Open Journal of Applied Sciences, 2013, 3, 74-78
doi:10.4236/ojapps.2013.31B1015 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
2-A Cosmological Model with Varying G and 
in General Relativity
Harpreet1, R. K.Tiwari2, H. S. Sahota1
1Department of Applied Sciences , Sant Baba Bhag Singh Institute of Engineering & Technology, Jalandhar, India
2Goverment Engineering College, Reva, M.P., India
Received 2013
ABSTRACT
Spatially homogeneous and anisotropic Cosmological models play a significant role in the description of the early
stages of evolution of the universe. The problem of the cosmological constant is still unsettled. The authors recently
considered time dependent G and  with Bianchi type–I Cosmological model .We considered in this paper homogene-
ous Bianchi type -I space-time with variable G and  containing matter in the form of a perfect fluid assuming the
cosmological term proportional to R-2 (where R is scale factor). Initially the model has a point type singularity, gravita-
tional constant G (t) is decreasing and cosmological constant  is infinite at this time. When time increases  decreases.
Unlike in some earlier works we have neither assumed equation of state nor particular form of G. The model does not
approach isotropy, if ‘t’ is small .The model is quasi-isotropic for large value of ‘t’.
Keywords: Bianchi Type-I Universe; Varying G and ; Cosmology
1. Introduction
Cosmology is the scientific study of large scale proper-
ties of the universe as a whole. Cosmology is study of mo-
tion of crystalline objects. The origin of universe is great-
est cosmological mystery even today. As we are aware
that the expansion of the universe is undergoing time
acceleration [Perlmutter et al., (1997,1998,1999), Riess
et al., (1998,2004), Allen et al., (2004), Peebles et al.,
(2003), Padmanabhan. (2003) &Lima. (2004)]. Present
universe is suitably represented by Fried-
mann-Robertson-Walker model which is isotropic and
homogeneous in nature. To resolve the problem of a
huge difference between the effective cosmological con-
stant observed today and the vacuum energy density pre-
dicted by the quantum field theory, several mechanisms
have been proposed by Weinberg (1989). A possible way
is to consider a varying cosmological term due to the
coupling of dynamic degree of freedom with the matter
fields of the universe. The cosmological constant is small
because the universe is old. Models with dynamically
decaying cosmological term representing the energy den-
sity of vacuum have been studied by R. G. Vishwakarma,
(2000,2001,2005), A. I. Arbab, (1998) and Berman
(1991,1991b). Cosmological scenarios with a time vary-
ing cosmological constant were proposed by several re-
searchers. A number of models with different decay laws
for the variation of cosmological term were investigated
during the last two decades[ Chen & Wu (1990); Pavan
(1991); Carvalho et al.,(1992); Lima & Maia(1994);
Lima & Trodden (1996); Arbab & Abdel-Rahman(1994);
Cunha & Santos (2004); Carneiro & Lima(2005)].
A lot of work has been done by Saha (2005a, 2005b,
2006a, 2006b), in studying the anisotropic Bianchi type-I
Cosmological Model in general relativity with varying G
and . In this paper we study homogeneous Bianchi type
-I space-time with variable G and  containing matter in
the form of a perfect fluid. We obtain solution of the
Einstein field equations assuming the cosmological term
proportional to R-2 (where R is scale factor).
2. The Metric and Field Equations
We consider the Bianchi type - I metric in the orthogonal
form
(1)
The non-zero components of the Ricci tensor Rij
We assume that cosmic matter is taken to be perfect
fluid given by the energy- momentum tensor
(2)
where p,
 are the isotropic pressure and energy density
of the fluid. We take equation of state

HARPREET ET AL. 75
(2
1)
vi is four velocity vector of the fluid satisfying
(3)
Einstein's field equations with time dependent G and 
are
(4)
For the metric (1) and energy-momentum tensor (2) in
commoving system of coordinates, the field equation (4)
yields.
(5)
(6)
(7)
(8)
In view of vanishing of the divergence of Einstein
tensor, we have
(9)
The usual energy conservation equation of general
relativity quantities is
(10)
Equation (9) together with (10) puts G and  in some
sort of coupled field given by
(11)
implying that  is a constant whenever G is constant.
Using equation(21) in equation (10) and then integrating,
we get k > 0
(11
1)
We define, R as the average scale factor of Bianchi type-
I universe.
(12)
The Hubble parameter H, volume expansion
 , shear 
and deceleration parameter q are given by
Einstein's field equations (5)-(8) can be also written in
terms of Hubble parameter H, shear
 and deceleration
parameter q as
(13)
(14)
On integrating (5) - (8), we obtain
(15)
and
(16)
where k1 and k2 are constants of integration. From (14),
we obtain
(17)
Implying that  0
Thus the presence of positive  lowers the upper limit
of anisotropy whereas a negative  contributes to the
anisotropy.
Equation (17) can also be written as
(18)
where
2
3
8
c
H
G


 is the critical density and 8
vG




is the vacuum density.
From (13) and (14), we get,
(19)
Thus the universe will be in decelerating phase for
negative  and for positive  universe will slow down
the rate of decrease,showing that the rate of volume ex-
pansion decreases during time evolution and presence of
positive , slows down the rate of this decrease whereas
a negative  would promote it.
3. Solution of the Field Equation
The system of equations (5)-(8) and (11) supply only five
equations in seven unknown parameters (A, B, C,, p, ,

HARPREET ET AL.
76
and G). Two extra equations are needed to solve the sys-
tem completely. For this purpose we take cosmological
term is proportional to R-2 , where ‘a’ is a positive con-
stant.i.e we take the decaying vacuum energy density
(20)
This variation law was proposed by Olson et al.,
(1987), Pavon (1991), Maia et al., (1994); Silveira et al.,
(1994,1997) and Torres et al.,(1996).Because observa-
tions suggest that  is very small in the present universe,
a decreasing functional form permits  to be large in
early universe.
Using equation (111) and equation (20) in eq (11) we
get
(21)
From equations (13), (14), (20) and (201) we get
(22)
Find the time evolution of Hubble parameter, integrate
(22), we get
(23)
where t0 is a constant of integration. The integration con-
stant is related to the choice of origin of time.
From eq (23),we obtain the scale factor
(24)
By using equation (24) in (15) and (16) in the metric
(1), we get
(25)
where m1,m2 and m3 are constants.
For the model (25), the spatial V, density , gravita-
tional constant G and cosmological constant  are
(26)
(27)
(28)
(29)
Expansion scalar  and shear  are
(30)
(31)
(32)
4. Observations and Conclusion
1) Thus we observe that as spatial volume V0 at t =
= and expansion scalar  is infinite,
which shows that universe starts evolving with zero
volume at t = infinite rate of expansion.
2) The scale factors also vanish at t = and hence
the model has a point type singularity at initial epoch.
Initially at t = the energy density ‘
 ‘, pressures ‘p’,
shear , cosmological term  tend all infinite but G is
finite.
3) As t increases the spatial volume increases but the
expansion rate decreases. Thus the rate of expansion
slows down with increase in time.
4) As t the spatial volume V becomes infinitely
large. All parameters
 ,  , p,  ,   0 asymptotically
but G is decreasing. Therefore at large value of t model
gives empty universe. The cosmic scenario starts from a
big bang at t = and continues until t =.If t =
then gravitation constant is zero and as t increase G also
increases.
The possibility of G increasing with time, at least in
some stages of the development of the universe, has been
investigated by Abdel-Rahman (1990), Chow (1981),
Levit (1980) and Milne (1935). 2
1
T
  Include Berman
(1990), Berman and Som (1990), Berman et al., (1989),
and Bertolami (1986b,1986a). This form of  is physi-
cally reasonable as observations suggest that  is very
small in the present universe. A decreasing functional
form permits  to be large in the early universe.
5) The ratio /0 as t∞ .So the model approach
isotropy for large value of t.
In summary, we have investigated the Bianchi type-I
cosmological model with variable G and  in presence of
perfect fluid with cosmological term proportional to R-2
(R is scale factor) suggested by Silveira et al., (1994,
1997) and others. Initially the model has a point type
singularity, gravitational constant G (t) is decreasing and
cosmological constant  is infinite at this time when time
increases  decrease.
It is interesting that Beesham (1994) , Lima and Car-
valho (1994), Kalligas, et al., (1995) and Lima (1996)
have also derived the Bianchi type I cosmological mod-
els with variable G and  assuming a particular from of
G and by taking equation of state .But we have neither
assumed equation of state nor particular form of G. The
model approach isotropy for large value of t, the model is
quasi-isotropic i.e. 0.




HARPREET ET AL. 77
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