Bianchi Type-II, VIII & IX Perfect Fluid Cosmological Models in Brans Dicke Theory of Gravitation

Field equations in the presence of perfect fluid distribution are obtained in a scalar tensor theory of gravitation proposed by Brans and Dicke [1] with the aid of Bianchi type-II, VIII & IX metrics. Exact prefect fluid Bianchi typeIX cosmological model is presented since other models doesn’t exist in Brans-Dicke scalar tensor theory of gravitation. Some physical and geometrical properties of the models are also discussed.


Introduction
Brans and Dicke [1] theory of gravitation is well known modified version of Einstein's theory.It is a scalar tensor theory in which the gravitational interaction is mediated by a scalar field  as well as the tensor field ij g of Einstein's theory.In this theory the scalar field  has the dimension of the inverse of the gravitational constant.In recent years, there has been a renewed interest of the gravitational constant.The latest inflationary models (Mathiazhagan and Johri [2]), possible "graceful exit" problem (Pimental [3]) and extended chaotic inflations (Linde [4]) are based on Brans and Dicke theory of gravitation.
Brans-Dicke field equations for combined scalar and tensor field are is an Einstein tensor, ij is the stress energy tensor of the matter, T  is the dimensionless coupling constant and comma and semicolon denote partial and covariant differentiation respecttively.The equation of motion is a consequence of the field Equations (1.1) and (1.2).Several aspects of Brans-Dicke cosmology have been extensively investigated by many authors.The work of Singh and Rai [5] gives a detailed discussion of Brans-Dicke cosmological models.In particular, spatially homogeneous Bianchi models in Brans-Dicke theory in the presence of perfect fluid with or with out radiation are quite important to discuss the early stages of evolution of the universe.
Chakraborty [19], Raj Bali and Dave [20], Raj Bali and Yadav [21] studied Bianchi type IX string as well as viscous fluid models in general relativity.Reddy, Patrudu and Venkateswarlu [22] studied Bianchi type-II, VIII & IX models in scale covariant theory of gravitation.Shanthi and Rao [23] studied Bianchi type-VIII & IX models in Lyttleton-Bondi Universe.Also Rao and Sanyasi Raju [24] and Sanyasi Raju and Rao [25] have studied Bianchi type-VIII & IX models in Zero mass scalar fields and self creation cosmology.Rahaman et al. [26] have investigated Bianchi type-IX string cosmological model in a scalar-tensor theory formulated by Sen [27] based on Lyra [28] manifold.Rao et al. [29][30][31] have studied Bianchi type-II, VIII & IX string cosmological models, perfect fluid cosmological models in SaezBallester scalar-tensor theory of gravitation and string cosmological models in general relativity as well as self creation theory of gravitation respectively.
In this paper we discuss Bianchi type-II, VIII & IX perfect fluid cosmological models in a scalar-tensor theory proposed by Brans and Dicke [1].

Metric and Energy Momentum Tensor
We consider a spatially homogeneous Bianchi type-II, VIII and IX metrics of the form where ( , , )      are the Eulerian angles, and are functions of t only.It represents

R S
Bianchi type-II if and The energy momentum tensor for perfect fluid distribution is given by where  is the density and is the pressure.p Also In the co moving coordinate system, we have from Equations (2.2) and (2.3) The quantities  and are functions of "t" only.p

Bianchi Type-II, VIII & IX Perfect
Fluidcosmological Models in Brans-Dicke Theory of Gravitation where "  " denotes differentiation with respect to "t".
When  = 0, - where " ′ " denotes differentiation with respect to "T".Since we are considering the Bianchi type-II, VIII and IX metrics, we have   for Bianchi type-II, VIII and IX metrics respectively.Therefore, from the Equation (3.10), we will consider the following possible cases with   0 where " ′ " denotes differentiation with respect to "T".From (3.14)The corresponding metric can be written in the form We can observe that the spatial volume decreases as time " T " increases, i.e., the model is contracting.Also the model has initial singularity at The scalar expansion  and shear  are given by

as
. So, the rate of expansion is rapid as time decreases and it becomes slow as time increases.The shear scalar as and as .Thus the shape of universe changes uniformly.The deceleration parameter q is obtained as   The corresponding metric can be written in the form The corresponding metric can be written in the form The corresponding metric can be written in the form where   1 f   , sinhθ and sinθ respectively.In the above expressions, the volume decreases as time increases if 1 4 m  i.e., the models are contracting, the volume increases as time increases if 1 4 m  i.e., the models are expanding and the volume is independent of time T if 1 4 m  . Also the models have initial singular--ity at T b a   , 0 a  .The expansion  and shear  are equal for all Bianchi type-II, VIII & IX perfect fluid radiating cosmological models in general relativity.Which are given by The deceleration parameter 2 2 ; It can be seen that for large " " the quantities T  and  will become zero if 1 4 m  .Also the quantities  and  tends to +∞ as if and tends to -∞ if .Thus the rate of expansion is rapid as time decreases, it becomes slow as time increases and the shape of universe changes uniformly.In the case of , we can see that the Spatial Volume "V" is independent of time "T" and , the models are not isotropic for large T. The negative value of the deceleration parameter q shows that the models inflate except for m = 1.CASE Now the expression for expansion  and shear  are given by for Bianchi type-IX perfect fluid radiating cosmological model in Brans-Dicke theory of gravitation.The spatial volume tends to zero as T→∞.Thus the model is contracting with the increase of time and also the model has no real singularity.The deceleration parameter q is obtained as .The negative value of q indicates that 2 q   the model is inflationary.Also, since lim 0   which confirms that the universe remains anisotropic throughout the evolution.

Conclusions
Bianchi type space-times play a vital role in understanding and description of the early stages of evolution of the universe.In particular, the study of Bianchi type-II, VIII & IX universes are important because familiar solutions like FRW universe with positive curvature, the desitter universe, the Taub-Nut solutions etc correspond of Bianchi type-II, VIII & IX space-times.In view of the importance of Bianchi type-II, VIII & IX space-times and also since exact solutions offer an alternative and complementary approach to study various cosmological models, in this paper we have presented Bianchi type-II, VIII & IX perfect fluid cosmological models in Brans-Dicke theory of gravitation.
In case of 0 , .Also established the non-existence of Bianchi type-II & VIII perfect fluid cosmological models in this theory.Since "a" is an arbitrary constant and "ω" is a coupling constant, it is always possible to assign specific values to "a" and "ω" to keep the pressure "p" (3.27) and density "ρ" (3.28) be always positive.

3 . 29 )
Thus (3.29) together with (3.27) and (3.28) constitutes an exact Bianchi type-IX perfect fluid cosmological model in Brans-Dicke scalar-tensor theory of gravitation.PHYSICAL AND GEOMETRICAL PROPERTIES: The volume element of the Bianchi type-IX perfect fluid cosmological model is given by

for
Bianchi type-IX perfect fluid cosmological model in Brans-Dicke theory of gravitation.The scalar expansion 0 .45) Thus (3.45) together with (3.43) & (3.44) constitutes Bianchi type-II Perfect fluid radiating cosmological models in general theory of relativity.FOR BIANCHI TYPE-VIII METRIC(

1 )
48) together with (3.46) & (3.47) constitutes Bianchi type-VIII Perfect fluid radiating cosmological models in general theory of relativity.FOR BIANCHI TYPE-IX METRIC ( 51) together with (3.49) & (3.50) constitutes Bianchi type-IX Perfect fluid radiating cosmological models in general theory of relativity.PHYSICAL AND GEOMETRICAL PROPERTIES: The volume element of the above three models [(3.45), (3.48) & (3.51)] are given by we can observe that the only Bianchi type-IX perfect fluid cosmological model exists in Brans-Dicke theory of gravitation.The model is anisotropic, inflationary and has initial singularity at T b a 0 a    we can observe that Bianchi type-II, VIII & IX perfect fluid radiating cosmological models of general relativity exist in this theory.The models have initial singularity at T b we have obtained only Bianchi type-IX anisotropic radiating perfect fluid cosmological model of general relativity with    .In this case also we have observed that Bianchi type-II & VIII cosmological models doesn't exist in this theory.