Modified Chaplygin gas and constraints on its B parameter from cold dark matter and unified dark matter energy cosmological models

P. Thakur 1 S. Ghose 0 B. C. Paul 0 0 Physics Department, North Bengal University , Darjeeling, 734 013 West Bengal, India 1 Physics Department, Alipurduar College , Jalpaiguri, 736122 West Bengal, India A B S T R A C T We study modified Chaplygin gas (MCG) as a candidate for dark energy and predict the values of parameters of the gas for a physically viable cosmological model. The equation of state of MCG (p = B A ) involves three parameters: B, A and . The permitted values of these parameters are determined with the help of a dimensionless age parameter (H 0t 0) and H (z) z data. Specifically, we study the allowed ranges of values of the B parameter in terms of and As (As is defined in terms of the parameters in the theory). We explore the constraints of the parameters in the cold dark matter and unified dark matter energy models, respectively. 1 I N T R O D U C T I O N Recent cosmological observations, such as high-redshift surveys of Type Ia supernovae (Perlmutter et al. 1997a,b; Riess et al. 1998; Tonry et al. 2003), cosmic microwave background radiation (Melchiorri et al. 2000; Lange et al. 2001; Jaffe et al. 2001; Halverson et al. 2002; Netterfield et al. 2002) and Wilkinson Microwave Anisotropy Probe (Bennet et al. 2003; Briddle et al. 2003; Hinshaw et al. 2003; Kogut et al. 2003; Spergel et al. 2003), predict that our present Universe is passing through an accelerated phase of expansion preceded by a period of deceleration. It is known that the ordinary matter and fields of the standard model are not sufficient to accommodate the present phase of acceleration (preceded by deceleration). Consequently, a modification of the matter sector of the Einstein gravity is essential to incorporate the recent predictions from observational cosmology. The notion of a new type of matter has emerged, which must have negative pressure. Recent astronomical data when interpreted in the context of the big bang model have provided some interesting information about the composition of the Universe. The analysis reveals that our Universe is spatially flat and consists of 70 per cent dark energy with negative pressure, the remaining 30 per cent dust matter [cold dark matter (CDM) plus baryons], and negligible radiation. It has been predicted that the dark energy may be responsible for the present acceleration of our Universe. The most simple candidate for this uniformly distributed (i.e. unclustered) dark energy is considered to be in the form of vacuum energy density or a cosmological constant (). The model with a cosmological constant is entangled with (i) the fine-tuning problem (the present amount of dark energy is so small compared with the fundamental scale) and (ii) the coincidence problem (the dark E-mail: (PT); (SG); (BCP) energy density is comparable with the critical density today). Alternatively, the other choices are (i) a light homogeneous scalar field , whose effective potential V () leads to an accelerated phase at a later stage of the universe (Caldwell, Steinhardt & Dave 1998; Saini et al. 2000), (ii) an X-matter component, which is characterized by an equation of state p = , where 1 < 0 (Peebles & Ratra 2002), (iii) effects from extra dimensions (Lue 2002; Sahni & Shtanov 2002), (iv) an exotic fluid, Chaplygin gas, etc. The motivation of this paper is to obtain a cosmological model constraining Chaplygin gas taking into account observational facts. Chaplygin gas was first introduced in aerodynamics in 1904. Recently, it has been shown that Chaplygin gas may be useful for describing dark energy because of its negative pressure. Although it has positive energy density, it carries a negative pressure for which it is referred to as an exotic fluid. In the context of string theory, the Chaplygin gas emerges from the dynamics of a generalized d-brane in a (d+1,1) spacetime. It can be described by a complex scalar field which is obtained from a generalized BornInfeld action. The equation of state is given by where A is a positive constant and p and are pressure and density, respectively. Subsequently, a modified form of the equation of state has been developed (Billic, Tupper & Viollic 2001; Bento, Berrolami & Sen 2002) of the form with 0 < 1 which is known as generalized Chaplygin gas (GCG). It has two free parameters: A (positive) and . In the GCG model, at low energy density the fluid pressure is negative and constant while at high energy density it behaves almost like a pressureless fluid. Thus, it smoothly interpolates between a non-relativistic matter phase in the past and a negative pressure dark energy regime at late times. Recently, a modified form of GCG has been considered in cosmology (Liu & Li 2005). The modified Chaplygin gas (MCG) is more general and contains three free parameters. The idea is to interpolate states of standard fluids at high pressures and high energy densities to a constant negative pressure at low energy densities (Debnath, Banerjee & Chakraborty 2004). In addition, it covers whole aspects of GCG. This model accommodates consistent (i) gravitational lensing test (Silva & Bertolami 2003; Dev & Alcaniz 2004), and (ii) gamma-ray bursts (Bertolami & Silva 2006). The equation of state for this MCG is given by where A, B and are arbitrary constants with 0 1. As there are three free parameters, unlike GCG, we look for a suitable range of B parameter for MCG for a viable cosmological model accommodating the observational evidence. The parameters are determined by (i) considering a dimensionless age parameter H 0t 0 (Dev, Alcaniz & Jain 2002) and (ii) H (z) z data analysis (Wu & Yu 2006). We perform the following analyses. Case 1. The age parameter (H 0t 0) is dimensionless and a constant irrespective of the model we are considering. For simplicity, we choose its standard value to be 0.95 (ignoring error). Imposing this constant age parameter, we determine the effective ranges of values of the free parameters in this model. As the parameters have some preferred range of values, one can ultimately constrain one of them, in particular the matter part B. Case 2. Using [H (z) z] data, we further verify the validity of the constraints on the parameters obtained in case 1. We use the Hubble parameter versus redshift relation given in Table 1. The 2 minimization technique has been used in this process. There are nine data points of H(z) at redshift z used to constrain the MCG model. We investigate both CDM and unified dark matter energy (UDME) models in the next sections. The UDME model refers to the model in which the MCG represents dark matter and dark energy as a whole, where the total energy density comprises radiation, baryon and MCG energy density. In the case of the CDM model, the constituents of our universe are radiation, CDM and MCG. This paper is organized as follows. In Section 2, we present the relevant field equations and introduce Hubble and deceleration parameters, respectively. In Sections 3 and 4, we explor (...truncated)