Cosmological models in conformal representations of Jordan theory

This paper is devoted to the investigation of conformally-related variants of the modified tensor-scalar Jordan theory on the example of determination of the comparative characteristics of the model Universe in "Einstein" and "proper" frames. Within the framework of this model, we consider the possibility for the accelerated expansion of the Universe at the recent epoch.


Introduction
The role of conformal transformations in gravitational theories has been discussed by H. Weyl [1], W. Pauli [2], A.Z. Petrov [3], R. Dicke [4], etc (for a recent review see [5]). In the Jordan-Brans-Dicke (JBD) theory the specificity of the scalar field is particularly evident in the conformal representations, where the scalar field is separated from the metric part of the gravitational field, and as it turns out, it may play a definite role in the phenomenon of acceleration of the expanding Universe. In the absence of matter fields, JBD theory is conformally-invariant [6]. Under certain conformal transformations it takes a standard "Einstein" form. In this case, the energy-momentum tensor for minimally or conformally coupled scalar field is added to the energy-momentum tensor of matter. There are clear advantages to have this conformal relation: one can use solutions already obtained for one representation to generate solutions of another, equivalent, representation.
The present paper is devoted to the theoretical study of the evolution of the Universe within the framework of various variants of JBD theory. We consider the possibility for the accelerated expansion of the Universe at the recent epoch. By taking into account that in General Relativity the arguments in favor of the cosmological constant are convincing enough, we introduce a similar quantity in JBD theory, assuming that the corresponding field is a scalar one. However, the latter is non-dynamical and is governed by the gravitational scalar.
2. The cosmological problem with the cosmological scalar ϕ = Λy/y 0 The modified version of the Jordan theory, in addition to the metric tensor, contains a scalar field y(x) in its gravitational sector. The corresponding action functional has the form [7] where k = 8π/c 2 , y 0 is a constant, ζ is the dimensionless parameter of the theory and we use the notation y μ = ∂ μ y. After the conformal transformation g μν = g μν y/y 0 , the action (1) takes the form [8] were the new scalar field Φ is defined by the relation The action (2) coincides with the Einstein action in the presence of a minimally-coupled scalar field ("Einstein" frame of the Jordan theory) [8]. The corresponding cosmological problem, described by the Friedmann-Roberston-Walker (FRW) line element has been discussed in [11]. In the present paper we compare the results obtained for the model with a minimally coupled scalar field with the model in the presence of a self-consistent scalar field in the case of action (1) and for the metric The time coordinates and the scale factors in two different representations are related by As in the paper [9], we solve the standard cosmological problem with the notations H = · R/R, ψ = · y/y, q = ·· RR/R 2 , where the dot stands for the derivative with respect to τ . The set of field equations reads By taking into account that in the problem under consideration ϕ = Λy/y 0 , the equation (7) is reduced to ·· y/y = −3ψH, or equivalentlẏ From here, for the evolution of the expanding Universe (H > 0) we get ·· y/y < 0 ⇒ψ/ψ 2 < −1.
Further, subtracting (9) from (8), we get Defining a new function · y = z(y), the latter is reduced to the Euler equation [10]: with the solution [11] After the substitution of (14) into (7) and (9), we obtain the relation between the constants of integration c 1 and c 2 : from which it follows that c 1 and c 2 have opposite signs, when Λ > 0, and one of these constants is equal to zero, when Λ = 0. By taking into account that we conclude that c 1 < 0, c 2 > 0. These constants are determined from the condition y = y 0 at t = 0: As a result one finds the following relations By taking into account that our ultimate aim is to compare the obtained results with the corresponding problem for the action (2), let us present (14) in the form After the integration of this we find the following expression Here we have introduced the notations From (10) one can see thatψ from which for the function R/R 0 one obtains For the numerical calculations of the parameters H, Ω Λ , Ω y , q it is convenient to use the following combination of equations (10) and (12): With the definition z(ψ) =ψ/ψ, the equation (24) is reduced to a known class of differential equations (see for example, [10]): zz (ψ) − z − (4 + 3ζ)ψ/2 = 0. Taking z = uψ, the variables are separated and we find In order to find the Einsteinian limit (ζ → ∞), we present (25) in the form from which it follows that lim σ→∞ ψ = 1/σ → 0.As a result, the solution of the equation (26) can be written as

Determination of the integral parameters
It is easy to see that all relevant functions are expressed in terms of the ratio ψ/H, which can be directly determined from (10): Then, from (12), for the deceleration parameter one finds From (9) we have where Ω Λ is the effective energy contribution of the cosmological scalar. It should be noted that (30) allows us to obtain ψ 0 /H 0 for y = y 0 : which is needed for the calculation of α = |c 1 | /c 2 from (18). For the evaluation of H/H 0 we use (28): and the behavior of R/R 0 is determined from (23). By taking into account that we consider the late stages of the Universe expansion, the expression (20) for large positive values of the argument can be presented in the form where x = (3/2) H 0 0 Ω Λ t + δ and we determine the relation between conformally related time coordinates by (6). As a result one gets τ = α 1/σ t.
In figures 1-4 we have plotted the time evolution of the functions H/H 0 , R/R 0 , q, Ω y , Ω Λ for the values of the parameter ζ = 50 and ζ = 1000.
Taking the line element in the FRW form (4), we introduce the cosmological functions The system of equations in convenient combinations is the following: In order to solve the equation (37) it is resonable to calculate the derivative of the expression By taking into account (37), we can see thaṫ Now, combining the equations (35) and (36), we obtain The solution of the equation (41) can be written in the form where A 2 = 2Λ (3 + 4ζ) / (3 + 2ζ). In the case (43), ya 3 = c sinh (At + α), the integration constants are determined from the following relations From the equation (37) we have From here it follows that Now, by using equations (38), (44) and (47), we get At the initial time t = 0 from (35) one can find the ratio ψ 0 / H 0 : From (44) and (49) it is easy to find the parameter α: The expression for the scale factor a (t) is obtained by integrating (48):

Conclusion
As a result, for the time dependence of the parameters in the model of the Universe similar qualitative features are obtained as in [8]. The quantitative differences are related to the form of the potential energy for the scalar field with general conclusion confirming the essential dependence of the influence of scalar field from the interaction parameter ζ [12]. On the example of the Universe acceleration (see figure 3) it is clearly seen that for large positive values of ζ (of the order 10 3 ) the acceleration practically coincides with that for General Relativity. For smaller values of ζ, the influence of the scalar field on the expansion rate is stronger and this leads to the corresponding decrease of q.     For nonminimally coupled scalar field in the presence of ϕ = Λ we consider the proper representation of the Jordan modified theory, taking into account the presence of a cosmological scalar ϕ = Λ. The results obtained for different values of the dimensionless parameter ζ (50, 500, 1000) are presented in figures 5-8. These results show that the behavior of the curves for ζ = 500 and ζ = 1000 is practically indistinguishable, in contrast to the case ζ = 50. This confirms the proximity of the Jordan theory and General Relativity for large values of ζ [14]. The results described above show that the role of scalar fields in variants with small ζ is different from that in General Relativity, providing a possibility to adjust the deceleration parameter q in comparing with the observational data. Figure 7 shows the time dependence of the deceleration parameter for different values of the energy contribution of the cosmological scalar. Figure 8 displays the energy contributions of the scalar field and the cosmological scalar typical for all versions of the theory. In comparing the obtained results with other models for a flat universe within tensor-scalar theory [15] a well-defined tendency of development is observed: the energy contributions of ordinary matter and the scalar field steadily converge to zero, whereas the contribution due to the presence of the cosmological parameter grows.
The work was supported by State Committee Science MES RA, within the frame of the research project No. SCS 13-1C040.