Cosmology and stability in scalar tensor bigravity with non-minimal kinetic coupling gravity
aa r X i v : . [ g r- q c ] A ug Cosmology and stability in scalar tensor bigravitywith non-minimal kinetic coupling gravity
F. Darabi ∗ and M. Mousavi † Department of Physics, Azarbaijan Shahid Madani University, Tabriz, 53714-161 Iran
We generalize the scalar tensor bigravity models to the non-minimal kinetic coupling scalar ten-sor bigravity models with two scalar fields whose kinetic terms are non-minimally coupled to twoEinstein tensors constructed by two metrics. We show that a broad class of expanding universescan be explained by some solutions of this model. Then, we study the stability issue of the solutionsby means of imposing homogeneous perturbation on the equations of motion and extract the stablesolutions.
PACS numbers: 95.36.+x, 12.10.-g, 11.10.Ef
I. INTRODUCTION
The infrared modifications of Einstein general relativity is receiving attention by two aspects: i) the theoreticalaspect from an effective field theory point of view that is one of the very natural choices to seek when declaring thediffeomorphism invariance in GR, and ii) the observational aspect by means of explaining the accelerating expansionof the Universe. In this regard, it seems that general relativity has to include a new mass term to validate suchattempts. Recently, a ghost free [1, 2] non linear massive gravity model (the dRGT theory) was constructed [3–7]involving two metric tensors, one dynamical g µν and one non-dynamical f µν appearing in a unique set of terms. Thesetwo metrics are used to add a new mass term to the original action (GR), namely the interaction term. For this modelwith a non-dynamical reference metric, it has been shown that the flat Friedmann-Robertson-Walker (FRW) Universedose not exist [8], and that the open FRW solutions are allowed [9] involving the problems of strong coupling [10] andghostlike instabilities [11]. Attempts to screen such problems led people to a natural way of extending the massivegravity theory and going beyond it toward a similar new model in which two dynamical symmetric tensors g µν and f µν appear as foreground and background metrics, respectively in a completely symmetric manner [12–15] which iscalled the bigravity theory. Obviously, the massive bigravity theory covers the massive gravity theory and treats withtwo metrics thoroughly in a symmetric way such that convinces one to get rid of the aether-like concept of referencemetric in massive gravity. Cosmology in massive gravity model has been studied in Refs. [16–19], whereas in bigravityseveral branches of regular cosmological solutions have been extracted [20–22]. Afterwards, the bigravity model and itsmodification with two independent scalar fields have been investigated in [23–25] presenting the models which certifythe stable solution describing the spatially flat FRW solution. In these models, the scalar fields are minimally coupledto the metric. As we know, there are varieties of cosmological models which contain scalar fields non-minimally coupledto gravity [26–28]. Moreover, one can go on the extension of scalar-tensor theories and construct coupling betweenthe derivatives of the scalar fields and the curvature, namely the non-minimal kinetic coupling scalar tensor theory[29]. In this work, we consider such non-minimal kinetic coupling scalar tensor bigravity model as a generalization ofthe model studied in [23].The organization of this paper is as follows. In section 2, we review the field equations of motions for the minimalbigravity action. In section 3, we construct bigravity models with two scalar fields forming the action of non-minimalkinetic coupling scalar tensor bigravity model and again obtain the equations of motions besides extracting the Bianchiconstraints. In section 4, we show that a broad class of the expansion history of the universe can be explained bysome solutions of the bigravity model, whereas these solutions do not have always stability against the perturbation.In section 5, we go through the stability issue of the solutions. The paper ends with a conclusion. ∗ Electronic address: [email protected] † Electronic address: [email protected]
II. BIGRAVITY THEORY
The action of Hassan-Rosen theory named bigravity has the following structure [30] S bi = M g Z d x p − det gR ( g ) + M f Z d x p − det f R ( f ) + 2 m M Z d x p − det g X n =0 β n e n (cid:16)p g − f (cid:17) . (1)Here g µν and f µν are two dynamical tensors in the gravity sector, and R ( g ) and R ( f ) are the scalar curvaturescorresponding to the metric tensors g µν and f µν , respectively. It should be noted that M eff is defined as1 M = 1 M g + 1 M f . (2)The tensor p g − f is the square root of g µρ f ρν which means, (cid:16)p g − f (cid:17) µ ρ (cid:16)p g − f (cid:17) ρ ν = g µρ f ρν = X µ ν . For thisdefined tensor, e n ( X )’s are given by e ( X ) =1 , e ( X ) = [ X ] , e ( X ) = 12 (cid:0) [ X ] − [ X ] (cid:1) ,e ( X ) = 16 (cid:0) [ X ] − X ][ X ] + 2[ X ] (cid:1) ,e ( X ) = 124 (cid:0) [ X ] − X ] [ X ] + 3[ X ] + 8[ X ][ X ] − X ] (cid:1) ,e k ( X ) =0 for k > , (3)where [ X ] means the trace of the tensor X µ ν . For simplicity, we take the minimal and non-trivial case as [31] S bi = M g Z d x p − det gR ( g ) + M f Z d x p − det f R ( f ) + 2 m M Z d x p − det g (cid:16) − tr p g − f + det p g − f (cid:17) , (4)which is apparently a direct result of using equation (3) in terms of e n as follows3 − tr p g − f + det p g − f = 3 e (cid:16)(cid:16)p g − f (cid:17) µ ν (cid:17) − e (cid:16)(cid:16)p g − f (cid:17) µ ν (cid:17) + e (cid:16)(cid:16)p g − f (cid:17) µ ν (cid:17) . (5)Considering non-minimal models leads to quite complicated calculations, whereas in the minimal model the interactionterm of two metrics g µν and f µν is just obtained by the trace of (cid:16)p g − f (cid:17) µ ν . It should be mentioned that thissimplification does not absolutely change our following results. Before starting the next section, it would be worthwhileto explain a little about the details of extracting the field equations of bigravity model. The variation of action (1) isgiven by δ g S bi = M g Z d xδ g (cid:16)p − det gR ( g ) (cid:17) − m M Z d xδ g (cid:16)p − det g (cid:16) − tr p g − f + det p g − f (cid:17)(cid:17) , (6)and δ f S bi = M f Z d xδ f (cid:16)p − det f R ( f ) (cid:17) − m M Z d xδ f (cid:16)p − det g (cid:16) − tr p g − f + det p g − f (cid:17)(cid:17) . (7)Clearly, the first terms of two above equations produce the known Einstein tensors for metrics g µν and f µν , respectivelyas δ g (cid:16)p − det gR ( g ) (cid:17) = p − det g (cid:18) R ( g ) µν − g µν R ( g ) (cid:19) , (8)and δ f (cid:16)p − det f R ( f ) (cid:17) = p − det f (cid:18) R ( f ) µν − f µν R ( f ) (cid:19) . (9)By considering that δ tr (cid:16)p g − f (cid:17) = tr (cid:16) g p g − f δg − (cid:17) = tr (cid:18) f (cid:16)p g − f (cid:17) − δg − (cid:19) beside the property p det( − g ) p det g − f = p det( − f ) we can conclude that0 = M g (cid:18) R ( g ) µν − R ( g ) g µν (cid:19) + m M (cid:26) g µν (cid:16) − tr p g − f (cid:17) + 12 f µρ (cid:16)p g − f (cid:17) − ρ ν + 12 f νρ (cid:16)p g − f (cid:17) − ρ µ (cid:27) , (10)and 0 = M f (cid:18) R ( f ) µν − R ( f ) f µν (cid:19) + m M p det ( f − g ) (cid:26) − f µρ (cid:16)p g − f (cid:17) ρ ν − f νρ (cid:16)p g − f (cid:17) ρ µ + f µν det (cid:16)p g − f (cid:17)(cid:27) . (11)Here, we are ended up with two independent equations of motion for metrics g µν and f µν in general model of bigravity. III. BIGRAVITY WITH TWO SCALAR FIELD KINETIC TERMS NON-MINIMALLY COUPLED TOCURVATURES
Let us review the original scalar tensor bigravity action S tot = M g Z d x p − det gR ( g ) + M f Z d x p − det f R ( f ) + S φ + S ξ +2 m M Z d x p − det g (cid:16) − tr p g − f + det p g − f (cid:17) , (12)where we have S φ = − M g Z d x p − det g { ǫg µν ∂ µ φ∂ ν φ + 2 V ( φ ) } + Z d x L matter ( g µν , Φ i ) , (13)and S ξ = − M f Z d x p − det f { λf µν ∂ µ ξ∂ ν ξ + 2 U ( ξ ) } , (14)where V ( φ ) and U ( ξ ) are scalar field potentials of metrics g µν and f µν , respectively. Now, we are interested in writingthe modified form of the above action by means of including the non-minimal kinetic derivative couplings to thecurvatures in the action as follows S tot = M g Z d x p − det g (cid:16) R ( g ) − h εg µν + κG ( g ) µν i ∂ µ φ∂ ν φ − V ( φ ) (cid:17) + M f Z d x p − det f (cid:16) R ( f ) − h λf µν + βG ( f ) µν i ∂ µ ξ∂ ν ξ − U ( ξ ) (cid:17) +2 m M Z d x p − det g (cid:16) − tr p g − f + det p g − f (cid:17) , (15)where we have ignored the matter contribution, and κ and λ are the g µν and f µν coupling parameters with dimensionof ( length ) , respectively. Varying the action (15) with respect to g µν and φ yields the field equations0 = − G ( g ) µν + M g T ( φ ) µν + κM g Θ ( g ) µν + m M (cid:26) g µν (cid:16) − tr p g − f (cid:17) + 12 f µρ (cid:16)p g − f (cid:17) − ρ ν + 12 f νρ (cid:16)p g − f (cid:17) − ρ µ (cid:27) , (16)and 0 = h εg µν + κG ( g ) µν i ∇ µ ∇ ν φ − V φ , (17)respectively, where V φ ≡ dV ( φ ) /dφ , ∇ µ ≡ ∇ µg , and T ( φ ) µν = ε (cid:20) ∇ µ φ ∇ ν φ − g µν ( ∇ φ ) (cid:21) − g µν V ( φ ) , (18)Θ ( g ) µν = − ∇ µ φ ∇ ν φR ( g ) + 2 ∇ α φ ∇ ( µ φR ( g ) α ν ) + ∇ α φ ∇ β φR ( g ) µανβ + ∇ µ ∇ α φ ∇ ν ∇ α φ − ∇ µ ∇ ν φ (cid:3) φ − G ( g ) µν ( ∇ φ ) + g µν (cid:20) − ∇ α ∇ β φ ∇ α φ ∇ β φ + 12 ( (cid:3) φ ) − ∇ α φ ∇ β R ( g ) αβ (cid:21) , (19)where (cid:3) ≡ (cid:3) g is the d’Alembertian with respect to the metric g . Using (15), and because f is dynamical as well as g , there is a symmetry between them so that we are allowed to write0 = − G ( f ) µν + M f T ( ξ ) µν + βM g Θ ( f ) µν + m M p det ( f − g ) (cid:26) − f µρ (cid:16)p g − f (cid:17) ρ ν − f νρ (cid:16)p g − f (cid:17) ρ µ + f µν det (cid:16)p g − f (cid:17)(cid:27) , (20)0 = h λf µν + βG ( f ) µν i ∇ µ ∇ ν φ − U ξ , (21)where U ξ ≡ dU ( ξ ) /dξ . Under the mentioned symmetry condition ∇ µg ↔ ∇ µf and (cid:3) g ↔ (cid:3) f we just have the changes T ( ξ ) µν = λ (cid:20) ∇ µ ξ ∇ ν ξ − f µν ( ∇ ξ ) (cid:21) − f µν U ( ξ ) , (22)Θ ( f ) µν = − ∇ µ ξ ∇ ν ξR ( f ) + 2 ∇ α ξ ∇ ( µ ξR ( f ) α ν ) + ∇ α ξ ∇ β ξR ( f ) µανβ + ∇ µ ∇ α ξ ∇ ν ∇ α ξ − ∇ µ ∇ ν ξ (cid:3) ξ − G ( f ) µν ( ∇ ξ ) + f µν (cid:20) − ∇ α ∇ β ξ ∇ α ξ ∇ β ξ + 12 ( (cid:3) ξ ) − ∇ α ξ ∇ β R ( f ) αβ (cid:21) . (23)According to Bianchi identity we have 0 = ∇ µg (cid:18) R ( g ) µν − g µν R ( g ) (cid:19) , (24)0 = ∇ µf (cid:18) R ( f ) µν − f µν R ( f ) (cid:19) . (25)By imposing the covariant derivative ∇ µg and ∇ µf on equations (16) and (20) and using the constraints ∇ µ (cid:16) M g T ( φ ) µν + κM g Θ ( g ) µν (cid:17) = 0, and ∇ µ (cid:16) M f T ( ξ ) µν + βM f Θ ( f ) µν (cid:17) = 0 , resulting from equations (17), (18), (19), (21),(22) and (23), besides refereing to (24) and (25), we can obtain two following constraints0 = − g µν ∇ µg (cid:16) tr p g − f (cid:17) + 12 ∇ µg (cid:26) f µρ (cid:16)p g − f (cid:17) − ρ ν + f νρ (cid:16)p g − f (cid:17) − ρ µ (cid:27) , (26)0 = ∇ µf (cid:20)p det( f − g ) (cid:26) − (cid:16)p g − f (cid:17) − ν σ g σµ − (cid:16)p g − f (cid:17) − µ σ g σν + f µν det (cid:16)p g − f (cid:17)(cid:27)(cid:21) , (27)which can be used to extract an important constraint on the metric coefficients implying that we can find a dynamicalcosmology in this model. IV. COSMOLOGICAL EQUATIONS WITH NON-MINIMAL KINETIC COUPLING BIGRAVITYMODEL
As an important step to test a theory as a real cosmological model, it would be fruitful to inquire whether it ispossible to have a model expressing the arbitrary evolution of the expanding universe. As a result, now we considerthe FRW universe for the g µν metric besides using the conformal time t = τ . We emphasize that in bigravity notonly the metric g µν but also f µν is dynamical, therefore we take the following form of metrics ds g = X µ,ν =0 g µν dx µ dx ν = a ( τ ) − dτ + X i =1 ( dx i ) ! , (28) ds f = X µ,ν =0 f µν dx µ dx ν = − c ( τ ) dτ + b ( τ ) X i =1 ( dx i ) . (29)Obviously we are not allowed to take c ( τ ) = 1 nor c ( τ ) = b ( τ ), because by this option we will have the Minkowskimetric for ds f that leads to Massive gravity non-dynamical cosmology [32]. Thus, for this case, the ( τ, τ ) and ( i, i )components of (16) are given by0 = − M g H − m M ( a − ab ) + ε ˙ φ V ( φ ) a ( τ ) ! M g − κM g H ˙ φ a , (30)0 = M g (cid:16) H + H (cid:17) + m M (3 a − ab − ac ) + ε ˙ φ − V ( φ ) a ( τ ) ! M g − κM g " ˙ H ˙ φ a − H ˙ φ a + 2 H ˙ φ ¨ φa . (31)Moreover, the ( τ, τ ) and ( i, i ) components of (20) yield0 = − M f K + m M c (1 − a b ) + λ ˙ ξ U ( ξ ) c ( τ ) ! M f − βM f K ˙ ξ a , (32)0 = M f (cid:16) − K − K + 2 KL (cid:17) + m M ( − a cb + c ) + − λ ˙ ξ U ( ξ ) a ( τ ) ! M f + βM f c (cid:20) ˙ ξ (cid:18) − ˙ K − K KL (cid:19) − K ˙ ξ ¨ ξ (cid:21) , (33)by definition of K = ˙ b/b and L = ˙ c/c . Applying two scalar fields for metrics (28) and (29) helps us to describe threemetric coefficients a ( τ ), b ( τ ) and c ( τ ) as three degrees of freedom; this is not possible just by one scalar field. It turnsout that the equations (26) and (27) carry important results for the variables defined in the metrics (28) and (29) asfollows cH = bK or c ˙ aa = ˙ b. (34)This is a constraint imposing on the metrics and relating them to each other. For the case ˙ a = 0 the above constraintgives c = a ˙ b/ ˙ a but for ˙ a = 0, we have ˙ b = 0 meaning that a and b are constant but c is arbitrary. Actually, thisconstraint is allowing us to go on and construct an expanding cosmology. Redefining scalar fields in accordance withconformal times η = ζ = τ as ϕ = ϕ ( η ) and ξ = ξ ( ζ ) with ω ( η ) = ϕ ′ ( η ) , ˜ V ( η ) = V ( ϕ ( η )), σ ( ζ ) = ξ ′ ( ζ ) (prime isthe derivative with respect to its conformal time) and ˜ U ( ζ ) = U ( ξ ( ζ )) facilitate us to rewrite equations (30)-(33) asfollows0 = − M g H − m M ( a − ab ) + (cid:18) εω ( τ )2 + V ( τ ) a ( τ ) (cid:19) M g − κM g H ω ( τ )2 a , (35)0 = M g (cid:16) H + H (cid:17) + m M (3 a − ab − ac ) + (cid:18) ǫω ( τ )2 − V ( τ ) a ( τ ) (cid:19) M g − κM g a h(cid:16) ˙ H − H (cid:17) ω ( τ ) + H ˙ ω ( τ ) i , (36)0 = − M f K + m M c (1 − a b ) + (cid:18) λσ ( τ )2 + U ( τ ) c ( τ ) (cid:19) M f − βM f K σ ( τ )2 a , (37)0 = M f (cid:16) − K − K + 2 KL (cid:17) + m M ( − a cb + c ) + (cid:18) − λσ ( τ )2 + U ( τ ) a ( τ ) (cid:19) M f + βM f c (cid:20) σ ( τ ) (cid:18) − ˙ K − K KL (cid:19) − K ˙ σ ( τ ) (cid:21) . (38)Adding and subtracting equations (35) and (36) give us two following equations0 = 2 M g (cid:16) ˙ H − H (cid:17) + m M ( ab − ac ) + ǫω ( τ ) M g + κM g a h ω ( τ ) (cid:16) − H + ˙ H (cid:17) + H ˙ ω ( τ ) i , (39)0 = M g (cid:16) H + 4 H (cid:17) + m M (6 a − ab − ac ) − M g a ( τ ) ˜ V ( τ ) + κM g a h(cid:16) ˙ H + 3 H (cid:17) ω ( τ ) + H ˙ ω ( τ ) i . (40)Again, by subtracting and adding equations (37) and (38) we will have0 = 2 M f (cid:16) − ˙ K + KL (cid:17) + m M ( − a cb + a c b ) − λσ ( τ ) M f + βM f c h σ ( τ ) (cid:16) − ˙ K − K + 3 KL (cid:17) − K ˙ σ ( τ ) i , (41)0 = M f (cid:16) − K + 2 KL − K (cid:17) + m M c (cid:18) − a b − a b c (cid:19) + 2 M f c ( τ ) ˜ U ( τ )+ βM f c h σ ( τ ) (cid:16) − K + 3 KL − ˙ K (cid:17) − K ˙ σ ( τ ) i . (42)Therefore, for arbitrary metric coefficients a ( τ ), b ( τ ) and c ( τ ), by choosing ω ( τ ), σ ( τ ), ˜ V ( τ ) and ˜ U ( τ ) satisfyingequations (39)-(42), we can reconstruct cosmological models with given evolutions of a ( τ ), b ( τ ) and c ( τ ). A. Conformal description of the power expanding universe
In the last section, in the equations (28) and (29), we have applied the conformal metric with a conformal time τ to express the evolution of the scalar field φ ( τ ). In the following, we are going to show how the conformal time candescribe the known cosmologies. As a result, again we remark the conformally flat FLRW metric as ds = a ( τ ) − dτ + X i =1 ( dx i ) ! . (43)According to the power expanding universe the scale factor is treating as a ( τ ) = a n τ n with n = 1. Considering n = 1can explain the de Sitter universe in the arbitrary model representing dark energy or inflation. Looking back to thepower expanding scale factor leads to the following redefinition of the time coordinate as d ˜ t = ± a n τ n dτ, (44)so we have ˜ t = ± a n n − τ − n . (45)By considering the above discussion, we will have the following form for the FLRW metric ds = − d ˜ t + (cid:20) ± ( n −
1) ˜ ta (cid:21) − n − n X i =1 ( dx i ) . (46)From this equation one finds that for the range 0 < n <
1, the metric describes the Phantom universe [33], for n > n < t → ˜ t − t on (46). By this shift, we have the BigRip at ˜ t = t , the present time at ˜ t < t , and the infinite past (˜ t → −∞ ) is equivalent to the limit of τ → ∞ . For thequintessence range ( n >
1) we again choose + sign of (44) or (45). Hence, the limit of τ → t → + ∞ and the limit of τ → + ∞ corresponds to ˜ t →
0, which may describe the Big Bang. Similarly, for the deceleratinguniverse we choose + sign. As a result, the limit of τ → + ∞ corresponds to ˜ t → + ∞ , and that of τ → t → n = 1) we have if τ → t → + ∞ and if τ → ±∞ ˜ t → −∞ . (47) B. Dark energy solution with a ( τ ) = b ( τ ) = c ( τ ) Our universe can be described by the metric g µν . As a result, we have freedom to choose the functions c ( τ ) and b ( τ )which are not directly describing the expansion of the universe since they are f µν degrees of freedom in the Einsteinframe. Any way, we choose the functions c ( τ ) and b ( τ ) equal to a ( τ ) to have more convenient calculation, however,we should not deduce that they do not have any physical meaning. Thus, we are allowed to take a ( τ ) = b ( τ ) = c ( τ )leading to K = H = L . As a result, the metric interaction terms in equations (39)-(42) vanish and also by supposing M f = M g , ǫ = λ and κ = β we can obtain κ ˙ ω ( τ ) + ω ( τ ) a ( τ ) ǫH + κ − H + ˙ HH !! + 2 a ( τ ) ˙ HH − H ! = 0 , (48) a ( τ ) ˜ V ( τ ) = 3 H + ω ( τ ) (cid:18) κH a ( τ ) − ǫ (cid:19) , (49) κ ˙ σ ( τ ) + σ ( τ ) a ( τ ) ǫH + κ − H + ˙ HH !! + 2 a ( τ ) ˙ HH − H ! = 0 , (50)and a ( τ ) ˜ U ( τ ) = 3 H + σ ( τ ) (cid:18) κH a ( τ ) − ǫ (cid:19) . (51)Obviously, we can deduce that σ ( τ ) = ω ( τ ) and so ˜ V ( τ ) = ˜ U ( τ ). By inserting a ( τ ) = a n τ n and a ( τ ) = e nτ into (48)and (49) and supposing that a = 1, ǫ = , κ = 1 and σ ( τ ) = ω ( τ ) we can find˙ ω ( τ ) + ω ( τ ) (cid:18) − nτ n − + 6 n − τ (cid:19) + 2 τ n − ( n −
1) = 0 , (52)˜ V ( τ ) = 3 n τ n − + ω ( τ ) (cid:18) n τ n − − τ n (cid:19) , (53)˙ ω ( τ ) + ω ( τ ) (cid:18) e nτ n − n (cid:19) − ne nτ = 0 , (54)˜ V ( τ ) = 3 n e − nτ + ω ( τ ) (cid:18) n e − nτ − (cid:19) . (55)Clearly, ω ( τ ) and ˜ V ( τ ) are indirectly coupled to each other, so the ω ’s which are obtained from equations (52) and(54) are influenced by the evolution of potential ˜ V ( τ ). In order to extract the cosmological constant in our model,we need that ˜ V ( τ ) become constant. Then, by differentiation both sides of (53) with respect to τ we can extract theexpression for ˙ ω ( τ ) satisfying that of equation (52). Thus, we are facilitated to write ω ( τ ) for the case a ( τ ) = τ n and˜ V ( τ ) = cte as ω ( τ ) = (cid:0) − n ) τ n − (cid:1) (cid:16) n τ n +1 − τ (cid:17) + 6 n ( n − (cid:0) n τ n +1 − τ (cid:1) (cid:18) n (1 − n ) τ n − + nτ n − n τ n − − τ n − nτ n − + n − τ (cid:19) . (56)Considering the above result besides referring to (53) makes it clear that ω ( τ ) vanishes and also ˜ V ( τ ) becomesconstant, provided that n = 1. This result is exactly the same as that of scalar tensor bigravity model explainingthe de Sitter model with a ( τ ) = τ as a dark energy universe. In the next section, we will seek the stability of allmentioned solutions, even the de Sitter one, under the homogeneous perturbation.Now, we consider the scale factor a ( τ ) = e nτ . Similar to the previous approach for constant potentials we have ω ( τ ) = 6 n e − + 2 ne nτ (cid:0) − n e − nτ (cid:1)(cid:0) n − e nτ n (cid:1) (cid:0) n e − nτ − (cid:1) + 2 ne − − n e − − nτ , (57)which should take just positive values not to conflict with the definition ω ( η ) = ϕ ′ ( η ) . Studying the three dimensionaldiagram of the above equation for ω ( τ ) with the variables τ and n lets us to write ω ( τ ) > → (100 < τ < + ∞ for all values of n) . (58)Thus, to be more precise, we may describe the cosmological constant solution for the case a ( τ ) = e nτ as followsfor ( n > , τ → + ∞ ) we have (cid:16) ˜ V → n and ω ( τ ) → n (cid:17) , for ( n < , τ → + ∞ ) we have (cid:16) ˜ V → ω ( τ ) → (cid:17) . (59) V. COSMOLOGY AND STABILITY OF SOLUTIONS
Following the last section, we shall discuss about the validity of these wide range of presented solutions to choosejust the cases which satisfy both the consistency condition for our model and the stability constraint. In the following,we want to investigate the stability of the solutions discussed in the previous section.Going through equations (48) and (50), shows us a way to find the positive ranges of ω ( τ ) and σ ( τ ) by means ofplotting their three dimensional diagrams extracted by equations (48) and (50) in which we put two mentioned scalefactors a ( τ ) = a n τ n and a ( τ ) = e nτ . As we know, ω ( τ ) and σ ( τ ) are ˙ φ and ˙ ξ , respectively, so they are not allowedto take negative values in order to avoid of ghost and inconsistency in the theory.By inserting a ( τ ) = a n τ n and a ( τ ) = e nτ into (48) and supposing that a = 1, ǫ = , κ = 1 and σ ( τ ) = ω ( τ ) we canfind ˙ ω ( τ ) + ω ( τ ) (cid:18) − nτ n − + 6 n − τ (cid:19) + 2 τ n − ( n −
1) = 0 , (60)˙ ω ( τ ) + ω ( τ ) (cid:18) e nτ n − n (cid:19) − ne nτ = 0 , (61)respectively. Two above equations have the following solutions ω ( τ ) = e (cid:16) τ − n (2 − n ) n − n −
1) ln( τ ) (cid:17) c − e (cid:16) τ − n (2 − n ) n − n −
1) ln( τ ) (cid:17) ( n − × n +21 − n τ Γ (cid:16) − n , τ − n n − n (cid:17) (cid:16) τ − n n − n (cid:17) n − n (384 n + 880 n − n − n − n + 15) + e τ − n n − n τ n − (cid:0) n (cid:0) n + 164 n + 116 n + 15 (cid:1) τ n + 2 n (2 n + 5) τ n +4 + 4 n (cid:0) n + 26 n + 15 (cid:1) τ n +2 + τ (cid:1) n (384 n + 1264 n + 764 n + 4 n − , (62) ω ( τ ) = e nτ − e nτ n c + 2 e nτ − e nτ n n e e nτ n (cid:18) − e − nτ n − e − nτ n (cid:19) + Ei (cid:16) e nτ n (cid:17) n , (63)where c and c are the integration constants. The allowed ranges of τ and n resulting in the positive valued ω aregiven by the Table.1 and Table.2 . Table 1: Intervals for τ and n resulting from the condition ω ( τ ) > a ( τ ) = τn . τ n (cid:0) , − (cid:1) (0 . , . ,
1) (1 , + ∞ ) (cid:0) , (cid:1) (0 , (cid:0) , + ∞ (cid:1) (0 . , . Table 2: Intervals for τ and n resulting from the condition ω ( τ ) > a ( τ ) = e nτ . To find the positive valued ω , we first plotted ω ( τ, n ) in the three dimensional diagrams with respect to τ (for three ranges of conformaltime within two classes of τ > τ <
0) and n (for three ranges given in subsection A). Then, we selected the allowed ranges of τ and n resulting in positive ω . τ n ( − ,
0) ( −∞ , . − , −
10) ( − . , − . ,
0) (1 , + ∞ )(0 ,
1) ( − . , ,
10) (0 . , . ,
1) (0 . , . ,
1) (1 , + ∞ )(100 , + ∞ ) ( −∞ , + ∞ ) Referring to equations (35)- (38) and rewriting them by including the conformal times η and ζ , gives us0 =2 M g (cid:16) ˙ H − H (cid:17) + m M ( a ( τ ) b ( τ ) − a ( τ ) c ( τ )) + ǫω ( η ) ˙ η M g + κM g a ( τ ) h ω ( η ) ˙ η (cid:16) − H + ˙ H (cid:17) + Hω ′ ( η ) ˙ η + 2 Hω ( η ) ˙ η ¨ η i , (64)0 = M g (cid:16) H + 4 H (cid:17) + m M (cid:0) a ( τ ) − a ( τ ) b ( τ ) − a ( τ ) c ( τ ) (cid:1) − M g a ( τ ) ˜ V ( η )+ κM g a ( τ ) h(cid:16) ˙ H + 3 H (cid:17) ω ( η ) ˙ η + Hω ′ ( η ) ˙ η + 2 Hω ( η ) ˙ η ¨ η i , (65)0 =2 M f (cid:16) − ˙ K + KL (cid:17) + m M (cid:18) − a ( τ ) c ( τ ) b ( τ ) + a ( τ ) c ( τ ) b ( τ ) (cid:19) − λσ ( ζ ) ˙ ζ M f + βM f c ( τ ) h σ ( ζ ) ˙ ζ (cid:16) − ˙ K − K + 3 KL (cid:17) − Kσ ′ ( ζ ) ˙ ζ − Kσ ( ζ ) ˙ ζ ¨ ζ i , (66)0 = M f (cid:16) − K + 2 KL − K (cid:17) + m M c ( τ ) (cid:18) − a ( τ ) b ( τ ) − a ( τ ) b ( τ ) c ( τ ) (cid:19) + 2 M f ˜ U ( ζ ) c ( τ ) + βM f c ( τ ) h σ ( ζ ) ˙ ζ (cid:16) − K + 3 KL − ˙ K (cid:17) − Kσ ′ ( ζ ) ˙ ζ − Kσ ( ζ ) ˙ ζ ¨ ζ i . (67)Additionally, we can write the scalar field equations as follows0 = ω ( η )¨ η + ω ′ ( η ) ˙ η Hω ( η ) ˙ η + ˜ V ′ ( η ) a − Ka (cid:20) H ω ′ ( η ) ˙ η H ω ( η )¨ η + 2 H ˙ H ˙ ηω ( η ) (cid:21) , (68)and0 = ˜ σ ( ζ )¨ ζ + σ ′ ( ζ ) ˙ ζ K − L ) σ ( ζ ) ˙ ζ + ˜ U ′ ( ζ ) c − βc " K σ ′ ( ζ ) ˙ ζ σ ( ζ )¨ ζ ! + σ ( ζ ) ˙ ζ (cid:16) K − K L + 6 K ˙ K (cid:17) . (69)Equations (48) and (49) will give us the metric coefficients provided that we specify these four equations with afunction f ( τ ) as ω ′ ( η ) + ω ( η ) (cid:20) ǫe f ( τ ) κf ′ ( η ) − f ′ ( η ) + f ′′ ( η ) f ′ ( η ) (cid:21) + 2( η ) e f ( τ ) κ (cid:18) f ′′ ( η ) f ′ ( η ) − f ′ ( η ) (cid:19) = 0 , (70)1˜ V ( η ) = 3 f ′ ( η ) e − f ( τ ) + ω ( η ) (cid:20) κf ′ ( η ) e − f ( τ ) − ǫ e − f ( τ ) (cid:21) , (71) σ ′ ( ζ ) + 2 e f ( ζ ) β (cid:18) f ′′ ( ζ ) f ′ ( ζ ) − f ′ ( ζ ) (cid:19) + σ ( ζ ) (cid:20) λe f ( ζ ) βf ′ ( ζ ) − f ′ ( ζ ) + f ′′ ( ζ ) f ′ ( ζ ) (cid:21) = 0 , (72)˜ U ( ζ ) = 3 f ′ ( ζ ) e − f ( ζ ) + ω ( ζ ) (cid:20) βf ′ ( ζ ) e − f ( ζ ) − λ e − f ( ζ ) (cid:21) . (73)As a result, we have a ( τ ) = b ( τ ) = c ( τ ) = e f ( τ ) , η = ζ = τ. (74)We intend to study the stability of solution (74), so we consider the following perturbation to reconstruct the pertur-bation equations ζ → ζ + δζ, η → η + δη, b → b (1 + δf b ) , a → a (1 + δf a ) , K → K + δK, H → H + δH. (75)Having more simplicity persuade us to consider the following terms: M f = M g = M M , (76) ǫ = λ = 1 / , κ = β. (77)After a troublesome calculation explained in Appendix A, we obtain ddτ δηδHδζδKδf a δf b = − b b − b b − b b − b b − h h + h b h b − h h + h b b b − h h − h h + h b h b − h h + h b h b − c c − c c − c c − c c − c c g g g g g g δηδHδζδKδf a δf b , (78)where the (6 ×
6) perturbation matrix is called M and its elements have been defined in Appendix A. The eigenvalueequation with eigenvalues λ has the following form0 = λ + q λ + q λ + q λ + q λ + q λ + q . (79)Since we are interested in finding stable solutions, we should consider the negative value of the eigenvalues λ . Asa result, all the eigenmodes gradually disappear and thus it ends up in a damped perturbation. Note that theequation (79) is respecting the general form of the eigenvalue function expansion of an arbitrary matrix (n × n) as f M ( λ ) = ( − λ ) n + trM( − λ ) n − + ... + detM, which can help us to conclude that this equation with negative eigenvaluesrequires q = − trM > , (80)or (see the Appendix B) trM = − h h + h b b b − b b − c c + g < . (81)2Similar to the previous argument in finding the allowed ranges of τ and n resulting in the positive valued ω , toinvestigate the stable solutions we find the allowed ranges of τ and n , given by Table 3 and Table 4, resulting in thenegative values of trM where use has been made of (62) and (63) with c = c = 0. Note that, we should be carefulthat the following results are obtained in accordance with the consistency ranges of this model mentioned in Table 1and Table 2. For instance, in the case a ( τ ) = τ n we have − trM > . < τ < . , < n < . ω ( τ ) and hence this interval is ruled out. Table 3: Intervals for τ and n resulting from the condition − trM > a ( τ ) = τn . τ n (0 . , .
37) [1 , . . , + ∞ ) [1 , ,
7) (0 , . ,
7) (0 . , . Table 4: Intervals for τ and n resulting from the condition − trM > a ( τ ) = e nτ . τ n (0 , .
02) ( − . , − . . ,
10) (0 . , . . ,
1) (0 . , . , .
35) (1 , . . , .
73) (1 . , . It would be fruitful to survey the stability of the dark energy solutions (56) and (57) for the presented scale factors a ( τ ) = τ and a ( τ ) = e nτ , respectively. Obviously, the solution (59) is not stable since the ranges of Table 4 do notcover it , thus, we do not have any dark energy solution for the case a ( τ ) = e nτ . For the de Sitter solution describedby (56) , we should plot trM by using (56) for the limit n → τ for n → . < τ < .
16 and 1 . < τ < . , (82)which is covered with the ranges of Table 3 and so the de Sitter solution of the case a ( τ ) = τ is stable. VI. CONCLUSIONS
In this paper, we have constructed non-minimal kinetic coupling bigravity models with two independent scalarfields. It has been shown that a wide range of expansion history of the universe can be explained by a solution of thebigravity model, particularly inflation or current accelerating expansion of the universe. This description is differentfrom the predictions of the models in massive gravity where the background metric is non-dynamical and does notlead to the spatially flat homogeneous FRW cosmological solution. In the original scalar-tensor bigravity theory [23] ithas been shown that in spite of the difficulty of finding stable cosmological solutions , one can find some explicit stablesolutions by studying the sign of the trace of perturbation matrix extracted from the perturbed field equations. In thiswork, we have followed the mentioned approach by studying the diagrams of the eigenvalue equation condition (81),which imposes a constraint on the trace of perturbation matrix M to take just negative values. We considered the deSitter universe in the non-minimal kinetic coupling bigravity model and showed that the current model thoroughlyendorses the de Sitter universe evolution as well as the scalar tensor bigravity with the choice of a ( τ ) = b ( τ ) = c ( τ )for the solution a ( τ ) = τ , and then obtained the cosmological constant ˜ V ( τ ) = ˜ V = Λ with ω ( τ ) = 0. For theaccelerating solution a ( τ ) = e nτ we got the dark energy universe for the ranges obtained in the relation (59).Also, we considered the constraint − trM > τ and n mentioned in Table 3 and Table 4. It should be noticed that these intervals are satisfying the mentionedallowed ranges for τ and n in Table 1 and Table 2. In the first place, we focused on the results of Table 3 for the case a ( τ ) = τ n . Seemingly, the first two couple ranges of τ and n in Table 3 outline roughly the solution interval τ > n > τ and n implying the interval 0 < n < a ( τ ) = τ and analyzed its stability in (82) accordingto the final results reported in Table 3 which surely shows that it is stable because the value n = 1 is included inthe ranges reported in Table 3 . Furthermore, in the case of other suggested scale factor e nτ with the correspondingvalues of τ and n mentioned in (59), showing the dark energy solution, we cannot distinguish any stability. As aresult, the solutions of non-minimal kinetic coupling bigravity model has some common stability intervals with thatof [35]. Throughout the paper, we just considered the homogeneous perturbation, which is independent of the spatialcoordinates. This is because the inhomogeneous perturbation and/or anisotropic background create ghost [34], andthe superluminal mode [35] in general leads to the violation of causality [36]. Appendix A: The calculations of Eqs.(78) and (81)
In the following, we extract equations Eqs. (78) and (81) first by using (34) L = K + ˙ KK − ˙ HH . (A1)Having plugged the equations (34) and (A1) into the equations (64)-(67) we can eliminate L and c as follows0 =2 M g (cid:16) ˙ H − H (cid:17) + m M a ( τ ) b ( τ ) (cid:18) − KH (cid:19) + ǫω ( η ) ˙ η M g + κM g a ( τ ) h ω ( η ) ˙ η (cid:16) − H + ˙ H (cid:17) + Hω ′ ( η ) ˙ η + 2 Hω ( η ) ˙ η ¨ η i , (A2)0 =2 M g (cid:16) ˙ H + 2 H (cid:17) + m M (cid:18) a ( τ ) − a ( τ ) b ( τ ) − a ( τ ) Kb ( τ ) H (cid:19) − M g a ( τ ) ˜ V ( η )+ κM g a ( τ ) h(cid:16) ˙ H + 3 H (cid:17) ω ( η ) ˙ η + Hω ′ ( η ) ˙ η + 2 Hω ( η ) ˙ η ¨ η i , (A3)0 =2 M f − K ˙ HH + K ! + 2 m M (cid:18) − a ( τ ) Kb ( τ ) H + a ( τ ) K b ( τ ) H (cid:19) − λσ ( ζ ) ˙ ζ M f +2 βM f H b ( τ ) K " σ ( ζ ) ˙ ζ K + 6 K − K ˙ HH ! − Kσ ′ ( ζ ) ˙ ζ − Kσ ( ζ ) ˙ ζ ¨ ζ , (A4)0 =2 M f − K − K ˙ HH ! + m M (cid:18) K b ( τ ) H − a ( τ ) K b ( τ ) H − a ( τ ) Kb ( τ ) H (cid:19) + 2 M f ˜ U ( ζ ) c ( τ ) + βM f c ( τ ) h σ ( ζ ) ˙ ζ (cid:16) − K + 3 KL − ˙ K (cid:17) − Kσ ′ ( ζ ) ˙ ζ − Kσ ( ζ ) ˙ ζ ¨ ζ i . (A5)By considering the equations (A2) and (A4), we find the following equation H − m M HM g a ( τ ) b ( τ ) (cid:18) − KH (cid:19) − ǫω ( η ) ˙ η H − κ Ha ( τ ) h ω ( η ) ˙ η (cid:16) − H + ˙ H (cid:17) + Hω ′ ( η ) ˙ η + 2 Hω ( η ) ˙ η ¨ η i = K + m M M f (cid:18) − KH (cid:19) a ( τ ) b ( τ ) H − λσ ( ζ ) ˙ ζ K + βH b ( τ ) K " σ ( ζ ) ˙ ζ × K + 6 K − K ˙ HH ! − Kσ ′ ( ζ ) ˙ ζ − Kσ ( ζ ) ˙ ζ ¨ ζ . (A6)Having eliminated ˙ H from (A2) and (A3) and then from (A4) and (A5), we calculate4 (cid:18) ω ( η ) ˙ η (cid:18) ǫ − κH a ( τ ) (cid:19) + a ( τ ) ˜ V ( η ) (cid:19) = 3 H + m M M g (cid:0) a ( τ ) − a ( τ ) b ( τ ) (cid:1) , (A7) (cid:18) σ ( ζ ) ˙ ζ (cid:18) λH K − βH b ( τ ) K (cid:19) + b ( τ ) ˜ U ( ζ ) (cid:19) = 3 H + m M M f (cid:18) a ( τ ) b ( τ ) − b ( τ ) (cid:19) . (A8)Combination of (A6), (A7) and (A8) gives us the following equation0 = ( K − H ) − ˜ U ( ζ ) b ( τ ) K H + ˜ V ( η ) a ( τ ) H − K a ( τ ) (cid:20) ω ( η ) ˙ η (cid:18) HH − H (cid:19) + ω ′ ( η ) ˙ η + 2 ω ( η ) ˙ η ¨ η (cid:21) + m M × " K H M f (cid:18) a ( τ ) b ( τ ) − b ( τ ) (cid:19) − HM g (cid:18) a ( τ ) b ( τ ) − b ( τ ) (cid:19) + (cid:18) − KH (cid:19) (cid:18) a ( τ ) H b ( τ ) H − a ( τ ) b ( τ )4 HM g (cid:19) − β b ( τ ) × " σ ˙ ζ KH K + 6 H K − H ˙ HK ! − σ ′ ( ζ ) ˙ ζ H K − H σ ( ζ ) ˙ ζ ¨ ζK . (A9)Regarding the independent equations (A3), (A7), (A8) and (A9) beside using (64), (65), (66) and (67) help us toextract the following four perturbation equations0 = δ ˙ H (cid:18) ω ( η ) κa ( τ ) (cid:19) + δH H + 2 m a ( τ ) H + 12 κHω ( η ) a ( τ ) − HH − ω ( η )2 H − κω ( η ) ˙ HHa ( τ ) ! − m a ( τ ) H δK + δf a (cid:18) m a ( τ ) − H + ω ( η ) (cid:18) − κH a ( τ ) + 36 κH a ( τ ) (cid:19) + 4 ˙ H (cid:19) − m a ( τ ) δf b + δ ˙ η ω ( η ) κ ˙ Ha ( τ ) + 12 ω ( η ) κH a ( τ ) + 2 H − H − ω ( η )2 ! + ω ( η ) δη × − H − κ ¨ Ha ( τ ) + κ ˙ HHa ( τ ) − κH ˙ Ha ( τ ) + 36 κH a ( τ ) + ˙ H H ! + δη − H ˙ H + 16 H − H + 2 ˙ H H ! , (A10)0 = δf a (cid:18) H + ω ( η ) (cid:18) κH a ( τ ) − (cid:19) − m a ( τ ) (cid:19) + δf b (cid:0) m a ( τ ) (cid:1) + δH (cid:18) − κHω ( η ) a ( τ ) − H (cid:19) + δ ˙ η (cid:18) − κH a ( τ ) (cid:19) × ω ( η ) + δη ω ( η ) κH ˙ Ha ( τ ) − κH a ( τ ) + H ! + 6 H ˙ H − H ! , (A11)0 = δf a (cid:0) − m a ( τ ) (cid:1) + δf b (cid:18) H − ω ( η )2 + 18 κω ( η ) H a ( τ ) + 6 m a ( τ ) (cid:19) + δ ˙ ζ × (cid:18) ω ( η )2 − κω ( η ) H a ( τ ) (cid:19) + δH (cid:18) ω ( η )2 H − κω ( η ) Ha ( τ ) − H (cid:19) + δK (cid:18) − ω ( η )2 H + 9 κω ( η ) Ha ( τ ) (cid:19) + δζ ω ( η ) κH ˙ Ha ( τ ) − κH a ( τ ) + H ! + 6 H ˙ H − H ! , (A12)0 = ( δK − δH ) ω ( η ) − κ a ( τ ) + κ ˙ H a ( τ ) H + 38 H ! −
32 + ˙ HH ! + ( δf a − δf b ) H − ˙ HH + ω ( η ) (cid:18) κH a ( τ ) − H (cid:19)! + ( δω ( η ) − δσ ( ζ )) 9 κH a ( τ )
5+ ( δη − δζ ) ω ( η ) κ ˙ Ha ( τ ) − κH a ( τ ) + 12 − ˙ H H + κ ¨ H a ( τ ) H − κ ˙ H a ( τ ) H ! + ¨ H H − ˙ H H + 7 ˙ H − H ! + (cid:16) δ ˙ η − δ ˙ ζ (cid:17) ω ( η ) H + 3 κH a ( τ ) − κ ˙ H a ( τ ) H ! + ˙ H H − H ! + (cid:16) δ ˙ H − δ ˙ K (cid:17) κω ( η )2 a ( τ ) H . (A13)Apparently, the equation (A11) determines δ ˙ η by which we can extract δ ˙ H from (A10) as δ ˙ η = − b b δf a − b b δf b − b b δH − b b δη. (A14) δ ˙ H = δH (cid:18) − h h + h b h b (cid:19) + δK (cid:18) − h h (cid:19) + δf a (cid:18) − h h + h b h b (cid:19) + δf b (cid:18) − h h + h b h b (cid:19) + δη (cid:18) − h h + h b h b (cid:19) . (A15)Equation (A12) gives us δ ˙ ζ as δ ˙ ζ = − c c δf a − c c δf b − c c δH − c c δK − c c δζ. (A16)Finally, in order to obtain δ ˙ K , we substitute the equations (A14)- (A16) into (A13) as follows δ ˙ K = g δH + g δK + g δf b + g δf a + g δη + g δζ, (A17)where g = − h h + h b h b − b d b d + c d c d − d d ,g = − h h + c d c d + d d ,g = − h h + h b h b − b d b d + c d c d − d d ,g = − h h + h b h b − b d b d + c d c d + d d ,g = − h h + h b h b − b d b d + d d ,g = c d c d − d d , (A18)and also h = 2 + κωa ,h = 10 H + 2 m a H + 12 κHωa − HH − ω H − κω ˙ Ha H ,h = − m a H ,h = 12 m a − H + ω (cid:18) − κH a (cid:19) + 4 ˙ H,h = − m a ,h = ω κ ˙ Ha + 12 κH a − ! + 2 H − H, (A19) b = 6 H + ω (cid:18) κH a − (cid:19) − m a , b = 6 m a ,b = − κωHa − H,b = ω (cid:18) − κH a (cid:19) ,b = ω H κH ˙ Ha − κH a ! , (A20) c = − b ,c = − b ,c = ω (cid:18) − κH a (cid:19) ,c = ω (cid:18) H − κHa (cid:19) − H,c = ω (cid:18) − H + 9 κHa (cid:19) ,c = b , (A21) d = ω − κ a + κ ˙ H a H + 38 H ! −
32 + ˙ HH ,d = 4 H − ˙ HH + ω (cid:18) κH a − H (cid:19) ,d = ω κ ˙ Ha − κH a + 12 − ˙ H H + κ ¨ H a H − κ ˙ H a H ! +¨ H H − ˙ H H + 7 ˙ H − H + 9 κH a F ( a ) ,d = ω H − κH a − κ ˙ H a H ! + ˙ H H − H , (A22)where δσ ( ζ ) = F ( a ) δζ , δω ( η ) = F ( a ) δη and F ( a ) is of the order a or H . Herein, looking back to the equations(48) and (50) shows that we cannot work with the analytical exact expression for ω ( τ ) and σ ( τ ) unless we specify thescale factor a ( τ ) by which we become able to solve (48). Any way, using the equations (A14)-(A17) beside the twofollowing equations δH = δ ˙ f a , δK = δ ˙ f b , (A23)we seek the range of stability of the solutions. It is worthwhile to note that ω ( τ ) appears in the final form of therelations (A14), (A15), (A16), (A17) and (A23) in which we replace it by (62) and (63) (it should be remarked thatwe have chosen c = c = 0) for a ( τ ) = τ n and a ( τ ) = e nτ , respectively. Appendix B: The expressions for tr M
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