Cosmological Solution Moduli of Bigravity

We construct the complete set of metric-configuration solutions of the ghost-free massive bigravity for the scenario in which the g-metric is the Friedmann-Lemaitre-Robertson-Walker (FLRW) one, and the interaction Lagrangian between the two metrics contributes an effective ideal fluid energy-momentum tensor to the g-metric equations. This set corresponds to the exact background cosmological solution space of the theory.


Introduction
The massive gravity theory which was constructed in [1,2], and which is a nonlinear generalization of the Fierz-Pauli massive gravity model [3] is Boulware-Deser-ghost-free [4,5]. An extension of this massive gravity model which originally admits a flat reference metric was also constructed in [6,7,8] for a general background or reference metric. Later on, a ghostfree massive bimetric theory in which the interaction term between the foreground, and the background metrics arises from the mass terms was proposed by introducing a copy of general relativity (GR) dynamics for the background metric [9,10,11,12]. A particular class of cosmological solutions of this massive bigravity theory have been extensively studied in recent years [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The solution space corresponding to an effective decoupling of the two metric sectors of the theory is constructed in [27]. The elements of the decoupling solution moduli that is derived in [27] give rise to self-accelerating cosmologies in the g−sector via the contribution of an effective cosmological constant. On the other hand, in this work we will construct the general cosmological solution space of the massive bigravity theory. Throughout our derivation, we will follow a parallel route with those of [28,29] in which cosmological solutions of massive gravity are derived in a formal fashion without explicit classification. On the contrary, in what follows likewise in [27], our main perspective will be to derive the entire set of metric couples (f, g) explicitly, which will consistently solve all the field equations of the theory, and which will enable a semi-decoupling of the f −metric from the g−metric sector. Since we will consider a homogeneous, and an isotropic scenario in the g−sector apart from assigning a Friedmann-Lemaitre-Robertson-Walker (FLRW) form for g which corresponds to the cosmological background metric of the universe, we will also take the overall contribution of the interaction Lagrangian to the g−metric equations as an effective ideal fluid energy-momentum tensor. In this respect, the Bianchi identity of the interaction terms in the g−metric equations will naturally transform to be the continuity equation of the effective fluid. We will show that when one proposes such a solution ansatz then the only remaining task to derive the set of metric couples which allow this picture is to find the general solutions of an inhomogeneous cubic matrix equation. We will derive the entire set of solutions of this matrix equation whose coefficients are functions of the elementary symmetric polynomials of the solutions themselves rather than being constants. By using the general solutions of this matrix equation one can construct the solution moduli of the metrics (f, g) which admit a cosmological scenario in the g−sector.
In Section one, starting from the bigravity dynamics by assuming a cosmological g−sector and thus, introducing the above-mentioned ansatz for the contribution of the interaction terms in the g−metric equations we will derive an algebraic matrix equation whose solutions will lead us to the metric couples which are compatible with the field equations of the cosmological picture. In Section two, we will derive the Jordan normal form solutions of this equation. By using the complete set of Jordan normal form solutions we will show in Section three that when a special form of similarity transformation is used one can obtain the general set of solutions of the above-mentioned ansatz-generated matrix equation. Therein, we will also discuss the completeness of this solution set. Then, again in Section three as a consequence of the completely-derived general solution space of the ansatz equation we will be able to define the cosmological solution moduli space of bigravity. We reserve Section four for the derivation of the equations of the cosmological dynamics in g−sector, as well as a discussion about the associated f −dynamics, and its solution methodology.

The set-up
The ghost-free bigravity action can be given as [9,10,11,12] where g is the foreground, and f is the background metric which are coupled to two types of matter via the actions S g M , S f M , respectively. Λ g , Λ f are the cosmological constants in each sector. R g , R f are the corresponding Ricci scalars. The Lagrangian which describes the interaction between the two metrics above reads where {e n } are the elementary symmetric polynomials corresponding to the square-root-matrix Originally, the interaction Lagrangian also contains the terms β 0 e 0 = β 0 , and β 4 e 4 = β 4 det √ Σ. However, we combine their contributions with the cosmological constants Λ g , and Λ f , respectively. Eq. (2.2) reduces to the Fierz-Pauli form in the weak-field limit when one chooses [9] β 1 + 2β 2 + β 3 = −1.
(2.5) From Eq. (2.1) one can obtain the field equations for the metric g as Whereas, the field equations of the metric f become The corresponding energy-momentum tensors arising from the interaction term Eq.(2.2) are T g µν = −g µρ τ ρ ν + L int g µν , (2.8) and The matrix τ with the entries {τ ρ ν } is defined to be [12] The effective energy-momentum tensors in Eqs. (2.6), and (2.7) are ought to satisfy the Bianchi identities If a solution configuration satisfies one of these equations then the other one is automatically satisfied [15,16]. Now, let us focus on the cosmological solutions in the g−sector of the action Eq. (2.1). Thus, we will take g as the FLRW metric g = −dt 2 + a 2 (t) 1 − kr 2 dr 2 + a 2 (t)r 2 dθ 2 + a 2 (t)r 2 sin 2 θdϕ 2 . (2.12) Let us also consider the solutions for which the effective energy-momentum tensor entering into the g−metric equations in Eq. (2.6) takes the form of an ideal fluid. For an ideal fluid the on-shell Lagrangian can be taken as [29] L IF =p, (2.14) so that is obtained by varying the Lagrangian Eq. (2.14) with respect to g, and by using the first law of thermodynamics [29]. Our effective fluid that is introduced in Eq. (2.13) will certainly obey the first law of thermodynamics as, it must satisfy the conservation equation in Eq. (2.11) which will result in an ordinary continuity or fluid equation when Eq. (2.13) is substituted in it. If we take the effective ideal fluid four-velocity vector as U µ = (1, 0, 0, 0) in the rest frame of the fluid, and use the FLRW metric Eq. (2.12) we obtain (2.17) Index raising on both sides of Eq. (2.8) by the metric g gives where (T g ) µ ν = [g −1 T g ] µ ν . By using Eqs. (2.16), and (2.17) in this expression we obtain the matrix equation where

Classification of the solutions
Next, we will derive and classify the Jordan canonical form solutions of the cubic matrix equation (2.19). This is a highly non-trivial matrix equation for two reasons: first, it is not in a polynomial form, and second, its coefficients are functions of the elementary symmetric polynomials e 1 , e 2 of its solutions √ Σ rather than being constants. For this reason, in this section we will derive the diagonal and the nondiagonal Jordan form solutions of it, and show in the next section that they can be used to generate the entire solution space. Firstly, let us define the polynomials and x(Ax 2 + Bx + C) = 0, (3.2) whose roots we will generally call α i , and λ j , respectively. Since, in general for any Jordan canonical form matrix J (diagonal or nondiagonal) when it is substituted in Eq. (2.19) the eigenvalues namely the diagonal elements of J must satisfy one copy of Eq. (3.1), and three copies of the polynomial in Eq. (3.2) the multiplicity of α i in the diagonal of J must be one. Therefore, Jordan form solutions of Eq. (2.19) can be partitioned as where H 3×3 is a three by three Jordan normal form matrix which satisfies the matrix polynomial equation as well as the ones, in which both of the roots λ 1 , λ 2 appear. By direct substitution, the reader may verify that these matrices do solve Eq. (2.19), and the corresponding nondiagonal Jordan forms that share the same eigenvalues can not satisfy Eq. (2.19) in this case when ∆ > 0. To find the explicit form of these solutions we have to know e 1 , e 2 which constitute both the coefficients given in Eq. (2.21) (of the matrix equation that these solutions must satisfy), and the entries of these solution matrices listed above. In other words, we have to solve e 1 , and e 2 in terms of the {β i }−parameters of the action Eq. (2.1), and the constituents of the solution ansatz Eq. (2.13) so that Eq. (2.19) is satisfied. We will first consider the cases N 1,2,3,4,5,6 . If we take the trace of these solutions we get where n = 3, 3, 2, 1, 2, 1 for N 1 , N 2 , N 3 , N 4 , N 5 , N 6 , respectively. We also have By using Eq. (3.7), and singling out e 2 from this expression we get If we substitute this result into Eq. (3.2) we obtain the relation Since from Eq. (3.7) we have α i = e 1 − nλ j , substituting this into Eq. (3.1), and successive usage of Eq. (3.10) leads us to the relation Finally, when we use Eqs. (3.12), and (3.9) back in Eq. (3.10) and we refer to the definitions in Eq. (3.11) we obtain an equation for e 1 solely in terms of the β i −coefficients. For the n = 3 cases this equation reads where we define (3.14) For the n = 2 cases we get For each real root of Eq. (3.13) we have the solutions N 1 , N 2 , and for each real root of Eq. (3.15) we have the solutions N 3 , N 5 of Eq. (2.19) with the corresponding entries that can be read from The domain of validity of these solutions are determined by the conditions and from Eq. (3.9) we have Now, by using Eq. (3.18) in Eq. (3.7) we get Substituting this result, together with Eq. (3.19) into Eq. (3.1) gives us The Eqs.
where we label the excess or the repeated root on the diagonal of the solution by λ j . By refereing to the definitions in Eq. (2.21) we find that We also have By using the identity in the above equation we see that for these solutions By using this result in Eq. (3.23) we obtain the equation for λ j . Its solutions are Using Eqs. (3.27), and (3.28) in Eq. (3.2) will enable us to write e 2 in terms of e 1 . After some algebra we get We note that, when the (+) solution is taken in Eq.
where we defined In summary, for these latest cases we find that  . The final solution we have to derive explicitly in this class is N 9 . If we take its trace we find that From this relation by referring to Eq. (2.21) we see that We see in this formulation that e 1 remains completely an arbitrary spacetime field. For a particular choice of it one can read e 2 from Eq. (3.40) The reader may again verify that these matrices satisfy Eq. (2.19) by direct substitution. If we take the trace of the matrices in Eq. (3.40) we find where n = 3, 1, 2, 3, 2, for N 10 , N 11 , N 12 , N 13 , N 14 , respectively. When the definitions in Eq. (2.21) are used this relation yields which can be written in the form For n = 1, this equation is reduced to a linear one and it has the solution (3.51) Having found e 1 now we can explicitly express the entries of the solutions N 12 , N 14 via  is

∆ < 0 solutions
where we define λ = R + Ii with which reduces to the condition We realize that, in this solution e 1 remains to be an arbitrary spacetime function. When one specifies e 1 one can express e 2 in terms of it from Eq.

The solution space
In the previous section, we have explicitly constructed the entire set of nontrivial Jordan canonical form solutions of Eq. (2.19). We have disregarded the trivial case of √ Σ = diag(α i , 0, 0, 0) which results in nonphysical f −metric solutions. Before defining the solution space of Eq. (2.19), let us discuss one last constraint on the solutions that we constructed in the previous section. In obtaining the Jordan normal form of the solutions, although we used the conditions on the elementary symmetric polynomials e 1 , and e 2 we did not refer to the e 3 −structure of the solutions. This is a necessary, and a crucial point, as our solution ansatz Eq. (2.13) brings the constraint Eq. Next, we introduce the matrix field where P (x µ ) is an invertible 3 × 3 matrix field, and m(x µ ) is a scalar field which can simply be taken as m(x µ ) = 1 without loss of generality. Since any element J ∈ J is a solution of Eq. where Σ is the square of any element in M, and g is the FLRW metric. However, not all elements of M which solve Eq. (2.19) will lead to symmetric results in Eq. (4.8) thus, physically acceptable background metrics. We have to impose the condition gΣ = Σ T g, (4.9) which guarantees the symmetry of f . Therefore, we define the cosmological solution moduli of the action Eq. (2.1) as the set Γ C = (g, f ) f = gX 2 X ∈ M, and gX 2 = (X T ) 2 g . (4.10) We should state that in special, when one chooses the diagonal elements in Eq. (4.2), then squares them, and substitutes the result in Eq. (4.8) one directly obtains the exact background metric solutions in a concise way without being obliged to concern the symmetry condition. On the other hand, for the more general cases one has to choose a special form for the matrix Eq. (4.3) to satisfy Eq. (4.9). Since, the symmetry requirement in Eq. (4.9) becomes a closer inspection denotes that for a particular choice of J ∈ J this equation brings three algebraic constraint conditions on the function-entries of the solution-generating P −matrix in Eq. (4.3) which enables us to determine three of the entries of P in terms of the other six entries which remain arbitrary. Next, we will give a summary of the cosmological dynamics.

Cosmological dynamics
In the g−sector beside the effective ideal fluid energy-momentum tensor that is introduced in Eq. (2.13), we will also take the physical matter as a perfect fluid with the energy-momentum tensor where p = p(t), and ρ = ρ(t) are the pressure, and the energy density of the g−matter fluid, respectively. Now, by using the physical g−matter, and the effective energy-momentum tensors together with the FLRW metric Eq.
(2.12) in the g−metric equations Eq. (2.6) leads us to the t−component equation as well as the three identical spatial-component equations where we have used Eq.

Concluding Remarks
For the massive bigravity theory [9,10,11,12] we constructed the complete solution moduli space of the (f, g) couples of metrics which admit a FLRW cosmology in the g−sector via the presence of an effective ideal fluid contribution coming from the interaction Lagrangian of the mass terms in addition to the matter one. We employed the cosmological solution ansatz by choosing the energy-momentum tensor of the interaction terms in the g−metric equations in the form of an effective ideal fluid one. This choice resulted in a cubic matrix equation for the building block matrix of the interaction Lagrangian that is composed of the two metrics. By deriving the general solution space of this nontrivial matrix equation (whose coefficients are also functions of the elementary symmetric polynomials of its solutions) we were able to construct and define the complete solution space of the (f, g) metric configurations which enable FLRW cosmologies in the g−sector that is modified by an effective ideal fluid whose contributions are proportional to the square of the graviton mass. Although, we obtained the general solutions of the ansatz matrix equation we also discussed that one still has to impose a symmetry condition on these solutions to construct a symmetric result for the f −metric. Therefore, in spite of the existence of a matrix field degree of freedom in constructing f −solutions out of the g−sector fields one has to render three out of nine function components of this arbitrary matrix field to satisfy the symmetry condition we mentioned. Furthermore, we also discussed that one might also have to fix the remaining degrees of freedom of the f −metric in satisfying the f −metric sector field equations in the presence of f −type matter. We have shown that, the cosmological solution moduli of bigravity that we constructed is composed of similarity equivalence classes which do not differ from each other only in their functional form but also in the equations of state that they impose on the associated effective ideal fluid they give rise to. Finally, in the last section, we presented the resulting cosmological equations of the g, and the f −metrics for which we shortly discussed the solution flow chart dictated by the semi-decoupling of the two metric sectors. The known exact solutions of bigravity can in general be divided into three groups [20]. There is a class of solutions in which both metrics are proportional to each other. There exists another class of spherically symmetric solutions which has a nondiagonal background metric. There are also solutions including both diagonal but not proportional f , and g metrics. In this work, we present the complete cosmological background solution space of the theory. Massive bigravity as a ghost-free massive gravity theory promises to possess cosmological solutions which can exhibit late time selfacceleration which could compensate the dark energy problem in standard cosmology. The background cosmological solutions [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and their perturbations and stability issues [14,22,23,30,31] arising from the above-listed known solutions have gained a considerable interest and they are under extensive inspection recently. It has been shown that although there are stability problems and the perturbations of these solutions differ from the ones of GR these problems can still be overcome by turning on the f −type matter which is heuristically interpreted as dark matter [14,22,23,30,31]. We believe that, apart from its mathematical legitimacy of completeness which presents an extensive amount of new cosmological solutions of the theory our derivation of the cosmological background solution space can also serve for the phenomenology of the theory. We have found that, in the general similarity equivalence class structure of the solutions there is a rich variety of functional relations between the spatial parts of the two metrics unlike the case in the particular cosmological solution which is widely studied in the literature. The behavior of the ratio of the g, and f −scale factors of this particular solution (which we believe must be related to the N 1 , or N 2 solutions we have discussed) causes early time instabilities of the perturbations which differ from the GR ones. Therefore, we hope that among the variety of complete background solutions we have derived there may exist ones which may admit acceptable perturbation behavior with respect to GR perturbations. To explicitly construct, and study the solutions in this direction one may follow two main routes, one may either inspect the solution behavior in the various similarity classes one by one or one may attempt to design particular form of cosmological solutions with or without f −matter which exhibit a stable nature of perturbations within the solution construction methodology we have discussed. However, we should also state that in our generally-constructed solution space, majority of the f −metric solutions may fail to exhibit homogeneity, and/or isotropy behavior. On the other hand, one may question the necessity of homogeneity, and isotropy in the f −sector since opposite cases may have acceptable results from the g−metric perturbation theory point of view, and in addition they may lead to interesting variety of dark matter scenarios. Finally, we point out a possible direction in which one can extend the results of the present work to study the cosmological solutions within the newly proposed formalism of ghost-free effective-metric-matter coupling [32,33,34].