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

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 by𝐺_𝑖𝑗=−8𝜋𝜙⁻¹𝑇_𝑖𝑗−𝜔𝜙⁻²𝜙^,𝑖𝜙_,𝑗−12𝑔^𝑖𝑗−𝜙₋₁𝜙_,𝑘𝜙^,𝑘𝜙_;𝑘,𝑘𝜙_𝑖;𝑗−𝑔_𝑖𝑗𝜙_;𝑘,𝑘,=8𝜋(3+2𝜔)⁻¹𝑇,(1.1) where 𝐺_𝑖𝑗=𝑅_𝑖𝑗−(1/2)𝑅𝑔_𝑖𝑗 is an Einstein tensor, 𝑇_𝑖𝑗 is the stress energy tensor of the matter, and 𝜔 is the dimensionless constant.

The equation of motion 𝑇^{𝑖𝑗,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. [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 𝑑𝑠²=𝑑𝑡²−𝑅²𝑑𝜃²+𝑓²(𝜃)𝑑𝜙²−𝑆²[𝑑𝜓+ℎ(𝜃)𝑑𝜙]²,(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 ℎ(𝜃)=𝜃,Bianchi type VIII if 𝑓(𝜃)=cosℎ𝜃 and ℎ(𝜃)=sinℎ𝜃,Bianchi type IX if 𝑓(𝜃)=sin𝜃 and ℎ(𝜃)=cos𝜃.

The energy momentum tensor for cosmic strings [17] is 𝑇_𝑖𝑗=𝜌𝑢_𝑖𝑢_𝑗−𝜆𝑥_𝑖𝑥_𝑗,(2.2) where 𝑢_𝑖 is the four-velocity of the string cloud, 𝑥_𝑖 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 𝑢^𝑖 and 𝑥^𝑖 is given as 𝑢^𝑖𝑢_𝑖=−𝑥^𝑖𝑥_𝑖=1,𝑢^𝑖𝑥_𝑖=0.(2.3) In the commoving coordinate system, we have from (2.2) and (2.3) 𝑇¹¹=𝑇²²=0,𝑇³³=𝜆,𝑇⁴⁴=𝜌,𝑇^𝑖𝑗=0for𝑖≠𝑗.(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 𝑅̈𝑅+𝑆𝑆̈+𝑅𝑅̇𝑆𝑆̇+4𝑆𝑅²⁴+𝜔𝜙̇²2𝜙²+𝜙̈2𝑅̈𝜙+𝑅𝑅̇𝜙𝜙̇+𝑆𝑆̇𝜙𝜙̇=0,𝑅+𝑅̇²+𝛿𝑅²−3𝑆²4𝑅⁴+𝜔𝜙̇²2𝜙²+𝜙̈2𝑅̇𝑆̇𝜙+2𝑅𝑅𝜙̇𝜙̇=8𝜋𝜙𝜆,𝑅𝑆−4𝑆𝑅²⁴+𝑅̇²+𝛿𝑅²−𝜔𝜙̇²2𝜙²+2𝑅̇𝜙̇𝑆̇𝑅𝜙+𝑆𝑆̇𝜙𝜙̇=8𝜙𝜋𝜌,𝜙̈+𝜙̇2𝑆𝑅̇−𝑅𝑅̇ℎ(𝜙𝜃)𝜙̇=0,𝜌̇+𝑅2𝜌+𝑅̇𝑆𝑆̇=3+8𝜋2𝜔(𝜆+𝜌),𝑅+(𝜌−𝜆)𝑆𝑆̇=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=𝑒^𝛼, 𝑆=𝑒^𝛽, 𝑑𝑡=𝑅²𝑆𝑑𝑇, (3.1) reduce to 𝛼^+𝛽^−𝛼^2−2𝛼^𝛽^+𝑒^4𝛽4+𝜔2𝜙𝜙^22−𝛼^𝜙^2𝛼_𝜙+𝜙𝜙^=0,(3.2)−𝛼^2−2𝛼^𝛽^+𝛿𝑒^{(2𝛼+2𝛽)}−34𝑒^4𝛽+𝜔𝜙^22𝜙²−𝛽^𝜙^𝜙+𝜙𝜙^=8𝜋𝜙𝜆𝑒^{(4𝛼+2𝛽)}2𝛼_,(3.3)𝛽^+𝛼^2+𝛿𝑒^{(2𝛼+2𝛽)}−14𝑒^4𝛽−𝜔𝜙^22𝜙²+𝛽^𝜙^𝜙+2𝛼^𝜙^𝜙=8𝜙𝜋𝜌𝑒^{(4𝛼+2𝛽)}𝛼,(3.4)_−𝛽^ℎ(𝜃)𝜙^𝜙𝜙=0,(3.5)_=8𝜋3+2𝜔(𝜆+𝜌)𝑒^{(4𝛼+2𝛽)}𝜌,(3.6)_+2𝜌𝛼^+(𝜌−𝜆)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 ℎ(𝜃)=𝜃, ℎ(𝜃)=sinh𝜃, and ℎ(𝜃)=cos𝜃 for Bianchi types II, VIII, and IX metrics, respectively. Therefore, from (3.5), we will consider the following possible cases with ℎ(𝜃)≠0: ((21))𝛼𝛼^_−𝛽^=0,𝜙^≠0,(3)𝛼_−𝛽^≠0,𝜙′=0,−𝛽^=0,𝜙′=0.(3.8)

Case 1 (for 𝛼^−𝛽^=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.(3.9) By using (3.9), (3.2) to (3.7) will reduce to 2𝛽^−3𝛽^2+𝑒^4𝛽4+𝜔2𝜙𝜙^22−𝛽^𝜙^2𝛽_𝜙+𝜙𝜙^=0,(3.10)−3𝛽^2+𝛿𝑒^4𝛽−34𝑒^4𝛽+𝜔𝜙^22𝜙²−𝛽^𝜙^𝜙+𝜙𝜙^=8𝜋𝜙𝜆𝑒^6𝛽3𝛽_2,(3.11)+𝛿𝑒^4𝛽−14𝑒^4𝛽−𝜔𝜙^22𝜙²+3𝛽^𝜙^𝜙=8𝜙𝜋𝜌𝑒^6𝛽𝜙,(3.12)_=8𝜋3+2𝜔(𝜆+𝜌)𝑒^6𝛽𝜌,(3.13)_+(3𝜌−𝜆)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^+2𝛿𝑒^4𝛽−𝑒^4𝛽+2𝛽^𝑇=0.(3.20)
For Bianchi Type II Metric (𝛿=0)
From (3.20), we get 𝑇𝛽^+𝛽^=12𝑇𝑒^4𝛽.(3.21) From (3.21), we get 𝑒^4𝛽=𝑐¹²𝑇²cosech²𝑐₁log𝑇+𝑐₂,(3.22) where 𝑐₁ and 𝑐₂ are integration constants.
Using (3.22) in (3.11) and (3.12), we have 8𝜋𝜌=−8𝜋𝜆=𝑇²2𝑐₁sinh𝑐₁log𝑇+𝑐₂.(3.23) The corresponding metric can be written in the form 𝑑𝑠²=𝑐¹³𝑇³cos𝑒𝑐ℎ³𝑐₁log𝑇+𝑐₂𝑑𝑇²−𝑐¹𝑇cos𝑒𝑐ℎ𝑐¹−𝑐₁log𝑇+𝑐₂𝑑𝜃²+𝑑𝜙²𝑇cos𝑒𝑐ℎ𝑐¹log𝑇+𝑐₂[𝑑𝜓+𝑑𝜙]².(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^=32𝑇𝑒^4𝛽.(3.25) From (3.25), we get 𝑒^4𝛽=𝑐³²3𝑇²cosech²𝑐₃log𝑇+𝑐₄,(3.26) where 𝑐₃ and 𝑐₄ are integration constants.
Using (3.26) in (3.11) and (3.12), we have 8𝜋𝜌=−8𝜋𝜆=√3𝑇²𝑐₃sinh𝑐₃log𝑇+𝑐₄.(3.27) The corresponding metric can be written in the form 𝑑𝑠²=𝑐³³9𝑇³cos𝑒𝑐ℎ³𝑐₃log𝑇+𝑐₄𝑑𝑇²−𝑐³√3𝑇cos𝑒𝑐ℎ𝑐³−𝑐₁log𝑇+𝑐₄𝑑𝜃²+cosh²𝑑𝜙²√3𝑇cos𝑒𝑐ℎ𝑐³log𝑇+𝑐₄[𝑑𝜓+sinh𝜃𝑑𝜙]².(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^+𝛽^=−12𝑇𝑒^4𝛽.(3.29) From (3.29), we get 𝑒^4𝛽=𝑐⁵²𝑇²secℎ²𝑐₅log𝑇+𝑐₆,(3.30) where 𝑐₅ and 𝑐₆ are integration constants.
Using (3.30) in (3.11) and (3.12), we have 8𝜋𝜌=−8𝜋𝜆=𝑇²2𝑐₅cosh𝑐₅log𝑇+𝑐₆.(3.31) The corresponding metric can be written in the form 𝑑𝑠²=𝑐⁵³𝑇³secℎ³𝑐₅log𝑇+𝑐₆𝑑𝑇²−𝑐⁵𝑇secℎ𝑐⁵−𝑐₅log𝑇+𝑐₆𝑑𝜃²+sin²𝜃𝑑𝜙²𝑇secℎ𝑐⁵log𝑇+𝑐₆[𝑑𝜓+cos𝜃𝑑𝜙]².(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=𝑐^13/2𝑇^3/2𝜃=𝑢_;𝑖𝑖cos𝑒𝑐ℎ^3/2𝑐₁log𝑇+𝑐₂,=3𝑐¹2𝑇coth𝑐¹𝜎₂log𝑇+𝑐₂,=3𝑐¹²8𝑇²coth²𝑐₁log𝑇+𝑐₂(3.33) for the Bianchi type II model, 𝑉=(−𝑔)^1/2=𝑐^33/2𝑇^3/2𝜃=𝑢_;𝑖𝑖cos𝑒𝑐ℎ^3/2𝑐₃log𝑇+𝑐₄cosh𝜃,=3𝑐³2𝑇coth𝑐³𝜎₂log𝑇+𝑐₄,=3𝑐³²8𝑇²coth²𝑐₃log𝑇+𝑐₄(3.34) for the Bianchi type VIII model, and 𝑉=(−𝑔)^1/2=𝑐^53/2𝑇^3/2𝜃=𝑢_;𝑖𝑖secℎ^3/2𝑐₅log𝑇+𝑐₆sin𝜃,=3𝑐⁵2𝑇tanh𝑐⁵𝜎₂log𝑇+𝑐₆,=3𝑐⁵²8𝑇²tanh²𝑐₅log𝑇+𝑐₆(3.35) for the Bianchi type IX model.

Case 3 (for 𝜆=𝜌 (geometric string)). From (3.14), we get 𝜌=𝑐₁𝑒^−2𝛽.(3.36) From (3.11) and (3.12), we get (𝛿−1)=8𝜋𝜌𝜙𝑒^2𝛽.(3.37) From (3.37) and (3.36), we get (𝛿−1)=8𝜋𝜙𝑐¹.(3.38)
For Bianchi Types II and VIII Metrics (𝛿=0and−1)
From (3.38), we have 𝜙=constantsay𝑐₂.(3.39) Using (3.39), the field equations (3.10) to (3.13) reduce to 2𝛽^−3𝛽^2+𝑒^4𝛽2𝛽_4=0,(3.40)−3𝛽^2+𝛿𝑒^4𝛽−34𝑒^4𝛽=8𝜋𝜌𝑐₂3𝛽_2𝑒^6𝛽,(3.41)+𝛿𝑒^4𝛽−14𝑒^4𝛽=8𝜋𝜌𝑐₂0=8𝜋𝑒^6𝛽,(3.42)3+2𝜔(2𝜌)𝑒^6𝛽.(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^+𝑒^4𝛽=0.(3.45) From (3.45), we get 𝑆²=𝑒^2𝛽=2𝑚₁secℎ2𝑚₁𝑇+𝑛₁.(3.46) The corresponding metrics can be written in the form 𝑑𝑠²=−8𝑚2𝑚₁₃₁secℎ³2𝑚₁𝑇+𝑛₁𝑑𝑇²−2𝑚₁secℎ2𝑚₁𝑇+𝑛₁𝑑𝜃²+𝑑𝜙²𝑑𝑠₂secℎ2𝑚₁𝑇+𝑛₁[𝑑𝜓+𝜃𝑑𝜙]²,=−8𝑚2𝑚₁₃₁secℎ³2𝑚₁𝑇+𝑛₁𝑑𝑇²−2𝑚₁secℎ2𝑚₁𝑇+𝑛₁𝑑𝜃²+cosh²𝜃𝑑𝜙²secℎ2𝑚₁𝑇+𝑛₁[𝑑𝜓+sinh𝜃𝑑𝜙]².(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 2𝛽^−3𝛽^2+𝑒^4𝛽4+𝜔2𝜙𝜙^22−𝛽^𝜙^3𝛽_2𝜙+𝜙𝜙^=0,(3.50)+𝛿𝑒^4𝛽−14𝑒^4𝛽−𝜔𝜙^22𝜙²+3𝛽^𝜙^𝜙𝜙=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^+𝑒^4𝛽+2𝛽^𝑇=0,(3.55) that is, 𝑇𝛽^+𝛽^=−12𝑇𝑒^4𝛽.(3.56) From (3.56), we get 𝑒^4𝛽=𝑐¹²𝑇²secℎ²𝑐₁log𝑇+𝑐₂.(3.57) The corresponding metric can be written in the form 𝑑𝑠²=𝑐¹³𝑇³secℎ³𝑐₁log𝑇+𝑐₂𝑑𝑇²−𝑐¹𝑇secℎ𝑐¹−𝑐₁log𝑇+𝑐₂𝑑𝜃²+sin²𝜃𝑑𝜙²𝑇secℎ𝑐¹log𝑇+𝑐₂[𝑑𝜓+cos𝜃𝑑𝜙]².(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=(−𝜃𝑔=)6^1/2𝑚=₁8𝑚₁₃secℎ³2𝑚₁𝑇+𝑛₁,𝜎₂tanh2𝑚₁𝑇+𝑛₁,=6𝑚²¹secℎ²2𝑚₁𝑇+𝑛₁(3.59) for the Bianchi type II cosmological model (𝛿=0), 𝑉𝜃==(−2𝑚𝑔)^1/2₁=8𝑚₁₃secℎ³2𝑚₁𝑇+𝑛₁cosh𝜃,𝜎tanh2𝑚₁𝑇+𝑛₁cosh𝜃+sinh𝜃₂,=2𝑚²¹tanh²2𝑚₁𝑇+𝑛₁cosh²𝜃+sinh²𝜃+4𝑚₁tanh2𝑚₁𝑇+𝑛₁cosh𝜃sinh𝜃(3.60) for the Bianchi type VIII cosmological model (𝛿=−1), and 𝑉=(−𝑔)^1/2=𝑐^13/2𝑇^3/2𝜃=𝑢_;𝑖𝑖secℎ^3/2𝑐₁log𝑇+𝑐₂,=3𝑐¹2𝑇tanh𝑐¹𝜎₂log𝑇+𝑐₂,=3𝑐¹²8𝑇²tanh²𝑐₁log𝑇+𝑐₂(3.61) for the Bianchi type IX cosmological model (𝛿=1).

Case 4 (for 𝛼^−𝛽^≠0 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 𝛼^−𝛽^=0 and 𝜙^=0). Here, we get 𝛼=𝛽+𝑐.
Without loss of generality by taking the constant of integration 𝑐=0, we get 𝜙=cons𝛼ta=nt𝛽,say𝑐₁.(3.62) Using (3.62), the field equations (3.2) to (3.7) will reduce to 2𝛽^−3𝛽^2+𝑒^4𝛽2𝛽_4=0,(3.63)−3𝛽^2+𝛿𝑒^4𝛽−34𝑒^4𝛽=8𝜋𝜆𝑐₁3𝛽_2𝑒^6𝛽,(3.64)+𝛿𝑒^4𝛽−14𝑒^4𝛽=8𝜋𝜌𝑐₁0=8𝜋𝑒^6𝛽,(3.65)3+2𝜔(𝜌+𝜆)𝑒^6𝛽𝜌,(3.66)_+(3𝜌−𝜆)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^+2𝛿𝑒^4𝛽−𝑒^4𝛽=0.(3.69)For Bianchi Type II Metric (𝛿=0)
From (3.69), we get 𝑒^𝛽=(𝑎𝑇+𝑏)^−1/2,where𝑎²=1.(3.70) From (3.64), (3.65), and (3.70), we have 8𝜋𝜆=−𝑐¹(𝑎𝑇+𝑏)2,8𝜋𝜌=𝑐¹(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)⁻¹𝑑𝜃²+𝑑𝜙²−(𝑎𝑇+𝑏)⁻¹[𝑑𝜓+𝜃𝑑𝜙]².(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 𝑒^𝛽=(𝑎𝑇+𝑏)^−1/2,where𝑎²=3.(3.73) From (3.64) and (3.65), we have 88𝜋𝜋𝜆𝜌==−𝑐₁𝑐₁(𝑎𝑇+𝑏),(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)⁻¹𝑑𝜃²+cosh²𝜃𝑑𝜙²−(𝑎𝑇+𝑏)⁻¹[𝑑𝜓+sinh𝜃𝑑𝜙]².(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^+𝑒^4𝛽=0.(3.77) From (3.77), we get 𝑆²=𝑒^2𝛽=2𝑚₁secℎ2𝑚₁𝑇+𝑛₁.(3.78) From (3.63) and (3.64), we get (𝛿−1)=8𝜋𝜆𝑐₁𝑒^2𝛽.(3.79) From (3.66) & (3.79), we get 𝜆=𝜌=0.(3.80) The corresponding metric can be written in the form 𝑑𝑠²=−8𝑚2𝑚₁₃₁secℎ³2𝑚₁𝑇+𝑛₁𝑑𝑇²−2𝑚₁secℎ2𝑚₁𝑇+𝑛₁𝑑𝜃²+sin²𝜃𝑑𝜙²secℎ2𝑚₁𝑇+𝑛₁[𝑑𝜓+cos𝜃𝑑𝜙]².(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, 𝜃=𝑉=tan(−h𝜃𝑔()𝑎𝑇^1/2=+𝑏(𝑎)𝑇+_−3/2𝑏)^−3/2cosh𝜃,−3𝑎(𝑎𝑇+𝑏)^−1/2𝜎₂,=tanh²𝜃(𝑎𝑇+𝑏)⁻³+3𝑎²(𝑎𝑇+𝑏)−𝑎tanh𝜃(𝑎𝑇+𝑏)(3.83) for the Bianchi type VIII model, and 𝑉𝜃==𝑢(−_;𝑖𝑖𝑔)^1/2=8𝑚₁₃secℎ³2𝑚₁𝑇+𝑛₁sin𝜃,𝜎=2𝑚₁tanh2𝑚₁𝑇+𝑛₁sin𝜃+sin2𝜃₂,=2𝑚¹²tanh²2𝑚₁ℌ+2cos²𝜃+4𝑚₁tanh2𝑚₁ℌcos𝜃3,(3.84) where ℌ=(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, 𝑎≠0, and the model (3.81) has no initial singularity.