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

14 Lorentz-Violating Massive Gravity

14.1 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 ghost-free theory of massive gravity, it was believed that no Lorentz invariant theories of massive gravity could evade the BD ghost and Lorentz-violating theories were thus the best hope; we refer the reader to [438*] for a thorough review on the field. A thorough analysis of Lorentz-violating 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 ghost-free massive gravity using three scalar fields, and Ref. [379] for a SO (3)-invariant ghost-free theory of massive gravity which can be formulated with three Stückelberg scalar fields and propagating five degrees of freedom.

In most theories of Lorentz-violating 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. Preferred-frame 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.

At the linearized level the general mass term that satisfies this rotation symmetry is

1( 2 2 2 2 2 2 2 i j 2 i) ℒSO (3) mass = 8 m 0h 00 + 2m 1h0i − m 2h ij + m 3hihj − 2m 4h00hi , (14.1 )
where space indices are raised and lowered with respect to the flat spatial metric δij. This extends the Lorentz invariant mass term presented in (2.39*). In the rest of this section, we will establish the analogue of the Fierz–Pauli mass term (2.44*) in this Lorentz violating case and establish the conditions on the different mass parameters m0,1,2,3,4.

We note that Lorentz invariance is restored when m1 = m2, m3 = m4 and 2 2 2 m 0 = − m 1 + m 3. The Fierz–Pauli structure then further fixes m1 = m3 implying m0 = 0, which is precisely what ensures the presence of a constraint and the absence of BD ghost (at least at the linearized level).

Out of these five mass parameters some of them have a direct physical meaning [200*, 437*, 438*]

  • The parameter m2 is the one that represents the mass of the helicity-2 mode. As a result we should impose m22 ≥ 0 to avoid tachyon-like instabilities. Although we should bear in mind that if that mass parameter is of the order of the Hubble parameter today m2 ≃ 10 −33 eV, then such an instability would not be problematic.
  • The parameter m1 is the one responsible for turning on a kinetic term for the two helicity-1 modes. Since m1 = m2 in a Lorentz-invariant theory of massive gravity, the helicity-1 mode cannot be turned off (m1 = 0) while maintaining the graviton massive (m2 ⁄= 0). This is a standard result of Lorentz invariant massive gravity seen so far where the helicity-1 mode is always present. For Lorentz breaking theories the theory is quite different and one can easily switch off at the linearized level the helicity-1 modes in a theory of Lorentz-breaking massive gravity. The absence of a ghost in the helicity-1 mode requires m21 ≥ 0.
  • If m0 ⁄= 0 and m1 ⁄= 0 and m4 ⁄= 0 then two scalar degrees of freedom are present already at the linear level about flat space-time and one of these is always a ghost. The absence of ghost requires either m0 = 0 or m1 = 0 or finally m4 = 0 and m2 = m3.

    In the last scenario, where m = 0 4 and m = 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 non-linearly 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 m1 = 0 (including when m0 = 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 is
    ( ) m44 + 2H2 3(m23 − m24) − m22 > m24(m21 − m24) if m0 = 0, (14.2)
    while if instead m1 = 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).

As shown in the case of the Fierz–Pauli mass term and its non-linear 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 Section 2.4 we reviewed how to restore the broken diffeomorphism invariance using four Stückelberg fields a ϕ using the relation (2.75*). When Lorentz invariance is broken, the Stückelberg trick has to be performed slightly differently. Performing an ADM decomposition, which is appropriate for the type of Lorentz breaking we are considering, we can use for Stückelberg scalar fields Φ = Φ0 and Φi, i = 1,⋅⋅⋅,3 to define the following four-dimensional scalar, vector and tensors [107*]

n = (− gμν∂ Φ ∂ Φ )−1∕2 (14.3 ) μ ν nμ = n ∂μΦ (14.4 ) Y μ = gμα∂ Φi∂ Φj δ (14.5 ) νμ μ α μ να iji j Γν = Y ν + n n ∂αΦ ∂νΦ δij. (14.6 )
n can be thought of as the ‘Stückelbergized’ version of the lapse and Γ μ ν as that of the spatial metric.

In the Lorentz-invariant case we are stuck with the combination X μν = Yνμ− n−2gμαn αnν, but this combination can be broken here and the mass term can depend separately on n,nμ 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.

14.2 Phase m1 = 0

14.2.1 Degrees of freedom on Minkowski

As already mentioned, the helicity-1 mode have no kinetic term at the linear level on Minkowski if m1 = 0. Furthermore, it turns out that the field v in (14.17*) is the Lagrange multiplier which removes the BD ghost (as opposed to the field ψ in the case m0 = 0 presented previously). It imposes the constraint ˙τ = 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 helicity-0 mode on Minkowski either [200*, 437*, 438*]. So in the case where m1 = 0 there are only 2 modes propagating in the graviton on Minkowski, the 2 helicity-2 modes as in GR.

In this case, the absence of the ghost can be seen to follow from the presence of a residual symmetry on flat space [200*, 438*]

xi → xi + ξi(t), (14.7 )
for three arbitrary functions ξi(t). In the Stückelberg language this implies the following internal symmetry
Φi → Φi + ξi(Φ ). (14.8 )

To maintain this symmetry non-linearly the mass term should be a function of n and Γ μν [438*]

m2M--P2l√ --- μ ℒmass = − 8 − gF (n,Γ ν). (14.9 )

The absence of helicity-1 and -0 modes while keeping the helicity-2 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*).

14.2.2 Non-perturbative 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 m1 = 0 and m0 = 0. However, as we shall see below, the case m0 = 0 propagates five degrees of freedom including a helicity-0 mode that behaves as a scalar it follows that on generic backgrounds the theory with m1 = 0 also propagates a helicity-0 mode. The helicity-0 mode is thus infinitely strongly coupled when considered perturbatively about Minkowski.

14.3 General massive gravity (m0 = 0)

In [107*] the most general mass term that extends (14.1*) non-linearly was considered. It can be written using the Stückelberg variables defined in (14.3*), (14.4*) and (14.5*),

m2M 2 √ --- ℒSO (3) mass = −-----Pl − gV (n,nμ,Y μν ) . (14.10 ) 8
Generalizing the Hamiltonian analysis for this mass term and requiring the propagation of five degrees of freedom about any background led to the Lorentz-invariant ghost-free theory of massive gravity presented in Part II as well as two new theories of Lorentz breaking massive gravity.

All of these cases ensures the absence of BD ghost by having m0 = 0. The case where the BD ghost is projected thanks to the requirement m1 = 0 is discussed in Section 14.2.

14.3.1 First explicit Lorentz-breaking example with five dofs

The first explicit realization of a consistent nonlinear Lorentz breaking model is as follows [107*]

[ ] V1(n,n μ,Yνμ) = n −1 n + ζ(Γ ) U(𝒦&tidle;) + n−1𝒞(Γ ), (14.11 )
( ) n2n μnα &tidle;𝒦μν = |( Γ μα − [--------]2-|) ∂αΦi ∂νΦjδij, (14.12 ) n + ζ(Γ )
and where U, C and ζ are scalar functions.

The fact that several independent functions enter the mass term will be of great interest for cosmology as one of these functions (namely 𝒞) can be used to satisfy the Bianchi identity while the other function can be used for an appropriate cosmological history.

The special case ζ = 0 is what is referred to as the ‘minimal model’ and was investigated in [108*]. In unitary gauge, this minimal model is simply

(minimal) m2M--2Pl√ ---( ik −1 ik ) ℒSO(3) mass = − 8 − g U (g δkj) + N 𝒞(γ δij) , (14.13 )
where γij is the spatial part of the metric and ij ij −2 i j g = γ − N N N, where N is the lapse and i N the shift.

This minimal model is of special interest as both the primary and secondary second-class constraints that remove the sixth degree of freedom can be found explicitly and on the constraint surface the contribution of the mass term to the Hamiltonian is

∫ √ -- H ∝ M 2Plm2 d3x γ𝒞 (γikδij), (14.14 )
where the overall factor is positive so the Hamiltonian is positive definite as long as the function 𝒞 is positive.

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 m0 = 0, so the BD ghost is projected out in a way similar as in ghost-free massive gravity.

Interestingly, if 𝒞 = 0, this theory corresponds to m = 0 1 (in addition to m = 0 0) which as seen earlier the helicity-1 mode is absent at the linearized level. However, they survive non-linearly and so the case 𝒞 = 0 is infinitely strongly coupled.

14.3.2 Second example of Lorentz-breaking with five dofs

Another example of Lorentz breaking SO (3) invariant theory of massive gravity was provided in [107*]. In that case the Stückelberg language is not particularly illuminating and we simply give the form of the mass term in unitary gauge (Φ = t and i i Φ = x),

[ ] V = c1 N⃗T (N 𝕀 + 𝕄 )− 1(𝔽 + N − 1𝕄 𝔽)(N 𝕀 + 𝕄 )−1 ⃗N (14.15 ) 2 2 + 𝒞 + N − 1𝒞 &tidle;.
where 𝔽 = {fij} is the spatial part of the reference metric (for a Minkowski reference metric fij = δij), c1 is a constant, and 𝒞 and &tidle;𝒞 are functions of the spatial metric γij, while 𝕄 is a rank-3 matrix which depends on γikfkj.

Interestingly, &tidle;𝒞 does not enter the Hamiltonian on the constraint surface. The contribution of this mass term to the on-shell Hamitonian is [107]

∫ [ ] H ∝ M 2m2 d3x √ γ-− c1 ⃗N T (N 𝕀 + 𝕄 )−1𝔽 𝕄 (N 𝕀 + 𝕄 )−1N⃗ + 𝒞 , (14.16 ) Pl 2
with a positive coefficient, which implies that 𝒞 should be bounded from below

14.3.3 Absence of vDVZ and strong coupling scale

Unlike in the Lorentz-invariant case, the kinetic term for the Stückelberg fields does not only arise from the mixing with the helicity-2 mode.

When looking at perturbations about Minkowski and focusing on the scalar modes we can follow the analysis of [437*],

ds2 = − (1 − ψ )dt2 + 2∂ivdxi dt + (δij + τδij + ∂i∂jσ) dxidxj, (14.17 )
when m = 0 0 ψ plays the role of the Lagrange multiplier for the primary constraint imposing
( ) σ = -2-− 3- τ, (14.18 ) m24 ∇
where ∇ is the three-dimensional Laplacian. The secondary constraint then imposes the relation
2-- v = m21 ˙τ, (14.19 )
where dots represent derivatives with respect to the time. Using these relations for v and σ, we obtain the Lagrangian for the remaining scalar mode (the helicity-0 mode) τ [437, 200],
2( [( ) ] 2 2 M--Pl -4- -4- m-2-−-m-3 2 ℒτ = 4 m24 − m21 ∇ τ − 3τ ¨τ − 2 m44 (∇ τ) (14.20 ) ( ) ) m22 2 2 + 4--2 − 1 τ ∇τ − 3m 2τ . (14.21 ) m 4
In terms of power counting this means that the Lagrangian includes terms of the form M 2 m2 ∇ ϕ¨ϕ Pl arising from the term going as −2 −2 (m 4 − m 1 )∇ τ¨τ (where ϕ designates the helicity-0 mode which includes a combination of σ and v). Such terms are not present in the Lorentz-invariant Fierz–Pauli case and its non-linear ghost-free extension since m4 = m1 in that case, and they play a crucial role in this Lorentz violating setting.

Indeed, in the small mass limit these terms M 2m2 ∇ ϕ¨ϕ Pl dominate over the ones that go as M 2m4 ϕϕ¨ Pl (i.e., the ones present in the Lorentz invariant case). This means that in the small mass limit, the correct canonical normalization of the helicity-0 mode ϕ is not of the form ˆϕ = ϕ∕MPlm2 but rather √ -- ϕˆ= ϕ∕MPlm ∇, which is crucial in determining the strong coupling scale and the absence of vDVZ discontinuity:

  • The new canonical normalization implies a much larger strong coupling scale that goes as 1∕2 Λ2 = (MPlm ) rather than 2 1∕3 Λ3 = (MPlm ) as is the case in DGP and ghost-free massive gravity.
  • Furthermore, in the massless limit the coupling of the helicity-0 mode to the tensor vanishes fasters than some of the Lorentz-violating kinetic interactions in (14.20*) (which is scales as ˆ 2 ˆ m h∂ ϕ). This means that one can take the massless limit m → 0 in such a way that the coupling to the helicity-2 mode disappears and so does the coupling of the helicity-0 mode to matter (since this coupling arises after de-mixing of the helicity-0 and -2 modes). This implies the absence of vDVZ discontinuity in this Lorentz-violating theory despite the presence of five degrees of freedom.

The absence of vDVZ discontinuity and the larger strong coupling scale Λ2 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 Λ2 ≪ MPl. This is similar to what happens for the Lorentz-invariant ghost-free 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 ghost-free 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 − 30 m ≲ 10 eV in which case −4 −1 Λ2 ∼ (10 mm ) [108, 257].

14.3.4 Cosmology of general massive gravity

The cosmology of general massive gravity was recently studied in [109*] and we summarize their results in what follows.

In Section 12, we showed how the Bianchi identity in ghost-free massive gravity prevents the existence of spatially flat FLRW solutions. The situation is similar in general Lorentz violating theories of massive gravity unless the function 𝒞 in (14.11*) is chosen so as to satisfy the following relation when the shift and ni vanish [109*]

( ) ′ 1- H 𝒞 − 2𝒞 = 0. (14.22 )
Choosing a function 𝒞 which satisfies the appropriate condition to allow for FLRW solutions, the Friedmann equation then depends entirely on the function U (&tidle;𝒦) also defined in (14.11*). In this case the graviton potential (14.10*) acts as an effective ‘dark fluid’ with respective energy density and pressure dictated by the function U [109*]
m2 m2 ρeff = ---U (&tidle;𝒦 ) peff = ---(2U ′(&tidle;𝒦) − U (&tidle;𝒦)), (14.23 ) 4 4
leading to an effective phantom-like behavior when 2U ′∕U < 0.

This solution is stable and healthy as long as the second derivative of 𝒞 satisfies some conditions which can easily be accommodated for appropriate functions 𝒞 and U.

Expanding U in terms of the scale factor for late time ∑ ¯ n U = n≥0 Un(a − 1) one can use CMB and BAO data from [2] to put constraints on the first terms of that series [109*]

¯ ¯ U1-= 0.12 ± 2.1 and U2-< 2 ± 3 at 95% C.L. (14.24 ) U¯0 ¯U0
Focusing instead on early time cosmology BBN data can similarly be used to constrains the function U, see [109] for more details.
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