Pis'ma v ZhETF, vol.94, iss. 12, pp.913-917
© 2011 December 25
Limiting energy density and a regular accelerating Universe in
Riemann-Cartan spacetime
A. V. Minkevich1}
Department of Theoretical Physics and Astrophysics, Belarussian State University, 220030 Minsk, Belarus Department of Physics and Computer Methods, Warmia and Mazury University in Olsztyn, 10-561 Olsztyn, Poland
Submitted 19 October 2011
Isotropic cosmology built in the Riemann-Cartan spacetime by using sufficiently general expression of gravitational Lagrangian is investigated. It is shown that cosmological equations obtained by certain restrictions on indefinite parameters of gravitational Lagrangian lead to limiting energy density at the beginning of cosmological expansion and all cosmological models filled with usual gravitating matter satisfying standard energy conditions are regular with respect to energy density, spacetime metrics with its time derivative and torsion functions. At asymptotics cosmological solutions of spatially flat models coincide with that of standard ACDAf-model for accelerating Universe.
1. Introduction. The problem of the beginning of the Universe in time in the past - the problem of cosmological singularity (PCS) - remains as one of the most principal problems of relativistic cosmology and general relativity theory (GR). In accordance with PenroseHawking theorems about gravitational singularities the most part of cosmological solutions of GR are singular, if gravitating matter satisfies standard energy conditions. The behaviour of cosmological solutions near cosmological singularity was investigated in works by Belinsky V.A., Lifshits E.M., and Khalatnikov I.M. (see [1,2] and Refs. herein). At the same time many attempts were undertaken with the purpose to solve the PCS in the frame of GR and existent candidates to quantum gravitation theory as well as of different generalizations of Einstein's gravitation theory, some particular regular cosmological solutions were obtained (see, for example [3, 4], review [5] and [6]). From physical point of view the appearance of gravitational singularities in gravitating systems with positive values of energy density and pressure is connected with the fact that the gravitational interaction in such systems in the frame of GR always has the character of attraction, which increases with the growth of energy density. Although the gravitational interaction in the case of gravitating systems with negative pressure in the frame of GR can be repulsive, the PCS can not be solved by considering corresponding models: the most part of cosmological solutions including inflationary solutions are singular.
As it was shown in a number of papers (see [7-13] and Refs. herein) the gravitation theory in 4-dimensional Riemann-Cartan spacetime U4 - the Poncare gauge the-
e-mail: minkav@bsu.by, awm8matman.uwm.edu.pl
ory of gravity (PGTG) - offers opportunities to solve the PCS and also to explain the acceleration of cosmological expansion at present epoch without introducing the notion of dark energy (DE). First of all it should be noted that in the framework of gauge approach to gravitation the PGTG is a necessary generalization of metric theory of gravity if the Lorentz group is included to gauge group, which corresponds to gravitational interaction2). Let us to remind the most important physical results obtained in the frame of isotropic cosmology built in the frame of PGTG based on the gravitational Lagrangian Cg of general type including both a scalar curvature and different invariants quadratic in gravitational gauge field strengths - the curvature (-Fq/3m") an(l torsion (SQMv) tensors. Any homogeneous isotropic model (HIM) in the frame of PGTG is described by means of three functions of time - the scale factor of RobertsonWalker metrics R and two torsion functions Si and S2 determining non-vanishing components of torsion tensor (unlike Si the torsion function S2 is pseudoscalar with respect to spatial inversions). Two types of HIM were built and investigated: HIM with the only torsion function Si and HIM with two torsion functions. Isotropic cosmology based on HIM of the first type offers opportunities to solve the PCS [7-9]: all cosmological models filled with usual matter satisfying standard energy conditions (including inflationary models) are regular with respect to energy density, scale factor R with its time derivatives. However, the situation with DE in the case of these HIM becomes the same as in GR. Isotropic cosmology based on HIM with two torsion functions allows to build regular inflationary HIM and makes possible to
2'Prom this point of view namely the PGTG but not metric theory of gravity corresponds to supergravity theory.
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А. V. Minkevich
explain accelerating cosmological expansion at present epoch without introducing DE [10-13]. It is because the physical spacetime in the vacuum has the structure of de Sitter spacetime with non-vanishing torsion [13]. However, cosmological equations used in [10-12] do not exclude singular cosmological solutions, and the behaviour of cosmological solutions for flat HIM at asymptotics can differ from that of standard cosmological ACDM-model in dependence on initial conditions [14]. As it is shown in this Letter, by certain restrictions on indefinite parameters of gravitational Lagrangian PGTG allows to build totally regular isotropic cosmology for accelerating Universe, which quantitatively is in agreement at asymptotics with theory of standard cosmological ACDM-model.
2. Homogeneous isotropic models in PGTG. We will consider the PGTG based on the following expression of gravitational Lagrangian corresponding to spacial parity conservation (definitions and notations of [10] are used below):
Cg = [foF+F^'ihF^+frF^+hF^) + + + f6Fvlt) + f6F2 +
+ (aiSaitv + a2Svm) + a3S«aSf] . (1)
The Lagrangian (1) includes the parameter f0 = = (WirG)^1 (G is Newton's gravitational constant, the light velocity c = 1) and a number of indefinite parameters: fi (i = 1,2, ...,6) and as, (k = 1,2,3). Gravitational equations for HIM with two torsion functions corresponding to gravitational Lagrangian (1) allow to obtain cosmological equations generalizing Friedmann cosmological equations of GR and equations for torsion functions given in general form in [13]. These equations contain five indefinite parameters:
a = 2ai + a2 + 3a3, b = a2 - ai, f = fi + /2/2 + /3 + /4 + /5 + 3/6,
Qi = f2 — 2/3 + /4 + /5 + 6/e, <12 = 2/, /2,
and their mathematical structure and physical consequences depend essentially on restrictions on these parameters. Unlike metric gravitation theory, quadratic in the curvature terms of Cg do not lead to higher derivatives of R in cosmological equations; higher derivatives can appear because of terms of Cg quadratic in the torsion tensor; in order to exclude higher derivatives of R from cosmological equations we have to put the restriction a = 0 [13,15]. It should be noted that isotropic cosmology with a ^ 0 possesses some principal problems: in particular, cosmological equations at physically available initial conditions lead in this case to not physical solutions [16] and do not exclude singular solutions;
moreover, the presence of the seconde derivative of the Hubble parameter in cosmological equations leads to its oscillating behaviour at asymptotics [17]. The second restriction concerns the parameter q2: ii q2 0, the equation for the torsion function S2 is differential equation of the second order that leads to oscillating behaviour of the Hubble parameter [14]; by putting q2 = 0 we will obtain physically necessary consequences. Below we will analyze the main relations of isotropic cosmology given in [13] in general case without using any restrictions on indefinite parameters by putting the following restrictions: a = 0 and q2 = 0.
Cosmological equations generalizing Friedmann cosmological equations of GR take the following form:
A
R2
(H - 2Si
51 =
dfoZ
12/oZ
[p^66S22 + |(p^3p^l2bS22)2], (2)
H + Я2 - 2/751 - 25, =
[p + 3p^|(p^3p^l26S2)2] , (3)
where p is the energy density, p is the pressure, H = = R/R is the Hubble parameter (a dot denotes the differentiation with respect to time), the parameter a = = //3/q (/ > 0) has inverse dimension of energy density, and Z = l + a(p — 12bS2). The torsion function Si is determined by the following way:
Si =
a
4Z
\p - 3p + 12fowHS2 - 12(2b - wf0)S2S2],
(4)
where dimensionless parameter w = (2/ — <Zi)// 0 is introduced. The torsion function S| satisfies algebraic quadratic equation, which gives the following root
5| =
р^Зр 1 — (Ь/2/о)(1 + VX)
12 b 12be(l-w/4) '
(5)
where X = l+w(/02/62)[1^6//0^2(l^w/4)a(p+3p)] 3l In order to reduce cosmological equations (2), (3) to closed form we have to specify the content of HIM and its equation of state. In connection with this it should be noted that the matter content and its equation of state change during cosmological evolution and the form of equation of state depends on coupling of matter with gravitational field. In the case of usual gravitating matter with energy density pm > 0 and pressure pro > 0
3'lt seems that the second root for S| with opposite sign before
VX in (5) does not lead to physically satisfactory theory.
coupled minimally with gravitation the equation of state can be written in usual form: pm = pm(pm) and the law of energy conservation takes the form as in GR:
ЗЯ (pm + pm) = 0.
(6)
We introduce at early stage of cosmological expansion the scalar field 4> with potential V = V(<f>) as component of gravitating matter with the purpose to investigate inflationary HIM. By minimal coupling with gravitation the equation for scalar field takes the usual form as in GR:
dV
Шф=- — .
(7)
Then the total energy density p and pressure p are the following:
p=l^ + v + pm (p > 0), p=\i>2-V+pm. (8)
Now by using the formula (5) for torsion function Sf and Eqs. (6)-(8) we transform the torsion function S\ defined
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