Stability Analysis of Some Reconstructed Cosmological Models in f(\mathcal{G},T) Gravity
aa r X i v : . [ g r- q c ] J un Stability Analysis of SomeReconstructed Cosmological Modelsin f ( G , T ) Gravity
M. Sharif ∗ and Ayesha Ikram † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
The aim of this paper is to reconstruct and analyze the stabilityof some cosmological models against linear perturbations in f ( G , T )gravity ( G and T represent the Gauss-Bonnet invariant and trace ofthe energy-momentum tensor, respectively). We formulate the fieldequations for both general as well as particular cases in the contextof isotropic and homogeneous universe model. We reproduce the cos-mic evolution corresponding to de Sitter universe, power-law solutionsand phantom/non-phantom eras in this theory using reconstructiontechnique. Finally, we study stability analysis of de Sitter as well aspower-law solutions through linear perturbations. Keywords:
Reconstruction; Stability analysis; Modified gravity.
PACS:
Modified theories of gravity have attained much attention after the discov-ery of expanding accelerated universe. The basic ingredient responsible for ∗ [email protected] † [email protected] n -dimensions which coincides with GR in 4-dimensions [1]. TheRicci scalar ( R ) is known as first Lovelock scalar while Gauss-Bonnet (GB)invariant is the second Lovelock scalar yielding Einstein-Gauss-Bonnet grav-ity in 5-dimensions [2]. The GB invariant is a linear combination with aninteresting feature that it is free from spin-2 ghost instabilities defined as [3] G = R − R αβ R αβ + R αβµν R αβµν , where R αβ and R αβµν are the Ricci and Riemann tensors, respectively. Thisquadratic curvature invariant is a topological term in 4-dimensions whichpossesses trivial contribution in the field equations. To discuss the dynamicsof GB invariant in 4-dimensions, there are two interesting scenarios either tocouple G with scalar field or to add generic function f ( G ) in the Einstein-Hilbert action. The first scheme naturally appears in the effective action instring theory which investigates singularity-free cosmological solutions [4].The second approach known as f ( G ) gravity is introduced as an alterna-tive for dark energy which successfully discusses the late-time cosmologicalevolution [5]. This modified theory of gravity is endowed with a quite richcosmological structure as well as consistent with solar system constraints [6].The current cosmic accelerated expansion has also been discussed in mod-ified theories of gravity involving the curvature-matter coupling. Harko et al.[7] established f ( R, T ) gravity to study the curvature-matter coupling. Re-cently, we introduced the curvature-matter coupling in f ( G ) gravity namedas f ( G , T ) theory of gravity [8]. This coupling yields non-zero covariant di-vergence of the energy-momentum tensor and an extra force appears due towhich massive test particles follow non-geodesic trajectories while geodesiclines of geometry are followed by the dust particles. Shamir and Ahmad[9] constructed some cosmologically viable models in f ( G , T ) gravity usingNoether symmetry approach. It is mentioned here that cosmic expansion canbe obtained from geometric as well as matter components in such coupling.2he reconstruction as well as stability of cosmic evolutionary models inmodified theories of gravity are the captivating issues in cosmology. In re-construction technique, any known cosmic solution is used in the modifiedfield equations to find the corresponding function which reproduces the givenevolutionary cosmic history. In stability analysis, the isotropic and homo-geneous perturbations are usually considered in which Hubble parameter aswell as energy density are perturbed to examine the background stability astime evolves [10]. Nojiri et al. [11] formulated the reconstruction schemeto reproduce some cosmological models in f ( R ) gravity. Elizalde et al. [12]applied the same scenario for ΛCDM cosmology (Λ denotes cosmological con-stant while CDM stands for cold dark matter) in f ( R, G ) gravity as well asin modified GB theories of gravity. The stability of power-law solutions arealso discussed in modified gravity theories [13].S´aez-G´omez [14] explored the cosmological solutions in f ( R ) Hoˇrava-Lifshitz gravity and analyzed their stability against first order perturbationsaround FRW universe. Myrzakulov and his collaborators [15] discussed thecosmological models and found that f ( G ) gravity could successfully explainthe cosmic evolutionary history. Jamil et al. [16] reconstructed the cosmolog-ical models in f ( R, T ) gravity and found that numerical analysis for Hubbleparameter is in good agreement with observational data for redshift parame-ter <
2. The stability of de Sitter, power-law solutions as well as ΛCDM areanalyzed in the context of f ( R, G ) gravity [17]. Salako et al. [18] studied thecosmological reconstruction, stability as well as thermodynamics includingfirst and second laws for ΛCDM model in generalized teleparallel theory ofgravity. Sharif and Zubair [19] demonstrated that f ( R, T ) gravity can repro-duce ΛCDM model, phantom or non-phantom eras, de Sitter universe andpower-law cosmic history. They also analyzed the stability of reconstructedde Sitter as well as power-law solutions.In this paper, we reconstruct various cosmological models including de Sit-ter universe, power-law solutions and phantom/non-phantom eras in f ( G , T )theory. We also analyze the stability against linear homogeneous perturba-tions for de Sitter as well as power-law solutions. The paper has the followingformat. In section , we formulate the modified field equations while section is devoted to reconstruct some known cosmological solutions in this gravity.Section analyzes the stability of specific solutions against linear perturba-tions around FRW universe model. The results are summarized in the lastsection. 3 f ( G , T ) Gravity
The action for f ( G , T ) gravity is defined as [8] I = Z (cid:18) R + f ( G , T )2 κ + L m (cid:19) √− gd x, (1)where κ , g and L m represent coupling constant, determinant of the met-ric tensor ( g αβ ) and Lagrangian associated with matter distribution, respec-tively. Varying Eq.(1) with respect to g αβ , we obtain the field equations κ T αβ − R αβ + 12 g αβ R + 12 g αβ f ( G , T ) − ( T αβ + Θ αβ ) f T ( G , T ) − [2 RR αβ − R µα R µβ − R αµβν R µν + 2 R µνξα R βµνξ ] f G ( G , T ) − [2 Rg αβ (cid:3) − R αβ (cid:3) − R ∇ α ∇ β + 4 R µβ ∇ α ∇ µ + 4 R µα ∇ β ∇ µ − g αβ R µν ∇ µ ∇ ν + 4 R αµβν ∇ µ ∇ ν ] f G ( G , T ) = 0 , (2)where f T ( G , T ) = ∂f ( G , T ) /∂T, f G ( G , T ) = ∂f ( G , T ) /∂ G , (cid:3) = ∇ α ∇ α ( ∇ α denotes a covariant derivative) and T αβ is the energy-momentum tensor. Theexpressions for T αβ and Θ αβ are [20] T αβ = g αβ L m − ∂ L m ∂g αβ , Θ αβ = − T αβ + g αβ L m − g µν ∂ L m ∂g αβ ∂g µν , where we have assumed that L m depends only on g αβ rather than its deriva-tives. The non-zero divergence of T αβ is given by ∇ α T αβ = 1 κ − f T ( G , T ) (cid:20)(cid:18) ∇ α Θ αβ − g αβ ∇ α T (cid:19) f T ( G , T ) + (Θ αβ + T αβ ) ∇ α f T ( G , T )] . (3)The above equations indicate that the complete dynamics of f ( G , T ) gravityis based on the suitable choice of L m .The energy-momentum tensor for perfect fluid is T αβ = ( ρ + p ) u α u β − pg αβ , (4)where u α , ρ and p represent the four velocity, energy density and pressureof matter distribution, respectively. In this case, the expression for Θ αβ becomes Θ αβ = − pg αβ − T αβ , (5)4here L m = − p . The line element for FRW universe model is given by ds = dt − a ( t )( dx + dy + dz ) , (6)where a ( t ) is the scale factor. Using Eqs.(4)-(6) in (2), we obtain the corre-sponding field equation as follows3 H = κ ρ + 12 f ( G , T ) + ( ρ + p ) f T ( G , T ) − H ( H + ˙ H ) f G ( G , T )+ 12 H ˙ f G ( G , T ) , (7)where H = ˙ aa , T = ρ − p, G = 24 H ( ˙ H + H ) and dot represents derivativewith respect to time. The non-zero continuity equation (3) takes the form˙ ρ +3 H ( ρ + p ) = − κ + f T ( G , T ) (cid:20)(cid:18) ˙ p + 12 ˙ T (cid:19) f T ( G , T ) + ( ρ + p ) ˙ f T ( G , T ) (cid:21) . (8)The standard conservation law holds if right hand side of this equation van-ishes. For equation of state p = ωρ ( ω is the equation of state parameter),Eq.(8) yields ˙ ρ = − H (1 + ω ) ρ, (9)with additional constraint12 ˙ ρ (1 − ω ) f T + ρ (1 + ω ) (cid:16) ˙ G f G T + ˙ T f
T T (cid:17) = 0 . (10)We rewrite the above equations in terms of new variable N known ase-folding instead of t which is also related with redshift parameter ( z ) as [11] N = − ln(1 + z ) = ln ( a/a ) . Using the above definition of N , Eqs.(7) and (8) become3 H = κ ρ + 12 f + ρ (1 + ω ) f T − H ( H + H ′ ) f G + 288 H × ( HH ′′ + 3 H ′ + 4 HH ′ ) f GG + 12 H T ′ f G T , (11) ρ ′ + 3(1 + ω ) ρ = − κ + f T (cid:20)(cid:18) ωρ ′ + 12 T ′ (cid:19) f T + ρ (1 + ω ) ( G ′ f G T + T ′ f T T ) (cid:21) , where H = d N /dt, d/dt = H ( d/d N ) and prime denotes derivative withrespect to N . The simplest choice of f ( G , T ) model is f ( G , T ) = F ( G ) + F ( T ) , (12)5hich possesses no direct non-minimally coupling between curvature andmatter. For this particular model, the field equation (11) splits into a set oftwo ordinary differential equations as288 H ( HH ′′ + 3 H ′ + 4 HH ′ ) F GG − H ( H + H ′ ) F G + 12 F ( G ) − H = 0 ,ρ (1 + ω ) F T + 12 F ( T ) + κ ρ = 0 , where F G = dF ( G ) /d G and F T = d F ( T ) /dT . The field equations for perfectfluid matter distribution in f ( G ) gravity is recovered if F ( T ) vanishes whileGR is achieved for f ( G , T ) = 0. In this section, we reproduce different cosmological scenarios including de Sit-ter universe, power-law solutions and phantom/non-phantom eras in f ( G , T )gravity. The de Sitter cosmic evolution is interesting and well-known as it elegantlydescribes current expansion of the universe. This solution is considered asthe universe in which the energy density of matter and radiation is negligibleas compared to vacuum energy (energy density for DE dominated era) andthus the universe expands forever at a constant rate. The scale factor ofthis evolutionary model grows exponentially with constant Hubble parameter H ( t ) = H , defined as [17] a ( t ) = a e H t , (13)where a is an integration constant. Equation (9) gives energy density of theform ρ = ρ e − ω ) H t , (14)where ω = − ρ is a constant. Using Eqs.(13) and (14) in (7), we obtain12 f ( G , T ) − H f G ( G , T ) + (cid:18) ω − ω (cid:19) T f T ( G , T ) − ω ) H T × f G T ( G , T ) + κ T − ω − H = 0 , (15)6here G = 24 H is the GB invariant at H ( t ) = H . The solution of theabove differential equation is f ( G , T ) = c c e c G T − (cid:18) (1 − c H − ω )1+ ω − c H
40 (1 − ω ) (cid:19) + c c T − (cid:18) − ω ω (cid:19) − κ − ω T + 6 H , (16)where c i ’s ( i = 1 ,
2) are integration constants. Since we have used the conti-nuity equation (9) in Eq.(15), so we must constrain its solution. Using theabove equation with Eq.(10), we obtain the following functions f ( G , T ) = c c Ξ e c G T − (cid:18) (1 − c H
40 )(1 − ω )1+ ω − c H − ω ) (cid:19) + 2 κ ω − ω T + 6 H , (17) f ( G , T ) = c c Ξ T − (cid:18) − ω ω (cid:19) + 2 κ − ω Ξ T + 6 H , (18)where Ξ j ’s ( j = 1 , ,
3) are constants in terms of ω and H given in Appendix A . For the model (12), we have3 H − F + 12 H F G = 0 , κ ρ + 12 F + (1 + ω ) ρ F T = 0 , (19)where the first equation corresponds to de Sitter universe in the absence ofmatter contents in f ( G ) gravity [6]. Using the constraint (10), the secondequation becomes κ (1 − ω ) T + 12 (1 − ω )(1 − ω ) F − ω ) T F T T = 0 . (20)The solution of Eqs.(19) and (20) leads to f ( G , T ) = ˆ c e G H + ˆ c T (cid:18) √ − ω +2 ω ω (cid:19) + ˆ c T (cid:18) − √ − ω +2 ω ω (cid:19) − κ T − ω + 6 H , (21)where ˆ c j ’s are constants of integration. Equations (16) and (21) indicate thatde Sitter expansion can also be described in f ( G , T ) gravity.7 .2 Power-law Solutions Power-law solutions have significant importance to discuss different evolu-tionary phases of the universe in modified theory. These solutions describethe decelerated as well as accelerated cosmic eras which are characterized bythe scale factor as [17] a ( t ) = a t λ , H = λt , λ > . (22)The cosmic decelerated phase is observed for 0 < λ < λ = ) as well as dust ( λ = ) dominated eras while λ > G = 24 λ t ( λ − . (23)Using Eqs.(9), (22) and (23), the field equation becomes12 f − G f G + (1 + ω ) T − ω f T − λ − G f GG − λ (1 + ω ) G T λ − f G T − λ (cid:18) Tρ (1 − ω ) (cid:19) λ (1+ ω ) + κ T − ω = 0 , (24)whose solution is given by f ( G , T ) = ˜ c ˜ c T ˜ c G ( γ + γ ) + ˜ c ˜ c T ˜ c G ( γ − γ ) + ˜ c ˜ c T γ + γ T + γ T γ , (25)where ˜ c j ’s are integration constants and γ ˆ j ’s (ˆ j = 1 ...
6) are given in Appendix A . Inserting Eq.(25) in (10), we obtain f ( G , T ) = ˜ c ˜ c ∆ T ˜ c G ( γ + γ ) + ˜ c ˜ c ∆ T γ + ∆ T + ∆ T γ , (26) f ( G , T ) = ˜ c ˜ c Ω T ˜ c G ( γ − γ ) + ˜ c ˜ c Ω T γ + Ω T + Ω T γ . (27)where ∆ k ’s and Ω k ’s ( k = 1 ...
4) are given in Appendix A .Now we find the expression of f ( G , T ) for the choice of model (12). Thedifferential equation (24) yields two ordinary differential equations in vari-ables G and T given by F − G F G − λ − G F GG = 0 , − ω ) (1 − ω )(1 − ω ) T F T T + 2 κ T − ω − λ (cid:18) Tρ (1 − ω ) (cid:19) λ (1+ ω ) = 0 . The solution of these equations provide f ( G , T ) model as f ( G , T ) = ¯ c G + ¯ c G − λ + ¯ c T (cid:18) √ − ω +2 ω ω (cid:19) + ¯ c T (cid:18) − √ − ω +2 ω ω (cid:19) − κ T − ω + 54 λ (1 − ω )(1 − ω )9 λ (1 − ω )(1 − ω ) − − λ (1 + ω )] × (cid:18) Tρ (1 − ω ) (cid:19) λ (1+ ω ) , (28)where ¯ c k ’s are integration constants. Thus, the power-law solutions are re-constructed which may be helpful to explore the expansion history of theuniverse in this modified theory of gravity. Here, we reconstruct f ( G , T ) model which can explain the system includingboth phantom and non-phantom eras. In the Einstein gravity, the Hubble pa-rameter describing the phantom as well as non-phantom matter distributionis given by [11] H = κ ρ p a b + ρ q a − b ) , (29)where b, ρ p and ρ q represent the model parameter, energy densities of phan-tom and non-phantom matter fluids, respectively. The violation of all fourenergy conditions leads to phantom phase and the energy density grows whileit decreases in a non-phantom regime. The phantom energy density wouldbecome infinite in finite time, causing a huge gravitational repulsion that theuniverse would lose all structure and end in a big-rip singularity [21]. Whenthe scale factor is large, the first term on right hand side dominates whichcorresponds to the phantom era of the universe with ω = − − b/ < − ω = − b/ > − H ( t ) in terms of a new function S ( N ) as H = S ( N ) so that Eq.(29) becomes S ( N ) = S p e b N + S q e − b N , (30)9here S p = κ ρ p a b and S q = κ ρ q a − b . The GB invariant takes the form G = 24 S ( N ) + 12 S ( N ) S ′ ( N ) . (31)Inserting Eq.(30) in (31), we obtain a quadratic equation in e b N whosesolution is given by e b N = − (48 S p S q − G ) ± q (48 S p S q − G ) − − b ) S p S q b ) S p , b = 2 . For the sake of simplicity, we consider b = 2 so that it reduces to e b N = G − S p S q S p . (32)Using Eqs.(30) and (32) in (7), we have12 f − G f G + (cid:18) ω − ω (cid:19) T f T + G f GG − ω ) G T G − S p S q ) f G T − s G G − S p S q + κ T − ω = 0 , which is a complicated partial differential equation whose analytical solutioncannot be found.To find the reconstructed f ( G , T ) model, we consider its particular form(12) which provides the following set of differential equations12 F − G F G + G F GG − s G G − S p S q = 0 , F − ω ) (1 − ω )(1 − ω ) T F T T + 2 κ T − ω = 0 , where we have used the additional constraint in the second equation. Solvingthese equations, it follows that f ( G , T ) = d G + d G + d T (cid:18) √ − ω +2 ω ω (cid:19) + d T (cid:18) − √ − ω +2 ω ω (cid:19) + 14 p S p S q " G tan − s G − S p S q ) S p S q ! − p S p S q G ln (cid:18) G − S p S q + q G ( G − S p S q ) (cid:19)(cid:21) − κ T − ω . (33)where d k ’s are constants of integration. Thus, phantom and non-phantomcosmic history can be discussed in f ( G , T ) gravity. In this section, we analyze stability of some cosmological evolutionary solu-tions about linear homogeneous perturbations in this modified gravity. Weconstruct the perturbed field as well as continuity equations using isotropicand homogeneous universe model for both general and particular cases in-cluding de Sitter and power-law solutions. We assume a general solution H ( t ) = H ∗ ( t ) , (34)which satisfies the basic field equations for FRW universe model in f ( G , T )gravity. In terms of the above solution, the expressions for G ∗ and T ∗ are G ∗ = 24 H ∗ ( H ∗ + ˙ H ∗ ) = 24 H ∗ ( H ∗ + H ′∗ ) , T ∗ = ρ ∗ ( t )(1 − ω ) . For any particular f ( G , T ) model that can regenerate the above solution(34), the following equation of motion as well as non-zero divergence of theenergy-momentum tensor must be satisfied3 H ∗ = κ ρ ∗ + (1 + ω ) ρ ∗ f ∗ T + 12 f ∗ − H ∗ ( H ∗ + H ′∗ ) f ∗G + 288 × H ∗ ( H ∗ H ′′∗ + 3 H ′ ∗ + 4 H ∗ H ′∗ ) f ∗GG + 12 H ∗ T ′∗ f ∗G T ,ρ ′∗ + 3(1 + ω ) ρ ∗ = − κ + f ∗ T (cid:20)
12 ( T ′∗ + 2 ωρ ′∗ ) f ∗ T + (1 + ω ) ρ ∗ (cid:0) G ′∗ f ∗G T + T ′∗ f ∗ T T )] , where superscript ∗ denotes that the function and its corresponding deriva-tives are calculated at G = G ∗ and T = T ∗ . If the conservation law holds, weget energy density in terms of H ∗ ( t ) as ρ ∗ ( t ) = ρ e − ω ) R H ∗ ( t ) dt . H ( t ) = H ∗ ( t )(1 + δ ( t )) , ρ ( t ) = ρ ∗ ( t )(1 + δ m ( t )) , (35)where δ ( t ) and δ m ( t ) are the perturbation parameters.In order to analyze first order perturbations about the solution (34), weapply the series expansion on the function f ( G , T ) as f ( G , T ) = f ∗ + f ∗G ( G − G ∗ ) + f ∗ T ( T − T ∗ ) + O , (36)where O involves the terms proportional to quadratic or higher powers of G and T while only the linear terms are considered. Using Eqs.(35) and (36)in (7), we obtain the following perturbed field equation χ ¨ δ + χ ˙ δ + χ δ + χ ˙ δ m + χ δ m = 0 , (37)where χ h ’s ( h = 1 ...
5) are given in Appendix A . Inserting these perturbationsin Eq.(8), the perturbed continuity equation isΥ δ + Υ ˙ δ + Υ ¨ δ + Υ δ m + Υ ˙ δ m = 0 , (38)where Υ h ’s are provided in Appendix A . If the conversation law holds in thismodified gravity, Eq.(38) reduces to˙ δ m + 3(1 + ω ) H ∗ δ = 0 . (39)The perturbed equations (37) and (38) are helpful to analyze the stabilityof any specific FRW cosmological evolutionary model in f ( G , T ) gravity. Forthe particular model (12), these perturbed equations reduce toˆ χ ¨ δ + ˆ χ ˙ δ + ˆ χ δ + ˆ χ δ m = 0 , ˆΥ δ + ˆΥ δ m + ˆΥ ˙ δ m = 0 , where the coefficients of ( δ, δ m ) and their derivatives are expressed in Ap-pendix A . In the following subsections, we investigate the stability of deSitter and power-law solutions. 12 t ∆ H t L t - - ∆ m H t L Figure 1: Evolution of perturbations δ ( t ) and δ m ( t ) for model (17) with ω = 0. Consider the de Sitter solution H ∗ ( t ) = H , the perturbed equation (37)takes the form288 H f GG ¨ δ + (cid:0) H f GG + 24 ρ ∗ H (1 + ω ) f G T − ρ ∗ H (1 − ω ) × (1 + ω ) f GG T (cid:1) ˙ δ + (cid:0) − H − H f GG + 12 ρ ∗ H (1 + ω )[8 H − H (1 − ω )] f G T − ρ ∗ H (1 − ω )(1 + ω ) f GG T (cid:1) δ + 12 ρ ∗ H × (1 − ω ) f G T ˙ δ m + (cid:18) κ ρ ∗ + 12 ρ ∗ (3 − ω ) f T + ρ ∗ (1 − ω )(1 + ω ) f T T − ρ ∗ H (1 − ω )[ H + 3(1 + ω ) H ] f G T − ρ ∗ H (1 − ω ) (1 + ω ) × f G T T (cid:1) δ m = 0 , (40)where the superscript 0 represents that the function and its correspondingderivatives are evaluated at G and T . We consider the conserved perturbedequation for stability analysis since the de Sitter solutions are constructedusing the constraint (10) in the previous section. The numerical techniqueis used to solve Eqs.(39) and (40) for the model (17). The evolution of δ ( t )and δ m ( t ) are shown in Figure . We consider H = 67 . κ = 1 through-out the stability analysis of de Sitter universe models whereas integrationconstants are c = 1 × − and c = − × − .Figure shows smooth behavior of δ ( t ) (left) and δ m ( t ) (right) which donot decay in late times indicating that de Sitter model (17) is unstable. Thestability analysis of model (18) with same integration constants is shown inFigure . In the left panel, it is observed that small oscillations are produced13 t ´ - ´ - ´ - ´ - ´ - ∆ H t L t ∆ m H t L Figure 2: Evolution of perturbations δ ( t ) and δ m ( t ) for model (18) with ω = 0. t - ´ - ´ - ´ - ´ - ´ - ´ - ∆ H t L t - - - - - ∆ m H t L Figure 3: Evolution of perturbations δ ( t ) and δ m ( t ) for model (21) with ω = 0.about t = 4 while it decays in late times, thus the model (18) shows stablebehavior against perturbations. For model (12), Eq.(40) becomes288 H F GG ¨ δ + 864 H F GG ˙ δ + (cid:0) − H − H F GG (cid:1) δ + (cid:18) κ ρ ∗ + 12 ρ ∗ (3 − ω ) F T + ρ ∗ (1 − ω )(1 + ω ) F T T (cid:19) δ m = 0 . (41)Figure represents the behavior of δ ( t ) and δ m ( t ) for model (21) with integra-tion constants ˆ c = − , ˆ c = 0 .
001 and ˆ c = 1. It is shown that oscillationsin perturbation parameters are produced initially as shown in Figure . Thisoscillating behavior is clearly observed in Figure which decays in future forboth δ ( t ) as well as δ m ( t ) and hence the solution becomes stable.14 .02 0.04 0.06 0.08 t - ´ - ´ - ´ - ∆ H t L t - - - - ∆ m H t L Figure 4: Evolution of perturbations δ ( t ) and δ m ( t ) for model (21) with ω = 0. Here we investigate the stability of power-law solutions. These solutions de-scribe the accelerated as well as decelerated cosmological evolutionary phasesin the background of FRW universe. We first consider the reconstructedpower-law solution (26) and numerically solve Eqs.(37) and (39). For thismodel, we choose integration constants ˜ c = 10 , ˜ c = − . c = − . Figure shows the oscillating behavior of perturbed parameters ( δ ( t ) , δ m ( t ))for the cosmic accelerated era with ω = − . λ = 2. The perturbationsaround the power-law solutions decay in future leading to stable results. Theradiation ( λ = 1 / ω = 1 /
3) as well as matter ( λ = 2 / ω = 0)dominated eras cannot be discussed for the model (26) because singular aswell as complex terms appear which lead to non-physical case.Secondly, we consider the model (27) and analyze its behavior againstlinear perturbations. Figure shows the fluctuating behavior of consideredperturbations in the cosmic accelerated phase with ω = − . λ = 2.Here, we choose ˜ c = 0 . , ˜ c = − .
61 and ˜ c = − . It is observedthat the oscillating behavior disappears in future while both perturbationparameters will not decay in late times leading to unstable cosmologicalsolutions. The considered model cannot explain the cosmological evolutioncorresponding to matter and radiation dominated eras like previous model(26).Lastly, we explore the stability of model (28) with integration constants¯ c = − , ¯ c = − . , ¯ c = 1000 and ¯ c = 0 .
01. Figure represents the evolu-tion of ( δ, δ m ) versus time for λ = 1 . ω = − .
5. The left panel shows15
10 15 20 25 30 t ∆ H t L t ∆ m H t L Figure 5: Evolution of perturbations δ ( t ) and δ m ( t ) for model (26) with ω = − . λ = 2. t - - - ∆ H t L t - - - - - ∆ m H t L Figure 6: Evolution of perturbations δ ( t ) and δ m ( t ) for model (27) with ω = − . λ = 2. t - ´ - ´ - ´ - ´ ´ ´ ´ ∆ H t L t - ´ - ´ - ´ ´ ∆ m H t L Figure 7: Evolution of perturbations δ ( t ) and δ m ( t ) for model (28) with ω = − . λ = 1 .
1. 16hat the oscillations of δ ( t ) decay in late times while fluctuations of δ m ( t )remain present in future. Since a complete perturbation against any cosmo-logical solution includes the matter perturbations therefore, the solutions areunstable. In this paper, we have employed the reconstruction scheme to f ( G , T ) gravityin the background of isotropic and homogeneous universe model to reproducesome important cosmological models. The basic aspect of this modified grav-ity is the coupling between curvature and matter components which yieldsnon-zero divergence of the energy-momentum tensor. We have imposed ad-ditional constraint to obtain the standard conservation equation which hasbeen used to explain the cosmic evolution in this gravity.The de Sitter and power-law solutions have been reconstructed for generalas well as particular cases which are of great interest and have significantimportance in cosmology. We have also reconstructed the f ( G , T ) modelwhich can explain cosmic history of the phantom as well as non-phantomphases of the universe. Similar reconstruction technique is carried out forΛCDM model and found that this gravity fails to reproduce it for both generalas well as particular f ( G , T ) forms. The results are summarized in Table .In this table, X and × represent that f ( G , T ) gravity reproduces and fails toreproduce the corresponding cosmological backgrounds, respectively.17 able 1: Cosmological evolution in f ( G , T ) gravity. Cosmological Backgrounds General f ( G , T ) Model Particular f ( G , T ) Model de Sitter Universe
X X
Power-law Solutions
X X
Phantom/non-Phantom Eras × X On physical grounds, the stability analysis of different forms of genericfunction leads to classify the modified theories of gravity. We have appliedthe first order perturbations to Hubble parameter and energy density toanalyze the stability of models which reproduce de Sitter and power-lawcosmic history. We have perturbed the field equation as well as conservationlaw whose numerical solutions provide the stable/unstable results. • For the de Sitter universe, the evolution of perturbation has been plot-ted against time as shown in Figures - . These indicate that models(18) and (21) are stable against linear perturbations. • For the power-law universe, the stability analysis is given in Figures - .It is found that f ( G , T ) gravity fails to reproduce matter and radiationdominated eras while stable results are obtained for accelerated phaseof the universe for model (26).We conclude that the cosmological reconstruction and stability analysis mightrestrict f ( G , T ) gravity in the background of FRW universe. It would be in-teresting to discuss ghost instabilities due to the presence of curvature-mattercoupling. Appendix A
The expressions for Ξ i ’s in Eqs.(17) and (18) areΞ = 18 c H [8 c H { ω ) − ω (5 − ω ) } − (1 − ω )(1 − ω )] × [1 + ω − c H (1 − ω )] − , Ξ = 18 c H [(1 − ω )(1 − ω ) − c H { ω ) − ω (5 − ω ) } ] × [(1 + ω )(1 − c H ) { ω − c H (5 − ω − ω ) } ] − , Ξ = − [18 c H (1 − c H ) − ω { − c H (3 − c H ) } ω { − c H (7 − c H ) } + ω { − c H (7 − c H ) } ] × [(1 − ω )(1 − c H ) { ω − c H (5 − ω − ω ) } ] − . The values for γ ˆ j ’s in Eq.(25) are γ = 12 [5 − λ { c (1 + ω ) } ] ,γ = (cid:20) λ ˜ c (1 + ω ) { c λ (1 + ω ) + 2( λ − − } + 14 ( λ − λ + 7) + 4+ 8˜ c ( λ − (cid:18) ω − ω (cid:19)(cid:21) , γ = − (cid:18) − ω ω (cid:19) , γ = 2 κ ω − ,γ = λ (1 − ω ) λ (1+ ω ) − λ (1+ ω ) λ (1 − ω ) + 4 ! ρ − λ (1+ ω ) , γ = 23 λ (1 + ω ) . The expressions for ∆ k ’s and Ω k ’s in Eqs.(26) and (27) are∆ = 1 − γ γ , ∆ = 1 − γ γ , ∆ = γ (cid:18) − γ (cid:19) , ∆ = γ (cid:18) − γ γ (cid:19) , Ω = 1 − γ γ , Ω = 1 − γ γ , Ω = γ (cid:18) − γ (cid:19) , Ω = γ (cid:18) − γ γ (cid:19) , where γ = ˜ c λ (cid:2) c λ (1 + ω ) − λ (1 + 5 ω + 2 ω ) + 2( γ + γ ) (cid:3) ,γ = ˜ c λ (cid:2) c λ (1 + ω ) − λ (1 + 5 ω + 2 ω ) + 2( γ − γ ) (cid:3) . The values of χ h ’s in Eq.(37) are given as follows χ = 288 H ∗ f ∗GG ,χ = 288 H ∗ (3 H ∗ + 5 ˙ H ∗ ) f ∗GG + 6912 H ∗ (4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ¨ H ∗ ) f ∗GGG + 24(1 + ω ) ρ ∗ H ∗ f ∗G T − ω )(1 − ω ) H ∗ ρ ∗ f ∗GG T ,χ = − H ∗ − H ∗ ˙ H ∗ f ∗G − H ∗ (4 H ∗ − H ∗ ˙ H ∗ −
11 ˙ H ∗ − H ∗ ¨ H ∗ ) f ∗GG + 6912 H ∗ (4 H ∗ + ˙ H ∗ )(4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ¨ H ∗ ) f ∗GGG + 12(1 + ω ) ρ ∗ H ∗ × [2(4 H ∗ + ˙ H ∗ ) − − ω ) H ∗ ] f ∗G T − ω )(1 − ω ) ρ ∗ H ∗ (4 H ∗ + ˙ H ∗ ) f ∗GG T , = 12(1 − ω ) ρ ∗ H ∗ f ∗G T ,χ = κ ρ ∗ −
12 ( ω − ρ ∗ f ∗ T + (1 − ω )(1 + ω ) ρ ∗ f ∗ T T − − ω ) ρ ∗ H ∗ × [(4 + 3 ω ) H ∗ + ˙ H ∗ ] f ∗G T + 288(1 − ω ) ρ ∗ H ∗ (4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ¨ H ∗ ) × f ∗GG T − ω )(1 − ω ) ρ ∗ H ∗ f ∗G T T . The expressions for Υ h ’s areΥ = 3(1 + ω ) ρ ∗ H ∗ ( κ + f T ) − ω ) ρ ∗ H ∗ h H ∗ (1 − ω )(4 H ∗ + ˙ H ∗ ) − H ∗ ˙ H ∗ + 6 ˙ H ∗ + 3 H ∗ ¨ H ∗ ) i f ∗G T − ρ ∗ H ∗ (1 − ω )(1 + ω ) (4 H ∗ + ˙ H ∗ ) f ∗G T T + 576 ρ ∗ H ∗ (1 + ω )(4 H ∗ + ˙ H ∗ )(4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + 4 H ∗ ¨ H ∗ ) f ∗GG T , Υ = − ρ ∗ H ∗ (1 + ω ) h H ∗ (1 − ω ) − H ∗ + 3 ˙ H ∗ ) i f ∗G T + 576 ρ ∗ H ∗ × (1 + ω )(4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ˙ H ∗ ) f ∗GG T − ρ ∗ H ∗ (1 − ω ) × (1 + ω ) f ∗G T T , Υ = 24 ρ ∗ H ∗ (1 + ω ) f ∗G T , Υ = − ρ ∗ H ∗ (1 + ω ) (cid:2) (1 − ω ) f ∗ T + 2 ρ ∗ (1 + ω )(1 − ω ) f ∗ T T T (cid:3) − ρ ∗ H ∗ × (1 − ω )(1 + ω ) f ∗ T T + 24 ρ ∗ H ∗ (1 + ω )(4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ¨ H ∗ ) × (cid:2) f ∗G T + ρ ∗ (1 − ω ) f ∗ T T G (cid:3) , Υ = ρ ∗ (cid:18) κ + 12 (3 − ω ) f ∗ T (cid:19) + (1 + ω )(1 − ω ) ρ ∗ f ∗ T T . For model (12), the coefficients of ( δ, δ m ) have the following expressionsˆ χ = 288 H ∗ F ∗GG , ˆ χ = 288 H ∗ (3 H ∗ + 5 ˙ H ∗ ) F ∗GG + 6912 H ∗ (4 H ∗ ˙ H ∗ + 2 ˙ H ∗ + H ∗ ¨ H ∗ ) F ∗GGG , ˆ χ = − H ∗ − H ∗ ˙ H ∗ F ∗G − H ∗ (4 H ∗ − H ∗ ˙ H ∗ −
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