Applied Mathematics, 2011, 2, 348-354
doi:10.4236/am.2011.23041 Published Online March 2011 (http://www.SciRP.org/journal/am)
Bianchi-Type VI0 Bulk Viscous Fluid Models with Variable
Gravitational and Cosmological Constants
Manoj K. Verma, Shri Ram
Department of Applied Mathema tics, Institute of Technology, Ba naras Hindu University, Varanasi, India
Received December 16, 2010; revised January 19, 2011; accepted January 22, 2011
Abstract
This paper deals with Bianchi type VI0 anisotropic cosmological models filled with a bulk viscous cosmic
fluid in the presence of time-varying gravitational and cosmological constant. Physically realistic solutions
of Einstein's field equations are obtained by assuming the conditions 1) the expansion scalar is proportional
to shear scalar 2) the coefficient of the bulk viscosity is a power function of the energy density and 3) the
cosmic fluid obeys the barotropic equation of state. We observe that the corresponding models retain the well
established features of the standard cosmology and in addition, are in accordance with recent type Ia super-
novae observations. Physical behaviours of the cosmological models are also discussed.
Keywords: Bianchi VI0, Cosmology, Bulk viscosity, Variable G and .
1. Introduction
The adequacy of spatially homogeneous and isotropic
Friedman-Robertson-Walker (FRW) models for describ-
ing the present state of the universe is no basis for ex-
pecting that they are equally suitable for describing the
early stages of evolution of the universe. Cosmological
models which are spatially homogeneous but anisotropic
have significant role in the description of the universe at
it’s early stages of evolution. Bianchi space I-IX are use-
ful tools in constructing models of spatially homogene-
ous cosmologies [1]. Considerable work has been done
for constructing various Bianchi type cosmological mod-
els and their inhomogeneous generalizations. Among
these Bianchi type I spaces are simplest which subsequ-
ent generalizations of zero-curvature FRW models are.
Bianchi type VI0 spaces are of particular interest since
they are sufficiently complex, while at the same time,
they are simple generalizations of Bianchi type I spaces.
Barrow [2] has poin ted out that Bianchi type VI0 models
of the universe give a better explanation of some of the
cosmological problems such as primordial helium abun-
dance and they also isotropize in a special sense.
Ellis and MacCallum [3] obtained solutions of Eins-
tein's field equations for a Bianchi type VI0 space-time in
the case of a stiff-fluid. Collins [4] and Ruban [5 ] presen-
ted some exact solutions of Bianchi type VI0 for perfect
fluid distributions satisfying specific equations of state.
Dunn and Tupper [6] investigated a class of Bianchi type
VI0 perfect fluid cosmological models associated with
electromagnetic field. Lorentz [7] has generalized the
dust model of Ellis and MacCallum [3]. Roy and Singh
[8] derived some exact solutions of Einstein-Maxwells
equations representing a free gravitational field of mag-
netic type with perfect fluid and incident magnetic field.
Shri Ram [9] presented an algorithm for generating exact
per- fect fluid solutions of Einstein's field equations, not
satis- fying the equation of state, for spatially homoge-
neous cosmological models of Bianchi type VI0.
Bulk viscosity is supposed to play a very important
role in the early evolution of the universe. There are many
circumstances during the evolution of the universe in
which bulk viscosity could arise. The bulk viscosity co-
efficient determines the magnitude of the viscous stress
relative to the expansion. Ribeiro and Sanyal [10] studied
Bianchi type VI0 models containing a viscous fluid in the
presence of an axial magnet i c fiel d. Pat el and Ko ppar [11]
presented some Bianchi type VI0 viscous fluid cosmo-
logical models with expansion and shear. Bali et al. [12]
studied a Bianchi type VI0 magnetized barotropic bulk
viscous fluid massive string universe. Bali et al. [13] ob-
tained some exact solutions for a homogeneous Bianchi
type VI0 space-time filled with a magnetized bulk visc-
ous fluid in the presence of a massive comic string. Bali
et al. [14] have also discussed the properties of the free
gravitational fields and their invariant characterizations

M. K. VERMA ET AL.
349
and imposing certain conditions over the free gravitatio-
nal fields.
The cosmological constant  and the gravitational
constant G are two parameters present in Einstein's field
equations. The Newtonian constant G plays the role of
coupling constant between geometry and matter in Eins-
tein's field equations. There have been numerous mod-
ifications of general relativity in which G varies with
time in order to achieve possible unification of gravita-
tion and elementary particle physics or to incorporate
Mach's principle in general relativity. From the point of
view of incorporating particle physics into Einstein's
theory of gravitation, the simplest approach is to in terpret
the cosmological constant  in terms of quantum me-
chanics and the physics of the vacuum [15]. The  term
has also been interpreted in terms of Higg's scalar field
[16]. Linde [17] proposed that  term is a function of
temperature and related it to the process of broken sym-
metries. The cosmological constant problem related to
the existence of  has been extensively discussed in lite-
rature. A phenomena logical solution to this problem is
suggested by considering  as a function of time, so that
it was large in the early universe and got reduces with the
expansion of the universe [18]. A number of authors e.g.
Kalligas et al. [19], Arbab [20], Abdussattar and Vish-
wakarma [21] proposed linking of variations of G and 
within the framework of general relativity. This approach
is appealing as it leaves the form of Einstein equations
formally unchanged by allowing a variation of G to be
accompanied by change in . Pradhan and Yadav [22]
investigated bulk viscous anisotropic cosmological mod-
els with variable G and . Pradhan et al. [23] derived
FRW universe with verying G and . Since Bianchi type
I spaces are subsequent generalization of zero curvature
FRW models, Singh et al. [24] obtained some Bianchi
type I models with variable G and . Singh et al. [25]
obtained early viscous universe with variable G and .
Singh and Tiwari [26] presented Bianchi type I models in
the presence of a perfect fluid with time varying G and 
in general relativity. Singh and Kotambkar [27] dis-
cussed cosmological models with variable G and  in
space-times of higher dimensions. Singh and Kale [28]
dealt with Bianchi type I, Kantowski-Sachs and Bianchi
type III anisotropic models of the universe filled with a
bulk viscous cosmic fluid in the presence of variable G
and . Bali and Tinker [29] investigated Bianchi type III
bulk viscous barotropic fluid cosmological model with
variab le G and  which leads to inflationary phase of the
universe. Recently, Verma and Shri Ram [30] obtained
Bianchi type III bulk viscous barotropic fluid cosmolog-
ical model with variable G and  in simple and syste-
matic way. Homogeneous cosmologies with Bianchi type
VI0 space filled with perfect fluids, satisfying specific
equation of state linking the pressure and matter energy
density are widely used to study different properties of
solutions of Einstein's field equations. Pradhan and Bali
[31] presented magnetized Bianchi type VI0 barotropic
massive string universe with decaying vacuum energy
density. Recently, a new class of LRS Bianchi type VI0
universe with free gravitational field and decaying va-
cuum energy is obtained by Pradhan et al. [32].
In this paper, we investigate Bianchi type VI0 bulk
viscous barotropic fluid cosmological models with time
varying gravitational and cosmological constants. The
paper is organized as follows, we present the metric and
Einstein's field equation for a viscous fluid with time-de-
pendent G and. We deal with solutions of the field eq-
uations and we obtain solutions of the field equations
under the assumptions that 1) the expansion scalar is pro-
portional to the shear scalar 2) the bulk viscosity coeffi-
cient is a power function of the energy de nsity and 3) the
cosmic fluid obeys the barotropic equation of state. The
corresponding models represent expanding, shearing and
non-rotating universe which give essentially space for
large time. We also discuss the physical and kinematical
behaviours of Bianchi type VI0 anisotropic cosmological
models. Some concluding remarks have also been given.
2. Field Equations and General Expressions
We consider Bianchi type VIo metric in the form
22 2222222mx mx
dsdt AdxBe dyCe dz
 
  (1)
where ,and
A
BCare function of cosmic time t and m is
a constant parameter.
The energy-momentum tensor for a bulk viscous fluid
distribution is given by


j
jj
iii
Tpvvpg   (2)
where
;
p
=p vii

 (3)
Here ,,andpp

 are respectively, energy-density of
matter, thermodynamic pressure, effective pressure and
bulk viscosity coefficient. The four-velocity vector of the
fluid satisfies.
1
i
i
vv

 (4)
A semicolon stands for covariant differentiation.
The Einstein’s field equations with time-dependent
Gand

are
18.
2
ijijij ij
RRg GTg    (5)
For the line-element (1) with a bulk viscous fluid dis-
tribution, the field Equation (5), in commoving frame,
give rise to the following equations :

M. K. VERMA ET AL.
350
2
28
BCBCmGp
BCBC A
  
    (6)
2
28
AC ACmGp
AC AC A
  
  (7)
2
28
ABABm Gp
ABABA
  
    (8)
2
28
ABACBC mG
AB AC BCA
  
  (9)
0
BC
BC




  (10)
where a dot denotes differentiation with respect to .t
An additional equation for time changes of G and

is obtained by the divergence of Einstein tensor, i.e.
;
1
2
jj
ii
j
RRg




, which leads to

;
80
jj
ii
j
GT g  
yielding

88 0
ABC
GGp
ABC
 


 





 
 (11)
The conservation of energy Equation (11), after using
Equation (3), splits into two equation

0
ABC
pABC 

 


 
 (12)
2
88
A
BC
GG
A
BC
 

 


 
  (13)
The average scale factor Sfor the metric (1) is de-
fined by
3
SABC (14)
The volume scale factor Vis given by
3
V SABC (15)
The generalized mean Hubble parameter H is given by

123
1
3
HHHH
(16)
where 123
,,
A
BC
HHH
A
BC


 The expansion scalar
.

and shear scalar

are given by
;
ii
A
BC
v
A
BC


 


  (17)
and
222
2
222
1
3
A
BCAB BCAC
A
BBCAC
ABC





 
(18)
An important observational quantity in cosmology is
the deceleration parameter q which is defined as
2
SS
qS 

 (19)
The sign of q indicates whether is model inflates or
not. The positive sign corresponds to the standard dece-
lerating model whereas the negative sign indicates infla-
tion.
3. Solution of the Field Equations
We are at liberty to make certain assumptions as we have
more unknown,,,,,,andABCp G

 with lesser num-
ber of field Equations (6)-(13). For complete determina-
tion of these field variables, we first assume that the ex-
pansion scalar

is proportional to the shear scalar

.
This condition leads to
n
A
B

(20)
where n is a positive constant.
Equation (10), on integration, yields
BlC

(21)
where l is an integration constant. Without loss of ge-
nerality we can take l. From Equations (6) and (7), we
obtain
2
2
20 BABBA m
BABBA A





    (22)
Substitution of Equations (20) and (21) in Equation
(22)
 
22
2
2
2
10
1
n
BBm
nB
Bn
B


 

  (23)
which reduce to
 
22
21
4
22 10
1
n
Bm
Bn B
Bn


 


 (24)
On assuming (),BfB
 takes the form



22221
21 n n
dffMB
dB B


 (25)
where 2
24,1.
1
m
Mn
n


 Equation (25) has the general
solution

24 2
22
22
4n
M
Ba
fB B


 (26)
where a is the constant of integration. From Equation
(26) we have
1
42 2
n
BdB M
dt
Ba


 (27)

M. K. VERMA ET AL.
351
The solution of Equation (27) is not valid for1n

.
We can obtain physically realistic models by choosing
the values of n for which Equation (27) is integrable.
3.1. Model I
When 2n Equation (27) reduces to
3
42 2
BdBMdt
Ba

 (28)
which, after integration, leads to

1/ 2
2
22
12
Bctca



(29)
where 1
c and 2 c are arbitrary constants.
Therefore, the metric (1) can be written in the form





1/ 2
2222222
2222mx mx
dsdtTa dxTa
edyedz

 
 
where
12
ct cT (30)
It is clear that, given

t

, we can find the physical
and kinematical parameters associated with metric (30).
The effect of bulk viscosity is to produce a change in the
cosmic fluid and therefore exhibits essential change on
character of the solution. In most of the investigations,
the bulk viscosity is assumed to be a simple power func-
tion of the energy density [33, 34]


0
t


  (31)
where 0

and


0   are constant. For small density,

may even be equal to unity [35]. The case 1


corresponds to a radiative fluid [34]. Near a big-bank,
012


 is a more appropriate assumption to obtain
realistic models [36].
For the specification of


t

, we also assume that the
fluid obeys the equation of state
p

  (32)
where


01 

 is a constant. From Equations (12)
and (32), we obtain


22
21 0 T
Ta






 (33)
where a dash denotes differentiation with respect to .T
Integration of Equation (33) yi elds




1
22
kT a



 (34)
where k is integration constant. Differentiating Equation
(34), we obtain




2
22
21kTTa



  (35)
Also, from Equation (9), we find that



22 22
2
22
54 4
84
mT ma
GTa 

  (36)
which on differentiation leads to








232222222
33
22 22
548 10458
88 22
m TmTmaaTma
GG Ta Ta  
 

 

(37)
Using Equations (13), (31) and (35) in Equation (37), we get
 
 



1
232222 2222
0
3(1)2 2
2222 22
548 1045816 1
32
2
mTmTmaaT makT
kT
GTaTa Ta

 



 



 










 



(38)
Using Equations (34) and (38) in Equation (36), we obtain
 









1
232222 2222
0
4212
2222 22
22 22
2
22
548 1045821 4
() 2
544 .
4
mTmTmaaT makTkT
tTaTa Ta
mT ma
Ta






 

 






(39)
The gravitational constant ()Gt is zero initially and gradually increases and tends to infinity at late time. We

M. K. VERMA ET AL.
352
also observe that the cosmological term  is initially
infinite. It is decreasing function of time and approaches
to zero at late time which is supported by recent result
from the observations of type supernova explosion (SNIa).
From Equations (31) and (34), we obtain




1
22
0
tkTa





 (40)
The physical and kinematical parameters of the model
(30) are given by the following expressions.

322
VTa
(41)


22
23HTTa
(42)


22
2TT a   (43)

22
23TTa   (44)


22 2
32qT a T (45)
The value of the deceleration parameter is positive for
all time which shows the decelerating behaviour of the
cosmological m odel.
For model (30), we observe that the spatial volume in-
creases with time T and it becomes infinite for large
value of T. At Ta, the spatial volume is zero and
,,,p


all are infinite but vanish for large T. Thus,
the model has a big-bank singularity at the finite time
Ta. The physical and kinematical parameters are all
well behaved for .aT . The bulk viscosity coeffi-
cient is infinite atTa and tends to zero for large time.
Since


constant, the anisotropy is maintained for all
times. It can be seen that the model is irrotational.
Therefore, the model describes a continuously expanding,
shearing and non-rotating universe with a big-bang start
at Ta.
3.2. Mode-II
From Equation (27), we get

1
42
2n BdB
dt
M
Ba

 (46)
with the help of Eq. (46), the line-element (1) reduces to
 
22
222222222
24 2
4nnmxmx BdB
dsdBBdxBedyedz
MB a

 

(47)
By a suitable transformation of coordinates, the
line-element (47) reduces to
 
22
222222222
24 2
4nnmxmx TdT
dsdTT dxTedyedz
MT a

 

(48)
For the model (48), the physical and kinematical pa-
rameters are given

23n
VT

 (49)


2n
T


 (50)


2
3
n
HT

 (51)


1n
T


 (52)



1
2
n
qn



(53)
Following the procedure as in Section (3.1), we obtain
the expression for energy density ,,G

and

as under
 
12
1n
kT




 (54)

  




 


34 2
32
1/2
42
3236 3
21
242
01 1
222 2
2124
21
28
82
821
4
nnn
nn
nnMTa
nM mnM
GTa TTT
kM nTakn
TT

 
 


 
 


 


 







(55)

 





 



1
1/22
422 42
101
1
12 2222
34 224 2
322
3236324 2
2
21
4
212421
21
28 2
nn n
nnnnn
kT akMnT a
kn
TT T
nnMTanMTa
nM mnMm
TTTTT

 



 
 




 



 


 


(56)

M. K. VERMA ET AL.
353

12
01 n
kT


 
 (57)
We observe that the spatial volume is zero at 0T

.
At this epoch the energy density

, expansion

, the
shear scalar

and the bulk viscosity coefficient are all
infinite. Therefore the model (48) starts evolving with a
big-bang at 0T. The spatial volume tends to infinite
and ,,,


become zero as for large time T. The gra-
vitational constant G is zero initially and tends to infi-
nity for large time T. The cosmological term

is infi-
nite at the beginning of the model and tends decreases
gradually to become zero at late time. The deceleration
parameter q is positive for 1n

and is negative 1n.
Therefore Equation (48) represents a model of a decele-
rating universe for 1n and a model of an accelerating
universe for 1n. The present day observations and
literature favour accelerating model of the universe. The
anisotropy in the models is maintained throughout.
4. Conclusions
In this paper we have studied Bianchi type VI0 space-time
models with bulk viscosity in the presence of time-de-
pendent gravitational and cosmological constants. We
have presented two physically viable anisotropic models
of the universe. For 2n, the model I evolves with a
big-bang start at the finite time Ta and does not ap-
proach isot ropy as T. For large T, energy density
becomes zero. The rate of expansion in the model slows
down tending to zero as T. Since the deceleration
parameter is positive for all time T, this model corres-
ponds to an expanding, shearing, non-rotating and dece-
lerating universe. For 1,on the model II represents
a decelerating universe whereas it represents an accele-
rating universe for 1n. Model II starts evolving with
a big-bang singularity at 0T and expands uniformly.
The model with negative deceleration parameter is com-
patible with the recent supernovae Ia observations that
the universe is undergoing a late time acceleration. The
anisotropy is maintained in both the models. The gravita-
tional constant ()Gt is zero initially and gradually in-
creases and tends to infinity at late time. The cosmologi-
cal term is infinite initially and approaches to zero at late
time. These are supported by recent results from the ob-
servations of the type Ia supernova explosion (SN Ia).
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