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)