Journal of Modern Physics, 2012, 3, 9-15
http://dx.doi.org/10.4236/jmp.2012.31002 Published Online January 2012 (http://www.SciRP.org/journal/jmp)
Totally Anisotropic Cosmological Models with Bulk
Viscosity for Variable G and 
Shri Ram, Manish K. Singh, Manoj K. Verma
Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi, India
Received July 24, 2011; revised September 1, 2011; accepted October 10, 2011
ABSTRACT
Einstein’s field equations with variable gravitational and cosmological constants are considered in the presence of bulk
viscous fluid for the totally anisotropic Bianchi type II space-time in such a way as to preserve the energy momentum
tensor. We have presented solutions of field equations which represent expanding, shearing and non-rotating cosmo-
logical models of the universe. The physical behaviours of the models are discussed. We observe that the results ob-
tained match with recent observations of SNIa.
Keywords: Bianchi II; Cosmology; Hubble Parameter; Bulk Viscosity; Variable G and 
1. Introduction
The simplest model of the observed universe is well rep-
resented by Friedmann-Robertson-Walker (FRW) mod-
els, which are both spatially homogeneous and isotropic.
These models in some sense are good global approxima-
tion of the present-day universe. But on smaller scales, the
universe is neither homogeneous and isotropic nor do we
expect the universe in its early stages to have these prop-
erties. At very early times in the evolution of the universe,
most of the radiations and matter currently observed are
believed to have been created during the inflation. Mod-
ern cosmology is concerned with nothing less than a tho-
rough understanding and explanation of the past history,
the present state and the future evolution of the universe.
In fact, these are theoretical arguments from the recent
experimental data which support the existence of an ani-
sotropic phase approaching to isotropic phase leading to
consider the models of the universe with anisotropic back-
ground. Spatially homogeneous and anisotropic cosmolo-
gical models play significant roles in the description of
large-scale behaviours of the universe. Bianchi spaces I-
IX play important roles in constructing models of spa-
tially homogeneous and anisotropic cosmologies. Here we
confine ourselves to totally anisotropic space-time of Bian-
chi type II space-time which have fundamental role in
constructing cosmological models suitable for describing
the early evolution of the universe. Much attention has been
focused towards the study of locally rotationally symme-
tric (LRS) Bianchi type II space-times. Guzman [1] ob-
tained the general vacuum solution of Brans-Dicke field
equations for the totally anisotropic Bianchi type II space-
time. Singh and Shri Ram [2] presented totally anisotro-
pic Bianchi type II cosmological models in scalar tensor-
theories of gravitation developed by Saez-Ballester [3],
Lau and Prokhovnik [4]. Singh et al. [5] obtained exact
solutions of Einstein’s field equations in vacuum and in
the presence of stiff matter for the totally anisotropic Bian-
chi type II space-time in normal gauge for Lyra’s geome-
try when the gauge function is time-dependent. Recently,
Yadav and Haque [6] obtained a spatially homogeneous
and totally anisotropic Bianchi type II cosmological mo-
del representing massive string in normal gauge for Lyra’s
manifold.
At the early stages of the universe when neutrinos de-
coupling occurred, the matter behaved like a viscous fluid.
The coefficient of viscosity decreases as the universe ex-
pands. Misner [7,8] studied the effect of viscosity on the
evolution of the universe and suggested that the strong
dissipation, due to the neutrino viscosity, may considera-
bly reduce the anisotropy of the black body radiation.
Murphy [9] developed a uniform cosmological model filled
with fluid which possesses pressure and bulk viscosity ex-
hibiting the interesting feature that the big-bang type sin-
gularity appears in the infinite past. Grn [10], Dunn and
Tupper [11], Coley and Tupper [12], Banerjee and San-
tos [13,14] etc. constructed and discussed cosmological
models under the influence of both bulk and shear vis-
cosities. Padmanabhan and Chitre [15] investigated the
effect of bulk viscosity on the evolution of the universe
at large.
The cosmological constant problem is one of the out-
standing problems in cosmology. In recent years there has
C
opyright © 2012 SciRes. JMP

S. RAM ET AL.
10
been a lot of interests in the study of the role of cosmo-
logical constant  at very early and the later stages of the
evolution of the universe. A wide range of observations
suggest that the universe possesses a non-zero cosmolo-
gical constant. The  term has been interpreted in terms
of the Higgs scalar field by Bergmann [16]. Drietlein [17]
suggested that the mass of Higgs boson is connected with
 being a function of temperature and is related to the
process of broken symmetries, and therefore it could be a
function of time in a spatially homogeneous expanding
universe. In quantum field theory, the cosmological con-
stant is considered as the vacuum energy density. The ge-
neral speculation is that the universe might have been created
from an excited vacuum fluctuation (absence of inflationary
scenario) followed by super cooling and reheating sub-
sequently due to the vacuum energy.
Dirac [18] first introduced the idea of a variable G what
he called Large Number Hypothesis and since then vari-
ous works have been carried out for a modified general
relativity theory with this variation in G. A number of au-
thors such as Beesham [19,20], Berman [21], Kalligas et a l .
[22], Abdussattar and Vishwakarma [23] proposed the
linking of variation of G and  within the frameworks of
general relativity and studied several models with the Fried-
mann-Robertson-Walker (FRW) metric. This approach is
appealing since it leaves the form of Einstein equations
formally unchanged by allowing a variation of G to be
accompanied by a change in . Arbab [24,25] and Singh
et al. [26] have considered cosmological models with vis-
cous fluid considering variable cosmological and gravita-
tional constants. Singh et al. [27] presented a number of
classes of solutions of Einstein’s field equations with va-
riable G,  and bulk viscosity coefficient in the frame-
work of non causal theory. Several authors investigated
anisotropic bulk viscous fluid cosmological models of
various Bianchi types time-dependent G and  (see Pradhan
and Kumhar [28], Verma and Shri Ram [29,30] and ref-
erences cited therein).
Bali and Tinker [31] investigated bulk viscous fluid
flow for Bianchi type III space-time model with variable
G and , and obtained solutions of the field equations
under certain physical and mathematical conditions. Mo-
tivated by this work, we present totally anisotropic Bi-
anchi type-II bulk viscous barotropic cosmological mod-
els with variable G and  by making the following as-
sumptions: 1) the conditions between the metric poten-
tials A, B, C as3
12
,,
nn
m
mm ABC
n
A
BC tt

 
t
; 2) the matter
energy density and isotropic pressure satisfy the equation
of state ,0 1p

 
; 3) the coefficient of bulk vis-
cosity 0


  where0

and

are constants. We pre-
sent the metric and field equations in Section 2. In Sec-
tion 3, we deal with the solutions of the field equations and
obtain two classes of solutions for and 1n1.n

We also discuss the physical features of the cosmological
models. Some concluding remarks are given in Section 4.
2. Field Equations and General Expressions
We consider the totally anisotropic Bianchi type-II metric
in the form

2
222 222
dsdtAdxzdyB dyCdz 2 (1)
where the metric potentials A, B and C are functions of
cosmic time t. Einstein’s field equations with time-de-
pendent cosmological and gravitational constants are
18π.
2
ijijij ij
RRgGTg (2)
The energy-momentum tensor for a bulk viscous
fluid distribution is given by
ij
T


iji jij
Tpvv   pg
(3)
where ;
i
pp vi   is the effective pressure,

is the
coefficient of bulk viscosity, is isotropic pressure,
p

is the energy density and is fluid four-velocity
vector satisfying
i
v
1.
i
i
vv


In commoving coordinates, Einstein’s field Equation
(2) for the metric (1) are
2
22
38π,
4
BCBC AGp
BCBC BC

 
 
 
(4)
2
22
18π,
4
AC ACAGp
AC ACBC

 

(5)
2
22
18π,
4
ABABAGp
A
BAB BC

 
   
(6)
2
22
18π
4
AB AC BCAG
A
BACBC BC


 
   (7)
where the overdot denotes differentiation with respect to
time t. Moreover, an additional equation for time changes
of G and

is obtained by taking the divergence of Ein-
stein tensor i.e.
1;
2
jj
ii
RRgj

0




(8)
which leads to


8π;
jj
ii
GTgj 0.

 (9)
A semicolon denotes covariant differentiation. Equa-
tion (9) readily yields

.
8π
ABC G
p
A
BCG G
 


 


 



0
(10)
The conservation equation for energy-momentum
;
j
i
Tj

gives

0.
ABC
pABC 





 
 (11)

S. RAM ET AL.
11
Using Equation (11), Equation (10) splits into the fol-
lowing equations

123
10
n
mmm
t
 
 
.




 (20)

0.
ABC
pABC 

 


 
 (12) Integration of Equation (20) yields




123 1
1
exp 1
n
mmm
dt
n




  









(21)
2
8π8π.
ABC
GG
ABC


 


 
 
(13)
where d is a constant of integration. Differentiation of
Equation (21) gives
3. Solutions of Field Equations




 
123
123
1
1
1
exp 1
n
n
mmm
dt
mmm
t
n







  










(22)
Here we have four independent field equations contain-
ing eight unknowns viz. ,, ,,,,,ABC pG



. So we
shall assume extra conditions to obtain unique solutions
of the field equations.
Now using Equations (16)-(19) into Equation (7), we
obtain
In most of the investigations in cosmology, the bulk
viscosity is assumed to be a simple power function of the
energy density i.e.



2
1223 31
22
123 1
1
8π
4
2
exp .
1
n
n
mmmm mma
Gtb
mmm
t
n



 







0


  (14) 2
c
(23)
where 0

and

are constants. Murphy [9] assumed
1

 in the case of small density which corresponds to
a radiative fluid. We also assume that the fluid obeys the
barotropic equation of state Differentiation of (23) gives


2
1223 31
21 22
123 123
1
8π8π
21
4
22
exp 1
n
n
n
GG
nmm mmmma
tbc
mmmmmm
t
n t








 











(24)
,0 1p

  . (15)
3.1. Model I
We assume that solutions of the scale factors of the forms
3 12
,,
nn
m mm ABC
ABC tt
 


n
t
(16)
Substituting Equations (13) and (16) into Equation (24),
we have
where n is a positive constant. On integration of Equation
(16), we obtain



2
123
2
1223 31
21
2
123 123 1
22
8π8π
2
22
1exp
41
n
n
n
n
mmm
GG t
nmm mm mm
t
mmm mmm
at
n
bc t








 






1
1
exp,
1
n
mt
Aa n






(17)
1
2
exp ,
1
n
mt
Bb n






(18)
1
3
exp 1
n
mt
Cc n









(19)
(25)
Using Equations (14) and (22) into Equation (25), we
find that
where a, b, c are constants of integration and 1.n

Using Equations (15)-(16) into Equation (12), we obtain
  
 
 
2
1223311 231 2 31 23
11
122
1
2123
1
0
123 123
21
1exp exp
41 1
11
4π
4π1exp
1
nn
n
n
n
nmm mmmmammmmmmmm m
Gt
nn tbc
mmm
d
dmmmmmmt
n t










.
t

  


 










 

 

 
 


 


 


(26)

S. RAM ET AL.
12
Again, from Equations (21), (23) and (26), we obtain
the value of as given in Equation (27).

The Gravitational constant G is zero at t = 0 and
gradually increases and tends to infinity at late times.
The cosmological term  is infinite at t = 0 and becomes
zero as . t¥
The scalar expansion


and shear scalar



are
given by
123
,
n
mmm
t


 (28)


222
123 122331
2
2.
3n
mmm mmmmmm
t

 
 (29)
The coefficient of bulk viscosity has the value given by



123 1
0
1
exp 1
n
mmm
dt



n





 





(30)
An important observational quantity is the deceleration
parameter q which is defined as
2
VV
qV


 (31)
where The sign of q indicates whether the
model inflates or not. The positive sign corresponds to
standard decelerating model whereas negative sign indi-
cates inflation. For the present solutions of A, B and C,
the decelerating parameter has the value given by
3.VABC

1
123
3
1
n
nt
qmmm

  . (32)
Clearly q is positive for

11
123
3
n
mmm
tn

 


and
is negative for

11
123
3
n
mmm
tn







. The decelera-
tion parameter indeed has a sign flip at

11
123
3
n
mmm
tn





. For

11
123
3
n
mmm
tn





,
the solution gives an accelerating model of the universe.
when
 11
123
3
n
mmm
tn

 




, our solution represents a de-
celerating model of the universe.
The spatial volume V of the model has the value given by




13 123
1
exp 31
n
mmm
Vabc t
n













. (33)
We observe that the spatial volume is constant at
0t

. At this epoch the energy density

is finite and
,

 are zero. For 0t

, the physical parameters
,,p,


n
and  are well behaved and are decreasing
functions of time. As , the spatial volume tends to
infinity if
t
1

and the physical parameters tend to zero.
Thus, for physical reality of the model, we must have
0n1

. The model essentially gives an empty space-
time for large time. We also find that


tends to a con-
stant limit as , which shows that the anisotropy in
the universe is maintained throughout. Since 0
t


 
and 0

, the model leads to the inflationary phase of
the universe [32].
3.2. Model II
We now obtain solution of the field Equations (4)-(7) for
1n

. For 1n

, the scale factors in Equation (16) are
given by
3 12
,,
m mm
ABC
A
tB tC t
 
 (34)
which, on integration, gives
3 12
12
,,
m mm
3
A
ktB ktCkt  (35)
where are constants of integration.
123
Substituting Equations (15) and (35) into Equation (12),
we obtain
,,kkk

123
10
mmm
t
 






(36)
which, on integration, leads to
 
12 3
1mmm
Mt


 


(37)
  



2
1223311 231 23
11
222
2
1223311 231 231 23
11
122
21
1exp 8πexp
41 1
21
1exp exp
411
nn
n
nn
n
mmmmmmm mmm mm
atd t
nn
tbc
nmmmmmmam mmm mmm mm
tt
nn
tbc






 
 
 
 
 


 
 



 





 


 
 
1
2123
1 0
123 123
11
4π
4π1exp
1
n
n
mmm
d
dmmm mmmt
n
t







.





 
  


 


  

 
(27)

S. RAM ET AL. 13
where M is a constant of integration.
The coefficient of bulk viscosity has the value given by
 
123
1
0
mmm
Mt



 


. (38)
The effect of bulk viscosity is to produce a change in
perfect fluid and hence exhibit essential influence on the
character of the solution. The effect is clearly visible in
isotropic pressure and energy density.
Using Equations (34) and (35) into Equation (7), we have


123
2
2
1223 311
222
23
1
8π
4
mmm
mm mm mmk
Gt
tkk




(39)
Equation (39), on differentiation, yields



123
1223 31
3
2
2 11 2 3
22
23
2
8π8π
1
2
mmm
mmmm mm
GG t
kmm mt
kk



1






(40)
Combining Equations (13), (34) (37), (38) and (40),
we obtain Equation (41).
Substituting for G and

in Equation (11), we obtain
Equation (42).
The expansion



and shear scalar


have values
given by

123
,
mmm
t


 (43)


222
123 122331
2
2.
3
mmm mmmmmm
t

 
 (44)
We observe that the gravitational constant G is zero at
0t

and gradually increases and tends to infinite as
. We also see that the cosmological term  is infi-
nite at
t
0t

and a decreasing function of time, and it
approaches a small positive value at late time which is
supported by recent results from the observations of the
type Ia supernova explosion (SNIa). Naturally a cosmo-
logical model is required to explain acceleration in the
present universe. Thus, this model is consistent with the
results of recent observations.
The deceleration parameter q has the value given by

123
3
1qmmm   . (45)
From Equation (45), we observe that
123
0if 3qmmm


and
123
03qifmmm.
Thus, our solution represents an accelerating model of
the universe if


123
3mmm

 and decelerating mo-
del if


3m
123
mm

.
The spatial volume V of the model is given by


123
3
123
mmm
Vkkkt



(46)
which is zero at 0.t

At the energy density0t

,
expansion

and shear scalar

all are infinite. Thus,
the model starts with a big-bang singularity at 0.t

The
above parameters decrease with passage of time. The
spatial volume increases as time increases and becomes
infinite at late time. As t ,,,p

 and

tend
to zero. Thus, the model represents an expanding shear-
ing and non-rotating universe which essentially gives an
empty space for large time. We also find that


does not

  




 
123 123
123
2
1 1223311 123
222
23
1
1
11 0123
123
1
4
4π1
mmmmmm
mmm
mmmmmmkmmm
Gt
tkk
Mmmm
Mm mmt
t





 




 
 









 





t
(41)


 


 
  

 
123
123 123
123 123
2
2 12 23 311
222
23
2
11223 31112 3
222
23
1
111 0123
123
1
4
1
8π
4
4π1
mmm
mmmmmm
mmmmmm
mmmm mmkt
tkk
mmmmmmkmmm
Mt t
tkk
Mmmm
tMmmm t
t






 


 


 




 









 






1
.
(42)

S. RAM ET AL.
14
tend to zero as . Therefore, the anisotropy in the
model is maintained throughout.
t
4. Conclusion
In this paper we have studied totally anisotropic Bianchi
type-II bulk viscous fluid cosmological models with time-
dependent gravitational and cosmological constants. We
have presented two classes of physically viable cosmo-
logical models for and We have obtained
expressions for physical parameter
1n1.n
,,,p

 G and

as functions of time t. For 1n

, the model evolves with
a finite volume at and does not approach isotropy
as For large time, the energy density becomes zero.
The model is accelerating for
0t
.t

 11n
12
3
mm
n
3
m
t








and is decelerating for

11
123 .
3
n
mmm
tn

 


For
, the model starts evolving with a big-bang singu-
larity at This model represents an accelerating or
decelerating universe according as is
greater than 3 or less than 3. The anisotropy is main-
tained throughout in the model. The cosmological term is
infinite initially and approaches to zero at late time. The
gravitational constant G is zero initially and gradually
increases and tends to infinity at late time. These are sup-
ported by recent results from the observations of the type
Ia supernova explosion (SNIa).
1n
0.t
12
tmmm
3
REFERENCES
[1] E. Guzman, “General Vacuum Solution for Brans-Dicke
Bianchi Type II,” Astrophysics and Space Science, Vol.
179, No. 2, 1996, pp. 331-334.
doi:10.1007/BF00646953
[2] J. K. Singh and S. Ram, “A Study on Totally Anisotropic
bianchi Type II Space-Time,” IL Nuova Cimento B, Vol.
111, 1996, pp. 1487-1494.
[3] D. Saez and V. J. Ballester, “A Simple Coupling with
Cosmological Implications,” Physics Letter A, Vol. 113,
No. 9, 1985, pp. 467-470.
doi:10.1016/0375-9601(86)90121-0
[4] Y. K. Lau and S. J. Prokhovnik, “The Large Numbers
Hypothesis and a Realistic Theory of Gravitation,” Aus-
tralian Journal Physics, Vol. 39, No. 3, 1986, pp. 339-
346. doi:10.1071/PH860339
[5] J. K. Singh, C. P. Singh and S. Ram, “Totally Anisotropic
Bianchi Type II Cosmological Models in Lyra’s Geome-
try,” Proceeding of Mathematical Society B.H.U, Vol. 11,
1996, pp. 83-88.
[6] A. K. Yadav and A. Haque, “Lyra’s Cosmology of Mas-
sive String in Anisotropic Bianchi-II Space-Time,” In-
ternational Journal of Theoretical Physics, Vol. 50, 2011,
pp. 2850-2863. doi:10.1007/s100773-011-0784-0
[7] C. W. Misner, “Transport Processes in the Primordial
Fireball,” Nature, Vol. 214, 1967, pp. 40-41.
doi:10.1038/214040a0
[8] C. W. Misner, “The Isotropy of the Universe,” Astro-
physics Journal, Vol. 151, 1968, pp. 431-457.
doi:10.1086/149448
[9] G. L. Murphy, “Bing-Bang without Singularities,” Physi-
cal Review D, Vol. 8, No. 12, 1973, pp. 4231-4233.
doi:10.1103/PhysRevD.8.4231
[10] I. Grin, “Viscous Inflationary Universe Models,” Astro-
physics and Space Science, Vol. 173, No. 2, 1990, pp.
191-225. doi:10.1007/BF00643930
[11] K. A. Dunn and B. O. J. Tupper, “Tilting and Viscous
Models in a Class of Type-VI0 Cosmologies,” Astro-
physical Journal, Vol. 222, No. 2, 1978, pp. 405-411.
doi:10.1086/156154
[12] A. A. Coley and B. O. J. Tupper, “Viscous-Fluid Col-
lapse,” Physical Review D, Vol. 29, No. 12, 1984, pp.
2701-2704. doi:10.1103/PhysRevD.29.2701
[13] A. Banerjee and N. O. Santos, “Solutions of Einstein-
Yang-Mills Equations with Plane Symmetry,” Journal of
Mathematical Physics, Vol. 24, No. 11, 1983, pp. 2635-
2636. doi:10.1063/1.525637
[14] A. Banerjee and N. O. Santos, “Spatially Homogeneous
Cosmological Models,” General Relativity and Gravita-
tion, Vol. 16, No. 3, 1984, pp. 217-224.
doi:10.1007/BF00762537
[15] T. Padmanabhan and S. M. Chitre, “Viscous Universe,”
Physics Letter A, Vol. 120, No. 9, 1987, pp. 433-436.
doi:10.1016/0375-9601(87)90104-6
[16] P. G. Bergmann, “Comments on the Scalar-Tensor The-
ory,” International Journal of Theoretical Physics, Vol. 1,
No. 1, 1968, pp. 25-36. doi:10.1007/BF00668828
[17] J. Drietlein, “Broken Symmetry and Cosmological Con-
stant,” Physical Review Letter, Vol. 33, No. 20, 1974, pp.
1243-1244. doi:10.1103/PhysRevLett.33.1243
[18] P. A. M. Dirac, “The Cosmological Constant,” Nature,
Vol. 139, 1937, pp. 323-323. doi:10.1038/139323a0
[19] A. Beesham, “Comment on the Paper ‘The Cosmological
Constant  as a Possible Link to Einstein’ Theory of
Gravity, the Problem of Hadronic and Creation,” IL
Nuovo Cimento, Vol. B96, No. 4, 1986, pp. 17-20.
[20] A. Beesham, “Variable-G Cosmology and Creation,”
International Journal of Theoretical Physics, Vol. 25, No.
12, 1986, pp. 1295-1298. doi:10.1007/BF00670415
[21] M. S. Berman, “Cosmological Models with Variable
Gravitation and Cosmological Constant,” General Rela-
tivity and Gravitation, Vol. 23, No. 4, 1991, pp. 465-469.
doi:10.1007/BF00756609
[22] D. Kalligas, P. Wesson and C. W. F. Everitt, “Flat FRW
Models with Variable-G and ,” General Relativity and
Gravitation, Vol. 24, 1992, pp. 351-357.
doi:10.1007/BF00760411
[23] Abdussattar and R. G. Vishwakarma, “Some FRW Mod-
els with Variable G and ,” Classical Quantum Gravity,
Vol. 14, No. 4, 1997, pp. 945-953.
doi:10.1088/0264-9381/14/4/011

S. RAM ET AL. 15
[24] A. I. Arbab, “Cosmological Models with Variable Cos-
mological and Gravitational Constants and Bulk Viscous
Fluid,” General Relativity and Gravitation, Vol. 29, No. 1,
1997, pp. 61-74. doi:10.1023/A:1010252130608
[25] A. I. Arbab, “Bianchi Type I Viscous Universe with Vari-
able G and ,” General Relativity and Gravitation, Vol.
30, No. 9, 1998, pp. 1401-1405.
doi:10.1023/A:1018856625508
[26] T. Singh, A. Beesam and W. S. Mbokazi, “Bulk Viscous
Cosmological Models with Variable G and ,” General
Relativity and Gravitation, Vol. 30, No. 4, 1991, pp. 573-
581. doi:10.1023/A:1018866107585
[27] C. P. Singh, S. Kumar and A. Pradhan, “Early Viscous
Universe with Variable Gravitational and Cosmological
Constants,” Classical Quantum Gravity, Vol. 24, No. 2,
2007, pp. 455-474. doi:10.1088/0264-9381/24/2/011
[28] A. Pradhan and S. S. Kumhar, “LRS Bianchi Type II
Bulk Viscous Fluid Universe with Decaying Vacuum En-
ergy Density ,” International Journal of Theoretical
Physics, Vol. 48, No. 5, 2009, pp. 1466-1477.
doi:10.1007/s10773-008-9918-4
[29] M. K. Verma and S. Ram, “Bulk Viscous Bianchi Type-
III Cosmological Model with Time-Dependent G and ,”
International Journal of Theoretical Physics, Vol. 49, No.
4, 2010, pp. 693-700. doi:10.1007/s10773-010-0248-y
[30] M. K. Verma and S. Ram, “Spatially Homogeneous Bulk
Viscous Fluid Models with Time-Dependent Gravita-
tional Constant and Cosmological Term,” Advanced
Studies in Theoretical Physics, Vol. 5, No. 8, 2011, pp.
387-398.
[31] R. Bali and S. Tinker, “Bianchi Type III Bulk Viscous
Barotropic Fluid Cosmological Models with Variable G
and ,” Chinese Physics Letter, Vol. 26, No. 2, 2009, pp.
029802-029806. doi:10.1088/0256-307X/26/2/029802
[32] S. Weinberg, “Nonlinear Realizations of Chiral Symme-
try,” Physical Review, Vol. 166, No. 5, 1968, pp. 1568-
1577. doi:10.1103/PhysRev.166.1568