"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

4 The Dvali–Gabadadze–Porrati Model

The idea behind the DGP model [209*, 208*, 207*] is to start with a four-dimensional braneworld in an infinite size-extra dimension. A priori gravity would then be fully five-dimensional, with respective Planck scale M5, but the matter fields localized on the brane could lead to an induced curvature term on the brane with respective Planck scale MPl. 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.

4.1 Gravity induced on a brane

We start with the five-dimensional action for the DGP model [209, 208, 207] with a brane localized at y = 0,

∫ ( 3∘ ----- [√ --- 2 ]) S = d4x dy M-5- − (5)g (5)R + δ (y ) − gM-PlR [g ] + ℒm (g,ψi) , (4.1 ) 4 2
where ψ i represent matter field species confined to the brane with stress-energy tensor T μν. This brane is considered to be an orbifold brane enjoying a ℤ2-orbifold symmetry (so that the physics at y < 0 is the mirror copy of that at y > 0.) We choose the convention where we consider − ∞ < y < ∞, reason why we have a factor or M 35∕4 rather than M 35∕2 if we had only consider one side of the brane, for instance y ≥ 0.

The five-dimensional Einstein equation of motion are then given by

M 35 (5)GAB = 2δ(y)(5)TAB (4.2 )
(5) ( 2 ) μ ν TAB = − M PlG μν + Tμν δAδB. (4.3 )
The Israel matching condition on the brane [323] can be obtained by integrating this equation over ∫ 𝜀 dy − 𝜀 and taking the limit 𝜀 → 0, so that the jump in the extrinsic curvature across the brane is related to the Einstein tensor and stress-energy tensor of the matter field confined on the brane.

4.1.1 Perturbations about flat spacetime

In DGP the four-dimensional graviton is effectively massive. To see this explicitly, we look at perturbations about flat spacetime

ds2 = (η + h (x,y)) dxA dxB. (4.4 ) 5 AB AB
Since at this level we are dealing with five-dimensional GR, we are free to set the five-dimensional gauge of our choice and choose five-dimensional de Donder gauge (a discussion about the brane-bending mode will follow)
A 1- A ∂Ah B = 2∂Bh A. (4.5 )

In this gauge the five-dimensional Einstein tensor is simply

( ) (5) 1- 1-C GAB = − 2□5 hAB − 2hC ηAB , (4.6 )
where 2 □5 = □ + ∂y is the five-dimensional d’Alembertian and □ is the four-dimensional one.

Since there is no source along the μy or yy directions ((5)T μy = 0 = (5)Tyy), we can immediately infer that

( □5h μy) = 0 ⇒ hμy = 0 (4.7 ) □5 hyy − hμμ = 0 ⇒ hyy = hμμ, (4.8 )
up to an homogeneous mode which in this setup we set to zero. This does not properly account for the brane-bending mode but for the sake of this analysis it will give the correct expression for the metric fluctuation hμν. We will see in Section 4.2 how to keep track of the brane-bending mode which is partly encoded in hyy.

Using these relations in the five-dimensional de Donder gauge, we deduce the relation for the purely four-dimensional part of the metric perturbation,

∂μhμ = ∂νh μ. (4.9 ) ν μ
Using these relations in the projected Einstein equation, we get
1- 3 [ 2] ( 2 ) 2 M 5 □ + ∂y (h μν − h ημν) = − δ(y ) 2Tμν + M Pl(□h μν − ∂μ∂νh ) , (4.10 )
where α μν h ≡ hα = η hμν is the four-dimensional trace of the perturbations.

Solving this equation with the requirement that hμν → 0 as y → ± ∞, we infer the following profile for the perturbations along the extra dimension

−|y|√ −□- hμν(x,y) = e hμν(x), (4.11 )
where the □ should really be thought in Fourier space, and h (x) μν is set from the boundary conditions on the brane. Integrating the Einstein equation across the brane, from − 𝜀 to + 𝜀, we get
1-lim M 3[∂ h (x,y) − h(x, y)η ]𝜀 + M 2(□h (x,0) − ∂ ∂ h (x,0)) 2 𝜀→0 5 y μν μν− 𝜀 Pl μν μ ν = − 2Tμν(x), (4.12 )
yielding the modified linearized Einstein equation on the brane
[ √ ---- ] M 2Pl (□h μν − ∂μ ∂νh) − m0 − □ (hμν − hη μν) = − 2T μν, (4.13 )
where all the metric perturbations are the ones localized at y = 0 and the constant mass scale m0 is given by
M 3 m0 = --5-. (4.14 ) M 2Pl
Interestingly, we see the special Fierz–Pauli combination h − hη μν μν appearing naturally from the five-dimensional nature of the theory. At this level, this corresponds to a linearized theory of massive gravity with a scale-dependent effective mass 2 √ ---- m (□ ) = m0 − □, which can be thought in Fourier space, m2 (k) = m0k. We could now follow the same procedure as derived in Section 2.2.3 and obtain the expression for the sourced metric fluctuation on the brane
2 1 ( 1 1 ) hμν = − --2---------√---- T μν − -T ημν + ---√----∂μ ∂νT , (4.15 ) M Pl□ − m0 − □ 3 3m − □
where T = η μνT μν is the trace of the four-dimensional stress-energy tensor localized on the brane. This yields the following gravitational exchange amplitude between two conserved sources Tμν and ′ T μν,
∫ ∫ fmassive 𝒜DGTPT′ = d4x hμνT ′μν = d4x T ′μν ----μναβ√-----T αβ, (4.16 ) □ − m0 − □
where the polarization tensor fmassive μνα β is the same as that given for Fierz–Pauli in (2.57*) in terms of m 0. In particular the polarization tensor includes the standard factor of − 1 ∕3Tη μν as opposed to − 1∕2T ημν as would be the case in GR. This is again the manifestation of the vDVZ discontinuity which is cured by the Vainshtein mechanism as for Fierz–Pauli massive gravity. See [165*] for the explicit realization of the Vainshtein mechanism in DGP which is where it was first shown to work explicitly.

4.1.2 Spectral representation

In Fourier space the propagator for the graviton in DGP is given by

G&tidle;maμsνsαiβve(k) = fmμaνsαsiβve(k, m0 )𝒢&tidle;(k), (4.17 )
with the massive polarization tensor massive f defined in (2.58*) and
&tidle;𝒢 (k ) = ----1----, (4.18 ) k2 + m0k
which can be written in the Källén–Lehmann spectral representation as a sum of free propagators with mass μ,
∫ ∞ 2 &tidle;𝒢(k) = -ρ(μ-)--dμ2, (4.19 ) 0 k2 + μ2
with the spectral density 2 ρ(μ )
ρ(μ2) = 1-m0-----1---, (4.20 ) π μ μ2 + m20
which is represented in Figure 1*. As already emphasized, the graviton in DGP cannot be thought of a single massive mode, but rather as a resonance picked about μ = 0.

We see that the spectral density is positive for any μ2 > 0, confirming the fact that about the normal (flat) branch of DGP there is no ghost.

Notice as well that in the massless limit m0 → 0, we see appearing a representation of the Dirac delta function,

1-m0-----1--- 2 lmim→0 π μ μ2 + m2 = δ(μ ), (4.21 ) 0
and so the massless mode is singled out in the massless limit of DGP (with the different tensor structure given by f massive ⁄= f(0) μναβ μναβ which is the origin of the vDVZ discontinuity see Section 2.2.3.)

4.2 Brane-bending mode

Five-dimensional gauge-fixing

In Section 4.1.1 we have remained vague about the gauge-fixing and the implications for the brane position. The brane-bending mode is actually important to keep track of in DGP and we shall do that properly in what follows by keeping all the modes.

We work in the five-dimensional ADM split with the lapse √ -yy 1 N = 1∕ g = 1 + 2hyy, the shift N μ = gμy and the four-dimensional part of the metric, gμν(x,y) = ημν + hμν(x,y). The five-dimensional Einstein–Hilbert term is then expressed as

(5) M-53√ --- ( 2 2) ℒ R = 4 − gN R [g] + [K ] − [K ] , (4.22 )
where square brackets correspond to the trace of a tensor with respect to the four-dimensional metric gμν and K μν is the extrinsic curvature
1 K μν = ----(∂ygμν − DμN ν − D νN μ), (4.23 ) 2N
and Dμ is the covariant derivative with respect to gμν.

First notice that the five-dimensional de Donder gauge choice (4.5*) can be made using the five-dimensional gauge fixing term

3 ( )2 ℒ (5) = − M-5- ∂AhA − 1∂BhA (4.24 ) Gauge−Fixing 8 B 2 A 3 [( )2 = − M-5- ∂μhμ − 1∂νh + ∂yN ν − 1∂ νhyy (4.25 ) 8 ν 2 2 ( )2 ] μ 1- 1- + ∂μN + 2∂yhyy − 2 ∂yh ,
where we keep the same notation as previously, h = ημνhμν is the four-dimensional trace.

After fixing the de Donder gauge (4.5*), we can make the addition gauge transformation A A A x → x + ξ, and remain in de Donder gauge provided A ξ satisfies linearly A □5ξ = 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).

Four-dimensional Gauge-fixing

Keeping the brane at the fixed position y = 0 imposes y ξ = 0 since we need y ξ (y = 0) = 0 and ξ should be bounded as y → ∞ (the situation is slightly different in the self-accelerating branch and this mode can lead to a ghost, see Section 4.4 as well as [361*, 98*]).

Using the bulk profile √--- hAB (x,y ) = e− −□ |y|hAB (x ) and integrating over the extra dimension, we obtain the contribution from the bulk on the brane (including the contribution from the gauge-fixing term) in terms of the gauge invariant quantity

2 2 &tidle;hμν = hμν + √-----∂(μN ν) = − √-----K μν (4.26 ) − □ − □
[ ] M 3 ∫ 1 √ ----( 1 ) 1 √ ----( 1 ) Sibnutelgkrated= --5- d4x − -&tidle;hμν − □ &tidle;hμν − -&tidle;h ημν + --hyy − □ &tidle;h − -hyy . 4 2 2 2 2
Notice again a factor of 2 difference from [389*] which arises from the fact that we integrate from y = − ∞ to y = + ∞ imposing a ℤ2-mirror symmetry at y = 0, rather than considering only one side of the brane as in [389*]. Both conventions are perfectly reasonable.

The integrated bulk action (4.27*) is invariant under the residual linearized gauge symmetry

h → h + 2∂ ξ (4.27 ) μν μν √ -(μ--ν) N μ → Nμ − − □ ξν (4.28 ) h → h (4.29 ) yy yy
which keeps both &tidle;hμν and hyy invariant. The residual gauge symmetry can be used to set the gauge on the brane, and at this level from (4.27*) we can see that the most convenient gauge fixing term is [389*]
2 ( )2 ℒ(4) = − M--Pl ∂μhμ − 1-∂νh + m0N ν , (4.30 ) Residual Gauge−Fixing 4 ν 2
with again 3 2 m0 = M 5∕M Pl, so that the induced Lagrangian on the brane (including the contribution from the residual gauge fixing term) is
∫ [ ( ) ] M P2l 4 1 μν 1 αμ 1 μ 2 μ Sboundary = --4- d x 2h □ (hμν − 2h ημν) − 2m0N μ ∂ αh − 2∂ h − m 0N μN . (4.31 )
Combining the five-dimensional action from the bulk (4.27*) with that on the boundary (4.31*) we end up with the linearized action on the four-dimensional DGP brane [389*]
2 ∫ [ S(lin)= M-Pl d4x 1-hμν [□ − m √ − □-] (h − 1hη ) − m N μ∂ h (4.32 ) DGP 4 2 0 μν 2 μν 0 μ yy ] μ [√---- ] m0 √ ---- − m0N − □ + m0 N μ − -4-hyy − □(hyy − 2h ) .
As shown earlier we recover the theory of a massive graviton in four dimensions, with a soft mass 2 √ ---- m (□ ) = m0 − □. This analysis has allowed us to keep track of the physical origin of all the modes including the brane-bending mode which is especially relevant when deriving the decoupling limit as we shall see below.

The kinetic mixing between these different modes can be diagonalized by performing the change of variables [389*]

1 ( ) hμν = ---- h′μν + πημν (4.33 ) MPl ----1---- ′ ---1--- N μ = MPl √m0--N μ + MPlm0 ∂μπ (4.34 ) √ ---- h = − 2---− □-π, (4.35 ) yy m0MPl
so we see that the mode π is directly related to h yy. In the case of Section 4.1.1, we had set h = 0 yy and the field π is then related to the brane bending mode. In either case we see that the extrinsic curvature K μν carries part of this mode.

Omitting the mass terms and other relevant operators, the action is diagonalized in terms of the different graviton modes at the linearized level h ′ μν (which encodes the helicity-2 mode), N ′ μ (which is part of the helicity-1 mode) and π (helicity-0 mode),

∫ [ √---- ] S(lin) = 1- d4x 1h ′μν□ (h′ − 1h ′ημν) − N ′μ − □N ′+ 3π□ π . (4.36 ) DGP 4 2 μν 2 μ

Decoupling limit

We will be discussing the meaning of ‘decoupling limits’ in more depth in the context of multi-gravity and ghost-free 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 helicity-0 mode without the complications from the standard helicity-2 interactions that already arise in GR. For this purpose we can take the limit MPl → ∞ while simultaneously sending m0 = M 35∕M 2Pl → 0 while keeping the scale Λ = (m2 MPl )1∕3 0 fixed. This is the scale at which the first interactions arise in DGP.

In DGP the decoupling limit should be taken by considering the full five-dimensional theory, as was performed in [389*]. The four-dimensional Einstein–Hilbert term does not give to any operators before the Planck scale, so in order to look for the irrelevant operator that come at the lowest possible scale, it is sufficient to focus on the boundary term from the five-dimensional action. It includes operators of the form

( h ′ )n ( N ′ )k ( ) ℓ ℒ (5) ⊃ m0M 2Pl∂ --μν √----μ--- --∂π--- , (4.37 ) boundary MPl m0MPl m0MPl
with integer powers n,k, ℓ ≥ 0 and n + k + ℓ ≥ 3 since we are dealing with interactions. The scale at which such an operator arises is
( )1∕(n+3k∕2+2 ℓ−3) Λn,k,ℓ = M Pnl+k+ℓ−2mk0∕2+ℓ−1 (4.38 )
and it is easy to see that the lowest possible scale is Λ3 = (MPlm20)1∕3 which arises for n = 0,k = 0 and ℓ = 3, it is thus a cubic interaction in the helicity-0 mode π which involves four derivatives. Since it is only a cubic interaction, we can scan all the possible ways π enters at the cubic level in the five-dimensional Einstein–Hilbert action. The relevant piece are the ones from the extrinsic curvature in (4.22*), and in particular the combination N ([K ]2 − [K2 ]), with
1 −√−□y N = 1 + 2-e hyy (4.39 ) 1 1 √--- K μν = − -(1 − -e− − □yhyy)(∂μN ν + ∂ νNμ ). (4.40 ) 2 2
Integrating m0M 2PlN ([K ]2 − [K2 ]) along the extra dimension, we obtain the cubic contribution in π on the brane (using the relations (4.34*) and (4.35*))
-1-- 2 ℒ Λ3 = 2Λ3 (∂π) □ π. (4.41 ) 3
So the decoupling limit of DGP arises at the scale Λ3 and reduces to a cubic Galileon for the helicity-0 mode with no interactions for the helicity-2 and -1 modes,
( ) ℒ = 1h′μν□ h′ − 1h ′η − 1N ′μ√ −-□N ′ (4.42 ) DL DGP 8 μν 2 μν 4 μ 3 1 + --π□ π + ---3(∂π )2□ π. 2 2Λ 3

4.3 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 self-accelerating solutions. The cosmology and phenomenology of DGP was first derived in [159*, 163*] (see also [388*, 385*, 387*, 386*]).

4.3.1 Friedmann equation in de Sitter

To get some intuition on how cosmology gets modified in DGP, we first look at de Sitter-like solutions and then infer the full Friedmann equation in a FLRW-geometry. We thus start with five-dimensional Minkowski in de Sitter slicing (this can be easily generalized to FLRW-slicing),

2 2 ( 2 (dS) μ ν) ds 5 = b(y ) dy + γμν dx dx , (4.43 )
where γ(μdνS) is the four-dimensional de Sitter metric with constant Hubble parameter H,
(dS) μ ν 2 2 2 γμν dx dx = − dt + a (t)dx, and the scale factor is given by a (t) = exp (Ht ). The metric (4.43*) is indeed Minkowski in de Sitter slicing if the warp factor b(y) is given by
𝜖H|y| b(y) = e , with 𝜖 = ±1, (4.44 )
and the mod y has be imposed by the ℤ 2-orbifold symmetry. As we shall see the branch 𝜖 = +1 corresponds to the self-accelerating branch of DGP and 𝜖 = − 1 is the stable, normal branch of DGP.

We can now derive the Friedmann equation on the brane by integrating over the 00-component of the Einstein equation (4.2*) with the source (4.3*) and consider some energy density T00 = ρ. The four-dimensional Einstein tensor gives the standard contribution G = 3H2 00 on the brane and so we obtain the modified Friedmann equation

M 3[ ∫ 𝜀 ] ---5 lim (5)G00 dy + 3M 2PlH2 = ρ, (4.45 ) 2 𝜀→0 − 𝜀
with (5)G = 3(H2 − b′′(y)∕b(y)) 00, so
∫ 𝜀 (5) l𝜀i→m0 G00dy = − 6𝜖H, (4.46 ) −𝜀
leading to the modified Friedmann equation,
H2 − 𝜖m0H = --1--ρ, (4.47 ) 3M 2Pl
where the five-dimensional nature of the theory is encoded in the new term − 𝜖m H 0 (this new contribution can be seen to arise from the helicity-0 mode of the graviton and could have been derived using the decoupling limit of DGP.)

For reasons which will become clear in what follows, the choice 𝜖 = − 1 corresponds to the stable branch of DGP while the other choice 𝜖 = +1 corresponds to the self-accelerating branch of DGP. As is already clear from the higher-dimensional perspective, when 𝜖 = +1, the warp factor grows in the bulk (unless we think of the junction conditions the other way around), which is already signaling towards a pathology for that branch of solution.

4.3.2 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 non-constant H. This amounts to account for ˙ H and other derivative corrections which might have been omitted in deriving this equation by assuming that H was constant. But the Friedmann equation corresponds to the Hamiltonian constraint equation and higher derivatives (e.g., H˙ ⊃ a¨ 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 five-dimensional 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 bi-gravity and the Friedmann equations (12.51*), (12.52*) and (12.54*) are indeed free of any derivatives of the Hubble parameter.

4.3.3 Observational viability of DGP

Independently of the ghost issue in the self-accelerating branch of the model, there has been a vast amount of investigation on the observational viability of both the self-accelerating 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.

Observational tests of DGP fall into the following two main categories:

  • 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 H term in the Friedmann equation of DGP, even in the normal branch where the late-time 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 −1 m0 ≲ 10 H0, where H0 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 late-time 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 N-body 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 five-dimensional study dedicated to DGP. The study of cosmological perturbations within the context of DGP was also performed in depth for instance in [367, 92].

4.4 Self-acceleration branch

The cosmology of DGP has led to a major conceptual breakthrough, namely the realization that the Universe could be ‘self-accelerating’. This occurs when choosing the 𝜖 = +1 branch of DGP, the Friedmann equation in the vacuum reduces to [159*, 163*]

H2 − m0H = 0, (4.48 )
which admits a non-trivial solution H = m0 in the absence of any cosmological constant nor vacuum energy. In itself this would not solve the old cosmological constant problem as the vacuum energy ought to be set to zero on its own, but it can lead to a model of ‘dark gravity’ where the amount of acceleration is governed by the scale m0 which is stable against quantum corrections.

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 self-accelerating branch as well as a few cosmological observations.


The existence of a ghost on the self-accelerating branch of DGP was first pointed out in the decoupling limit [389*, 411*], where the helicity-0 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 self-accelerating 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 five-dimensional theory by K. Koyama in [360*] (see also [263, 361] and [98*]).

When perturbing about Minkowski, it was shown that the graviton has an effective mass 2 √ ---- m = m0 − □. When perturbing on top of the self-accelerating solution a similar analysis can be performed and one can show that in the vacuum the graviton has an effective mass at precisely the Higuchi-bound, m2eff = 2H2 (see Ref. [307*]). When matter or a cosmological constant is included on the brane, the graviton mass shifts either inside the forbidden Higuchi-region 0 < m2 < 2H2 eff, or outside m2 > 2H2 eff. We summarize the three case scenario following [360, 98]

  • In [307*] it was shown that when the effective mass is within the forbidden Higuchi-region, the helicity-0 mode of graviton has the wrong sign kinetic term and is a ghost.
  • Outside this forbidden region, when m2 > 2H2 eff, the zero-mode of the graviton is healthy but there exists a new normalizable brane-bending mode in the self-accelerating branch8 which is a genuine degree of freedom. For 2 2 m eff > 2H the brane-bending mode was shown to be a ghost.
  • Finally, at the critical mass m2eff = 2H2 (which happens when no matter nor cosmological constant is present on the brane), the brane-bending mode takes the role of the helicity-0 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.

In summary, independently of the matter content of the brane, so long as the graviton is massive 2 m eff > 0, the self-accelerating branch of DGP exhibits a ghost. See also [210] for an exact non-perturbative argument studying domain walls in DGP. In the self-accelerating branch of DGP domain walls bear a negative gravitational mass. This non-perturbative solution can also be used as an argument for the instability of that branch.

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 spin-0 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 self-accelerating solution on the brane would be stable.

Finally, a stable self-acceleration was also shown to occur in the massless case m2 = 0 eff by relying on Gauss–Bonnet terms in the bulk and a self-source AdS5 solution [156]. The five-dimensional theory is then similar as that of DGP (4.1*) but with the addition of a five-dimensional Gauss–Bonnet term 2 ℛ GB in the bulk and the wrong sign five-dimensional Einstein–Hilbert term,

[ ∫ ∘ -----( M 3 M 3ℓ2 ) S = d5x − (5)g − --5(5)R [(5)g] − --5--(5)ℛ2GB [(5)g] (4.49 ) 4 4 [ 2 ] ] + δ(y) √ − g-M-PlR + ℒ (g,ψ ) . 2 m i
The idea is not so dissimilar as in new massive gravity (see Section 13), where here the wrong sign kinetic term in five-dimensions is balanced by the Gauss–Bonnet term in such a way that the graviton has the correct sign kinetic term on the self-sourced AdS5 solution. The length scale ℓ is related to this AdS length scale, and the self-accelerating branch admits a stable (ghost-free) de Sitter solution with − 1 H ∼ ℓ.

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 bi-gravity 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 self-accelerating solution.

In what follows, we go back to the standard DGP model be it the self-accelerating branch (𝜖 = 1) or the normal branch (𝜖 = − 1).

4.5 Degravitation

One of the main motivations behind modifying gravity in the infrared is to tackle the old cosmological constant problem. The idea behind ‘degravitation’ [211*, 212*, 26*, 216*] is if gravity is modified in the IR, then a cosmological constant (or the vacuum energy) could have a smaller impact on the geometry. In these models, we would live with a large vacuum energy (be it at the TeV scale or at the Planck scale) but only observe a small amount of late-acceleration due to the modification of gravity. In order for a theory of modified gravity to potentially tackle the old cosmological constant problem via degravitation it needs to have the two following properties:

First, gravity must be weaker in the infrared and effectively massive [216*] so that the effect of IR sources can be degravitated.
Second, there must exist some (nearly) static attractor solutions towards which the system can evolve at late-time for arbitrary value of the vacuum energy or cosmological constant.

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 five-dimensional spacetime is not Lorentz invariant, but in a way which would not (at this background level) be observed when confined on the four-dimensional 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.

Inspired by the form of the graviton in DGP, m2 (□ ) = m √−-□- 0, we can generalize the form of the graviton mass to

( ) 2 2 − □ α m (□) = m 0 ---2 , (4.50 ) m 0
with α a positive dimensionless constant. α = 1 corresponds to a modification of the kinetic term. As shown in [153*], any such modification leads to ghosts, so we do not consider this case here. α > 1 corresponds to a UV modification of gravity, and so we focus on α < 1.

In the decoupling limit the helicity-2 decouples from the helicity-0 mode which behaves (symbolically) as follows [216*]

( ) 3□π − -----1-----□ □1 −απ 2 + ⋅⋅⋅ = − -1--T, (4.51 ) MPlm4 (01−α) MPl
where T is the trace of the stress-energy tensor of external matter fields. At the linearized level, matter couples to the metric 1 gμν = ημν + MPl(h′μν + πημν). We now check under which conditions we can still recover a nearly static metric in the presence of a cosmological constant Tμν = − ΛCCg μν. In the linearized limit of GR this leads to the profile for the helicity-2 mode (which in that case corresponds to a linearized de Sitter solution)
′ ΛCC ρ σ hμν = − -----ηρσx x ημν. (4.52 ) 6MPl
One way we can obtain a static solution in this extended theory of massive gravity at the linear level is by ensuring that the solution for π cancels out that of h ′μν so that the metric gμν remains flat. ΛCC- μ ν π = + 6MPl ημνx x is actually the solution of (4.51*) when only the term 3□π contributes and all the other operators vanish for π ∝ xμxμ. This is the case if α < 1∕2 as shown in [216*]. This explains why in the case of DGP which corresponds to border line scenario α = 1∕2, one can never fully degravitate a cosmological constant.


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 α 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).

Note that α ≡ 0 corresponds to a hard mass gravity. Within the context of DGP, such a model with an ‘auxiliary’ extra dimension was proposed in [235, 133*] where we consider a finite-size large extra dimension which breaks five-dimensional Lorentz invariance. The five-dimensional action is motivated by the five-dimensional gravity with scalar curvature in the ADM decomposition (5)R = R [g] + [K ]2 − [K2 ], but discarding the contribution from the four-dimensional curvature R [g]. Similarly as in DGP, the four-dimensional curvature still appears induced on the brane

M 2Pl∫ ℓ ∫ 4 √ ---( ( 2 2 ) ) S = ---- dy d x − g m0 [K ] − [K ] + δ(y)R[g] , (4.53 ) 2 0
where ℓ is the size of the auxiliary extra dimension and gμν is a four-dimensional metric and we set the lapse to one (this shift can be kept and will contribute to the four-dimensional Stückelberg field which restores four-dimensional invariance, but at this level it is easier to work in the gauge where the shift is set to zero and reintroduce the Stückelberg fields directly in four dimensions). Imposing the Dirichlet conditions gμν(x,y = 0) = fμν, we are left with a theory of massive gravity at y = 0, with reference metric fμν and hard mass m0. Here again the special structure ([K ]2 − [K2 ]) inherited (or rather inspired) from five-dimensional gravity ensures the Fierz–Pauli structure and the absence of ghost at the linearized level. Up to cubic order in perturbations it was shown in [138*] that the theory is free of ghost and its decoupling limit is that of a Galileon.

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 ghost-free theory of massive gravity derived in [137*, 144*]

4.5.1 Cascading gravity

Deficit angle

It is well known that a tension on a cosmic string does not cause the cosmic strong to inflate but rather creates a deficit angle in the two spatial dimensions orthogonal to the string. Similarly, if we consider a four-dimensional brane embedded in six-dimensional gravity, then a tension on the brane leads to the following flat geometry

( Δ 𝜃 ) ds26 = ημν dxμ dxν + dr2 + r2 1 − --- d𝜃2, (4.54 ) 2π
where the two extra dimensions are expressed in polar coordinates {r,𝜃} and Δ 𝜃 is a constant which parameterize the deficit angle in this canonical geometry. This deficit angle is related to the tension on the brane ΛCC and the six-dimensional Planck scale (assuming six-dimensional gravity)
ΛCC Δ 𝜃 = 2π ---4. (4.55 ) M 6
For a positive tension ΛCC > 0, this creates a positive deficit angle and since Δ 𝜃 cannot be larger than 2π, the maximal tension on the brane is M 4 6. For a negative tension, on the other hand, there is no such bound as it creates a surplus of angle, see Figure 2*.
View Image
Figure 2: Codimension-2 brane with positive (resp. negative) tension brane leading to a positive (resp. negative) deficit angle in the two extra dimensions.

This interesting feature has lead to many potential ways to tackle the cosmological constant by considering our Universe to live in a 3 + 1-dimensional 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 large-extra-dimension 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.

Spectral representation

Furthermore in n-extra dimensions the gravitational potential is diluted as V (r) ∼ r−1−n. If the propagator has a Källén–Lehmann spectral representation with spectral density 2 ρ (μ ), the Newtonian potential has the following spectral representation

∫ ∞ 2 − μr V (r) = ρ(μ-)e----dμ2. (4.56 ) 0 r
In a higher-dimensional DGP scenario, the gravitation potential behaves higher dimensional at large distance, − (1+n) V(r) ∼ r which implies 2 n− 2 ρ(μ ) ∼ μ in the IR as depicted in Figure 1*.

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.

A brane on a brane

Both the spectral representation and the fact that codimension-two (and higher) branes can accommodate for a cosmological constant while remaining flat has made the field of higher-codimension 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]).

To avoid these issues, one can consider simply applying the DGP procedure step by step and consider a 4 + 1-dimensional DGP brane embedded in six dimension. Our Universe would then be on a 3 + 1-dimensional DGP brane embedded in the 4 + 1 one, (note we only consider one side of the brane here which explains the factor of 2 difference compared with (4.1*))

M 4∫ √ ---- M 3 ∫ √ ---- S = ---6 d6x − g6(6)R + --5- d5x − g5(5)R (4.57 ) 2 ∫ ∫2 M-2Pl 4 √ ---(4) 4 + 2 d x − g4 R + d xℒmatter(g4,ψ ).
This model has two cross-over scales: 3 2 m5 = M 5∕M Pl which characterizes the scale at which one crosses from the four-dimensional to the five-dimensional regime, and m6 = M 64∕M 35 yielding the crossing from a five-dimensional to a six-dimensional behavior. Of course we could also have a simultaneous crossing if m = m 5 6. In what follows we focus on the case where M > M > M Pl 5 6.

Performing the same linearized analysis as in Section 4.1.1 we can see that the four-dimensional theory of gravity is effectively massive with the soft mass in Fourier space

∘ -------- πm5 m2 − k2 m2(k ) = ---------∘6------. (4.58 ) 4 arcth mm6−+kk- 6
We see that the 4 + 1-dimensional brane plays the role of a regulator (a divergence occurs in the limit m5 → 0).

In this six-dimensional model, there are effectively two new scalar degrees of freedom (arising from the extra dimensions). We can ensure that both of them have the correct sign kinetic term by

  • Either smoothing out the brane [240, 148] (this means that one should really consider a six-dimensional curvature on both the smoothed 4 + 1 and on the 3 + 1-dimensional branes, which is something one would naturally expect9).
  • 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 six-dimensional 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 no-go theorem [484*].

The extension of cascading gravity to more than two extra dimensions was considered in [149*]. It was shown in that case how the 3 + 1 brane remains flat for arbitrary values of the cosmological constant on that brane (within the regime of validity of the weak-field approximation). See Figure 3* for a picture on how the scalar potential adapts itself along the extra dimensions to accommodate for a cosmological constant on the brane.

View Image
Figure 3: Seven-dimensional cascading scenario and solution for one the metric potential Φ on the (5 + 1)-dimensional brane in a 7-dimensional cascading gravity scenario with tension on the (3 + 1)-dimensional brane located at y = z = 0, in the case where M 4∕M 3= M 5∕M 4 = m 6 5 7 6 7. y and z represent the two extra dimensions on the (5 + 1 )-dimensional brane. Image reproduced with permission from [149], copyright APS.

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