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.
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 ) with the perturbation expansion being good for distances small relative to the curvature radius of the geometry:31
The generic form for the helicity-0 mode which preserves isotropy near is[139*] this ansatz was used to derive the existence of both self-accelerating and screening solutions.
We start with the decoupling limit Lagrangian given in (8.38*). Following the same notation as in Ref.  we set , where the coefficients are given in terms of the ’s in (8.47*). The self-accelerating branch of solutions then corresponds to the ansatz
For this ansatz, the background equations of motion reduce to
The stability of these solutions can be analyzed by looking at the Lagrangian for the quadratic fluctuations
Another way to solve the system of Eqs. (12.7*) and (12.8*) is to consider instead flat solutions . 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 , and the background configuration for parametrized by satisfies
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 . 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 , and as soon as 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.
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 , the mini-superspace Lagrangian takes the schematic form7* for a summary of these different approaches.
While the previous argument rules out the possibility of spatially flat and closed FLRW solutions, open ones are allowed . To see this we make the ansatz , where expressed in the form6.3*) takes the form
Focusing on the other branch and assuming , the general solution determines in terms of takes the form where is a constant determined by the quadratic equation[285, 125*, 464].
As pointed out in , 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 . Rather, it is the existence of a new physical length scale in massive gravity, which cause the dynamics to be inhomogeneous at cosmological scales. If this scale 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 . Thus, as long as , 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.  and Ref. . 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 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
Although it is possible to find solutions in which the two metrics are proportional to each other [478*], these solutions require in addition that the stress energies of matter sourcing and 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 ansatz12.4.2. The square root may be easily taken to give
For instance, from the above form we may determine that the effective contribution to the stress energy tensor sourcing arising from the mass term is of the form
As usual, we can take the limit to recover solutions for massive gravity on Minkowski (if ) or more generally if the scaling of the parameters is chosen so that and and hence 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
Thus, for example, assuming no additional matter couples to the metric, both bi-gravity and massive gravity on a fixed reference metric admit exact cosmological solutions for which the metric is de Sitter or anti-de Sitter
In particular, in the open universe case , , , , we recover the open universe solution of massive gravity considered in Section 12.2.2, where for comparison , and .
Let us reiterate again that there are a large class of inhomogeneous but isotropic cosmological solutions for which the effective Friedmann equation for the 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 , and their stability.
Closely related to this, we may consider solutions which maintain homogeneity, but are anisotropic [284*, 393*, 123*]. In  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.
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 as in (6.23*), then the Friedmann equation takes the form
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 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 takes the form
This generalized bound was worked out in  and takes the form
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.
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 ’s where the conversion between the ’s and the ’s is given in (6.28*)) 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 and solving for the effective Friedmann equation for the metric which matter couples to is [8, 9] with generally positive results. Growth histories of the bi-gravity cosmological solutions have been considered in . 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 .
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.
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  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 , extensions with terms either in massive gravity  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 is given in (9.15*) and depends explicitly on the dynamical quasi-dilaton field .
The action takes the familiar form with an additional kinetic term introduced for the quasi-dilaton which respects the global symmetry9.14*).
The background ansatz is taken as
The analysis of  shows that these self-accelerating cosmological solutions are ghost free provided that[119, 116] has a scalar (Higuchi type) ghost. The analysis of  confirms these properties in a more general extension of this model.