Massive gravity, the elasticity of space-time and perturbations in the dark sector
MMassive gravity, the elasticity of space-time and perturbations in the dark sector
Richard A. Battye ∗ Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy,The University of Manchester, Manchester M13 9PL, U.K
Jonathan A. Pearson † Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, U.K andJodrell Bank Centre for Astrophysics, School of Physics and Astronomy,The University of Manchester, Manchester M13 9PL, U.K (Dated: November 3, 2018)We consider a class of modified gravity models where the terms added to the standard Einstein-Hilbert Lagrangian are just a function of the metric only. For linearized perturbations around anisotropic space-time, this class of models is entirely specified by a rank 4 tensor that encodes possiblytime-dependent masses for the gravitons. This tensor has the same symmetries as an elasticitytensor, suggesting an interpretation of massive gravity as an effective rigidity of space-time. If wechoose a form for this tensor which is compatible with the symmetries of FRW and enforce fullreparameterization invariance, then the only theory possible is a cosmological constant. However, inthe case where the theory is only time translation invariant, the ghost-free massive gravity theory isequivalent to the elastic dark energy scenario with the extra Lorentz violating vector giving rise to2 transverse and 1 longitudinal degrees of freedom, whereas when one demands spatial translationinvariance one is left with scalar field theory with a non-standard kinetic term.
I. INTRODUCTION
The realization that the Universe appears to be accelerating has fuelled the search to alternative theories of gravity[1] as a possible explanation for what has become called the dark sector . In this paper we will focus on what is thesimplest subset of such theories, in which the dark sector Lagrangian is only a function of the metric (and no extraderivatives thereof) following the approach discussed in [2]. The action for this type of theory is given by S = (cid:90) d x √− g (cid:20) R + 16 πG L m − L d ( g µν ) (cid:21) . (1.1)If T µν is the energy-momentum tensor for the matter sector Lagrangian L m and U µν is that associated with thedark sector Lagrangian L d , then the Einstein equation is G µν = 8 πGT µν + U µν . Typically, we will be interestedin spacetimes which are isotropic where U µν = ρu µ u ν + P γ µν can specified in terms of a density, ρ , and pressure, P = wρ . There are two classes of theories which can be described by (1.1): elastic dark energy and theories of massivegravitons. Typically, theories of massive gravitons have been considered as a fundamental theory around Minkowskispace-time, but it is also possible to think about masses for the gravitons as being induced by a some unknowneffective physics encoded by L d .The study of massive gravity theories (see, for example, [3, 4]) has a long history. This started with the linearizedtheories of Pauli and Feirz [5], progressing to the studies of Boulware and Deser [6]. It has received a new lease oflife in recent times with the proposal of non-linear dRGT massive gravity theory [7–10] and its connections to theVainshtein screening mechanism [11–14]. Massive gravity theories are built upon the pretext that the resulting theoryshould be “ghost-free” [9, 15–21], and have begun to be studied in cosmological backgrounds [22–34]. Such theoriesare usually presumed to be Lorentz invariant which leads to the Pauli-Fierz tuning. Giving up Lorentz-invariance isanother way to remove ghosts from the theory, as pointed out by [35, 36] and further studied in [3, 37–40].Elastic dark energy (EDE) is an idea which has been developed from relativistic elasticity theory [41–47]. Thebasic concept is that the stress-energy component responsible for the dark energy has rigidity which stabililizesperturbations that would, if modelled as those of a perfect fluid, give rise to exponential growth in the densitycontrast. The framework was adapted for cosmological purposes in [48, 49] in order to provide a phenomenologicalmodel for domain wall as an explanation for accelerated expansion which have P/ρ = w dw = − /
3. However, in ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] J a n principle the equation of state parameter, w , is allowed to take “any” value so long as the rigidity modulus, µ ,is sufficiently large. Indeed, the theory is well-defined in the limit w → − w → µ → II. THE GENERAL “METRIC ONLY” THEORY
The action which will give linearized field equations for the perturbed field variables is given by S { } = (cid:90) d x √− g (cid:20) ♦ R + 16 πG ♦ L m − L { } (cid:21) . (2.1)We use ♦ Q to denote the second measure-weighted variation of the quantity Q , defined as ♦ Q ≡ √− g δ ( √− g Q ). L { } is the Lagrangian for dark sector perturbations, given by L { } = 18 W µναβ δg µν δg αβ , (2.2)where δg µν is the metric fluctuation. This clearly looks like a mass term for the metric perturbations. The tensor W µναβ = W ( µν )( αβ ) = W αβµν (2.3)is the mass matrix determining how the components of δg µν mix to provide the mass; all linearized massive gravitiesare encoded by choices of W . Hence, the complete linearized theory we study is S { } = (cid:90) d x √− g (cid:20) ♦ R + 16 πG ♦ L m − W µναβ δg µν δg αβ (cid:21) . (2.4)This theory contains metric fluctuations which have a kinetic term, and a mass term. We will see that the spatialcomponents of W can be interpreted as an elasticity tensor.To isolate the degrees of freedom in the theory, δg µν can be decomposed as δg µν = h µν + 2 ∇ ( µ ξ ν ) . (2.5)In the parlance of [42, 47, 53, 54] h µν is the Eulerian metric perturbation and ξ µ is a vector field representing possiblecoordinate transformations. In standard General Relativity, the action is independent of ξ µ , but in more generaltheories this can become a physical field. This formulation is equivalent to what is sometimes called the Stuckelbergtrick [4, 18, 55]. Inserting (2.5) into the action (2.4) and integrating by parts reveals that S { } = (cid:90) d x √− g (cid:20)(cid:0) δ E G µν − πGδ E T µν − δ E U µν (cid:1) h µν + 2 ξ ( µ δ E ( ∇ ν ) U µν ) (cid:21) , (2.6)where the variational operator “ δ E ” denotes that the quantity is evaluated with the metric perturbation variable h µν (rather than δg µν ), and where we have defined the perturbed dark energy-momentum tensor δ E U µν = − (cid:0) W µναβ + U µν g αβ (cid:1) δg αβ − ξ α ∇ α U µν + 2 U α ( µ ∇ α ξ ν ) . (2.7)It is now a simple matter to obtain the functional derivatives of the action with respect to the perturbed metric h µν and ξ µ -fields, ˆ δ ˆ δh µν S { } = δ E G µν − πGδ E T µν − δ E U µν = 0 , (2.8a)ˆ δ ˆ δξ µ S { } = δ E ( ∇ ν U µν ) = 0 . (2.8b)The variational principle was used to demand that these expressions vanish, yielding the perturbed gravitational fieldequations and perturbed conservation equation respectively. Using (2.7) to evaluate (2.8b) yields L µναβ ∇ µ ∇ α ξ β + ( ∇ µ W µναβ ) ∇ α ξ β + ( ∇ µ ∇ α U µν ) ξ α = − (cid:0) ( ∇ µ W µναβ ) h αβ + P µναβ ∇ µ h αβ (cid:1) , (2.9)where we defined the effective metric L µναβ and derivative-coupling P µναβ terms, L µναβ ≡ W µναβ + U µν g αβ − U α ( µ g ν ) β , P µναβ ≡ W µναβ + U αβ g µν − g νβ U αµ . (2.10)We impose spatial isotropy upon the background with the (3+1) decomposition and in doing so we will obtain themost general linearized massive gravity Lagrangian compatible with spatial isotropy of the background. We foliatethe 4D spacetime by 3D sheets with a time-like unit vector, u µ , being everywhere orthogonal to the sheets. The4D spacetime has metric g µν , and the 3D sheets have metric γ µν . The (3+1) decomposition of the 4D metric is g µν = γ µν − u µ u ν , where u µ u µ = − γ µν u µ = 0. This structure provides an extrinsic curvature tensor K µν = K ( µν ) on the 3D sheets, given by K µν = ∇ µ u ν and satisfying u µ K µν = 0 (the extrinsic curvature tensor is given by K µν = Kγ µν ). We define “time” and “space” differentiation as the derivative operator projected along the time andspace directions, ˙ ψ ≡ u µ ∇ µ ψ, ∇ µ ψ ≡ γ νµ ∇ ν ψ. (2.11)Using this technology we decompose the gradient of a scalar into two orthogonal terms, ∇ µ ψ = − ˙ ψu µ + ∇ µ ψ. (2.12)This enables us to find the values of two useful “kinetic scalars”, ∇ µ ψ ∇ µ ψ = − ˙ ψ + ∇ µ ψ ∇ µ ψ, (2.13a) (cid:3) ψ ≡ ∇ µ ∇ µ ψ = − ¨ ψ + ∇ µ ∇ µ ψ. (2.13b)The last term of each expression simply selects out the spatial derivatives of the scalar field. Another useful applicationof the (3+1) decomposition is to find all the freedom in a tensor which is compatible with spatial isotropy of thebackground spacetime.We use the (3+1) decomposition to isolate the components of the perturbed metrics by writing δg µν = 2Φ u µ u ν + 2 N ( µ u ν ) + ¯ H αβ γ αµ γ βν , (2.14a) h µν = 2 φu µ u ν + 2 n ( µ u ν ) + ¯ h αβ γ αµ γ βν , (2.14b)and we isolate the time-like and space-like components of the vector field via ξ µ = − χu µ + ω µ , (2.14c)where N µ u µ = n µ u µ = 0 , u µ ¯ H µν = u µ ¯ h µν = 0 and u µ ω µ = 0.In [2] we showed that the general decomposition of the mass-matrix W compatible with spatial isotropy is W µναβ = A W u µ u ν u α u β + B W (cid:0) u µ u ν γ αβ + u α u β γ µν (cid:1) + 2 C W (cid:0) γ α ( µ u ν ) u β + γ β ( µ u ν ) u α (cid:1) + D W γ µν γ αβ + 2 E W γ µ ( α γ β ) ν , (2.15)where there are only 5 free functions which only depend on time. Using the mass matrix (2.15) and (2.14a) in theLagrangian (2.2) yields8 L { } = 4 A W Φ + 4 B W Φ ¯ H + 2 C W N α N α + D W ¯ H + 2 E W ¯ H αβ ¯ H αβ . (2.16)This can be written in terms of “graviton masses” (see e.g., [3]) where the free functions { A W , . . . , E W } are given by2 L { } = m ( δg ) + 2 m ( δg i ) − m ( δg ij ) + m ( δg ii ) − m δg δg ii (2.17)with A W = m , B W = − m , C W = 4 m , D W = 4 m and E W = − m . One should keep in mind, therefore,that when we talk about the { A W , . . . , E W } we are actually talking about the graviton masses m i , albeit ones thatdepend on time. The values of these masses for a “Goldstone” theory are given in [56] and those which are inducedby perturbations in scalar fields in [2].Using (2.14) to evaluate (2.5) yieldsΦ = φ + ˙ χ, N α = n α − ˙ (cid:36) α − ∇ α χ, ¯ H αβ = ¯ h αβ + 2 ∇ ( α ω β ) − Kγ αβ χ − Kω ( α u β ) , (2.18)where ˙ (cid:36) α ≡ (cid:0) ˙ ω α − Kω α (cid:1) . Substituting (2.14a) into the action (2.4) one finds the absence of ˙ φ and ˙ n α terms inthe kinetic part of the Einstein-Hilbert Lagrangian; this can also be seen in results given by [3, 36, 39] and in theADM formulation [57]. φ and n α are now Lagrange multipliers whose equations of motion are constraint equations,allowing them to be eliminated. Using re-definitions of the coefficients, we can effectively set φ = 0 and n α = 0 whichis equivalent to choice of the synchronous gauge. We will make this choice in what follows.The two independent components of the equation of motion (2.9), after inserting the (3+1)-decomposition, aregiven by (cid:2) A W + ρ (cid:3) ¨ χ + (cid:2) ˙ A W + H (4 A W + ρ − P ) (cid:3) ˙ χ + (cid:2) P + C W (cid:3) ∇ χ − (cid:2) H (3 ˙ P + 2 ˙ ρ − ˙ A W + 3 ˙ B W ) − (2 A W + 5 ρ + 3 P − B W − D W − E W ) H +(2 ρ + 3 B W − A W ) ¨ aa + ¨ ρ (cid:3) χ − (cid:2) ˙ B W + H (3 B W + 3 D W + 2 E W − P ) (cid:3) ∂ i ω i + (cid:2) B W + C W (cid:3) ∂ i ˙ ω i = 12 (cid:2) ˙ B W + H (3 B W + 3 D W + 2 E W − P ) (cid:3) h + 12 (cid:2) B W − P (cid:3) ˙ h, (2.19a) (cid:2) ρ − C W (cid:3) ¨ ω i − (cid:2) ˙ C W + H (4 C W − ρ + 3 P ) (cid:3) ˙ ω i − (cid:2) E W − P (cid:3) ∇ ω i − (cid:2) E W + D W (cid:3) ∂ i ∂ k ω k + (cid:2) ˙ C W − H (3 D W − B W + 2 E W − C W − P ) (cid:3) ∂ i χ + (cid:2) B W + C W (cid:3) ∂ i ˙ χ = − (cid:2) P − E W (cid:3) ∂ j h ij + 12 (cid:2) P + D W (cid:3) ∂ i h. (2.19b) ρ and P are the density and pressure coming from the dark fluid (i.e. the components of the background dark energymomentum tensor U µν ). The benefit of using the (3+1) decomposition has become apparent: we are able to identifythe degrees of freedom. There are the tensor degrees of freedom h + and h × which are present in standard GeneralRelativity, a vector degree of freedom ω i , that can be split into a longitudinal (scalar) and two transverse (vector)degrees of freedom, and a scalar degree of freedom χ . Therefore, prima facie there are 6 extra degrees of freedom. Aswe will discuss below either χ or ω i can be a ghost and therefore the coefficients must be chosen to suppress one orboth of them. In the case where χ is the ghost then there are 5 degrees of freedom with those in ω i being split into alongitudinal (scalar) and two transverse (vector) degrees of freedom. If ω i is the ghost then there are only 3 degreesof freedom. III. MECHANISMS FOR THE ELIMINATION OF GHOSTS
From (2.6) we see that the kinetic terms of the ω µ and χ fields enter the theory via12 S { } ⊃ (cid:90) d x √− g (cid:20) ( A W + ρ ) ˙ χ + ( C W + P ) ∇ µ χ ∇ µ χ +( C W − ρ ) ˙ ω µ ˙ ω µ + ( D W + P )( ∇ µ ω µ ) + 2( E W − P ) ∇ µ ω ν ∇ µ ω ν (cid:21) . (3.1)Let us now focus on the standard scenario of perturbations around Minkowski space-time when both ρ = P = 0. If A W >
0, then C W < χ to have a kinetic term with the “proper sign”. But if this is the case then ω µ has a kinetic term with the “wrong sign” (the same is true if A W < χ, ω µ must be a ghost. Thereare a few ways to get out of this.First, one can make the coefficient of ˙ χ vanish by setting A W = 0, which removes χ as a propagating mode andthe equation of motion is a constraint that can be enforced in a Lorentz invariant theory. Secondly, one could set C W = 0, since that would remove ω µ as a propagating mode. Finally, one could set χ ≡ A W = 0, there is no ˙ χ term in the Lagrangian and the equation of motion simply becomes a constraintequation specifying the value of χ from the other field variables. This can be back-substituted into the action sothat the theory explicitly does not contain the χ -field. From our presentation it is clear that in this case the ghostcan be identified with the time-like degree of freedom χ . We have been able to deduce this since we used a (3+1)decomposition. Performing a transverse-longitudinal decomposition does not aid the identification of the ghost.If we choose the 5 parameters in (2.15) to be given by A W = X + Y , B W = − X , C W = − Y , D W = X , E W = Y ,then the theory is Lorentz invariant and the mass-matrix is be given by W µναβ = Xg µν g αβ + Y g µ ( α g β ) ν , (3.2)where X, Y are two parameters that are dependent on background field variables only. The standard route for isolatingthe ghost in the Lorentz invariant theory [55] decomposes ξ µ into its transverse and longitudinal modes as ξ µ = ζ µ + ∇ µ κ, (3.3)where ∇ µ ζ µ = 0 and κ is a scalar field. For Minkowski background spacetime, inserting (2.5), (3.3) and the mass-matrix (3.2) into the Lagrangian (2.2), whilst assuming that X, Y are constants, yields L { } = 18 ( Xh + Y h µν h µν ) + 12 Y h µν ∂ ( µ ζ ν ) + 12 ( Xh (cid:3) κ + Y h µν ∂ µ ∂ ν κ ) + 12 Y ∂ µ ζ ν ∂ ( µ ζ ν ) + Y ∂ ( µ ζ ν ) ∂ µ ∂ ν κ + 12 ( X + Y )( (cid:3) κ ) . (3.4)This expression has made the ghost problem manifest in the Lorentz invariant language. The existence of the lastterm, ( X + Y )( (cid:3) κ ) , means that ghosts are inevitable (see e.g. [15, 58, 59]). The cure is to set X = − Y , which removesthe problematic kinetic term, and leaves the Pauli-Feriz mass-term, L { } ⊃ L PF = h − h µν h µν . The parameter choice X = − Y is called the Pauli-Feirz tuning , and will render massive gravitons ghost-free at linearized order on Minkowskibackgrounds. In this case A W = 0, B W = − X , C W = X , D W = X , E W = − X which is a special case of the moregeneral situation discussed earlier.Rather than retain Lorentz invariance and be forced to use the Pauli-Feirz tuning to remove the ghost, it has beensuggest that one can just fix the field u µ ξ µ = 0 which implies that χ = 0, removing it as a physical degree of freedom.Of course, this is not really a solution to the problem of the ghost, since we have just set the field to zero. However,as we will see in the next section that it is possible to impose a symmetry which is equivalent to this. The condition u µ ξ µ = 0 imposes an interesting structure upon the fields in theory when we use the transverse-longitudinal splitlanguage. Using this and (3.3) implies that ˙ κ = − u µ ζ µ which can be differentiated to yield ¨ κ = − u µ ˙ ζ µ . This shows usthat the κ -field (i.e. the longitudinal component of the ξ µ -field) does not propagate. Instead, the n th time derivativeof κ is replaced by the ( n − th time derivative of ζ µ . Evaluating the “kinetic scalars” (2.13) for the scalar κ , yields ∇ µ κ ∇ µ κ = − u µ u ν ζ µ ζ ν + ∇ µ κ ∇ µ κ, (3.5a) (cid:3) κ = u µ ˙ ζ µ + ∇ µ ∇ µ κ. (3.5b)Using (3.5b), the previously offensive term in (3.4) becomes( X + Y )( (cid:3) κ ) = ( X + Y )( u µ u ν ˙ ζ µ ˙ ζ ν + 2 u α ˙ ζ α ∇ µ ∇ µ κ + ∇ µ ∇ µ κ ∇ α ∇ α κ ) , (3.6)and we observe that the multiple derivatives of κ that are present are entirely spatial. The upshot is that there areno time-derivatives of the scalar κ left, and, crucially no second time-derivatives of κ in (cid:3) κ . The term ( X + Y )( (cid:3) κ ) in (3.4) is now longer problematic, and does not require removal. IV. IMPOSING REPARAMETERIZATION INVARIANCE
A key aspect of the theories under consideration here is the spontaneous violation of reparameterization invariance.It is interesting to see under what conditions it can be be reimposed on the theory. Therefore, we consider howthe vector field ξ µ sources the perturbed gravitational field equations, and under what circumstances its componentsdecouples from the field equations. From (2.1) the field equations for the metric are δ E G µν = 8 πGδ E T µν + δ E U µν ,where δ E U µν is the dark energy momentum tensor and contains contributions to the field equations from the darksector. In [2] we showed that δ E U µν = − (cid:0) W µναβ + U µν g αβ (cid:1) δg αβ − ξ α ∇ α U µν + 2 U α ( µ ∇ α ξ ν ) . (4.1)It is useful to note the terms in δ E U µν which are present due to the background dark energy-momentum tensor U µν .The components of δ E U µν are written as perturbed fluid variables, δ E U µν = δρu µ u ν + 2( ρ + P ) v ( µ u ν ) + δP γ µν + P Π µν , (4.2)where one can obtain δρ = (cid:20) ˙ ρ + H (2 ρ − A W + 3 B W ) (cid:21) χ − ( A W + ρ ) ˙ χ + ( ρ + B W ) (cid:18) h + ∂ i ω i (cid:19) , (4.3a) δP = − (cid:20) ˙ P + H (2 P − B W + 3 D W + 2 E W ) (cid:21) χ + ( B W − P ) ˙ χ − ( P + 3 D W + 2 E W ) (cid:18) h + ∂ i ω i (cid:19) , (4.3b)( ρ + P ) v i = ( P + C W ) ∂ i χ + ( ρ − C W ) ˙ ω i , (4.3c) P Π ij = 2( P − E W ) (cid:18) h ij + ∂ ( i ω j ) − δ ij ( h + ∂ k ω k ) (cid:19) . (4.3d)These effective fluid variables define how the components of the vector field ξ µ sources the gravitational field equations.If one, or both, of the fields χ and ω i does not appear in (4.3) then that field is not dynamical and hence can becompletely ignored. It is clear that particular choices of the free functions in the mass-matrix it will be possible toachieve this. When one or both does not appear, it means that the theory is invariant under the symmetry associatedwith that field. Therefore, we can impose reparameterization invariance in three natural ways: • ξ µ -field decouples from the system when ˙ ρ + 3 H ( ρ + P ) = 0, ˙ P + H ( P + 3 D W + 2 E W ) = 0, A W = − ρ , B W = P , C W = − P , ρ = − B W , ρ = C W , P = E W , D W = − P . Hence, in the “fully” reparameterization invariant case,where the theory is forced to be invariant under x µ → x µ + ξ µ , the only values of ρ, P that are allowed are thosewhich are provided by a cosmological constant, ρ = − P , and all perturbed fluid variables vanish. Neither the χ - nor the ω i -field propagate. • χ = u µ ξ µ field decouples from the system when the parameters satisfy A W = − ρ , B W = P , C W = − P ,˙ ρ + H (2 ρ − A W + 3 B W ) = 0, ˙ P + H (2 P − B W + 3 D W + 2 E W ) = 0 from which we can deduce that ˙ ρ + 3 H ( ρ + P ) = 0 and ˙ P + H ( P + 3 D W + 2 E W ) = 0. These equations appear to leave two coefficients, D W and E W ,unspecified. If we now define two parameters β and µ via D W = β − P − µ and E W = µ + P then we find that β = ( ρ + P ) d P d ρ which is the definition of the relativistic bulk modulus and µ can then be interpreted as a rigiditymodulus of an elastic medium. Hence, in the case where we impose time translation invariance, t → t + χ , butnot spatial translation invariance, then we find that the theory must be EDE. The equations of motion (2.19)become − H (cid:2) ˙ P + 3 β H (cid:3) χ = 0 , (4.4a) (cid:2) ρ + P (cid:3) ¨ ω i + (cid:2) ˙ P + H ( P + ρ ) (cid:3) ˙ ω i − (cid:2) E W − P (cid:3) ∇ ω i − (cid:2) E W + D W (cid:3) ∂ i ∂ k ω k = − (cid:2) P − E W (cid:3) ∂ j h ij + (cid:2) P + D W (cid:3) ∂ i h. (4.4b)Note that (4.4a) vanishes for arbitrary values of χ since β = ( ρ + P ) ˙ P / ˙ ρ , and that there is a propagating vectordegree of freedom, ω i . (4.4b) is the equation of motion presented in [60]. In this case the mass-term for thegravitons is L { } = ρ (cid:20) ( w − ˆ µ )¯ h + 2( w + ˆ µ )¯ h µν ¯ h µν (cid:21) , (4.5)where w = P/ρ, ˆ µ ≡ µ/ρ . This case is equivalent to setting A W = 0 in the Minkowski space case. Since χ doesnot appear in (4.3) it is no longer a physical degree and there is no ghost. • ω i decouples when ρ + B W = 0, P + 3 D W + 2 E W = 0, ρ = C W , P = E W from which we can deduce that B W = − C W = − ρ and D W = − E W = − P . Therefore, we see that in the case where we impose spatialtranslation invariance, x i → x i + ξ i , but not time translation invariance, then the perturbations have some ofthe characteristics of massive scalar field theory as explained in [2]. The equations of motion (2.19) become (cid:2) A W + ρ (cid:3) ¨ χ + (cid:2) ˙ A W + H (4 A W + ρ − P ) (cid:3) ˙ χ + (cid:2) ρ + P (cid:3) ∇ χ + (cid:2) ( ˙ A W + 3( A W − ρ − P ) H ) H + ( A W + 4 ρ + 3 P ) ˙ H (cid:3) χ = − (cid:2) ρ + P (cid:3) ˙ h, (4.6a) (cid:2) ˙ ρ + 3 H ( ρ + P ) (cid:3) ∂ i χ = 0 . (4.6b)Note that (4.6b) vanishes for arbitrary values of χ due to the background conservation equation. If we definethe entropy w Γ = (cid:18) δPδρ − w (cid:19) δ , (4.7)and set A W = [ w + (cid:15) (1 + w )] ρ we find that w Γ = (cid:18)
11 + (cid:15) − w (cid:19) (cid:20) δ − H (cid:18) w (1 + (cid:15) ) + 1 w (1 + (cid:15) ) − (cid:19) (1 + w ) θ (cid:21) . (4.8)This implies that this theory is a scalar field theory with a non-standard kinetic term. The mass-term is givenby L { } = − wρ (cid:20) ¯ h − h µν ¯ h µν (cid:21) , (4.9)which we note does not satisfy the Pauli-Feirz tuning. This case is equivalent to setting C W = 0 in the Minkowskispace case. Since ω i does not appear in (4.3) it is no longer a physical degree and there is no ghost. V. DISCUSSION
It is well established in the literature, that can be ghosts in general massive gravity theories, and various methodshave been devised to remove them. If one imposes Lorentz invariance, then one is forced to use the Pauli-Feirz tuningto excise the ghost. If one is willing to give up Lorentz invariance, then certain parameter choices allow for ghost-freemassive gravity theories. We have shown that the theory (2.2) with the (3+1)-decomposition of the mass-matrix(2.15) imposed with just time-translation invariance constitutes a linearized theory with healthy Lorentz-violatingmassive gravitons with 5 physical degrees of freedom. That theory is exactly the EDE model previously discussed inthe literature [60]. In addition, if one imposes spatial translation invariance then there is another ghost-free massivegravity theory with 3 degrees of freedom.In terms of the EDE parameters and graviton masses, A W = m = − ρ, B W = − m = P, C W = 4 m = − P, D W =4 m = β − P − µ, E W = − m = µ + P . The masses can be conveniently parameterized by some overall mass scale M ≡ ρ ∼ H (since we wrote U µν = ρu µ u ν + P γ µν , ρ has units of mass squared), m = − M , m = − wM , m = − (ˆ µ + w ) M , m = ( w − ˆ µ ) M , m = − wM . (5.1)We defined ˆ µ ≡ µ/ρ in analogy with w = P/ρ . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) w Μ (cid:96) Figure 1: The shaded region denotes the range of values of the equation of state w and shear modulus ˆ µ which yield soundspeeds less than unity, which is where the inequalities (5.3) are satisfied. The red (solid) lines denote lines of constant m = M (0 . , . ,
0) from left to right and the blue (dashed) lines of constant m = M (0 . , . , − . To connect to dark energy, we note that the fraction of the total energy density that is dark energy is linked tothe mass scale M via Ω de = M / (3 H ). Hence, we see that the “natural” scale for the masses in order for themodification of gravity to act as a source of cosmic acceleration is of the order the Hubble parameter, and are allmultiplied by order unity “corrections” defined by two parameters which encode the properties of the elastic medium:its equation of state parameter w and shear modulus ˆ µ . The longitudinal and transverse sound speeds of EDE are[60] c s = w + ˆ µ w , c v = ˆ µ w . (5.2)Stability and subluminality require that 0 ≤ c i ≤
1, so that we have the following constraints on the possible valuesof ˆ µ − w (1 + w ) ≤ ˆ µ ≤ (1 − w ) , ≤ ˆ µ ≤ w. (5.3)In Figure 1 we plot the allowed values of ( w, ˆ µ ) which satisfy (5.3), and some lines of constant m and m . Observa-tionally the values of w, M and ˆ µ (only ˆ µ is the “new” parameter) can be constrained and then (5.1) used to obtainthe graviton masses.Elastic dark energy and massive gravity share two common features. First, they are both constructed from rank-4tensors, (the elasticity tensor and mass-matrix, respectively) and these tensors have identical symmetries in theirindices. Secondly, they both have five propagating degrees of freedom. The extra degrees of freedom in elasticdark energy may have a different fundamental origin to those in massive gravity, but they enable an interestinginterpretation to be extracted from massive gravities. Our interpretation is that massive gravity is the manifestationof rigidity of spacetime . Acknowledgements
We have appreciated conversations with Niayesh Afshordi, Stephen Appleby, Ruth Gregoryand Kurt Hinterbichler. JAP is supported by the STFC Consolidated Grant ST/J000426/1. This research wassupported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and by the Province of Ontario through the Ministry of EconomicDevelopment and Innovation. [1] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis,
Modified Gravity and Cosmology , Phys. Rept. (2012) 1–189,[ arXiv:1106.2476 ].[2] R. A. Battye and J. A. Pearson,
Effective action approach to cosmological perturbations in dark energy and modifiedgravity , JCAP (2012) 019, [ arXiv:1203.0398 ].[3] V. A. Rubakov and P. G. Tinyakov,
Infrared-modified gravities and massive gravitons , Phys. Usp. (2008) 759–792,[ arXiv:0802.4379 ].[4] K. Hinterbichler, Theoretical Aspects of Massive Gravity , Rev.Mod.Phys. (2012) 671–710, [ arXiv:1105.3735 ].[5] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field , Proc.Roy. Soc. Lond.
A173 (1939) 211–232.[6] D. Boulware and S. Deser,
Can gravitation have a finite range? , Phys.Rev. D6 (1972) 3368–3382.[7] C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action , Phys.Rev.
D82 (2010) 044020,[ arXiv:1007.0443 ].[8] C. de Rham, G. Gabadadze, and A. J. Tolley,
Resummation of Massive Gravity , Phys. Rev. Lett. (2011) 231101,[ arXiv:1011.1232 ].[9] S. Hassan and R. A. Rosen,
Resolving the Ghost Problem in non-Linear Massive Gravity , Phys.Rev.Lett. (2012)041101, [ arXiv:1106.3344 ].[10] S. Hassan and R. A. Rosen,
On Non-Linear Actions for Massive Gravity , JHEP (2011) 009, [ arXiv:1103.6055 ].[11] A. Vainshtein,
To the problem of nonvanishing gravitation mass , Phys. Lett. B (1972) 393–394.[12] E. Babichev, C. Deffayet, and R. Ziour, The Recovery of General Relativity in massive gravity via the Vainshteinmechanism , Phys.Rev.
D82 (2010) 104008, [ arXiv:1007.4506 ].[13] A. De Felice, R. Kase, and S. Tsujikawa,
Vainshtein mechanism in second-order scalar-tensor theories , Phys.Rev.
D85 (2012) 044059, [ arXiv:1111.5090 ].[14] F. Sbisa, G. Niz, K. Koyama, and G. Tasinato,
Characterising Vainshtein Solutions in Massive Gravity , Phys.Rev.
D86 (2012) 024033, [ arXiv:1204.1193 ].[15] P. Creminelli, A. Nicolis, M. Papucci, and E. Trincherini,
Ghosts in massive gravity , JHEP (2005) 003,[ hep-th/0505147 ].[16] L. Alberte, A. H. Chamseddine, and V. Mukhanov, Massive Gravity: Exorcising the Ghost , JHEP (2011) 004,[ arXiv:1011.0183 ].[17] L. Alberte, A. H. Chamseddine, and V. Mukhanov,
Massive Gravity: Resolving the Puzzles , JHEP (2010) 023,[ arXiv:1008.5132 ].[18] C. de Rham, G. Gabadadze, and A. Tolley,
Ghost free Massive Gravity in the St´uckelberg language , Phys.Lett.
B711 (2012) 190–195, [ arXiv:1107.3820 ].[19] S. Hassan, R. A. Rosen, and A. Schmidt-May,
Ghost-free Massive Gravity with a General Reference Metric , JHEP (2012) 026, [ arXiv:1109.3230 ].[20] S. F. Hassan and R. A. Rosen,
Bimetric Gravity from Ghost-free Massive Gravity , JHEP (2012) 126,[ arXiv:1109.3515 ].[21] M. F. Paulos and A. J. Tolley, Massive Gravity Theories and limits of Ghost-free Bigravity models , arXiv:1203.4268 .[22] L. Grisa and L. Sorbo, Pauli-Fierz Gravitons on Friedmann-Robertson-Walker Background , Phys.Lett.
B686 (2010)273–278, [ arXiv:0905.3391 ].[23] F. Berkhahn, D. D. Dietrich, and S. Hofmann,
Self-Protection of Massive Cosmological Gravitons , JCAP (2010)018, [ arXiv:1008.0644 ].[24] F. Berkhahn, D. D. Dietrich, and S. Hofmann,
Consistency of Relevant Cosmological Deformations on all Scales , JCAP (2011) 024, [ arXiv:1104.2534 ].[25] F. Berkhahn, D. D. Dietrich, and S. Hofmann,
Cosmological Classicalization: Maintaining Unitarity under RelevantDeformations of the Einstein-Hilbert Action , Phys. Rev. Lett. (2011) 191102, [ arXiv:1102.0313 ].[26] G. D’Amico et. al. , Massive Cosmologies , Phys. Rev.
D84 (2011) 124046, [ arXiv:1108.5231 ].[27] A. E. Gumrukcuoglu, C. Lin, and S. Mukohyama,
Open FRW universes and self-acceleration from nonlinear massivegravity , JCAP (2011) 030, [ arXiv:1109.3845 ].[28] A. E. Gumrukcuoglu, C. Lin, and S. Mukohyama,
Cosmological perturbations of self-accelerating universe in nonlinearmassive gravity , JCAP (2012) 006, [ arXiv:1111.4107 ].[29] M. Crisostomi, D. Comelli, and L. Pilo,
Perturbations in Massive Gravity Cosmology , JHEP (2012) 085,[ arXiv:1202.1986 ].[30] P. Gratia, W. Hu, and M. Wyman,
Self-accelerating Massive Gravity: Exact solutions for any isotropic matterdistribution , Phys.Rev.
D86 (2012) 061504, [ arXiv:1205.4241 ].[31] A. De Felice, A. E. Gumrukcuoglu, S. Mukohyama, A. E. Gumrukcuoglu, and S. Mukohyama,
Massive gravity: nonlinearinstability of the homogeneous and isotropic universe , Phys.Rev.Lett. (2012) 171101, [ arXiv:1206.2080 ].[32] M. S. Volkov,
Cosmological solutions with massive gravitons in the bigravity theory , JHEP (2012) 035,[ arXiv:1110.6153 ].[33] M. S. Volkov,
Exact self-accelerating cosmologies in the ghost-free massive gravity – the detailed derivation , arXiv:1207.3723 .[34] G. D’Amico, Cosmology and perturbations in massive gravity , Phys.Rev.
D86 (2012) 124019, [ arXiv:1206.3617 ]. [35] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, Ghost condensation and a consistent infraredmodification of gravity , JHEP (2004) 074, [ hep-th/0312099 ].[36] V. Rubakov,
Lorentz-violating graviton masses: Getting around ghosts, low strong coupling scale and VDVZdiscontinuity , hep-th/0407104 .[37] S. Dubovsky, Phases of massive gravity , JHEP (2004) 076, [ hep-th/0409124 ].[38] G. Gabadadze and L. Grisa,
Lorentz-violating massive gauge and gravitational fields , Phys.Lett.
B617 (2005) 124–132,[ hep-th/0412332 ].[39] D. Blas, D. Comelli, F. Nesti, and L. Pilo,
Lorentz Breaking Massive Gravity in Curved Space , Phys.Rev.
D80 (2009)044025, [ arXiv:0905.1699 ].[40] D. Comelli, M. Crisostomi, F. Nesti, and L. Pilo,
Degrees of Freedom in Massive Gravity , arXiv:1204.1027 .[41] B. Carter, Elastic perturbation theory in general relativity and a variation principle for a rotating solid star , Communications in Mathematical Physics (1973) 261–286. 10.1007/BF01645505.[42] B. Carter and H. Quintana, Gravitational and acoustic waves in an elastic medium , Phys.Rev
D16 (1977), no. 10 16.[43] B. Carter,
Rheometric structure theory, convective differentiation and continuum electrodynamics , Proceedings of theRoyal Society of London. A. Mathematical and Physical Sciences (1980), no. 1749 169–200.[44] B. Carter,
Speed of sound in a high-pressure general-relativistic solid , Phys. Rev. D (Mar, 1973) 1590–1593.[45] B. Carter and H. Quintana, Foundations of general relativistic high-pressure elasticity theory , Proceedings of the RoyalSociety of London. A. Mathematical and Physical Sciences (1972), no. 1584 57–83.[46] B. Carter,
Interaction of gravitational waves with an elastic solid medium , gr-qc/0102113 .[47] J. L. Friedman and B. F. Schutz, Erratum: ”On the stability of relativistic systems” [Astrophys. J., Vol. 200, p. 204 - 220(1975)]. , .[48] M. Bucher and D. N. Spergel,
Is the dark matter a solid? , Phys. Rev.
D60 (1999) 043505, [ astro-ph/9812022 ].[49] R. A. Battye, M. Bucher, and D. Spergel,
Domain wall dominated universes , astro-ph/9908047 .[50] R. A. Battye and A. Moss, Anisotropic perturbations due to dark energy , Phys. Rev.
D74 (2006) 041301,[ astro-ph/0602377 ].[51] R. Battye and A. Moss,
Anisotropic dark energy and CMB anomalies , Phys.Rev.
D80 (2009) 023531, [ arXiv:0905.3403 ].[52] J. A. Pearson,
Effective field theory for perturbations in dark energy and modified gravity , arXiv:1205.3611 .[53] R. Battye and B. Carter, Gravitational perturbations of relativistic membranes and strings , Phys.Lett.
B357 (1995)29–35, [ hep-ph/9508300 ].[54] R. A. Battye and B. Carter,
Second order Lagrangian and symplectic current for gravitationally perturbedDirac-Goto-Nambu strings and branes , Class. Quant. Grav. (2000) 3325–3334, [ hep-th/9811075 ].[55] N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effective field theory for massive gravitons and gravity in theory space , Ann. Phys. (2003) 96–118, [ hep-th/0210184 ].[56] S. Dubovsky, P. Tinyakov, and I. Tkachev,
Cosmological attractors in massive gravity , Phys.Rev.
D72 (2005) 084011,[ hep-th/0504067 ].[57] R. L. Arnowitt, S. Deser, and C. W. Misner,
The Dynamics of general relativity , gr-qc/0405109 .[58] F. de Urries and J. Julve, Degrees of freedom of arbitrarily higher derivative field theories , gr-qc/9506009 .[59] F. de Urries and J. Julve, Ostrogradski formalism for higher derivative scalar field theories , J.Phys.A
A31 (1998)6949–6964, [ hep-th/9802115 ].[60] R. A. Battye and A. Moss,
Cosmological Perturbations in Elastic Dark Energy Models , Phys. Rev.
D76 (2007) 023005,[ astro-ph/0703744astro-ph/0703744