Reconstruction of GR cosmological solutions in modified gravity theories
RReconstruction of GR cosmological solutions in modified gravitytheories
I. V. Fomin ∗ and S. V. Chervon † Bauman Moscow State Technical University, 2-nd Baumanskaya street, 5, Moscow, 105005, Russia Kazan Federal University, Kremlevskaya street 18, Kazan, 420008, Russia
March 12, 2019
Abstract
We study the special class of the exact solutions in cosmological models based on theGeneralized Scalar-Tensor Gravity with non-minimal coupling of a scalar field to the Ricciscalar and to the Gauss-Bonnet scalar in 4D Friedmann universe corresponding to similarones in GR. The parameters of cosmological perturbations in such models correspond to thecase of Einstein gravity with a high precision. As the example of proposed approach, weobtain the exact solutions for the power-law and exponential power-law inflation.
An inflationary scenario was invented to solve many problems in big-bang cosmology, in-cluding the explanation for the origin of large scale structure of the universe. The first modelsof cosmological inflation as a rule were built on the basis of General Relativity (GR) in 4DFriedmann-Robertson-Walker (FRW) space-time under the assumption of the existence of self-interacting scalar field which is the source of the accelerated expansion of the universe [1–5].After the discovery of the second accelerated expansion of the universe at the present time,cosmological models with various modifications of Einstein gravity were proposed for its expla-nation or, in other words, to explain the nature of dark energy which include, among others,scalar-tensor gravity theories and Einstein-Gauss-Bonnet gravity [6–8].Nevertheless, recent observations of gravitational waves from the merger of neutron starslead to serious restrictions on the modification of Einstein’s theory of gravity. These restrictionsare related to the velocity of the propagation of gravitational waves which is equal to speed oflight in vacuum and which corresponds to the case of GR with the high accuracy 10 − [9].Scalar-tensor gravity (STG) with non-minimal coupling of a scalar field to curvature areimportant extensions of GR explaining the initial inflationary evolution, as well as the lateaccelerating expansion of the universe [10,11]. The standard method for analysis of STG modelsand studying correspondence to GR is conformal transformations from Jordan frame to Einsteinone [10, 11]. The connection between STG and GR cosmological solutions without conformaltransformations was considered in [12]. Also, the influence of non-minimal coupling on thedeviation from de Sitter expansion is studied in [13] and it was shown that this approach leadsto the models which have a good concordance to the observational constraints. ∗ [email protected] † [email protected] We have to stress that Starobinsky model [1] was related to modified f(R) gravity. a r X i v : . [ g r- q c ] M a r or the very early universe approaching the Planck scale one can consider Einstein gravitywith some corrections as the effective theory of the quantum gravity. The effective supergravityaction from superstrings induces correction terms of higher order in the curvature, which mayplay a significant role in the early Universe. The one of such correction is the Gauss-Bonnet(GB) term in the low-energy effective action of the heterotic strings [14]. Also, the GB termarises in the second order of the Lovelock gravity which is the generalization of the Einsteingravity [15].The exact solutions for cosmological models on the basis of Einstein-Gauss-Bonnet gravitywith non-minimal coupling of a scalar field with GB-scalar in 4D Friedmann universe were con-sidered in [16–34]. When such cosmological models are considering the two important problemscan be noted: the lack of conformal transformations to Einstein frame and the dependence ofthe propagation velocity of gravitational waves on time.To avoid the first problem, the connection between cosmological models based on standardGR inflation and EGB inflation was found in papers [35–37] without conformal transformations.Therefore, one has the possibility to compare these two types of models and evaluate the impactof non-minimal interaction on the character of cosmological inflation.To solve the second problem we suggest, in the present paper, the connection between cos-mological dynamics of EGB and standard GR inflation which allows rapid transform to generalrelativity case during inflationary epoch. Based on suggested connection it is possible to con-struct EGB inflationary models with fast approaching to GR inflation.To cover more types of modified gravity models considering in literature it is useful togeneralize developed approach on models with non-minimal interaction of a scalar field withRicci and Gauss-Bonnet scalars simultaneously, which can be considered both together andseparately. We will call such models as Generalized Scalar-Tensor Gravity (GSTG) theories.Since the most common scalar-tensor theory of gravity is the Hordeski gravity [38–42], it isalso necessary to determine the relationship between the types of gravity under considerationand GSTG theory. Such a connection was presented in the works [41, 42].One more issue we discuss in the present article is an appearance of various normalizationsof the tensor of gravitational waves in the literature. We discuss the influence of these normal-izations on the values of the parameters of cosmological perturbations and the verification ofinflationary models.Our paper is organized as follows. In Section 2 we represent the equations of cosmologicaldynamics for GSTG in spatially-flat 4D Friedmann universe and we introduce the deviationparameters which characterize the departure of non-minimal coupling functions from zero (i.e.the absence of non-minimal interactions). It is shown there the way of obtaining GSTG modelparameters which reconstruct any exact solution from GR cosmology. With suggested connec-tions between deviation parameters and the scale factor of the universe expansion it is provedfast approaching GSTG inflation to GR one with high accuracy. The Section 3 is devoted toconsideration of GSTG cosmological parameters of cosmological perturbations and comparisonof them to observational constraints. Also possible normalizations of tensor perturbations arediscussed as well there. In Section 4 we apply proposed approach to generate new exact infla-tionary solutions for GSTG from known solutions in scalar GR cosmology: the power-law andexponential power-law scale factor. The corresponding of obtained GSTG power-law solution toobservational constraints is performed as well. In Section 5 we study the connection of GSTGto Horndeski gravity and find the relation between Horndeski gravity parameters and ones forGSTG. In Section 6 we briefly discuss the restrictions on the deviations from GR for gravitymodels under consideration. Finally, Section 7 contains our summary.2 The conformity between GR and GSTG motivated inflation
We study GSTG theory with the action S = 12 (cid:90) d x √− g [ F ( φ, R ) − ω ( φ ) g µν ∂ µ φ∂ ν φ − V ( φ )] , (1)where F ( φ, R ) = R + f ( φ ) R + ξ ( φ ) R GB (2)Here f ( φ ) defines a non-minimal coupling of a scalar field with Ricci scalar and ξ ( φ ) defines a non-minimal coupling of a scalar field with Gauss-Bonnet scalar R GB = R µνρσ R µνρσ − R µν R µν + R . The equations of cosmological dynamics at the stage of inflation in spatially flat 4D Fried-mann universe (in the system of units m P = c = 1) ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) , (3)can be written as E ≡ f ) H + 3 H ˙ f − ω φ − V ( φ ) − H ˙ ξ = 0 , (4) E ≡ (1 + f )(3 H + 2 ˙ H ) + 2 H ˙ f + ¨ f + ω φ − V ( φ ) − H ˙ ξ − H ˙ H ˙ ξ − H ¨ ξ = 0 , (5) E ≡ ω ¨ φ + 3 ωH ˙ φ + 12 ˙ φ ω (cid:48) φ + V (cid:48) φ − H f (cid:48) φ − HF (cid:48) φ + 12 H ξ (cid:48) φ + 12 H ˙ Hξ (cid:48) φ = 0 , (6)with the additional condition [43]˙ φE + ˙ E + 3 H ( E − E ) = 0 . (7)Taking into account (7) we conclude that two equations from (4)–(6) are independent only.Our choice is to represent the cosmological dynamic equations as V ( φ ) = 3(1 + f ) H + (1 + f ) ˙ H + 52 H ˙ f + 12 ¨ f − H ˙ ξ − H ¨ ξ − H ˙ H ˙ ξ, (8) ω ( φ ) ˙ φ = H ˙ f − f ) ˙ H − ¨ f − H ˙ ξ + 8 H ˙ H ˙ ξ + 4 H ¨ ξ. (9)As one can see, the constant coupling of the scalar field and the Gauss-Bonnet scalar ξ = const doesn’t change the equations of cosmological dynamics for scalar-tensor gravity.The case of GR corresponds to the choice f = 0, ω = 1 and ξ = 0 . Then from the (1) onederives the action S E = (cid:90) d x √− g (cid:20) R − g µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (10)which leads to the following cosmological dynamic equations3 H = 12 ˙ φ + V ( φ ) , (11) − H − H = 12 ˙ φ − V ( φ ) , (12)¨ φ + 3 H ˙ φ + V (cid:48) φ = 0 , (13)We can rewrite them in terms of the cosmic time and the scalar field as arguments, in theform V E ( φ ( t )) = 3 H + ˙ H, V E ( φ ) = 3 H − H (cid:48) φ , (14)˙ φ E = − H, ˙ φ E = − H (cid:48) φ . (15)3o characterize the difference between cosmological inflation based on Einstein gravity andGeneralized Scalar-Tensor Gravity, we introduce the deviation parameters ∆ ST = ∆ ST ( t ) and∆ GB = ∆ GB ( t ) which are connected to the coupling functions as f ( φ ) = − ∆ ST , (16)˙ ξ = − ∆ GB H . (17)In terms of these parameters one has the following background dynamic equations V ( φ ) = 3(1 − ∆ ST ) H + (1 − ∆ ST ) ˙ H − H ˙∆ ST −
12 ¨∆ ST + ˙∆ GB + 5 H ∆ GB , (18) ω ( φ ) ˙ φ = − − ∆ ST ) ˙ H − H ˙∆ ST + ¨∆ ST − GB + 2 H ∆ GB . (19)For the case ∆ ST = 0 and ∆ GB = 0 the equations (18) and (19) are transformed to (14)–(15).To analyze the variety of exact solutions of cosmological inflation models based on the GSTGwith the dynamic equations (8)–(9) or (18)–(19) we prove the following assertion on a specialclass of the exact cosmological solutions: In four dimensional Friedman-Robertson-Walker space-time for each exact background cos-mological solution in GR cosmology, where the scalar field φ (cid:54) = const, exists the same solutionin the GSTG with nontrivial in general case coupling functions f (cid:54) = 0 , ξ (cid:54) = const and kineticfunction ω (cid:54) = 1 . To prove this assertion we define the kinetic function and the deviation parameters as follows ω ( φ ) = 1 + 3 (cid:15) (cid:18) ∆ ST + 2 ∆ GB H (cid:19) , (20)∆ ST ( t ) = β ST a − ( t ) , (21)∆ GB ( t ) = α GB a − ( t ) , (22)where (cid:15) = − ˙ H/H is the slow-roll parameter, β ST and α GB are the coupling constants of thescalar field with the Ricci scalar and Gauss-Bonnet scalar, respectively.It is not difficult to check that after substituting the functions (20)–(22) into the backgrounddynamics equations (18)–(19) we have the same equations as (14)–(15) V ( φ ) = 3 H + ˙ H, (23)˙ φ = − H. (24)Further, from (16) and (17) we derive the expressions for non-minimal coupling functionsand the kinetic function f ( t ) = − β ST a ( t ) , (25)˙ ξ = − α GB a H , (26) ω ( t ) = 1 + 3 (cid:15)a (cid:18) β ST + 2 α GB Ha (cid:19) . (27)Therefore, for each exact solution of the system (23)–(24) one can find corresponding func-tions (25)-(27) which characterise the type of GSTG model.4ow we want to represent the equations (23)–(24) and the functions (25)-(27) in terms of ascalar field φ as the argument on the basis of the following relations˙ H = − H (cid:48) φ , (28)˙ ξ = ξ (cid:48) φ ˙ φ = − ξ (cid:48) φ H (cid:48) φ , (29) a ( φ ) = a exp (cid:32) − (cid:90) HH (cid:48) φ dφ (cid:33) . (30)As the result, equations (23)–(24) are transformed to we have the Ivanov-Salopek-Bond equa-tions [44, 45] V ( φ ) = 3 H − H (cid:48) φ , (31)˙ φ = − H (cid:48) φ , (32)with corresponding non-minimal coupling and kinetic functions f ( φ ) = − β ST a exp (cid:32)(cid:90) HH (cid:48) φ dφ (cid:33) , (33) ξ (cid:48) φ = α GB a H (cid:48) φ H exp (cid:32) (cid:90) HH (cid:48) φ dφ (cid:33) , (34) ω ( φ ) = 1 + 32 (cid:32) HH (cid:48) φ (cid:33) exp (cid:32)(cid:90) HH (cid:48) φ dφ (cid:33) (cid:34) β ST + 2 α GB H exp (cid:32) (cid:90) HH (cid:48) φ dφ (cid:33)(cid:35) . (35)Also, using the structure of (35) one can represent the coupling function ξ ( φ ) as ξ ( φ ) = Ψ( φ ) exp (cid:32) (cid:90) HH (cid:48) φ dφ (cid:33) , (36)where the function Ψ( φ ) is defined from the equation (cid:18) Ψ (cid:48) φ H (cid:48) φ + 52 Ψ H (cid:19) H = α GB a . (37)For minimal coupling of a scalar field and the Gauss-Bonnet term α GB = 0 from the equation(37) one has Ψ( φ ) = const × exp (cid:32) − (cid:90) HH (cid:48) φ dφ (cid:33) , (38)and, therefore, from (36) we obtain ξ = const .We will consider the equation (37) as an explicit integrability condition of the system (4)–(6)for GR-like cosmological models based on GST gravity.Further, we consider the evolution of the deviation parameters in terms of the e -foldsnumbers N = ln ( a/a ) = − (1 / (cid:82) ( H/H (cid:48) ) dφ . From the expressions (21)–(22) we obtain∆ ST ∝ exp( − N ) and ∆ GB ∝ exp( − N ). Thus, the initial deviations between GR and GSTgravity rapidly decrease with the expansion of the universe in these models. The value of the e -folds numbers at the end of inflation is estimated as N = 50 −
60 and, therefore, we have∆ ST ( N = 60)∆ ST ( N = 0) = e − ≈ . × − , ∆ GB ( N = 60)∆ GB ( N = 0) = e − ≈ . × − , (39)where ∆ ST ( N = 0), ∆ GB ( N = 0) and ∆ GB ( N = 60), ∆ GB ( N = 60) correspond to the valuesof the deviation parameters at the beginning and at the end of inflation. Therefore, for thecase ∆ ST ( N = 0) (cid:28) e (for example, for conformal coupling with ( β ST /a ) = 1 /
6) and∆ GB ( N = 0) (cid:28) e at the end of inflation one has ∆ ST (cid:39) GB (cid:39) The parameters of cosmological perturbations
One of the main methods of verification of cosmological models is the comparison of theobtained parameters of cosmological perturbations with observational constraints. Now, weconsider the parameters of cosmological perturbations in the inflationary models based on mod-ified gravity [42, 43] in terms of the deviation parameters for decreasing slow-roll parameterssuch that at the crossing of the Hubble radius one has the conditions (cid:15) (cid:28) δ (cid:28) ≤ (cid:15) < (cid:15) = 1).To calculate the parameters of cosmological perturbations we define the following functions w = 1 − ∆ ST + 2 ∆ GB H , (40) w = 2(1 − ∆ ST ) H − ˙∆ ST + 6∆ GB , (41) w = − − ∆ ST ) H − − ∆ ST ) ˙ H + 32 ¨∆ ST − GB + (cid:18)
152 ˙∆ ST − GB (cid:19) H, (42) w = 1 − ∆ ST − HH (cid:32) GB H − ˙∆ GB ˙ H (cid:33) . (43)The velocities of scalar and tensor perturbations are c S ≡ w w H − w w + 4 w ˙ w w − w ˙ w ) w (4 w w + 9 w ) , (44) c T ≡ w w . (45)If we consider a small deviation from GR, i.e. ∆ ST (cid:39) GB (cid:39) w = 1 , w = 2 H, w = − H − H, w = 1 , (46)and from (44)–(45) one can obtain c S (cid:39) c T (cid:39)
1, therefore the conditions for the crossingof the Hubble radius c S k = aH and c T k = aH , where k is the wave number, are reduced to k = aH .The power spectra which are given by the expressions P S = H π Q S c S , P T = H π Q T c T , (47)with the following functions Q S ≡ w (4 w w + 9 w )3 w , Q T ≡ w s , (48)for ∆ ST (cid:39) GB (cid:39) P S = 12 (cid:15) (cid:18) H π (cid:19) , P T = 2 s (cid:18) H π (cid:19) . (49)The indices of scalar and tensor perturbations and tensor-to-scalar ratio for ∆ ST (cid:39) GB (cid:39) n S − ≡ d ln P S d ln k = ˙ P S H (1 − (cid:15) ) P S = 2 (cid:18) δ − (cid:15) − (cid:15) (cid:19) , (50) n T ≡ d ln P T d ln k = ˙ P T H (1 − (cid:15) ) P T = − (cid:15) − (cid:15) , (51)6 = P T P S = 4 Q S Q T (cid:18) c S c T (cid:19) = 4 s(cid:15), (52)where δ = (cid:15) − ˙ (cid:15) H(cid:15) = − ¨ H H ˙ H is the second slow-roll parameter.In most works the parameters of cosmological perturbations are calculated for the value s = 4(for example, see [4, 5] and a lot of the other works on cosmological inflation). However, in somepapers [34, 48–51] one can find the other results which correspond to s = 1. This difference canbe easily explained by considering the normalization of the tensor of gravitational waves (tensorperturbations) which can be defined as [5, 49]ˆ h ij ( η, x ) = (cid:90) d k (2 π ) / (cid:88) λ =1 , (cid:104) h k ( η ) e ij ( k , λ ) ˆ a k ,λ e i k · x + h ∗ k ( η ) e ∗ ij ( k , λ ) ˆ a + k ,λ e − i k · x (cid:105) , (53)by choosing the amplitude h k ( η ) = √ s × h k ( η ), where h k ( η ) is the normalized amplitude ofgravitational waves. In the expression (53) k is the wave vector, η is the conformal time, e ij ( k , λ ) are polarization tensors, ˆ a k ,λ and ˆ a + k ,λ are the creation and annihilation operators, λ = 1 , h k ( η ) and h k ( η ) are related as (cid:88) λ (cid:104) | h ∗ k,λ h k (cid:48) ,λ | (cid:105) = s × (cid:88) λ (cid:104) | h ∗ k,λ h k (cid:48) ,λ | (cid:105) , (54)and, therefore, from the connection between two-point correlation and power spectrum of tensorperturbations [5, 49] (cid:88) λ (cid:104) | h ∗ k,λ h k (cid:48) ,λ | (cid:105) ≡ (2 π ) πk P T ( k ) δ ( k − k (cid:48) ) , (55)one has P T ( k ) = s × P T ( k ) which gives the different values of tensor-to-scalar ratio r for s = 1and s = 4. To summarize the various results we will consider the parameters of cosmologicalperturbations in terms of the constant parameter s = 1 , P S = 2 . × − , n s = 0 . ± . , (56) r < . , r < .
065 (Planck 2018/BICEP2/Keck-Array) . (57)Thus, the parameters of cosmological perturbations for the case of GST gravity under theconditions (20)–(22) coincide with ones in general relativity with high accuracy. Therefore, theproposed approach leads to the generalisation of the inflationary models based on the Einsteingravity into the same ones in generalized scalar-tensor gravity. Now, we consider the the proposed approach on the example of the inflation with exponentialpotentials, namely the power-law inflation and it’s generalization as the exponential power-lawinflation on the basis of the following Hubble parameter H ( φ ) = µ exp( − µ φ ) + µ , (58)where µ , µ and µ are some constants. 7 .1 The power-law inflation For the case of the power-law inflation we consider µ (cid:54) = 0, µ (cid:54) = 0 and µ = 0.The exact solutions of the equations (31)–(32) are V ( φ ) = µ (3 − µ ) exp( − µ φ ) , (59) φ ( t ) = 1 µ ln (cid:2) µ µ t + c (cid:3) , (60) H ( t ) = µ µ µ t + c , (61) a ( t ) = a (2 µ µ t + c ) / µ , (62)where c is the constant of integration.Further, we calculate the exact explicit expression for coupling and kinetic functions fromthe equations (33)–(37) f ( φ ) = − β ST a exp (cid:18) − φµ (cid:19) , (63) ξ ( φ ) = − α GB a µ (6 µ −
5) exp (cid:20) (6 µ − µ φ (cid:21) + const, µ (cid:54) = ± (cid:114)
56 (64) ω ( φ ) = 1 + 3 exp (cid:16) − φ µ (cid:17) µ µ (cid:20) β ST µ + 2 α GB exp (cid:18) − (3 − µ )2 µ φ (cid:19)(cid:21) . (65)Therefore, we have generalized cosmological solutions which are reduced to GR ones for α GB = 0 and β ST = 0. For the case of the exponential power-law inflation we consider µ (cid:54) = 0, µ (cid:54) = 0, µ (cid:54) = 0.The exact solutions of the equations (31)–(32) are V ( φ ) = µ (3 − µ ) exp( − µ φ ) + 6 µ µ exp( − µ φ ) + 3 µ , (66) φ ( t ) = 1 µ ln (cid:2) µ µ t + c (cid:3) , (67) H ( t ) = µ µ µ t + c + µ , (68) a ( t ) = a exp( µ t )(2 µ µ t + c ) / µ . (69)This model implies the exit from inflation, also for t → ∞ the Hubble parameter H = µ which corresponds to the second accelerated exponential expansion of the universe.One can find the exact explicit expression for coupling and kinetic functions for µ = (cid:112) / f ( φ ) = − β ST a exp (cid:32)(cid:114) − φ − µ µ e (cid:113) φ (cid:33) , (70) ξ ( φ ) = α GB a µ exp (cid:18) − µ µ e (cid:113) φ (cid:19) µ (cid:18) µ + µ e (cid:113) φ (cid:19) − e µ Ei (cid:18) , µ µ e (cid:113) φ + 2 (cid:19) + const, (71)8 ( φ ) = 1 + 6 (cid:18) µ + µ e (cid:113) φ (cid:19) µ exp (cid:32) − (cid:114) φ − µ µ e (cid:113) (cid:33) × (cid:40) α GB exp (cid:34)(cid:32)(cid:114) − √ (cid:33) φ − µ µ e (cid:113) φ (cid:35) + β ST (cid:18) µ + µ e (cid:113) φ (cid:19)(cid:41) . (72)where Ei is the exponential integral function [46].Also, the parameter µ = (cid:112) / f ( φ ) = − β ST a exp (cid:32) − (cid:114) φ − µ µ e (cid:113) φ (cid:33) , (73) ξ ( φ ) = α GB a µ µ exp (cid:18) µ µ e (cid:113) φ (cid:19) µ + 3 µ e (cid:113) φ + Ei (cid:18) , µ µ e (cid:113) φ (cid:19) + 2 e µ Ei (cid:18) , µ µ e (cid:113) + 3 (cid:19) + const, (74) ω ( φ ) = 1 + 9 (cid:18) µ + µ e (cid:113) φ (cid:19) µ exp (cid:32) − (cid:114) φ − µ µ e (cid:113) (cid:33) × (cid:40) α GB exp (cid:34)(cid:32)(cid:114) − √ (cid:33) φ − µ µ e (cid:113) φ (cid:35) + β ST (cid:18) µ + µ e (cid:113) φ (cid:19)(cid:41) . (75)In this case, we have sufficiently complex expressions for coupling and kinetic functions toprovide GR solutions for exponential power-law inflation in the case of GST gravity. Further, we calculate the parameters of cosmological perturbations for this model for thearbitrary value of µ and will consider µ = (cid:112) / µ = (cid:112) / µ = 0 as the partial cases.Firstly, we obtain the expressions for slow-roll parameters (cid:15) = − ˙ HH = 2 µ µ (2 µ µ µ t + cµ + µ ) , (76) δ = − ¨ H H ˙ H = 2 µ µ (2 µ µ µ t + cµ + µ ) . (77)For the case of the power-law inflation with µ = 0 from expressions (76)–(77) we obtain (cid:15) = δ = 2 µ .From the dependence of the e-folds number from cosmic time N = ln (cid:18) aa (cid:19) = 12 µ ln(2 µ µ t + c ) (78)we obtain the inverse relationship t = t ( N ), namely t ( N ) = 12 µ µ (cid:16) e µ − c (cid:17) . (79)9fter substituting the dependence (79) into the expression for power spectrum of the scalarperturbations on the crossing on the Hubble radius (49) we have the condition P S ( N ) = 12 (cid:15) (cid:18) H π (cid:19) = µ π µ exp (cid:0) − µ N (cid:1) = 2 . × − . (80)The different choice of parameters µ and µ (cid:54) = (cid:112) / N = 60 can satisfy this condition.From the expressions (50) and (52) one has the following dependence r = r ( n S ) for power-lawinflation r = 4 s ( n S − n S − . (81)s=4 s=1Figure 1: The dependences r = r ( n S ) for exponential power-law inflation with µ = (cid:112) / µ = (cid:112) / s = 4, s = 1.For exponential power-law inflation with µ (cid:54) = 0 we have the condition (cid:15) = 12 (cid:18) δµ (cid:19) . (82)10urther, from the dependence of the e-folds number from cosmic time N = ln (cid:18) aa (cid:19) = 12 µ ln(2 µ µ t + c ) + µ t (83)we obtain the inverse relationship t = t ( N ), namely t ( N ) = − µ µ µ (cid:20) cµ − µ W (cid:18) µ µ exp(2 µ µ N + cµ µ ) (cid:19)(cid:21) , (84)where W is denote the Lambert function [46].After substituting the dependence (84) into the expression for power spectrum of the scalarperturbations on the crossing on the Hubble radius (49) we have the condition P S ( N ) = 12 (cid:15) (cid:18) H π (cid:19) = µ π µ (cid:104) W (cid:16) µ µ exp(2 N µ µ + cµ µ ) (cid:17) + 1 (cid:105) (cid:104) W (cid:16) µ µ exp(2 N µ µ + cµ µ ) (cid:17)(cid:105) = 2 . × − . (85)As one can see, the different choice of four model’s parameters c , µ , µ , µ for N = 60 cansatisfy this condition.Secondly, from (50) and (52) we obtain the dependence r = 16 n S − n S − ∓ √ µ (cid:16) −√ µ + (cid:113) µ + 4 n S − n S + 12 (cid:17) n S − , (86)where we will consider the solution with a negative sign.From dependence r = r ( n S ) on the Fig. 1 it can be concluded that the model of exponentialpower-law inflation corresponds restrictions (56)–(57) for both normalizations s = 4, s = 1and power-law inflation is not consistent with these restrictions for s = 4. Consequently, thenormalization of the gravitational wave tensor is critical for the verification of the power-lawinflationary model on the spectral parameters of cosmological perturbations. Now, we consider the connection between the inflationary models which based on the action(1) and Horndeski gravity [38] defined by the action S = (cid:90) d x √− g ( L + L + L + L ) , (87) L = K ( φ, X ) , L = − G ( φ, X ) (cid:3) φ, (88) L = G ( φ, X ) R + G ,X (cid:2) ( (cid:3) φ ) − ( ∇ µ ∇ ν φ )( ∇ µ ∇ ν φ ) (cid:3) , (89) L = G ( φ, X ) G µν ∇ µ ∇ ν φ − G ,X [( (cid:3) φ ) −− (cid:3) φ )( ∇ µ ∇ ν φ )( ∇ µ ∇ ν φ ) + 2( ∇ µ ∇ α φ )( ∇ α ∇ β φ )( ∇ β ∇ µ φ )] , (90)where X = −∇ µ φ ∇ µ φ/ (cid:3) φ = ∇ µ ∇ µ φ , K , G , G , G is some functions of φ and X , G j,X ( φ, X ) = ∂G j ( φ, X ) /∂X with j = 4 , K ( φ, X ) = ω ( φ ) X − V ( φ ) − ξ (cid:48)(cid:48)(cid:48)(cid:48) φ X (3 − ln X ) , (91) G ( φ, X ) = − ξ (cid:48)(cid:48)(cid:48) φ X (7 − X ) , (92) G ( φ, X ) = 12 (1 + f ( φ )) − ξ (cid:48)(cid:48) φ X (2 − ln X ) , (93) G ( φ, X ) = 2 ξ (cid:48) φ ln X, (94)11ne can obtain the dynamic equations similar (4)–(6) and, therefore, we have the same cosmo-logical models based on the Horndeski gravity.Thus, to reconstruct the parameters of this type of gravity it is enough to calculate thederivatives of the function ξ ( φ ) and substitute the solutions of the equations (4)–(6) in theexpressions (91)–(94).For example, for the power-law inflation with const = 0 in the expression (64) one has ∂ ( k ) ξ ( φ ) ∂φ k = (cid:20) (6 µ − µ (cid:21) k ξ ( φ ) , where k = 1 , , , . (95)After substituting the solutions (59), (63)–(65) and (95) into the expressions (91)–(94) weobtain the parameters of the the Horndeski gravity corresponding to power-law inflation. Also,one can reconstruct the type of the Horndeski gravity for exponential power-law inflation in thesimilar way. Now we will consider the scenario of the evolution of the universe, based on a special classof cosmological solutions for the generalized scalar-tensor theory of gravity.At the beginning of the inflationary stage, the values of the deviation parameters ∆ ST ,∆ GB and, respectively, the functions determining the non-minimal coupling of the scalar fieldand the curvature f ( φ ) and ξ ( φ ) may be large enough to provide a connection with string andsuperstring theory [14] which inspired the additional terms associated with modifications of theEinstein gravity.During inflation, these deviations rapidly decrease according to the laws ∆ ST ∝ a − ( t ) and∆ GB ∝ a − ( t ), thus, at the end of the inflationary stage, the part (2) of the action (1) whichdetermines the type of gravity is F ( φ, R ) ≡ R + f ( φ ) R + ξ ( φ ) R GB = R + O (∆ ST , ∆ GB ) , (96)where O (∆ ST , ∆ GB ) (cid:28) ω ( φ ) ≡ (cid:15) (cid:18) ∆ ST + 2 ∆ GB H (cid:19) = 1 + 3 (cid:15) O (∆ ST , ∆ GB ) , O (∆ ST , ∆ GB ) (cid:28) . (97)Further, after completion of the inflation stage, the dynamics of the universe at the stages ofthe dominance of radiation and matter are determined by the Friedmann solutions a ( t ) ∝ t / and a ( t ) ∝ t / , and the deviations with Einstein’s gravity continues to decrease with theexpansion of the universe.At the present stage of the universe’s evolution, i.e. for a larger number of e-folds, gravityin these models coincides with GR with even higher accuracy than in previous eras. In theevent that the stage of the second accelerated expansion of the Universe in the present epochis described by ΛCDM model with exponential expansion ( ˙ H = 0) one has (cid:15) = 0 and ω → ∞ ,which corresponds to Einstein’s gravity. If dark energy is determined by a quintessence field witha different corresponding dynamics ˙ H (cid:54) = 0 ( ˙ H ≈ (cid:15) (cid:54) = 0 and the observational constraint | ω | > − in determining the propagation velocity of gravitationalwaves from the merger of neutron stars and black holes [9].12 Conclusion
Investigations of modified gravity theories, including that with non-minimal coupling of ascalar field to Ricci and Gauss-Bonnet scalars, are connected with early inflationary epoch andwith nowadays accelerated expansion of the universe (dark energy).In the present paper, we studied the connection of exact solutions in GR cosmology withthat dictated by modified 4D gravity with the self-interacting scalar field coupled to Ricci andGauss-Bonnet scalars by minimal and non-minimal manners.It was proved the existence for each exact GR cosmology solution the same solution in theGSTG cosmology for the special choice of their parameters (33)–(35). Namely, we introducedthe deviation parameters ∆ ST and ∆ GB (16)–(17) as qualitative characteristic of differencebetween the models and found the connection of the functional parameters of GSTG with thescale factor of GR cosmology solutions. By this approach a new class of exact cosmologicalsolutions in models based on generalized scalar-tensor theories of gravity was obtained. We alsoexamined the connection of such models with the Hordeski gravity.It was shown that the initially large deviation between Einstein’s gravity and its consideredmodifications rapidly decreases with the expansion of the Universe. The parameters of cosmo-logical perturbations in the proposed models coincide with ones in the case of general relativitywith high accuracy.We gave the algorithm of reconstruction for any exact solution obtained in GR [52, 53] thesame solution in GSTG cosmology with the same parameters of cosmological perturbations.As an example of this approach we obtained the exact cosmological solutions for power-lawand exponential power-law inflation. The correspondence of these models to modern observa-tional data on the spectral parameters of cosmological perturbations was considered. The effectof the normalization of the gravitational wave tensor on the verification of cosmological modelswas also discussed. This work has been partially supported by the RFBR grant 18-52-45016 IND a. S.V.C. isgrateful for support by the Program of Competitive Growth of Kazan Federal University.
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