A Toy Model of Universe

J. Naji, R. Darabi, S. Heydari

Iran Department of Physics, Ilam Branch, Islamic Azad University, Ilam, Iran.
Physics Department, Ilam Branch, Islamic Azad University, Ilam, Iran.
Young Researchers and Elite Club, Islamic Azad University, Mahabad, Iran.

DOI: 10.4236/ojm.2014.41001                

Abstract

In this paper, we suggest that a toy model of our universe is based on FRW bulk viscous cosmology in presence of modified Chaplygin gas. We obtain modified Friedman equations due to bulk viscosity and Chaplygin gas and calculate time-dependent energy density for the special case of flat space.

Keywords

FRW Cosmology; Modified Chaplygin Gas; Bulk Viscosity

On the other hand, we know that the viscosity plays an important role in the cosmology [20]. In another word, the presence of viscosity in the fluid introduces many interesting pictures in the dynamics of homogeneous cosmological models, which is used to study the evolution of universe. In Ref. [21], the exact solutions of the field equations for a five-dimensional space-time with viscous fluid were obtained. Also in Ref. [22] a cosmological model with viscous fluid in higher-dimensional spacetime was constructed. Then, in Ref. [23] the exact solutions of the field equations for a five-dimensional cosmological model with variable bulk viscosity were obtained. The isotropic homogeneous spatially flat cosmological model with bulk viscous fluid was constructed in Ref. [24]. The bulk viscous cosmological models with constant bulk viscosity coefficient were constructed in Ref. [25]. In the recent work [26] the FRW bulk viscous cosmology was considered and bulk viscous coefficient was obtained in the flatspace, and then extended to non-flat space [27]. In this work, we consider both bulk viscous effect and Chaplygin gas in FRW cosmology in flat space.

2. Equations

The Friedmann-Robertson-Walker (FRW) universe in four-dimensional space-time is described by the following metric [28,29],

(1)

where, and represents the scale factor. The and are the usual azimuthal and polar angles of spherical coordinates. Also, constant denotes the curvature of the space. In this paper we consider the case of only, which is corresponding to flat space. In that case the Einstein equation is given by,

(2)

where we assumed and. Also the energy-momentum tensor corresponding to the bulk viscous fluid and modified Chaplygin gas [30-35] is given by the following relation,

(3)

where is the energy density and is the velocity vector with normalization condition. Also, the total pressure and the proper pressure involve bulk viscosity coefficient and Hubble expansion parameter are given by the following equations [36-42],

(4)

and,

(5)

with and. The equation of state is one of the most important quantity to describe the features of dark energy models. It is clear that the parameter shows bulk viscosity and B shows effect of Chaplygin gas. In the Ref. [43] the dynamics of FRW cosmology with modified Chaplygin gas as the matter formulated. Then the nature of the critical points are studied by evaluating the eigenvalues of the linearized Jacobi matrix for the special case of. In this paper we consider special case with and extend the Ref. [43] to including bulk viscous coefficient.

In that case the Friedmann equations are given by,

(6)

and,

(7)

where dot denotes derivative with respect to cosmic time. The energy-momentum conservation law obtained as the following,

(8)

In the next section we try to obtain time-dependent density by using above equations.

3. Solutions

Using the Equations (4)-(6) in the conservation relation (8) we have,

(9)

If we set, then one can extract energy density depend on scale factor [43],

(10)

where c is an integration constant. Here we also consider bulk viscous coefficient and would like to obtain energy density depend on time. In order to solve Equation (9) we use the following ansatz,

(11)

where constants, , , and should be determined. Substituting relation (11) in the Equation (9) gives us the following coefficients,

(12)

(13)

(14)

(15)

If we neglect both bulk viscosity and presence of Chaplygin gas then,

(18)

which is agree with results of the Refs. [27,43] where established. On the other hand for the large bulk viscosity coefficient one can find that and hence obtained. Also for the case of infinitesimal

(16) where, (17)

one can obtain constant negative energy density. In the general case, Equation (11) with coefficients (12)-(16) tells us that the energy density is decreasing function of time. Such behavior happen for the Hubble expansion parameter which is discussed below.

By using time-dependent density in the relation (6) one can obtain Hubble expansion parameter. In that case we draw plot of Hubble expansion parameter in the Figure 1 for.

In that case the modified Chaplygin gas model describes the evolution of the universe from the radiation regime to the Λ-cold dark matter scenario, where the fluid behaves as a cosmological constant, so there is an accelerated expansion of the universe.

It is possible to study deceleration parameter of this theory which obtained by the following relation,

(19)

Numerically we draw deceleration parameter in terms of time in the Figure 2.

4. Conclusions

In this work, we studied the FRW bulk viscous cosmology with modified Chaplygin gas as the matter contained. We obtained the modified Friedmann equations due to bulk viscous and Chaplygin gas coefficients. Then tried to solve equations and found time-dependent energy density. Therefore, we could extract Hubble expansion and deceleration parameters.

Figure 1. Hubble expansion parameter in terms of time for, , and.

Figure 2. Deceleration parameter in terms of time for and. Solid, dotted, dashed and dash dotted lines represent respectively.

For the future work, it is possible to repeat calculation of this paper for the case of arbitrary α or non-flat universe where. In that case one deals with the following equation,

(20)

where.

Conflicts of Interest

The authors declare no conflicts of interest.

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