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Cosmological Constant and Gravitational Repulsion Effect: 1. Homogeneous models with radiation

Within the framework of the minimum quadratic Poincare gauge theory of gravity in the Riemann-Cartan spacetime we study the influence of gravitational vacuum energy density (a cosmological constant) on the dynamics of various gravitating systems. It is shown that the inclusion of the cosmological term can lead to gravitational repulsion. For some simple cases of spatially homogeneous cosmological models with radiation we obtain non-singular solutions in form of elementary functions and elliptic integrals.


I. INTRODUCTION
In the Poincare gauge theory of gravity (PGT) the gravity is described by the field of tetrads h i µ and the Ricci rotation coefficients A ik ν [1][2][3][4]. Due to the dynamical independence of these gauge fields the spacetime has both the curvature F ik µν and the torsion S i µν . Usually, one connects the torsion with the spin angular momentum of matter. In the simplest PGT -the Einstein-Cartan theory (ECT) [1][2] -by virtue of the algebraic dependence between the spin angular momentum and the torsion, we recover Greneral Relativity (GR) for the spinless gravitating systems.
In the PGT, the choice of gravitational Lagrangian is very important. The principle of local gauge invariance itself gives only the form of the gauge fields and their strengths but does not allow us to determine the explicit form of the Lagrangian for the gauge field. As was shown in [4,5] the simplest generalization of the ECT, which satisfies the restrictions following from the simultaneous consideration of the quantization problem (theory without "ghosts" and "tachyons") [6], the Birkhoff's theorem [7], and the problem of avoiding the metric singularity in the homogeneous isotropic cosmological models [8,9], is the minimum quadratic gauge theory of gravity (MQGT) with the gravitational Lagrangian: , F is the scalar curvature of the Riemann-Cartan spacetime, f 0 = (16πG) −1 , (G is the Newton's gravitational constant), and α < 0 is a dimensionless coefficient.
In [5] the basic field equations of MQGT were derived and in [10][11][12][13][14] some of their physical consequences were investigated. In this paper we will study the influence of the cosmological term on the dynamics of some spatially homogeneous gravitating systems. This work is a direct continuation of [10] and we will use all notations in [10], unless otherwise stated.

II. COSMOLOGICAL TERM AND GRAVITATIONAL REPULSION EFFECT.
According to the modern viewpoint, the cosmological term is connected with the energy density of the gravitating vacuum. Taking into account the cosmological term χ in the Lagrangian (1.1), two field equations of MQGT are obtained by independent variation of (1.1) with respect to h i µ and A ik ν : In this work we restrict ourselves to consider ones the so-called conformaly-invariant gravitating systems of spinless matter with trace-free energy-momentum tensor: It then follows from (2.1) that F = 2χ f 0 = const, and consequently, from (2.2) we find After some simple manipulation from (2.1) we obtain the MQGT equation for the considered systems in the form of the Einstein's equation of GR with the effective energy-momentum tensor as: . It is obvious that if χ = 0 we recover ordinary Einstein's equation (cf. refs [5,13]). If χ < 1 4β the solutions of eq. (2.5) cannot provide any qualitatively new properties in comparison with the corresponding solutions of GR.
However, under the condition the effective energy-momentum tensor t λµ ef f in (2.5) is negative, that leads to the gravitational repulsion effect. This effect is induced by a large cosmological constant (high gravitating vacuum energy density) and by the square curvature term in gravitational Lagrangian (1.1), although the torsion is vanishing. Due to this repulsion effect, the Penrose-Hawking energy conditions [15] might be broken and we can find a series of non-singular solutions, which are of certaint interest for astrophysical and cosmological applications. Note that in the very early stage of the cosmological evolution the vacuum energy density χ was , in fact, very high.
In the following sections on the basis of (2.5) we will study the gravitational repulsion effect in some simple spatialy homogeneous cosmological models with ultrarelativistic perfect fluid (isotropic radiation) with the following energy-momentum tensor and equation of state: We consider the homogeneous isotropic cosmological models with the Friedmann-Robertson-Walker metric: where R(t) is scale factor, and k = −1, 0, +1 for open, flat and closed models correspondingly. Then, the 00 -component of (2.5) takes the form: where a dot denotes differentiation with respect to time t. The "energy-momentum conservation law" has the standard forṁ It follows from Eqs. (3.3) and (2.8) that Subtituting (3.4) into (3.2) we obtain after simple manipulations the differential equation In the case considered, when χ > 1 4β it is easy to find the following regular solutions In fact, the scale factor (3.6) has the positive minimum value not only for closed and flat models (k = 1, 0), but also for open models, when k = -1 by virtue 4βχ − 1 > 0. Consequently, the energy density is limited ρ ≤ ρ max = C 2 0 /R 4 min . On the basis of the obtained solution we can construct the regular homogeneous isotropic (inflationary) cosmological models and the regular models of superdense isotropic radiation systems in the framework of MQGT (cf. refs. [16,17]).

IV. REGULAR BIANCHI TYPE-I MODELS WITH RADIATION
We now study the homogeneous Bianchi type-I radiation models with the metric given in form: It follows from kk-components of eq. (2.5) (k = 1, 2, 3) the following relationships: where r = ( r 1 r 2 r 3 ) 1/3 ; 3 k=1 s k = 0. Then, after some transformations the 00-component of eq. (2.5) and "energy-momentum conservation law" can be written as: where l 2 = 1 6 3 k=1 s 2 k , is a measure of the anisotropy. By using (4.4) it is easy to find the exact solution of (4.3) in the following analytic form where x = r 2 > 0. It is clear that the integral (4.5) makes sense only if P I (x) > 0.
Note that for all x ≥ x 1 , where x 1 is the greatest real root of the cubic polynomial P I (x), P I (x) ≥ 0 holds. If x 1 ≤ 0, P I (x) ≥ 0 for all x ≥ 0, and therefore, the solution (4.5) is singular because the function x(t) = 0 at a finite time t = t 0 . However, if x 1 > 0 the solution (4.5) is regular: x ≥ x 1 > 0 for all t ∈ (−∞, +∞). Thus, in order to get non-singular solution from (4.5) we have to find conditions for the existence of positive real roots of the cubic polynomial P I (x). Detailed analysis shows, that under the conditions (2.6) and the cubic polynomial P I (x) posseses three real roots x 1 ≥ x 2 > 0 > x 3 . Then, for x ≥ x 1 the solution (4.5) is regular and can be reduced to the eliptic integrals Π, F (see Ref. [10]) The minimum value of the universe "volume" is positive V min = r 3 min = x 3/2 1 > 0, where and the maximum value of energy density ρ is limited: ρ max = C 2 I /x 2 1 . Thus, under the conditions (2.6) and (4.7) the vacuum (repulsion) effect dominates over ordinary gravitational contraction effect and anisotropic effect (that is sufficiently weak), so permits avoiding the singularity in the Bianchi type-I cosmological models with radiation.

V. REGULAR BIANCHI TYPE-V MODELS WITH RADIATION
In this section we consider the Bianchi type-V models that are a direct generalization of open Friedmann-Robertson-Walker models. Taking ( x 1 , x 2 , x 3 ) as local coordinates we can write the interval of Bianchi type-V models in the diagonal form [18,19] For the models considered we can find the following non-vanishing components of the Einstein's tensor where r = (r 1 r 2 r 3 ) 1/3 . It follows from G 0 1 = 0 thaṫ and, consequently r 1 = r (by rescaling). From (2.5) we get G 1 1 = G 2 2 = G 3 3 and by virtue of (5.3) we can find where s is an integration constant and has the sense of a measure of the anisotropy. Then, 00-component of (2.5) can be reduced tȯ The "energy-momentum conservation law" has the ordinary form By analogy with the foregoing section we obtain the exact solution of (5.5) in an analytic form where x = r 2 > 0.
By analogy with section 4, together with condition (2.6) we can find the similar to (4.7) condition for the existence of positive real root of the cubic polynomial P V (x) as where It is easy to show that A V (C V , β, χ) > 0, and consequently, the condition (5.9) makes sense. Note that condition (5.9) gives stronger a restriction on the anisotropy than (4.7). In the case considered, the vacuum (repulsion) effect dominates over the ordinary gravitational attraction effect, the anisotropic effect, and the effect of negative curvature.
Thus, under the conditions (2.6) (large cosmological constant) and (5.9) (small anisotropy) the solution (5.7) is regular and has the same form as (4.8). Then, we can construct the non-singular Bianchi type-V radiation cosmological models with the limiting energy density ρ ≤ ρ max < ∞, and nonzero metric functions r i (t) > 0 at any time.

VI. CONCLUSIONS
In this paper we have shown that the gravitating vacuum effect can be of importance for avoiding the singularity in some homogeneous cosmological models with the isotropic radiation. We obtain regular solutions for Friedmann-Robertson-Walker, Bianchi type-I, and Bianchi type-V models. These solutions have been expressed in terms of elementary functions and elliptic integrals. Note that the gravitational repulsion effect can play an important role in the case of conformally-invariant systems such as: massless conformallyinvariant scalar field, electromagnetic field... On the basis of eq. (2.5) following [20] we can find exact non-singular solutions for various cosmological models with radiation, scalar, and electromagnetic fields, as well as for spherically-symmetric conformally-invariant systems.