Journal of Modern Physics
Vol.4 No.8(2013), Article ID:35793,4 pages DOI:10.4236/jmp.2013.48139

LRS Bianchi Type-I Cosmological Model with Anisotropic Dark Energy and Special Form of Deceleration Parameter

Kishor Shankarrao Adhav, Rajesh Purushottam Wankhade, Abhijit Shankarrao Bansod

Department of Mathematics, Sant Gadge Baba Amravati University, Amravati, India


Copyright © 2013 Kishor Shankarrao Adhav et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received April 24, 2013; revised June 2, 2013; accepted July 9, 2013

Keywords: LRS Bianchi Type-I Space-Time; Special Form of Deceleration Parameter; Anisotropic Fluid; Dark Energy; Isotropization


We have studied Locally Rotationally Symmetric (LRS) Bianchi type-I cosmological model filled with anisotropic fluid in general theory of relativity. The solutions of the field equations are obtained by using special form of deceleration parameter which gives early deceleration and late time accelerating cosmological model. The geometrical and physical aspect of the model is also studied.

1. Introduction

Our universe is undergoing a late-time accelerating expansion which has been evidenced by Riess et al. [1], Bahcall et al. [2], Bennett et al. [3], Spergel et al. [4], Cunha [5]. We live in a spatially flat universe composed of (approximately) 4% baryonic matter, 22% dark matter and 74% dark energy. Recently, Li et al. [6] studied the present acceleration of the universe by analyzing the sample of baryonic acoustic oscillation (BAO) with cosmic microwave background (CMB) radiation and concluded that such sample of BAO with CMB increases the present cosmic acceleration which has been further explained by plotting graphs for change of deceleration parameter q with redshift.

Many authors suggested a number of ideas to explain the current accelerating universe, such as scalar field model, exotic equation of state (EoS), modified gravity, and the inhomogeneous cosmology model. The dark energy EoS parameter is, where p is the darkenergy pressure and is its energy density. The value is necessary for comic acceleration. The simplest candidate for dark energy is the cosmological constant () for which. The matter with gives rise to Big Rip singularity (Caldwell [7]). Elizalde et al. [8] and Nojiri et al. [9] proposed several ideas to prevent the Big Rip singularity by introducing quantum effect terms in the action. Recently, Astashenok et al. [10] studied phantom cosmology without Big Rip singularity.

In the present paper, we have considered a LRS spatially homogeneous and anisotropic Bianchi type-I cosmological model with special form of deceleration parameter in general relativity. The physical and geometrical aspects of the model are also discussed. To have a general description of an anisotropic dark energy component, we consider a phenomenological parameterization of dark energy in terms of its equation of state and skewness parameter. Some features of the evolution of the metric and the dynamics of the anisotropic dark energy fluid have been also examined. This paper is organized as follows. In Section 2, we have given the line element, energy momentum tensor and its parametrization and the field equations. In Section 3, isotropization and solutions are given. The physical and geometrical parameters such as the anisotropy parameter of expansion, the energy density, the deviation-free EoS parameter and the deviation parameter etc are also studied with proper interpretation.

2. Metric and Field Equations

The LRS Bianchi type-I line element is given by


where and are the scale factors (metric potential) and functions of the cosmic time t only (nonstatic case).

Here we are dealing only with an anisotropic fluid whose energy-momentum tensor is in the following form


We parametrize it as follows:


where is the energy density of the fluid, is the equation of state (EoS) parameters of the fluid and is the skewness parameter.

Here and are not necessarily constants and can be taken as functions of the cosmic time t.

The Einstein field equations, in natural limits (and) are


The Einstein’s field Equations (2.3) for metric (2.1) with the help of Equations (2.2) give




where dot indicates the derivative with respect to t.

3. Isotropization and the Solutions

We have three linearly independent Equations (2.4)-(2.6) with five unknowns and. In order to solve this system completely, we use a special form of deceleration parameter as


where is mean scale factor of the universe and (>0) is constant.

This form has been proposed by Singha and(2.3)

The Einstein’s field Equations (2.3) for metric (2.1) with the help of Equations (2.2) giveal equation of state (EoS) parameter.

From Equation (3.1) after integrating, we obtain the Hubble parameter as


where m is an arbitrary constant of integration.

Integrating twice Equation (3.1), we get and the average scale factor as


The spatial volume is given by




here we consider.

The mean Hubble parameter for LRS Bianchi type-I metric may be given by


The directional Hubble parameters in the direction, and respectively can be defined as


Subtracting Equation (2.5) from Equation (2.6), we get


Now, from Equations (3.4) and (3.8), we get


Integrating, this gives


= constant of integration.

In order to solve above Equation (3.10), we use the condition


Using Equation (3.11) in Equation (3.10), we obtain


Now integrating Equation (3.12) and using Equation (3.5), we obtain the scale factors as



Using Equations (3.13) and (3.14) the directional Hubble parameters are found as



The mean Hubble parameter for LRS Bianchi type-I metric may be given by


The anisotropy parameter of the expansion is defined as

where represent the directional Hubble parameters in the directions of x, y and z axis respectively and is found as


The shear scalar , defined by is found as


Using Equations (3.15), (3.16) and (2.4) we obtain the energy density as


Using Equations (3.16) and (3.20) in Equation (2.5), we obtain the deviation-free parameter as


Now using Equation (3.11), we obtain the deviation parameter as


The expansion scalar is found to be


4. Discussion

1) From Figure 1, one can verify that q decreases from to during evolution of the universe.

2) The dynamics of energy density for LRS Bianchi type-I metric is illustrated in Figure 2. Here one can observe that all models [for different] start with Big Bang having infinite density and as time increases (for finite time), the energy density tends to a finite value. Hence, after some finite time, the models approach to a steady state.

3) In Figure 3, we have plotted anisotropy parameter of expansion (D) against cosmic time t. For LRS Bianchi type-I model, it is observed that anisotropy parameter decreases to zero after some time. Hence, the model reaches to isotropy after some finite time which matches with the recent observations as the universe is isotropic at large scale.

4) The evolution of expansion scalar for is shown in Figure 4. It is observed that the expansion is infinite at. As cosmic time t increases, it decreases to a finite value after some finite value of t.

5. Conclusions

We have verified that the energy density of the fluid, the deviation-free equation of state parameter and the deviation parameter are all dynamical.

Figure 1. The variation of vs for.

Figure 2. Energy density vs cosmic time.

Figure 3. Anisotropy parameter vs cosmic time for.

Figure 4. Expansion scalar vs cosmic time.

It is observed that when, we get, and.

Which is mathematically equivalent to the cosmological constant (L CDM) model.

The SNe Ia data (Riess et al. [13], Astier et al. [14], Riess et al. [15]), the SDSS data (Eisenstein et al. [16]), and the three year WMAP data (Spergel et al. [17]) all indicate that the model or the model that reduces to model is described as a standard excellent model in cosmology to describe the cosmological evolution. Hence, one can conclude that LRS Bianchi type-I cosmological model is the best fitted model as it reduces to model.

6. Acknowledgements

Authors are thankful to UGC, New Delhi for financial assistance through Major Research Project.


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