Effective Matter Cosmologies of Massive Gravity I: Non-Physical Fluids

For the massive gravity, after decoupling from the metric equation we find a broad class of solutions of the Stuckelberg sector by solving the background metric in the presence of a diagonal physical metric. We then construct the dynamics of the corresponding FLRW cosmologies which inherit effective matter contribution through the decoupling solution mechanism of the scalar sector.


Introduction
The Fierz-Pauli [1] massive gravity theory has been extended to a non-linear Boulware-Deser (BD) [2,3] ghost-free level in [4,5]. Later this theory is upgraded to include a general background metric [6,7,8]. The cosmological solutions of this non-linear and ghost-free theory has been an active topic of research in recent years [9].
In [10] we have studied the Einstein solutions of the so-called minimal sector of the massive gravity. On the other hand [11] was devoted to develop a methodology again for the minimal theory which would exactly solve the Stückelberg sector by first constructing the solution generating background (fiducial) metric. The approach in both of these works was to construct a solution ansatz which would decouple the metric and the scalar sectors. This enables the determination of the background metric satisfying the ansatz constraint which leads to the solution of the scalar sector. In the following work we adopt a similar formalism for the most general massive gravity theory. Our objective will be to construct an ansatz which will function in the same direction. As a solution to this ansatz constraint we will determine the background metric which will lead us to a class of solutions of the scalar sector for a given diagonally-formed physical metric. Such a formulation will replace solving the scalars from the dynamical field equations by a semi-algebraic solution of the corresponding ansatz equation. As a physical consequence of such a solution method admitting decoupling of the field equations of the metric and the scalar sectors we will be able to construct the FLRW cosmological dynamics associating the above-mentioned scalar moduli and the background metric. We will show that the metric sector thus the cosmological equations get contributions from an effective matter energy-momentum tensor which parametrizes the ansatz we consider and which enters into the metric equation as a remainder of the act of decoupling the scalars from it. We will also discuss the conservation relation the effective matter must satisfy. This is a modified version of the usual energy-momentum conservation. Therefore the effective ideal fluid appearing in the cosmological equations must be considered as a non-physical one.
In Section two we construct the necessary ansatz mentioned above which will decouple the Stückelberg scalars from the metric sector by introducing an effective energy-momentum tensor. In Section three we will present the explicit solutions of the ansatz equation for the background metric and the scalars of the theory. We will also discuss the constraint equation to be satisfied by the effective matter so that the scalar solutions obtained become also the solutions of the theory. Finally Section four will be reserved for the discussion of the dynamics of the corresponding cosmological solutions of the general massive gravity which possess effective matter terms as modifications in the Friedmann and the cosmic acceleration equations.

The ansatz
The ghost-free massive gravity action with a general background metric which is coupled to matter can be given as [6] of the four by four matrix functional √ Σ in which with g µν being the inverse physical metric, {φ a (x µ )} for a = 0, 1, 2, 3 the Stückelberg scalars, andf ab (φ c ) the background metric. The square root matrix is defined via √ Σ √ Σ = Σ. The metric equation corresponding to (2.1) can be derived as [6] where T M µν is the physical matter energy-momentum tensor and for later convenience we have written compactly the contribution of the Stückelberg sector as T S µν which is derived in [6] as with 1 4 being the four by four unit matrix. Now if we define the matrix [T S ] µ ν ≡ T S µν then in matrix notation we can write (2.5) as Now by using the symmetries [12] g( for any integer n and also by referring to the definitions of the symmetric polynomials (2.2) we can write (2.7) as β n e n g . (2.9) To decouple the Stückelberg sector from the metric one and then to solve for the background metric and the scalars of the theory we now introduce the solution ansatz where C 1 , C 2 are arbitrary constants and in matrix sense [T ] µ ν ≡T µν is completely an arbitrary symmetric tensor parametrizing the solution moduli for which later we will show that it enters into the metric sector as an effective energy-momentum tensor source. We have defined its trace as (2.11) In (2.10) we have also introduced the matrix [τ ] µ ν ≡τ µν with the definitioñ 12) by assuming that the effective energy-momentum tensor will depend explicitly on the physical metric which will be the case for the cosmological solutions. Our fundamental motive in proposing this form of a solution ansatz has a cosmological perspective. The introduction of the first two terms in (2.10) is for designing solutions in which the massive sector of the theory contributes an effective cosmological constant and a non-matter originated energy-momentum tensor to the metric equations. In this way the cosmological equations of the theory can be formulated in the canonical form of the general relativity (GR) ones with additional effective cosmological constant and matter contribution which can work as a cure for the dark energy problem of GR cosmology by admitting self-accelerating solutions. However as we will discuss next the last two terms in (2.10) are needed for mathematical consistency so that (2.10) becomes a soluble ansatz. At this stage there is no guarantee that such an ansatz may lead to solutions however below we will show that it admits solutions both in the Stückelberg and the metric sectors thus it is a legitimate one. To generate solutions firstly we have to make the observation that on the right hand side in (2.9) there appears  (2.14) show that L OS must satisfy the inverse-metric variation Since for a general Lagrangian the variation solely with respect to the inverse metric leads to we can deduce that We conclude that the solutions which satisfy (2.10) at the Lagrangian level must lead to L S = L OS . Therefore for the symmetric polynomials of the matrix √ Σ we have the on-shell relation This analysis shows the necessity of adding the last two terms to the ansatz (2.10). If one plans to have the second generic term in (2.10) then (2.17) is the simplest 1 Lagrangian level ansatz that contains the effective energymomentum tensor explicitly in it and produces the second term in (2.10) upon variation with respect to the inverse metric. However (2.17) also produces the last two terms of (2.10). Hence they must be included in the solution ansatz when one fixes the on-shell Lagrangian as (2.17) in its simplest form.

The Stückelberg Sector
In this section we will focus on solving the Stückelberg scalars and the background metric in the solution ansatz (2.10). First let us take the trace of (2.10). If we multiply both sides by g and then take the trace of the matrix equation in (2.10) by also using (2.2) after some algebra we find where we introduce the tracẽ Using (2.18) in the above equation will eliminate e 3 and then we can solve for e 2 . The computation reads where we have also used e 0 = 1. Now that we have found e 2 in terms of e 1 = tr √ Σ we can turn our attention to finding solutions to the ansatz (2.10). Substituting (2.9) in (2.10) then multiplying both sides by g and using (2.18) 2 lead us to the cubic matrix equation for where we have introduced the four by four matrices Although on-shell we have derived e 2 in terms of e 1 which is not specified yet we will keep the compact notation of e 2 in the following formulation for the sake of simple appearance of the equations. To be able to solve (3.4) we will assume that the physical metric g, Σ (thus √ Σ), the effective energymomentum tensorT , andτ are all diagonal matrices so that (3.4) becomes a diagonal matrix equation. Our results in the following analysis will justify that there exist diagonal Σ solutions to (3.4) and we are free to specify the form ofT and we restrict ourselves to the diagonal metric solutions of the metric sector. This scheme is also conformal with the cosmological solutions we will consider later. We should state here that the solution ansatz (2.10) is independent of the diagonality assumption we will choose for the equation (3.4) in the following analysis. However for a non-diagonal choice of √ Σ equation (3.4) will give a set of coupled cubic algebraic equations for the entries of the matrix √ Σ. Also when solved from (3.4) the non-diagonal form of √ Σ will lead us to a set of coupled first-order partial differential equations. In this more general picture of solutions one can also choose non-diagonal physical metrics g, and the tensorsT ,τ . Thus more general set of nondiagonal solutions to (3.4) can be derived by choosing various non-diagonal solution forms of √ Σ, g, andT but in this case one has to face the difficulty of algebraic and later differential coupling of equations. On the other hand the diagonality assumption of the ingredients of (3.4) decouples the algebraic equations for the entries of √ Σ first and then the partial differential equations for the Stückelberg fields later. As a final remark in this direction: assuming diagonality for √ Σ but not for g will put extra constraints on the effective energy-momentum tensorT via (3.4). Now let us define If ∆ > 0 (3.4) has three distinct real roots, if ∆ < 0 then it has two complex and one real root, also if ∆ = 0 then there are three real roots again with a two-fold degeneracy. The general solutions of (3.4) can be given as where and For the general solutions (3.7) of (3.4) the elementary symmetric polynomial function e 1 namely tr √ Σ is left undetermined as a solution parametrizing function. The basic reason for this degree of freedom in the solution generating scheme we have constructed is the following: originally there exit two independent parameters namely e 2 , and e 1 in the solution ansatz (2.10) as we can eliminate e 3 by the introduction of the on-shell Lagrangian (2.18), when one takes the trace of (2.10) which is equivalent to (3.4) one obtains (3.3) which enables us to express e 2 in terms of e 1 , on the other hand one may expect to determine e 1 from the explicit solution (3.7) by taking the trace of it as well, however (3.7) is algebraically another way of writing (3.4) which it satisfies as a solution thus taking its trace would lead us to no additional identity than (3.3). Therefore we can conclude that in the solution (3.7) e 1 remains as a free function to parametrize the solutions. Below we will see that demanding the reality of the solutions may bring restrictions on e 1 or fix and determine its functional form in terms of the other free function parameters of the solutions (i.e. the effective energy-momentum tensor). Now by assuming that we focus on the real solutions of (3.4) squaring both sides of (3.7) yields where we introduce the pull-back of the background metricf by using the Stückelberg co-ordinate transformations {φ a (x µ )} as From this equation we deduce that f must be a diagonal matrix as also it was obvious from our construction. Now let us also assume that the background metricf is also of the diagonal form. To find solutions to these set of equations we will follow the method developed in [10,11]. Firstly we choosē f asf = diag(f 00 , f ii ), for i = 1, 2, 3. (3.14) With this choice therein and bearing in mind that the right hand side is a diagonal matrix, from (3.13) we obtain Note that if we propose the condition then we can satisfy the second set of equations in (3.15). Furthermore these equations namely (3.16) can be solved by demanding meaning that φ a = φ a (x a ) only. On the other hand as discussed in [10,11] when (3.16) is used in the first set of equations in (3.15) one can simplify the first set to the form By taking square root on both sides these equations become Finally if we choose the diagonal components of the background metricf as become the solutions of (3.18). Here we have introduced the completely arbitrary integrable functions F a (x a )'s. These Stückelberg scalar field solutions when the background metric is chosen as (3.21) are the solutions of the ansatz (2.10). We should also state that to be able to construct (3.21) explicitly one first has to specify the effective energy-momentum tensorT (which has to obey a conservation equation as we will discuss below) then one has to solve the diagonal metric from the metric sector. We see that the form of our solutions justifies the diagonal matrix assumption we have made for (2.10) all through our analysis. On the other hand for more general non-diagonal forms of √ Σ, g, andT one would have a mixed-differential term-wise coupling in the partial differential equations (3.13) which can not be so easily brought to a decoupled form. Those equations may also admit solutions in principle however the reader should appreciate that solving them would be more involved than the simpler decoupling solution method we have discussed above. Another essential loss of generality was choosing the background metric diagonal for decoupling and consequently generating solutions to these partial differential equations. Assuming non-diagonal background metric forms would cause similar coupling complications. We further note that all the conditions on ∆ needed to obtain the real roots of (3.4) contain the matrix D in them. For this reason in general the sign conditions on ∆ for the reality of the solutions (as ∆ is a functional ofT via D) depending on the particular choice ofT may or may not bring restrictions on its domain. Similarly as √ Σ is a functional ofT too the valid domain of the solutions of the background metric and the Stückelberg fields may also be restricted for certain class of solutions of (3.4) and the choice ofT . However as we have mentioned before there also exists the freedom of assigning e 1 = tr √ Σ as a compensation in the most general set-up to design solutions which are free of these restrictions. Now on the other hand if we set which puts no restrictions on the regions of validity of the solutions and the effective source 3 then it is guaranteed that there is one real matrix solution to (3.4) and it is where due to the condition (3.23) we have We remark that since all the matrices in the above relations are diagonal the power operations can directly be applied on the diagonal entries. Now unlike the more general solutions we have discussed the condition (3.23) will determine e 1 . To see this first note that from (3.23) we get

Substituting (3.3) into this equation we obtain
which is a quadratic equation for e 1 = tr √ Σ. The discriminant of this equation is In order to have a real root for (3.27) we must have This brings a constraint on the trace of the effective energy-momentum tensor T as well as the one forτ . We will see in the cosmological solution scheme of the next section that choosing equality in (3.29) will fix the equation of state of the effective ideal fluid. In spite of this restriction on the other hand this solution has physical advantages as it will not cause additional constraints on the building blocks of the cosmological solutions. However if equality is not chosen close inspection shows that the free parameters in (3.28) can still be tuned to give solutions in physically sensible domains. For example to enlarge the domain of validity of the solutions when C 1 > 0, β 2 > 0, C 2 < 0 one can tune β 2 to small values and when C 1 < 0, β 2 < 0, C 2 > 0 one can tune the norm of β 2 to high values to release the restrictions onT µ µ . On the other hand choosing the discriminant as zero will relax all the domain restrictions but fix the form ofT in return. Now assuming (3.29) is satisfied, the real solutions to (3.27) become where we define the matrix (3.33) Following the same track of solution route we have introduced earlier in this section we find that the background metricf becomes where H ′ = gH 2 . We can also write down the Stückelberg scalar solutions again as Although by now we can explicitly construct the scalar solutions for the more general case in (3.22) or the special one in (3.35) their being the solutions of the Stückelberg sector of (2.1) is not guaranteed yet. To guarantee this we must focus on the scalar field equations of (2.1). It can be directly deduced from the metric equation (2.4) that the scalar field equations must be equivalent to the covariant constancy condition where ∇ µ is the covariant derivative of the Levi-Civita connection of g. If we substitute our ansatz (2.10) in this equation we get Then by using the metric compatibility ∇ µ g αβ = 0 we obtain a constraint which can be considered as a modified conservation or continuity equation for the effective energy-momentum tensorT . Thus finally we conclude that if one choosesT in (2.10) as a solution of (3.38) then for the background metric (3.21) or (3.34) the Stückelberg scalar solutions (3.22) or (3.35) of (2.10) respectively become the scalar field solutions of the massive gravity action (2.1) together with the diagonal metric g to be solved from the metric sector which we will inspect for cosmological cases next.

FLRW Dynamics
Now we turn our attention to the metric sector. If we substitute the ansatz (2.10) into the metric equation (2.4) we get the on-shell equation where we have defined the effective cosmological constant We see that upon using the ansatz (2.10) the metric sector is completely decoupled from the scalars whose contribution is truncated to the presence of an effective cosmological constant and an energy-momentum tensor. Let us consider (4.1) for the cosmological FLRW metric in the spatially spherical coordinates {t, r, θ, ϕ}. We should note at this point that the cosmological metric is diagonal in this frame so that our analysis in the previous section is applicable. For consistency with the isotropy and homogeneity in (4.3) we also choose the physical and the effective sources in (4.1) as ideal fluids so that From these definitions we can deduce the trace of T M andT as Also referring to its definition in (2.12) via (4.4)τ can be computed as for which we have takenρ,p to be linearly independent with g µν . Its trace becomesτ µ µ = −4p. If now we use the metric (4.3) in (4.1) a standard computation which is slightly modified due to the extra terms in (4.1) gives the tt-component equa- which is the modified Friedmann equation. The three spatial component ii-equations lead to the same form of equation By using (4.8) in (4.9) we obtain the modified cosmic acceleration equation On the other hand, in the more general picture of solutions (at least for the ones again diagonal in { √ Σ, g,f}) one may follow a different but a more challenging track to generate a broader class of solutions. In this method for example for the cosmological solutions one can first solve {ρ,p} (in other general cases, the components ofT ) from the Friedmann equations (modified Einstein equations), the physical matter equations, and the conservation equation ofT , and now one can independently choose a background metric f , then one can solve the coupled partial differential equations in (3.13) to obtain the Stückelberg scalars in terms of {g,f,ρ,p}. Hence, in this more general solution scenario which we have not considered here the Stückelberg scalars become functions of the thermodynamic state of the effective ideal fluid. We should also remark another important point here. Although the equation of state of the physical matter is subject to natural constraints we are completely free to choose any formp = f(ρ) of it for the effective matter case. Even non-physical effective ideal fluid choices are possible provided they satisfy (4.13) which is different than the universal energy-momentum conservation law (4.12) of physical matter. However as we have discussed in the previous section in spite of this large freedom we have bounds on the effective energy-momentum tensor to have real solutions of the reference metric and the scalar sector. On the other hand in the special solution case if (3.29) is saturated that is to say if ∆ ′ = 0 then the solutions are valid for the entire coordinate span but now we have to fix the equation of state of the effective matter asp = 1 11ρ + C ′ , (4.14) where C ′ = 1 11 We have obtained (4.14) by using (4.5) and (4.7) in (3.28) then by equating the result to zero. Another way of obtaining real solutions in (3.27) is to equate the constant coefficient to zero. This leads to an equation of statẽ with where we have used (4.5) and (4.7). Now before we conclude let us turn attention to the Stückelberg sector solutions of the special type which satisfy (3.23) and which accompany the cosmological metric solutions we have discussed here. We have from (4.4)  Also via (4.6) (4.20) Therefore by inspecting (3.32) and (3.33) under these identifications we find that where where {f ii (x µ )} are arbitrary and then one can still satisfy (3.19). On the other hand beside the above mentioned somewhat trivial cases we can find more general solutions of {φ a } for a completely specified background metric by solving the equations in (3.15) where one should replace G ′ by (4.21). Finally we should remark at this point that for the more general case of background metric and the Stückelberg scalar solutions defined via the equations (3.7)-(3.22) there appears no problem of triviality occurring in the form of bothf and {φ a }'s like the one we have encountered above for the solutions satisfying the condition (3.23).

Conclusion
We have constructed a specially chosen solution ansatz which has enabled us to device a method to find solutions to the Stückelberg scalar sector of the massive gravity theory. We have done this by decoupling the scalar sector from the metric one and by reducing the task to finding the solutions of a mutual constraint equation of the background metric and the scalars of the theory. In this way we were able to find a broad solution class of the Stückelberg sector by specifying the associated background metric in terms of the physical one and the ansatz parametrizing effective matter energymomentum tensor. In this solution scheme the metric sector only sees the scalars as a collective effect in the presence of an effective energy-momentum tensor whose conservation equation is modified by a variation and a trace term. Later we constructed the associated modified FLRW cosmological dynamics by assigning an ideal fluid nature to both the ordinary and the effective matter. We should emphasize the point here that although the equations of different sectors are decoupled from each other the solutions are intimately related. One should first solve the physical metric and the effective energy-momentum tensor from the metric and the modified conservation equations then one should use these solutions to construct the background metric which admits such a solution scheme. Finally the solutions of the Stückelberg scalars trivially follow this construction up to a set of integrable functions.
The cosmological solutions of the massive gravity is an active topic of research which has reached some positive and negative conclusions. For example it has been shown that for a flat background metric there exist open FLRW solutions [13] however there are no flat or closed FLRW solutions [14]. The open FLRW solutions with the flat background metric have stability problems [15,16]. These facts led to the idea of choosing different background metrics for the cosmological solutions such as the de Sitter [17] and the FLRW type [18] ones. However these solutions have non-physical consequences regarding the Higuchi bound. On the other hand the inhomogeneous and anisotropic cosmological solutions of the ghost-free massive gravity have also been studied giving physically sensible results in certain regions of the parameter space of the theory [19,20]. Contrary to the mainstream of the corresponding literature we should remark that our approach does not pre-determine the background metric but solves it for the particular physical scenario in inspection. Therefore the formalism of the present work addresses not only to a very rich sector of the solution moduli of the background metric and the scalars of the theory but also to a wide range of physical applications via the choice of the effective matter. We certainly owe this richness to the large solution space of the massive gravity. We hope that the solution methodology achieved here will provide an appropriate framework to generate new and physically acceptable cosmological or astrophysical solutions of the theory. However in this direction both the stability issues and the Higuchi bound behavior of the solutions found here need to be inspected in detail elsewhere.
We observe that the effective and collective contribution of the scalars and the reference metric to the FLRW dynamics is suppressed by a squared graviton mass coefficient. However there also appear the C 1 and C 2 factors which possibly contain the degree of freedom of tuning the solutions for agreement with the physical needs. Such a scale generation may physically justify our construction as an effective theory of sub-solutions of the the exact theory. Both in the more general (which admits a wide range of solutions depending on specifying the effective ideal fluid) and the special cases we have mentioned, the modified cosmological dynamics can separately be studied and their testable consequences can further be classified as well. In this direction we should remark that the solutions we have constructed do contain self-accelerating ones. In particular, when we set C 2 to zero then we have the solutions in which the contribution of the mass sector to the metric one becomes solely an effective cosmological constant (where still the richness of the corresponding Stückelberg scalar and the background metric solutions remains intact) which is similar to many of such solutions constructed in the literature. However the more interesting class of self-accelerating solutions can be obtained by allowing the existence of effective fluids. The reader may appreciate the possibility of a wide range of such solutions as a consequence of the free choice of the state equation for these fluids which in general is completely arbitrary. For this reason we leave a detailed examination of this issue for a later work. More generally the point which deserves to be mentioned, and emphasized on, is that, the present construction enables the freedom of choosing any ordinary or exotic form of effective matter which will admit a non-physical dynamics in generating solutions. This is a consequence of the fact that although we have specified the effective energymomentum tensor in the ideal fluid form, while we construct the on-shell Lagrangian corresponding to the solution ansatz we did not take the usual Lagrangian of the ideal fluid which would be just the effective pressure upon using the first law of thermodynamics and which would generate the perfect fluid energy-momentum tensor with no extra terms in the metric equation. Thus our choice of the effective fluid does not obey the first law of thermodynamics as can be obviously seen also from the modified energy-momentum relation it satisfies and for this reason it can be called non-physical. This brings the opportunity of generating a rich class of solutions which would especially emerge from effective fluid choices which can not have physical correspondents. Furthermore the existence of the implicit solution relation we have discussed above between the physical, the background metrics and the effective matter also suggests a coupling dynamics among them. Such a construction which could include a dynamical nature for the background metric and a reasonable origin for the effective matter can be searched within the context of bi-metric gravity cosmological solutions [21,22,23,24]. One can also separately consider to extend the ansatz we have used to generate similar solutions of bigravity. Finally, we should point out the possibility of modifying our ansatz to other forms which may lead to various other solutions not necessarily cosmological within the same line of reasoning.