on the ratio of shear, a, to Hubble constant, H, in the neighbourhood of our Galaxy ... 8~rp - 3K22 [3~0p TM (g3 + 3n + 4k2e -16~"t) + 2~/(2/£3 + 5k2e-X6"vt)].
Viscous fluid cosmological models in LRS Bianchi type V universe with varying A ANIRUDH PRADHAN*) Department of Mathematics, Hindu Post-graduate College, Zamania-232 331, Ghazipur, India LALLAN YADAV, ANIL KUMAR YADAV Department of Physics, K.N. Post-graduate College, Sant Ravidas Nagar (Gyanpur), Bhadohi - 221 304, India Received 13 October 2003 Some LRS Bianchi type V viscous-fluid cosmological models are investigated, in which the coefficient of shear viscosity is considered as proportional to the scale of expansion in the model. This leads to A = B '~, where A and B are metric potentials, n being a constant. The coefficient of bulk viscosity is also assumed to be a power function of mass density. The cosmological constant is found to be a decreasing function of time, which is supported by results from recent type Ia supernovae observations. Some physical aspects of the models are also discussed. PACS: 98.80.Es, 98.80.-k Key words: cosmology, variable cosmological constant, viscous-fluid models, early universe 1
Introduction
The homogeneous and isotropic Friedman-Robertson-Walker (FRW) cosmological models, which are used to describe standard cosmological models, are particular case of the Bianchi type I, V, and IX space-time, according to whether the constant curvature of the physical three-space, t = constant, is zero, negative or positive. In these models, neutrino viscosity explains the large radiation entropy in the universe and the degree of isotropy of the cosmic background radiation. The standard cosmological models are too restrictive because of the insistence on the isotropy of the physical three-space and several attempts have been made to study non-standard cosmological models e. g. MacCallum [1], Narlikar and Kembhavi [2], and Narlikar [3]. It is therefore interesting to carry out detailed studies of gravitational fields which are described by space-time of various Bianchi types. When the Bianchi type V space-time expands equally in two spatial directions, it is called locally rotationally symmetric Bianchi type V space-time. The LRS type V anisotropic spacetimes are spatially homogeneous, that is to say, there exists a three-parameter Lie group of isometrics acting transitively on space-like hyper-surface. In addition, they admit a one-parameter isotropy group, whose orbits are closed and lie in the space-like hyper-surfaces. This is why these space-times are called locally rotationally symmetric. It should be noted that the full (four-parameter) isometry group thus acts multiply transitively on the space-like hyper-surfaces. These hyper-surfaces can be * ) E-mail: acpradhan©yahoo, corn, pradhan@iucaa, ernet, in
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labeled by a cosmic time t and are also referred to as homogeneous hyper-surfaces. The LRS Bianchi type V cosmologies have commanded the attention of many researchers during the past decades. The reason is that, while their dynamical behaviour can be far more general than that of the simple Robertson-Walker and Bianchi type I models, they are still less complicated than Bianchi type VI-IX. These kinds of models are also interesting, because Lidsey [4] showed that they are equivalent to a flat (FRW) universe with a self-interacting scalar field and a free massless scalar field, but produced no explicit example. Some explicit solutions were pointed out by Aguirregabiria et al. [5, 6]. For simplification and description of the large-scale behaviour of the actual universe, LRS Bianchi V spacetime have been widely studied by many researchers [7-13]. By this motivation we intend to investigate an LRS Bianchi V cosmological model in the presence of viscous fluid. Models with a dynamic cosmological term A(t) are becoming popular as they solve the cosmological-constant problem in a natural way. There is significant observational evidence for the detection of Einstein's cosmological constant, A or a component of material content of the universe, that varies slowly with time and space and so acts like A. Recent cosmological observations by High-z Supernova Team and Supernova Cosmological Project (Gaxnavich et al. [14], Perlmutter et al. [15], Riess et al. [16], Schmidt et al. [17]) suggest the existence of a positive cosmological constant A with the magnitude A ( G 5 / c 3) ~ 10 -123. These observations on magnitude and red-shift of type Ia supernova suggest that our universe may be an accelerating one with a large function of the cosmological density in the form of the cosmological A-term. Earlier researches on this topic, are contained in Zeldovich [18], Weinberg [19], Dolgov [20], Sertolami [21], Ratra and Peebles [22], Carroll, Press and Turner [23]. Some of the recent discussions on the cosmological-constant "problem" and consequence on cosmology with a time-varying cosmological-constant have been discussed by Dolgov [24], Tsagas and Maartens [25], Sahni and Starobinsky [26], Peebtes [27], Padmanabhan [28], Vishwakarma [29], and Pradhan et al. [30]. This motivates us to study the cosmological models, where A varies with time. In this paper, motivated by the situations discussed above, we shall focus upon the problem with varying cosmological constant in presence of bulk and shear viscous fluid. We do this by extending the work of Bali and Yadav [12] by including varying cosmological constant and the coefficient of bulk viscosity as function of time. This paper is organized as follows. The metric and the field equations are presented in Section 2. In Section 3, we deal with the solution of the field equations in presence of viscous fluid. Section 4 includes the solution of some other models, whereas in Section 5, we deal with the model in absence of viscosity. In Section 6, we give the concluding remarks. 2
Field e q u a t i o n s
We consider LRS Bianchi type V metric in the form ds 2 = - d t 2 + A2dx 2 + BVe2X(dy2 + dz2),
(1)
where A and B are functions of t only. 488
Czech. J. Phys. 54 (2004)
Viscous fluid cosmological models
.
.
.
The Einstein's field equations (in gravitational units c -- 1, G = 1) read
R{ - -~R~ + Ag{ : -8~T~,
(2)
where R~ is the Ricci tensor, R = giJR/j is the Ricci scalar, and T~ is the stress energy tensor in the presence of bulk stress given by Landau and Lifshitz [31]:
T/ : (p + ;)v~vJ + p~ - ,(v~; + ~J;, + ~ v % , + v~v'vJ;L)
- (~- ~,7) o(g~, + v~v~) .
(3)
Here p, p, 7/ and ~ are the energy density, isotropic pressure, coefficient of shear viscosity and bulk viscous coefficient respectively and vi is the flow vector satisfying the relations gijviV j = --1. (4) The semicolon (;) indicates covariant differentiation. We choose the coordinates to be comoving, so that v l = 0 = v 2 = v 3, v 4 = 1 . (5) The Einstein's field equations (2) for the line element (1) has been set up as
o =-y+.2
A2+A'
0 =
A2+A' 2AB
8rp =
AB
B2 + B2
(6) (7)
3 A ~ + A'
(8)
where the dot at the symbols A and B denotes ordinary differentiation with respect to t and 0 is the scalar of expansion given by 0 = v ;i. 3 3.1
General
The
(9)
model
features
Equations (6)-(8) are three independent equations in seven unknowns A, B, p, p, rh ~ and A. For the complete determination of the system, we need four extra conditions. Referring to Thorne [32], current observations of the velocity-redshift relation for extragalactic sources suggest that Hubble expansion of the universe is isotropic today to within ~ 30 per cent [33, 34]. Put more precisely, redshift studies place the limit (7
--
H-
Czech. J. Phys. 54 (2004)
