions are very similar to the Galileontype of interactions introduced by Nicolis, Rattazzi and Trincherini in Ref. [412] as a generalization of the decoupling limit of DGP. For simplicity we shall focus most of the discussion on the Vainshtein mechanism with Galileons as a special example, and then mention in Section 10.1.3 peculiarities that arise in the special case of massive gravity (see for instance Refs. [58, 57]).
We now first review how the Vainshtein mechanism works more explicitly in a static and spherically symmetric configuration before applying it to other systems. Note that the Vainshtein mechanism relies on irrelevant operators. In a standard EFT this cannot be performed without going beyond the regime of validity of the EFT. In the context of Galileons and other very specific derivative theories, one can reorganize the EFT so that the operators considered can be large and yet remain within the regime of validity of the reorganized EFT. This will be discussed in more depth in what follows.
12.1.2 Static and spherically symmetric configurations in Galileons
12.1.2.1 Suppression of the force
Considering the EarthMoon system, the force mediated by at the surface of the Moon is suppressed by 13 orders of magnitude compared to the Newtonian one in the cubic Galileon. While small, this is still not far off from the possible detectability from the lunar laser ranging space experiment [488], as will be discussed further in what follows. Note that in the quartic Galileon, that force is suppressed instead by 17 orders of magnitude and is there again very negligible.
When applying this naive estimate (10.16) to the HulseTaylor system for instance, we would infer a suppression of 15 orders of magnitude compared to the standard GR results. As we shall see in what follows this estimate breaks down when the time evolution is not negligible. These points will be discussed in the phenomenology Section 11, but before considering these aspects we review in what follows different aspects of massive gravity from a field theory perspective, emphasizing the regime of validity of the theory as well as the quantum corrections that arise in such a theory and the emergence of superluminal propagation.
12.1.2.2 Perturbations

First, we recover \(Z \sim \sqrt {{r_*}/r} \gg 1\) for r ≪ r_{*}, which is responsible for the redressing of the strong coupling scale as we shall see in (10.24). On the notrivial background the new strong coupling scale is \({\Lambda _*} \sim \sqrt Z \Lambda \gg \Lambda\) for r ≪ r_{*}. Similarly, on top of this background the coupling to external matter no longer occurs at the Planck scale but rather at the scale \(\sqrt Z {M_{{\rm{Pl}}}} \sim {10^7}{M_{{\rm{Pl}}}}\).

Second, we see that within the regime of validity of the classical calculation, the modes propagating along the radial direction do so with a superluminal phase and group velocity \(c_r^2 = 4/3 > 1\) and the modes propagating in the orthoradial direction do so with a subluminal phase and group velocity \(c_\Omega ^2 = 1/3\). This result occurs in any Galileon and multiGalileon theory which exhibits the Vainshtein mechanism [412, 129, 246]. The subluminal velocity is not of great concern, not even for Cerenkov radiation since the coupling to other fields is so much suppressed, but the superluminal velocity has been source of many questions [1]. It is definitely one of the biggest issues arising in these kinds of theories see Section 10.6.
Before discussing the biggest concerns of the theory, namely the superluminalities and the low strongcoupling scale, we briefly present some subtleties that arise when considering static and spherically symmetric solutions in massive gravity as opposed to a generic Galileon theory.
12.1.3 Static and spherically symmetric configurations in massive gravity
The Vainshtein mechanism was discussed directly in the context of massive gravity (rather than the Galileon larger family) in Refs. [363, 365, 99, 440] and more recently in [58, 455, 57]. See also Refs. [478, 105, 61, 413, 277, 160, 38, 37, 39] for other spherically symmetric solutions in massive gravity.

First if the parameters of the ghostfree theory of massive gravity are such that α_{3} + 4α_{4} ≠ 0, there is a mixing \({h^{\mu v}}X_{\mu v}^{(3)}\) between the helicity0 and 2 modes of the graviton that cannot be removed by a local field redefinition (unless we work in an special types of backgrounds). The effects of this coupling were explored in [99, 57] and it was shown that the theory does not exhibit any stable static and spherically symmetric configuration in presence of a localized pointlike matter source. So in order to be phenomenologically viable, the theory of massive gravity needs to be tuned with α_{3} + 4α_{4} = 0. Since these parameters do not get renormalized this is a tuning and not a finetuning.
 When α_{3}+4α_{4} = 0 and the previous mixing \({h^{\mu v}}X_{\mu v}^{(3)}\) is absent, the decoupling limit of massive gravity resembles a specific quartic Galileon, where the coefficient of the cubic Galileon is related to quartic coefficient (and if one vanishes so does the other one),where we have set α_{2} = 1 and the Galileon Lagrangians \({\mathcal L}_{{\rm{Gal}}}^{(3,4)}[\pi ]\) are given in (8.44) and (8.45). Note that in this decoupling limit the graviton mass always enters in the combination \(\alpha/\Lambda _3^3\), with α = (1 + 3/2α_{3}). As a result this decoupling limit can never be used to directly probe the graviton mass itself but rather of the combination \(\alpha/\Lambda _3^3\) [57]. Beyond the decoupling limit however the theory breaks the degeneracy between and m.$$\begin{array}{*{20}c} {{\mathcal{L}_{{\rm{Helicity  0}}}} =  {3 \over 4}{{(\partial \pi)}^2} + {{3\alpha} \over {4\Lambda _3^3}}\mathcal{L}_{({\rm{Gal}})}^{(3)}[\pi ]  {1 \over 4}{{\left({{\alpha \over {\Lambda _3^3}}} \right)}^2}\mathcal{L}_{({\rm{Gal}})}^{(4)}[\pi ]} \\ {+ {1 \over {{M_{{\rm{Pl}}}}}}\left({\pi T + {\alpha \over {\Lambda _3^3}}{\partial _\mu}\pi {\partial _\nu}\pi {T^{\mu \nu}}} \right),\quad \quad} \\ \end{array}$$(10.21)
Not only is the cubic Galileon always present when the quartic Galileon is there, but one cannot prevent the new coupling to matter ∂_{ μ }π∂_{ ν }∂πT^{ μν } which is typically absent in other Galileon theories.
Interestingly, when performing the perturbation analysis on this solution, the modes along all directions are subluminal, unlike what was found for the Galileon in (10.20). It is yet unclear whether this is an accident to this specific solution or if this is something generic in consistent solutions of massive gravity.
12.2 Validity of the EFT
The Vainshtein mechanism presented previously relies crucially on interactions which are important at a low energy scale Λ ≪ M_{Pl}. These interactions are operators of dimension larger than four, for instance the cubic Galileon (∂π)^{2}□π is a dimension7 operator and the quartic Galileon is a dimension10 operator. The same can be seen directly within massive gravity. In the decoupling limit (8.38), the terms \({h^{\mu v}}X_{\mu v}^{(2,3)}\) are respectively dimension7 and10 operators. These operators are thus irrelevant from a traditional EFT viewpoint and the theory is hence not renormalizable.
This comes as no surprise, since gravity itself is not renormalizable and there is thus no reason to expect massive gravity nor its decoupling limit to be renormalizable. However, for the Vainshtein mechanism to be successful in massive gravity, we are required to work within a regime where these operators dominate over the marginal ones (i.e., over the standard kinetic term ∂π)^{2} in the strongly coupled region where ∂^{2}π ≫ Λ^{3}). It is, therefore, natural to wonder whether or not one can ever use the effective field description within the strong coupling region without going outside the regime of validity of the theory.
 1.
First, as we shall see in what follows, the Galileon interactions or the interactions that arise in the decoupling limit of massive gravity and which are essential for the Vainshtein mechanism do not get renormalized within the decoupling limit (they enjoy a nonrenormalization theorem which we review in what follows).
 2.
The nonrenormalization theorem together with the shift and Galileon symmetry implies that only higher operators of the form (∂^{ ℓ }π)^{ m }, with ℓ, m ≥ 2 are generated by quantum corrections. These operators differ from the Galileon operators in that they always generate terms that more than two derivatives on the field at the level of the equation of motion (or they always have two or more derivatives per field at the level of the action).
This means that there exists a regime of interest for the theory, for which the operators generated by quantum corrections are irrelevant (nonimportant compared to the Galileon interactions). Within the strong coupling region, the field itself can take large values, π ∼ Λ, ∂π ∼ Λ^{2}, ∂^{2}π ∼ Λ^{3}, and one can still rely on the Galileon interactions and take no other operator into account so long as any further derivative of the field is suppressed, d^{ n }π ≪ Λ^{n+1} for any n ≥ 3.
This is similar to the situation in DBI scalar field models, where the field operator itself and its velocity is considered to be large π ∼ Λ and ∂π Λ^{2}, but the field acceleration and any higher derivatives are suppressed ∂^{ n }π ≪ Λ^{n+1} for n ≥ 2 (see [157]). In other words, the Effective Field expansion should be reorganized so that operators which do not give equations of motion with more than two derivatives (i.e., Galileon interactions) are considered to be large and ought to be treated as the relevant operators, while all other interactions (which lead to terms in the equations of motion with more than two derivatives) are treated as irrelevant corrections in the effective field theory language.
Finally, as mentioned previously, the Vainshtein mechanism itself changes the canonical scale and thus the scale at which the fluctuations become strongly coupled. On top of a background configuration, interactions do not arise at the scale Λ but rather at the rescaled strong coupling scale \({\Lambda _*} = \sqrt Z \Lambda\), where Z is expressed in (10.7). In the strong coupling region, Z ≫ 1 and so Λ_{*} ≫ Λ. The higher interactions for fluctuations on top of the background configuration are hence much smaller than expected and their quantum corrections are therefore suppressed.
12.3 Nonrenormalization
The nonrenormalization theorem mentioned above states that within a Galileon theory the Galileon operators themselves do not get renormalized. This was originally understood within the context of the cubic Galileon in the procedure established in [411] and is easily generalizable to all the Galileons [412]. In what follows, we review the essence of nonrenormalization theorem within the context of massive gravity as derived in [140].
12.4 Quantum corrections beyond the decoupling limit
As already emphasized, the consistency of massive gravity relies crucially on a very specific set of allowed interactions summarized in Section 6. Unlike for GR, these interactions are not protected by any (known) symmetry and we thus expect quantum corrections to destabilize this structure. Depending on the scale at which these quantum corrections kick in, this could lead to a ghost at an unacceptably low scale.
Furthermore, as discussed previously, the mass of the graviton itself is subject to quantum corrections, and for the theory to be viable the graviton mass ought to be tuned to extremely small values. This tuning would be technically unnatural if the graviton mass received large quantum corrections.
 1.
Destabilization of the potential:
At oneloop, matter fields do not destabilize the structure of the potential. Graviton loops on the hand do lead to new operators which do not belong to the ghostfree family of interactions presented in (6.9–6.13), however they are irrelevant below the Planck scale.
 2.
Technically natural graviton mass:
As already seen in (10.30), the quantum corrections for the graviton mass are suppressed by the graviton mass itself, δm^{2} ≲ m^{2}(m/M_{Pl})^{2/3} this result is confirmed at oneloop beyond the decoupling limit and as result a small graviton mass is technically natural.
12.4.1 Matter loops
The essence of these arguments go as follows: Consider a ‘covariant’ coupling to matter, ℒ_{matter}(g_{ μν }, ψ_{ i }), for any species ψ_{ i } be it a scalar, a vector, or a fermion (in which case the coupling has to be performed in the vielbein formulation of gravity, see (5.6)).
At one loop, virtual matter fields do not mix with the virtual graviton. As a result as far as matter loops are concerned, they are ‘unaware’ of the graviton mass, and only lead to quantum corrections which are already present in GR and respect diffeomorphism invariance. So the only potential term (i.e., operator with no derivatives on the metric fluctuation) it can lead to is the cosmological constant.
This result was confirmed at the level of the oneloop effective action in [146], where it was shown that a field of mass M leads to a running of the cosmological constant δΛ_{CC} ∼ M^{4}. This result is of course wellknown and is at the origin of the old cosmological constant problem [484]. The key element in the context of massive gravity is that this cosmological constant does not lead to any ghost and no new operators are generated from matter loops, at the oneloop level (and this independently of the regularization scheme used, be it dimensional regularization, cutoff regularization, or other.) At higher loops we expect virtual matter fields and graviton to mix and effect on the structure of the potential still remains to be explored.
12.4.2 Graviton loops
As see in Section 10.1 (see also Section 10.2), massive gravity is phenomenologically viable only if it has an active Vainshtein mechanism which screens the effect of the helicity0 mode in the vicinity of dense environments. This Vainshtein mechanisms relies on having a large background for the helicity0 mode, π = π_{0} + δπ with \({\partial ^2}{\pi _0} \gg \Lambda _3^3 = {m^2}{M_{{\rm{Pl}}}}\), which in unitary gauge implies h = h_{0} + (δh, with h_{0} ≫ M_{Pl}.
The resolution to this issue lies within the Vainshtein mechanism itself and its implementation not only at the classical level as was done to estimate the mass of the ghost in (10.33) but also within the calculation of the quantum corrections themselves. To take the Vainshtein mechanism consistently into account one needs to consider the effective action redressed by the interactions themselves (as was performed at the classical level for instance in (10.9)).
As a result, at the oneloop level the quantum corrections destabilize the structure of the potential but in a way which is irrelevant below the Planck scale.
12.5 Strong coupling scale vs cutoff
Whether it is to compute the Vainshtein mechanism or quantum corrections to massive gravity, it is crucial to realize that the scale Λ = (m^{2} M_{Pl})^{1/3} (denoted as Λ in what follows) is not necessarily the cutoff of the theory.
The cutoff of a theory corresponds to the scale at which the given theory breaks down and new physics is required to describe nature. For GR the cutoff is the Planck scale. For massive gravity the cutoff could potentially be below the Planck scale, but is likely well above the scale Λ, and the redressed scale Λ_{*} computed in (10.24). Instead Λ (or Λ_{*} on some backgrounds) is the strongcoupling scale of the theory.
When hitting the scale Λ or Λ_{*} perturbativity breaks down (in the standard field representation of the theory), which means that in that representation loops ought to be taken into account to derive the correct physical results at these scales. However, it does not necessarily mean that new physics should be taken into account. The fact that treelevel calculations do not account for the full results does in no way imply that theory itself breaks down at these scales, only that perturbation theory breaks down.
Massive gravity is of course not the only theory whose strong coupling scale departs from its cutoff. See, for instance, Ref. [31] for other examples in chiral theory, or in gravity coupled to many species. To get more intuition on these types of theories and on the distinction between strong coupling scale and cutoff, consider a large number N ≫ 1 of scalar fields coupled to gravity. In that case the effective strong coupling scale seen by these scalars is \({M_{{\rm{eff}}}} = {M_{{\rm{Pl}}}}/\sqrt N \ll {M_{{\rm{Pl}}}}\), while the cutoff of the theory is still M_{Pl} (the scale at which new physics enters in GR is independent of the number of species living in GR).
“In effective field theories it is common to identify the onset of new physics with the violation of treelevel unitarity. However, we show that this is parametrically incorrect in the case of chiral perturbation theory, and is probably theoretically incorrect in general. In the chiral theory, we explore perturbative unitarity violation as a function of the number of colors and the number of flavors, holding the scale of the “new physics” (i.e., QCD) fixed. This demonstrates that the onset of new physics is parametrically uncorrelated with treeunitarity violation. When the latter scale is lower than that of new physics, the effective theory must heal its unitarity violation itself, which is expected because the field theory satisfies the requirements of unitarity. (…) A similar example can be seen in the case of general relativity coupled to multiple matter fields, where iteration of the vacuum polarization diagram restores unitarity. We present arguments that suggest the correct identification should be connected to the onset of inelasticity rather than unitarity violation.” [31].
12.6 Superluminalities and (a)causality
 1.Phase Velocity: For a wave of constant frequency, the phase velocity is the speed at which the peaks of the oscillations propagate. For a wave [77]the phase velocity v_{phase} is given by$$f(t,x) = A\sin (\omega t  kx) = A\sin \left({\omega \left({t  {x \over {{v_{{\rm{phase}}}}}}} \right)} \right),$$(10.35)$${v_{{\rm{phase}}}} = {\omega \over k}.$$(10.36)
 2.Group Velocity: If the amplitude of the signal varies, then the group velocity represents the speed at which the modulation or envelop of the signal propagates. In a medium where the phase velocity is constant and does not depend on frequency, the phase and the group velocity are the same. More generally, in a medium with dispersion relation ω (k), the group velocity isWe are familiar with the notion that the phase velocity can be larger than speed of light c (in this review we use units where c = 1.) Similarly, it has been known for now almost a century that$${v_{{\rm{group}}}} = {{\partial \omega (k)} \over {\partial k}}.$$(10.37)
“(…) the group velocity could exceed c in a spectral region of an anomalous dispersion” [399].
 3.
Signal Velocity “yields the arrival of the main signal, with intensities of the order of magnitude of the input signal ” [77]. Nowadays it is common to define the signal velocity as the velocity from the part of the pulse which has reached at least half the maximum intensity. However, as mentioned in [399], this notion of speed rather is arbitrary and some known physical systems can exhibit a signal velocity larger than c.
 4.
Front Velocity: Physically, the front velocity represents the speed of the front of a disturbance, or in other words “Front velocity (…) correspond[s] to the speed at which the very first, extremely small (perhaps invisible) vibrations will occur.” [77]. The front velocity is thus the speed at which the very first piece of information of the first “forerunner” propagates once a front or a “sudden discontinuous turnon of a field ” is turned on [399].
“The front is defined as a surface beyond which, at a given instant in time the medium is completely at rest ” [77],where θ (t) is the Heaviside step function.$$f(t,x) = \theta (t)\sin (\omega t  kx),$$(10.38)In practise the front velocity is the large (high frequency) limit of the phase velocity.
The front velocity, on the other hand, is the real ‘measure’ of the speed of propagation of new information, and the front velocity is always (and should always be) (sub)luminal. As shown in [445], “the ‘speed of light’ relevant for causality is v_{ph}(∞), i.e., the highfrequency limit of the phase velocity. Determining this requires a knowledge of the UV completion of the quantum field theory.” In other words, there is no sense in computing a classical version of the front velocity since quantum corrections always dominate.

In Galileons theories the presence of superluminal group velocity has been established for all the parameters which exhibit an active Vainshtein mechanism. These are present in spherically symmetric configurations near massive sources as well as in selfsourced plane waves and other configurations for which no special kind of matter is required.

Since massive gravity reduces to a specific Galileon theory in some limit, we expect the same result to be true there well and to yield solutions with superluminal group velocity. However, to date no fully consistent solution has yet been found in massive gravity which exhibits superluminal group velocity (let alone superluminal front velocity which would be the real signal of acausality). Only local configurations have been found with superluminal group velocity or finite frequency phase velocity but it has not been proven that these are stable global solutions. Actually, in all the cases where this has been checked explicitly so far, these local configurations have been shown not to be part of global stable solutions.
It is also worth noting that the potential existence of superluminal propagation is not restricted to theories which break the gauge symmetry. For instance, massless spin3/2 are also known to propagate superluminal modes on some nontrivial backgrounds [306].
12.6.1 Superluminalities in Galileons
Superluminalities in Galileon and other closely related theories have been pointed out in several studies for more a while [412, 1, 262, 220, 115, 129, 246]. Note also that Ref. [313] was the first work to point out the existence of superluminal propagation in the higherdimensional picture of DGP rather than in its purely fourdimensional decoupling limit. See also Refs. [112, 110, 311, 312, 218, 219] for related discussions on super versus subluminal propagation in conformal Galileon and other DBIrelated models. The physical interpretation of these superluminal propagations was studied in other nonGalileon models in [199, 43] and see [206, 469] for their potential connection with classicalization [214, 213, 205, 11].
In all the examples found so far, what has been pointed out is the existence of a superluminal group velocity, which is the regime inspected is the same as the phase velocity. As we will see below (see Section 10.7), in the one example where we can compute the phase velocity for momenta at which loops ought to be taken into account, we find (thanks to a dual description) that the corresponding front velocity is exactly luminal even though the lowenergy group velocity is superluminal. This is no indication that all Galileon theories are causal but it comes to show how a specific Galileon theory which exhibits superluminal group velocity in some regime is dual to a causal theory.
In most of the cases considered, superluminal propagation was identified in a spherically symmetric setting in the vicinity of a localized mass as was presented in Section 10.1.2. To convince the reader that these superluminalities are independent of the coupling to matter, we show here how superluminal propagation can already occur in the vacuum in any Galileon theories without even the need of any external matter.
12.6.2 Superluminalities in massive gravity
The existence of superluminal propagation directly in massive gravity has been pointed out in many references in the literature [87, 276, 192, 177] (see also [496] for another nice discussion). Unfortunately none of these studies have qualified the type of velocity which exhibits superluminal propagation. On closer inspection it appears that there again for all the cases cited the superluminal propagation has so far always been computed classically without taking into account quantum corrections. These results are thus always valid for the low frequency group velocity but never for the front velocity which requires a fully fledged calculation beyond the treelevel classical approximation [445].
 1.
Argument: Some background solutions of massive gravity admit superluminal propagation.
Limitation of the argument: the solutions inspected were not physical.
Ref. [276] was the first work to point out the presence of superluminal group velocity in the full theory of massive gravity rather than in its Galileon decoupling limit. These superluminal modes ride on top of a solution which is unfortunately unrealistic for different reasons. First, the solution itself is unstable. Second, the solution has no rest frame (if seen as a perfect fluid) or one would need to perform a superluminal boost to bring the solution to its rest frame. Finally, to exist, such a solution should be sourced by a matter source with complex eigenvalues [142]. As a result the solution cannot be trusted in the first place, and so neither can the superluminal propagation of fluctuations about it.
 2.
Argument: Some background solutions of the decoupling limit of massive gravity admit superluminal propagation.
Limitation of the argument: the solutions were only found in a finite region of space and time.
In Ref. [87] superluminal propagation was found in the decoupling limit of massive gravity. These solutions do not require any special kind of matter, however the background has only be solved locally and it has not (yet) been shown whether or not they could extrapolate to sensible and stable asymptotic solutions.
 3.
Argument: There are some exact solutions of massive gravity for which the determinant of the kinetic matrix vanishes thus massive gravity is acausal.
Limitation of the argument: misuse of the characteristics analysis — what has really been identified is the absence of BD ghost.
Ref. [192] presented some solutions which appeared to admit some instantaneous modes in the full theory of massive gravity. Unfortunately the results presented in [192] were due to a misuse of the characteristics analysis.
The confusion in the characteristics analysis arises from the very constraint that eliminates the BD ghost. The existence of such a constraint was discussed in length in many different formulations in Section 7 and it is precisely what makes ghostfree (or dRGT) massive gravity special and theoretically viable. Due to the presence of this constraint, the characteristics analysis should be performed after solving for the constraints and not before [326].
In [192] it was pointed out that the determinant of the time kinetic matrix vanished in ghostfree massive gravity before solving for the constraint. This result was then interpreted as the propagation of instantaneous modes and it was further argued that the theory was then acausal. This result is simply an artefact of not properly taking into account the constraint and performing a characteristics analysis on a set of modes which are not all dynamical (since two phase space variables are constrained by the primary and secondary constrains [295, 294]). In other word it is precisely what wouldhavebeen the BD ghost which is responsible for canceling the determinant of the time kinetic matrix. This does not mean that the BD ghost propagates instantaneously but rather that the BD ghost is not present in that theory, which is the very point of the theory.
One can show that the determinant of the time kinetic matrix in general does not vanish when computing it after solving for the constraints. In summary the results presented in [192] cannot be used to deduce the causality of the theory or absence thereof.
 4.
Argument: Massive gravity admits shock wave solutions which admit superluminal and instantaneous modes.
Limitation of the argument: These configurations lie beyond the regime of validity of the classical theory.
Shock wave local solutions on top of which the fluctuations are superluminal were found in [177]. Furthermore, a characteristic analysis reveals the possibility for spacelike hypersurfaces to be characteristic. While interesting, such configurations lie beyond the regime of validity of the classical theory and quantum corrections ought to be included.
Having said that, it is likely that the characteristic analysis performed in [177] and then in [178] would give the same results had it been performed on regular solutions.^{27} This point is discussed below.
 5.
Argument: The characteristic analysis shows that some field configurations of massive gravity admit superluminal propagation and the possibility for spacelike hypersurfaces to be characteristic.
Limitation of the argument: Same as point 2. Putting this limitation aside this result is certainly correct classically and in complete agreement with previous results presented in the literature (see point 2 where local solutions were given).
Even though the characteristic analysis presented in [177] used shock wave local configurations, it is also valid for smooth wave solutions which would be within the regime of validity of the theory. In [178] the characteristic analysis for a shock wave was presented again and it was argued that CTCs were likely to exist.
To better see the essence behind the general characteristic analysis argument, let us look at the (simpler yet representative) case of a Proca field with an additional quartic interaction as explored in [420, 467],The idea behind the characteristic analysis is to “replace the highest derivative terms ∂^{ N } A by k^{ N }Ã ” [420] so that one of the equations of motion is$$\mathcal{L} =  {1 \over 4}F_{\mu \nu}^2  {1 \over 2}{m^2}{A^\mu}{A_\mu}  {1 \over 4}\lambda {({A^\mu}{A_\mu})^2}.$$(10.48)When λ ≠ 0, one can solve this equation maintaining k^{ μ }Ã_{ μ } ≠ 0. Then there are certainly field configurations for which the normal to the characteristic surface is timelike and thus the mode with k^{ μ }Ã_{ μ } ≠ 0 can propagate superluminally in this Proca field theory. However, as we shall see below this very combination \({\mathcal Z} = [({m^2} + \lambda {A^v}{A_v}){k^\alpha}{k_\alpha} + 2\lambda {({A^v}{k_v})^2}] = 0\) with k^{2} timelike (say k^{ μ } = (1,0,0,0)) is the coefficient of the timelike kinetic term of the helicity0 mode. So one can never have [(m^{2} + λA^{ ν } A_{ ν }) k^{ α }k_{ α } + 2λ(A^{ ν }k_{ ν })^{2}] = 0 with k^{ μ } = (1, 0, 0, 0) (or any timelike direction) without automatically having an infinitely strongly helicity0 mode and thus automatically going beyond the regime of validity of the theory (see Ref. [87] for more details.)$$\left[ {({m^2} + \lambda {A^\nu}{A_\nu}){k^\alpha}{k_\alpha} + 2\lambda {{({A^\nu}{k_\nu})}^2}} \right]{k^\mu}{\tilde A_\mu} = 0.$$(10.49)To see this more precisely, let us perform the characteristic analysis in the Stückelberg language. An analysis performed in unitary gauge is of course perfectly acceptable, but to connect with previous work in Galileons and in massive gravity the Stückelberg formalism is useful.
In the Stückelberg language, A_{ μ } → A_{ μ } + m^{−1} ∂_{ μ }π, keeping track of the terms quadratic in π, we have@$$\mathcal{L}_\pi ^{(2)} =  {1 \over 2}{Z^{\mu \nu}}{\partial _\mu}\pi {\partial _\nu}\pi ,$$(10.50)It is now clear that the combination found in the characteristic analysis \({\mathcal Z}\) is nothing other than$${\rm{with}}\qquad {Z^{\mu \nu}}[{A_\mu}] = {\eta ^{\mu \nu}} + {\lambda \over {{m^2}}}{A^2}{\eta ^{\mu \nu}} + 2{\lambda \over {{m^2}}}{A^\mu}{A^\nu}.$$(10.51)where Z^{ μν } is the kinetic matrix of the helicity0 mode. Thus, a configuration with \({\mathcal Z} = 0\) with k^{ μ } = (1, 0, 0, 0) implies that the Z^{00} component of helicity0 mode kinetic matrix vanishes. This means that the conjugate momentum associated to cannot be solved for in this timeslicing, or that the helicity0 mode is infinitely strongly coupled.$$\mathcal{Z} \equiv {Z^{\mu \nu}}{k_\mu}{k_\nu},$$(10.52)This result should sound familiar as it echoes what has already been shown to happen in the decoupling limit of massive gravity, or here of the Proca field theory (see [43, 468] for related discussions in that case). Considering the decoupling limit of (10.48) with m → 0 and \(\hat \lambda = \lambda/{m^4} \to\) const, we obtain a decoupled massless gauge field and a scalar field,$${\mathcal{L}_{{\rm{DL}}}} =  {1 \over 4}F_{\mu \nu}^2  {1 \over 2}{(\partial \pi)^2}  {{\hat \lambda} \over 4}{(\partial \pi)^4}.$$(10.53)For fluctuations about a given background configuration π = π_{0}(x) + δπ, the fluctuations see an effective metric \({\tilde Z^{\mu v}}({\pi _0})\) given by$${\tilde Z^{\mu \nu}}({\pi _0}) = \left({1 + \hat \lambda {{(\partial {\pi _0})}^2}} \right){\eta ^{\mu \nu}} + 2\hat \lambda {\partial ^\mu}{\pi _0}{\partial ^\nu}{\pi _0}.$$(10.54)Of course unsurprisingly, we find \({\tilde Z^{\mu v}}({\pi _0}) \equiv {m^{ 2}}{Z^{\mu v}}[{m^{ 1}}{\partial _\mu}{\pi _0}]\). The fact that we can find superluminal or instantaneous propagation in the characteristic analysis is equivalent to the statement that in the decoupling limit there exists classical field configurations for π_{0} for which the fluctuations propagate superluminally (or even instantaneously). Thus, the results of the characteristic analysis are in agreement with previous results in the decoupling limit as was pointed out for instance in [1, 412, 87].
Once again, if one starts with a field configuration where the kinetic matrix is well defined, one cannot reach a region where one of the eigenvalues of crosses zero without going beyond the regime of validity of the theory as described in [87]. See also Refs. [318, 445] for the use of the characteristic analysis and its relation to (micro)causality.
The presence of instantaneous modes in some (selfaccelerating) solutions of massive gravity was actually pointed out from the very beginning. See Refs. [139] and [364] for an analysis of selfaccelerating solutions in the decoupling limit, and [125] for selfaccelerating solutions in the full theory (see also [264] for a complementary analysis of selfaccelerating solutions.) All these analysis had already found instantaneous modes on some selfaccelerating branches of massive gravity. However, as pointed out in all these analysis, the real question is to establish whether or not these solutions lie within the regime of validity of the EFT, and whether one could reach such solutions with a finite amount of energy and while remaining within the regime of validity of the EFT.
This aspect connects with Hawking’s chronology protection argument which is already in effect in GR [302, 303], (see also [472] and [473] for a comprehensive review). This argument can be extended to Galileon theories and to massive gravity as was shown in Ref. [87].
It was pointed out in [87] and in many other preceding works that there exists local backgrounds in Galileon theories and in massive gravity which admit superluminal and instantaneous propagation. (As already mentioned, in point 2. above in massive gravity it is however unclear whether these localized backgrounds admit stable and consistent global realizations). The worry with superluminal propagation is that it could imply the presence of CTCs (closed timelike curves). However, when ‘cranking up’ the background sufficiently so as to reach a solution which would admit CTCs, the Galileon or the helicity0 mode of the graviton becomes inevitably infinitely strongly coupled. This means that the effective field theory used breaks down and the background becomes unstable with arbitrarily fast decay time before any CTC can ever be formed.

No stable global solutions have been found with the same properties.

No CTCs can been constructed within the regime of validity of the theory. As shown in Ref. [87] CTCs constructed with these configurations always lie beyond the regime of validity of the theory. Indeed in order to create a CTC, a mode needs to become instantaneous. As soon as a mode becomes instantaneous, the regime of validity of the classical theory is null and classical considerations are thus obsolete.

Finally, and most importantly, all the results presented so far for Galileons and massive gravity (including the ones summarized here), rely on classical configurations. As was explained at the beginning of this section causality is determined by the front velocity for which classical considerations break down. Therefore, no classical calculations can ever prove or disprove the (a)causality of a theory.
12.6.3 Superluminalities vs BoulwareDeser ghost vs Vainshtein
The plane wave solutions provided in (10.40) is still a vacuum solution in this case. Following the same analysis as that provided in Section 10.6.1, one can easily find modes propagating with superluminal group and phase velocity for appropriate choices of functions F (x^{1} − t) (while keeping within the regime of validity of the theory.)
As a result the presence of local solutions in massive gravity which admit superluminalities is not connected to the constraint that removes the BD ghost. Rather it is likely that the presence of superluminalities could be tied to the Vainshtein mechanism (with flat asymptotic boundary conditions), which as we have seen is crucial for these types of theories (see Refs. [1, 313] and [129] for a possible connection.) More recently, the presence of superluminalities has also been connected to the idea of classicalization which is tied to the Vainshtein mechanism [206, 469]. It is possible that the only way these superluminalities could make sense is through this idea of classicalization. Needless to say this is very much speculative at the moment. Perhaps the Galileon dualities presented below could help understanding these open questions.
12.7 Galileon duality
The low strong coupling scale and the presence of superluminalities raises the question of how to understand the theory beyond the redressed strong coupling scale, and whether or not the superluminalities are present in the front velocity.
A nontrivial map between the conformal Galileon and the DBI conformal Galileon was recently presented in [113] (see also [55]). The conformal Galileon side admits superluminal propagation while the DBI side of the map is luminal. Since both sides are related by a ‘simple’ field redefinition which does not change the physics, and cannot change the causality of the theory, this suggests that the superluminalities encountered in that example must be in the group velocity rather than the front velocity.
Recently, another Galileon duality was proposed in [115] and [136] by use of simple Legendre transform. First encountered within the decoupling limit of bigravity [224], the duality can be seen as being related to the freedom in how to introduce the Stückelberg fields. However, the duality survives independently from bigravity and could be significant in the context of massive gravity.
What was computed in these examples for a nontrivial Galileon theory (and in all the examples known so far in the literature) is only the treelevel group velocity valid till the (redressed) strong coupling scale of the theory. Once hitting the (redressed) strong coupling scale the loops need to be included. In the dual free theory however there are no loops to account for, and thus the result of luminal velocity in that free theory is valid at all scale and has to match the front velocity. This is strongly suggestive that the front velocity in that example of nontrivial Galileon theory is luminal and the theory is causal even though it exhibits a superluminal group velocity.
It is clear at this point that a deeper understanding of this class of theories is required. We expect this will be the subject of further studies. In the rest of this review, we focus on some phenomenological aspects of massive gravity before presenting other theories of massive gravity.
13 Part III Phenomenological Aspects of Ghostfree Massive Gravity
14 Phenomenology
Below, we summarize some of the phenomenology of massive gravity and DGP. Many other interesting results have been derived in the literature, including the implication for the very Early Universe. For instance false vacuum decay and the HartleHawking noboundary proposal was studied in the context of massive gravity in [498, 439, 499] where it was shown that the graviton mass could increase the rate. The implications of massive gravity to the cyclic Universe were also studied in Ref. [91] with a regular bounce. We emphasize that in massive gravity the reference metric has to be chosen once and for all and cannot be modified, no matter what the background configuration is (no matter whether we are interested in cosmology, or in spherically symmetry configurations or other). This is a consistent procedure since massive gravity has been shown to be free of the BD ghost for any choice of reference metric independently of the background configuration. Theories with different reference metrics represent different independent theories.
14.1 Gravitational waves
14.1.1 Speed of propagation
If the photon had a mass it would no longer propagate at ‘the speed of light’, but at a lower speed. For the photon its speed of propagation is known with such an accuracy in so many different media that it can be used to put the most stringent constraints on the photon mass to [68] m_{ γ } < 10^{−18} eV. In the rest of this review we will adopt the viewpoint that the photon is massless and that light does indeed propagate at the ‘speed of light’.
The earliest bounds on the graviton mass were based on the same idea. As described in [487], (see also [394]), if the graviton had a mass, gravitational waves would propagate at a speed different than that of light, \(\upsilon _g^2 = 1  {m^2}/{E^2}\) (assuming a speed of light c = 1). This different velocity between the light and gravitational waves would manifest itself in observations of supernovae. Assuming the emission of a gravitational wave with frequency larger than the graviton mass, this could lead to a bound on the graviton mass of m < 10^{−23} eV considering a frequency of 100 Hz and a supernovae located 200 Mpc away [487] (assuming that the photon propagates at the speed of light).
Alternatively, another way to test the speed of gravitational waves and bound the graviton mass without relying on any assumptions on the photon is through the observation of inspiralling compact objects which allows to derive the frequencydependence of GWs. The detection of GWs in Advanced LIGO could then bound the graviton mass potentially all the way down to < 10^{−29} eV [487, 486, 71].
The graviton mass is also relevant for the production of primordial gravitational waves during inflation. Following the analysis of [282] it was shown that the graviton mass opens up the production of gravitational waves during inflation with a sharp peak with a height and position which depend on the graviton mass. See also [403] for the study of exact plane wave solutions in massive gravity.
Nevertheless, these bounds on the graviton mass are relatively weak compared to the typical value of m ∼ 10^{−30} − 10^{−33} eV considered till now in this review. The reason for this is because these bounds do not take into account the effects arising from the additional polarization in the gravitational waves which would be present if the graviton had a mass in a Lorentzinvariant theory. For the photon, if it had a mass, the additional polarization would decouple and would therefore be irrelevant (this is related to the absence of vDVZ discontinuity at the classical level for a Proca theory.) In massive gravity, however, the helicity0 mode of the graviton couples to matter. As we shall see below, the bounds on the graviton mass inferred from the absence of fifth forces are typically much more stringent.
14.1.2 Additional polarizations
One of the predictions of GR is the existence of gravitational waves (GW) with two transverse independent polarizations.
While GWs have not been directly detected via interferometer yet, they have been detected through the spindown of binary pulsar systems [322, 457, 485]. This detection via binary pulsars does not count as a direct detection, but it matches expectations from GWs with such an accuracy, and for now so many different systems of different relative masses that it seems unlikely that the spindown could be due to something different than the emission of GWs.
As emphasized in the first part of this review, and particularly in Section 2.5, the sixth excitation, namely the longitudinal one, represents a ghost degree of freedom. Thus, if that mode is observed, it cannot be arising from a Lorentzinvariant massive graviton. Its presence could be linked for instance to new scalar degrees of freedom which are independent from the graviton itself. In massive gravity, only five polarizations are expected. Notice however that the helicity1 mode does not couple directly to matter or external sources, so it is unlikely that GWs with polarizations which mix the transverse and longitudinal directions would be produced in a natural process.
Furthermore, any physical process which is expected to produce GWs would include very dense sources where the Vainshtein mechanism will thus be expected to be active and screen the effect of the helicity0 mode. As a result the excitation of the breathing mode is expected to be suppressed in any theory of massive gravity which includes an active Vainshtein mechanism.

The helicity2 modes are produced in the same way as in GR and would be indistinguishable if they travel distances smaller than the graviton Compton wavelength

The helicity1 modes are not produced

The breathing or conformal mode is produced but suppressed by the Vainshtein mechanism and so the magnitude of this mode is suppressed compared to the helicity2 polarization by many orders of magnitudes.

The longitudinal mode does not exist in a ghostfree theory of massive gravity. If such a mode is observed it must be arise from another field independent from the graviton.
We will also discuss the implications for indirect detection of GWs via binary pulsar spindown in Section 11.4. We will see that already in these setups the radiation in the breathing mode is suppressed by 8 orders of magnitude compared to that in the helicity2 mode. In more relativistic systems such as blackhole mergers, this suppression will be even bigger as the Vainshtein mechanism is stronger in these cases, and so we do not expect to see the helicity0 mode component of a GW emitted by such systems.
To summarize, while additional polarizations are present in massive gravity, we do not expect to be able to observe them in current interferometers. However, these additional polarizations, and in particular the breathing mode can have larger effects on solarsystem tests of gravity (see Section 11.2) as well as for weak lensing (see Section 11.3), as we review in what follows. They also have important implications for black holes as we discuss in Section 11.5 and in cosmology in Section 12.
14.2 Solar system
A lot of the phenomenology of massive gravity can be derived from its decoupling limit where it resembles a Galileon theory. Since the Galileon was first encountered in DGP most of the phenomenology was first derived for that model. The extension to massive gravity is usually relatively straightforward with a few subtleties which we mention at the end. We start by reviewing the phenomenology assuming a cubic Galileon decoupling limit, which is directly applicable for DGP and then extend to the quartic Galileon and ghostfree massive gravity.
Within the context of DGP, a lot of its phenomenology within the solar system was derived in [388, 386] using the full higherdimensional picture as well as in [215]. In these work the effect from the helicity0 mode in the advanced of the perihelion were computed explicitly. In particular in [215] it was shown how an infrared modification of gravity could have an effect on small solar system scales and in particular on the Moon. In what follows we review their approach.
14.2.1 DGP and cubic Galileon
As pointed out in [215] and [388, 386], the effect could be bigger for the advance of the perihelion of Mars around the Sun, but at the moment the accuracy is slightly less.
14.2.2 Massive gravity and quartic Galileon:
14.3 Lensing
As mentioned previously, one peculiarity of massive gravity not found in DGP nor in a typical Galileon theory (unless we derive the Galileons from a higherdimensional brane picture [157]) is the new disformal coupling to matter of the form ∂_{ μ }π∂_{ ν }πT^{ μν }, which means that the helicity0 mode also couples to conformal matter.
In the vacuum, for a static and spherically symmetric configuration the coupling ∂_{ μ }π∂_{ ν }πT^{ μν } plays no role. So to the level at which we are working when deriving the Vainshtein mechanism about a pointlike mass this additional coupling to matter does not affect the background configuration of the field (see [140] for a discussion outside the vacuum, taking into account for the instance the effect of the Earth atmosphere). However, it does affect this disformal coupling does affect the effect metric seen by perturbed sources on top of this configuration. This could have some implications for structure formation is to the best of our knowledge have not been fully explored yet, and does affect the bending of light. This effect was pointed out in [490] and the effects to gravitational lensing were explored. We review the key results in what follows and refer to [490] for further discussions (see also [448]).
At the level of galaxies or clusters of galaxy, the effect might be more tangible. The reason for that is that for the mass of a galaxy, the associated strong coupling radius is not much larger than the galaxy itself and thus at the edge of a galaxy these effects could be stronger. These effects were investigated in [490] where it was shown a few percent effect on the tangential shear caused by the helicity0 mode of the graviton or of a disformal Galileon considering a NavarroFrenkWhite halo profile, for some parameters of the theory. Interestingly, the effect peaks at some specific radius which is the same for any halo when measured in units of the viral radius. Even though the effect is small, this peak could provide a smoking gun for such modifications of gravity.
Recently, another analysis was performed in Ref. [407], where the possibility to testing theories of modified gravity exhibiting the Vainshtein mechanism against observations of cluster lensing was explored. In such theories, like in massive gravity, the second derivative of the field can be large at the transition between the screened and unscreened region, leading to observational signatures in cluster lensing.
14.4 Pulsars
One of the main predictions of massive gravity is the presence of new polarizations for GWs. While these new polarization might not be detectable in GW interferometers as explained in Section 11.1.2, we could still expect them to lead to detectable effects in the binary pulsar systems whose spindown is in extremely good agreement with GR. In this section, we thus consider the power emitted in the helicity0 mode of the graviton in a binarypulsar system. We use the effective action approach derived by Goldberger and Rothstein in [254] and start with the decoupling limit of DGP before exploring that of ghostfree massive gravity and discussing the subtleties that arise in that case. We mainly focus on the monopole and quadrupole radiation although the whole formalism can be derived for any multipoles. We follow the derivation of Refs. [158, 151], see also Refs. [100, 18] for related studies.
In order to account for the Vainshtein mechanism into account we perform a similar backgroundperturbation split as was performed in Section 10.1. The source is thus split as T = T_{0} + δT where T_{0} is a static and spherically source representing the total mass localized at the center of mass and captures the motion of the companions with respect to the center of mass.
This matter profile leads to a profile for the helicity0 mode (here mimicked as a cubic Galileon which is the case for DGP) as in (10.3) as π = π_{0}(r) + π, where the background π_{0}(r) has the same static and spherical symmetry as T_{0} and so has the same profile as in Section 10.1.2.
Without the Vainshtein mechanism, the mode functions would be the same as for a standard freefield in flat spacetime, \({\upsilon _\ell} \sim {1 \over {r\sqrt {\pi \omega}}}\,{\rm{cos(}}\omega {\rm{r)}}\) and the power emitted in the monopole would be larger than that emitted in GR, which would be clearly ruled out by observations. The Vainshtein mechanism is thus crucial here as well for the viability of DGP or ghostfree massive gravity.
14.4.1 Monopole
We see that the radiation in the monopole is suppressed by a factor of (Ω_{ P }r_{*})^{−3/2} compared with the GR result. For the HulseTaylor pulsar this is a suppression of 10 orders of magnitudes which is completely unobservable (at best the precision of the GR result is of 3 orders of magnitude).
Notice, however, that the suppression is far less than what was naively anticipated from the static approximation in Section 10.1.2.
The same analysis can be performed for the dipole emission with an even larger suppression of about 19 orders of magnitude compared the PetersMathews formula.
14.4.2 Quadrupole
The quadrupole emission in the field π is slightly larger than the monopole. The reason is that energy conservation makes the nonrelativistic limit of the monopole radiation irrelevant and one needs to take the first relativistic correction into account to emit in that channel. This is not so for the quadrupole as it does not correspond to the charge associated with any Noether current even in the nonrelativistic limit.
14.4.3 Quartic Galileon
When extending the analysis to more general Galileons or to massive gravity which includes a quartic Galileon, we expect a priori by following the analysis of Section 10.1.2, to find a stronger Vainshtein suppression. This result is indeed correct when considering the power radiated in only one multipole. For instance in a quartic Galileon, the power emitted in the field π via the quadrupole channel is suppressed by 12 orders of magnitude compared the GR emission.
However, this estimation does not account for the fact that there could be many multipoles contributing with the same strength in a quartic Galileon theory [151].
In situations where there is a large hierarchy between the mass of the two objects (which is the case for instance within the solar system), perturbation theory can be seen to remain under control and the power emitted in the quartic Galileon is completely negligible.
14.5 Black holes
As in any gravitational theory, the existence and properties of black holes are crucially important for probing the nonperturbative aspects of gravity. The celebrated blackhole theorems of GR play a significant role in guiding understanding of nonperturbative aspects of quantum gravity. Furthermore, the phenomenology of black holes is becoming increasingly important as understanding of astrophysical black holes increases.
Massive gravity and its extensions certainly exhibit blackhole solutions and if the Vainshtein mechanism is successful then we would expect solutions which look arbitrary close to the Schwarzschild and Kerr solutions of GR. However, as in the case of cosmological solutions, the situation is more complicated due to the absence of a unique static spherically symmetric solution that arises from the existence of additional degrees of freedom, and also the existence of other branches of solutions which may or may not be physical. There are a handful of known exact solutions in massive gravity [413, 363, 365, 277, 105, 56, 477, 90, 478, 455, 30, 357], but the most interesting and physically relevant solutions probably correspond to the generic case where exact analytic solutions cannot be obtained. A recent review of blackhole solutions in bigravity and massive gravity is given in [478].
An interesting effect was recently found in the context of bigravity in Ref. [41]. In that case, the Schwarzschild solutions were shown to be unstable (with a GregoryLaflamme type of instability [268, 269]) at a scale dictated by the graviton mass, i.e., the instability rate is of the order of the age of the Universe. See also Ref. [42] where the analysis was generalized to the nonbidiagonal. In this more general situation, spherically symmetric perturbations were also found but generically no instabilities. Blackhole disappearance in massive gravity was explored in Ref. [401].
One immediate consequence of working with bigravity is that since the g metric is sourced by polynomials of \(\sqrt {\mathbb X} = \sqrt {{g^{ 1}}f}\) whereas the f metric is sourced by polynomials of \(\sqrt {{f^{ 1}}g}\). We, thus, require that \({\mathbb X}\) is invertible away from curvature singularities. This is equivalent to saying that the eigenvalues of g^{−1} and ^{−1} should not pass through zero away from a curvature singularity. This in turn means that if one metric is diagonal and admits a horizon, the second metric if it is diagonal must admit a horizon at the same place, i.e., two diagonal metrics have common horizons. This is a generic observation that is valid for any theory with more than one metric [167] regardless of the field equations. Equivalently, this implies that if f is a diagonal metric without horizons, e.g., Minkowski spacetime, then the metric for a black hole must be nondiagonal when working in unitary gauge. This is consistent with the known exact solutions. For certain solutions it may be possible by means of introducing Stückelberg fields to put both metrics in diagonal form, due to the Stückelberg fields absorbing the offdiagonal terms. However, for the generic solution we would expect that at least one metric to be nondiagonal even with Stückelberg fields present.
15 Cosmology
One of the principal motivations for considering massive theories of gravity is their potential to address, or at least provide a new perspective on, the issue of cosmic acceleration as already discussed in Section 3. Adding a mass for the graviton keeps physics at small scales largely equivalent to GR because of the Vainshtein mechanism. However, it inevitably modifies gravity in at large distances, i.e., in the infrared. This modification of gravity is thus most significant for sources which are long wavelength. The cosmological constant is the most infrared source possible since it is build entirely out of zero momentum modes and for this reason we may hope that the nature of a cosmological constant in a theory of massive gravity or similar infrared modification is changed.
There have been two principal ideas for how massive theories of gravity could be useful for addressing the cosmological constant. On the one hand, by weakening gravity in the infrared, they may weaken the sensitivity of the dynamics to an already existing large cosmological constant. This is the idea behind screening or degravitating solutions [211, 212, 26, 216] (see Section 4.5). The second idea is that a condensate of massive gravitons could form which act as a source for selfacceleration, potentially explaining the current cosmic acceleration without the need to introduce a nonzero cosmological constant (as in the case of the DGP model [159, 163], see Section 4.4). This idea does not address the ‘old cosmological constant problem’ [484] but rather assumes that some other symmetry, or mechanism exists which ensures the vacuum energy vanishes. Given this, massive theories of gravity could potential provide an explanation for the currently small, and hence technically unnatural value of the cosmological constant, by tying it to the small, technically natural, value of the graviton mass.
Thus, the idea of screening/degravitation and selfacceleration are logically opposites to each other, but there is some evidence that both can be achieved in massive theories of gravity. This evidence is provided by the decoupling limit of massive gravity to which we review first. We then go on to discuss attempts to find exact solutions in massive gravity and its various extensions.
15.1 Cosmology in the decoupling limit
15.1.1 Friedmann equation in the decoupling limit
15.1.2 Screening solution
In this branch of solution, the strong coupling scale for fluctuations on top of this configuration becomes of the same order of magnitude as that of the screened cosmological constant. For a large cosmological constant the strong coupling scale becomes to large and the helicity0 mode would thus not be sufficiently Vainshtein screened.
Thus, while these solutions seem to indicate positively that there are selfscreening solutions which can accommodate a continuous range of values for the cosmological constant and still remain flat, the range is too small to significantly change the old cosmological constant problem. Nevertheless, the considerable difficulty in attacking the old cosmological constant problem means that these solutions deserve further attention as they also provide a proof of principle on how Weinberg’s no go could be evaded [484]. We emphasize that what prevents a large cosmological constant from being screened is not an issue in the theoretical tuning but rather an observational bound, so this is already a step forward.
These two classes of solutions are both maximally symmetric. However, the general cosmological solution is isotropic but inhomogeneous. This is due to the fact that a nontrivial time dependence for the matter source will inevitably source B (t), and as soon as Ḃ ≠ 0 the solutions are inhomogeneous. In fact, as we now explain in general, the full nonlinear solution is inevitably inhomogeneous due to the existence of a nogo theorem against spatially flat and closed FLRW solutions.
15.2 FLRW solutions in the full theory
15.2.1 Absence of flat/closed FLRW solutions
15.2.2 Open FLRW solutions
15.3 Inhomogenous/anisotropic cosmological solutions
As pointed out in [117], the absence of FLRW solutions in massive gravity should not be viewed as an observational flaw of the theory. On the contrary, the Vainshtein mechanism guarantees that there exist inhomogeneous cosmological solutions which approximate the normal FLRW solutions of GR as closely as desired in the limit m → 0. Rather, it is the existence of a new physical length scale 1/m in massive gravity, which cause the dynamics to be inhomogeneous at cosmological scales. If this scale 1/m is comparable to or larger than the current Hubble radius, then the effects of these inhomogeneities would only become apparent today, with the universe locally appearing as homogeneous for most of its history in the local patch that we observe.
One way to understand how the Vainshtein mechanism recovers the prediction of homogeneity and isotropy is to work in the formulation of massive gravity in which the Stückelberg fields are turned on. In this formulation, the Stückelberg fields can exhibit order unity inhomogeneities with the metric remaining approximately homogeneous. Matter that couples only to the metric will perceive an effectively homogeneous and anisotropic universe, and only through interaction with the Vainshtein suppressed additional scalar and vector degrees of freedom would it be possible to perceive the inhomogeneities. This is achieved because the metric is sourced by the Stückelberg fields through terms in the equations of motion which are suppressed by m^{2}. Thus, as long as R ≫ m^{2}, the metric remains effectively homogeneous and isotropic despite the existence of nogo theorems against exact homogeneity and isotropy.
In this regard, a whole range of exact solutions have been studied exhibiting these properties [364, 474, 363, 97, 264, 356, 456, 491, 476, 334, 478, 265, 124, 123, 125, 455, 198]. A generalization of some of these solutions was presented in Ref. [404] and Ref. [266]. In particular, we note that in [475, 476] the most general exact solution of massive gravity is obtained in which the metric is homogeneous and isotropic with the Stückelberg fields inhomogeneous. These solutions exist because the effective contribution to the stress energy tensor from the mass term (i.e., viewing the mass term corrections as a modification to the energy density) remains homogeneous and isotropic despite the fact that it is build out of Stückelberg fields which are themselves inhomogeneous.
15.3.1 Special isotropic and inhomogeneous solutions
15.3.1.1 Effective cosmological constant
15.3.1.2 Massive gravity limit
In particular, in the open universe case \({\Lambda _f} = 0,\,k = 0,\,q = u,\,T = ua\sqrt {1 + \vert k\vert{r^2}}\), we recover the open universe solution of massive gravity considered in Section 12.2.2, where for comparison \(f(t) = {1 \over {\sqrt {\vert k\vert}}}ua(t),\,{\phi ^0}(t,r) = T(t,r)\) and ϕ^{ r } = U (t, r).
15.3.2 General anisotropic and inhomogeneous solutions
Let us reiterate again that there are a large class of inhomogeneous but isotropic cosmological solutions for which the effective Friedmann equation for the g metric is the same as in GR with just the addition of a cosmological constant which depends on the graviton mass parameters. However, these are not the most general solutions, and as we have already discussed many of the exact solutions of this form considered so far have been found to be unstable, in particular through the absence of kinetic terms for degrees of freedom which implies infinite strong coupling. However, all the exact solutions arise from making a strong restriction on one or the other of the metrics which is not expected to be the case in general. Thus, the search for the ‘correct’ cosmological solution of massive gravity and bigravity will almost certainly require a numerical solution of the general equations for Q, R, N, n, c, b, and their stability.
Closely related to this, we may consider solutions which maintain homogeneity, but are anisotropic [284, 393, 123]. In [393] the general Bianchi class A cosmological solutions in bigravity are studied. There it is shown that the generic anisotropic cosmological solution in bigravity asymptotes to a selfaccelerating solution, with an acceleration determined by the mass terms, but with an anisotropy that falls off less rapidly than in GR. In particular the anisotropic contribution to the effective energy density redshifts like nonrelativistic matter. In [284, 123] it is found that if the reference metric is made to be of an anisotropic FLRW form, then for a range of parameters and initial conditions stable ghost free cosmological solutions can be found.
These analyses are ongoing and it has been uncovered that certain classes of exact solutions exhibit strong coupling instabilities due to vanishing kinetic terms and related pathologies. However, this simply indicates that these solutions are not good semiclassical backgrounds. The general inhomogeneous cosmological solution (for which the metric is also inhomogeneous) is not known at present, and it is unlikely it will be possible to obtain it exactly. Thus, it is at present unclear what are the precise nonlinear completions of the stable inhomogeneous cosmological solutions that can be found in the decoupling limit. Thus the understanding of the cosmology of massive gravity should be regarded as very much work in progress, at present it is unclear what semiclassical solutions of massive gravity are the most relevant for connecting with our observed cosmological evolution.
15.4 Massive gravity on FLRW and bigravity
15.4.1 FLRW reference metric
One straightforward extension of the massive gravity framework is to allow for modifications to the reference metric, either by making it cosmological or by extending to bigravity (or multigravity). In the former case, the nogo theorem is immediately avoided since if the reference metric is itself an FLRW geometry, there can no longer be any obstruction to finding FLRW geometries.
However, in practice, the generalization of the Higuchi consideration [307] to this case leads to an unacceptable bound (see Section 8.3.6).
This latter problem which is severe for massive gravity with dS or FLRW reference metrics,^{33} gets resolved in bigravity extensions, at least for a finite regime of parameters.
15.4.2 Bigravity
We should stress again that just as in massive gravity, the absence of FLRW solutions should not be viewed as an inconsistency of the theory with observations, also in bigravity these solutions may not necessarily be the ones of most relevance for connecting with observations. It is only that they are the most straightforward to obtain analytically. Thus, cosmological solutions in bigravity, just as in massive gravity, should very much be viewed as a work in progress.
15.5 Other proposals for cosmological solutions
Finally, we may note that more serious modifications the massive gravity framework have been considered in order to allow for FLRW solutions. These include massvarying gravity and the quasidilaton models [119, 118]. In [281] it was shown that massvarying gravity and the quasidilaton model could allow for stable cosmological solutions but for the original quasidilaton theory the selfaccelerating solutions are always unstable. On the other hand, the generalizations of the quasidilaton [126, 127] appears to allow stable cosmological solutions.
In addition, one can find cosmological solutions in nonLorentz invariant versions of massive gravity [109] (and [103, 107, 108]). We can also allow the mass to become dependent on a field [489, 375], extend to multiple metrics/vierbeins [454], extensions with f (R) terms either in massive gravity [89] or in bigravity [416, 415] which leads to interesting selfaccelerating solutions. Alternatively, one can consider other extensions to the form of the mass terms by coupling massive gravity to the DBI Galileons [237, 19, 20, 315].
As an example, we present here the cosmology of the extension of the quasidilaton model considered in [127], where the reference metric \({\bar f_{\mu v}}\) is given in (9.15) and depends explicitly on the dynamical quasidilaton field σ.
16 Part IV Other Theories of Massive Gravity
17 New Massive Gravity
17.1 Formulation
Independently of the formal development of massive gravity in four dimensions described above, there has been interest in constructing a purely three dimensional theory of massive gravity. Three dimensions are special for the following reason: for a massless graviton in three dimensions there are no propagating degrees of freedom. This follows simply by counting, a symmetric tensor in three dimensions has six components. A massless graviton must admit a diffeomorphism symmetry which renders three of the degrees of freedom pure gauge, and the remaining three are nondynamical due to the associated first class constraints. On the contrary, a massive graviton in three dimensions has the same number of degrees of freedom as a massless graviton in four dimensions, namely two. Combining these two facts together, in three dimensions it should be possible to construct a diffeomorphism invariant theory of massive gravity. The usual massless graviton implied by diffeomorphism invariance is absent and only the massive degree of freedom remains.
17.2 Absence of BoulwareDeser ghost
As a result of the introduction of the new gauge symmetries, we straightforwardly count the number of nonperturbative degrees of freedom. The total number of fields are 16: six from g_{ μν }, six from q_{ μν }, three from A_{ μ } and one from π. The total number of gauge symmetries are 7: three from diffeomorphisms, three from linear diffeomorphisms and one from the U (1). Thus, the total number of degrees of freedom are 16 − 7 (gauge) − 7 (constraint) = 2 which agrees with the linearized analysis. An independent argument leading to the same result is given in [317] where NMG including its topologically massive extension (see below) are presented in Hamiltonian form using EinsteinCartan language (see also [176]).
17.3 Decoupling limit of new massive gravity
The decoupling limit clarifies one crucial aspect of NMG. It has been suggested that NMG could be power counting renormalizable following previous arguments for topological massive gravity [196] due to the softer nature of divergences in threedimensional and the existence of a dimensionless combination of the Planck mass and the graviton mass. This is in fact clearly not the case since the above cubic interaction is a nonrenormalizable operator and dominates the Feynman diagrams leading to perturbative unitarity violation at the strong coupling scale Λ_{5/2}(see Section 10.5 for further discussion on the distinction between the breakdown of perturbative unitarity and the breakdown of the theory).
17.4 Connection with bigravity
The existence of the NMG theory at first sight appears to be something of an anomaly that cannot be reproduced in higher dimensions. There also does not at first sight seem to be any obvious connection with the diffeomorphism breaking ghostfree massive gravity model (or dRGT) and multigravity extensions. However, in [425] it was shown that NMG, and certain extensions to it, could all be obtained as scaling limits of the same 3dimensional bigravity models that are consistent with ghostfree massive gravity in a different decoupling limit. As we already mentioned, the key to seeing this is the auxiliary formulation where the tensor f_{μν} is related to the missing extra metric of the bigravity theory.
17.5 3D massive gravity extensions
17.6 Other 3D theories
17.6.1 Topological massive gravity
The AdS/CFT in the context of topological massive gravity was also studied in Ref. [449].
17.6.2 Supergravity extensions
Moreover, N = 2 supergravity extensions of TMG were recently constructed in Ref. [370] and its N = 3 and N = 4 supergravity extensions in Ref. [371].
17.6.3 Critical gravity
Finally, let us comment on a special case of three dimensional gravity known as log gravity [65] or critical gravity in analogy with the general dimension case [384, 183, 16]. For a special choice of parameters of the theory, there is a degeneracy in the equations of motion for the two degrees of freedom leading to the fact that one of the modes of the theory becomes a ‘logarithmic’ mode.
Indeed, at the special point μℓ = 1, (where ℓ is the AdS length scale, Λ = −1/ℓ^{2}), known as the ‘chiral point’ the leftmoving (in the language of the boundary CFT) excitations of the theory become pure gauge and it has been argued that the theory then becomes purely an interacting theory for the right moving graviton [93]. In Ref. [377] it was earlier argued that there was no massive graviton excitations at the critical point μℓ = 1 however Ref. [93] found one massive graviton excitation for every finite and nonzero value of μℓ, including at the critical point μℓ = 1.
This case was further analyzed in [273], see also Ref. [274] for a recent review. It was shown that the degeneration of the massive graviton mode with the left moving boundary graviton leads to logarithmic excitations.
As usual, it is apparent that this theory describes one massless graviton (with no propagating degrees of freedom) and one massive one whose mass is given by \({M^2} =  {m^2}\bar \sigma\). However, by choosing \(\bar \sigma = 0\) the massive mode becomes degenerate with the existing massless one.
Based on this result as well as on the finiteness and conservation of the stress tensor and on the emergence of a Jordan cell structure in the Hamiltonian, the correspondence to a logarithmic CFT was conjectured in Ref. [273], where the to be dual log CFTs representations have degeneracies in the spectrum of scaling dimensions.
Strong indications for this correspondence appeared in many different ways. First, consistent boundary conditions which allow the log modes were provided in Ref. [272], were it was shown that in addition to the BrownHenneaux boundary conditions one could also consider more general ones. These boundary conditions were further explored in [305, 397], where it was shown that the stressenergy tensor for these boundary conditions are finite and not chiral, giving another indication that the theory could be dual to a logarithmic CFT.
Then specific correlator functions were computed and compared. Ref. [449] checked the 2point correlators and Ref. [275] the 3point ones. A similar analysis was also performed within the context of NMG in Ref. [270] where the 2point correlators were computed at the chiral point and shown to behave as those of a logarithmic CFT.
Further checks for this AdS/log CFT include the 1loop partition function as computed in Ref. [241]. See also Ref. [274] for a review of other checks.
It has been shown, however, that ultimately these theories are nonunitary due to the fact that there is a nonzero inner product between the log modes and the normal models and the inability to construct a positive definite norm on the Hilbert space [432].
17.7 Black holes and other exact solutions
A great deal of physics can be learned from studying exact solutions, in particular those corresponding to black hole geometries. Black holes are also important probes of the nonperturbative aspects of gravitational theories. We briefly review here the types of exact solutions obtained in the literature.
17.8 New massive gravity holography
One of the most interesting avenues of exploration for NMG has been in the context of Maldacena’s AdS/CFT correspondence [396]. According to this correspondence, NMG with a cosmological constant chosen so that there are asymptotically antide Sitter solutions is dual to a conformal field theory (CFT). This has been considered in [67, 381, 380] where it was found that the requirements of bulk unitarity actually lead to a negative central charge.
17.9 Zweidreibein gravity
As we have seen, there is a conflict in NMG between unitarity in the bulk, i.e., the requirement that the massive gravitons are not ghosts, and unitarity in dual CFT as required by the positivity of the central charge. This conflict may be resolved, however, by replacing NMG with the 3dimensional bigravity extension of ghostfree massive gravity that we have already discussed. In particular, if we work in the EinsteinCartan formulation in three dimensions, then the metric is replaced by a ‘dreibein’ and since this is a bigravity model, we need two ‘dreibeins’. This gives us the Zweidreibein gravity [63].
These results potentially have an impact on the higher dimensional case. We see that in three dimensions we potentially have a diffeomorphism invariant theory of massive gravity (i.e., bigravity) which at least for AdS solutions exhibits unitarity both in the bulk and in the boundary CFT for a finite range of parameters in the theory. However, these bigravity models are easily extended into all dimensions as we have already discussed and it is similarly easy to find AdS solutions which exhibit bulk unitarity. It would be extremely interesting to see if the associated dual CFTs are also unitary thus providing a potential holographic description of generalized theories of massive gravity.
18 LorentzViolating Massive Gravity
18.1 SO(3)invariant mass terms
The entire analysis performed so far is based on assuming Lorentz invariance. In what follows we briefly review a few other potentially viable theories of massive gravity where Lorentz invariance is broken and their respective cosmology.
Prior to the formulation of the ghostfree theory of massive gravity, it was believed that no Lorentz invariant theories of massive gravity could evade the BD ghost and Lorentzviolating theories were thus the best hope; we refer the reader to [438] for a thorough review on the field. A thorough analysis of Lorentzviolating theories of massive gravity was performed in [200] and more recently in [107]. See also Refs. [236, 73, 114] for other complementary studies. Since this field has been reviewed in [438] we only summarize the key results in this section (see also [74] for a more recent review on many developments in Lorentz violating theories.) See also Ref. [378] for an interesting spontaneous breaking of Lorentz invariance in ghostfree massive gravity using three scalar fields, and Ref. [379] for a SO (3)invariant ghostfree theory of massive gravity which can be formulated with three Stückelberg scalar fields and propagating five degrees of freedom.
In most theories of Lorentzviolating massive gravity, the SO (3, 1) Poincaré group is broken down to a SO (3) rotation group. This implies the presence of a preferred time. Preferredframe effects are, however, strongly constrained by solar system tests [487] as well as pulsar tests [54], see also [494, 493] for more recent and even tighter constraints.
We note that Lorentz invariance is restored when m_{1} = m_{2}, m_{3} = m_{4} and \(m_0^2 =  m_1^2 + m_3^2\). The FierzPauli structure then further fixes m_{1} = m_{3} implying m_{0} = 0, which is precisely what ensures the presence of a constraint and the absence of BD ghost (at least at the linearized level).

The parameter m_{2} is the one that represents the mass of the helicity2 mode. As a result we should impose \(m_2^2 \ge 0\) to avoid tachyonlike instabilities. Although we should bear in mind that if that mass parameter is of the order of the Hubble parameter today m_{2} ≃ 10^{−33} eV, then such an instability would not be problematic.

The parameter m_{1} is the one responsible for turning on a kinetic term for the two helicity1 modes. Since m_{1} = m_{2} in a Lorentzinvariant theory of massive gravity, the helicity1 mode cannot be turned off (m_{1} = 0) while maintaining the graviton massive (m_{2} ≠ 0). This is a standard result of Lorentz invariant massive gravity seen so far where the helicity1 mode is always present. For Lorentz breaking theories the theory is quite different and one can easily switch off at the linearized level the helicity1 modes in a theory of Lorentzbreaking massive gravity. The absence of a ghost in the helicity1 mode requires \(m_1^2 \ge 0\).

If m_{0} ≠ 0 and m_{1} ≠ 0 and m_{4} ≠ 0 then two scalar degrees of freedom are present already at the linear level about flat spacetime and one of these is always a ghost. The absence of ghost requires either m_{0} = 0 or m_{1} = 0 or finally m_{4} = 0 and m_{2} = m_{3}.
In the last scenario, where m_{4} = 0 and m_{2} = 3, the scalar degree of freedom loses its gradient terms at the linear level, which means that this mode is infinitely strongly coupled unless no gradient appears fully nonlinearly either.
The case m_{0} has an interesting phenomenology as will be described below. While it propagates five degrees of freedom about Minkowski it avoids the vDVZ discontinuity in an interesting way.
Finally, the case m_{1} = 0 (including when m_{0} = 0) will be discussed in more detail in what follows. It is free of both scalar (and vector) degrees of freedom at the linear level about Minkowski and thus evades the vDVZ discontinuity in a straightforward way.
 The analogue of the Higuchi bound was investigated in [72]. In de Sitter with constant curvature H, the generalized Higuchi bound iswhile if instead m_{1} = 0 then no scalar degree of freedom are propagating on de Sitter either so there is no analogue of the Higuchi bound (a scalar starts propagating on FLRW solutions but it does not lead to an equivalent Higuchi bound either. However, the absence of tachyon and gradient instabilities do impose some conditions between the different mass parameters).$$m_4^4 + 2{H^2}\left({3(m_3^2  m_4^2)  m_2^2} \right) > m_4^2(m_1^2  m_4^2)\quad {\rm{if}}\quad {m_0} = 0\,,$$(14.2)
As shown in the case of the FierzPauli mass term and its nonlinear extension, one of the most natural way to follow the physical degrees of freedom and their health is to restore the broken symmetry with the appropriate number of Stückelberg fields.
In the Lorentzinvariant case we are stuck with the combination \(X_{\,\, v}^\mu = Y_{\,\, v}^\mu  {n^{ 2}}{g^{\mu \alpha}}{n_\alpha}{n_\nu}\), but this combination can be broken here and the mass term can depend separately on n, n_{ μ } and Y. This allows for new mass terms. In [107] this framework was derived and used to find new mass terms that exhibit five degrees of freedom. This formalism was also developed in [200] and used to derive new mass terms that also have fewer degrees of freedom. We review both cases in what follows.
18.2 Phase m_{1} = 0
18.2.1 Degrees of freedom on Minkowski
As already mentioned, the helicity1 mode have no kinetic term at the linear level on Minkowski if m_{1} = 0. Furthermore, it turns out that the field υ in (14.17) is the Lagrange multiplier which removes the BD ghost (as opposed to the field ψ in the case m_{0} = 0 presented previously). It imposes the constraint \(\dot \tau = 0\) which in turns implies τ = 0. Using this constraint back in the action, one can check that there remains no time derivatives on any of the scalar fields, which means that there are no propagating helicity0 mode on Minkowski either [200, 437, 438]. So in the case where m_{1} = 0 there are only 2 modes propagating in the graviton on Minkowski, the 2 helicity2 modes as in GR.
The absence of helicity1 and 0 modes while keeping the helicity2 mode massive makes this Lorentz violating theory of gravity especially attractive. Its cosmology was explored in [201] and it turns out that this theory of massive gravity could be a candidate for cold dark matter as shown in [202].
Moreover, explicit black hole solutions were presented in [438] where it was shown that in this theory of massive gravity black holes have hair and the Stückelberg fields (in the Stückelberg formulation of the theory) do affect the solution. This result is tightly linked to the fact that this theory of massive gravity admits instantaneous interactions which is generic to any action of the form (14.9).
18.2.2 Nonperturbative degrees of freedom
Perturbations on more general FLRW backgrounds were then considered more recently in [72]. Unlike in Minkowski, scalar perturbations on curved backgrounds are shown to behave in a similar way for the cases m_{1} = 0 and m_{0} = 0. However, as we shall see below, the case m_{0} = 0 propagates five degrees of freedom including a helicity0 mode that behaves as a scalar it follows that on generic backgrounds the theory with m_{1} = 0 also propagates a helicity0 mode. The helicity0 mode is thus infinitely strongly coupled when considered perturbatively about Minkowski.
18.3 General massive gravity (m_{0} = 0)
All of these cases ensures the absence of BD ghost by having m_{0} = 0. The case where the BD ghost is projected thanks to the requirement m_{1} = 0 is discussed in Section 14.2.
18.3.1 First explicit Lorentzbreaking example with five dofs
The fact that several independent functions enter the mass term will be of great interest for cosmology as one of these functions (namely \({\mathcal C}\)) can be used to satisfy the Bianchi identity while the other function can be used for an appropriate cosmological history.
At the linearized level about Minkowski (which is a vacuum solution) this theory can be parameterized in terms of the mass scales introduces in (14.1) with m_{0} = 0, so the BD ghost is projected out in a way similar as in ghostfree massive gravity.
Interestingly, if \({\mathcal C} = 0\), this theory corresponds to m_{1} = 0 (in addition to m_{0} = 0) which as seen earlier the helicity1 mode is absent at the linearized level. However, they survive nonlinearly and so the case \({\mathcal C} = 0\) is infinitely strongly coupled.
18.3.2 Second example of Lorentzbreaking with five dofs
18.3.3 Absence of vDVZ and strong coupling scale
Unlike in the Lorentzinvariant case, the kinetic term for the Stückelberg fields does not only arise from the mixing with the helicity2 mode.

The new canonical normalization implies a much larger strong coupling scale that goes as Λ_{2} = (M_{Pl}m)^{1/2} rather than Λ_{3} = (M_{Pl}m^{2})^{1/3} as is the case in DGP and ghostfree massive gravity.

Furthermore, in the massless limit the coupling of the helicity0 mode to the tensor vanishes fasters than some of the Lorentzviolating kinetic interactions in (14.20) (which is scales as \(m\hat h{\partial ^2}\hat \phi\). This means that one can take the massless limit m → 0 in such a way that the coupling to the helicity2 mode disappears and so does the coupling of the helicity0 mode to matter (since this coupling arises after demixing of the helicity0 and 2 modes). This implies the absence of vDVZ discontinuity in this Lorentzviolating theory despite the presence of five degrees of freedom.
The absence of vDVZ discontinuity and the larger strong coupling scale Λ_{2} makes this theory more tractable at small mass scales. We emphasize, however, that the absence of vDVZ discontinuity does prevent some sort of Vainshtein mechanism to still come into play since the theory is still strongly coupled at the scale Λ_{2} ≪ M_{Pl}. This is similar to what happens for the Lorentzinvariant ghostfree theory of massive gravity on AdS (see Section 8.3.6 and [154]). Interestingly, however, the same redressing of the strong coupling scale as in DGP or ghostfree massive gravity was explored in [109] where it was shown that in the vicinity of a localized mass, the strong coupling scale gets redressed in such a way that the weak field approximation remains valid till the Schwarzschild radius of the mass, i.e., exactly as in GR.
In these theories, bounds on the graviton comes from the exponential decay in the Yukawa potential which switches gravity off at the graviton’s Compton wavelength, so the Compton wavelength ought to be larger than the largest gravitational bound states which are of about 5 Mpc, putting a bound on the graviton mass of m ≲ 10^{−30} eV in which case Λ_{2} ∼ (10^{−4} mm)^{−1} [108, 257].
18.3.4 Cosmology of general massive gravity
The cosmology of general massive gravity was recently studied in [109] and we summarize their results in what follows.
This solution is stable and healthy as long as the second derivative of \({\mathcal C}\) satisfies some conditions which can easily be accommodated for appropriate functions \({\mathcal C}\) and U.
19 Nonlocal massive gravity
The ghostfree theory of massive gravity proposed in Part II as well as the Lorentzviolating 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 bigravity as well and in the zwei dreibein extension of new massive gravity.
One of the motivations of nonlocal theories of massive gravity is to formulate the theory without any reference metric.^{34} This is the main idea behind the nonlocal theory of massive gravity introduced in [328].^{35}
The theory propagates what looks like a ghostlike instability irrespectively of the exact formulation chosen in (15.2). However, it was recently argued that the wouldbe 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 ≃ 0.67H_{0}, where H_{0} is the Hubble parameter today, reproduces the observed amount of dark energy. The mass term acts as a dark fluid with effective timedependent equation of state ω_{eff}(a) ≃ −1.04 − 0.02(1 − a), where a is the scale factor, and is thus phantomlike.
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 nonlocal 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.
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.
20 Outlook
The past decade has witnessed a revival of interest in massive gravity as a potential alternative to GR. The original theoretical obstacles that came in the way of deriving a consistent theory of massive gravity have now been overcome, but with them comes a new set of challenges that will be decisive in establishing the viability of such theories. The presence of a low strong coupling scale on which the Vainshtein mechanism relies has opened the door to a new way of thinking about these types of effective field theories. At the moment, it is yet unclear whether these types of theories could lead to an alternative to UV completion. The superluminalities that also arise in many cases with the Vainshtein mechanism should also be understood in more depth. At the moment, its real implications are not well understood and no case of true acausality has been shown to be present within the regime of validity of the theory. Finally, the difficulty in finding fullyfledged cosmological and blackhole solutions in many of these theories (both in ghostfree massive gravity and bigravity, and in other extensions or related models such as cascading gravity) makes their full phenomenology still evasive. Nevertheless, the well understood decoupling limits of these models can be used to say a great deal about phenomenology without going into the complications of the full theories. These represent many open questions in massive gravity, which reflect the fact that the field is yet extremely young and many developments are still in progress.
Footnotes
 1.
In the case of ‘discretization’ or ‘deconstruction’ the higher dimensional approach has been useful only ‘after the fact’. Unlike for DGP, in massive gravity the extra dimension is purely used as a mathematical tool and the formulation of the theory was first performed in four dimensions. In this case massive gravity is not derived per se from the higherdimensional picture but rather one can see how the structure of general relativity in higher dimensions is tied to that of the mass term.
 2.
The equation of motion with respect to \({\chi}\) gives \(\square \tilde{\chi}=0\), however this should be viewed as a dynamical relation for which should not be plugged back into the action. On the other hand, when deriving the equation of motion with respect to \(\tilde {\mathcal X}\), we obtain a constraint equation for \(\tilde {\mathcal X}:\tilde {\mathcal X} = 2{\square_{\mathcal X}}\) which can be plugged back into the action (and χ is then treated as the dynamical field).
 3.
This is already a problem at the classical level, well before the notion of particle needs to be defined, since classical configurations with arbitrarily large ϕ_{1} can always be constructed by compensating with a large configuration for ϕ_{2} at no cost of energy (or classical Hamiltonian).
 4.
In this review, the notion of fully nonlinear coordinate transformation invariance is equivalent to that of full diffeomorphism invariance or covariance.
 5.
Up to other Lovelock invariants. Note however that f (R) theories are not exceptions, as the kinetic term for the spin2 field is still given by \(\sqrt { g} R\). See Section 5.6 for more a more detailed discussion in the case of massive gravity.
 6.
Strictly speaking, the notion of spin is only meaningful as a representation of the Lorentz group, thus the theory of massive spin2 field is only meaningful when Lorentz invariance is preserved, i.e., when the reference metric is Minkowski. While the notion of spin can be extended to other maximally symmetric spacetimes such as AdS and dS, it loses its meaning for nonmaximally symmetric reference metrics f_{ μν }.
 7.
 8.
In the normal branch of DGP, this branebending mode turns out not to be normalizable. The normalizable branebending mode which is instead present in the normal branch fully decouples and plays no role.
 9.
Note that in DGP, one could also consider a smooth brane first and the results would remain unchanged.
 10.
The local gauge invariance associated with covariance leads to d first class constraints which remove 2d degrees of freedom, albeit in phase space. For global symmetries such as Lorentz invariance, there is no firstclass constraints associated with them, and that global symmetry only removes d (d − l)/2 degrees of freedom. Technically, the counting should be performed in phase space, but the results remains the same. See Section 7.1 for a more detailed review on the counting of degrees of freedom.
 11.
Discretizing at the level of the metric leads to a mass term similar to (2.83) which as we have seen contains a BD ghost.
 12.
This special fully nonlinear and Lorentz invariant theory of massive gravity, which has been proven in all generality to be free of the BD ghost in [295, 296] has since then be dubbed ‘dRGT’ theory. To avoid any confusion, we thus also call this ghostfree theory of massive gravity, the dRGT theory.
 13.
The analysis performed in Ref. [95] was unfortunately erroneous, and the conclusions of that paper are thus incorrect.
 14.
In the previous section we obtained directly a theory of massive gravity, this should be seen as a trick to obtain a consistent theory of massive gravity. However, we shall see that we can take a decoupling limit of bi (or even multi)gravity so as to recover massive gravity and a decoupled massless spin2 field. In this sense massive gravity is a perfectly consistent limit of bigravity.
 15.
The field redefinition is local so no new degrees of freedom or other surprises hide in that field redefinition.
 16.
 17.
Some dofs may ‘accidentally’ disappear about some special backgrounds, but dofs cannot disappear nonlinearly if they were present at the linearized level.
 18.
More recently, Alexandrov impressively performed the full analysis for bigravity and massive gravity in the vielbein language [15] determining the full set of primary and secondary constraints, confirming again the absence of BD ghost. This resolves the potential sources of subtleties raised in Refs. [96, 351, 349, 348].
 19.
We stress that multiplying with the matrix λ is not a projection, the equation (7.52) contains as much information as the equation of motion with respect to λ, multiplying the with the matrix λ on both sides simply make the rank of the equation more explicit.
 20.
If only M vielbein of the N vielbein are interacting there will be (N − M + 1) copies of diffeomorphism invariance and M − 1 additional Hamiltonian constraints, leading to the correct number of dofs for (N − M + 1) massless spin2 fields and M − 1 massive spin2 fields.
 21.
Technically, only one of them generates a first class constraint, while the N − 1 others generate a secondclass constraint. There are, therefore, (N − 1) additional secondary constraints to be found by commuting the primary constraint with the Hamiltonian, but the presence of these constraints at the linear level ensures that they must exist at the nonlinear level. There is also another subtlety in obtaining the secondary constraints associated with the fact that the Hamiltonian is pure constraint, see the discussion in Section 7.1.3 for more details.
 22.
 23.
 24.
Note that the Vainshtein mechanism does not occur for all parameters of the theory. In that case the massless limit does not reproduce GR.
 25.
This result has been checked explicitly in Ref. [146] using dimensional regularization or following the log divergences. Taking power law divergences seriously would also allow for a scalings of the form \((\Lambda _{{\rm{Cutoff}}}^4/M_{{\rm{Pl}}}^n){h^n}\), which are no longer suppressed by the mass scale m (although the mass scale m would never enter with negative powers at one loop.) However, it is well known that power law divergences cannot be trusted as they depend on the measure of the path integral and can lead to erroneous results in cases where the higher energy theory is known. See Ref. [84] and references therein for known examples and an instructive discussion on the use and abuses of power law divergences.
 26.
If c_{3} = 0, we can easily generalize the background solution to find other configurations that admit a superluminal propagation.
 27.
 28.
 29.
The minimal model does not have a Vainshtein mechanism [435] in the static and spherically symmetric configuration so in the limit α_{3} → −1/3, or equivalently → → 0, we indeed expect an order one correction.
 30.
In taking this limit, it is crucial that the second metric f_{ μν } be written in a locally inertial coordinate system, i.e., a system which is locally Minkowski. Failure to do this will lead to the erroneous conclusion that massive gravity on Minkowski is not a limit of bigravity.
 31.
In the context of DGP, the Friedmann equation was derived in Section 4.3.1 from the full fivedimensional picture, but one would have obtained the correct result if derived instead from the decoupling limit. The reason is the main modification of the Friedmann equation arises from the presence of the helicity0 mode which is already captured in the decoupling limit.
 32.
See [478] for a recent review and more details. The convention on the parameters b_{ n } there is related to our β’s here via \({b_n} =  {1 \over 4}(4  n)!{\beta _n}\).
 33.
Notice that this is not an issue in massive gravity with a flat reference metric since the analogue Friedmann equation does not even exist.
 34.
Notice that even if massive gravity is formulated without the need of a reference metric, this does not change the fact that one copy of diffeomorphism invariance in broken leading to additional degrees of freedom as is the case in new massive gravity.
 35.
Notes
Acknowledgements
CdR wishes to thank Denis Comelli, Matteo Fasiello, Gregory Gabadadze, Daniel Grumiller, Kurt Hinterbichler, Andrew Matas, Shinji Mukohyama, Nicholas Ondo, Luigi Pilo and especially Andrew Tolley for useful discussions. CdR is supported by Department of Energy grant DESC0009946.
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