Modified Chaplygin gas cosmology with observational constraints

Abstract

The spatially homogeneous and totally anisotropic Bianchi type-II space-time models with modified Chaplygin gas having the equation of state $p=A\rho -\frac{B}{{\rho }^{\alpha }},$ 0 ≤ A ≤ 1, 0 ≤ α ≤ 1, where A, α and B are positive constants, have been investigated. It has been shown that the equation of state for such modified model is valid from the radiation era to the ΛCDM. The statefinder, which is the cosmological diagnostic pair {r, s} has been adopted to characterize different phases of the universe. The physical and geometrical properties of the corresponding cosmological models have been discussed. The observational constraints, essentially dependent on the hubble parameter H₀ and deceleration parameter q₀ have been investigated using 28 data points of H(z), SNe Ia and H(z)+ SNe Ia [57]. It has been seen that the average scale factor a(t) can be expanded in terms of an infinite convergent series around the current value of the average scale factor a₀ using Taylor’s theorem, in which the current value of the deceleration parameter q₀, the dimensionless jerk parameter j₀, the snap parameter $s_0^{*}$ and the lerk parameter ℓ₀ appeared.

Introduction

The recent observational data [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] on the late-time acceleration of the universe and the existence of dark matter have posed a fundamental theoretical challenge to gravitational theories. Astronomical and cosmological observations, such as large-scale redshift surveys structure [4], [5] , the cosmic microwave background (CMB) [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35] and Wilkinson Microwave Anisotropy Probe (WMAP) [8], [9] indicate that the observable universe experiences an accelerated expansion. The source of this acceleration is usually attributed to an exotic type of fluid with negative pressure called commonly dark energy (DE). Various kinds of dark energy models have been proposed such as cosmological constant [10], quintessence [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], k-essence [14], [15], [16], tachyon [17], [18], [19], phantom [20], [21], [22], dilatonic ghost condensate [23], quartessence [24], Chaplygin gas [25], quintom [26], holographic dark energy [27] and extra dimensions [28]. Various other types of DE cosmological models have been discussed by several authors [29], [30], [31]. The nature of the dark sector of the universe (i.e., dark energy and dark matter) remains a mystery. An economical and attractive idea to unify the dark sector of the universe is to consider it as a single component that acts as both dark energy and dark matter. One way to achieve the unification of dark energy and dark matter is by using the so-called Chaplygin gas. The pure Chaplygin gas or generalized Chaplygin gas (GCG) is a perfect fluid which behaves like a pressureless fluid at an early stage and a cosmological constant at a later stage. Therefore, it would be important if a unified dark matter and dark energy scenario could be found, in which these two components are different candidates of a single fluid [32], [33], [34].

The GCG model [35] is an interesting candidate for the unification of the dark matter and dark energy. In the GCG approach an exotic background fluid is considered with the equation of state $p=-B/{\rho }^{\alpha },$ where B and α are positive constant, 0 ≤ α ≤ 1. The case $\alpha =1$ corresponds to the Chaplygin gas. The GCG model has been successfully confronted with many phenomenological tests such as Supernovae data, CMB data etc. The energy density behaves as a matter and as a cosmological constant at a later stage, is an attractive feature of these models at early times. In this work, it is assumed that the isotropic pressure p of the cosmological fluid obeys the modified Chaplygin gas equation of state [36],

$$\begin{array}{c}\hfill p=A\rho -\frac{B}{{\rho }^{\alpha }},0\le A\le 1,0\le \alpha \le 1,\end{array}$$

where A, B and α are universal positive constants. The equation of state with $A=0$ and $\alpha =1$ known as Chaplygin gas (CG) was first introduced by Chaplygin to study the lifting forces upon the wing of the aeroplane in aerodynamics. Its generalized form with $A=0$ and α > 0 known as generalized Chaplygin gas (GCG) was first introduced by Kamenshchik et al. [37] and Bento et al. [35]. The equation of state (1) corresponds to the radiation-dominated era when $A=\frac{1}{3}$ and ρ → ∞. The equation of state corresponds to ΛCDM model i.e., DE cosmological fluid with negative pressure when ρ → 0. Generally the modified Chaplygin equation of state corresponds to a mixture of ordinary matter and DE. For $\rho ={\left(\frac{B}{A}\right)}^{\frac{1}{\alpha +1}}$ the matter content is pure dust with $p=0$.

In order to differentiate between various dark energy models, a geometrical diagnostic pair {r, s} called statefinder, has been introduced in Sahni et al. [55]. The statefinder probes the expansion dynamics of the universe through the higher derivatives of the expansion factor $\ddot{a}$ and $\stackrel{⃛}{a}$. The statefinder pair defines two new cosmological parameters r and s in addition to the the Hubble parameter H and the deceleration parameter q, which is defined as

$$\begin{array}{c}\hfill r=\frac{\stackrel{⃛}{a}}{a{H}^3},\end{array}$$

$$\begin{array}{c}\hfill s=\frac{r-1}{3(q-\frac{1}{2})},\end{array}$$

where $q\ne \frac{1}{2}$.

Statefinder is a geometrical diagnostic, because it depends only on the expansion factor a. Its important property is that $\{r,s\}=\{1,0\}$ is a fixed point for the flat ΛCDM FRW cosmological model. Departure of a given DE model from this fixed point is a good way of establishing the ‘distance’ of this model from flat ΛCDM. Sami et al. [39] and Myrzakulov et al. [40] have also used statefinder to distinguish the different models of dark energy.

The spatially homogeneous and anisotropic cosmological models have a significant role in the description of the universe in the early stages of its evolution. The Bianchi type models have been studied by several authors [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52] in an attempt to understand better the observed small amount of anisotropy in the universe. The Bianchi type-II models have also been used to examine the role of certain anisotropic sources during the formation of the large-scale structure as we see in the universe today. Some of the Bianchi type cosmologies are the natural hosts of large scale magnetic fields, and therefore their study can shed light on the implications of cosmic magnetism for galaxy formation. The simplest Bianchi types model that contains the flat FRW universe may be taken as a special case of the Bianchi type-I space time.

In the present paper, a spatially homogeneous and anisotropic Bianchi type-II cosmological models with modified chaplygin gas have been discussed in two cases. The statefinder diagnostic pair {r, s} has been adopted to characterize different phases of the universe. The geometrical and physical behaviors of the models have also been discussed. The observational constraints, which are essentially dependent on the hubble parameter H₀ and deceleration parameter q₀ have been investigated using 28 data points of H(z), SNe Ia and H(z)+ SNe Ia [57]. The Taylor’s expansions of the average scale factors a(t) around a₀ have also been investigated in both cases.