1. Introduction

Brans and Dicke [1] introduced a scalar-tensor theory of gravitation involving a scalar function in addition to the familiar general relativistic metric tensor. In this theory the scalar field has the dimension of inverse of the gravitational constant, and its role is confined to its effects on gravitational field equations.

Brans-Dicke field equations for combined scalar and tensor field are given by where is an Einstein tensor, is the stress energy tensor of the matter, and is the dimensionless constant.

The equation of motion is a consequence of the field equation (1.1).

Several aspects of Brans-Dicke cosmology have been extensively investigated by many authors. The work of Singh and Rai [2] gives a detailed survey of Brans-Dicke cosmological models discussed by several authors. Nariai [3], Belinskii and Khalatnikov [4], Reddy and Rao [5], Banerjee and Santos [6], Ram [7], Ram and Singh [8], Berman et al. [9], Reddy [10], Reddy and Naidu [11], Adhav et al. [12], and Rao et al. [13] are some of the authors who have investigated several aspects of this theory.

In recent years, there has been a considerable interest in cosmological models in Einstein’s theory and in several alternative theories of gravitation with cosmic string source. Cosmic strings and domain walls are the topological defects associated with spontaneous symmetry breaking whose plausible production site is cosmological phase transitions in the early universe (Kibble [14]). The gravitational effects of cosmic strings have been extensively discussed by Vilenkin [15], Gott [16], Latelier [17], and Stachel [18] in general relativity. Relativistic string models in the context of Bianchi space times have been obtained by Krori et al. [19], Banarjee et al. [20], Tikekar and Patel [21], and Bhattacharjee and Baruah [22]. String cosmological models in scalar-tensor theories of gravitation have been investigated by Sen [23], Barros et al. [24], Banerjee et al. [25], Gundlach and Ortiz [26], Barros and Romero [27], Pradhan [28], Mohanty et al. [29], and others.

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 types II, VIII, and IX universes is important because familiar solutions like FRW universe with positive curvature, the de Sitter universe, the Taub-NUT solutions, and so forth correspond to Bianchi types II, VIII, and IX space-times. Chakraborty [30], Bali and Dave [31], and Bali and Yadav [32] studied Bianchi type IX string as well as viscous fluid models in general relativity. Reddy et al. [33] studied Bianchi types II, VIII, and IX models in scale covariant theory of gravitation. Shanthi and Rao [34] studied Bianchi types VIII and IX models in Lyttleton-Bondi universe. Also Rao and Sanyasiraju [35] and Sanyasirajuand Rao [36] have studied Bianchi types VIII and IX models in Zero mass scalar fields and self-creation cosmology. Rahaman et al. [37] have investigated Bianchi type IX string cosmological model in a scalar-tensor theory formulated by Sen [38] based on Lyra [39] manifold. Rao et al. [40–42] have obtained Bianchi types II, VIII, and IX string cosmological models, perfect fluid cosmological models in Saez-Ballester theory of gravitation, and string cosmological models in general relativity as well as self-creation theory of gravitation, respectively.

In this paper we will discuss Bianchi types II, VIII, and IX string cosmological models in a scalar-tensor theory proposed by Brans and Dicke [1].

2. Metric and Energy Momentum Tensor

We consider a spatially homogeneous Bianchi types II, VIII, and IX metrics of the form where are the Eulerian angles, and are functions of only. It represents Bianchi type II if and ,Bianchi type VIII if and ,Bianchi type IX if and .

The energy momentum tensor for cosmic strings [17] is where is the four-velocity of the string cloud, is the direction of anisotropy, and are the rest energy density and the tension density of the string cloud, respectively. The string source is along the -axis which is the axis of symmetry. Orthonormalisation of and is given as In the commoving coordinate system, we have from (2.2) and (2.3) The quantities and the scalar field in the theory depend on only.

3. Bianchi Types II, VIII, and IX String Cosmological Models in Brans-Dicke Theory of Gravitation

The field equations (1.1), (1.2) for the metric (2.1) with the help of (2.2), (2.3), and (2.4) can be written as where “.” denotes differentiation with respect to “”.

When & , the field equation (3.1) correspond to the Bianchi types II, VIII, and IX universes, respectively.

Using the transformation , , , (3.1) reduce to where “” denotes differentiation with respect to “”.

Since we are considering the Bianchi types II, VIII, and IX metrics, we have , , and for Bianchi types II, VIII, and IX metrics, respectively. Therefore, from (3.5), we will consider the following possible cases with :

Case 1 (for and ). Here, we get .
Without loss of generality by taking the constant of integration , we get By using (3.9), (3.2) to (3.7) will reduce to where “” denotes differentiation with respect to “”.

Latelier [17] discussed, in general, the following equations of state: and Reddy [10] obtained inflationary string cosmological models in Brans-Dicke scalar-tensor theory of gravitation for Here we will present string cosmological models corresponding to and .

Case 2 (for (Reddy string)). From (3.13), we get then Without loss of generality by taking the constants of integration and , we get Now from (3.11), (3.12), and (3.19), we get
For Bianchi Type II Metric
From (3.20), we get From (3.21), we get where and are integration constants.
Using (3.22) in (3.11) and (3.12), we have The corresponding metric can be written in the form Thus, (3.24) together with (3.23) constitutes the Bianchi type II string cosmological model in Brans-Dicke theory of gravitation.For Bianchi Type VIII Metric
From (3.20), we get From (3.25), we get where and are integration constants.
Using (3.26) in (3.11) and (3.12), we have The corresponding metric can be written in the form Thus, (3.28) together with (3.27) constitutes the Bianchi type VIII string cosmological model in Brans-Dicke theory of gravitation. For Bianchi Type IX Metric
From (3.20), we get From (3.29), we get where and are integration constants.
Using (3.30) in (3.11) and (3.12), we have The corresponding metric can be written in the form Thus, (3.32) together with (3.31) constitutes the Bianchi type IX string cosmological model in Brans-Dicke theory of gravitation.

3.1. Physical and Geometrical Properties

The volume element , expansion , and shear for the models (3.24), (3.28), and (3.32) are given by for the Bianchi type II model, for the Bianchi type VIII model, and for the Bianchi type IX model.

Case 3 (for (geometric string)). From (3.14), we get From (3.11) and (3.12), we get From (3.37) and (3.36), we get
For Bianchi Types II and VIII Metrics
From (3.38), we have Using (3.39), the field equations (3.10) to (3.13) reduce to From (3.43), we get and, since , we will get .
From (3.40) to (3.42), we get From (3.45), we get The corresponding metrics can be written in the form Thus, (3.47) together with (3.44) constitutes an exact Bianchi types II and VIII vacuum cosmological models, respectively, in general relativity. For Bianchi Type IX Metric
From (3.38), we have Using (3.48) in (3.36), we get Using (3.49) the field equations (3.10) to (3.14) reduce to From (3.52), we get Without loss of generality by taking the constants of integration and , we get From (3.50), (3.51), and (3.54), we get that is, From (3.56), we get The corresponding metric can be written in the form Thus, (3.58) together with (3.49) constitutes an exact Bianchi type IX vacuum cosmological model in Brans-Dicke theory of gravitation.

3.2. Physical and Geometrical Properties

The spatial volume , expansion , and the shear for the models (3.47) and (3.58) are given by for the Bianchi type II cosmological model , for the Bianchi type VIII cosmological model , and for the Bianchi type IX cosmological model .

Case 4 (for and ). In this case, we get Bianchi types II, VIII, and IX string cosmological models in general relativity as obtained and presented by Rao et al. [42].

Case 5 (for and ). Here, we get .
Without loss of generality by taking the constant of integration , we get Using (3.62), the field equations (3.2) to (3.7) will reduce to From (3.66), we get From (3.63) to (3.65) and (3.68), we have For Bianchi Type II Metric
From (3.69), we get From (3.64), (3.65), and (3.70), we have From (3.71) we get .
The corresponding metric can be written in the form Thus, (3.72) together with (3.71) constitutes an exact Bianchi type II string cosmological model in general theory of relativity.For Bianchi Type VIII Metric
From (3.69), we get From (3.64) and (3.65), we have Therefore, from (3.74), we have The corresponding metric can be written in the form Thus, (3.76) together with (3.74) constitutes an exact Bianchi type VIII string cosmological model in general theory of relativity.For Bianchi Type IX Metric
From (3.69), we get From (3.77), we get From (3.63) and (3.64), we get From (3.66) & (3.79), we get The corresponding metric can be written in the form Thus, (3.81) together with (3.80) constitutes an exact Bianchi type IX vacuum cosmological model in general theory of relativity.

3.3. Physical and Geometrical Properties

The spatial volume , expansion , and the shear for the models (3.72), (3.76), and (3.81) are given by for the Bianchi type II model, for the Bianchi type VIII model, and where for the Bianchi type IX model.

4. Conclusions

In view of the importance of Bianchi types II, VIII, and IX space times and cosmic strings in the study of relativistic cosmology and astrophysics, in this paper we have studied and presented Bianchi types II, VIII, and IX string cosmological models in Brans-Dicke theory of gravitation.

In case of (1.1), for the equation of state , the models (3.24), (3.28), and (3.32) represent, respectively, Bianchi types II, VIII, and IX string cosmological models in Brans-Dicke theory of gravitation. The spatial volume of the models (3.24), (3.28), and (3.32) are decreasing as ; that is, the models are contacting with the increase of time. Also, the models have no initial singularity.

In Case of 3, for the equation of state , we will get only Bianchi type IX vacuum cosmological model in Brans-Dicke theory of gravitation. Also, in this case, we established the nonexistence of Bianchi types II and VIII geometric string cosmological models in Brans-Dicke theory of gravitation and hence presented only vacuum cosmological models of general relativity. The volume of all the models is decreasing as , and also the models are free from singularities.

In Case 5, we obtained only Bianchi types II and VIII string cosmological models of general relativity with and also got Bianchi type IX vacuum cosmological model of general relativity, since the scalar field is constant. The spatial volume of the models (3.72), (3.76), and (3.81) are decreasing as ; that is, the models are contracting with the increase of time. Also the models (3.72) and (3.76) have initial singularity at , , and the model (3.81) has no initial singularity.