Advances in High Energy Physics

Volume 2018, Article ID 6306848, 9 pages

https://doi.org/10.1155/2018/6306848

## Anisotropic Cosmological Models with Two Fluids

^{1}Department of Mathematics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad 500 078, India^{2}Centre of Theoretical Physics, Jamia Millia Islamia, New Delhi 110 025, India

Correspondence should be addressed to B. Mishra; moc.oohay@attuduvib

Received 25 July 2017; Revised 21 October 2017; Accepted 26 December 2017; Published 23 January 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 B. Mishra 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. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Anisotropic dark energy cosmological models have been constructed in a Bianchi V space-time, with the energy momentum tensor consisting of two noninteracting fluids, namely, bulk viscous fluid and dark energy fluid. Two different models are constructed based on the power law cosmology and de Sitter universe. The constructed model was also embedded with different pressure gradients along different spatial directions. The variable equation of state (EoS) parameter and skewness parameters for both models are obtained and analysed. The physical properties of the models obtained with the use of scale factors of power law and de Sitter law are also presented.

#### 1. Introduction

The most popular problem in modern cosmology has been invoked by the current discovery of accelerated expansion of the universe. This has been confirmed as an established fact through different observational data, such as Type Ia Supernovae (SNIa) [1, 2], CMB radiation [3–5], and gravitational lensing [6, 7]. This development is explained at the backdrop of general relativity (GR) through the introduction of an unknown energy source termed as dark energy (DE). This DE provides a repulsive gravity that helps in driving the acceleration by generating a strong negative force leading to an antigravity effect. In Friedman-Robertson-Walker (FRW) universes, viscosity appears as the only dissipate phenomenon, so a considerable amount of interest is seen in the study of cosmological models with bulk viscous fluid. In the inflationary phase, the contribution of bulk viscosity is well recognized, which gives rise to a negative pressure that simulates a repulsive gravity. The equation of state (EoS) parameter of the viscous fluid, having value lower than −1, is generally considered to be significant in the context of DE cosmology, which is also indictaed by observational results [3–5]. However, DE crossing phantom divide line having greater than or equal to 1 is lightly favoured. In consistency with the observational results, Copeland et al. [8] and Li et al. [9] have used the scalar field approach with an introduction of time-dependent EoS parameter to obtain the acceptable range for . Another way to achieve this result is to reveal the solutions of Einstein’s field equations by incorporating some kinematical assumptions, which are in consistency with the observed kinematics of the universe. As a testimony to this, Hubble parameter has been widely used to obtain explicit accelerating cosmological models in the framework of spatially homogeneous space-time [10].

It can be noted that dominance of an anisotropic stress gives rise to an anisotropic expansion. This dominance will have a considerable impact via anisotropic stress on cosmological evolutions such as magnetic fields, hydrodynamic shear viscosity, and collisionless relativistic particles [11, 12]. However, researches on DE with homogeneous and anisotropic space-time with time-varying EoS parameter observed that, at late time of cosmic evolution, DE yields isotropic pressure [13, 14]. On the other hand, several researchers [15–17] focused on the fact that Wilkinson Microwave Anisotropy Probe (WMAP) data [18, 19] requires Bianchi type morphology instead of Friedman-Robertson-Walker (FRW) type for better accurate explanation of the anisotropic universe. Campanelli et al. [20] revealed that, irrespective of the level of anisotropy in geometry of the universe and dark energy EoS, the SNIa data are always more consistent with standard isotropic universe.

Mishra et al. [21] have constructed the cosmological model based on pressure anisotropy in the presence of a gauge function, whereas Mishra et al. [22] have studied the anisotropic universe with general forms of scale factor. Several cosmological models were obtained with constant deceleration parameter, where the matter is in the form of perfect fluid or ordinary matter. However, many of those matters are not enough to describe the dynamics of an accelerating universe relating to anisotropy. This motivates us to consider the model of the accelerating universe filled with noninteracting fluids [23–25]. Akarsu and Kilinc [26, 27] have assumed constant deceleration parameter to construct and investigate DE models in Bianchi I and Bianchi III space-time. Yadav et al. [28] have assumed variable EoS parameter but constant deceleration parameter to construct the DE cosmological model in a locally rotationally symmetric Bianchi V space-time. Theoretical models of interacting and noninteracting DE have been discussed widely in the literature [29–31]. The paper is arranged as follows: in Section 2, a mathematical formalism of an anisotropic DE universe is presented along with the relevant physical parameters. Two dark energy cosmological models, one with power law cosmology and the other with de Sitter universe, have been constructed and analysed in Section 3. The summary is given in Section 4.

#### 2. Formalism

From an observational viewpoint, one of the most important results is the theorem of Wald [32], which states that universe with accelerating expansion tends towards isotropy at late phase. As a matter of fact, if the universe undergoes an early period of inflation, the present day universe will seem to be highly isotropic. Further, since the universe has now started accelerating, any kind of anisotropy will remain small in the late phase of cosmic acceleration. Bianchi universes are the class of cosmological models that are homogeneous but not necessarily isotropic on spatial slices. It contains, as a subclass, the standard isotropic model known as FRW universe. Calculations of nucleosynthesis and microwave background anisotropies in Bianchi models have been compared against data from the real universe, typically giving null results which can be translated into upper limits on anisotropy. Tentative detections of nonzero anisotropic shear by Jaffe et al. [33] are currently believed to be in consistency with other known cosmological parameters [34] and with polarization of the microwave background [35]. However, these models remain widely studied for their pedagogical value, mainly making them exact tractable solutions of Einstein’s field equation.

In the present paper, we are interested in studying the behaviour of anisotropy universe in the DE cosmological model. The standard FRW universe is homogeneous and isotropic. But, in order to address the small-scale anisotropy nature of the universe, Bianchi space-time is well accepted as it represents a globally hyperbolic spatially homogeneous but not isotropic space-time. Among all 9 space-times of Bianchi, Bianchi V space-time is very intuitive as it has more degrees of freedom characterized by Lie groups and generates pseudospherical space. Hence, in order to construct an anisotropic DE cosmological model in GR, we have considered here Bianchi V space-time in the formwhere , are the directional scale factors considered to be different along three orthogonal directions and thereby provide a source for anisotropic expansion. Here, we choose , and , with being a nonzero arbitrary constant. Assuming that GR is well defined at cosmic scales, we incorporate Einstein’s field equations,where , , , and , respectively, denote the Einstein tensor, Ricci tensor, Ricci scalar, and total effective energy momentum tensor (EMT) and . is the Newtonian gravitational constant and is the speed of light with . Here, EMT consists of two different components: the barotropic bulk viscous fluid and DE fluid . In case of barotropic cosmic fluid, the proper pressure is given as . The pressure with the contribution from bulk viscosity is also directly proportional to energy density; that is, , where is the proportionality constant [36, 37] and . Hence, the effects of both proper pressure and barotropic bulk viscous pressure together can be expressed aswhere is the bulk viscous coefficient. One can infer that such a relation bears similarity to the pressure term with the contribution from perfect fluid , where is the equation of state parameter for perfect fluid. However, a major part of the EoS in of the present model comes from barotropic bulk viscosity. So, in no viscosity condition, the pressure term reduces to the pressure of perfect fluid. Hence, the EMT for viscous fluid is given aswhere are the four velocity vectors of the fluid. It may be noted that there are no observational reasons to conclude that pressure is isotropic in DE. However, since the fluids are comoving, one may get this isotropic pressure in DE. Subsequently both the DE fluid and EoS parameter are direction-dependent. Hence, the EMT of DE fluid is considered in the formwhere is the EoS parameter of the DE fluid along the dimensional axes , , and . is the dark energy density. The deviations of from -axis, -axis, and -axis, respectively, denote the skewness parameters , , and . In the presence of EMT, Einstein’s field equations (2) corresponding to Bianchi type V space-time (1) lead to the following:where an over dot over the field variable represents the derivatives with respect to the cosmic time . Moreover, the unit of cosmic time is considered as follows: 1 unit of cosmic time = 10 billion years. The average scale factor and volumetric scale factor for the model are, respectively, and . The generalized mean Hubble parameter can be expressed as , where , and are the directional Hubble parameters in the directions of , , and , respectively. Now, the field equations (6)–(10) can be framed in Hubble terms as

The energy conservation equation for viscous fluid, , yieldsThe energy conservation equation for dark energy fluid, , yieldsFrom (16), incorporating the relation between Hubble parameter and average scale factor, we getwhere is the integration constant or rest energy density.

From (14), we haveIn order to solve (17), we split the conservation equation into two parts: one corresponds to the deviation of equation of the state parameters as and other is the deviation-free part as [26]. It can be observed that the behaviour of energy density is controlled by the deviation-free part of EoS parameter, whereas anisotropic pressure along different directions can be obtained from second part of the above conservation equation as it corresponds to the conservation of matter field with equal pressure along all directions. Hence, we obtained the dark energy density as

Now, from (13), incorporating the value of , we get

Again, from (10), with the choice of integrating constant to be unity, we get . Moreover, for an anisotropic relation, we assume that , where is the average anisotropy parameter [38]. Hence,

Now, the dark energy density and effective EoS parameter with the function can be reformulated, respectively, as

With the help of (18)–(21), (11)–(13) can be expressed in functional form aswhere .

#### 3. Cosmological Models and Their Behaviours

From the above formalism, it is quite clear that obtaining an exact solution to the field equations is a cumbersome process. Therefore, without violating any physical meaning of the expression and in order to study the cosmological model in this formalism, we have assumed two scale factors: one leads to power law expansion and the other leads to de Sitter expansion.

##### 3.1. Power Law Expansion Model

Recently, many observational results as well as experiments predict a tensor-to-scalar ratio that provides convincing results for standard inflationary scenario even though the value of the ratio contradicts the limits from Planck data. During a power law expansion, the inflationary scenario predicts the generation of gravitational waves. In this model, the scale factor for power law cosmology can be represented as , where is a positive constant and . Also, and are positive constants. Now, the volume scale factor and Hubble parameter . It is obvious that, for , the model will be an accelerating one. Now, subsequently, the directional Hubble parameters can be obtained as , , and and consequently the mean deceleration parameter becomes . The deceleration parameter is a negative constant quantity for , since and are positive constants and this is in good agreement with the present observational data that predicts an accelerating universe; therefore, in order to get an accelerating model with this power law scale factor, the exponent if ; otherwise it has to be decided from .

The universe, in general, is isotropic; but, according to the observational results of CMB temperature anisotropy, a small amount of anisotropy in the universe cannot be ruled out. However, any anisotropy in spatial expansion must be considered as a little perturbation of the isotropic behaviour, which suggests that the exponent must be close to 1. In fact, according to the present result from the analysis of anisotropy as predicted from Planck data [39, 40] and from our earlier work, [21, 38]. The power law model is quite successful in the sense that it neither encounters the horizon problem nor witnesses the flatness problem with . The energy density contribution coming from the usual cosmic fluid for the power law model reduces to

Now, with the help of (26), the dark energy density and dark energy EoS parameter as described in (23) and (24) can be, respectively, reduced towhere . The skewness parameters , , and reduce towhere . It is seen that both and decrease with the increase in time. The decrease in is decided by three different factors, that is, in the first term, in the second term, and in the third term. The role of bulk viscous cosmic fluid comes through the third term. One may note that if , even though the contribution coming from the usual cosmic fluid does not vanish, it does not contribute to the time variation of the dark energy density. For , the time variations of second and third terms can be clubbed together. Consequently, for this choice, the skewness parameters become constant quantity and appear to be a simple time-independent deviation from the usual isotropic pressure.

The figures in the manuscript have been drawn for different physical quantities which are expressed in Planckian unit system . Also, 1 unit of cosmic time = 10 billion years. In Figures 1 and 2, we have observed, respectively, that the matter energy density and the dark energy density remain positive during the cosmic evolution for the representative value of the constants (). Hence, it indicates that both weak energy condition (WEC) and null energy condition (NEC) are satisfied in the derived model. Further, both and decrease with increase in time and slowly reach small positive values in the present epoch. The value of dark energy density comes closer to zero and then smoothly approaches small positive value, which indicates that the considered two fluids affect the dark energy density. It is worth noting here that, irrespective of the value of the viscous coefficient , the behaviour of remains alike. So, in Figure 2, we have chosen the value of the viscous coefficient to be . However, a small effect of viscous fluid in dark energy density cannot be ruled out. Figure 3 represents the variation of with cosmic time for different values of viscous coefficients . The range value of EoS parameter suggested by combination of SNIa data with CMB anisotropy and galaxy clustering statistics is [10], whereas the range suggested by recent observations is reduced to more stringent constraints around −1 [34, 41, 42]. However, we consider here the earlier data range, since power law behaviour dominates the cosmic dynamics in early phase of cosmic evolution [38]. For , Figure 3 clearly shows that evolves within a range, which is almost aligned with SNIa and CMB observations. Moreover, it is observed that when the bulk viscous coefficient increases, the EoS parameter gradually converges to at late time.