"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

8 Decoupling Limits

8.1 Scaling versus decoupling

Before moving to the decoupling of massive gravity and bi-gravity, 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.

Imagine the following gauge field theory

1- 2 μ ℒ = − 2m A μA , (8.1 )
i.e., the Proca mass term without any kinetic Maxwell term for the gauge field. Since there are no dynamics in this theory, there is no degrees of freedom. Nevertheless, one could still proceed and use the same split A μ = A ⊥μ + ∂μχ∕m as performed previously,
ℒ = − 1-m2A ⊥ A⊥μ + m (∂ A ⊥μ)χ − 1(∂χ )2, (8.2 ) 2 μ μ 2
so as to introduce what appears to be a kinetic term for the mode χ. At this level the theory is still invariant under χ → χ + m ξ and ⊥ ⊥ A μ → A μ − ∂μξ, and so while there appears to be a dynamical degree of freedom χ, the symmetry makes that degree of freedom unphysical, so that (8.2*) still propagates no physical degree of freedom.

Now consider the m → 0 scaling limit of (8.2*) while keeping A⊥ μ and χ finite. In that scaling limit, the theory reduces to

1 ℒm →0 = − --(∂χ)2, (8.3 ) 2
i.e., one degree of freedom with no symmetry which implies that the theory (8.3*) propagates one degree of freedom. This is correct and thus means that (8.3*) is not a consistent decoupling limit of (8.2*) since the number of degrees of freedom is different already at the linear level. In the rest of this review, we will call a decoupling limit a specific type of scaling limit which preserves the same number of physical propagating degrees of freedom in the linear theory. As suggested by the name, a decoupling limit is a special kind of limit in which some of the degrees of freedom of the original theory might decouple from the rest, but the total number of degrees of freedom remains identical. For the theory (8.2*), this means that the scaling ought to be taken not with ⊥ A μ fixed but rather with &tidle;⊥ ⊥ A μ = Aμ ∕m fixed. This is indeed a consistent rescaling which leads to finite contributions in the limit m → 0,
1-&tidle;⊥ &tidle;⊥μ &tidle;⊥μ 1- 2 ℒm →0 = − 2A μA + (∂μA )χ − 2 (∂ χ) , (8.4 )
which clearly propagates no degrees of freedom.

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 that they are inaccessible from another sector.

Before looking at the massive gravity limit of bi-gravity and other decoupling limits of massive and bi-gravity, let us start by describing the different scaling limits that can be taken. We start with a bi-gravity theory where the two spin-2 fields have respective Planck scales Mg and Mf and the interactions between the two metrics arises at the scale m. In order to stick to the relevant points we perform the analysis in four dimensions, but the following arguments extend trivially to arbitrary dimensions.

  • Non-interacting 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 Mg,f remain fixed as m → 0, we then recover two massless non-interacting spin-2 fields (carrying both 2 helicity-2 modes), in addition to a decoupled sector containing a helicity-0 mode and a helicity-1 mode. In bi-gravity matter fields couple only to one metric, and this remains the case in the limit m → 0, so that the two massless spin-2 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 spin-2 fields (say f μν) decouples. This can be studied by sending its respective Planck scale to infinity. The resulting limit corresponds to a massive spin-2 field (carrying five dofs) and a decoupled massless spin-2 field carrying 2 dofs. This is nothing other than the massive gravity limit of bi-gravity (which includes a fully decoupled massless sector).

    If one considers matter coupling to the metric fμν which scales in such a way that a non-trivial solution for fμν survives in the Mf → ∞ limit f μν → f¯μν, we then obtain a massive gravity sector on an arbitrary non-dynamical reference metric ¯fμν. The dynamics of the massless spin-2 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 Mf ,Mg → ∞ compared to how fast m → 0. For instance if one takes the limit Mf ,Mg → ∞ and m → 0, while keeping both Mg ∕Mf and Λ3 = Mgm2 3 fixed, then we obtain what is called the Λ3-decoupling limit of bi-gravity (derived in Section 8.4), where the dynamics of the two helicity-2 modes (which are both massless in that limit), and that of the helicity-1 and -0 modes can be followed without keeping track of the standard non-linearities of GR.

    If on top of this Λ3-decoupling limit one further takes Mf → ∞, then one of the massless spin-2 fields fully decoupled (no communication between that field and the helicity-1 and -0 modes). If, on the other hand, we take the additional limit m → 0 on top of the Λ3-decoupling limit, then the helicity-0 and -1 modes fully decouple from both helicity-2 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 bi-gravity theory itself.

Similar decoupling limit could be taken in multi-gravity and out of N interacting spin-2 fields, we could obtain for instance N decoupled massless spin-2 fields and 3(N − 1) decoupled dofs in the helicity-0 and -1 modes.

In what follows we focus on massive gravity limit of bi-gravity when Mf → ∞.

8.2 Massive gravity as a decoupling limit of bi-gravity

8.2.1 Minkowski reference metric

In the following two sections we review the decoupling arguments given previously in the literature, (see for instance [154*]). We start with the theory of bi-gravity presented in Section 5.4 with the action (5.43*)

M 2g√ --- M 2f∘ ---- 1 √ --- ℒbi−gravity = ---- − gR [g] + ---- − f R [f ] +-m2M 2Pl − gℒm (g,f) 2√ --- 2 ∘ ----4(matter) + − gℒ(gmatter)(gμν,ψg ) + − f ℒf (fμν,ψf ), (8.5 )
with ∑4 ℒm (g,f ) = n=0 αnℒn [𝒦 (g,f )] as defined in (6.3*) and where μ μ √ -μα---- 𝒦 ν = δν − g f αν. We also allow for the coupling to matter with different species ψg,f living on each metrics.

We now consider matter fields ψf such that fμν = ημν is a solution to the equations of motion (so for instance there is no overall cosmological constant living on the metric fμν). In that case we can write that metric fμν as

1 fμν = ημν + ----χμν, (8.6 ) Mf
We may now take the limit Mf → ∞, while keeping the scales Mg and m and all the fields χ, g,ψ f,g fixed. We then recover massive gravity plus a completely decoupled massless spin-2 field χ μν, and a fully decoupled matter sector ψf living on flat space
M →∞ ℒbi−gravity −−f−−→ ℒMG (g,η) + √−-gℒ (matter)(gμν,ψg) (8.7 ) g + 1χμνℰˆαβχ + ℒ (matter)(η ,ψ ), 2 μν αβ f μν f
with the massive gravity Lagrangian ℒMG is expressed in (6.3*). That massive gravity Lagrangian remains fully non-linear in this limit and is expressed in terms of the full metric gμν and the reference metric ημν. While the metric fμν is ‘frozen’ in this limit, we emphasize however that the massless spin-2 field χ μν is itself not frozen – its dynamics is captured through the kinetic term μν ˆαβ χ ℰμν χ αβ, but that spin-2 field decouple from its own matter sector ψf, (although this can be accommodated for by scaling the matter fields ψf accordingly in the limit Mf → ∞ so as to maintain some interactions).

At the level of the equations of motion, in the limit Mf → ∞ we obtain the massive gravity modified Einstein equation for gμν, the free massless linearized Einstein equation for χμν which fully decouples and the equation of motion for all the matter fields ψf on flat spacetime, (see also Ref. [44]).

8.2.2 (A)dS reference metric

To consider massive gravity with an (A)dS reference metric as a limit of bi-gravity, we include a cosmological constant for the metric f into (8.5*)

∫ 2 4 ∘ ---- ℒCC,f = − M f d x − fΛf . (8.8 )
There can also be in principle another cosmological constant living on top of the metric g μν but this can be included into the potential 𝒰 (g,f). The background field equations of motion are then given by
( ) 2 m2M P2l δ √ --- 2 M fG μν[f ] +--√----- ---μν − g𝒰 (g, f) = T μν(ψf) − M fΛf fμν (8.9 ) 4 − f (δf ) 2 m2M--P2l -δ--√ --- M PlG μν[g] + 4√ −-g δgμν − g𝒰 (g, f) = T μν(ψg). (8.10 )
Taking now the limit Mf → ∞ while keeping the cosmological constant Λf fixed, the background solution for the metric fμν is nothing other than dS (or AdS depending on the sign of Λf). So we can now express the metric fμν as
-1-- fμν = γμν + Mf χμν, (8.11 )
where γμν is the dS metric with Hubble parameter ∘ ----- H = Λf ∕3. Taking the limit Mf → ∞, we recover massive gravity on (A)dS plus a completely decoupled massless spin-2 field χμν,
∫ 2 4 ∘ ---- Mf →∞ M--2Pl√ --- m2- ℒbi−gravity − M f d x − f Λf −−−−→ 2 − gR + 4 𝒰 (g,γ) (8.12 ) 1 + -χ μν ˆℰαμβν χ αβ, 2
where once again the scales MPl and m are kept fixed in the limit Mf → ∞. γμν now plays the role of a non-trivial reference metric for massive gravity. This corresponds to a theory of massive gravity on a more general reference metric as presented in [296]. Here again the Lagrangian for massive gravity is given in (6.3*) with now 𝒦 μ(g) = δμ − √g-μαγαν- ν ν. The massive gravity action remains fully non-linear in the limit M → ∞ f and is expressed solely in terms of the full metric g μν and the reference metric γ μν, while the excitations χ μν for the massless graviton remain dynamical but fully decouple from the massive sector.

8.2.3 Arbitrary reference metric

As is already clear from the previous discussion, to recover massive gravity on a non-trivial reference metric as a limit of bi-gravity, 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 f this is in principle always possible. For instance one can consider a point source of mass MBH living on the metric f. Then, taking the limit Mf ,MBH → ∞ while keeping the ratio MBH ∕Mf 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 f is included.

As soon as the dynamics of the matter field is considered, one has to send the scale of that field to infinity so that it maintains some nonzero effect on f in the limit Mf → ∞, i.e.,

√ ---- (matter) -1-- μν ----1----δ--−-fℒ-f----- Mlfim→ ∞ M 2 T = Mlfim→∞ √ − fM 2 δfμν → finite. (8.13 ) f f
Nevertheless, this can be achieved in such a way that the fluctuations of the matter fields remain finite and decouple in the limit Mf → ∞. We note that this scaling is the key difference between the decoupling limit of bi-gravity on a Minkowski reference metric derived in section 8.2.1 where the matter field scale as -1- μν limMf →∞ M2f T → 0 and the decoupling limit of bi-gravity on an arbitrary reference metric derived here.

As an example, suppose that the Lagrangian for the matter (for example a scalar field) sourcing the f metric is

∘ ----( ( )) ℒ (matter)= − f − 1-fμν∂ χ ∂ χ − V F χ- (8.14 ) f 2 μ ν 0 λ
where F(X ) is an arbitrary dimensionless function of its argument. Then choosing χ to take the form
χ = Mf ¯χ + δχ, (8.15 )
and rescaling V0 = M 2fV¯0 and λ = Mf ¯λ, then on taking the limit Mf → ∞ keeping χ¯, δχ, λ¯ and ¯V 0 fixed, since
( ) (matter) 2∘ ---- 1-μν ¯ (χ¯) ℒf → M f − f − 2f ∂ μ¯χ∂νχ¯− V0F λ¯ + fluctuations, (8.16 )
we find that the background stress energy blows up in such a way that 1 μν M2fT remains finite and nontrivial, and in addition the background equations of motion for ¯χ remain well-defined and nontrivial in this limit,
¯ ( ) □ ¯χ = V0-F ′ ¯χ- . (8.17 ) f ¯λ ¯λ
This implies that even in the limit M → ∞ f, f μν can remain consistently as a nontrivial sourced metric which is a solution of some dynamical equations sourced by matter. In addition the action for the fluctuations δχ asymptotes to a free theory which is coupled only to the fluctuations of fμν which are themselves completely decoupled from the fluctuations of the metric g and matter fields coupled to g.

As a result, massive gravity with an arbitrary reference metric can be seen as a consistent limit of bi-gravity in which the additional degrees of freedom in the f metric and matter that sources the background decouple. Thus all solutions of massive gravity may be seen as Mf → ∞ decoupling limits of solutions of bi-gravity. 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 bi-gravity, we could of course do the same for any multi-metric theory. For instance, out of N-interacting 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 spin-2 field.

8.3 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 Fierz–Pauli at the linear level, the first irrelevant interactions typically arise at the scale Λ5 = (m4MPl )1∕5. For the setups we have in mind, m ≪ Λ ≪ M 5 Pl. But we shall see that interactions arising at such a low-energy scale are always pathological (reminiscent to the BD ghost [111*, 173*]), and in ghost-free massive gravity the first (irrelevant) interactions actually arise at the scale Λ3 = (m3MPl )1∕3.

We start by deriving the decoupling limit in the absence of vectors (helicity-1 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 = α = 0 0 1 in (6.3*).

8.3.1 Interaction scales

In GR, the interactions of the helicity-2 mode arise at the very high energy scale, namely the Planck scale. In massive gravity a new scale enters and we expect some interactions to arise at a lower energy scale given by a geometric combination of the Planck scale and the graviton mass. The potential term 2 2√ --- M Plm − g ℒn[𝒦 [g,η ]] (6.3*) includes generic interactions between the canonically normalized helicity-0 (π), helicity-1 (Aμ), and helicity-2 modes (hμν) introduced in (2.48*)

( h )j ( ∂A )2k ( ∂2π ) ℓ ℒj,k,ℓ = m2M 2Pl ---- ------ --2---- MPl mMPl m MPl = Λ− 4+(j+4k+3ℓ)hj (∂A )2k (∂2π)ℓ, (8.18 ) j,k,ℓ
at the scale
( ) 2k+2ℓ− 2 j+2k+ℓ−2 1∕(j+4k+3ℓ−4) Λj,k,ℓ = m M Pl , (8.19 )
and with j,k,ℓ ∈ ℕ, and j + 2k + ℓ > 2.

Clearly ,the lowest interaction scale is Λj=0,k=0,ℓ=3 ≡ Λ5 = (MPlm4 )1∕5 which arises for an operator of the form (∂2π )3. If present such an interaction leads to an Ostrogradsky instability which is another manifestation of the BD ghost as identified in [173*].

Even if that very interaction is absent there is actually an infinite set of dangerous interactions of the form (∂2 π)ℓ which arise at the scale Λj=0,k=0,ℓ≥3, with

4 1∕5 2 1∕3 Λ5 = (MPlm ) ≤ (Λj=0,k=0,ℓ≥3) < Λ3 = (MPlm ) . (8.20 )
with Λj=0,k=0,ℓ→ ∞ = Λ3.

Any interaction with j > 0 or k > 0 automatically leads to a larger scale, so all the interactions arising at a scale between Λ5 (inclusive) and Λ3 are of the form 2 ℓ (∂ π ) and carry an Ostrogradsky instability. For DGP we have already seen that there is no interactions at a scale below Λ3. In what follows we show that same remains true for the ghost-free 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 (∂2π ).

8.3.2 Operators below the scale Λ3

We now express the potential term √ --- M 2Plm2 − g ℒn[𝒦 ] introduced in (6.3*) using the metric in term of the helicity-0 mode, where we recall that the quantity 𝒦 is defined in (6.7*), as ∘ ----- μ &tidle; μ ( −1 &tidle;)μ 𝒦 ν[g, f] = δν − g f ν, where &tidle;f is the ‘Stückelbergized’ reference metric given in (2.78*). Since we are interested in interactions without the helicity-2 and -1 modes (j = k = 0), it is sufficient to follow the behaviour of the helicity-0 mode and so we have

| ) &tidle; | --2--- --1--- 2 } μ fμν|h=A|=0 = η μν − MPlm2Π μν + M2Plm4Π μν =⇒ 𝒦μ | = --Π-ν--, (8.21 ) gμν|| = η μν ) ν h=A=0 MPlm2 h=0
with again Π μν = ∂μ∂νπ and Π2 := η αβΠμαΠ νβ μν.

As a result, we infer that up to the scale Λ3 (excluded), the potential in (6.3*) is

4 | m2M--2Pl√ ---∑ &tidle; | ℒmass = 4 − g αnℒn [𝒦 [g,f]]|h=A=0 (8.22 ) n=2 m2M 2 ∑4 [ Π μν ] = -----Pl αnℒn -----2- (8.23 ) 4 n=2 MPlm 1 μναβ ( α2 μ′ν′ α3 μ′ ν′ α4 μ′ ν′) α′ β′ = -𝜖 𝜖μ′ν′α′β′ -2-δν δν + ------4δν Π ν + --2--6Π ν Πν Π α Πβ , 4 m MPlm M Plm
where as mentioned earlier we focus on the case without a cosmological constant and tadpole i.e., α0 = α1 = 0. All of these interactions are total derivatives. So even though the ghost-free theory of massive gravity does in principle involve some interactions with higher derivatives of the form (∂2 π)ℓ it does so in a very precise way so that all of these terms combine so as to give a total derivative and being harmless.22

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 (∂2π)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 Λ3 or higher.

8.3.3 Λ3-decoupling limit

Since there are no operators all the way up to the scale Λ3 (excluded), we can take the decoupling limit by sending MPl → ∞, m → 0 and maintaining the scale Λ3 fixed.

The operators that arise at the scale Λ3 are the ones of the form (8.18*) with either j = 1,k = 0 and arbitrary ℓ ≥ 2 or with j = 0,k = 1 and arbitrary ℓ ≥ 1. The second case scenario leads to vector interactions of the form 2 2 ℓ (∂A ) (∂ π) and will be studied in the next Section 8.3.4. For now we focus on the first kind of interactions of the form h(∂2π )ℓ,

ℒdmeacss = hμνX¯μν, (8.24 )
with [144] (see also refs. [137*] and [143])
δ | ¯X μν = ----ℒmass|| (8.25 ) δhμν h=A=0 2 2 ( √ ---∑4 ) | = M-Plm----δ-- − g αnℒn [𝒦 [g, &tidle;f]] || . 4 δhμν n=2 h=A=0
Using the fact that
| δ𝒦n | n ( ) ----|| = -- Πnμ−ν1− Πnμν , (8.26 ) δhμν|h=A=0 2
we obtain
3 4 ( ) ¯ Λ-3∑ 4 −-n- (n) ---n--- (n−1) X μν = 8 αn Λ33n X μν [Π ] + Λ3 (n− 1)X μν [Π] , (8.27 ) n=2 3
where the tensors X (μnν) are constructed out of Πμν, symbolically, X (n) ∼ Π (n) but in such a way that they are transverse and that their resulting equations of motion never involve more than two derivatives on each fields,
(0)μ μναβ X μ′[Q ] = 𝜀 𝜀μ′ναβ (8.28 ) X (1)μ′[Q ] = 𝜀μναβ𝜀 ′′ Qν′ (8.29 ) μ μν αβ ν ′ ′ X (2)μμ′[Q ] = 𝜀μναβ𝜀μ′ν′α′β Q νν Q αα (8.30 ) (3)μ μναβ ν′ α ′ β′ X μ′[Q ] = 𝜀 𝜀μ′ν′α′β′ Qν Qα Q β (8.31 ) X (n≥4 )μ [Q ] = 0, (8.32 ) μ′
where we have included X (0) and X (n≥4) for completeness (these become relevant for instance in the context of bi-gravity). The generalization of these tensors to arbitrary dimensions is straightforward and in d-spacetime dimensions there are d such tensors, symbolically (n) n d− n−1 X = 𝜀𝜀Π δ for n = 0,⋅⋅⋅,d − 1.

Since we are dealing with the decoupling limit with MPl → ∞ the metric is flat −1 g μν = η μν + M Pl hμν → ημν and all indices are raised and lowered with respect to the Minkowski metric. These tensors (n) X μν can be written more explicitly as follows

X (0)[Q ] = 3!ημν (8.33 ) μν X (μ1ν) [Q ] = 2!([Q ]ημν − Q μν) (8.34 ) (2) 2 2 2 X μν [Q ] = ([Q ] − [Q ])ημν − 2([Q ]Qμν − Q μν) (8.35 ) X (μ3ν) [Q ] = ([Q ]3 − 3[Q][Q2] + 2[Q3])ημν (8.36 ) ( 2 2 2 3 ) − 3 [Q ] Qμν − 2[Q ]Q μν − [Q ]Q μν + 2Q μν .
Note that they also satisfy the recursive relation
(n) 1 ( α β αβ ) (n−1) X μν = 4-−-n- − nΠ μδν + Π ημν X αβ , (8.37 )
with (0) X μν = 3!ημν.

Decoupling limit

From the expression of these tensors Xμν in terms of the fully antisymmetric Levi-Cevita tensors, it is clear that the tensors (n) X μν are transverse and that the equations of motion of μν ¯ h X μν 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 ghost-lie 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 (∂2π )ℓ which could have arisen at an energy between Λ5 = (MPlm4 )1∕5 and Λ3 , but it also ensures that the interactions that arise at the scale Λ 3 are healthy.

As already mentioned, in the decoupling limit MPl → ∞ the metric reduces to Minkowski and the standard Einstein–Hilbert term simply reduces to its linearized version. As a result, neglecting the vectors for now the full Λ3-decoupling limit of ghost-free massive gravity is given by

1 1 ( 2α2 + 3α3 α3 + 4α4 ) ℒ Λ3 = − -h μν ˆℰαμβν hαβ +-hμν 2α2X μ(1ν)+ -----3----X (2μ)ν + ----6----Xμ(3ν) (8.38 ) 4 8 Λ 3 Λ3 1 ∑3 an = − -h μν ˆℰαμβν hαβ + h μν -3(n−1)X (nμ)ν , 4 n=1 Λ3
with a1 = α2 ∕4, a2 = (2α2 + 3α3)∕8 and a3 = (α3 + 4α4)∕8 and the correct normalization should be α2 = 1.

Unmixing and Galileons

As was already the case at the linearized level for the Fierz–Pauli theory (see Eqs. (2.47*) and (2.48*)) the kinetic term for the helicity-0 mode appears mixed with the helicity-2 mode. It is thus convenient to diagonalize these two modes by performing the following shift,

2α2 + 3α3 hμν = &tidle;h μν + α2 πημν −------3---∂μπ ∂νπ, (8.39 ) 2Λ 3
where the non-linear term has been included to unmix the coupling hμνX (μ2ν), leading to the following decoupling limit [137]
[ ∑ 5 ] ℒ = − 1-h&tidle;μν ˆℰαβ&tidle;h + ---cn--ℒ (n) [π ] − 2-(α3-+-4α4)&tidle;hμνX (3) , (8.40 ) Λ3 4 μν αβ Λ3 (n− 2) (Gal) Λ63 μν n=2 3
where we introduced the Galileon Lagrangians (n) ℒ(Gal)[π] as defined in Ref. [412*]
(n) ---1---- 2 ℒ(Gal)[π] = (6 − n)!(∂π) ℒn −2[Π ] (8.41 ) 2 = − ---------πℒn −1[Π ], (8.42 ) n(5 − n)!
where the Lagrangians ℒn [Q ] = 𝜀𝜀Qn δ4−n for a tensor Qμ ν are defined in (6.9*) – (6.13*), or more explicitly in (6.14*) – (6.18*), leading to the explicit form for the Galileon Lagrangians
ℒ (2) [π] = (∂ π)2 (8.43 ) (Gal) ℒ (3) [π] = (∂ π)2[Π ] (8.44 ) (Gal) ( ) ℒ ((4)Gal)[π] = (∂ π)2 [Π ]2 − [Π2 ] (8.45 ) (5) 2( 3 2 3) ℒ (Gal)[π] = (∂ π) [Π ] − 3[Π][Π ] + 2 [Π ] , (8.46 )
and the coefficients cn are given in terms of the αn as follows,
c2 = 3 α22, c3 = 32α2(2α2 + 3α3), 1 2 2 5 (8.47 ) c4 = 4(4α2 + 9α3 + 16α2 (α3 + α4)), c5 = 8(2α2 + 3α3 )(α3 + 4 α4).
Setting α2 = 1, we indeed recover the same normalization of − 3∕4(∂π )2 for the helicity-0 mode found in (2.48*).

X (3)-coupling

In general, the last coupling &tidle;h μνXμ(3ν) between the helicity-2 and helicity-0 mode cannot be removed by a local field redefinition. The non-local field redefinition

&tidle; &tidle; massless (3)αβ hμν → hμν + G μναβ X , (8.48 )
where Gmassless μναβ is the propagator for a massless spin-2 field as defined in (2.64*), fully diagonalizes the helicity-0 and -2 mode at the price of introducing non-local interactions for π.

Note however that these non-local interactions do not hide any new degrees of freedom. Furthermore, about some specific backgrounds, the field redefinition is local. Indeed focusing on static and spherically symmetric configurations if we consider π = π0(r) and &tidle;h μν given by

&tidle; μ ν 2 2 h μν dx dx = − ψ (r)dt + ϕ(r)dr , (8.49 )
so that
&tidle;hμνX (3)= − ψ ′(r)π ′(r)3. (8.50 ) μν 0
The standard kinetic term for ψ sets ψ′(r) = ϕ(r)∕r as in GR and the X (3) coupling can be absorbed via the field redefinition, ϕ → ¯ϕ − 2(α3 + 4α4 )π′(r)3∕r Λ−6 0 3, leading to the following new sextic interactions for π,
1 &tidle;hμνX (μ3ν)→ − -2π0′(r)6, (8.51 ) r
interestingly this new order-6 term satisfy all the relations of a Galileon interaction but cannot be expressed covariantly in a local way. See [61*] for more details on spherically symmetric configurations with the X (3)-coupling.

8.3.4 Vector interactions in the Λ3-decoupling limit

As can be seen from the relation (8.19*), the scale associated with interactions mixing two helicity-1 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 helicity-1 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.

In addition to the Stückelberg fields associated with local covariance, in the vielbein formulation one also needs to introduce 6 additional Stückelberg fields ωab associated to local Lorentz invariance, ωab = − ωba. These are non-dynamical since they never appear with derivatives, and can thus be treated as auxiliary fields which can be integrated. It is however useful to keep them in the decoupling limit action, so as to retain a closes-form expression. In terms of the Lorentz Stückelberg fields, the full decoupling limit of massive gravity in four dimensions at the scale Λ3 is then (before diagonalization) [419*]

1 1 ∑3 a ℒ (0Λ) = − -h μν ˆℰαμβν hαβ +-hμν ----n--X (μnν) (8.52 ) 3 4 2 n=1Λ33(n− 1) 3β ( 1 ) + --1δaαbβcγδd δaα δbβFcγωdδ + 2[ωbβ ωcγ + -δbβωcμω μγ ](δ + Π )dδ 8 2 β2 αβγδ a( b c d b c b c μ d) + --δabcd (δ + Π )α 2 δβF γω δ + [ω βω γ + δβω μω γ](δ + Π )δ 8 ( ) + β3δαabβcγδd (δ + Π )a(δ + Π )b 3F cγωdδ + ωcμ ωμγ(δ + Π )dδ , 48 α β
(the superscript (0) indicates that this decoupling limit is taken with Minkowski as a reference metric), with F = ∂ A − ∂ A ab a b b a and the coefficients β n are related to the α n as in (6.28*).

The auxiliary Lorentz Stückelberg fields carries all the non-linear mixing between the helicity-0 and -1 modes,

∫ ∞ a′ b′ ωab = due −2ue−uΠa Fa′b′e−uΠb (8.53 ) 0 ∑ --(n-+-m-)!- n+m n m = 21+n+mn!m! (− 1) (Π F Π )ab. (8.54 ) n,m
In some special cases these sets of interactions can be resummed exactly, as was first performed in [139*], (see also Refs. [364*, 456*]).

This decoupling limit includes non-linear combinations of the second-derivative tensor Π μν and the first derivative Maxwell tensor Fμν. Nevertheless, the structure of the interactions is gauge invariant for A μ, and there are no higher derivatives on A in the equation of motion for A, so the equations of motions for both the helicity-1 and -2 modes are manifestly second order and propagating the correct degrees of freedom. The situation is more subtle for the helicity-0 mode. Taking the equation of motion for that field would lead to higher derivatives on π itself as well as on the helicity-1 field. Since this theory has been proven to be ghost-free 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 helicity-1 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 helicity-1 mode is no longer manifestly well-behaved, 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.

8.3.5 Beyond the decoupling limit

Physical degrees of freedom

In Section 8.3, we have introduced four Stückelberg fields a ϕ which transform as scalar fields under coordinate transformation, so that the action of massive gravity is invariant under coordinate transformations. Furthermore, the action is also invariant under global Lorentz transformations in the field space,

μ μ a a b x → x , gμν → gμν, and ϕ → &tidle;Λbϕ . (8.55 )
In the DL, taking MPl → ∞, all fields are living on flat space-time, so in that limit, there is an additional global Lorentz symmetry acting this time on the space-time,
xμ → ¯Λ μxν, h → ¯Λ α¯Λ βh , and ϕa → ϕa. (8.56 ) ν μν μ ν αβ
The internal and space-time Lorentz symmetries are independent, (the internal one is always present while the space-time one is only there in the DL). In the DL we can identify both groups and work in the representation of the single group, so that the action is invariant under,
μ μ ν α β a a b x → Λν x , h μν → Λ μΛνh αβ, and ϕ → Λbϕ . (8.57 )
The Stückelberg fields ϕa then behave as Lorentz vectors under this identified group, and π defined previously behaves as a Lorentz scalar. The helicity-0 mode of the graviton also behaves as a scalar in this limit, and π captures the behavior of the graviton helicity-0 mode. So in the DL limit, the right requirement for the absence of BD ghost is indeed the requirement that the equations of motion for π remain at most second order (time) in derivative as was pointed out in [173*], (see also [111*]). However, beyond the DL, the helicity-0 mode of the graviton does not behave as a scalar field and neither does the π in the split of the Stückelberg fields. So beyond the DL there is no reason to anticipate that π captures a whole degree of freedom, and it indeed, it does not. Beyond the DL, the equation of motion for π will typically involve higher derivatives, but the correct requirement for the absence of ghost is different, as explained in Section 7.2. One should instead go back to the original four scalar Stückelberg fields a ϕ and check that out of these four fields only three of them be dynamical. This has been shown to be the case in Section 7.2. These three degrees of freedom, together with the two standard graviton polarizations then gives the correct five degrees of freedom and circumvent the BD ghost.

Recently, much progress has been made in deriving the decoupling limit about arbitrary backgrounds, see Ref. [369].

8.3.6 Decoupling limit on (Anti) de Sitter

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 ghost-free 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.

At the linearized level about dS, ghost-free massive gravity reduces to the Fierz–Pauli action with &tidle; gμν = γμν + hμν = γ μν + h μν∕MPl, where γ μν is the dS metric with constant Hubble parameter H0,

2 ℒ (2) = − 1h μν(ˆℰdS)α βhαβ − m--γμνγ αβ (H μαH νβ − HμνH αβ) , (8.58 ) MG,dS 4 μν 8
where H μν is the tensor fluctuation as introduced in (2.80*), although now considered about the dS metric,
∇ (μAν) Π μν H μν = hμν + 2------- + 2---2 (8.59 ) [ m m ][ ] − -1-- ∇μA-α-+ Πμα- ∇-νAβ-+ Π-νβ γαβ, MPl m m2 m m2
with Π μν = ∇ μ∇ νπ, ∇ being the covariant derivative with respect to the dS metric γμν and indices are raised and lowered with respect to this same metric. Similarly, ˆℰdS is now the Lichnerowicz operator on de Sitter,
[ (ˆℰdS)αμβνhαβ = − 1-□h μν − 2∇ (μ∇ αhαν) + ∇ μ∇ νh (8.60 ) 2 ( ) αβ 2 1- ] − γμν(□h − ∇ α∇ βh ) + 6H 0 hμν − 2h γμν .
So at the linearized level and neglecting the vector fields, the helicity-0 and -2 mode of massive gravity on dS behave as
(2) 1 m2 ( ) 1 ℒMG,dS = − --hμν(ˆℰdS)αμβνhαβ − --- h2μν − h2 − -F 2μν (8.61 ) 4 8 ( 8 ) − 1-hμν (Π μν − [Π ]γμν) −-1-- [Π2] − [Π ]2 . 2 2m2
After integration by parts, [Π2 ] = [Π ]2 − 3H2 (∂π )2. The helicity-2 and -0 modes are thus diagonalized as in flat space-time by setting h = ¯h + πγ μν μν μν,
(2) 1 μν αβ m2 ( 2 2) 1 2 ℒ MG,dS = − -¯h (ℰˆdS)μν¯h αβ − --- ¯hμν − ¯h − --Fμν (8.62 ) 4( ( )2) 8 8 3- H-- ( 2 2¯ 2 2) − 4 1 − 2 m (∂ π) − m hπ − 2m π .

The most important difference from linearized massive gravity on Minkowski is that the properly canonically normalized helicity-0 mode is now instead

∘ -------2- ϕ = 1 − 2H-- π. (8.63 ) m2
For a standard coupling of the form -1-πT MPl, where T is the trace of the stress-energy tensor, as we would infer from the coupling 1 μν MPlh μνT after the shift hμν = ¯h μν + π γμν, this means that the properly normalized helicity-0 mode couples as
matter -------m2------- ℒ helicity−0 = M √m2--−--2H2-ϕT , (8.64 ) Pl
and that coupling vanishes in the massless limit. This might suggest that in the massless limit m → 0, the helicity-0 mode decouples, which would imply the absence of the standard vDVZ discontinuity on (Anti) de Sitter [358, 430], unlike what was found on Minkowski, see Section 2.2.3, which confirms the Newtonian approximation presented in [186].

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 helicity-0 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 m2 > 2H2, which corresponds to the well-know Higuchi bound [307*, 190*]. In d spacetime dimensions, the Higuchi bound is m2 > (d − 2)H2. In other words, on dS there is a forbidden range for the graviton mass, a theory with 2 2 0 < m < 2H or with 2 m < 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 2 2 0 < m < 2H, the ghost is the helicity-0 mode, while for 2 m < 0, the ghost is he helicity-1 mode (at quadratic order the helicity-1 mode comes in as m2 2 − -4-Fμν). Furthermore, when 2 m < 0, both the helicity-2 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 2 2 m = 2H (or 2 2 m = (d − 2 )H in d spacetime dimensions), represents the partially massless case where the helicity-0 mode disappears. As we shall see in Section 9.3, this is nothing other than a linear artefact and non-linearly the helicity-0 mode always reappears, so the PM case is infinitely strongly coupled and always pathological.

A summary of the different bounds is provided below as well as in Figure 4*:

  • m2 < 0: Helicity-1 modes are ghost, helicity-2 and -0 are tachyonic, sick theory
  • 2 m = 0: General Relativity: two healthy (helicity-2) degrees of freedom, healthy theory,
  • 2 2 0 < m < 2H: One “Higuchi ghost” (helicity-0 mode) and four healthy degrees of freedom (helicity-2 and -1 modes), sick theory,
  • m2 = 2H2: Partially Massless Gravity: Four healthy degrees (helicity-2 and -1 modes), and one infinitely strongly coupled dof (helicity-0 mode), sick theory,
  • 2 2 m > 2H: Massive Gravity on dS: Five healthy degrees of freedom, healthy theory.
View Image
Figure 4: Degrees of freedom for massive gravity on a maximally symmetric reference metric. The only theoretically allowed regions are the upper left Green region and the line m = 0 corresponding to GR.

Massless and decoupling limit

  • As one can see from Figure 4*, in the case where H2 < 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 helicity-0 mode decouples from external matter sources and there is no vDVZ discontinuity. Notice however that the helicity-0 mode is nevertheless still strongly coupled at a low energy scale.

    When considering the decoupling limit m → 0, MPl → ∞ 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, MPl → ∞ 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.

Decoupling limit

When taking the decoupling limit of massive gravity on dS, there are two additional contributions to take into account:

  • First, as mentioned in Section 8.3.5, care needs to be applied to properly identify the helicity-0 mode on a curved background. In the case of (A)dS, the formalism was provided in Ref. [154*] by embedding a d-dimensional 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)
    -1--H μν = -1--hμν + -2-Πμν − -1-Π2 (8.65) MPl MPl Λ33 Λ63 μν 1 H2 ( 2 1 ) + -3--2 (∂π)2(γμν − --3Πμν) − --6Π μαΠνβ∂ απ∂β π Λ3 m Λ 3 Λ 3 2H2- (∂π)4- +H m2 Λ9 + ⋅⋅⋅. 3
    Any corrections in the third line vanish in the decoupling limit and can thus be ignored, but the corrections of order H2 in the second line lead to new non-trivial contributions.
  • Second, as already encountered at the linearized level, what were total derivatives in Minkowski (for instance the combination 2 2 [Π ] − [Π ]), now lead to new contributions on de Sitter. After integration by parts, m −2([Π2] − [Π]2) = m −2Rμν∂ μπ∂νπ = 12H2 ∕m2 (∂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 ℒ3,4(Π ), where ℒ3,4 are defined in (6.12*, 6.13*) or more explicitly in (6.17*, 6.18*).

Taking these two effects into account, we obtain the full decoupling limit for massive gravity on de Sitter,

5 (dS) (0) H2- ∑ --λn--- (n) ℒ Λ3 = ℒΛ3 + m2 Λ3(n−1)ℒ (Gal)[π], (8.66 ) n=2 3
where ℒ(Λ03) is the full Lagrangian obtained in the decoupling limit in Minkowski and given in (8.52*), and (n) ℒ (Gal) are the Galileon Lagrangians as encountered previously. Notice that while the ratio H ∕m remains fixed,this decoupling limit is taken with H, m → 0, so all the fields in (8.66*) live on a Minkowski metric. The constant coefficients λ n depend on the free parameters of the ghost-free theory of massive gravity, for the theory (6.3*) with α1 = 0 and α2 = 1, we have
3- 3- 1- -3- λ2 = 2, λ3 = 4 (1 + 2α3 ), λ4 = 4 (− 1 + 6 α4), λ5 = − 16 (α3 + 4α4). (8.67 )
At this point we may perform the same field redefinition (8.39*) as in flat space and obtain the following semi-diagonalized decoupling limit,
dS) 1 μν αβ α3 + 4α4 μν (3) ∑ 5 &tidle;cn (n) ℒΛ3 = − -h ˆℰμν hαβ +------9--h Xμν + --3(n−-2)ℒ (Gal)[π ] (8.68 ) 4 8Λ 3 n=2Λ 3 + Contributions from the helicity-1 modes,
where the contributions from the helicity-1 modes are the same as the ones provided in (8.52*), and the new coefficients &tidle;cn = − cn∕4 + H2 ∕m2 λn cancel identically for m2 = 2H2, α3 = − 1 and α4 = − α3 ∕4 = 1∕4, as pointed out in [154*], and the same result holds for bi-gravity as pointed out in [301*]. Interestingly, for these specific parameters, the helicity-0 loses its kinetic term, and any self-mixing as well as any mixing with the helicity-2 mode. Nevertheless, the mixing between the helicity-1 and -0 mode as presented in (8.52*) are still alive. There are no choices of parameters which would allow to remove the mixing with the helicity-1 mode and as a result, the helicity-0 mode generically reappears through that mixing. The loss of its kinetic term implies that the field is infinitely strongly coupled on a configuration with zero vev for the helicity-1 mode and is thus an ill-defined theory. This was confirmed in various independent studies, see Refs. [185*, 147*].

8.4 Λ3-decoupling limit of bi-gravity

We now proceed to derive the Λ3-decoupling limit of bi-gravity, 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 Λ3-decoupling of bi-gravity 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.

We start with the action (5.43*) for bi-gravity, with the interaction

∫ M 2Plm2 4 √--- ∑4 ℒg,f = ---4--- d x − g αn ℒn [𝒦 [g,f]] (8.69 ) ∫ n=0 M 2Plm2 [β0 a b c d β1 a b c d = − ------𝜀abcd --e ∧ e ∧ e ∧ e + --f ∧ e ∧ e ∧ e (8.70 ) 2 4! 3! ] + β2-f a ∧ fb ∧ ec ∧ ed + β3f a ∧ f b ∧ fc ∧ ed + β4fa ∧ fb ∧ fc ∧ f d , 2!2! 3! 4!
where the relation between the α’s and the β’s is given in (6.28*).

We now introduce Stückelberg fields a a a ϕ = x − χ for diffs and a Λ b for the local Lorentz. In the case of massive gravity, there was no ambiguity in how to perform this ‘Stückelbergization’ but in the case of bi-gravity, one can either ‘Stückelbergize the metric fμν or the metric gμν. In other words the broken diffs and local Lorentz symmetries can be restored by performing either one of the two replacements in (8.69*),

a a a b c fμ → f&tidle;μ = Λ bfc(ϕ(x))∂μϕ . (8.71 )
or alternatively
eaμ → &tidle;eaμ = Λabebc(ϕ (x))∂μϕc. (8.72 )
For now we stick to the first choice (8.71*) but keep in mind that this freedom has deep consequences for the theory, and is at the origin of the duality presented in Section 10.7.

Since we are interested in the decoupling limit, we now perform the following splits, (see Ref. [419] for more details),

a a --1-- a a a --1-- a eμ = ¯eμ + 2M hμ, fμ = ¯eμ + 2M vμ Pl f Λab = eˆωab = δab + ˆωab + 1-ˆωacˆωcb + ⋅⋅⋅ 2 a -ωab-- ˆω b = mMPl ( a a ) ∂μ ϕa = ∂μ xa + --A--- + ∂--π (8.73 ) mMPl Λ33
and perform the scaling or decoupling limit,
M → ∞, M → ∞, m → 0 (8.74 ) Pl f
while keeping
1 Λ3 = (m2MPl )3 → constant, MPl ∕Mf → constant, (8.75 ) and β → constant. n
Before performing any change of variables (any diagonalization), in addition to the kinetic term for quadratic h, v and A, there are three contributions to the decoupling limit of bi-gravity:

Mixing of the helicity-0 mode with the helicity-1 mode A μ, as derived in (8.52*),
Mixing of the helicity-0 mode with the helicity-2 mode a h μ, as derived in (8.40*),
Mixing of the helicity-0 mode with the new helicity-2 mode vaμ,

noticing that before field redefinitions, the helicity-0 mode do not self-interact (their self-interactions are constructed so as to be total derivatives).

As already explained in Section 8.3.6, the first contribution arising from the mixing between the helicity-0 and -1 modes is the same (in the decoupling limit) as what was obtained in Minkowski (and is independent of the coefficients βn or αn). 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 v, i.e., unaffected by the bi-gravity nature of the theory. This statement simply follows from scaling considerations. In the decoupling limit there cannot be any mixing between the helicity-1 and neither of the two helicity-2 modes. As a result, the helicity-1 modes only mix with themselves and the helicity-0 mode. Hence, in the scaling limit (8.74*, 8.75*) the helicity-1 decouples from the massless spin-2 field.

Furthermore, the first line of (8.52*) which corresponds to the dynamics of a hμ and the helicity-0 mode is also unaffected by the bi-gravity nature of the theory. Hence, the second contribution is the also the same as previously derived. As a result, the only new ingredient in bi-gravity is the mixing between the helicity-0 mode and the second helicity-2 mode va μ, given by a fixing of the form μν h Xμν.

Unsurprisingly, these new contributions have the same form as , with three distinctions: First the way the coefficients enter in the expressions get modified ever so slightly (β1 → β1∕3 and β3 → 3β3). Second, in the mass term the space-time index for va μ ought to dressed with the Stückelberg field,

a a b a b b 3 vμ → vb∂ μϕ = vb(δμ + Πμ∕Λ 3). (8.76 )
Finally, and most importantly, the helicity-2 field μ va (which enters in the mass term) is now a function of the ‘Stückelbergized’ coordinates ϕa, which in the decoupling limit means that for the mass term
a a μ μ 3 a v b = vb[x + ∂ π∕Λ 3] ≡ vb[&tidle;x]. (8.77 )
These two effects do not need to be taken into account for the v that enters in its standard curvature term as it is Lorentz and diff invariant.

Taking these three considerations into account, one obtains the decoupling limit for bi-gravity,

ℒ (bi−gravity) = ℒ (0)− 1vμν[x]ˆℰαβvαβ[x] (8.78 ) Λ3 Λ3 4 μν 1 M ( Πν ) ∑3 &tidle;β − ----Plvμβ[&tidle;x] δνβ + -β3- -3n(+n1−1)Xμ(nν)[Π], 2 Mf Λ3 n=0 Λ3
with &tidle; βn = βn∕(4 − n)!(n − 1)!. Modulo the non-trivial dependence on the coordinate 3 &tidle;x = x + ∂π ∕Λ3, this is a remarkable simple decoupling limit for bi-gravity. Out of this decoupling limit we can re-derive all the DL found previously very elegantly.

Notice as well the presence of a tadpole for v if β1 ⁄= 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 spin-2 field.

In the presence of a cosmological constant for both metrics (and thus a tadpole in this framework), we can also take the limit Mf → ∞ and recover straight away the decoupling limit of massive gravity on (A)dS, as obtained in (8.66*).

This illustrates the strength of this generic decoupling limit for bi-gravity (8.78*). In principle we could even go further and derive the decoupling limit of massive gravity on an arbitrary reference metric as performed in [224*]. To obtain a general reference metric we first need to add an external source for vμν that generates a background for V¯μν = Mf ∕MPlU¯μν. The reference metric is thus expressed in the local inertial frame as

-1--¯ --1--¯ ¯ αβ -1-- −2 fμν = ημν + Mf Vμν + 4M f2VμαVβνη + Mf vμν + 𝒪 (M f ) (8.79 ) = ημν + --1-U¯μν + -1-vμν + 𝒪 (MPl, Mf )−2. (8.80 ) MPl Mf
The fact that the metric f looks like a perturbation away from Minkowski is related to the fact that the curvature needs to scale as 2 m in the decoupling limit in order to avoid the issues previously mentioned in the discussion of Section 8.2.3.

We can then perform the scaling limit Mf → ∞, while keeping the β’s and the scale Λ = (M m2 )1∕3 3 Pl fixed as well as the field v μν and the fixed tensor ¯U μν. The decoupling limit is then simply given by

( ν) 3 ℒ(¯U)= ℒ(0)− 1¯U μβ[&tidle;x ] δν + Π-β ∑ --&tidle;βn+1-X (n)[Π ] (8.81 ) Λ3 Λ3 2 β Λ33 Λ3 (n− 1) μν n=0 3 − 1vμνℰˆαβv , 4 μν αβ
where the helicity-2 field v fully decouples from the rest of the massive gravity sector on the first line which carries the other helicity-2 field as well as the helicity-1 and -0 modes. Notice that the general metric ¯U has only an effect on the helicity-0 self-interactions, through the second term on the first line of (8.81*) (just as observed for the decoupling limit on AdS). These new interactions are ghost-free and look like Galileons for conformally flat ¯ U μν = λημν, with λ constant, but not in general. In particular, the interactions found in (8.81*) would not be the covariant Galileons found in [166, 161, 157*] (nor the ones found in [237*]) for a generic metric.
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