International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 573967 | https://doi.org/10.5402/2012/573967

V. U. M. Rao, M. Vijaya Santhi, "Bianchi Types II, VIII, and IX String Cosmological Models in Brans-Dicke Theory of Gravitation", International Scholarly Research Notices, vol. 2012, Article ID 573967, 8 pages, 2012. https://doi.org/10.5402/2012/573967

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

Academic Editor: W.-H. Steeb
Received07 Aug 2011
Accepted20 Sep 2011
Published28 Nov 2011

Abstract

Bianchi types II, VIII, and IX string cosmological models are obtained and presented in a scalar-tensor theory of gravitation proposed by Brans and Dicke (1961) for 𝜆+𝜌=0. We also established the existence of only Bianchi type IX vacuum cosmological model for 𝜆=𝜌, where 𝜆 and 𝜌 are tension density and energy density of strings, respectively. Some physical and geometrical features of the models are also discussed.

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 by𝐺𝑖𝑗=8𝜋𝜙1𝑇𝑖𝑗𝜔𝜙2𝜙,𝑖𝜙,𝑗12𝑔𝑖𝑗𝜙,𝑘𝜙,𝑘𝜙1𝜙𝑖;𝑗𝑔𝑖𝑗𝜙;𝑘,𝑘,𝜙;𝑘,𝑘=8𝜋(3+2𝜔)1𝑇,(1.1) where 𝐺𝑖𝑗=𝑅𝑖𝑗(1/2)𝑅𝑔𝑖𝑗 is an Einstein tensor, 𝑇𝑖𝑗 is the stress energy tensor of the matter, and 𝜔 is the dimensionless constant.

The equation of motion 𝑇𝑖𝑗,𝑗=0(1.2) 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. [4042] 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 𝑑𝑠2=𝑑𝑡2𝑅2𝑑𝜃2+𝑓2(𝜃)𝑑𝜙2𝑆2[]𝑑𝜓+(𝜃)𝑑𝜙2,(2.1) where (𝜃,𝜙,𝜓) are the Eulerian angles, 𝑅 and 𝑆 are functions of 𝐭 only. It represents Bianchi type II if 𝑓(𝜃)=1 and (𝜃)=𝜃,Bianchi type VIII if 𝑓(𝜃)=cos𝜃 and (𝜃)=sin𝜃,Bianchi type IX if 𝑓(𝜃)=sin𝜃 and (𝜃)=cos𝜃.

The energy momentum tensor for cosmic strings [17] is 𝑇𝑖𝑗=𝜌𝑢𝑖𝑢𝑗𝜆𝑥𝑖𝑥𝑗,(2.2) where 𝑢𝑖 is the four-velocity of the string cloud, 𝑥𝑖 is the direction of anisotropy, 𝜌 and 𝜆 are the rest energy density and the tension density of the string cloud, respectively. The string source is along the 𝑍-axis which is the axis of symmetry. Orthonormalisation of 𝑢𝑖 and 𝑥𝑖 is given as 𝑢𝑖𝑢𝑖=𝑥𝑖𝑥𝑖=1,𝑢𝑖𝑥𝑖=0.(2.3) In the commoving coordinate system, we have from (2.2) and (2.3) 𝑇11=𝑇22=0,𝑇33=𝜆,𝑇44=𝜌,𝑇𝑖𝑗=0for𝑖𝑗.(2.4) The quantities 𝜌,𝜆 and the scalar field 𝜙 in the theory depend on 𝑡 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 ̈𝑅𝑅+̈𝑆𝑆+̇𝑅̇𝑆+𝑆𝑅𝑆24𝑅4+𝜔̇𝜙22𝜙2+̈𝜙𝜙+̇𝑅̇𝜙+̇𝑆̇𝜙𝑅𝜙2̈𝑅𝑆𝜙=0,𝑅+̇𝑅2+𝛿𝑅23𝑆24𝑅4+𝜔̇𝜙22𝜙2+̈𝜙𝜙+2̇𝑅̇𝜙=𝑅𝜙8𝜋𝜆𝜙,2̇𝑅̇𝑆𝑆𝑅𝑆24𝑅4+̇𝑅2+𝛿𝑅2𝜔̇𝜙22𝜙2+2̇𝑅̇𝜙+̇𝑆̇𝜙𝑅𝜙=𝑆𝜙8𝜋𝜌𝜙,̇𝑆𝑆̇𝑅𝑅̇𝜙(𝜃)𝜙̈̇𝜙2̇𝑅=0,𝜙+𝑅+̇𝑆𝑆=8𝜋̇𝑅3+2𝜔(𝜆+𝜌),̇𝜌+2𝜌𝑅̇𝑆+(𝜌𝜆)𝑆=0,(3.1) where “.” denotes differentiation with respect to “𝑡”.

When 𝛿=0,1 & +1, the field equation (3.1) correspond to the Bianchi types II, VIII, and IX universes, respectively.

Using the transformation 𝑅=𝑒𝛼, 𝑆=𝑒𝛽, 𝑑𝑡=𝑅2𝑆𝑑𝑇, (3.1) reduce to 𝛼+𝛽𝛼22𝛼𝛽+𝑒4𝛽4+𝜔𝜙22𝜙2𝛼𝜙𝜙+𝜙𝜙=0,(3.2)2𝛼𝛼22𝛼𝛽+𝛿𝑒(2𝛼+2𝛽)34𝑒4𝛽+𝜔𝜙22𝜙2𝛽𝜙𝜙+𝜙𝜙=8𝜋𝜆𝜙𝑒(4𝛼+2𝛽),(3.3)2𝛼𝛽+𝛼2+𝛿𝑒(2𝛼+2𝛽)14𝑒4𝛽𝜔𝜙22𝜙2+𝛽𝜙𝜙+2𝛼𝜙𝜙=8𝜋𝜌𝜙𝑒(4𝛼+2𝛽)𝛼,(3.4)𝛽(𝜃)𝜙𝜙𝜙=0,(3.5)=8𝜋3+2𝜔(𝜆+𝜌)𝑒(4𝛼+2𝛽)𝜌,(3.6)+2𝜌𝛼+(𝜌𝜆)𝛽=0,(3.7) where “” denotes differentiation with respect to “𝑇”.

Since we are considering the Bianchi types II, VIII, and IX metrics, we have (𝜃)=𝜃, (𝜃)=sinh𝜃, and (𝜃)=cos𝜃 for Bianchi types II, VIII, and IX metrics, respectively. Therefore, from (3.5), we will consider the following possible cases with (𝜃)0: (1)𝛼𝛽=0,𝜙0,(2)𝛼𝛽0,𝜙=0,(3)𝛼𝛽=0,𝜙=0.(3.8)

Case 1 (for 𝛼𝛽=0 and 𝜙0). Here, we get 𝛼=𝛽+𝑐.
Without loss of generality by taking the constant of integration 𝑐=0, we get 𝛼=𝛽.(3.9) By using (3.9), (3.2) to (3.7) will reduce to 2𝛽3𝛽2+𝑒4𝛽4+𝜔𝜙22𝜙2𝛽𝜙𝜙+𝜙𝜙=0,(3.10)2𝛽3𝛽2+𝛿𝑒4𝛽34𝑒4𝛽+𝜔𝜙22𝜙2𝛽𝜙𝜙+𝜙𝜙=8𝜋𝜆𝜙𝑒6𝛽,(3.11)3𝛽2+𝛿𝑒4𝛽14𝑒4𝛽𝜔𝜙22𝜙2+3𝛽𝜙𝜙=8𝜋𝜌𝜙𝑒6𝛽𝜙,(3.12)=8𝜋3+2𝜔(𝜆+𝜌)𝑒6𝛽𝜌,(3.13)+(3𝜌𝜆)𝛽=0,(3.14) where “” denotes differentiation with respect to “𝑇”.

Latelier [17] discussed, in general, the following equations of state: 𝜌=𝜆(geometricstring),𝜌=(1+𝜔)𝜆(𝑝-string),(3.15) and Reddy [10] obtained inflationary string cosmological models in Brans-Dicke scalar-tensor theory of gravitation for 𝜌+𝜆=0(Reddystring).(3.16) Here we will present string cosmological models corresponding to 𝜌+𝜆=0 and 𝜌=𝜆.

Case 2 (for 𝜆+𝜌=0 (Reddy string)). From (3.13), we get 𝜙=0,(3.17) then 𝜙=𝑎𝑇+𝑏.(3.18) Without loss of generality by taking the constants of integration 𝑎=1 and 𝑏=0, we get 𝜙=𝑇.(3.19) Now from (3.11), (3.12), and (3.19), we get 2𝛽+2𝛿𝑒4𝛽𝑒4𝛽+2𝛽𝑇=0.(3.20)
For Bianchi Type II Metric (𝛿=0)
From (3.20), we get 𝑇𝛽+𝛽=12𝑇𝑒4𝛽.(3.21) From (3.21), we get 𝑒4𝛽=𝑐12𝑇2cosech2𝑐1log𝑇+𝑐2,(3.22) where 𝑐1 and 𝑐2 are integration constants.
Using (3.22) in (3.11) and (3.12), we have 𝑇8𝜋𝜌=8𝜋𝜆=22𝑐1𝑐sinh1log𝑇+𝑐2.(3.23) The corresponding metric can be written in the form 𝑑𝑠2=𝑐13𝑇3cos𝑒𝑐3𝑐1log𝑇+𝑐2𝑑𝑇2𝑐1𝑇𝑐cos𝑒𝑐1log𝑇+𝑐2𝑑𝜃2+𝑑𝜙2𝑐1𝑇𝑐cos𝑒𝑐1log𝑇+𝑐2[]𝑑𝜓+𝑑𝜙2.(3.24) 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 (𝛿=1)
From (3.20), we get 𝑇𝛽+𝛽=32𝑇𝑒4𝛽.(3.25) From (3.25), we get 𝑒4𝛽=𝑐323𝑇2cosech2𝑐3log𝑇+𝑐4,(3.26) where 𝑐3 and 𝑐4 are integration constants.
Using (3.26) in (3.11) and (3.12), we have 8𝜋𝜌=8𝜋𝜆=3𝑇2𝑐3𝑐sinh3log𝑇+𝑐4.(3.27) The corresponding metric can be written in the form 𝑑𝑠2=𝑐339𝑇3cos𝑒𝑐3𝑐3log𝑇+𝑐4𝑑𝑇2𝑐3𝑐3𝑇cos𝑒𝑐3log𝑇+𝑐4𝑑𝜃2+cosh2𝑑𝜙2𝑐1𝑐3𝑇cos𝑒𝑐3log𝑇+𝑐4[]𝑑𝜓+sinh𝜃𝑑𝜙2.(3.28) 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 (𝛿=1)
From (3.20), we get 𝑇𝛽+𝛽1=2𝑇𝑒4𝛽.(3.29) From (3.29), we get 𝑒4𝛽=𝑐52𝑇2sec2𝑐5log𝑇+𝑐6,(3.30) where 𝑐5 and 𝑐6 are integration constants.
Using (3.30) in (3.11) and (3.12), we have 𝑇8𝜋𝜌=8𝜋𝜆=22𝑐5𝑐cosh5log𝑇+𝑐6.(3.31) The corresponding metric can be written in the form 𝑑𝑠2=𝑐53𝑇3sec3𝑐5log𝑇+𝑐6𝑑𝑇2𝑐5𝑇𝑐sec5log𝑇+𝑐6𝑑𝜃2+sin2𝜃𝑑𝜙2𝑐5𝑇𝑐sec5log𝑇+𝑐6[]𝑑𝜓+cos𝜃𝑑𝜙2.(3.32) 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 𝑉, expansion 𝜃, and shear 𝜎 for the models (3.24), (3.28), and (3.32) are given by 𝑉=(𝑔)1/2=𝑐13/2𝑇3/2cos𝑒𝑐3/2𝑐1log𝑇+𝑐2,𝜃=𝑢;𝑖𝑖=3𝑐1𝑐2𝑇coth1log𝑇+𝑐2,𝜎2=3𝑐128𝑇2coth2𝑐1log𝑇+𝑐2(3.33) for the Bianchi type II model, 𝑉=(𝑔)1/2=𝑐33/2𝑇3/2cos𝑒𝑐3/2𝑐3log𝑇+𝑐4cosh𝜃,𝜃=𝑢;𝑖𝑖=3𝑐3𝑐2𝑇coth3log𝑇+𝑐4,𝜎2=3𝑐328𝑇2coth2𝑐3log𝑇+𝑐4(3.34) for the Bianchi type VIII model, and 𝑉=(𝑔)1/2=𝑐53/2𝑇3/2sec3/2𝑐5log𝑇+𝑐6sin𝜃,𝜃=𝑢;𝑖𝑖=3𝑐5𝑐2𝑇tanh5log𝑇+𝑐6,𝜎2=3𝑐528𝑇2tanh2𝑐5log𝑇+𝑐6(3.35) for the Bianchi type IX model.

Case 3 (for 𝜆=𝜌 (geometric string)). From (3.14), we get 𝜌=𝑐1𝑒2𝛽.(3.36) From (3.11) and (3.12), we get (𝛿1)=8𝜋𝜌𝜙𝑒2𝛽.(3.37) From (3.37) and (3.36), we get (𝛿1)=8𝜋𝜙𝑐1.(3.38)
For Bianchi Types II and VIII Metrics (𝛿=0and1)
From (3.38), we have 𝜙=constantsay𝑐2.(3.39) Using (3.39), the field equations (3.10) to (3.13) reduce to 2𝛽3𝛽2+𝑒4𝛽4=0,(3.40)2𝛽3𝛽2+𝛿𝑒4𝛽34𝑒4𝛽=8𝜋𝜌𝑐2𝑒6𝛽,(3.41)3𝛽2+𝛿𝑒4𝛽14𝑒4𝛽=8𝜋𝜌𝑐2𝑒6𝛽,(3.42)0=8𝜋3+2𝜔(2𝜌)𝑒6𝛽.(3.43) From (3.43), we get 𝜌=0,(3.44) and, since 𝜆=𝜌, we will get 𝜆=0.
From (3.40) to (3.42), we get 2𝛽+𝑒4𝛽=0.(3.45) From (3.45), we get 𝑆2=𝑒2𝛽=2𝑚1sec2𝑚1𝑇+𝑛1.(3.46) The corresponding metrics can be written in the form 𝑑𝑠2=8𝑚13sec32𝑚1𝑇+𝑛1𝑑𝑇22𝑚1sec2𝑚1𝑇+𝑛1𝑑𝜃2+𝑑𝜙22𝑚1sec2𝑚1𝑇+𝑛1[]𝑑𝜓+𝜃𝑑𝜙2,𝑑𝑠2=8𝑚13sec32𝑚1𝑇+𝑛1𝑑𝑇22𝑚1sec2𝑚1𝑇+𝑛1𝑑𝜃2+cosh2𝜃𝑑𝜙22𝑚1sec2𝑚1𝑇+𝑛1[]𝑑𝜓+sinh𝜃𝑑𝜙2.(3.47) 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 (𝛿=1)
From (3.38), we have 𝑐1=0.(3.48) Using (3.48) in (3.36), we get 𝜌=0.(3.49) Using (3.49) the field equations (3.10) to (3.14) reduce to 2𝛽3𝛽2+𝑒4𝛽4+𝜔𝜙22𝜙2𝛽𝜙𝜙+𝜙𝜙=0,(3.50)3𝛽2+𝛿𝑒4𝛽14𝑒4𝛽𝜔𝜙22𝜙2+3𝛽𝜙𝜙𝜙=0,(3.51)=0.(3.52) From (3.52), we get 𝜙=𝑐3𝑇+𝑐4.(3.53) Without loss of generality by taking the constants of integration 𝑐3=1 and 𝑐4=0, we get 𝜙=𝑇.(3.54) From (3.50), (3.51), and (3.54), we get 2𝛽+𝑒4𝛽+2𝛽𝑇=0,(3.55) that is, 𝑇𝛽+𝛽1=2𝑇𝑒4𝛽.(3.56) From (3.56), we get 𝑒4𝛽=𝑐12𝑇2sec2𝑐1log𝑇+𝑐2.(3.57) The corresponding metric can be written in the form 𝑑𝑠2=𝑐13𝑇3sec3𝑐1log𝑇+𝑐2𝑑𝑇2𝑐1𝑇𝑐sec1log𝑇+𝑐2𝑑𝜃2+sin2𝜃𝑑𝜙2𝑐1𝑇𝑐sec1log𝑇+𝑐2[]𝑑𝜓+cos𝜃𝑑𝜙2.(3.58) 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 𝑉, expansion 𝜃, and the shear 𝜎 for the models (3.47) and (3.58) are given by 𝑉=(𝑔)1/2=8𝑚13sec32𝑚1𝑇+𝑛1,𝜃=6𝑚1tanh2𝑚1𝑇+𝑛1,𝜎2=6𝑚21sec22𝑚1𝑇+𝑛1(3.59) for the Bianchi type II cosmological model (𝛿=0), 𝑉=(𝑔)1/2=8𝑚13sec32𝑚1𝑇+𝑛1cosh𝜃,𝜃=2𝑚1tanh2𝑚1𝑇+𝑛1,𝜎cosh𝜃+sinh𝜃2=2𝑚21tanh22𝑚1𝑇+𝑛1cosh2𝜃+sinh2𝜃+4𝑚1tanh2𝑚1𝑇+𝑛1cosh𝜃sinh𝜃(3.60) for the Bianchi type VIII cosmological model (𝛿=1), and 𝑉=(𝑔)1/2=𝑐13/2𝑇3/2sec3/2𝑐1log𝑇+𝑐2,𝜃=𝑢;𝑖𝑖=3𝑐1𝑐2𝑇tanh1log𝑇+𝑐2,𝜎2=3𝑐128𝑇2tanh2𝑐1log𝑇+𝑐2(3.61) for the Bianchi type IX cosmological model (𝛿=1).

Case 4 (for 𝛼𝛽0 and 𝜙=0). 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 𝛼𝛽=0 and 𝜙=0). Here, we get 𝛼=𝛽+𝑐.
Without loss of generality by taking the constant of integration 𝑐=0, we get 𝛼=𝛽,𝜙=constantsay𝑐1.(3.62) Using (3.62), the field equations (3.2) to (3.7) will reduce to 2𝛽3𝛽2+𝑒4𝛽4=0,(3.63)2𝛽3𝛽2+𝛿𝑒4𝛽34𝑒4𝛽=8𝜋𝜆𝑐1𝑒6𝛽,(3.64)3𝛽2+𝛿𝑒4𝛽14𝑒4𝛽=8𝜋𝜌𝑐1𝑒6𝛽,(3.65)0=8𝜋3+2𝜔(𝜌+𝜆)𝑒6𝛽𝜌,(3.66)+(3𝜌𝜆)𝛽=0.(3.67) From (3.66), we get 𝜌+𝜆=0.(3.68) From (3.63) to (3.65) and (3.68), we have 2𝛽+2𝛿𝑒4𝛽𝑒4𝛽=0.(3.69)For Bianchi Type II Metric (𝛿=0)
From (3.69), we get 𝑒𝛽=(𝑎𝑇+𝑏)1/2,where𝑎2=1.(3.70) From (3.64), (3.65), and (3.70), we have 8𝜋𝜆=𝑐1(𝑎𝑇+𝑏)2𝑐,8𝜋𝜌=1(𝑎𝑇+𝑏)2.(3.71) From (3.71) we get 𝜆+𝜌=0.
The corresponding metric can be written in the form 𝑑𝑠2=(𝑎𝑇+𝑏)3𝑑𝑇2(𝑎𝑇+𝑏)1𝑑𝜃2+𝑑𝜙2(𝑎𝑇+𝑏)1[]𝑑𝜓+𝜃𝑑𝜙2.(3.72) 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 (𝛿=1)
From (3.69), we get 𝑒𝛽=(𝑎𝑇+𝑏)1/2,where𝑎2=3.(3.73) From (3.64) and (3.65), we have 8𝜋𝜆=𝑐1(𝑎𝑇+𝑏),8𝜋𝜌=𝑐1(𝑎𝑇+𝑏).(3.74) Therefore, from (3.74), we have 𝜆+𝜌=0.(3.75) The corresponding metric can be written in the form 𝑑𝑠2=(𝑎𝑇+𝑏)3𝑑𝑇2(𝑎𝑇+𝑏)1𝑑𝜃2+cosh2𝜃𝑑𝜙2(𝑎𝑇+𝑏)1[]𝑑𝜓+sinh𝜃𝑑𝜙2.(3.76) 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 (𝛿=1)
From (3.69), we get 2𝛽+𝑒4𝛽=0.(3.77) From (3.77), we get 𝑆2=𝑒2𝛽=2𝑚1sec2𝑚1𝑇+𝑛1.(3.78) From (3.63) and (3.64), we get (𝛿1)=8𝜋𝜆𝑐1𝑒2𝛽.(3.79) From (3.66) & (3.79), we get 𝜆=𝜌=0.(3.80) The corresponding metric can be written in the form 𝑑𝑠2=8𝑚13sec32𝑚1𝑇+𝑛1𝑑𝑇22𝑚1sec2𝑚1𝑇+𝑛1𝑑𝜃2+sin2𝜃𝑑𝜙22𝑚1sec2𝑚1𝑇+𝑛1[]𝑑𝜓+cos𝜃𝑑𝜙2.(3.81) 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 𝑉, expansion 𝜃, and the shear 𝜎 for the models (3.72), (3.76), and (3.81) are given by 𝑉=(𝑔)1/2=(𝑎𝑇+𝑏)3/2,𝜃=3𝑎(𝑎𝑇+𝑏),𝜎2=3𝑎22(𝑎𝑇+𝑏)2(3.82) for the Bianchi type II model, 𝑉=(𝑔)1/2=(𝑎𝑇+𝑏)3/2cosh𝜃,𝜃=tanh𝜃(𝑎𝑇+𝑏)3/23𝑎(𝑎𝑇+𝑏)1/2,𝜎2=tanh2𝜃(𝑎𝑇+𝑏)3+3𝑎2(𝑎𝑇+𝑏)𝑎tanh𝜃(𝑎𝑇+𝑏)(3.83) for the Bianchi type VIII model, and 𝑉=(𝑔)1/2=8𝑚13sec32𝑚1𝑇+𝑛1sin𝜃,𝜃=𝑢;𝑖𝑖=2𝑚1tanh2𝑚1𝑇+𝑛1,𝜎sin𝜃+sin2𝜃2=2𝑚12tanh22𝑚1+2cos2𝜃+4𝑚1tanh2𝑚1cos𝜃3,(3.84) where =(𝑇+𝑛1) 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 𝜆+𝜌=0, 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 𝑇; 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 𝜆=𝜌, 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 𝑇, 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 𝜆+𝜌=0 and also got Bianchi type IX vacuum cosmological model of general relativity, since the scalar field 𝜙 is constant. The spatial volume of the models (3.72), (3.76), and (3.81) are decreasing as 𝑇; that is, the models are contracting with the increase of time. Also the models (3.72) and (3.76) have initial singularity at 𝑇=𝑏/𝑎, 𝑎0, and the model (3.81) has no initial singularity.

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Copyright © 2012 V. U. M. Rao and M. Vijaya Santhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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