Tilted Bianchi Type VI 0 Cosmological Model in Saez and Ballester Scalar Tensor Theory of Gravitation

In recent years, there has been a considerable interest in the investigation of cosmological models in which the matter does not move orthogonally to the hyper surface of homogeneity. These are called tilted cosmological models. The general behaviors of tilted cosmological models have been studied by King and Ellis [1], Ellis and King [2], Collins and Ellis [3], Bali and Sharma [4,5], Bali and Meena [6]. Bradely and Sviestins [7] investigated that heat flow is expected for cosmological models. Following the development of inflationary models, the importance of scalar fields (mesons) has become well known. Saez and Ballester [8] have developed a new scalar tensor theory of gravitation in which the metric is coupled with a dimensionless scalar field in a simple manner. This coupling gives satisfactory description of the weak fields. In spite of the dimensionless character of the scalar field, an anti-gravity regime appears. This theory suggests a possible way to solve the missing matter problem in non-flat FRW cosmologies. Sing and Agrawal [9], Reddy and Rao [10], Reddy et al. [11], Mohanty and Sahu [12,13], Adhav et al. [14], Tripathy et al. [15] are some of the authors who have studied the various aspects of Saez and Ballester [8] scalar tensor theory. We derived the field equations for Bianchi type VI0 metric in Section 2. We solved the field equations in Section 3. We mentioned some physical and geometrical properties of the solutions in Section 4 and also mentioned the concluding remark in Section 5.


Introduction
In recent years, there has been a considerable interest in the investigation of cosmological models in which the matter does not move orthogonally to the hyper surface of homogeneity.These are called tilted cosmological models.The general behaviors of tilted cosmological models have been studied by King and Ellis [1], Ellis and King [2], Collins and Ellis [3], Bali and Sharma [4,5], Bali and Meena [6].
Bradely and Sviestins [7] investigated that heat flow is expected for cosmological models.Following the development of inflationary models, the importance of scalar fields (mesons) has become well known.Saez and Ballester [8] have developed a new scalar tensor theory of gravitation in which the metric is coupled with a dimensionless scalar field in a simple manner.This coupling gives satisfactory description of the weak fields.In spite of the dimensionless character of the scalar field, an anti-gravity regime appears.This theory suggests a possible way to solve the missing matter problem in non-flat FRW cosmologies.Sing and Agrawal [9], Reddy and Rao [10], Reddy et al. [11], Mohanty and Sahu [12,13], Adhav et al. [14], Tripathy et al. [15] are some of the authors who have studied the various aspects of Saez and Ballester [8] scalar tensor theory.
We derived the field equations for Bianchi type VI 0 metric in Section 2. We solved the field equations in Section 3. We mentioned some physical and geometrical properties of the solutions in Section 4 and also mentioned the concluding remark in Section 5.

Field Equations
Here we consider the Bianchi type VI 0 metric in the form where A, B and C are functions of cosmic time t only and q is a non-zero constant.
The field equations given by Saez and Ballester [8] for the combined scalar and tensor fields are (2) and the scalar field satisfies the equation is the Einstein tensor; n, an arbitrary exponent; and , a dimensionless coupling constant; T is the stress tensor of the matter.The energy momentum tensor for a perfect fluid distribution with heat conduction given by Ellis [2] as where p is the pressure,  is the energy density, i q is the heat conduction vector orthogonal to i u .The fluid vector i u has the components satisfying Equation ( 5) and  is the tilt angle.Here comma and semicolon denote ordinary and co-variant differentiation respectively.With the help of Equations (3-7), the field Equation (2) for the metric (1) in the commoving co-ordinate system take the following explicit forms: Hereafterwards the suffix 4 after a field variable represents ordinary differentiation with respect to time.

Solutions
Equations (8-13) are six equations with eight unknowns A, B, C, p,  , V,  and 1 q , therefore, we require two more conditions.
First we assume that the model is filled with stiff fluid which leads to We also assume that In order to derive exact solutions of the field Equations (8-13) easily, we use the following scale transformations: The field Equations (8-13) reduce to In view of Equation ( 14), Equation (17) and Equation (21), Equations (18,19), yield where   K are arbitrary constants.Thus the corresponding metric of our solution can be written as

Some Physical and Geometrical Properties of the Solutions
On integration Equation (22) yields where K are arbitrary constants.Using Equations (23,24) and Equation (26) in Equations (19, 20), we get Substituting Equations ( 23),( 26) and (27) in Equation (17) we get Further substituting Equations ( 23), ( 27) and (28) in Equation ( 21), we get K q e e e K qe The spatial volume for the model ( 25) is given by From Equations (27-29) we find that the pressure, energy density, tilt angle, heat conduction vector of the fluid distribution are constants at time T=0 and gradually decreases in the course of evolution.Equation (26) shows that the scalar field V changes with time and at time T=0, the scalar filed is found to be a constant.Equation (30) implies the anisotropic expansion of the universe with time.It is interesting to note that the model does not admit singularity throughout evolution.

Conclusions
In this paper we have solved Saez and Ballester field equations for the tilted Bianchi VI 0 cosmological model.It is observed that the pressure, energy density, tilt angle, heat conduction vector of the fluid distribution are constants at time T=0 and gradually decrease with the increase of the age of the universe.It is interesting to note that the models we have constructed here is free from singularity at time T=0 and for 0   the Saez and Ballester [8] theory approaches general relativity.This supports the analysis that the introduction of scalar field avoids initial singularity.