The flat Friedmann universes filled by stiff fluid and a nonminimally
coupled material scalar field with polynomial potentials of the
fourth degree are considered in the framework of the
Einstein-Cartan theory. Exact general solution is obtained for arbitrary positive values of the coupling constant
Recent cosmic observations [
The ECT finds applications in cosmology [
At present, the considerably increased precision of measurements in the modern observational cosmology stipulated its essential progress. In this connection, the exact cosmological solutions, which makes it possible to elucidate the detailed picture of an evolution of models, are of great interest. It is well known that in GR the exact solutions of the Friedmann-Robertson-Walker (FRW) cosmological models constitute a basis for comparison of theoretical predictions with observations. On the other hand, the problem of the cosmological perturbations may be correct, if it is constructed against a background of the exact solution.
Other problems of the cosmology, singularities, horizons, and so forth, remain no less acute problems. The main aim of this paper is the investigation of the possibility of the solution of some problems of the modern cosmology in the framework of the ECT. Furthermore, the exact integration of the ECT equations for the different combination of sources and the comparative analysis of the corresponding cosmological models are of interest in their own right, since they allows to elucidate the role of the sources of the gravitational field in cosmology.
In the framework of the ECT with a nonminimally coupled material scalar field that has polynomial potentials of the fourth degree, we here study isotropic spatially flat cosmologies in which stiff fluid is taken into account. The motivation to investigate a nonminimally coupled scalar field is based on the results obtained within the framework of the torsionless theories of gravity in connection with the inflationary cosmology (see, e.g., [
The interest in a scalar field potential
A stiff cosmological fluid, with pressure equal to the energy density, can be described by a massless scalar field, which is predicted by the string theory. On the other hand, the stiff fluid is an important component because, at early times, it could describe the shear dominated phase of a possible initial anisotropic scenario and is dominating the remaining components of the model [
In [
The exact general solution of an analogous problem for models containing only a nonminimally coupled scalar field with an arbitrary coupling constant For For There are the specific values of the parameter
The exact general solutions for spatially flat FRW cosmologies with a nonminimally coupled scalar field that has polynomial potentials of the fourth degree For For
The paper is organized as follows. In the next section we present the model and corresponding field equations. In Section
The Lagrangian
The metric
and
One can note that the Lagrangian (
Varying the action with the Lagrangian (
It is not difficult to verify that the effective scalar-torsion energy-momentum tensor
For spatially flat isotropic and homogeneous models with the metric
Equations (
For the stiff fluid we have
Adding (
The substitution of (
From (
we obtain
It must be noted here that the requirement of renormalization of the quantum theory of a scalar field results in the introduction of a nonminimal coupling and the potential in the form of (
In this paper, we will restrict our discussion to the material scalar field
The exact general solution to the Einstein-Cartan equations for
An analysis has shown that, depending on the choice of the gravitational field sources, different types of models are possible.
Let us consider the models without the scalar field potential
From (
For
The asymptotic behavior of
For
It should be observed here that we cannot express
It is easy to see that Hubble's parameter
It is interesting to observe that the models (
For
It follows from (
It is easy to verify that models (
As pointed out in [ stiff fluid dominated era
de Sitter regime
It follows from (
For minima of the scale factor the square of the trace of torsion is
It must be noted that, for
For
It is easy to verify that for
Thus, for the Big Bang singularity of the form
An analysis has shown that for
It is not difficult to show that for
Let us consider the models for which all sources of gravitational field from the Lagrangian (
From (
An analysis has shown that this solution with
For
It should be pointed out that, for
For
It is easy to see that the case for
It should be observed here that the accelerated expansion of models with the power-law asymptotic at late stages of the cosmological evolution has its origin in “Scalar-torsion field” + “
For
The analytic solution in elementary functions for
Formulae (
The solution in a parametric form for
For
(a)
(b)
For (
It is easy to see that for
It is not difficult to verify that for
For
The minima of the scale factor,
The asymptotic behavior of
The value of
An analysis has shown that with
In this paper, exact general solution for spatially flat isotropic and homogeneous cosmological models in ECT with stiff fluid and a nonminimally coupled material scalar field with a potential
In order to elucidate the role of the stiff fluid in the Einstein-Cartan cosmology, we should carry out a comparative analysis of the corresponding cosmological models obtained in this paper and those in [
By comparing these models, we arrive at the conclusion that for the cosmological models with “Scalar-torsion field” + “Stiff fluid” and “Scalar-torsion field” + “ integrability of the Einstein-Cartan equations for the existence of singular expanding models with the power-law the existence of nonsingular (not bouncing) expanding models with de Sitter asymptotic at late times, the existence of the specific value of the coupling constant
Furthermore, for the models with the “Scalar-torsion field” + “Stiff fluid” the presence of stiff fluid leads to the existence of asymmetrically bouncing models with the asymptotics
Additionally, stiff fluid in the mixture with the “Scalar-torsion field” + “
It should be observed that the existence of the power-law
Comparing the models obtained in this work with the DE models that could admit similar behaviors we note the following. The behavior of the scale factor by the law
The author is grateful to the unknown reviewers for the useful remarks.