Vol.3, No.7, 513-516 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.37072
Interacting generalized chaplygin gas model in bianchi
type-I universe
Raghavendra Chaubey
Applied Mathematics, DST-Centre for Interdisciplinary Mathematical Sciences, Faculty of Science, Banaras Hindu University,
Varanasi, India; *Corresponding Author:
Received 15 February 2011; revised 3 April 2011; accepted 18 April 2011.
ABSTRACT
In this paper, we have studied the generalized
chaplygin gas of interacting dark energy to ob-
tain the equation of state for the generalized
chaplygin gas energy density in anisotropic
Bianchi type-I cosmological model. For negative
value of B in equation of state of generalized
chaplygin gas, we see that <1
eff

 , that cor-
responds to a universe dominated by phantom
dark energy.
Keyw ords: Cosmological Models; Chaplygin Gas;
Cosmological Parameters
1. INTRODUCTION
One of the most important problems of cosmology, is
the problem of so-called dark energy (DE). The type Ia
supernova observations suggests that the universe is
dominated by dark energy with negative pressure which
provides the dynamical mechanism of the accelerating
expansion of the universe [1-3]. The strength of this ac-
celeration is presently matter of debate, mainly because
it depends on the theoretical model implied when inter-
preting the data. Most of these models are based on dy-
namics of a scalar or multi-scalar fields. Primary scalar
field candidate for dark energy was quintessence sce-
nario [4,5], a fluid with the parameter of the equation of
state lying in the range, 1<<1 3

.
In a very interesting paper Kamenshchik, Moschella
and Pasquier [6] have studied a homogeneous model
based on a single fluid obeying the Chaplygin gas equa-
tion of state
=
A
p


(1.1)
where p and

are respectively pressure and energy
density in comoving reference frame, with >0

;
A
is a positive constant. This equation of state has raised a
certain interest [7] because of its many interesting and,
in some sense, intriguingly unique features. Some possi-
ble motivations for this model from the field theory
points of view are investigated in [8]. The Chaplygin gas
emerges as an effective fluid associated with d-branes [9]
and can also be obtained from the Born-infield action
[10].
Inserting the equation of state (1.1) into the relativistic
energy conservation equation, leads to a density evolv-
ing as
2
=
A
BV 
 (1.2)
where B is an integration constant.
There exist a wide class of anisotropic cosmological
models, which also often studying in cosmology [11].
There are theoretical arguments that sustain the existence
of an anisotropic phase that approaches an isotropic case
[12]. Also, anisotropic cosmological models are found a
suitable candidate to avoid the assumption of specific
initial conditions in FRW models. The early universe
could also characterized by irregular expansion mecha-
nism. Therefore, it would be useful to explore cosmo-
logical models in which anisotropic, existing at early
stage of expansion, are damped out in the course of evo-
lution. Interest in such models have been received much
attention since 1978 [13].
Setare [14] has obtained the equation of state for the
generalized Chaplygin gas energy density in non-flat
universe. Chaubey [15] has obtained the role of modified
chaplygin gas in Bianchi type - I universe. In the present
paper, using the generalized Chaplygin gas model of
dark energy, we obtain equation of state for interacting
Chaplygin gas energy density in anisotropic Bianchi
type-I cosmological model. For negative value of B in
equation of state of generalized chaplygin gas, we see
that <1
eff



, that corresponds to a universe dominated
by phantom dark energy.
2. INTERACTING GENERALIZED
CHAPLYGIN GAS
In this section we obtain the equation of state for the
generalized Chapligin gas when there is an interaction

R. Chaubey / Natural Science 3 (2011) 513-516 514
between generalized Chaplygin gas energy density


and a Cold Dark Matter (CDM) with =0
m

.
The continuity equations for dark energy and CDM
are

31 =
H
Q 

 
 (2.1)
3=
mm
H
Q

 
. (2.2)
The interaction is given by the quality =Q


. This
is a decaying of the generalized Chaplygin gas compo-
nent into CDM with the decay rate . Taking a ratio of
two energy densities as =m
r


, the above equa-
tions lead to
1
=3 3
r
rHrrH







 (2.3)
Following [3], if we define
=3
eff
H



, 1 =3
eff
mrH


 (2.4)
Then, the continuity equations can be written in their
standard form

31 =0
eff
H 


 (2.5)

31 =0
eff
mmm
H  
 (2.6)
We consider the homogeneous anisotropic Bianchi
type-I cosmological model with line element
22222222
123
d=d ddd
s
taxayaz (2.7)
where 12
,aa and 3 a are function of t only.
The Einstein field equations for the metric (2.1) are
written in the form

323 2
2323
=.
m
aaa a
aaaa



 
   (2.8)

313 1
1313
=.
aaa a
aaaa



 
   (2.9)

1212
1212
=.
aa aa
aa aa



 
  (2.10)

23 31 12
1223 31
=
aa aa aa
aaa aaa



 
   (2.11)
where
  8G/c4 is constant.
We define
123
=Vaaa (2.12)
By using the method of Singh et al. [16-19], we obtain
 
1/3
11 1
d
=exp t atDVX Vt




 (2.13)
 
1/3
22 2
d
=exp t atDVX Vt




 (2.14)
 
1/3
333
d
=exp t atDVX Vt




 (2.15)
where i D (i = 1, 2, 3) and i
X
(i = 1, 2, 3) satisfy the
relation 123
=1DDD and 123 =0.XX X
Now, adding Eqs.2.9, 2.10 and 2.11 and three times
Eq.2.8, we get

32331 12 12
123 122331
2
3
=.
2m
aaaaa aa aa
aa aaa aaaa
p  

 
 
 
 



  

(2.16)
From Eqs.2.12 and 2.16, we have

3
=.
2m
Vp
V
 


 



(2.17)
Define as usual
2
2
3
== ;
mm
m
cr
V
V



2
2
3
==
cr
V
V




 (2.18)
From above, we obtain following relation for ratio of
energy densities r as
=m r


 (2.19)
In the generalized Chaplygin gas approach [10], the
equation of state to (1.1) is generalized to
=
A
p




 (2.20)
The above equation of state leads to a density evolu-
tion as

1
1
1
=B AV











 (2.21)
Taking derivatives in both sides of above equation
with respect to cosmic time, we obtain
 
11
1
=V BV ABV
V





 

 

 


 (2.22)
Substituting this relation into Eq.2.1 and using defini-
tion =Q



, we obtain
 
11
=1
()
[]
B
VV
VABV





 (2.23)
Here as in Ref. [20], we choose the following relation
for decay rate

2
=1 V br
V


(2.24)
with the coupling constant 2
b. Using Eq.2.14, the

R. Chaubey / Natural Science 3 (2011) 513-516 515
above decay rate take following form
2
=m
V
bV


  

 
 

(2.25)
Substituting this relation into Eq.2.23, one finds the
generalized Chaplygin gas energy equation of state
 
2
11
=1.
m
Bb
VABV



 
 



 


(2.26)
Now using the definition generalized Chaplygin gas
energy

, and using 
, we can rewrite the above
equation as

2
1
2
3
=1
1
m
Bb
V
V






 









 




(2.27)
From Eqs.2.4, 2.25 and 2.27, we have the effective
equation of state as

1
2
3
=1
1
eff B
V
V









 



(2.28)
By choosing a negative value for B we see that
<1
eff

, that corresponds to a universe dominated by
phantom dark energy, Eq.2.28, for =1

, is the effec-
tive parameter of state for Chaplygin gas. In this case, in
the expression for energy density (1.2), term under
square root should be positive, i.e. 2>VBA, then
the minimal value of the volume factor is given by
1
2
=
min
B
V
A




(2.29)
Now, from Eqs.2.13-2.15 and 2.29, we have find the
minimal value of the scale factors are given by
11
62
11 1
=exp
min
BB
aDX t
AA



 



 


 


(2.30)
11
62
22 2
=exp
min
BB
aDX t
AA



 



 


 


(2.31)
11
62
333
=exp
min
BB
aDX t
AA



 



 


 


(2.32)
According to this model we have a bouncing universe.
Generally for this model >0, <0AB
and 1>0


.
From Eq.2.21, we can realize that the cosmic scalar fac-
tors take values in the interval <<
imin i
aa(for i = 1,
2, 3) which corresponds to

1
1
0<<2A


,
where
1
1
=
min
B
VA






(2.33)
and
 
11
31 1
min =exp,=1, 2, 3. ii i
BB
aDX ti
AA




 


 

 

(2.34)
Using Eq.1.2, one can see that the Chaplygin gas in-
terpolates between dust at small i a and a cosmological
constant at large i a, but choosing a negative value of
B, this quartessence idea lose. Following [6] if we con-
sider a homogeneous scalar field ()t

and a potential
(V

) to describe the Chaplygin cosmology, we find
2
22
=B
VABV


 (2.35)
Now, by choosing a negative value for B we see
that 2<0 
, then we can write
=i

 (2.36)
In this case the lagrangian of scalar field


t

can
rewritten as
 
22
11
==
22
LV Vi
  
 
 (2.37)
The energy density and the pressure corresponding to
the scalar field

are as respectively

2
1
=2Vi



 (2.38)

2
1
=2 pVi



 (2.38)
Therefore, the scalar field

is a phantom field. This
implies that one can generate phantom-like equation of
state from an interacting generalized Chaplygin gas dark
energy model in anisotropic universe.
3. CONCLUSIONS
We have studied the generalized chaplygin gas of in-
teracting dark energy to obtain the equation of state for
the generalized chaplygin gas energy density in anisot-
ropic Bianchi type-I cosmological model. By choosing a
negative value for B we see that <1
eff

, that cor-

R. Chaubey / Natural Science 3 (2011) 513-516 516
responds to a universe dominated by phantom dark en-
ergy.
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