Stable and unstable cosmological models in bimetric massive gravity
Nonlinear, ghost-free massive gravity has two tensor fields; when both are dynamical, the mass of the graviton can lead to cosmic acceleration that agrees with background data, even in the absence of a cosmological constant. Here the question of the stability of linear perturbations in this bimetric theory is examined. Instabilities are presented for several classes of models, and simple criteria for the cosmological stability of massive bigravity are derived. In this way, we identify a particular self-accelerating bigravity model, infinite-branch bigravity (IBB), which exhibits both viable background evolution and stable linear perturbations. We discuss the modified gravity parameters for IBB, which do not reduce to the standard CDM result at early times, and compute the combined likelihood from measured growth data and type Ia supernovae. IBB predicts a present matter density and an equation of state . The growth rate of structure is well-approximated at late times by . The implications of the linear instability for other bigravity models are discussed: the instability does not necessarily rule these models out, but rather presents interesting questions about how to extract observables from them when linear perturbation theory does not hold.
Testing gravity beyond the limits of the Solar System is an important task of present and future cosmology. The detection of any modification of Einstein’s gravity at large scales or in past epochs would be an extraordinary revolution and change our view of the evolution of the Universe.
A theory of a massless spin-2 field is either described by general relativity Gupta (1954); Weinberg (1965); Deser (1970); Boulware and Deser (1975); Feynman et al. (1996); Deser (2010) or unimodular gravity Barceló et al. (2014a, b).
Consequently, most modifications of gravity proposed so far introduce one or more new dynamical fields, in addition to the massless metric tensor of standard gravity. This new field is usually a scalar field, typically through the so-called Horndeski Lagrangian Horndeski (1974); Deffayet et al. (2011), or a vector field, such as in Einstein-aether models (see Refs. Jacobson (2007); Solomon and Barrow (2014) and references therein). A complementary approach which has gained significant attention in recent years is, rather than adding a new dynamical field, to promote the massless spin-2 graviton of general relativity to a massive one.
The history of massive gravity is an old one, dating back to 1939, when the linear theory of Fierz and Pauli was published Fierz and Pauli (1939). We refer the reader to the reviews Hinterbichler (2012); de Rham (2014) for a reconstruction of the steps leading to the modern approach, which has resulted in a ghost-free, fully nonlinear theory of massive gravity de Rham et al. (2011) (see also Refs. de Rham and Gabadadze (2010a, b); Hassan and Rosen (2011, 2012); Maggiore and Mancarella (2014)). A key element of these new forms of massive gravity is the introduction of a second tensor field, or “reference metric,” in addition to the standard metric describing the curvature of spacetime. When this reference metric is fixed (e.g., Minkowski), this theory propagates the five degrees of freedom of a ghost-free massive graviton.
However, the reference metric can also be made dynamical, as proposed in Refs. Hassan et al. (2012); Hassan and Rosen (2012). This promotes massive gravity to a theory of bimetric gravity. This theory is still ghost free and has the advantage of allowing cosmologically viable solutions. The cosmology of bimetric gravity has been studied in several papers, e.g., in Refs. von Strauss et al. (2012); Akrami et al. (2013a, b); Comelli et al. (2012); De Felice et al. (2013); Comelli et al. (2012); Volkov (2012). The main conclusion is that bimetric gravity allows for a cosmological evolution that can approximate the CDM universe and can therefore be a candidate for dark energy without invoking a cosmological constant. Crucially, the parameters and the potential structure leading to the accelerated expansion are thought to be stable under quantum corrections de Rham et al. (2013), in stark contrast to a cosmological constant, which would need to be fine-tuned against the energy of the vacuum Weinberg (1989); Martin (2012).
Bimetric gravity has been successfully compared to background data (cosmic microwave background, baryon acoustic oscillations, and type Ia supernovae) in Refs. von Strauss et al. (2012); Akrami et al. (2013a), and to linear perturbation data in Refs. Könnig and Amendola (2014); Solomon et al. (2014a). The comparison with linear perturbations has been undertaken on subhorizon scales assuming a quasistatic (QS) approximation, in which the potentials are assumed to be slowly varying. This assumption makes it feasible to derive the modification to the Poisson equation and the anisotropic stress, two functions of scale and time which completely determine observational effects at the linear level.
The quasistatic equations are, however, a valid subhorizon approximation only if the full system is stable for large wave numbers. Previous work Comelli et al. (2012); De Felice et al. (2014); Comelli et al. (2014) has identified a region of instability in the past.111This should not be confused with the Higuchi ghost instability, which affects most massive gravity cosmologies and some in bigravity, but is, however, absent from the simplest bimetric models which produce CDM-like backgrounds Fasiello and Tolley (2013). Here we investigate this problem in detail. We reduce the linearized Einstein equations to two equations for the scalar modes, and analytically determine the epochs of stability and instability for all the models with up to two free parameters which have been shown to produce viable cosmological background evolution. The behavior of more complicated models can be reduced to these simpler ones at early and late times.
We find that several models which yield sensible background cosmologies in close agreement with the data are in fact plagued by an instability that only turns off at recent times. This does not necessarily rule these regions of the bimetric parameter space out, but rather presents a question of how to interpret and test these models, as linear perturbation theory is quickly invalidated. Remarkably, we find that only a particular bimetric model — the one in which only the and parameters are nonzero (that is, the linear interaction and the cosmological constant for the reference metric are turned on) — is stable and has a cosmologically viable background at all times when the evolution is within a particular branch. This shows that a cosmologically viable bimetric model without an explicit cosmological constant (by which we mean the constant term appearing in the Friedmann equation) does indeed exist, and raises the question of how to nonlinearly probe the viability of other bimetric models.
This paper is part of a series dedicated to the cosmological perturbations of bimetric gravity and their properties, following Ref. Solomon et al. (2014a).
Ii Background equations
We start with the action of the form Hassan and Rosen (2012)
where are elementary symmetric polynomials and are free parameters. Here is the standard metric coupled to the matter fields in the matter Lagrangian, , while is a new dynamical tensor field with metric properties. In the following we express masses in units of and absorb the mass parameter into the parameters . The graviton mass is generally of order . The action then becomes
There has been some discussion in the literature over how to correctly take square roots. We will find solutions in which becomes zero at a finite point in time (and only at that time), and so it is important to determine whether to choose square roots to always be positive, or to change sign on either side of the point. This was discussed in some detail in Ref. Gratia et al. (2013) (see also Ref. Gratia et al. (2014)), where continuity of the vielbein corresponding to demanded that the square root not be positive definite. We will take a similar stance here, and make the only choice that renders the action differentiable at all times, i.e., such that the derivative of with respect to and is continuous everywhere. In particular, using a cosmological background with , this choice implies that we assume , where with is the -metric Hubble rate. This is important because, as we will see later on, it turns out that in the cosmologically stable model, the metric bounces, so changes sign during cosmic evolution. Consequently the square roots will change sign as well, rather than develop cusps. Note that sufficiently small perturbations around the background will not lead to a different sign of this square root.
Varying the action with respect to , one obtains the following equations of motion:
Here the matrices are defined as, setting ,
where is the identity matrix and is the trace operator. Varying the action with respect to we find
where the overbar indicates the curvature of the metric.
The -metric Planck mass, , is a redundant parameter and can be freely set to unity Berg et al. (2012). To see this, consider the rescaling . The Ricci scalar transforms as , so the full Einstein-Hilbert term in the action becomes
The other term in the action that depends on is the mass term, which transforms as
where in the last equality we used the fact that the elementary symmetric polynomials are of order . Therefore, by additionally redefining the interaction couplings as , we end up with the original bigravity action but with 222Recall that we are expressing masses in units of the Planck mass, . In more general units, the redundant parameter is ..Consequently we set in the following.
Let us now consider the background cosmology of bimetric gravity. We assume a spatially flat FLRW metric,
where is conformal time and an overdot represents the derivative with respect to it. The second metric is chosen as
where is the conformal-time Hubble parameter associated with the physical metric, . The particular choice for the -metric lapse, , ensures that the Bianchi identity is satisfied (see, e.g., Ref. Hassan et al. (2012)).
Inserting the FLRW ansatz for into Eq. (5) we get
where we define an effective massive-gravity energy density as
while is the density of all other matter components (e.g., dust and radiation). The total energy density follows the usual conservation law,
It is useful to define the density parameter for the mass term (which will be the effective dark energy density):
where for matter and radiation.
The background dynamics depend entirely on the the -metric Hubble rate, , and the ratio of the two scale factors, Akrami et al. (2013a). Moreover, by using as time variable, with denoting derivatives with respect to , the background equations can be conveniently reformulated as a first-order autonomous system Koennig et al. (2014):
and denotes the equation of state corresponding to the sum of matter and radiation density parameter . We can define the effective equation of state
from which we obtain
Another useful relation gives the Hubble rate in terms of without an explicit dependence,
The background evolution of will follow Eq. (21) from an initial value of until , unless hits a singularity. In Ref. Koennig et al. (2014) it was shown that cosmologically viable evolution can take place in two distinct ways, depending on initial conditions: when evolves from 0 to a finite value (we call this a finite branch) and when evolves from infinity to a finite value (infinite branch). In all viable cases, the past asymptotic value of corresponds to while the final point corresponds to a de Sitter stage with (see Fig. 1 for an illustrative example).
In the following, we consider only pressureless matter, or dust, with . The reason is that we are interested only in the late-time behavior of bigravity when the Universe is dominated by dust. We also assume , although in principle nothing prevents a negative value of .
Iii Perturbation equations
In this section we study linear cosmological perturbations. We define our perturbed metrics in Fourier space by
where and are the background metrics with line elements
while and are perturbations around the backgrounds and , respectively, whose line elements are
After transforming to gauge-invariant variables Comelli et al. (2012),
and using as the time variable, the perturbation equations for the metric read:
while the corresponding equations for are
where and the coefficients are defined as
These equations are in agreement with those presented in Refs. Berg et al. (2012); Comelli et al. (2012); Solomon et al. (2014a) (for a more detailed derivation see, e.g., Ref. Khosravi et al. (2012)).
The matter equations are
Note that enters the equations only with derivatives; one could then define a new variable to lower the degree of the equations.444 only appears without derivatives in the mass terms, specifically in differences with , and so all appearances of are accounted for by the separate gauge-invariant variable . One could also adopt the gauge-invariant variables
to bring the matter conservation equations into the standard form of a longitudinal gauge but since this renders the other equations somewhat more complicated we will not employ them.
Iv Quasistatic limit
Large-scale structure experiments predominantly probe modes within the horizon. Conveniently, in the subhorizon and quasistatic limit, the cosmological perturbation equations simplify dramatically. In this section we consider this QSlimit of subhorizon structures in bimetric gravity.
The subhorizon limit is defined by assuming , while the QS limit assumes that modes oscillate on a Hubble timescale: for any variable .555Recall that we are using the dimensionless as our time variable. Concretely, this means that we consider the regime where for each field . We additionally take . In this limit we obtain the system of equations
where we have used the momentum constraints, Eqs. (40) and (44), to replace time derivatives of and . The above set of equations can be solved for , and in terms of (see also Ref. Solomon et al. (2014a)):
The QS limit is, however, only a good approximation if the full set of equations produces a stable solution for large . In fact, if the solutions are not stable, the derivative terms we have neglected are no longer small (as their mean values vary on a faster timescale than Hubble), and the QS limit is never reached. We therefore need to analyze the stability of the full theory.
Let us go back to the full linear equations, presented in section III. While we have ten equations for ten variables, there are only two independent degrees of freedom, corresponding to the scalar modes of the two gravitons. The degrees-of-freedom counting goes as follows (see Ref. Lagos et al. (2014) for an in-depth discussion of most of these points): four of the metric perturbations (, , , and ) and are nondynamical, as their derivatives do not appear in the second-order action. These can be integrated out in terms of the dynamical variables and their derivatives. We can further gauge fix two of the dynamical variables. Finally, after the auxiliary variables are integrated out, one of the initially dynamical variables becomes auxiliary (its derivatives drop out of the action) and can itself be integrated out.666We thank Macarena Lagos and Pedro Ferreira for discussions on this point.
This leaves us with two independent dynamical degrees of freedom. The aim of this section is to reduce the ten linearized Einstein equations to two coupled second-order equations, and then ask whether the solutions to that system are stable. We will choose to work with and as our independent variables, eliminating all of the other perturbations in their favor.
We can begin by eliminating , , , and their derivatives using the , , and components of the -metric perturbation equations. We will herein refer to these equations as , , and so on for the sake of conciseness. Doing this we see also that the and equations are linearly related. Then we can replace and with the help of the and equations. Finally, one can find a linear combination of the and equations which allows one to express as a function of , , and their derivatives. In this way, we can write our original ten equations as just two second-order equations for with the following structure:
where and are complicated expressions that depend only on background quantities and on . The eigenfrequencies of these equations can easily be found by substituting , assuming that the dependence of on time is negligibly small.777The criterion for this WKB approximation to hold is . We find that for large this approximation is almost always valid. For instance, assuming that only is nonzero, in the limit of large we find Könnig and Amendola (2014)
plus two other solutions that are independent of and are therefore subdominant. One can see then that real solutions (needed to obtain oscillating, rather than growing and decaying, solutions for ) are found only for , which occurs for , i.e., . At any epoch before this, the perturbation equations are unstable for large . In other words, we find an imaginary sound speed. This behavior invalidates linear perturbation theory on subhorizon scales and may rule out the model, if the instability is not cured at higher orders, for instance by a phenomenology related to the Vainshtein mechanism Vainshtein (1972); Babichev and Deffayet (2013).
Now let us move on to more general models. Although the other one-parameter models are not viable in the background888With the exception of the model, which is simply CDM. (i.e., none of them have a matter dominated epoch in the asymptotic past and produce a positive Hubble rate) Koennig et al. (2014), it is worthwhile to study the eigenfreqencies in these cases too, particularly because they will tell us the early time behavior of the viable multiple-parameter models. For simplicity, from now on we refer to a model in which, e.g., only and are nonzero as the model, and so on.
At early times, every viable, finite-branch, multiple-parameter model reduces to the single-parameter model with the lowest-order interaction. For instance, the , , and models all reduce to , the model reduces to , and so on. Similarly, in the early Universe, the viable, infinite-branch models reduce to single-parameter models with the highest-order interaction. Therefore, in order to determine the early time stability, we need to only look at the eigenfrequencies of single-parameter models, for which we find
Therefore, the only single-parameter models without instabilities at early times are the and models. Using the rules discussed above, we can now extend these results to the rest of the bigravity parameter space.
Since much of the power of bigravity lies in its potential to address the dark energy problem in a technically natural way, let us first consider models without an explicit -metric cosmological constant, i.e., . On the finite branch, all such models with reduce, at early times, to the model, which has an imaginary eigenfrequency for large (69) and is therefore unstable in the early Universe. Hence the finite-branch model and its subsets with are all plagued by instabilities. All of these models have viable background evolution Koennig et al. (2014). This leaves the model; this is stable on the finite branch as long as , but its background is not viable. We conclude that there are no models with which live on a finite branch, have a viable background evolution, and predict stable linear perturbations at all times.
This conclusion has two obvious loopholes: either including a cosmological constant, , or turning to an infinite-branch model. We first consider including a nonzero cosmological constant, although this may not be as interesting theoretically as the models which self accelerate. Adding a cosmological constant can change the stability properties, although it turns out not to do so in the finite-branch models with viable backgrounds. In the model, the eigenfrequencies,
are unaffected by at early times and therefore still imply unstable modes in the asymptotic past. This result extends (at early times) to the rest of the bigravity parameter space with . No other finite-branch models yield viable backgrounds. Therefore, all of the solutions on a finite branch, for any combination of parameters, are either unviable (in the background) or linearly unstable in the past.
Let us now turn to the infinite-branch models. In this case, it turns out that there exists a small class of viable models which has stable cosmological evolution: models where the only nonvanishing parameters are , , and , as well as the self-accelerating model. Here, evolves from infinity in the past and asymptotes to a finite de Sitter value in the future. As mentioned in Ref. Koennig et al. (2014), a nonvanishing or would not be compatible with the requirement . This can be seen directly from Eq. (22) in the limit of large . For these models we perform a similar eigenfrequency analysis and obtain