Cosmological Stability Bound in Massive Gravity and Bigravity
We give a simple derivation of a cosmological bound on the graviton mass for spatially flat FRW solutions in massive gravity with an FRW reference metric and for bigravity theories. This bound comes from the requirement that the kinetic term of the helicity zero mode of the graviton is positive definite. The bound is dependent only on the parameters in the massive gravity potential and the Hubble expansion rate for the two metrics.
We derive the decoupling limit of bigravity and FRW massive gravity, and use this to give an independent derivation of the cosmological bound. We recover our previous results that the tension between satisfying the Friedmann equation and the cosmological bound is sufficient to rule out all observationally relevant FRW solutions for massive gravity with an FRW reference metric. In contrast, in bigravity this tension is resolved due to different nature of the Vainshtein mechanism. We find that in bigravity theories there exists an FRW solution with late-time self-acceleration for which the kinetic terms for the helicity-2, helicity-1 and helicity-0 are generically nonzero and positive making this a compelling candidate for a model of cosmic acceleration.
We confirm that the generalized bound is saturated for the candidate partially massless (bi)gravity theories but the existence of helicity-1/helicity-0 interactions implies the absence of the conjectured partially massless symmetry for both massive gravity and bigravity.
Prepared for submission to JCAP
Cosmological Stability Bound in Massive Gravity and Bigravity
CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USA
- 1 Introduction
- 2 Deriving the bound in Massive Gravity on FRW
- 3 Generalizing the Bound to Bigravity
- 4 Decoupling Limit of Bigravity
- 5 Bigravity Decoupling Limit Derivation of Stability Bound
- 6 Partially Massless (Bi)Gravity
- 7 Conclusions
Attempts at modifying general relativity have a rich and interesting history. In recent years, this effort has been fueled by the observation [1, 2] that the universe is currently accelerating: deviations from GR at large scales, a weaker gravity, or gravitationally induced self-acceleration, might account for such a dynamics. What one hopes to gain in having an accelerated universe through modified gravity is something perhaps less mundane and certainly more appealing than a cosmological-constant-driven acceleration [3, 4]. The cosmological constant problem manifests itself in all theories of modified gravity, in massive gravity (MG) it translates directly into the requirement of a small mass . It has been argued that a graviton mass can be kept small in a technically natural way [5, 6].
The first theory formulation of massive gravity (dRGT) in which the number of degrees of freedom nonlinearly is 5 was given in [7, 8] (see [9, 10, 11, 12, 13, 14, 15, 16, 17] for proofs of the absence of the Boulware-Deser ghostly 6th mode ), and this theory was subsequently generalized to arbitrary reference metric  and to bigravity  where both metrics are dynamical. In the following we shall be concerned with spatially flat FRW solutions in these two theories, i.e. solutions where both the dynamical and the reference metric are homogenous and isotropic.
Finding observationally relevant cosmological solutions in these theories has received considerable attention [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. In massive gravity with a Minkowski reference metric, it was shown that no homogeneous and isotropic spatially flat solutions arise . However, it was argued in  that observationally consistent inhomogenous solutions should exist (see also [49, 50, 51, 52, 53, 54, 55, 56] for different approaches). Unfortunately these inhomogenous/anisotropic solutions have been hard to find except in special cases and many of these special cases have been shown to be infinitely strongly coupled. One simple resolution is to look at massive gravity theories with an FRW reference metric  or for bigravity theories in which FRW solutions are allowed [21, 24, 30, 36, 47, 24, 25]. It is these classes of solution that we shall be interested in for this work.
We will see that generically the FRW solutions in bigravity and massive gravity with an FRW reference metric (FRW massive gravity) are theoretically well defined in the sense that cosmological perturbations admit nonzero positive kinetic terms for all 5 degrees of freedom. However, for FRW massive gravity, when we go into the observationally relevant Vainshtein region [58, 59, 60] for which the normal Friedmann equation is recovered, the helicity-0 mode becomes a ghost. This result follows because these solutions begin to violate a generalization of a cosmological bound on the mass of graviton first discovered by Higuchi  (see also [62, 63, 64])
This generalization of the Higuchi bound to FRW massive gravity was derived in  and in this manuscript we will give two independent simpler derivations of this result. The first derivation makes use of the minisuperspace action to look for the instability, the second utilizes the massive (bi)gravity decoupling limit. The result obtained in  was that the Higuchi bound in massive gravity for an arbitrary FRW reference metric and arbitrary matter coupled to the dynamical metric is
where the dressed mass is given by
is the dynamical Hubble parameter, and is that of the reference metric . The ’s are the parameters in the mass potential.
The problem is that it is essentially impossible to satisfy this bound in the Vainshtein region in which the modified gravity contributions to the Friedmann equation are subdominant. This arises because both the Friedmann equation and the bound are polynomials of the same order in . On the one hand, recovery of standard cosmology requires that the mass is small compared to a polynomial in ; on the other hand the bound requires that the mass is large compared to another a polynomial of the same order in .
In this manuscript we shall show that this tension between the Higuchi bound and the Vainshtein mechanism is generically resolved in bigravity theories, even when the Planck mass for the metric is much larger than the usual Planck mass . This result is at first surprising, since the corrections to the action for the helicity-zero mode come suppressed by the ratio . The resolution, is that even these small corrections can actually dominate the dynamics of the helicity-zero mode in the Vainshtein region because we simultaneously have .
Using the two new methods of derivation, we show that in bigravity, by giving full dynamics to what is the fixed metric , the bound is relaxed down to:
where the same dressed mass enters. This is equivalent to saying that in bigravity the ‘dressed mass’, i.e. the dynamical mass of the massive graviton, is given by . Just as in the massive gravity case, this bound is valid for arbitrary matter content and for arbitrary FRW (i.e. it is not specific to de Sitter). Special cases of these results have been discussed in FRW perturbation analyses of [25, 34]), however by giving an independent derivation of them, we shall also elucidate the dynamics of the helicity-0 mode in the Vainshtein region.
The above condition allows one to shift the burden of satisfying the bound onto the value of Hubble rate for the metric, . Enriching massive gravity with the dynamics then proves crucial for relaxing the Higuchi bound. As expected, the massive gravity Higuchi bound is recovered in the decoupling limit. This bound may also be expressed in a more symmetric way which makes manifest the symmetry of the bigravity action under and and :
An intriguing, special, case of the Higuchi bound is what characterizes the so-called partially massless (PM) theory of massive gravity [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. There the Higuchi inequality is actually saturated for all values of regardless of the matter coupling to the metric and the Higuchi-Vainshtein tension is resolved in that massive gravity parameters act in such a way that one of the two scale factors becomes pure gauge.
The special parameter values for the potentially partially massless theory were first identified in massive gravity  and they are the same parameters that have also been used in bigravity . They correspond to , , , . For this special case, the bound becomes:
but, using the Friedmann equation for the metric, we have in this specific case that
Substituting in we find that the bound becomes an identity regardless of the value of and . This represents a nontrivial check on the consistency of the derived bound.
We shall give two independent derivations of the new bounds. The first is the simplest and utilizes the minisuperspace action alone. We present the reasoning here since it may be useful in similar analyses for more general theories. The idea is that since the bound comes from requiring the absence of ghosts for the helicity-0 mode, and the helicity-0 mode being a scalar already enters into the minisuperspace Lagrangian111Minisuperspace is the truncation of a gravity theory to the case where the metric and all fields are functions of time alone. In other words it the compactification down to dimensions., it should be possible to identify the sign of the kinetic term for the helicity-zero mode by analyzing the minisuperspace action alone. Performing this analysis for massive gravity we recover our result of  which was obtained by a much more laborious Hamiltonian analysis. This method easily generalizes to the bigravity case and for generic matter couplings.
The second derivation utilizes the decoupling limit of massive gravity and the decoupling limit of bigravity which we derive here in full, including the vector degrees of freedom following the approach of  (see also ). It was not guaranteed that this approach would agree with the exact answer since in principle it is possible that the exact bound contains terms which vanish in the decoupling limit. However, this is not the case. Taking the bigravity bound, and scaling , keeping the ratios , , and the parameters fixed, we find that the bound remains the same since it can be expressed entirely in terms of the fixed ratios
Since this is the limit used in deriving the massive gravity and bigravity decoupling limit, it follows that the cosmological bound can be entirely determined by the decoupling limit. The decoupling limit also allows us to go beyond the linearized analysis of  since it captures the leading nonlinear interactions that will be relevant in cosmological perturbation theory.
Finally, our derivation of the decoupling limit for bigravity leads to a new surprise, a dual formulation of Galileons. In massive gravity, the ‘Galileon’  arises as the helicity-0 mode in the map between the coordinate systems of the metric () and the metric () , namely
we will prove that this transformation admits an inverse of the form
is the dual Galileon field which is related to by a nonlocal field redefinition. It is the Galileon field viewed from the perspective of the metric. The decoupling limit of bigravity must be symmetric with respect to interchange of the two metrics, and hence it is symmetric with respect to the interchange of and . We shall discuss this duality in more detail elsewhere .
The paper is organized as follows. In Section 2 we introduce the dRGT theory of massive gravity and determine the bound using the minisuperspace approach. In Section 3 we generalize the stability bound to bigravity using the minisuperspace method and find that it is much easier to satisfy than in massive gravity. In Section 4 we derive the decoupling limit of bigravity and by extension that of massive gravity on a generic reference metric. In Section 5 we use the bigravity decoupling limit to give an independent derivation of the cosmological bounds. Finally, we discuss the conjectured partially massless gravity and bigravity in Section 6, establish that they saturate the bound, and discuss their validity as candidate partially massless theories. In the Appendix Appendix: Fluctuations in de Sitter Bigravity we give the details of the bound for fluctuations around de Sitter.
2 Deriving the bound in Massive Gravity on FRW
We now give a new derivation of the cosmological bound in massive gravity with a spatially flat FRW reference metric. This confirms the result obtained in  and we present it here as a warm up to the analogous bigravity calculation in the next section.
The theory of massive gravity defined on an arbitrary reference metric  is just a straightforward generalization of the theory proposed in . The Lagrangian takes the form of Einstein gravity with matter plus a potential that is a scalar function of the two metrics
The most general potential that has no ghosts  is built out of characteristic (symmetric) polynomials of the eigenvalues of the tensor
where the are free parameters, and
where represents the trace of a tensor with respect to the metric . The are the symmetric polynomials which are generally defined in dimensions by the determinant relation
so that and . The Lagrangian may also be written in the form
where . The relationship between the coefficients and the is given in Eq.(2.1). The mass term is invariant under the simultaneous interchange and .
As we shall see in detail, the generalization of the Higuchi bound can already be seen at the level of the minisuperspace action, in particular in the representation of massive gravity which includes the Stückelberg fields. The Stückelberg fields for diffeomorphisms are introduced by replacing the reference metric by its representation in an arbitrary coordinate system
The are the Stückelberg fields (or Goldstone modes for the broken diffemorphisms) and essentially encode the additional degrees of freedom that a massive graviton has over a massless one. These additional degrees of freedom are, in the high energy limit, decomposable into 2 helicity-1 modes and 1 helicity-0 mode. Explicitly, writing , then has the interpretation of the helicity-1 mode in the high energy limit, and is the additional helicity-0 mode. Since the bound comes from identifying the sign of the kinetic term of the helicity-0 mode, it is essentially enough to keep track of the term (we may for instance choose a gauge for which to aid in this identification).
However as is well known, in massive gravity, either part or all of the kinetic term for the helicity-0 mode actually comes from a mixing of and the metric . In perturbations, this shows up in the fact that a scalar part of couples to and hence . In the minisuperspace limit, the only scalar part of the metric is the scale factor itself 222Although the lapse is also a scalar, it is a Lagrange multiplier for a constraint and drops out of the action if the constraint is solved.. Thus we can identify the sign of the kinetic term for perturbations around a background that identifies kinetic term for , the mixing between and , and any independent kinetic term for the helicity-0 which enters as . This method is significantly simpler than our previous complete calculation  and leads quite quickly to the same result.
2.1 Minisuperspace Derivation
Thus, we begin with the action for minisuperspace for the dRGT model on a given FRW background written in an arbitrary gauge by means of the introduction of a Stueckelberg field for the broken time diffeomorphisms. The reference metric is given by
where is the Stückelberg field which will keep track of the helicity-0 mode kinetic term . In this section, for the sake of simplicity, we will limit ourselves to the case of a de Sitter reference metric. The more general case is dealt with in the bigravity setup of Section 3. The dynamical metric is similarly expressed as
The dRGT square root combination takes now the form
where is the volume factor and is the matter density.
The coefficients and are subject to the relation and in terms of the and , these coefficients are given by
In massive gravity, we can set since this term is a total derivative. In bigravity this term enters as the cosmological constant sourcing the metric and thus should in general be maintained.
The Friedmann equation, which can be obtained by varying the action with respect to , is:
where the extra ‘dark energy’ contribution from the massive gravity action is
Varying the action with respect to imposes, as a consequence of the special relation between the coefficients and , the constraint
At first sight there appear to be two branches of solutions to this equation. However, if we choose the term in the first parenthesis to vanish, the kinetic term for the helicity-1 mode vanishes and all such solutions are infinitely strongly coupled. Thus the only acceptable solution is
From now on, we will work in the normal branch for which this relation is true (branch 2 in the language of ). This allows us to write the contribution to the Friedmann equation in the form
To determine the kinetic term for the helicity zero mode we will utilize some of the properties of the minisuperspace action. Starting from the action
we now perturb it to second order in . As appears only algebraically in the action, it can be integrated out. After these steps the action takes the form:
where . We see that, with the sign of the kinetic term of the helicity zero mode in mind, we need only worry about the coefficient (it is that is carrying the helicity zero mode, ) in the above : integrating out removes any -proportional term in the action for fluctuations so that the last two terms in (2.1) need not be specified, i.e. we do not need to diagonalize.
One can now make use of the following set of relations: the background equations of motion imply and we have , so that
Using these in Eq.(2.1) leads to a further simplified action, whose helicity-0 kinetic term reads
and we have defined the dressed mass parameter as
Thus, the Higuchi bound, which amounts to the statement that the kinetic term for the helicity zero mode is positive, reads
Upon neglecting the branch of solutions , which inevitably gives a ghost in the vector sector, the bound translates into the more familiar statement:
where is the dressed mass, a generalization of the bare . This is the generalization of the Higuchi bound first derived in . Once more, let us stress that this result follows regardless of the matter content and is independent of . This result is consistent with that derived in  by an entirely different analysis. We now have a more clear explanation of how it arises and, as we shall see, a quicker route to its extensions and applications.
2.2 Higuchi versus Vainshtein tension
We take now a moment to briefly reproduce the reasoning that lead in  to the conclusion that FRW on FRW solutions in massive gravity are ruled out. The crucial point is that the bound in Eq. (2.28) must be complemented with a condition on derived from consistency of the expansion history with observations. Departures from the General Relativity (GR) expansion history are negligible at large redshifts and hence large ; thus the dependent modifications to the normal Friedmann equation must be small, at least for most of the history of the universe. In massive gravity, continuity with GR is achieved through the Vainshtein mechanism and it is for this reason that we refer to these two opposing requirements as the Higuchi-Vainshtein tension.
In order for the dependent contribution in the Friedmann equation (2.16) to be subleading (Vainshtein regime) we require:
This is immediately at odds with the stability (Higuchi) condition. We can see why by combining the two inequalities into the single statement
Eq. (2.30) is essentially impossible to satisfy. If the term dominates the inequality is violated, if the dominates it is saturated and if dominates it is only just satisfied since . However, this is not enough, there needs to be a large hierarchy between the two sides otherwise there will be modifications to the Friedmann equation . Considering that the combined inequality must hold over different cosmological epochs, one comes to the realization that there is no room in the parameter space of FRW massive gravity for it to simultaneously satisfy the requirements of stability and consistency with observations.
We stress that this does not rule out massive gravity as a theory of current cosmological expansion. There are many paths one can follow in the search for solutions to massive gravity that, if not exactly FRW, at least resemble FRW in appropriate regions, see e.g. . We choose here to instead demand exactly FRW solutions and move on to give full dynamics to the reference metric , thus entering into the realm of bigravity theories.
3 Generalizing the Bound to Bigravity
The action for bigravity models which are free from the Boulware-Deser ghost is a simple extension of that for massive gravity 
It is straightforward to generalize the previous argument to the case of bigravity. We denote the now dynamical second metric as
The minisuperspace action is now
The Friedmann equation for the scale factor , which we assume to correspond to the metric with which our matter is coupled, is given by
In the following we will assume that the only matter sourcing the second metric is a cosmological constant and, as such, can be absorbed into a definition of . In this case the Friedmann equation for the second metric is simply
For convenience let us denote
where we have utilized the following change of variables . Let us also employ a simple yet generic matter Lagrangian coupled only with the metric:
so that we add to the minisuperspace action the term
We shall see that all details of the precise nature of the matter Lagrangian will drop out of the final answer. We now pass to the canonical phase space formulation by defining the canonically conjugate momenta
As a consequence, the canonical action can be written as
Varying with respect to and imposes the analogues of the Hamiltonian constraints which can be solved to remove (we will choose the negative solution corresponding to an expanding geometry)
and, using , we obtain
or, after solving for and integrating out,
As it stands, this action is time reparametrization invariant, and we may utilize this fact to choose the gauge , so that
We may now perturb this action, and , to second order so as to obtain the quadratic action for perturbations around a given background. We also note that the contribution from fluctuations of the potential will at most contribute a mixing term ; it does not therefore add up to the kinetic term whose sign we are after and can thus be ignored in what follows333Note that the variable here is the scalar in the matter Lagrangian, not to be confused with the Stueckelberg field in the massive gravity analysis of the previous section.
Up to an unimportant positive overall factor, the quadratic fluctuation action reads:
where we have defined the dimensionless variable out of as . We have also employed Eq. (2.16) and (2.19) to simplify the coefficients: this provides a more compact expression and is of course allowed as coefficients are purely background quantities.
Since carries the information on , the helicity zero mode, , we proceed to diagonalize the above expression so as to read off the coefficient. If we posit
in the new variables the mixing term vanishes and, requiring the coefficient be positive, amounts to:
that is, the generalized Higuchi bound.
We stress again that simple algebraic manipulations coupled with the properties of the (or, equivalently, the ) coefficients allow Eq. (3.18) to be expressed in its most symmetric and pleasing form, Eq.(1.4), which we reproduce here:
Let us reiterate that this bound was derived for fairly generic matter content, but matter itself enters into the final expression only indirectly through . Thus the bound is entirely one on the Hubble rate, regardless of the matter content of the universe, and may thus be universally applied at all cosmological epochs.
3.1 Higuchi versus Vainshtein Resolution
As stressed in , one should not simply declare cosmological solutions viable by merely looking for regions of the parameter space where Eq. (3.19) is satisfied. Rather, we ought to perform a joint analysis of the stability condition and the requirement stemming from observations that, for most of the history of the universe, the Friedmann equation is well approximated by its GR prediction. In essence this requires us to be in the Vainshtein regime for which the cosmological effects of the helicity-0 mode are negligible and are suppressed by powers of . Unfortunately these two requirements oppose each other, thereby creating the Higuchi-vs-Vainshtein tension that effectively rules out the viability of spatially flat FRW solutions in FRW massive gravity.
Let us see how bigravity resolves the Higuchi-Vainshtein tension; the ingredients are the generalized bound and the Friedmann equations, which we report below:
Now, the inequality in Eq. (3.18) is certainly satisfied, without requiring much on the part of , if we posit . In particular if we assume that is nonzero, as is generically the case, then the second relation in (3.1) and the definition (see 2.26) are, at leading order in , respectively:
From here, we take the ratio of the two expressions, solve for and plug it back into Eq. (3.18), to obtain:
which is, evidently, consistent with the initial assumption.
As for the first Friedmann equation in (3.1), we see that, by itself, it does not put too much strain on . The fact is that the dressed mass is actually allowed to be small because precisely the lower bound on that in the massive gravity case comes from the stability condition, is now taken care of by our being in the region.
We have thus shown that there exists a large region in the parameter space of the bigravity theory, , which does relax the stability bound in a way which is consistent with observations. The stability-vs-observation conflict is therefore resolved. This resolution requires that in the cosmological bound, the term dominates over . It is clear that in the FRW massive gravity limit, this could never be satisfied since that limit requires us to take for finite and finite and so inevitably the former is subdominant to the latter. Thus it is precisely the naively suppressed bigravity interactions, which are crucial in resolving the conflict.
3.2 Ghost-free Self-Accelerating Cosmology
Having established the regime we should be in to satisfy the bound, we can infer the effective Friedmann equation by solving
and then substituting back into the Friedmann equation to give
This gives a self-accelerating cosmology in which asymptotically in the future the Hubble parameter tends to a constant de Sitter value of magnitude
Thus we see that quite in general the condition , which is implicitly assumed in having dominate, is essentially satisfied even in the asymptotics 444Note that in the asymptotics it is acceptable to have , this is because, even if it implies via (3.18) that , the latter is not an issue since at late times we desire order unity departures from GR to account for cosmic acceleration.:
Also note that, since scales as , it becomes easier and easier to satisfy the condition as we go back in time.
To make this discussion more precise, we can specify to the model with , and (, ) . In this case the Friedmann equation is exactly
and the stability constraint is exactly
which is easily seen to be always satisfied since it simplifies to the trivially satisfied relation
Thus we automatically generate a ghost-free late-time self-accelerating cosmology whose Friedmann equation may be put in the form
and asymptotes to de Sitter with
This represents one subset of the parameter space for which the bound is satisfied at all times. This specific solution, as well as similar ones with , have been studied in [85, 86]. There the authors employ statistical methods to asses the viability of bigravity solutions as compared to that of the standard cosmology. Quite interestingly, the solution discussed, which we have shown to satisfy the stability bound, is on par with in their analysis.
4 Decoupling Limit of Bigravity
We shall now derive the complete decoupling limit for bigravity including vector degrees of freedom. This decoupling limit will allow us to give an independent derivation of the generalized Higuchi bound in bigravity, and also allow us to determine the stability of the vectors. Just as in the massive gravity decoupling limit, the bigravity decoupling limit focuses on the interactions at the lowest energy scale, those of the helicity-zero mode.
4.1 Vierbein formulation
Our derivation will utilize the vierbein formalism [87, 88] (see also [89, 90]) and will follow closely the notation of . The starting point is to introduce Stückelberg fields for the broken Lorentz invariance and diffeomorphism (diff) invariance. Just as adding a mass term to gravity breaks diffeomorphism invariance, adding a mass term to bigravity breaks two independent copies of the diffeomorphism group down to the single diagonal subgroup. In the Einstein-Cartan (vierbein) formalism, it similarly breaks two copies of local Lorentz transformations (LLTs) down to a single copy. It is helpful to reintroduce these symmetries, and this can be easily achieved using the Stückelberg trick.
Beginning in unitary gauge, the action for bigravity
can be expressed in vierbein form as