Scalar-metric quantum cosmology with Chaplygin gas and perfect fluid

The European Physical Journal C, Jan 2018

In this paper we consider the flat FRW cosmology with a scalar field coupled with the metric along with generalized Chaplygin gas and perfect fluid comprising the matter sector. We use the Schutz’s formalism to deal with the generalized Chaplygin gas sector. The full theory is then quantized canonically using the Wheeler–DeWitt Hamiltonian formalism. We then solve the WD equation with appropriate boundary conditions. Then by defining a proper completeness relation for the self-adjointness of the WD equation we arrive at the wave packet for the universe. It is observed that the peak in the probability density gets affected due to both fluids in the matter sector, namely, the Chaplygin gas and perfect fluid.

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Scalar-metric quantum cosmology with Chaplygin gas and perfect fluid

Eur. Phys. J. C Scalar-metric quantum cosmology with Chaplygin gas and perfect fluid Saumya Ghosh 1 Sunandan Gangopadhyay 0 1 Prasanta K. Panigrahi 1 0 S.N. Bose National Centre for Basic Sciences , Block JD, Sector III, Salt Lake, Kolkata 700106 , India 1 Indian Institute of Science Education and Research Kolkata , Mohanpur, Nadia, West Bengal 741246 , India In this paper we consider the flat FRW cosmology with a scalar field coupled with the metric along with generalized Chaplygin gas and perfect fluid comprising the matter sector. We use the Schutz's formalism to deal with the generalized Chaplygin gas sector. The full theory is then quantized canonically using the Wheeler-DeWitt Hamiltonian formalism. We then solve the WD equation with appropriate boundary conditions. Then by defining a proper completeness relation for the self-adjointness of the WD equation we arrive at the wave packet for the universe. It is observed that the peak in the probability density gets affected due to both fluids in the matter sector, namely, the Chaplygin gas and perfect fluid. 1 Introduction The Chaplygin gas model has proved to be an interesting model due to its ability to describe the accelerated expansion of our universe [ 1 ]. The generalized version of this model [ 1,2 ] has also been studied extensively [ 3,4 ]. Interestingly it was found that this model emerges from the Nambu–Goto action of relativistic strings in the light cone coordinates [ 5 ]. Quantum mechanical analysis of an FRW cosmological model with generalized Chaplygin gas was discussed in [ 6,7 ]. Such an analysis is important in its own right because the universe must surely be governed quantum mechanically when the linear size of the universe was very close to the Plank scale (10−33cm). Further one can make a positive expectation that a quantum mechanical description of our universe may result in a singularity free birth of the universe. In [ 8 ], apart from the Chaplygin gas coupled to gravity, the perfect fluid was also included. The early and late time behavior of the expectation of the scale factor was then obtained analytically using the Schutz formalism [ 9,10 ]. This formalism has also been used extensively in [ 8,11–13 ]. In [ 13 ] the quantum dynamics of the spatially flat FRW model with Chaplygin gas and a scalar field coupled to the metric has been investigated. The wave-packet was found from the linear superposition of the wave-functions of the Schrodinger– Wheeler–DeWitt (SWDW) equation. It was found to show two distinct peaks. Similar studies has also been carried out in various other cosmological models [ 14–16 ]. In this paper, motivated by the works discussed above [ 8,13 ], we study the quantum dynamics of the FRW model with Chaplygin gas, scalar field coupled to the metric together with the inclusion of the perfect fluid. We apply the Schutz formalism together with the canonical approach [17] to obtain the super Hamiltonian. This is then quantized canonically to get the SWDW equation. This is then solved to obtain the quantum cosmological wave functions of the universe which in turn is used to construct the wave packet. This paper is organized as follows. In the Sect. 2 we have discussed the basic set up for the canonical quantization of a gravity model and also the Schutz’s formalism for dealing with the matter sector. In Sect. 3 we have carried out the quantization of the scalar metric theory of gravity in the presence of a Chaplygin gas and perfect fluid. We conclude in Sect. 4. 2 Basic set up for quantization We start by writing down the action which includes gravity with a scalar field coupled to it together with the perfect fluid and the generalized Chaplygin gas representing the matter sector d4x √−g R − F (φ)gμν φ,μφ,ν 3 √ d x hhi j K i j + M d4x √−g( p f + pc) S = M + 2 ∂ M ≡ Sg + Sm (1) where K i j is the extrinsic curvature tensor, hi j is the induced metric on the time-like hypersurface and F (φ) is an arbitrary function of the scalar field. The matter sector consists of two fluids, namely, the perfect fluid with pressure p f = ωρ f and the Chaplygin gas with pressure A pc = − ρα . c In the subsequent discussion we shall follow the Schutz formalism [ 9 ] to deal with only the Chaplygin gas of the matter sector. In this formalism, the four velocity of the fluid can be written down in terms of four potentials h, , θ and S as 1 uν = h ( ,ν + θ S,ν ) where h and S are the specific enthalpy and specific entropy respectively. The other two (θ and ) do not have any physical significance. The normalization condition of the four velocity reads uν uν = −1. In this paper, we shall work with the flat FRW (k = 0) metric given by ds2 = −N 2(t )dt 2 + a2(t )[dr 2 + r 2(dϑ 2 + Si n2ϑ dϕ2)] where N (t ) is the lapse function and a(t ) is the scale factor. The Ricci scalar R for this metric is given by 1 R = N 3a2 [−6aa˙ N˙ + 6N a˙ 2 + 6N aa¨ ]. The gravity part of the action can now be evalua (...truncated)


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Saumya Ghosh, Sunandan Gangopadhyay, Prasanta K. Panigrahi. Scalar-metric quantum cosmology with Chaplygin gas and perfect fluid, The European Physical Journal C, 2018, pp. 41, Volume 78, Issue 1, DOI: 10.1140/epjc/s10052-018-5521-x