Anisotropic Universes Sourced by Modified Chaplygin Gas

Abstract

In this work, we perform a comparative study of the Kantowski–Sachs (KS) and Bianchi-I anisotropic universes with Modified Chaplygin gas (MCG) as matter source. We obtain the volume and scale factors as solutions to the Einstein Field Equations (EFEs) for the anisotropic universes, and check whether the initial anisotropy is washed out or not for different values of the MCG parameters present in the solution by obtaining the anisotropy parameters for each solution. The deceleration parameter is also obtained for each solution, the significance of which is discussed in the concluding section. Interestingly there are a number of notable results that appear from our study which help us to compare and contrast the two different anisotropic models along with proper understanding of the role of MCG as matter source in these models.

1. Introduction

The conditions of homogeneity and isotropy are observed to be satisfied for the present universe on a large scale. The standard cosmological model expects that the observed large scale structure has grown under gravity in the sea of dark matter and with an initial Gaussian random density fluctuation of almost scale-invariant power spectrum. Over a period of time, the standard model and the cosmic isotropy have been tested through the Cosmic Microwave Background (CMB) radiation and the Large Scale Structure (LSS) data.

However, in the very early universe, when the energy densities were considerably higher, it is generally believed by cosmologists that the universe was highly anisotropic. Also, certain anomalies such as the lack of correlations on large angular scales, a statistically significant alignment and planarity of the CMB quadrupole and octopole modes and the observed large scale alignment in the quasar polarization vectors imply a possible violation of the statistical isotropy and the break down of the cosmological principle (see [1,2,3,4,5]). Nevertheless, there is no conclusive evidence for the cosmological isotropic hypothesis [4]. Also, there is no consensus on the physical nature and origin of the scale-invariance of the primordial perturbations [6]. However, it is believed that, with the growth of cosmic time, the anisotropy may be washed out to leave the universe isotropic to a greater extent. This implies that there must have been some mechanism of washing out this initial anisotropy with time [7]. Also, it may be inferred that the initial conditions of the universe do not influence it is present state.

A statically isotropic universe is usually modelled through an FRW metric. In order to account for the possible anisotropy in the space time, we will be concerned with two such anisotropic frameworks, namely the KS [8] and Bianchi-I. There are a number of Bianchi models in cosmology, in accordance with the corresponding classification of 3D spaces by Bianchi [9]. Bianchi’s classification was first applied to GR by Ellis and others [10,11,12]. The cosmology of Bianchi-I models has been studied in [5,13,14,15,16,17] whereas KS cosmology has been investigated in [18,19,20,21].

In the present work, along with the anisotropic spatial expansion, we consider the MCG as the source of matter field. The MCG is a generalized and modified form of the Chaplygin gas (CG), which was first proposed as an alternative to quintessence for explaining the accelerated expansion of the universe [22] and later on carried out work by several scientists [23,24,25]. The CG is a perfect fluid having Equation of State (EOS)

$$p=-\frac{A}{\rho },$$

(1)

where A is a positive constant and p and $\rho$ denote the usual pressure and energy density, respectively.

The fact that CG is connected to the d-branes appearing in String theory makes it even more interesting. The origin of CG in the universe has been addressed in [26]. The EOS for CG can be generalized to

$$p=-\frac{A}{{\rho }^{\alpha }},$$

(2)

where $\alpha$ can have any value between 0 and 1.

The above EOS describes an exotic fluid known as Generalized Chaplygin Gas (GCG) which behaves like pressureless dust in the early universe and like a Cosmological constant $\Lambda$ in the late universe [23]. Even this EOS was further modified to

$$p=B\rho -\frac{A}{{\rho }^{\alpha }},$$

(3)

where both A and B are positive constants. This is known as the modified Chaplygin gas [27,28].

It noteworthy that in the past two decades, the MCG has been used extensively in cosmology. Not only can it be used to describe the accelerated expansion of the universe in recent times [29,30,31], but also has been used to construct unified models of dark matter and dark energy [32,33,34,35], and also to explain early universe phenomena [36,37]. MCG has also been found to be consistent with observational tests like gravitational lensing test [38,39] and gamma ray bursts [40]. The observational constraints on MCG have also been studied in some detail [41,42,43,44,45,46]. In a recent work, Benaoum et al. [47] have constructed some modified class of MCG models using geometrothermodynamics those show internal thermodynamic interactions. In another work, Benaoum and collaborators have described the MCG models as FRW models mediated through scalar fields and obtained the self interacting potential through which they could able to discuss the evolution of the tachyon field [48]. Extended Chaplygin equations of state have been used in recent times to address some of the issues in cosmology and astrophysics [49,50,51]. Abelian et al. [52] have used a double Chaplygin model to investigate astrophysical systems in Newtonian regime using anisotropic matter.

We organize the paper as follows. In Section 2 we consider the solutions for KS universe sourced by MCG followed by the solutions for Bianchi-I universe with MCG in Section 3. We conclude with discussion on the results obtained.

2. KS Model

The line element for the KS universe is given as [8]

$$d{s}^2=-d{t}^2+a_1{\left(t\right)}^{2}d{r}^2+a_2{\left(t\right)}^2(d{\theta }^2+si{n}^2\theta d{\varphi }^2).$$

(4)

The EFE for the above line element with MCG as source can be written as (following the geometrized unit we take $8\pi G=c=1$ throughout the paper)

$$\frac{{\dot{a_2}}^2}{a_2^2}+2\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}+\frac{1}{a_2^2}=\rho ,$$

(5)

$$2\frac{\ddot{a_2}}{a_2}+\frac{{\dot{a_2}}^2}{a_2^2}+\frac{1}{a_2^2}=\frac{A}{{\rho }^{\alpha }}-B\rho ,$$

(6)

$$\frac{\ddot{a_1}}{a_1}+\frac{\ddot{a_2}}{a_2}+\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}=\frac{A}{{\rho }^{\alpha }}-B\rho .$$

(7)

Equation (7) can be rewritten as

$$2\frac{\ddot{a_1}}{a_1}+2\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}-\frac{{\dot{a_2}}^2}{a_2^2}-\frac{1}{a_2^2}=\frac{A}{{\rho }^{\alpha }}-B\rho .$$

(8)

From Equations (5) and (8), we obtain

$$\frac{d\left(V{H}_1\right)}{dt}=\frac{(1-B)}{2}\rho V+\frac{AV}{2{\rho }^{\alpha }},$$

(9)

where V is the volume given by $V=a_1{a}_2^2$ and $H_1$ is one of the directional Hubble parameters given as $H_1=\frac{\dot{a_1}}{a_1}$.

The conservation equation has the usual form

$$\dot{\rho }+3H(p+\rho )=0,$$

(10)

where the Hubble parameter H is given by $H=\frac{1}{3}(H_1+2H_2)$, $H_2$ being the second directional derivative $H_2=\frac{\dot{a_2}}{a_2}$.

Equation (10) gives the solution for the energy density as

$$\rho ={\rho }_0{\left[A_s+\frac{1-A_s}{V^{\left(1+B\right)(1+\alpha )}}\right]}^{\frac{1}{1+\alpha }},$$

(11)

where $A_s=\frac{A}{1+B}\frac{1}{{\rho }_0^{\left(\alpha +1\right)}}$ and $B\ne -1$. As we are considering anisotropic space-time, we must be in the very early epoch, when the volume is such that $V\ll 1$. Since B and $\alpha$ are both positive, the factor in the denominator of the second term of RHS of Equation (11) is much less than 1, making the second term dominant at early times. So, we obtain approximate $\rho$ from Equation (11) of the form

$$\rho \simeq \frac{C}{V^{1+B}},$$

(12)

where $C={\rho }_0{(1-A_s)}^{\frac{1}{1+\alpha }}$.

Using this form of $\rho$, Equation (9) can be written as

$$\frac{d\left(V{H}_1\right)}{dt}=\frac{(1-B)}{2}C{V}^{-B}+\frac{A}{2C^{\alpha }}{V}^{\alpha (1+B)+1}.$$

(13)

Similarly, from Equations (5), (6) and (12), we obtain

$$\frac{d\left(V{H}_2\right)}{dt}=\frac{(1-B)}{2}C{V}^{-B}+\frac{A}{2C^{\alpha }}{V}^{\alpha (1+B)+1}-a_1.$$

(14)

In the above Equations (13) and (14) are the EFE in terms of the volume.

From these equations, we obtain:

$$H_1=H-\frac{2K}{3V},H_2=H+\frac{K}{3V},$$

(15)

where $K=\int a_{1}dt$.

The third EFE in alternative form may be written using Equations (5)–(7) as

$$\begin{array}{c}\hfill 3\dot{H}+H_1^2+2H_2^2=\frac{3A{V}^{(1+B)\alpha }}{{\rho }_0^{\alpha }{(1-A_s)}^{\frac{\alpha }{1+\alpha }}}-{\rho }_0(3B+1){(1-A_s)}^{\frac{1}{1+\alpha }}{V}^{-(1+B)}.\end{array}$$

(16)

From the EFE in terms of volume, we obtain

$$\ddot{V}=\frac{3}{2}(1-B)C{V}^{-B}+\frac{3}{2}\frac{A}{C^{\alpha }}{V}^{\alpha (1+B)+1}-2a_1.$$

(17)

This gives $\dot{V}$ of the form

$$\dot{V}=\sqrt{3C{V}^{1-B}+3\frac{A}{C^{\alpha }}\frac{V^{\alpha (1+B)+2}}{\alpha (1+B)+2}-4\int a_{1}dV}.$$

(18)

From the above expressions, it can be observed that the solutions for the volume and scale factors depend on the MCG parameters $\alpha$, A, and B. However, the dependence on the B-parameter is found to be the strongest, so we choose to vary it for obtaining different solutions to check the anisotropic behaviour and accelerated expansion behaviour of the model. We will now consider three different values of the parameter B to assess its effect on the exact solutions.

Case: $B=1$.

We can write

$${\dot{V}}^2=3C+3\frac{A}{C^{\alpha }}\frac{V^{2\alpha +2}}{2\alpha +2}-4\int a_{1}dV.$$

(19)

Using small V approximation, i.e., considering the early universe, we obtain the solution as

$$V=C_1+\sqrt{3C+b}t,$$

(20)

where b is a constant and $C_1$ is integration constant.

Using Equation (15), we now obtain the solutions for the scale factors as

$$a_1=C_2{[C_1+\sqrt{3C+b}t]}^{\frac{1}{3}-\frac{2K}{3\sqrt{3C+b}}},$$

(21)

$$a_2=\frac{1}{\sqrt{C_2}}{[C_1+\sqrt{3C+b}t]}^{\frac{1}{3}+\frac{K}{3\sqrt{3C+b}}},$$

(22)

where $C_2$ is a constant.

We now consider

$$\Delta H_1=H_1-H,\Delta H_2=H_2-H.$$

(23)

For an anisotropic universe like the KS universe, we have directional scale factors representing different expansion rates along different spatial directions. This incorporates anisotropy in the expansion rates and, correspondingly, we have directional Hubble rates $H_1$ and $H_2$, leading to an anisotropy parameter $\mathrm{\Gamma }$ defined as [53,54,55]

$$\mathrm{\Gamma }=\frac{1}{3}\left[{\left(\frac{\Delta H_1}{H}\right)}^2+2{\left(\frac{\Delta H_2}{H}\right)}^2\right].$$

(24)

Using the above obtained scale factors, we find the anisotropy parameter to be

$$\mathrm{\Gamma }=\frac{2b}{(3C+b)},$$

(25)

which is a constant.

The average scale factor can be written as

$$\mathcal{R}={\left(a_1{a}_2^2\right)}^{\frac{1}{3}},$$

(26)

which, for the present case, becomes $\mathcal{R}={\left[C_1+\sqrt{3C+b}t\right]}^{\frac{1}{3}}.$ Assuming that the volume of the universe is negligibly small at the beginning, we may set $C_1=0$, so that the scale factor becomes $\mathcal{R}={[3C+b]}^{\frac{1}{6}}{t}^{1/3}$, for which the Hubble rate becomes $H=\frac{1}{3t}$.

Hence, we can define the deceleration parameter q as

$$q=-\frac{\mathcal{R}\ddot{\mathcal{R}}}{{\dot{\mathcal{R}}}^2}.$$

(27)

Alternatively, it may also be computed as

$$q=-1-\frac{\dot{H}}{H^2}.$$

(28)

Using either of the above relations, we obtain $q=2$ for the present model. This is the same value of deceleration parameter as obtained for Petrov Type-D anisotropic cosmological solutions by Alvarado [56] and, in the case of an anisotropic dark energy cosmological model, under the framework of a generalised Brans–Dicke theory by Tripathy et al. [57].

Case:  $B=0$.

In this case, we obtain

$${\dot{V}}^2=3CV+3\frac{A}{C^{\alpha }}\frac{V^{\alpha +2}}{\alpha +2}-4\mathit{\int }{a}_{1}dV.$$

(29)

In the small V approximation, it is not possible to obtain the solutions in the analytical form, however, in the asymptotic limit

$$V\propto {(t-t_0)}^2,a_1\propto {(t-t_0)}^{\frac{2}{3}},a_2\propto {(t-t_0)}^{\frac{2}{3}},$$

(30)

where $V\left(t_0\right)=0$. The scale factor may be expressed for this as $\mathcal{R}\simeq {(t-t_0)}^{\frac{2}{3}}$, and the Hubble parameter as $H=\frac{2}{3t}$. The deceleration parameter may be obtained as $q=0.5$.

In the asymptotic limit, the anisotropy parameter turns out to be zero.

Case:  $B=\frac{1}{2}$.

In this case, we have

$${\dot{V}}^2=3C{V}^{\frac{1}{2}}+3\frac{A}{C^{\alpha }}\frac{V^{\frac{3\alpha }{2}+2}}{\frac{3\alpha }{2}+2}-4\int a_{1}dV.$$

(31)

We obtain the solutions by using a small V approximation as

$$V={\beta }^{\frac{4}{3}}{(t-t_0)}^{\frac{4}{3}},$$

(32)

where $\beta =\frac{3}{4}\sqrt{3C}$.

The scale factors are obtained as

$$a_1=\frac{{\beta }^{\frac{4}{3}}}{g^2}{(t-t_0)}^{\frac{4}{9}},$$

(33)

$$a_2=g{(t-t_0)}^{\frac{4}{9}},$$

(34)

where g is integration constant.

The scale factor may be obtained as

$$\mathcal{R}={\beta }^{\frac{4}{9}}{(t-t_0)}^{\frac{4}{9}},$$

(35)

which provides a power law behaviour for the scale factor. The corresponding Hubble rate becomes $H=\frac{4}{9}{(t-t_0)}^{-1}.$

The anisotropy parameter has zero value whereas the deceleration parameter has value $q\sim 1.25.$ Here, in Figure 1, the evolution of the scale factor within the KS model is given for the three choices of the MCG parameter B. The parameter space used for plotting the figure include $\sqrt{3C}=4/3$, $b=-7/9$, $\beta =1$. The present epoch is considered as $t=1$. It is observed that the stiffness in the scale factor depends on the choice of the parameter B. While in initial epochs, the scale factor appears to be more stiff for a higher value of B; in a future epoch, the behaviour is the reverse, i.e., a lower value in B leads to a stiffer scale factor.

3. Bianchi-I Model

The line element for the Bianchi-I representing anisotropic flat space is given by

$$d{s}^2=-d{t}^2+{\left[R_i\left(t\right)d{x}^i\right]}^2,$$

(36)

where i = 1, 2, 3. We choose $R_1=a_1$ and $R_2=R_3=a_2$.

The EFE for this metric with MCG as source can be written as

$$\frac{{\dot{a_2}}^2}{a_2^2}+2\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}=\rho ,$$

(37)

$$\frac{{\dot{a_2}}^2}{a_2^2}+2\frac{\ddot{a_2}}{a_2}=\frac{A}{{\rho }^{\alpha }}-B\rho ,$$

(38)

$$\frac{\ddot{a_1}}{a_1}+\frac{\ddot{a_2}}{a_2}+\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}=\frac{A}{{\rho }^{\alpha }}-B\rho .$$

(39)

After some algebraic manipulation, from the above equations, we obtain

$$2\frac{\ddot{a_1}}{a_1}+2\frac{\dot{a_1}\dot{a_2}}{a_1{a}_2}-\frac{{\dot{a_2}}^2}{a_2^2}=\frac{A}{{\rho }^{\alpha }}-B\rho .$$

(40)

From Equations (30) and (33), we obtain

$$\frac{d\left(V{H}_1\right)}{dt}=\frac{(1-B)}{2}C{V}^{-B}+\frac{A}{2C^{\alpha }}{V}^{\alpha (1+B)+1}.$$

(41)

Again, from Equations (30) and (31), we obtain

$$\frac{d\left(V{H}_2\right)}{dt}=\frac{(1-B)}{2}C{V}^{-B}+\frac{A}{2C^{\alpha }}{V}^{\alpha (1+B)+1}.$$

(42)

From the above obtained EFEs in terms of volume, we find

$$\ddot{V}=\frac{3}{2}(1-B)C{V}^{-B}+\frac{3}{2}\frac{A}{C^{\alpha }}{V}^{\alpha (1+B)+1}.$$

(43)

After integration, this gives

$$\dot{V}=\sqrt{3C{V}^{1-B}+3\frac{A}{C^{\alpha }}\frac{V^{\alpha (1+B)+2}}{\alpha (1+B)+2}}.$$

(44)

It can be observed that, for the BI universe as well, the solutions for the volume and scale factors depend on the MCG parameters $\alpha$, A and B and, as such, the dependence on the B-parameter is substantial. Here, also, we consider the three different values of the parameter B to assess its effect on the exact solutions.

Case:  $B=1$.

In a small V limit, the above differential equation reduces to

$$\dot{V}=\sqrt{3C}.$$

(45)

This gives the solution

$$V=a{e}^{\sqrt{3C}t},a_1=\frac{a}{b^2}{e}^{\sqrt{3C}t},a_2=b{e}^{\sqrt{3C}t},$$

(46)

where both a and b are constants.

The corresponding scale factor is given by

$$\mathcal{R}=a^{1/3}{e}^{\sqrt{3C}t},$$

(47)

and the Hubble parameter becomes $H=\sqrt{3C}$. The expansion in this case is of de Sitter kind.

Here, the anisotropy parameter is evaluated to be zero. On the other hand, the deceleration parameter has a value $q=-1$.

Case:  $B=0$.

In a small V limit, Equation (37) reduces to

$${\dot{V}}^2=3CV.$$

(48)

The solution is of the form

$$V=\frac{3C}{4}{(t-t_0)}^2.$$

(49)

The scale factors can be found to be

$$a_1=\frac{3C}{4c^2}{(t-t_0)}^{\frac{2}{3}},a_2=c{(t-t_0)}^{\frac{2}{3}},$$

(50)

where c is constant of integration. In this case, the scale factor becomes $\mathcal{R}={\left(\frac{3C}{4}\right)}^{1/3}$${(t-t_0)}^{\frac{2}{3}}$, so that the Hubble rate is $\frac{2}{3}{(t-t_0)}^{-1}$. The scale factor increases with a power law behaviour time after an initial epoch $t_0$. The anisotropy and the deceleration parameters, respectively, turn out to be zero and $q=0.5$.

Case:  $B=\frac{1}{2}$.

In this case, we have

$${\dot{V}}^2=3C{V}^{\frac{1}{2}}+3\frac{A}{C^{\alpha }}\frac{V^{\frac{3\alpha }{2}+2}}{\frac{3\alpha }{2}+2}.$$

(51)

We obtain the solutions by using a small V approximation as

$$V={\beta }^{\frac{4}{3}}{(t-t_0)}^{\frac{4}{3}},$$

(52)

where $\beta =\frac{3}{4}\sqrt{3C}$.

The directional scale factors are obtained as

$$a_1=\frac{{\beta }^{\frac{4}{3}}}{g^2}{(t-t_0)}^{\frac{4}{9}},$$

(53)

$$a_2=g{(t-t_0)}^{\frac{4}{9}},$$

(54)

where g is integration constant.

The scale factor may be obtained as

$$\mathcal{R}={\beta }^{\frac{4}{9}}{(t-t_0)}^{\frac{4}{9}},$$

(55)

which provides a power law behaviour for the scale factor. The corresponding Hubble rate becomes $H=\frac{4}{9}{(t-t_0)}^{-1}.$

The anisotropy parameter is found to have zero value, whereas the deceleration parameter turns out to be $q\sim 1.25$ For the present case in Figure 2, we show the evolution of the scale factor for the Bianchi-I model considering three different choices of the parameter B. Out of the three choices, $B=1$ corresponds to a de Sitter kind of expansion and the other two represent a power law kind of expansion. In this case, the choice of B affects the stiffness in the scale factor, in the sense that a higher value of B provides a stiffer scale factor for a substantial time zone.

4. Discussion and Conclusions

In this paper, we have studied two anisotropic universes with MCG as matter source. We see that the solutions for the volume and scale factors depend on the MCG parameters $\alpha$, A and B. The dependence on the B-parameter is found to be the strongest, so we choose to vary it for obtaining different solutions.

The solutions are identical for $B=\frac{1}{2}$ for both the universes. However, for the rest of the two cases, the solutions differ significantly. For both the models, there is a large initial anisotropy, which washes out over a small period of time as isotropization starts quickly to produce the homogeneity and thus an isotropic universe observed on large scales.

For $B=0$, we find that the KS universe does not give an analytical solution while, for Bianchi-I universe, we obtain an analytical solution. However, we can obtain an approximate behaviour of the volume and scale factors in the asymptotic limit which is similar to that for the Bianchi-I universe. The initial anisotropy decays out with time and rate of isotropization for Bianchi-I model is almost identical to the rate for KS model in asymptotic limit. It may, however, be noted that after elapsing of some time this isotropy has been disappeared, but again regained at later times to give the observed isotropic universe. This may be physically interesting, as the losing and regaining of isotropy at the intermediate and later times, respectively, may correspond to the process of structure formation.

The most significant difference in behaviour between the two models is for $B=1$ case. In this case, for the KS universe we find that unlike all other cases, the anisotropy parameter is a non-zero constant, indicating that it does not vanish at any later time. The large initial anisotropy is retained significantly to prevent isotropization. On the contrary, for the Bianchi-I model, not only do we obtain a vanishing anisotropic parameter indicating isotropic universe, but also the solutions with de Sitter-like behaviour. Interestingly, although we have considered presence of no Cosmological Constant $\Lambda$ in our study, the constant C involving all the three MCG parameters behaves effectively as $\frac{\Lambda }{9}\sim {\Lambda }_{effective}$.

For $B=1$, in the case of the KS universe, besides remaining anisotropy, we obtain $q=2$, which gives a closed universe decelerating fast enough to collapse, whereas for the Bianchi-l universe, we obtain $q=-1$, providing an accelerating early universe in confirmity with the effective $\Lambda$ behaviour. On the other hand, for $B=0$, we obtain a flat matter dominated Bianchi-l universe with $q=+0.5$, while for $B=1/2$, we obtain both the KS and Bianchi-l solutions with q value 1.25 giving open, forever expanding universes. It is to be noted that, theoretically, Mishra et al. [58], in their model, have obtained the lowest values of q at −0.495 and −0.558, whereas highest values were 0.4 and 0.2. The other theoretical reported value of the deceleration parameter lies nearly $q=-1$ [59,60] to $q>-1$ [61]. However, the observational constraints, as set upon the parameter in the present epoch from type Ia supernova and X-ray cluster gas mass fraction measurements, are $q=-0.81\pm 0.14$ [62], whereas that from $H\left(z\right)$ and SN-Ia data are $q=-0.34\pm 0.05$ [63]. Therefore, our obtained value seems acceptable one as far as the accelerating early universe is concerned.

Thus, there are a number of very interesting results that appear from our study, which help us to compare and contrast the two different anisotropic models, along with proper understanding of the role of MCG as matter source in these models.