On the 6th Mode in Massive Gravity

Generic massive gravity models in the unitary gauge correspond to a self-gravitating medium with six degrees of freedom. It is widely believed that massive gravity models with six degrees of freedom have an unavoidable ghost-like instability; however, the corresponding medium has stable phonon-like excitations. The apparent contradiction is solved by the presence of a non-vanishing background pressure and energy density of the medium that opens up a stability window. The result is confirmed by looking at linear stability on an expanding Universe, recovering the flat space stability conditions in the small wavelength limit. Moreover, one can show that under rather mild conditions, no ghost-like instability is present for any wavelength. As a result, exploiting the medium interpretation, a generic massive gravity model with six degrees of freedom is perfectly viable.


Introduction
In recent years there has been a renewed interest in massive gravity, see for instance the reviews [1,2,3]. One the main issue of massive gravity is that a randomly picked model propagates six degrees of freedom (DoF) and among the two scalar modes one leads to ghost instabilities [3]. A lot of effort has been devoted in finding suitable models in which only five healthy DoF are present. At the non-perturbative level, when a residual Lorentz symmetry is imposed in the massive gravity action, only the dRGT model [4] has such a feature; if only only rotational symmetry is imposed,many choices are available [5,6,7]. Unfortunately in both cases the special choice of the ghost free action is not protected by symmetries. In the Lorentz invariant case there is a number of additional phenomenological difficulties related with the absence of a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological solutions, the presence of the vDVZ discontinuity [9,10,11] and serious difficulties with unitarity [8]. In general, the recent detection of gravitational waves [12,13] poses strong constraints on modified gravity theories relevant for dark energy, see for instance [14,15], disfavouring theories in which gravitational propagation velocity is modified.
In this paper we reconsider the faith of the sixth mode in the light of the interpretation of massive gravity as a self-gravitating medium [16,17,18]. The crucial point is that the energy density and pressure of the medium when present can open up a region of stability for massive gravity models with healthy six DoF. The very same mechanism guaranties that that two scalar phonon-like modes of perfect fluid are stable; the argument can be extended to a general medium. Technically we follow the following path • A generic massive gravity models with six DoF around Minkowski is intrinsically plagued by a scalar ghost.
• In the medium interpretation of the massive gravity models we find that in the absence of dynamical gravity, the phonon like excitations of such a models are stable.
• For consistency, the presence of the medium requires a non zero energy density ρ and pressure p, which, when gravity is switched on, leads to the departure from flat space as a consistent background.
• Analysing perturbations around a FLRW background in the short wave length limit, the same stability conditions for a medium on Minkowski spacetime are recovered.
This shows that massive gravity models with six DoF can be intrinsically stable. The outline of the paper is the following. In section 2 we study the stability of a generic medium in Minkowski space. In section 3 we relate massive gravity with a self-gravitating medium b exploiting the unitary gauge. Section 4 is devoted to the analysis of stability for a self-gravitating medium around a FLRW background. Our conclusions are given in section 6.

A Medium in Minkowski Space
Let us start from a scalar field theory of the form where U is the Lagrangian for four derivatively coupled scalar fields ϕ A , A = 0, 1, 2, 3 invariant under shift symmetry ϕ A → ϕ A + c A with ∂ µ c A = 0. As discussed in detail in [16,19,20,17,18] such an action can be used to give an effective description of the dynamics of self-gravitating media coupled with gravity. The low energy excitations (phonons) can be described by suitable fluctuations of the ϕ A , see appendix A for details. In this chapter we consider the perturbations π A around flat space It is easy to see that in order to have a EMT constant, the background value of the scalar fields is ϕ A = x A . The presence of the fluid breaks spontaneously the Lorentz group. Notice that settingφ A = 0 it would imply that there is no background fluid (ρ = p = 0). The Lagrangian U gives the following background EMT where ρ and p are constant that gives the energy density and the pressure evaluated on flat space by (A.9), measured by observers with 4-velocity u µ = (1, 0 ). The first thing to be noticed is that if one requires the background EMT to be Lorentz invariant 1 then we have to impose ρ + p = 0, i.e. T µν Mink = p η µν , i.e. the fluid has the same equation of state of a cosmological constant. From the media/fluid interpretation it is rather natural that the vacuum state breaks Lorentz by its presence, that is precisely what happen in any solid state lab. In what follows we will consider the case ρ + p = 0, while the exotic case ρ + p = 0 will be studied elsewhere. Decomposing the π A fields according to SO(3) vector and scalar perturbations: ϕ 0 = t + π 0 and ϕ i = , we get that for the scalar modes the quadratic Lagrangian is 2 where we have introduced the mass paratemeters M i defined in terms of the derivatives of the medium Lagrangian (2.1), their explicitly expression (A.11) is given in appendix A; such parameters are quite useful in order to classify the various media with respect of their mechanical and thermodynamical properties 3 . Moreover it is convenient to setM a = M 2 Pl M j , j = 0, 1, 2, 3, 4. The linear expansion of the medium action gives where σ is the background entropy per particle. By integration by parts, the linear action is zero upon the constancy in space and in time of σ. Thus no condition on p and ρ needs to be imposed in sharp contrast with the case where dynamical gravity is present, as we shall see later.
We consider the case of generic M i values, thus no tuning and/or additional symmetries are imposed. From the kinetic terms in (2.4) it is clear that there are two critical pointsM 1 + p + ρ = 0 andM 0 = 0 where at least one DoF can be integrated and the stability analysis has to be redone. As soon as we are away from such a points we can study the stability looking to the total energy, or equivalently the Hamiltonian (see appendix B). In the scalar sector the total energy is given by Imposing that the the energy is bounded from below in both the scalar and vector sectors leads to (see appendix B) Clearly, when ρ + p > 0, stability is possible, as it should be. Actually the conditions of dynamical and thermodynamical stability for a perfect fluid are the same [21]. It is interesting to note that the previous stability conditions do not hold in the Lorentz invariant case 4 . Indeed, requiring that that T µν M ink to be proportional to η µν implies that p + ρ = 0; thus only the barotropic equation of state w = −1 is viable and the stability windows closes down 5 .
Though Stückelberg fields are perfectly suitable for studying stability, they are not invariant under the shift symmetries of the action and their physical interpretation is not direct. It is convenient to introduce [21,18] as observables: the energy density perturbation δρ, the pressure perturbation δp and the entropy per particle perturbation δσ given by The equations of motion derived from (2.4) can be written as (2.13) The evolution equation (2.11) has been derived in the hypothesis that M 1 = 0. If M 1 = 0 we simply have δσ ′ = 0 . (2.14) It is clear from the equations of motion that the second scalar mode, beside δρ is the entropy per particle δσ. The stability condition that we have found imply that the speed of sound c 2 ρ of δρ and c 2 σ of δσ are both positive. Generically, the two scalar perturbations of a media are the energy density δρ and the entropy per particle δσ and their dynamics is stable.

Self-gravitating Medium and Massive Gravity
Let us now include dynamical gravity in our picture. The medium becomes self-gravitating by simply coupling minimally the scalar fields with gravity. Namely, one consider the following action where R is the Ricci scalar, as before, U , is the medium Lagrangian containing four derivatively coupled scalar fields minimally coupled with gravity (see appendix A for more details). Perturbations around flat space are studied by taking µν . In flat space, the expansion of the action (2.1) generates a linear term proportional to the EMT of the medium T µν Mink (2.3). At the quadratic level there is a contribution coming from both the Einstein-Hilbert Lagrangian, L (U ) g and the medium action, the later can be parametrised in the form [5,22,1,23] whose structure is the origin of the name massive gravity theories. Thus in the unitary gauge a selfgravitating medium is equivalent to a general massive gravity theory. The parameters λ i are given in terms of suitable first and second derivatives of the function U computed on Minkowski background and their expression can be found in (A.10) and are simply related to the parameters M i used to study the dynamics of a medium [17,18] by (3.5) To be as general as possible we imposed on the mass terms only rotational invariance. Such a choice is rather natural in the medium picture the spontaneous breaking of Lorentz symmetry originates simply by the presence of the medium. Strictly speaking only the presence of the term λ 2 2 corresponds to a mass term for the spin two component of the graviton, however we will generically refer to massive gravity when at least one of the λ 2 i is non-vanishing. Insisting on having a Lorentz symmetric background configuration constraints the value the λ a and the quadratic expansion 3.4 reduces to Of course in this case the corresponding medium is rather special; Lorentz invariance forces to have p = −ρ and thus T µν Mink = −ρ η µν . Before proceeding let us point out that a consistent study of gravity perturbations around flat space conflicts with the fluid/medium picture. Take for instance the unitary gauge; flat space is a solution of the background equations only if ρ = p = 0, as a result no fluid is present. Physically it simply means that when gravity matters, the consistent background is not flat: the presence of a non-vanishing p and ρ wants to warp the background. To study perturbations in a consistent way, keeping p and ρ non-vanishing, one has to look carefully at the scale involved.
1. If the curvature radius is large compared to the typical wavelength of the perturbations we are considering, the flat space picture is adequate and the fluctuations of the spacetime metric can be neglected.
2. The curvature radius is comparable with the typical wavelength of the perturbations, as a result the background solution has to amended and the metric fluctuations are important.
In case 1, at very large momentum and energy, curvature is negligible together with the mixing of the Stückelberg fields with the metric; much like in a spontaneously broken gauge theory where the ultraviolet behaviour is captured by πs alone. Thus, one can forget about gravity and simply study the effective quadratic Lagrangian for the πs obtained by expanding U and the stability analysis is the one given in the previous section. When case 2 applies we need to consider a consistent background, the natural one is an expanding FLRW Universe that will be discussed in the next section.
One may be tempted to simply forget about the restriction imposed by the background equations, however this is not an option and it leads to a contradiction. Indeed, as shown in appendix B, the quadratic Lagrangian in unitary gauge does not depend on p and ρ and for generic values of the masses two DoF are present and one is a ghost. On the other hand, employing the Newtonian gauge, the quadratic action depends on both p and ρ but this time, though the kinetic matrix for the relevant fields is not positive definite, three DoF seems to be present. As expected, to reconcile the two gauges one has that ρ = p = 0. The bottom line is that, a self-gravitating medium cannot be consistently studied in flat space unless there is no medium ! We stress again that the medium picture is not available when Lorentz invariance is imposed on the background EMT, then p + ρ = 0. In this case, the stability windows closes down, making clear, once again, the key role played by the background pressure and energy density of the medium which have to be different from zero to avoid instabilities in the presence of six DoF.

Stability in an Expanding Universe
Let now consider the case when the perturbation wavelength is comparable with the curvature scale; in this case the background pressure and energy density makes the Universe expand and we enter in the realm of cosmology. To study perturbations we consider a spatially flat FLRW perturbed solution in the conformal Newtonian gauge, namely while for the Stückelberg fields in the scalar sector we have The mass paramaters λ a and M a in a FLRW background the are similarly defined as in (3.4); it is convenient to introduce an overall factor a 4 6 in the quadratic Lagrangian corresponding to the square root of minus the determinant of the background metric; namely The expressions of M a are given in appendix A. The conservation of the background EMT is equivalent to the equation of motion forφ 0 = φ The quadratic Lagrangian for scalar perturbation in the Fourier basis can be unambiguously written as (see appendix B) where Q t = (Ψ, Φ, k 2 π l , k π 0 ) and ′ denotes the derivative with respect of the conformal time. In our case we have that (4.7) We denoted by H = a ′ /a the Hubble parameter in conformal time. The mass matrix M M M is not very illuminating. Differently from the analysis involving the Stückelberg fields in flat space, this time the effect of the background on the FLRW metric is consistently taken into account and The field Ψ has an algebraic equation of motion and can be integrated out getting; the resulting Lagrangian, after suitable integration by parts can be again put in the form (4.5). As a final step, a second field can be integrated out. Indeed, defining when the quadratic action is expressed in terms of q, π l and π 0 , it turns out that π 0 can be integrated out. The action has the very same general form (4.5) but with 2×2 matrices: The expression of M M M is not particularly illuminating and will not be given here. What is important is to find the condition for which the total energy is positive definite in the large k limit, namely when k 2 ≫ H 2 , to confirm the pure Stückelberg analysis in flat space. Proceeding as for the case of flat space, in the large k limit, we find the kinetic energy is positive if Finally, looking at vector modes it is easy to see that [18] stability requireŝ As a result, the stability conditions are the same of the ones derived with the previous Stückelberg analysis; in addition whenM 0 < ρ, ghost instability are forbidden at any k. The only instability found is a Jeans instability when, for k < k J , one of the eigenvalues of the mass matrix of the total energy becomes negative. Such gravitational instability is typical of any self-gravitating system. As a technical comment, we note that taking the limit H → 0, via the background equations, one is led again to p = ρ = 0 and we are back to well known problems discussed in the previous section.
Let us consider a number of concrete examples.
• Perfect fluid This is the simplest case one can immagine. Flat space stability actually coincide with thermodynamical stability [21].
which can be easily satisfied. The only exception is when the null energy condition is violated.
• Superfluid A superfluid is the simplest example of a medium with M 1 = 0. Being still a fluid, M 2 = 0. The Lagrangian has the general form U (b, Y, X), see appendix A and [18] for the definition of the relevant operators. Clearly the most stringent stability conditions are the ones involving M 1 that for a superfluid is given, together with the background value of X on FLRW, by Moreover for superfluids [18], see appendix A, we have that Then the stability condition involving M 1 reads simply remarkably there is a perfect cancellation of U X between M 1 and ρ + p rendering such condition easy to satisfy. Notice that a −1 φ ′ U Y − a −3 U b is just p + ρ for the normal component of the superfluid, see appendix A.
• Supersolid Take a supersolid (not the most general one) described by the Lagrangian U (b, Y, X, τ n ) with n = 1, 2, 3. The presence of τ n turns on M 2 , however the cancellation of U X in the conditions involving M 1 still takes place and we have that which again is not very restrictive. The condition is just ρ + p for a finite temperature solid; thus, as for superfluids, the effect of M 1 is to cancel the contribution from the superfluid component, see appendix A.
• Most general supersolid Consider now the case of the moste general solid whose Lagrangian U contains all ten operators rotational invariant operators, see appendix A. While terms of the form U ym , m = 0, 1, 2, 3, contribute to M 1 , there is no contribution in p + ρ being of the form y m U ym and y m = 0 at the background level. As a result the stability conditions can be rather restrictive for the most general supersolid.
The bottom line is that the stability conditions are easily satisfied with possible exception of the most general supersolid where the restriction on M 1 is non-trivial. It is also worth to stress the we have studied the stability conditions in the worse case where the medium itself is dominating the Universe and thus the scale of ρ and p are the same ofM a . For instance, when radiation is dominant we expect that very similar stability conditions should hold with p and ρ associated with the dominant component (ultra-relativistic particles) andM a ≪ p + ρ, making stability not an issue.

Massive Gravity Phases
Here we briefly describe what happens once the self-gravitating nature of the medium has been taken into account. The special case of a de Sitter background, where p + ρ = 0, which incidentally is naturally the one selected by the Lorentz invariant (LI) quadratic Lagrangians, will be analysed in a forthcoming paper; here we just report, for completeness, the main results.
• The most interesting and new case is when basically no tuning on U is required and this also applies to the resulting masses. 7 The vacuum state breaks Lorentz invariance which in turn allows the existence of non-trivial pressure and energy density background; stability can be achieved when p + ρ = 0. The full six DoF of the theory can be stable at quadratic order .
There are two critical points in the parameter space where the kinetic matrix is degenerate and the above analysis has to be repeated (we always require p, ρ = 0) -In the Lorentz breaking (LB) case we have only one propagating scalar whose kinetic term is proportional to k 2 H 2 (1 + w) and mass to (M 2 − M 3 ) (in the large k limit). Thus when w = −1 the kinetic term for such a mode vanishes.
-For a LI quadratic Lagrangian and with ρ+p = 0, the requirement of M 4 = 0, for the existence of an FLRW background, gives As a result, a LI self gravitating medium in such a phase has all masses M i zero and the parameters A and B are fine tuned according with: M 2 Pl A = ρ, M 2 Pl B = −ρ/2. and no scalar and vector propagates at the quadratic order. Only two massless tensor modes are present. In Minkowski space, keeping M 4 = 0 also M 1 = 0 and then tensors and transverse vectors propagate and we get 5 DoF. In the contest of the decoupling limit of Pauli-Fierz massive gravity, the vanishing of the kinetic term in the scalar sector is dealt by changing the Stückelberg content by adding an additional scalar [24,2].
• WhenM 1 + ρ + p = M 2 Pl λ 2 1 + ρ = 0, only a single scalar and two tensor modes propagate for a total of ≤ 3 DoF; such a phase can be implemented in a non-perturbative way and can be protected by a symmetry [1,22].
-In the LB theory kinetic term for the only propagating scalar mode is given by -In the LI case being λ 2 1 = −A implies A = ρ and, being p + ρ = 0, we have From (5.2) we get K = 0 and then no scalar mode propagates. Likewise in GR, only massless tensor mode are present.
Let us comment on the Higuchi bound [25,26]. Such a bound on the Pauli-Fierz mass in dS spacetime is derived when the massive spin 2 action gives no contribution to the background [27,28], namely no linear term is present. In the dRGT massive gravity model, a dS background exists when a non-flat timedependent reference metric [29] (basically the same of the one found in the contest of bigravity [30,31]) is considered; in this case the Eguchi bound is recovered. In our approach, by definition, the medium is self-gravitating and the above considerations do not apply.

Conclusions
We have analysed the low energy phonon-like excitations of generic media (fluids, superfluids, solids and supersolids) on Minkowski space showing that typically no instability is present. It turns out that when dynamical gravity is added to the game, in the unitary gauge, self-gravitating media are equivalent to massive gravity and six degrees of freedom are present. On the other hand, massive gravity theories with six DoF on Minkowski are plagued by ghost instabilities and a great effort has been devoted trying to find a non-perturbative way to project out the unwanted (ghost) sixth mode. We reconcile the two apparent contradictory facts by taking into account the pivotal role played by the pressure and energy density of the medium. The point is that flat space is not a consistent background for a self-gravitating medium and for massive gravity, unless the background pressure and energy density is set to zero. In such case the generic stability conditions for a medium are badly violated. This is not surprising, when p = ρ = 0 the medium picture breaks down simply because there is no medium. To consistently study the dynamics of a self-gravitating media in presence of dynamical gravity one has to change background taking a FLRW Universe; the stability analysis around such background again shows that generically no ghost instability is present and actually for modes with wavelength much smaller of the curvature scale H −1 the flat space results with gravity switched off are recovered. Interestingly, the fluid picture is incompatible with the requirement of Lorentz invariance of the medium energy-momentum tensor and leads again to p = ρ = 0 and inevitably the Lagrangian has to be tuned to get less than six DoF. As a result, the spontaneous breaking of Lorentz invariance is instrumental to open up the stability window for massive gravity/self-gravitating medium with six DoF. Such a breaking is an emergent lowenergy phenomenon triggered by the non-vanishing homogenous background value of the Stückelberg scalar fields due to the presence of a medium with non zero energy density and pressure. One of the main theoretical and phenomenological features of such a models is that the Einstein-Hilbert part of the action is untouched keeping the dynamics of the spin two graviton very close to GR, except for a massive deformed dispersion relation For a typical frequency of the order some Hertz, observations [12,13] require that the graviton velocity is very close to c = 1: Such a bound on the mass is not as stringent as requiring that the mass scale is of the order of the present Hubble parameter ∼ 10 −33 eV if massive gravity is relevant for dark energy.

A Self-Gravitating Media
The operators entering the Lagrangian describing the low energy dynamics of a general isotropic medium minimally coupled with gravity can be given in terms of matrix [17,18] The 4-vector u µ is the fluid's velocity and u µ ∂ µ ϕ a = 0; thus ϕ a can be interpreted as the spatial Lagrangian (comoving) coordinates of the fluid, while ϕ 0 represents the fluid's clock's. Together with b, and Y , X, {τ n } and {y m } form a set of ten independent rotational invariant operators, thus the Lagrangian of a general medium will have the form U (b, Y, X, τ a , y n ). The EMT tensor for a general medium as the form T µν = ρ u µ u ν + q µ u ν + q ν u µ + P µν ; where h µν = u µ u ν + g µν is the projector orthogonal to u µ and has the form of the heat flow vector while splitting In the case of a perfect fluid we simply have that For explicit expressions for the λ a are the following 2 n a −2n U Y τn . (A.10) The mass parameters {M a } are defined separating the contribution from {λ a } the part that is related to the background pressure and energy density, this is done by (3.5) which leads to In Minkowski space the mass parameters are obtained form the above expression by setting a = φ ′ = 1.

B Unitary vs Newtonian gauge in FRW space
The quadratic action for scalar perturbations in a Fourier basis can be unambiguously written in matrix notation as the relevant metric and scalar field perturbations are collected in Q, a vector with 4 components. Then we have the 4×4 matrices: 8 The matrix K K K is the kinetic matrix and its symmetric by construction as the mass matrix M M M . If a non dynamical field is present, it can be integrated out and the resulting will be still of the form (B.1) with a new Q and K K K, D D D and M M M of lower dimension. In order to study the stability of system described by (B.1) we simply impose the positivity of the Hamiltonian. The conjugate momenta are When K K K is non-degenerate we are sure that no constraint is present. The total energy E , basically the Hamiltonian, expressed as a quadratic form in Q ′ and Q is Equivalently, the Hamiltonian in terms of Π and Q can be written as The expression of the mass matrix is not relevant for us. Clearly, there are two propagating DoF and one is a ghost.

B.2 Newton Gauge
In the Newtonian gauge the scalar perturbations are both in the metric and in the scalar fields, namely h (N ) µν = 2 Ψ δ 0 µ δ 0 ν + 2 Φ δ i µ δ i ν ; (B.9) and ϕ 0 = φ(t) + π 0 , ϕ a = x a + ∂ a π l ; (B.10) In the Newtonian gauge Q (N ) = (Ψ, Φ, k 2 π l , k π 0 ) and Notice that this time the quadratic Lagrangian depends on p and ρ if the background equations are not imposed. Now K K K is rank three and there are three DoF in sharp contrast with the unitary gauge computation. The only way to recover the same result if to impose p = ρ = 0; then also π 0 can be integrated out and the we end up with two DoF and one is a ghost.