#### 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 evaluated and after
dropping the surface terms reads
Sg =
dt −6 a˙ N2a + N1 F (φ)a3φ˙ 2 .
The Hamiltonian for this action therefore reads (upto a factor
of N )
pa2 1 pφ2
Hg = − 24a + 4F (φ) a3
where pa = − ∂∂La˙ and pφ = − ∂φ˙
∂ L are the canonically
conjugate momenta corresponding to a and φ. To proceed further,
we shall now concentrate on the Chaplygin gas part of the
matter sector. We shall first write down the basic
thermodynamic relations following from the thermodynamic
description in [
17
]. These read
ρc = ρ0(1 +
), h = 1 +
τ d S = d
+ pcd(1/ρ0)
+ pc/ρ0,
where ρc is the total mass energy density, τ is the
temperature, ρ0 is the rest mass density and is the specific internal
energy. The next step is to obtain an expression for pc in
(10)
(11)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
terms of h and S. This can be done with the help of the
thermodynamic relations above and reads [
8
]
pc = − A
pc = − A
We now make an assumption that we shall be considering
the early universe case. Hence we assume
Sp1+α
Aa3(1+α).
Therefore Hamiltonian for the matter sector takes the form
(upto a factor of N )
1
Hm = −a3ωρ f + S 1+α p .
The super-Hamiltonian for the full theory now reads
H = Hg + Hm
pa2 1 pφ2 1
= − 24a + 4F (φ) a3 − a3ωρ f + S 1+α p .
To simplify this Hamiltonian further one uses the canonical
transformations
α
T = −(1 + α) p−1 S 1+α pS
α
pT = S 1+α p
along with the explicit form of the perfect fluid energy density
ρ f = a3(1B+ω) . This leads to
H = − 2p4a2a + 4F1(φ) apφ32 − ω Ba−3ω + pT
where pT is the canonical variable corresponding to the
matter sector along the direction of the cosmic time.
3 Quantization of the scalar-metric cosmology
We now proceed to write down the Wheeler–De Witt (WD)
equation for the Hamiltonian written down in the earlier
section. For this we replace pa = −i ∂∂a , pT = −i ∂∂T and
pφ = −i ∂∂φ in the Hamiltonian and assume that the
Hamiltonian Hˆ annihilates the wave function. This leads to the WD
equation which reads (setting h¯ = 1)
∂2 6 ∂2
∂a2 − F (φ)a2 ∂φ2
− 24a
Bωa−3ω + i ∂∂T
This can be written in the form
∂
Hˆ ψ = i ∂t ψ
where
ψ (a, φ, T ) = 0.
∂2 6 ∂2
Hˆ = ∂a2 − F (φ)a2 ∂φ2 − 24a Bωa−3ω .
To solve this equation we now make the following ansatz1
ψ (a, φ, t ) = ei Et (a, φ).
This gives
∂2 (a, φ)
∂a2
6
− F (φ)a2
∂2 (a, φ)
∂φ2
− 24a Bωa−3ω − E
(a, φ) = 0.
We note that for Hˆ to be a self-adjoint operator, the inner
product between any two wave functions ψ1 and ψ2 must
satisfy
(a,φ)
(ψ1, ψ2) =
a F (φ)ψ1∗ψ2dadφ
with the boundary conditions
ψ (0, φ, t ) = ψ (a, 0, t ) =
=
∂ψ (a, φ, t )
∂φ
∂ψ (a, φ, t )
∂a
Note that the inner product given in [
13
] does not make Hˆ
self-adjoint since the operator ∂φ acts on F (φ) appearing in
the denominator of the φ-term in the Hamiltonian.
Applying the method of separation of variables once again
by setting
(a, φ) = η(a)ζ (φ)
leads to the following
1 Note that T = t corresponds to the time coordinate in the above
equation.
(30)
(31)
(32)
(33)
(34)
(35)
(36)
where κ2 is the separation constant. Solving Eq. (30) for
ω = 1, we get
η(a) = √a√2
Note that the above solution reduces to an Airy function if
96B − 4κ2 = 0. This is the solution that one gets in the dust
dominated universe (ω = 0) in [
8
] with Chaplygin gas and
perfect fluid coupled to gravity.
Considering F (φ) = 6λφm , (m = − 2, λ > 0), we get
ζ (φ) = (κλ) m+2 (m + 2) m−+12 φ
1
× C3 (l − 1) J−l
+ C4 (l + 1) Jl
2κλ φ m2+2
m + 2
2κλ φ m2+2
m + 2
= (κλ) m+2 (m + 2) m−+12 φ C3 J−l
1
+ C4 Jl
2κλ φ m2+2
m + 2
2κλ φ m+22
m + 2
where
1
r = 3
√
C1 = C1 (1 − r ) 2
√
C2 = C2 (1 + r ) 2
1 + 96B − 4κ2
We observe that the boundary conditions are satisfied if we
set C1 = 0 = C3. The wave function therefore becomes
With the above solution in hand, we now construct a wave
packet by superposing all the eigenfunctions. We do this by
first integrating over all possible values of E
ψκ = C5κ m+12 aφ Jl 2κλ φ m2+2 (1 + r )
m + 2
∞
A( )e 3322 t 43 Jr ( a 2 )d ,
3
= 4
4E
3
=0
where A( ) is a weight factor that needs to be chosen properly
so that the integration can be performed. Taking A( ) =
e−γ 2 r− 13 , we obtain
1
ψκ = C5κ m+2 aφ Jl
2κλ φ m2+2
m + 2
3i t −(1+r)
2γ − 16 .
(37)
(38)
The next step to construct the wave packet is to integrate ψκ
over all possible values of κ. This yields
×
ψwp = C5 aφe 32−γ8−a33it
41 +24B
κ=0
1
G(κ)κ m+2 Jl
× m2κ+λ2 φ m2+2
where G(κ) is a weight factor. Choosing G(κ) = κ, yields
3r 3i t −(1+r)
a 2 2γ − 16
(1 + r )dκ
We now plot the probability densities at two different times
for two different values of B. From the plots, we observe
that the height of the peaks increases when the value of B
increases. A small increase in the value of B results in a large
increase in the peak heights. Further, it can be easily seen that
the height of the peaks decreases with the increase in time T .
The plots show the behaviour of the probability density function, that is |ψwp(a, φ, T )|2 for two
different time with two different values of B.
Using the above quantum wave packet, we now calculate
the expectation values of a(t ) and φ (t ). Using the definition
of the inner product between two wave functions (27), we
have
then constructed from the solution of the Wheeler–DeWitt
equation and exhibits two peaks as in [
13
]. The height of
these peaks gets enhanced due to the presence of the perfect
fluid.
a (t ) =
∞ ∞
a=0 φ=−∞
∞ ∞
a=0 φ=−∞
a F (φ)ψw∗ p a ψwpdadφ
a F (φ)ψw∗ pψwpdadφ
.
Substituting Eq. (39) in the above relation yields
1
a (t ) = 2
16γ + 64γ
.
From the above result, we observe that the expectation value
of the scale factor gets affected due to the presence of the
perfect fluid.
The expectation value for the field φ reads
∞ ∞
a=0 φ=−∞
∞ ∞
a=0 φ=−∞
a F (φ)ψw∗ p φ ψwpdadφ
a F (φ)ψw∗ pψwpdadφ
φ (t ) =
which gives
φ (t ) = Constant.
4 Conclusion
(40)
(41)
(42)
(43)
In this paper, we have studied the quantum cosmology of a
scalar field coupled to a flat FRW spacetime in the presence
of a generalized Chaplygin gas and perfect fluid. We observe
that the inclusion of the perfect fluid in the matter sector
has interesting consequences. We have followed the Schutz’s
formalism to deal with the Chaplygin gas sector of the theory.
The full theory is then quantized using the Wheeler–DeWitt
approach. The Wheeler–DeWitt equation is solved for ω = 1
using appropriate boundary conditions. The solution shows
that there exists a choice of the constant appearing in the
density of the perfect fluid for which the solutions reduce to
the ω = 0 solutions appearing in [
8
]. The wave packet is
Acknowledgements SG acknowledges the support by DST SERB
under Start Up Research Grant (Young Scientist), File No.YSS/2014/00
0180. SG also acknowledges the support of IUCAA, Pune for the
Visiting Associateship.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. A.Y. Kamenshchik , U. Moschella , V. Pasquier , Phys. Lett. B 511 , 265 ( 2001 )
2. M.C. Bento , O. Bertolami , A.A. Sen , Phys. Rev. D 66 , 043507 ( 2002 )
3. V. Gorini , A. Kamenshchik , U. Moschella , V. Pasquier , arXiv:gr-qc/ 0403062
4. O. Bertolami , arXiv:astro-ph/0403310
5. R. Jackiw , arXiv:physics/0010042
6. M. Bouhmadi-Lpez , P.V. Moniz , Phys. Rev. D 71 , 063521 ( 2005 )
7. P. Pedram , S. Jalalzadeh , S.S. Gousheh , arXiv: 0705 . 3587
8. P. Pedram , S. Jalalzadeh, Phys. Lett. B 659 , 6 ( 2008 )
9. B.F. Schutz , Phys. Rev. D 2 , 2767 ( 1970 )
10. B.F. Schutz , Phys. Rev. D 4 , 3559 ( 1971 )
11. F.G. Alvarenga , J.C. Fabris , N.A. Lemos , G.A. Monerat , Gen. Relativ. Grav. 34 , 651 ( 2002 )
12. F.G. Alvarenga , A.B. Batista , J.C. Fabris , S.V.B. Gonsalves , Gen. Relativ. Gravity 35 , 1659 ( 2003 )
13. B. Majumder , Phys. Lett. B 697 , 101 ( 2011 )
14. B. Majumder , N. Banerjee , Gen. Relativ. Gravity 45 , 1 ( 2013 )
15. S. Pal , N. Banerjee , Phys. Rev. D 90 , 104001 ( 2014 )
16. S. Pal , N. Banerjee , Phys. Rev. D 91 , 044042 ( 2015 )
17. V.G. Lapchinskii , V.A. Rubakov , Theor. Math. Phys. 33 , 1076 ( 1977 )