Doubly coupled matter fields in massive bigravity

In the context of massive (bi-)gravity non-minimal matter couplings have been proposed. These couplings are special in the sense that they are free of the Boulware-Deser ghost below the strong coupling scale and can be used consistently as an effective field theory. Furthermore, they enrich the phenomenology of massive gravity. We consider these couplings in the framework of bimetric gravity and study the cosmological implications for background and linear tensor, vector, and scalar perturbations. Previous works have investigated special branch of solutions. Here we perform a complete perturbation analysis for the general background equations of motion completing previous analysis.


Introduction
The high precision cosmological observations made it possible to test the underlying fundamental theory of gravity. Together with the assumption of General Relativity (GR) as being the right theory and the cosmological principle, the universe is well described by the ΛCDM model. It constitutes a predominant amount of dark energy in form of a cosmological constant and dark matter. Aside from negligible reported anomalies [1], the model is still the best fit to current cosmological data [2][3][4]. In spite of its observational triumph, the model suffers from serious theoretical problems, the most persisting one being the cosmological constant problem [5].
It is an unavoidable question to pursue whether the graviton could be massive which would correspond to a natural infrared modification of gravity since the mediated force by a massive graviton would be suppressed on large scales. The weakening of graviton could be put on equal footing with recent cosmological acceleration. At the linear level the theory is described by the Fierz and Pauli mass term [29] without introducing ghostly sixth mode. This linear model however suffers from the vDVZ discontinuity [30,31] when the mass of the graviton is sent to zero since General Relativity is not recovered in that limit. Actually, very soon after that Vainshtein realized that the linear approximation breaks down at some distance far from the source and that non-linear interactions become appreciable close to the source [32]. Usually, these non-linear interactions reintroduce the ghostly six mode, the Boulware-Deser ghost [33], and it was a challenging task to construct these potential interactions which would propagate only five physical degrees of freedom [34][35][36][37][38]. This ghost free theory of massive gravity is also technically natural and does not obtain strong renormalization by quantum corrections [39,40].
In the context of quantum stability of the theory, new ways of coupling the matter fields have been explored [41][42][43]. The classical potential interactions had to be tuned in a very specific way to maintain the Boulware-Deser ghost absent and if one wants to keep this property also at the quantum level, only very restricted matter couplings through an effective composite metric are allowed. This effective metric is built out of the two metrics in such a way that the matter quantum loops would only introduce a running of the cosmological constant for the effective metric which in other words correspond exactly to the allowed potential interactions. This doubly coupled matter fields introduce already at the classical level the Boulware-Deser ghost [41,44], but the coupling through the effective metric is special in the sense that the decoupling limit of the theory below the strong coupling scale is maintained ghost free [45,46]. Therefore, this coupling can be used as a consistent effective field theory. In the unconstrained vielbein formulation of the theory one can construct yet other type of effective metrics to which the matter fields can couple as well and the decoupling limit would still be free of the Boulware-Deser ghost [47]. Actually, the hope using the unconstrained vielbein formulation was to preserve the ghost freedom fully non-linearly with the original effective vielbein [48]. Unfortunately, this resulted in a negative result and also in this formulation the Boulware-Deser ghost is reintroduced [49]. However, it is worth mentioning that if one is willing to break the local Lorentz symmetry, one can indeed achieve this fully non-linearly [50]. The inclusion of the doubly coupled matter fields has very important implications for cosmological applications [41,[51][52][53][54][55][56][57][58][59][60] as well as for dark matter phenomenology [61][62][63].
The analysis of cosmological perturbations of the doubly coupled matter fields in massive gravity revealed that ghost and gradient instabilities can be successfully avoided together with the strong coupling issues since the vector and scalar perturbations maintain their kinetic terms [52]. The application to the massive bimetric gravity yielded gradient instability in the vector sector and ghost instability in the scalar sector for one of the branch of solutions, whereas the other branch of solutions was free of any ghost instability. It is still an open question whether this second branch of solutions is also free from any gradient instabilities. The main purpose of the present work is to investigate the perturbation analysis of the bimetric gravity theory in the presence of the doubly coupled matter fields on top of a general background equations of motion, without specifying the branch and providing also the full quadratic action for the scalar perturbations. Thus, our work complete the analysis started in [56].

Dynamical composite metric
A consistent coupling of some extra scalar field φ to both metrics simultaneously was introduced in [41] through a composite metricg µν We consider the same action as in [56] and R[f ] are Ricci scalar for g µν and f µν , respectively. As in [56], in this work we consider the matter contents of g µν and f µν metrics to be two cosmological constant: S pot denotes the non-derivative potential interactions S pot of the two metrics, where X stands for X µ ν and for a matrix M µ ν , e n (M ) are the elementary symmetric polynomials defined by where the antisymmetrization is unnormalized. In (2.7),X denoting the canonical kinetic term of φ in terms of the composite metric In the following we will study this action on FLRW background and establish our parametrization for linear perturbations.

Cosmological parametrization
We parametrize the two metrics g µν and f µν to be where N g , a g , N f and a f are functions of time only, and the matrix exponentials are defined Throughout this paper, spatial indices are raised and lowered by δ ij and δ ij .
Accordingly, it is convenient to parametrize the composite metric to bẽ Similar to (3.3)-(3.6), we may also decomposẽ At the linear order, we have, for the scalar modes,Ã (3.14) for the vector modes,S and for the tensor modesh The background equations of motion can be determined by requiring the vanishing of the first order action of A, ζ, ϕ, ψ and δφ, which is given by The set of equations of motion are thus given by where P ,X is the shorthand for ∂P/∂X, the H g and H f are the Hubble parameter associated with the two metrics respectively, i.e., In the above, where we have introduced for later convenience. The equation of motion for the scalar field is given by

Cosmological perturbations
The quadratic action for the two tensor perturbations h ij and γ ij is given by where a dot denotes derivative with respect to t, The quadratic action for the four vector modes S i , F i , σ i and ξ i is given by where M 2 is given in (4.2) and we also introduce with b 2 given in (3.30) for short. Since the vector modes S i and σ i have no dynamics in (4.3), we may solve them in terms of F i and ξ i and arrive at the reduced action for F i and ξ i , which is given by (4.6) From (4.5) it is transparent that there are two vectorial degrees of freedom giving that β = 0, which can be identified as F i − ξ i . For the stability condition we have to impose G v > 0.
We study now the linear stability of the scalar modes in our model. Initially we have 9 scalar modes, of which four (A, B, ζ and E) are from g µν , four (ϕ, ω, ψ and χ) are from f µν , and one is the perturbation of the scalar field δφ. In order to simplify the calculation, we choose a gauge in which δφ = χ = 0. In the residual 7 modes, only 2 modes are dynamical, which can be conveniently chosen to be After some manipulations, the final quadratic action for these two scalar modes takes the following general structure (in matrix form) where G mn and W mn are symmetric while F mn is antisymmetric, which are given by with m, n = 1, 2. In (4.10)-(4.11), we have (4.14) and where Ξ ij with i, j = 1, · · · , 6 are given in Appendix A. Up to now, no approximation is made in deriving the above expressions. Unlike the tensor and vector modes, the lengthy expressions in the above make the analysis for the scalar modes rather cumbersome. In the following, we analyze the instabilities in the small scale limit k → ∞. For the kinetic terms, we have and and It can also be verified that G 12 ∼ O(k 0 ). Thus in the large k limit, the no ghost condition concerning the kinetic terms requires that P ,X + 2XP ,XX > 0 as well as These results can be compared with those derived in [56]. As for the gradient terms, in the large k limit we have and W 12 ∼ O(k 2 ), whereŴ In (4.30),D is given in (4.26), and and Thus, in the large k limit, the absence of gradient instability requireŝ The propagating speeds of the two scalar modes are given by the eigenvalues of G −1 W, which correspond to in the same limit.

Conclusion
In this work, we investigate the cosmological perturbation analysis of the bimetric theory with a scalar field coupled simultaneously to both metrics in terms of a composite metric. The scalar field represents the matter field that lives on both metrics. The ghost and gradient instabilities of the tensor and vector modes as well as the ghost instabilities of the scalar modes of the same model have been analyzed in [56] for some concrete background evolution, while in this work we complete the analysis by presenting the full quadratic action for the scalar modes (4.9) as well as the conditions for the absence of gradient instabilities as in (4.34) on general background evolution in the presence of matter fields. Although in this work we focus on the small scale limit k → 0 due to the lengthy expressions, the results presented in this work enable one to make further analysis in different limits as well as upon concrete background solutions.
Moreover, we consider only the coupling of the scalar field to the composite metric in a minimal way, while in principle one may consider non-minimal derivative couplings as was pointed in [64]. This bimetric model with doubly coupled matter fields offers an interesting cosmological framework. In one branch of solutions, in which the Hubble rates are proportional to each other, this interesting phenomenology is plagued by the ghost and gradient instabilities as was shown in [56]. However, in the other branch of background cosmology with the algebraical ratio between the scale factors of the two metrics there are no ghost instabilities associated with the vector and scalar perturbations. Here, we also show the conditions for the absence of the gradient instabilities for the scalar perturbations, which were lacking in the literature. Fulfilling all these instability conditions, this branch of solutions still offers promising dark energy model, which has a very rich phenomenology [65].