Cosmology

12 Cosmology

One of the principal motivations for considering massive theories of gravity is their potential to address, or at least provide a new perspective on, the issue of cosmic acceleration as already discussed in Section 3. Adding a mass for the graviton keeps physics at small scales largely equivalent to GR because of the Vainshtein mechanism. However, it inevitably modifies gravity in at large distances, i.e., in the infrared. This modification of gravity is thus most significant for sources which are long wavelength. The cosmological constant is the most infrared source possible since it is build entirely out of zero momentum modes and for this reason we may hope that the nature of a cosmological constant in a theory of massive gravity or similar infrared modification is changed.

There have been two principal ideas for how massive theories of gravity could be useful for addressing the cosmological constant. On the one hand, by weakening gravity in the infrared, they may weaken the sensitivity of the dynamics to an already existing large cosmological constant. This is the idea behind screening or degravitating solutions [211 , 212 , 26 , 216 ] (see Section 4.5). The second idea is that a condensate of massive gravitons could form which act as a source for self-acceleration, potentially explaining the current cosmic acceleration without the need to introduce a non-zero cosmological constant (as in the case of the DGP model [159 , 163 ], see Section 4.4). This idea does not address the ‘old cosmological constant problem’ [484 *] but rather assumes that some other symmetry, or mechanism exists which ensures the vacuum energy vanishes. Given this, massive theories of gravity could potential provide an explanation for the currently small, and hence technically unnatural value of the cosmological constant, by tying it to the small, technically natural, value of the graviton mass.

Thus, the idea of screening/degravitation and self-acceleration are logically opposites to each other, but there is some evidence that both can be achieved in massive theories of gravity. This evidence is provided by the decoupling limit of massive gravity to which we review first. We then go on to discuss attempts to find exact solutions in massive gravity and its various extensions.

12.1 Cosmology in the decoupling limit

A great deal of understanding about the cosmological solutions in massive gravity theories can be learned from considering the ‘decoupling limit’ of massive gravity discussed in Section 8.3. The idea here is to recognize that locally, i.e., in the vicinity of a point, any FLRW geometry can be expressed as a small perturbation about Minkowski spacetime (about $⃗x = 0$ ) with the perturbation expansion being good for distances small relative to the curvature radius of the geometry:

[ ] [ 1 ] ds2 = − 1 − (H˙ + H2 )⃗x2 dt2 + 1 − -H2 ⃗x2 d⃗x2 (12.1 ) ( ) 2 1 FLRW μ ν = ημν + M---hμν dx dx . (12.2 ) Pl

In the decoupling limit $MPl → ∞$ , $m → 0$ we keep the canonically normalized metric perturbation $h μν$ fixed. Thus, the decoupling limit corresponds to keeping $H2MPl$ and $H˙MPl$ fixed, or equivalently $H2 ∕m2$ and $H˙ ∕m2$ fixed. Despite the fact that $H → 0$ vanishes in this limit, the analogue of the Friedmann equation remains nontrivial if we also scale the energy density such that $ρ ∕MPl$ remains finite. Because of this fact, it is possible to analyze the modification to the Friedmann equation in the decoupling limit.³¹

The generic form for the helicity-0 mode which preserves isotropy near $⃗x = 0$ is

In the specific case where $˙ B (t) = 0$ , this also preserves homogeneity in a theory in which the Galileon symmetry is exact, as in massive gravity, since a translation in $⃗x$ corresponds to a Galileon transformation of $π$ which leaves invariant the combination $∂μ∂νπ$ . In Ref. [139 *] this ansatz was used to derive the existence of both self-accelerating and screening solutions.

Friedmann equation in the decoupling limit

We start with the decoupling limit Lagrangian given in (8.38 *). Following the same notation as in Ref. [139 ] we set $an = − cn∕2$ , where the coefficients $cn$ are given in terms of the $αn$ ’s in (8.47 *). The self-accelerating branch of solutions then corresponds to the ansatz

1- 3 μ π = 2q0Λ3x μx + ϕ (12.4 ) 1 hμν = − --H2dSx μxμ + χμν (12.5 ) 2 Tμν = − λημν + τμν, (12.6 )

where $π,χ$ and $τ$ correspond to the fluctuations about the background solution.

For this ansatz, the background equations of motion reduce to

( 2) HdS a1 + 2a2q0 + 3a3q0 = 0 (12.7 ) 2 --λ-- 2Λ33- 2 3 H dS = 3M 2 + MPl (a1q0 + a2q0 + a3q0). (12.8 ) Pl

In the ‘self-accelerating branch’ when $HdS ⁄= 0$ , the first constraint can be used to infer $q0$ and the second one corresponds to the effective Friedmann equation. We see that even in the absence of a cosmological constant $λ = 0$ , for generic coefficients we have a constant $HdS$ solution which corresponds to a self-accelerating de Sitter solution.

The stability of these solutions can be analyzed by looking at the Lagrangian for the quadratic fluctuations

Thus, we see that the helicity zero mode is stable provided that

However, these solutions exhibit a peculiarity. To this order, the helicity-0 mode fluctuations do not couple to the matter perturbations (there is no kinetic mixing between $π$ and $χμν$ ). This means that there is no Vainshtein effect, but at the same time there is no vDVZ discontinuity for the Vainshtein effect to resolve!

Screening solution

Another way to solve the system of Eqs. (12.7 *) and (12.8 *) is to consider instead flat solutions $HdS = 0$ . Then (12.7 *) is trivially satisfied and we see the existence of a ‘screening solution’ in the Friedmann equation (12.8 *), which can accommodate a cosmological constant without any acceleration. This occurs when the helicity-0 mode ‘absorbs’ the contribution from the cosmological constant $λ$ , and the background configuration for $π$ parametrized by $q0$ satisfies

Perturbations about this screened configuration then behave as

In this case the perturbations are stable, and the Vainshtein mechanism is present which is necessary to resolve the vDVZ discontinuity. Furthermore, since the background contribution to the metric perturbation vanishes $hμν = 0$ , they correspond to Minkowski solutions which are sourced by a nonzero cosmological constant. In the case where $a3 = 0$ , these solutions only exist if $33a21- λ < MPl Λ 32a2$ . In the case where $a3 ⁄= 0$ , there is no upper bound on the cosmological constant which can be screened via this mechanism.

In this branch of solution, the strong coupling scale for fluctuations on top of this configuration becomes of the same order of magnitude as that of the screened cosmological constant. For a large cosmological constant the strong coupling scale becomes to large and the helicity-0 mode would thus not be sufficiently Vainshtein screened.

Thus, while these solutions seem to indicate positively that there are self-screening solutions which can accommodate a continuous range of values for the cosmological constant and still remain flat, the range is too small to significantly change the old cosmological constant problem. Nevertheless, the considerable difficulty in attacking the old cosmological constant problem means that these solutions deserve further attention as they also provide a proof of principle on how Weinberg’s no go could be evaded [484 ]. We emphasize that what prevents a large cosmological constant from being screened is not an issue in the theoretical tuning but rather an observational bound, so this is already a step forward.

These two classes of solutions are both maximally symmetric. However, the general cosmological solution is isotropic but inhomogeneous. This is due to the fact that a nontrivial time dependence for the matter source will inevitably source $B(t)$ , and as soon as $˙ B ⁄= 0$ the solutions are inhomogeneous. In fact, as we now explain in general, the full nonlinear solution is inevitably inhomogeneous due to the existence of a no-go theorem against spatially flat and closed FLRW solutions.

12.2 FLRW solutions in the full theory

12.2.1 Absence of flat/closed FLRW solutions

A nontrivial consequence of the fact that diffeomorphism invariance is broken in massive gravity is that there are no spatially flat or closed FLRW solutions [117 *]. This result follows from the different nature of the Hamiltonian constraint. For instance, choosing a spatially flat form for the metric $ds2 = − dt2 + a(t)2 d⃗x2$ , the mini-superspace Lagrangian takes the schematic form

Consistency of the constraint equation obtained from varying with respect to $N$ and the acceleration equation for $¨a$ implies

In GR, since $F1 (a ) = 0$ , there is no analogue of this equation. In the present case, this equation can be solved either by imposing $˙a = 0$ which implies the absence of any dynamic FLRW solutions, or by solving $∂F1(a)= 0 ∂a$ for fixed $a$ which implies the same thing. Thus, there are no nontrivial spatially flat FLRW solutions in massive gravity in which the reference metric is Minkowski. The result extends also to spatially closed cosmological solutions. As a result, different alternatives have been explored in the literature to study the cosmology of massive gravity. See Figure 7 * for a summary of these different approaches.

Figure 7: Alternative ways in deriving the cosmology in massive gravity.

12.2.2 Open FLRW solutions

While the previous argument rules out the possibility of spatially flat and closed FLRW solutions, open ones are allowed [283 ]. To see this we make the ansatz $2 2 2 2 ds = − dt + a(t) dΩ H3$ , where $2 d ΩH3$ expressed in the form

is the metric on a hyperbolic space, and express the reference metric in terms of Stückelberg fields $μ ν a b f&tidle;μν dx dx = ηab∂μϕ ∂ νϕ$ with

then the mini-superspace Lagrangian of (6.3 *) takes the form

aa˙2 [ ∘ --- ℒmGR = − 3M 2Pl|k|N a − 3M P2l----+ 3m2M 2Pl2a2 𝒳(2N a − f˙a − N |k|f N ∘ --- ] + α3a2𝒳 2(4N a − 3f˙a − N |k |f ) + 4α4a3 𝒳 3(N − f˙) ,

with $√ -- 𝒳 = 1 − --|k|f a$ . In this case, the analogue additional constraint imposed by consistency of the Friedmann and acceleration (Raychaudhuri) equation is

The solution for which $𝒳 = 0$ is essentially Minkowski spacetime in the open slicing, and is thus uninteresting as a cosmology.

Focusing on the other branch and assuming $𝒳 ⁄= 0$ , the general solution determines $f(t)$ in terms of $a(t)$ takes the form $1 f(t) = √-|k|ua(t)$ where $u$ is a constant determined by the quadratic equation

The resulting Friedmann equation is then

where

Despite the positive existence of open FLRW solutions in massive gravity, there remain problems of either strong coupling (due to absence of quadratic kinetic terms for physical degrees of freedom) or other instabilities which essentially rule out the physical relevance of these FLRW solutions [285 , 125 *, 464 ].

12.3 Inhomogenous/anisotropic cosmological solutions

As pointed out in [117 ], the absence of FLRW solutions in massive gravity should not be viewed as an observational flaw of the theory. On the contrary, the Vainshtein mechanism guarantees that there exist inhomogeneous cosmological solutions which approximate the normal FLRW solutions of GR as closely as desired in the limit $m → 0$ . Rather, it is the existence of a new physical length scale $1∕m$ in massive gravity, which cause the dynamics to be inhomogeneous at cosmological scales. If this scale $1∕m$ is comparable to or larger than the current Hubble radius, then the effects of these inhomogeneities would only become apparent today, with the universe locally appearing as homogeneous for most of its history in the local patch that we observe.

One way to understand how the Vainshtein mechanism recovers the prediction of homogeneity and isotropy is to work in the formulation of massive gravity in which the Stückelberg fields are turned on. In this formulation, the Stückelberg fields can exhibit order unity inhomogeneities with the metric remaining approximately homogeneous. Matter that couples only to the metric will perceive an effectively homogeneous and anisotropic universe, and only through interaction with the Vainshtein suppressed additional scalar and vector degrees of freedom would it be possible to perceive the inhomogeneities. This is achieved because the metric is sourced by the Stückelberg fields through terms in the equations of motion which are suppressed by $2 m$ . Thus, as long as $2 R ≫ m$ , the metric remains effectively homogeneous and isotropic despite the existence of no-go theorems against exact homogeneity and isotropy.

In this regard, a whole range of exact solutions have been studied exhibiting these properties [364 , 474 *, 363 , 97 , 264 , 356 , 456 , 491 , 476 *, 334 , 478 *, 265 , 124 , 123 *, 125 , 455 , 198 ]. A generalization of some of these solutions was presented in Ref. [404 ] and Ref. [266 ]. In particular, we note that in [475 *, 476 ] the most general exact solution of massive gravity is obtained in which the metric is homogeneous and isotropic with the Stückelberg fields inhomogeneous. These solutions exist because the effective contribution to the stress energy tensor from the mass term (i.e., viewing the mass term corrections as a modification to the energy density) remains homogeneous and isotropic despite the fact that it is build out of Stückelberg fields which are themselves inhomogeneous.

Let us briefly discuss how these solutions are obtained.³² As we have already discussed, all solutions of massive gravity can be seen as $M → ∞ f$ decoupling limits of bi-gravity. Therefore, we may consider the case of inhomogeneous solutions in bi-gravity and the solutions of massive gravity can always be derived as a limit of these bi-gravity solutions. We thus begin with the action

2 ∫ 2 ∫ ∘ ---- S = M-Pl d4x√ −-gR [g] + M-f- d4x − f R[f] (12.22 ) 2 2 m2M 2 ∫ √ ---∑ βn √ -- + -----eff d4x − g --ℒn ( 𝕏 ) + Matter, 4 n n!

where $−2 −2 −2 M eff = M Pl + M f$ and $√ -- ∘ -−-1- 𝕏 = g f$ and we may imagine matter coupled to both $f$ and $g$ but for simplicity let us imagine matter is either minimally coupled to $g$ or it is minimally coupled to $f$ .

12.3.1 Special isotropic and inhomogeneous solutions

Although it is possible to find solutions in which the two metrics are proportional to each other $2 fμν = C gμν$ [478 *], these solutions require in addition that the stress energies of matter sourcing $f$ and $g$ are proportional to one another. This is clearly too restrictive a condition to be phenomenologically interesting. A more general and physically realistic assumption is to suppose that both metrics are isotropic but not necessarily homogeneous. This is covered by the ansatz

ds2g = − Q (t,r)2 dt2 + N (t,r)2dr2 + R2 d2ΩS2, (12.23 ) 2 2 2 dsf = − (a(t,r)Q(t,r) dt + c(t,r)N (t,r) dr) (12.24 ) + (c(t,r)Q (t,r)dt − b(t,r)n(t,r)N (t,r) dr)2 + u (t,r)2R2 d2ΩS2,

and $2 d ΩS2$ is the metric on a unit 2-sphere. To put the $g$ metric in diagonal form we have made use of the one copy of overall diff invariance present in bi-gravity. To distinguish from the bi-diagonal case we shall assume that $c(t,r ) ⁄= 0$ . The bi-diagonal case allows for homogeneous and isotropic solutions for both metrics which will be dealt with in Section 12.4.2. The square root may be easily taken to give

( ) a cN ∕Q 0 0 √ -- | − cN∕Q b 0 0 | 𝕏 = |( |) , (12.25 ) 0 0 u 0 0 0 0 u

which can easily be used to determine the contribution of the mass terms to the equations of motion for $f$ and $g$ . This leads to a set of partial differential equations for $Q, R, N, n,c,b$ which in general require numerical analysis. As in GR, due to the presence of constraints associated with diffeomorphism invariance, and the Hamiltonian constraint for the massive graviton, several of these equations will be first order in time-derivatives. This simplifies matters somewhat but not sufficiently to make analytic progress. Analytic progress can be made however by making additional more restrictive assumptions, at the cost of potentially losing the most physically interesting solutions.

Effective cosmological constant

For instance, from the above form we may determine that the effective contribution to the stress energy tensor sourcing $g$ arising from the mass term is of the form

If we make the admittedly restrictive assumption that the metric $g$ is of the FLRW form or is static, then this requires that $T0r$ =0 which for $c ⁄= 0$ implies

This should be viewed as an equation for $u (t,r)$ whose solution is

Then conservation of energy imposes further

2 ( ) ( ) Tmass00 − Tmass𝜃𝜃=− m2 M-eff- 1β2 + 1-β3u (u − a)(u − b) + c2 M P2l 2 4 = 0, (12.29 )

since $u$ is already fixed we should view this generically as an equation for $c(t,r)$ in terms of $a(t,r)$ and $b(t,r)$

With these assumptions the contribution of the mass term to the effective stress energy tensor sourcing each metric becomes equivalent to a cosmological constant for each metric $T μ (g) = − Λ δμ mass ν g ν$ and $μ μ Tmass ν(f) = − Λf δ ν$ with

2 2 ( ) Λ = − m--M-eff 6β + 2β u + 1-β u2 , (12.31 ) g M 2Pl 0 1 2 2 2 2 ( ) Λf = − m--M-eff 1β2 + 1-β3u + 1β4u2 . (12.32 ) M f2u2 2 2 4

Thus, all of the potential dynamics of the mass term is reduced to an effective cosmological constant. Let us stress again that this rather special fact is dependent on the rather restrictive assumptions imposed on the metric $g$ and that we certainly do not expect this to be the case for the most general time-dependent, isotropic, inhomogeneous solution.

Massive gravity limit

As usual, we can take the $Mf → 0$ limit to recover solutions for massive gravity on Minkowski (if $Λf → 0$ ) or more generally if the scaling of the parameters $βn$ is chosen so that $Λf$ and $( 1 2) 6β0 + 2β1u + 2β2u$ and hence $Λg$ remains finite in the limit then these will give rise to solutions for massive gravity for which the reference metric is any Einstein space for which

For example, this includes the interesting cases of de Sitter and anti-de Sitter reference metrics.

Thus, for example, assuming no additional matter couples to the $f$ metric, both bi-gravity and massive gravity on a fixed reference metric admit exact cosmological solutions for which the $f$ metric is de Sitter or anti-de Sitter

( ) 2 2 2 dr2 2 2 dsg = − dt + a(t) ------2-+ r dΩ (12.34 ) (1 − kr ) 2 2 -dU-2- 2 2 dsf = − Δ (U )dT + Δ (U) + U d Ω , (12.35 )

where $Λf Δ (U ) = 1 − -3 U 2$ , and the scale factor $a(t)$ satisfies

where $ρM (t)$ is the energy density of matter minimally coupled to $g$ , $H = ˙a∕a$ , and $U, T$ can be expressed as a function of $r$ and $t$ and comparing with the previous representation $U = ur$ . The one remaining undetermined function is $T (t,r)$ and this is determined by the constraint that $2 (u − a)(u − b) + c = 0$ and the conversion relations

∘ ------ 1 Δ(U )dT = a(t,r)dt + c(t,r)√-------2-dr, (12.37 ) 1 − kr ---1---- ----1----- ∘ Δ (U ) dU = c(t,r)dt − b(t,r)n(t,r)√ 1 − kr2 dr, (12.38 ) U (t,r) = ur, (12.39 )

which determine $b(t,r)$ and $c(t,r)$ in terms of $T˙$ and $T′$ . These relations are difficult to solve exactly, but if we consider the special case $Λf = 0$ which corresponds in particular to massive gravity on Minkowski then the solution is

where $q$ is an integration constant.

In particular, in the open universe case $Λf = 0$ , $k = 0$ , $q = u$ , $∘ --------- T = ua 1 + |k |r2$ , we recover the open universe solution of massive gravity considered in Section 12.2.2, where for comparison $f (t) = √-1-ua (t) |k|$ , $ϕ0(t,r) = T(t,r)$ and $ϕr = U(t,r)$ .

12.3.2 General anisotropic and inhomogeneous solutions

Let us reiterate again that there are a large class of inhomogeneous but isotropic cosmological solutions for which the effective Friedmann equation for the $g$ metric is the same as in GR with just the addition of a cosmological constant which depends on the graviton mass parameters. However, these are not the most general solutions, and as we have already discussed many of the exact solutions of this form considered so far have been found to be unstable, in particular through the absence of kinetic terms for degrees of freedom which implies infinite strong coupling. However, all the exact solutions arise from making a strong restriction on one or the other of the metrics which is not expected to be the case in general. Thus, the search for the ‘correct’ cosmological solution of massive gravity and bi-gravity will almost certainly require a numerical solution of the general equations for $Q,R, N, n,c,b$ , and their stability.

Closely related to this, we may consider solutions which maintain homogeneity, but are anisotropic [284 *, 393 *, 123 *]. In [393 ] the general Bianchi class A cosmological solutions in bi-gravity are studied. There it is shown that the generic anisotropic cosmological solution in bi-gravity asymptotes to a self-accelerating solution, with an acceleration determined by the mass terms, but with an anisotropy that falls off less rapidly than in GR. In particular the anisotropic contribution to the effective energy density redshifts like non-relativistic matter. In [284 , 123 ] it is found that if the reference metric is made to be of an anisotropic FLRW form, then for a range of parameters and initial conditions stable ghost free cosmological solutions can be found.

These analyses are ongoing and it has been uncovered that certain classes of exact solutions exhibit strong coupling instabilities due to vanishing kinetic terms and related pathologies. However, this simply indicates that these solutions are not good semi-classical backgrounds. The general inhomogeneous cosmological solution (for which the metric is also inhomogeneous) is not known at present, and it is unlikely it will be possible to obtain it exactly. Thus, it is at present unclear what are the precise nonlinear completions of the stable inhomogeneous cosmological solutions that can be found in the decoupling limit. Thus the understanding of the cosmology of massive gravity should be regarded as very much work in progress, at present it is unclear what semi-classical solutions of massive gravity are the most relevant for connecting with our observed cosmological evolution.

12.4 Massive gravity on FLRW and bi-gravity

12.4.1 FLRW reference metric

One straightforward extension of the massive gravity framework is to allow for modifications to the reference metric, either by making it cosmological or by extending to bi-gravity (or multi-gravity). In the former case, the no-go theorem is immediately avoided since if the reference metric is itself an FLRW geometry, there can no longer be any obstruction to finding FLRW geometries.

The case of massive gravity with a spatially flat FLRW reference metric was worked out in [223 *], where it was found that if using the convention for which the massive gravity Lagrangian is (6.5 *) with the potential given in terms of the coefficient $β′s$ as in (6.23 *), then the Friedmann equation takes the form

Here the dynamical and reference metrics in the form

and the Hubble constants are related by

Ensuring a nonzero ghost-free kinetic term in the vector sector requires us to always solve this equation with

so that the Friedmann equation takes the form

where $Hf$ is the Hubble parameter for the reference metric. By itself, this Friedmann equation looks healthy in the sense that it admits FLRW solutions that can be made as close as desired to the usual solutions of GR.

However, in practice, the generalization of the Higuchi consideration [307 ] to this case leads to an unacceptable bound (see Section 8.3.6).

It is a straightforward consequence of the representation theory for the de Sitter group that a unitary massive spin-2 representation only exists in four dimensions for $m2 ≥ 2H2$ as was the case in de Sitter. Although this result only holds for linearized fluctuations around de Sitter, its origin as a bound comes from the requirement that the kinetic term for the helicity zero mode is positive, i.e., the absence of ghosts in the scalar perturbations sector. In particular, the kinetic term for the helicity-0 mode $π$ takes the form

Thus, there should exist an appropriate generalization of this bound for any cosmological solution of nonlinear massive gravity for which there an FLRW reference metric.

This generalized bound was worked out in [223 ] and takes the form

Again, by itself this equation is easy to satisfy. However, combined with the Friedmann equation, we see that the two equations are generically in conflict if in addition we require that the massive gravity corrections to the Friedmann equation are small for most of the history of the Universe, i.e., during radiation and matter domination

This phenomenological requirement essentially rules out the applicability of FLRW cosmological solutions in massive gravity with an FLRW reference metric.

This latter problem which is severe for massive gravity with dS or FLRW reference metrics,³³ gets resolved in bi-gravity extensions, at least for a finite regime of parameters.

12.4.2 Bi-gravity

Cosmological solutions in bi-gravity have been considered in [474 , 479 , 104 , 106 *, 8 *, 475 , 478 , 9 *, 7 , 62 *]. We keep the same notation as previously and consider the action for bi-gravity as in (5.43 *) (in terms of the $β$ ’s where the conversion between the $β$ ’s and the $α$ ’s is given in (6.28 *))

2 √ --- M 2∘ ---- ℒbi−gravity = M-Pl − gR [g] +--f- − f R[f] 2 2 m2M 2 ∑4 βn √ -- + -----Pl ---ℒn[ 𝕏 ] + ℒmatter[g,ψi], (12.50 ) 4 n=0 n!

assuming that matter only couples to the $g$ metric. Then the two Friedmann equations for each Hubble parameter take the respective form

( ) 2 ∑3 3(4 − n)m2 βn H n 1 H = − -------------- --- + ---2-ρ (12.51 ) n=0 [ 2n! Hf 3]M Pl 2 ∑3 2 ( )n− 3 H2f = − M-Pl 3m--βn+1- H-- . (12.52 ) M f2 n=0 2n! Hf

Crucially, the generalization of the Higuchi bound now becomes

The important new feature is the last term in square brackets. Although this tends to unity in the limit $Mf → ∞$ , which is consistent with the massive gravity result, for finite $Mf$ it opens a new regime where the bound is satisfied by having $( ) HfMPl-2 ≫ 1 HMf$ (notice that in our convention the $β$ ’s are typically negative). One may show [224 ] that it is straightforward to find solutions of both Friedmann equations which are consistent with the Higuchi bound over the entire history of the universe. For example, choosing the parameters $β2 = β3 = 0$ and solving for $Hf$ the effective Friedmann equation for the metric which matter couples to is

and the generalization of the Higuchi bound is

which is trivially satisfied at all times. More generally, there is an open set of such solutions. The observationally viability of the self-accelerating branch of these models has been considered in [8 , 9 ] with generally positive results. Growth histories of the bi-gravity cosmological solutions have been considered in [62 ]. However, while avoiding the Higuchi bound indicates absence of ghosts, it has been argued that these solutions may admit gradient instabilities in their cosmological perturbations [106 ].

We should stress again that just as in massive gravity, the absence of FLRW solutions should not be viewed as an inconsistency of the theory with observations, also in bi-gravity these solutions may not necessarily be the ones of most relevance for connecting with observations. It is only that they are the most straightforward to obtain analytically. Thus, cosmological solutions in bi-gravity, just as in massive gravity, should very much be viewed as a work in progress.

12.5 Other proposals for cosmological solutions

Finally, we may note that more serious modifications the massive gravity framework have been considered in order to allow for FLRW solutions. These include mass-varying gravity and the quasi-dilaton models [119 *, 118 ]. In [281 ] it was shown that mass-varying gravity and the quasi-dilaton model could allow for stable cosmological solutions but for the original quasi-dilaton theory the self-accelerating solutions are always unstable. On the other hand, the generalizations of the quasi-dilaton [126 *, 127 *] appears to allow stable cosmological solutions.

In addition, one can find cosmological solutions in non-Lorentz invariant versions of massive gravity [109 *] (and [103 , 107 *, 108 *]). We can also allow the mass to become dependent on a field [489 , 375 ], extend to multiple metrics/vierbeins [454 ], extensions with $f(R )$ terms either in massive gravity [89 ] or in bi-gravity [416 , 415 ] which leads to interesting self-accelerating solutions. Alternatively, one can consider other extensions to the form of the mass terms by coupling massive gravity to the DBI Galileons [237 , 19 , 20 , 315 ].

As an example, we present here the cosmology of the extension of the quasi-dilaton model considered in [127 *], where the reference metric $¯ fμν$ is given in (9.15 *) and depends explicitly on the dynamical quasi-dilaton field $σ$ .

The action takes the familiar form with an additional kinetic term introduced for the quasi-dilaton which respects the global symmetry

∫ √ ---[ 1 ω S = M 2Pl d4x − g -R − Λ − ----2(∂ σ)2 2 2M Pl ] m2 ( ) + --- ℒ2[&tidle;𝒦] + α3ℒ3 [𝒦&tidle;] + α4 ℒ4[&tidle;𝒦 ] , (12.56 ) 4

where the tensor $𝒦&tidle;$ is given in (9.14 *).

The background ansatz is taken as

2 2 2 2 2 ds = − N (t)dt + a(t) d⃗x , (12.57 ) ϕ0 = ϕ0(t), (12.58 ) i i ϕ = x, (12.59 ) σ = σ(t), (12.60 )

so that

f dx μdx ν = − n (t)2dt2 + d⃗x2, (12.61 ) μν n(t)2 = (ϕ˙0(t))2 + --ασ---˙σ2. (12.62 ) M 2Plm2

The equation that for normal massive gravity forbids FLRW solutions follows from varying with respect to $0 ϕ$ and takes the form

where

As the universe expands $X (1 − X )J ∼ 1∕a4$ which for one branch of solutions implies $J → 0$ which determines a fixed constant asymptotic value of $X$ from $J = 0$ . In this asymptotic limit the effective Friedmann equation becomes

where

defines an effective cosmological constant which gives rise to self-acceleration even when $Λ = 0$ (for $ω < 6$ ).

The analysis of [127 ] shows that these self-accelerating cosmological solutions are ghost free provided that

where

In particular, this implies that $α σ > 0$ which demonstrates that the original quasi-dilaton model [119 , 116 ] has a scalar (Higuchi type) ghost. The analysis of [126 ] confirms these properties in a more general extension of this model.

	Abstract
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
	Acknowledgments
	References
	Footnotes
	Figures