Stability of Einstein static universe in gravity theory with a non-minimal derivative coupling
aa r X i v : . [ g r- q c ] J a n Stability of Einstein static universe in gravity theory with anon-minimal derivative coupling
Qihong Huang , , Puxun Wu , and Hongwei Yu ∗ Department of Physics and Synergetic InnovationCenter for Quantum Effects and Applications,Hunan Normal University, Changsha, Hunan 410081, China School of Physics and Electronic Science,Zunyi Normal College, Zunyi 563006, China Center for High Energy Physics, Peking University, Beijing 100080, China
Abstract
The emergent mechanism provides a possible way to resolve the big bang singularity problem byassuming that our universe originates from the Einstein static (ES) state. Thus, the existence of astable ES solution becomes a very crucial prerequisite for the emergent scenario. In this paper, westudy the stability of an ES universe in gravity theory with a non-minimal coupling between thekinetic term of a scalar field and the Einstein tensor. We find that the ES solution is stable underboth scalar and tensor perturbations when the model parameters satisfy certain conditions, whichindicates that the big bang singularity can be avoided successfully by the emergent mechanism inthe non-minimally kinetic coupled gravity.
PACS numbers: 98.80.Cq, 04.50.Kd ∗ Corresponding author: [email protected] . INTRODUCTION Although the standard cosmological model achieves great success, it still suffers from sev-eral theoretical problems. The attempt to resolve theses problems leads to the invention ofthe inflation theory [1–3], which settles successfully most of the problems in the standardcosmological model, but leaves the big bang singularity problem open. To avoid this prob-lem, some theories, such as the pre-big bang [4], the cyclic scenario [5] and the emergentscenario [6], have been proposed. The emergent scenario, proposed by Ellis et al. , in theframework of general relativity [6], assumes that the universe originates from an Einsteinstatic (ES) state rather than a big bang singularity. So, it requires that the universe canstay in an ES state past eternally, and exit this static state naturally and then evolve toa subsequent inflationary era. Apparently, a very crucial prerequisite for the emergent sce-nario is the existence of a stable ES solution under various perturbations, such as quantumfluctuations. However, in the framework of general relativity, the emergent mechanism isnot as successful as one expected in avoiding the big bang singularity since there is no stableES solution in a Friedmann universe with a scalar field minimally coupled with gravity [7].A natural generalization of the minimally coupled gravity is to assume a non-minimalcoupling between the scalar field and the curvature, which can be generated naturally whenquantum corrections are considered and is essential for the renormalizability of the scalarfield theory in curved space. This non-minimally coupled scalar field has been suggested to beresponsible for both the early cosmic inflation [8] and the present accelerated expansion [9].If the coupling is a general function of the scalar field, the resulting theory is called thescalar-tensor theory [10]. The popular modified gravity, f ( R ) gravity [11], can be cast intoa special form of the Brans-Dicke theory, which is a particular example of the scalar-tensortheory [12], with a potential for the effective scalar-field degree of freedom. Let us alsonote that a pioneering inflation model was constructed in a f ( R ) theory by Starobinsky [1],which allows a graceful exit from inflation to the subsequent radiation dominated stage andproduces a very good fit to existing CMB observational data [13].It has been found that in f ( R ) theory the inhomogeneous scalar perturbations break thestability of the ES solution which is stable under homogeneous scalar perturbations [14, 15].Recently, Miao et. al [16] found that there is no stable ES solution when scalar perturbationsand tensor ones are considered together in the scalar-tensor theory of gravity with a normal2erfect fluid, such as radiation or pressureless matter. It is worthy to note that the stabilityof ES solutions have also been analyzed in some other theories [17–31].Except for the coupling between the scalar field and the curvature, there are many othercoupling, such as the coupling between the kinetic term of the scalar field and the Einsteintensor. This non-minimal derivative coupling has been discussed extensively in cosmology.For example, it can provide an inflationary mechanism [32–36], explain both a quasi-deSitter phase and an exit from it without any fine-tuned potential [37], and behave as adark matter [38] or a dark energy [39, 40]. Recently, the stability of ES solutions in thenon-minimally derivative coupled gravity have been studied in [28]. However, in [28] onlya very special case of ˙ φ = 0 is considered, which is not a general result derived from theconditions of static state solution, where ˙ φ = dφ/dt with φ and t being the scalar field andthe cosmic time, respectively. In addition, only the homogeneous scalar perturbations andtensor perturbations are considered in [28]. Thus, for the non-minimally kinetic coupledgravity, it is unclear whether the ES solution remain to be stable against inhomogeneousscalar perturbations and what the effect the special condition ˙ φ = 0 has on the stableregions, and this motivates us to the present work.The paper is organized as follows. In Section 2, we give the field equations of gravitytheory with a non-minimal derivative coupling and the ES solution. In Section 3, we ana-lyze the stability of ES solution under tensor perturbations. In Section 4, the homogeneousand inhomogeneous scalar perturbations are considered. Finally, our main conclusions arepresented in Section 5. Throughout this paper, unless specified, we adopt the metric signa-ture ( − , + , + , +). Latin indices run from 0 to 3 and the Einstein convention is assumed forrepeated indices. II. THE FIELD EQUATIONS AND EINSTEIN STATIC SOLUTION
The action of the non-minimally derivative coupled gravity has the form [32, 41] S = Z d x √− g h R πG − ( g µν + κG µν ) ∇ µ φ ∇ ν φ − V ( φ ) i + S m , (1)where R is the Ricci curvature scalar, G is the Newtonian gravitational constant, g µν is themetric tensor with g being its trace, G µν is the Einstein tensor, V ( φ ) is the potential ofthe scalar field φ , κ stands for the coupling parameter with dimension of (length) , and S m g µν and the scalar field φ , respec-tively, one can obtain two independent equations: G µν = 8 πG [ T ( m ) µν + T ( φ ) µν + κϑ µν ] (2)and ( g µν + κG µν ) ∇ µ ∇ ν φ = V ,φ , (3)where V ,φ = dVdφ , T ( m ) µν is the energy-momentum tensor of the perfect fluid, and T ( φ ) µν = ∇ µ φ ∇ ν φ − g µν ( ∇ φ ) − g µν V, (4) ϑ µν = − ∇ µ φ ∇ ν φR + 2 ∇ α φ ∇ ( µ φR αν ) + ∇ α φ ∇ β φR µανβ + ∇ µ ∇ α φ ∇ ν ∇ µ φ − ∇ µ ∇ ν φ (cid:3) φ −
12 ( ∇ φ ) G µν + g µν (cid:20) − ∇ α ∇ β φ ∇ α ∇ β φ + 12 ( (cid:3) φ ) − ∇ α φ ∇ β φR αβ (cid:21) . (5)Here, ( ∇ φ ) = ∇ α φ ∇ α φ and (cid:3) φ = ∇ α ∇ α φ .To find an ES solution, we consider a homogeneous and isotropic universe described bythe Friedmann-Lemaˆ ı tre-Robertson-Walker metric ds = a ( η ) [ − dη + γ ij dx i dx j ] , (6)where η is the conformal time, a ( η ) denotes the conformal scale factor, and γ ij representsthe metric on the three-sphere γ ij dx i dx j = dr − Kr + r ( dθ + sin θdφ ) . (7)Here K = +1 , , − ij ) components of Eq. (2) give H + K = 8 πG a h ρ + 12 a φ ′ + V − κ a (3 H + K ) φ ′ i , (8)2 H ′ + H + K = − πGa h p + 12 a φ ′ − V + κ a (cid:0) H ′ − H − K + 4 H φ ′′ φ ′ (cid:1) φ ′ i , (9)where ρ and p are the energy density and the pressure of the perfect fluid, respectively, p = wρ with w being a constant, H = a dadη and ′ denotes the derivative with respect to theconformal time η . From Eq. (3) we obtain the dynamical equation of the scalar field1 a ( φ ′′ + 2 H φ ′ ) − κ a (cid:2) ( H + K ) φ ′′ + 2 HH ′ φ ′ (cid:3) = − V ,φ . (10)4 . Einstein static solution The ES solution requires that the conditions of a = a = constant and a ′ = a ′′ = 0 shouldbe satisfied. Then, Eq. (8) can be reduced to Ka = 8 πG (cid:16) ρ + 12 a φ ′ + V − a κKφ ′ (cid:17) , (11)where the subscript 0 represents the value at the ES state. It is easy to see that to obtainan ES state ρ , V and φ ′ must be constant. From Eqs. (9) and (10) we have Ka = − πG (cid:16) p + 12 a φ ′ − V − a κKφ ′ (cid:17) , (12) dVdφ | φ = φ = 0 . (13)Thus, the scalar field with a constant speed moves on a constant potential in the ES state.However, in [28], a special case φ ′ = 0 was considered and φ = 0 was assumed. CombiningEqs. (11) and (12) leads to Ka = 4 πGρ (1 + w ) + 4 πGφ ′ a − πGφ ′ κ Ka , (14)which indicates that K = 0 for the existence of an ES solution since a φ ′ = (cid:0) dφdt (cid:1) . Afterintroducing two new constants F = 4 πGφ ′ a = 4 πG ˙ φ , ρ s = 4 πGρ , (15)Eq. (14) can be re-expressed as 1 a = F + (1 + w ) ρ s (1 + 2 κF ) K . (16)Since a and ρ should take positive values, the existence conditions of ES solutions are a > ρ s >
0. For the case κ <
0, when K = 1, we find that the existence of ESsolutions requires w < − , < F < − κ , < ρ s < − F w ,w < − , F > − κ , ρ s > − F w ,w = − , < F < − κ , ρ s > ,or w > − , ≤ F < − κ , ρ s > . (17)5hile, for K = −
1, the conditions become w < − , ≤ F < − κ , ρ s > − F w ,w < − , F > − κ , < ρ s < − F w ,or w ≥ − , F > − κ , ρ s > . (18)For the case of κ > a > ρ s > w < − , F > , < ρ s < − F w ,w = − , F > , ρ s > ,or w > − , F ≥ , ρ s > , (19)when K = 1, and w < − , F ≥ , ρ s > − F w , (20)when K = − III. TENSOR PERTURBATIONS
For the tensor perturbations, the perturbed metric has the following form [42] ds = a ( η ) [ − dη + ( γ ij + 2 h ij ) dx i dx j ] . (21)For convenience, we perform a harmonic decomposition for the perturbed variable h ij h ij = H T,nlm ( η ) Y ij,nlm ( θ k ) , (22)where summations over n , m , l are implied. The quantum numbers m and l will be sup-pressed hereafter as they do not enter the differential equation for the perturbations. Theharmonic function Y n = Y nlm ( θ i ) satisfies [43]∆ Y n = − k Y n = ( − n ( n + 2) Y n , n = 0 , , , ..., K = +1 − n Y n , n ≥ , K = 0 − ( n + 1) Y n , n ≥ , K = − K = 1, while it is continuous for K = 0 or − (cid:0) πGφ ′ κa (cid:1) H ′′ T + (cid:0) H + 8 πGφ ′ φ ′′ κa (cid:1) H ′ T + (cid:0) − πGφ ′ κa (cid:1) ( k + 2 K ) H T = 0 . (24)Under the ES background, this equation can be simplified as H ′′ T + BH T = 0 , B ≡ (1 − κF )(1 + κF ) ( k + 2 K ) . (25)To obtain the stable ES solution against tensor perturbations, B > k . For the case of κ <
0, when K = 1, we find that the restriction condition B > k ≥ , ≤ F < − κ . (26)While, when K = −
1, the stable ES solution requires k > , ≤ F < − κ ,or ≤ k < , F > − κ . (27)For the case of κ > B > k ≥ , ≤ F < κ , (28)when K = 1, and k > , ≤ F < κ ,or ≤ k < , F > κ . (29)when K = −
1. It is easy to see that the stable ES solution exists only in the case ofspatially closed universe ( K = 1). Thus, in the following analysis K = 1 is considered.Combing the existence conditions given in Eqs. (17,19) and the stability conditions undertensor perturbations, we obtain that w , F and ρ s should satisfy w < − , < F < − κ , < ρ s < − F w ,w < − , − κ < F < − κ , ρ s > − F w ,w = − , < F < − κ , ρ s > ,or w > − , ≤ F < − κ , ρ s > , (30)7or κ <
0, and w < − , < F < κ , < ρ s < − F w ,w = − , < F < κ , ρ s > ,or w > − , ≤ F < κ , ρ s > , (31)for κ > IV. SCALAR PERTURBATIONS
To analyze the stability of ES solutions under scalar perturbations, we consider the per-turbed metric: ds = a ( η ) [ − (1 + 2Ψ) dη + (1 + 2Φ) γ ij dx i dx j ] , (32)where the Newton gauge has been used, Ψ is the Bardeen potential and Φ denotes theperturbation to the spatial curvature.Using the above perturbed metric and the field equations given in Eqs. (2, 3), we obtainthe following perturbation equations14 πGa ( ∇ Φ + 3Φ) = − δρ + 1 a ( φ ′ Ψ − φ ′ δφ ′ )+ κ a [3 φ ′ δφ ′ − φ ′ ∇ Φ − φ ′ ] , (33) − (Ψ + Φ) = 4 πGφ ′ κ a (Ψ − Φ) , (34)3 a ( − Φ ′′ + Φ) + 1 a ∇ (Ψ + Φ) = 4 πG (cid:26) δp − a ( φ ′ Ψ − φ ′ δφ ′ )+ κ a [ φ ′ ∇ (Φ − Ψ) + 3 φ ′ Φ ′′ + 3(Ψ + Φ) φ ′ − φ ′ δφ ′ ] (cid:27) (35) (cid:18) − κ a (cid:19) δφ ′′ − (cid:18) − κ a (cid:19) ∇ δφ − (cid:20) (Ψ ′ − ′ ) − κ a (Ψ ′ − Φ ′ ) (cid:21) φ ′ = 0 . (36)Here, the perturbation of the scalar field φ → φ + δφ is considered. For the perfect fluid,the perturbation of its energy-momentum tensor can be expressed as δT µ ( m ) ν = δρu µ u ν + u µ D ν q + u ν D µ q + δpP µν , (37)8here u µ is the four-velocity of matter and q is related to the perturbation of the spatialcomponent of this four-velocity. The projection tensor P µν and the derivative D µ are definedas P µν = δ µν + u µν , (38) D µ = P αµ ∂ α = ∂ µ + u µ u α ∂ α . (39)The relation between the density and pressure perturbations is δp = c s ρ δ, (40)where, δ = δρ/ρ and c s = w is the sound speed.Similar to the case of tensor perturbations, we perform a harmonic decomposition for theperturbed variablesΨ = Ψ n ( η ) Y n ( θ i ) , Φ = Φ n ( η ) Y n ( θ i ) , q = q n ( η ) Y n ( θ i ) ,δ = δ n ( η ) Y n ( θ i ) , δφ = δφ n ( η ) Y n ( θ i ) . (41)Combining Eqs (33), (34), (35) and (36) gives two independent perturbed equationsΦ ′′ n + b Φ n + a δφ ′ n = 0 , (42) δφ ′′ n + b δφ n + a Φ ′ n = 0 , (43)with b = wk − κF (1 − κF )( w − κa (1 + κF ) − κ F (3 w −
1) + (1 + κF )(3 w + 1)(1 + κF ) ,a = − F [(1 − w ) κa + ( w − a (1 + κF ) 1 φ ′ ,b = 1 − κa − κa k ,a = 2 (cid:16) − κa + 11 + κF (cid:17) φ ′ . (44)Introducing two new variables δϕ = δφ ′ and Υ = Φ ′ , Eqs. (42) and (43) can be rewritten9s Φ ′ n − Υ n = 0 , Υ ′ n + b Φ n + a δϕ n = 0 ,δφ ′ n − δϕ n = 0 ,δϕ ′ n + b δφ n + a Υ n = 0 . (45)The stability of ES solutions is determined by the eigenvalues of the coefficient matrix, whichis µ = − M ± √ N , (46)where M = b + b − a a , N = − b b + ( b + b − a a ) . (47)If µ <
0, a small perturbation from the ES state will result in an oscillation around thisstate rather than an exponential deviation. Thus, the corresponding ES solution is stable.Otherwise, it is unstable. µ < M > , N > , M > N. (48)Since b = 0 and M = N when k = 0, the homogeneous scalar perturbations require µ = − M − √ N − M = − b + a a < . (49) A. Stability
For the scalar perturbations, the analysis of the stability of ES solutions is very compli-cated. To simplify discussions, we will consider the constraints from the tensor perturbationsand the existence conditions obtained in the previous sections, in which it is found that theES solution is stable under the conditions of K = 1 and Eq. (30, 31).10 . κ < From Eq. (49), we obtain that the stability conditions under homogeneous scalar pertur-bations are F = 0 , − < w < − , ρ s > ,or < F < − κ , − < w < − κF κF , ρ s > λ + , (50)where λ ± = 1 + 3 w + κF [ −
11 + 3 w − κF (1 + 3 w )]6 κ (1 + w )[1 + 3 w + κF ( − w )] ± s (1 + 2 κF ) [1 + (22 − κF ) κF + 6 w + 6 κF (10 + 13 κF ) w + 9( − κF ) w ] κ (1 + w ) [1 + 3 w + κF ( − w )] . (51)For 0 < F < − κ , one can obtain − < − κF κF < − , which means that w is negative.For the inhomogeneous scalar perturbations, the physical modes have n ≥ k ≥ n = 1 mode corresponds to a gauge degree of freedom related to a globalrotation. For F = 0 and k = 8, we obtain that the region of w and ρ s < w ≤ , ρ s > , < w < , < ρ s < − w − κ + 4 κw + 15 κw , < w < , ρ s > − w − κ + 4 κw + 15 κw ,or w ≥ , ρ s > , (52)While, when 0 < F < − κ and k = 8, we find that w and ρ s need to satisfy0 < w < κF − κ F κF + 2 κ F , < ρ s < − F w [1 + 3 κF + κ F ](1 + w )[ − w + κF ( − w ) + 2 κ F (1 + w )] ,or w ≥ κF − κ F κF + 2 κ F > , ρ s > . (53)Obviously, a positive w is required for the stable ES solution under the inhomogeneousscalar perturbations, which conflicts with the conditions given by the homogeneous scalarperturbations and the tensor ones. Thus, there is no stable ES solution in the case of κ < ABLE I: Summary of the combinations of the stability conditions under homogeneous scalarperturbations and that given in Eq. (31) with K = 1 and κ > w F ρ s − < w < − F = 0 ρ s > < F < ξ λ − < ρ s < λ + < F < κ ρ s > − κF κ +3 κw w = − < F < κ ρ s > − κF κ − < w < < F < w κ − κw − κF κ +3 κw < ρ s < λ + F = κw κ − κw − κF κ +3 κw < ρ s < ζ κw κ − κw < F ≤ ξ − κF κ +3 κw < ρ s < λ − w κ − κw < F < ξ ρ s > λ + F = ξ ρ s > λ − ξ < F < κ ρ s > − κF κ +3 κw ≤ w < < F < w κ − κw − κF κ +3 κw < ρ s < λ + F = w κ − κw − κF κ +3 κw < ρ s < ζ w κ − κw < F < κ − κF κ +3 κw < ρ s < λ − w κ − κw < F < κ ρ s > λ + w ≥ < F < κ − κF κ +3 κw < ρ s < λ + κ > When κ >
0, the results are summarized in Tab. (I) where the conditions shown in Eq. (31)have been considered together. The constants ξ and ζ are defined as ξ = − − w + 9 w κ ( −
23 + 78 w + 9 w ) + 12 s w + 3 w − w κ ( −
23 + 78 w + 9 w ) ,ζ = 4 F (1 + κF + κ F )(1 + w )[ − − w + κF (11 − w + 2 κF + 6 κF w )] . Now we consider the contribution from inhomogeneous scalar perturbations. We find thatthere is no stable ES solution for w ≤ M > M − N > k = 8. Since the expressions are complicated, we do not show them12ere.When 0 < w < , from Tab. (I) one can see that there are four different kinds of stabilityconditions under homogeneous scalar perturbations. We will analyze inhomogeneous scalarperturbations under these conditions, respectively.(i) 0 < F < w κ − κw and − κF κ +3 κw < ρ s < λ + . The stability condition M > ρ s to satisfy0 < ρ s < − κF κ + 3 κw , or ρ s > ̟, (54)where ̟ = 7 + [19 + κF (2 − w ) − w ] κF + 5 w κ (1 + w )[5 + 15 w + 2 κF (7 + 3 w )]+ 12 s (1 + 2 κF ) [49 + 3 κF (50 + 19 κF ) + 70 w − κF (34 + 57 κF ) w + (5 + 11 κF ) w ] κ (1 + w ) [5 + 15 w + 2 κF (7 + 3 w )] , (55)where k = 8 is taken. Since ̟ > λ + for 0 < w < and 0 < F < w κ − κw , there is no overlapfor the allowed regions of ρ s from homogeneous and inhomogeneous scalar perturbations,which indicates that the ES solution is unstable.(ii) F = w κ − κw and − κF κ +3 κw < ρ s < ζ . The inhomogeneous perturbations require ρ s tosatisfy Eq. (54) when k = 8. Since ̟ > ζ , there is no stable ES solution in this case too.(iii) w κ − κw < F < κ and − κF κ +3 κw < ρ s < λ − . When k = 8, ρ s is also required to satisfyEq. (54). We find that ̟ > λ − . Thus, the ES solution is unstable.(iv) w κ − κw < F < κ and ρ s > λ + . In this case, since the analytical results for thestability regions under inhomogeneous scalar perturbations are very complicated, we do notshow them here and but resort to a numerical discussion. We find that when n = 2 thesmallest stability regions is obtained. With the increase of the value of n , the stabilityregions become larger and larger, which can be seen from Fig. (1). In this Figure, n is takento be n = 0 , , , , ,
6, respectively, where n = 0 corresponds to the case of homogeneousscalar perturbations. When n → ∞ , b reduces to b ≃ wk . The stable ES solution13 .00 0.02 0.04 0.06 0.08 0.100.80.91.01.11.21.31.41.5 wF n = = = = = = wF n = = = = = = FIG. 1: The stability regions in the F − w plane under homogeneous and inhomogeneous pertur-bations with n taken to be n = 0 , , , , , n = 0 corresponds to the results from homogeneousperturbations. The left panel is plotted with ρ s = 15 and κ = , while the right one for ρ s = 15and κ = 1. requires M ≃ (cid:16) w + 1 − κa − κa (cid:17) k > , (56) M − N ≃ w (1 − κa )1 − κa k > . (57)The above two equations give − F w < ρ s < − κF κ + 3 κw , ρ s > κFκ + κw . (58)where 0 < w < and w κ − κw < F < κ are considered together. Since κFκ + κw < λ + is alwayssatisfied, the stability regions are larger than what are obtained under homogeneous scalarperturbations. Thus, in this case the ES solution is stable under both homogeneous andinhomogeneous scalar perturbations and the stability regions are given by taking n = 2.When w ≥ , we find that M > M − N > . CONCLUSION
In this paper, we have analyzed the stability of an ES universe under both scalar andtensor perturbations in gravity theory with a coupling between the kinetic term of thescalar field and the Einstein tensor. Homogeneous and inhomogeneous perturbations areconsidered together and inhomogeneous perturbations will compress the allowed regionsof model parameters significantly. We find that the stable ES solution exists only in thespatially closed universe ( K = 1) and it requires the coupling constant κ > < w < ,which indicates that if this perfect fluid is the pressureless matter or radiation the stableES solution does not exist, although it does under homogeneous perturbations. Thus, inthe non-minimally kinetic coupled gravity with the perfect fluid satisfying 0 < w < thestable ES solution can exist under both scalar and tensor perturbations and the emergentmechanism can be used to avoid the big bang singularity.When F = 0, our results reduce to what were obtained in [28] where a special condition˙ φ = 0 was considered. In this special case our analyses show that inhomogeneous scalarperturbations will break the stability of an ES solution although the solution is stable undertensor and homogeneous scalar ones. Therefore, the big bang singularity problem can notbe solved successfully if ˙ φ = 0 is taken.Finally, a few comments are now in order for the emergent scenario proposed for avoidingthe big-bang singularity. The emergent scenario assumes the existence of a stable Einsteinstatic state and its past eternity. But usually such a state only exists under certain condi-tions. So, there is a question as to how this particular state comes into being in the firstplace, and in this regard, let us note that one possibility might be the creation of this statefrom “nothing” through quantum tunneling[44, 45]. Another issue is that even this state isstable classically, one still needs to address the question as to whether it is stable quantummechanically, possibly by calculating the characteristic decay time in a quantum theory ofcosmology when this state was formed. 15 cknowledgments This work was supported by the National Natural Science Foundation of China underGrants No. 11775077, No. 11435006, No.11690034, and No. 11375092. [1] A. A. Starobinsky, Phys. Lett. B , 99 (1980).[2] A. H. Guth, Phys. Rev. D , 347 (1981).[3] A. D. Linde, Phys. Lett. B , 389 (1982).[4] M. Gasperini and G. Veneziano, Phys. Rep. , 1 (2003); J. E. Lidsey, D. Wands, and E. J.Copeland, Phys. Rep. , 343 (2000).[5] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D , 123522 (2001); P.J. Steinhardt and N. Turok, Science , 1436 (2002); Phys. Rev. D , 126003 (2002); J.Khoury, P. J. Steinhardt, and N. Turok, Phys. Rev. Lett. , 031302 (2004).[6] G. F. R. Ellis and R. Maartens, Class. Quant. Grav. , 223 (2004); G. F. R. Ellis, J. Murugan,and C. G. Tsagas, Class. Quant. Grav. , 233 (2004).[7] J. D. Barrow, G. F. R. Ellis, R. Maartens and C. G. Tsagas, Class. Quant. Grav. , L155(2003); A. S. Eddington, Mon. Not. Roy. Astron. Soc. , 668 (1930); G. W. Gibbons, Nucl.Phys. B , 784 (1987); G. W. Gibbons, Nucl. Phys. B , 636 (1988).[8] L. F. Abbott, Nucl. Phys. B , 233 (1981); T. Futamase and K. i. Maeda, Phys. Rev. D , 399 (1989).[9] V. Sahni and S. Habib, Phys. Rev. Lett. , 1766 (1998); J. P. Uzan, Phys. Rev. D , 123510(1999); F. Perrotta, C. Baccigalupi and S. Matarrese, Phys. Rev. D , 023507 (1999).[10] P.G. Bergmann, Int. J. Theor. Phys. , 25 (1968); K. Nordtvedt, Astrophys. J. , 1059(1970); R. Wagoner, Phys. Rev. D , 3209 (1970).[11] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. , 451 (2010); A. De Felice and S. Tsujikawa,Living Rev. Relativity , 3 (2010).[12] C. Brans and R. H. Dicke, Phys. Rev. , 925 (1961); R. H. Dicke, Phys. Rev. , 2163(1962).[13] Planck Collaboration: P. A. R. Ade, N. Aghanim, M. Arnaud, et al., Astron. Astrophys. ,A20 (2016).
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