Open Journal of Applied Sciences, 2013, 3, 89-93
doi:10.4236/ojapps.2013.31B1018 Published Online April 2013 (
1-A Cosmological Model with Varying G and
in General Relativity
Harpreet, R. K. Tiwari, H. S. Sahota
Department of Applied Sciences , Sant Baba Bhag Singh Institute of Engineering & Technology, Khiala, Padhiana,
Jalandhar- 144030,Punjab, India
Received 2013
In this paper homogeneous Bianchi type -I space-time with variable G and containing matter in the form of a perfect
fluid assuming the cosmological term proportional to H2 (where H is Hubble Parameter). 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. The model does not approach isotropy, if it is small. The model is quasi-isotropic for large
value of it.
Keywords: Bianchi Type-I Universe; Varying G and ; Cosmology; Hubble Parameter
1. Introduction
Cosmology is study of large scale structures of universe.
The simplest homogeneous and anisotropic model is
Bi-anchi type –I cosmological model which is gener-
aliza-tion of Friedmann-Robertson-Walker (FRW) model.
Cosmological model with cosmological constant are se-
rious participants to describe the dynamics of the uni-
verse. The origin of universe is greatest cosmological
mystery even today. As we are aware that the expansion
of the universe is undergoing time acceleration Perlmut-
ter et al.,(1997,1998,1999), Riess et al.,(1998,2004), Al-
len et al.,(2004), Peebles et al.,(2003), Padmanabhan.
(2003) & Lima.(2004).To resolve the problem of a huge
differ-ence between the effective cosmological constant
ob-served today and the vacuum energy density predicted
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 uni-
verse is old. Models with dynamically decaying cosmo-
logical term representing the energy density of vacuum
have been studied by Vishwakarma, R. G. (2000,
2001,2005), Arbab, A. I.(1998), Berman, M. S.(1991a,
1991b). Cosmological scenarios with a time varying
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
number of authers investigated Bianchies models, using
the approach that there is link between variation of
gravitational constant and cosmological constant [Abdel-
Rahman 1990;Berman 1991a ;Kalligas et al.1992; Ab-
dussattar and Vishkarma 2005;Pradhan et al 2006;Singh
and Tiwari 2007 ]. Lot of work has been done by Saha
(2005a, 2005b, 2006a, 2006b), in studying the anisot-
ropic 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 H2 (where H is Hubble Parameter).
1.1. The Metric and Field Equations
We consider the Bianchi type - I metric in the orthogonal
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
where p,
are the isotropic pressure and energy density
of the fluid. We take equation of state
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vi is four velocity vector of the fluid satisfying
Einstein's field equations with time dependent G and
For the metric (1) and energy-momentum tensor (2) in
commoving system of coordinates, the field equation (4)
In view of vanishing of the divergence of Einstein
tensor, we have
The usual energy conservation equation of general
relativity quantities is
Equation (9) together with (10) puts G and in
some sort of coupled field given by
implying that is a constant whenever G is constant.
Using equation (21) in equation (10) and then integrating,
we get k> 0
We define, R as the average scale factor of Bianchi
type- I universe.
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 decelera-
tion parameter q as
On integrating (5) - (8), we obtain
Where k1 and k2 are constants of integration. From
(14), we obtain
Implying that 0
Thus the presence of positive lowers the upper limit
of anisotropy whereas a negative contributes to the
Equation (17) can also be written as
is the critical density and
is the vacuum density.
From (13) and (14), we get,
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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.
1.2. 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, ,
and G). Two extra equations are needed to solve the sys-
tem completely. For this purpose we take cosmological
term is proportional to H2 .
This variation law was proposed by Olson, et al.,
(1987), Pavon(1991), Maia, et al., (1994); Silveira, et al.,
(1994, 1997), Torres, et al.,(1996).Because observations
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 equation (11)
we get
From equations (13), (14), (20) and (201) we get
Find the time evolution of Hubble parameter, integrate
(22), we get
where t0 is a constant of integration. The integration con-
stant is related to the choice of origin of time.
From equation (23), we obtain the scale factor
By using equation (24) in (15) and (16), the metric (1),
we get
where M = and m1,m2 and m3 are constants. For the
model (25), the spatial V, density , gravitational con-
stant G and cosmological constant are
Expansion scalar and shear are
2. Observations and Conclusions
1. Thus we observe that as spatial volume V0 at  t =
0 and expansion scalar
is infinite, which shows that
universe starts evolving with zero volume at t = 0 infinite
rate of expansion. Hence the model has a point type sin-
gularity at initial epoch.
2. Initially at t = 0 the energy density ‘
’, pres-sures
p’, shear ,cosmological term
tend all infinite.
3. As t increases the spatial volume increases but the
expansion rate decreases. Thus the rate of expansion
slows down with increase in time and tend to zero.
4. As t the spatial volume V becomes infi-nitely
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 = 0 and continues until t=.
5. The ratio
0 as t. So the model ap-proach
isotropy for large value of t.
6. We also see for model (25), if t then q =2 im-
plies universe is decelerating as q is positive.
7. 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
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Berman (1990), Ber-man and Som (1990), Berman et al.
(1989), and Bertolami (1986b, 1986a). This form of is
physically reasonable as observations suggest that is
very small in the present universe. A decreasing func-
tional form permits to be large in the early universe.
8. In summary, we observed on investigation that Bi-
anchi type-I cosmological model with variable G and
in presence of perfect fluid with cos-mological term is
proportional to H2 (H is Hub-ble Parameter) 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 in-
finite at this time when time increases decrease.The
model approach isotropy for large value of t ,the model is
quasi-isotropic i.e. 0.
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