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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 573967, 16 pages
http://dx.doi.org/10.5402/2012/573967
Research Article

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

Department of Applied Mathematics, Andhra University, Andhra Pradesh Visakhapatnam 530 003, India

Received 7 August 2011; Accepted 20 September 2011

Academic Editors: J. BičÑk, G. Cleaver, and W.-H. Steeb

Copyright Β© 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.

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 πœ†+𝜌=0. We also established the existence of only Bianchi type IX vacuum cosmological model for πœ†=𝜌, where πœ† 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=βˆ’8πœ‹πœ™βˆ’1π‘‡π‘–π‘—βˆ’πœ”πœ™βˆ’2ξ‚€πœ™,π‘–πœ™,π‘—βˆ’12π‘”π‘–π‘—πœ™,π‘˜πœ™,π‘˜ξ‚βˆ’πœ™βˆ’1ξ€·πœ™π‘–;π‘—βˆ’π‘”π‘–π‘—πœ™;π‘˜,π‘˜ξ€Έ,πœ™;π‘˜,π‘˜=8πœ‹(3+2πœ”)βˆ’1𝑇,(1.1) where 𝐺𝑖𝑗=π‘…π‘–π‘—βˆ’(1/2)𝑅𝑔𝑖𝑗 is an Einstein tensor, 𝑇𝑖𝑗 is the stress energy tensor of the matter, and πœ” 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. [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=𝑑𝑑2βˆ’π‘…2ξ€Ίπ‘‘πœƒ2+𝑓2(πœƒ)π‘‘πœ™2ξ€»βˆ’π‘†2[]π‘‘πœ“+β„Ž(πœƒ)π‘‘πœ™2,(2.1) where (πœƒ,πœ™,πœ“) are the Eulerian angles, 𝑅 and 𝑆 are functions of 𝐭 only. It represents Bianchi type II if 𝑓(πœƒ)=1 and β„Ž(πœƒ)=πœƒ,Bianchi type VIII if 𝑓(πœƒ)=cosβ„Žπœƒ and β„Ž(πœƒ)=sinβ„Žπœƒ,Bianchi type IX if 𝑓(πœƒ)=sinπœƒ and β„Ž(πœƒ)=cosπœƒ.

The energy momentum tensor for cosmic strings [17] is 𝑇𝑖𝑗=πœŒπ‘’π‘–π‘’π‘—βˆ’πœ†π‘₯𝑖π‘₯𝑗,(2.2) where 𝑒𝑖 is the four-velocity of the string cloud, π‘₯𝑖 is the direction of anisotropy, 𝜌 and πœ† 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 𝑒𝑖 and π‘₯𝑖 is given as 𝑒𝑖𝑒𝑖=βˆ’π‘₯𝑖π‘₯𝑖=1,𝑒𝑖π‘₯𝑖=0.(2.3) In the commoving coordinate system, we have from (2.2) and (2.3) 𝑇11=𝑇22=0,𝑇33=πœ†,𝑇44=𝜌,𝑇𝑖𝑗=0for𝑖≠𝑗.(2.4) The quantities 𝜌,πœ† and the scalar field πœ™ in the theory depend on 𝑑 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 Μˆπ‘…π‘…+Μˆπ‘†π‘†+̇𝑅̇𝑆+𝑆𝑅𝑆24𝑅4+πœ”Μ‡πœ™22πœ™2+Μˆπœ™πœ™+Μ‡π‘…Μ‡πœ™+Μ‡π‘†Μ‡πœ™π‘…πœ™2Μˆπ‘…π‘†πœ™=0,𝑅+̇𝑅2+𝛿𝑅2βˆ’3𝑆24𝑅4+πœ”Μ‡πœ™22πœ™2+Μˆπœ™πœ™+2Μ‡π‘…Μ‡πœ™=π‘…πœ™8πœ‹πœ†πœ™,2Μ‡π‘…Μ‡π‘†βˆ’π‘†π‘…π‘†24𝑅4+̇𝑅2+𝛿𝑅2βˆ’πœ”Μ‡πœ™22πœ™2+2Μ‡π‘…Μ‡πœ™+Μ‡π‘†Μ‡πœ™π‘…πœ™=π‘†πœ™8πœ‹πœŒπœ™,ξ‚΅Μ‡π‘†π‘†βˆ’Μ‡π‘…π‘…ξ‚ΆΜ‡πœ™β„Ž(πœƒ)πœ™ΜˆΜ‡πœ™ξ‚΅2̇𝑅=0,πœ™+𝑅+̇𝑆𝑆=8πœ‹Μ‡π‘…3+2πœ”(πœ†+𝜌),Μ‡πœŒ+2πœŒπ‘…Μ‡π‘†+(πœŒβˆ’πœ†)𝑆=0,(3.1) where β€œ.” denotes differentiation with respect to β€œπ‘‘β€.

When 𝛿=0,βˆ’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, 𝑑𝑑=𝑅2𝑆𝑑𝑇, (3.1) reduce to π›Όξ…žξ…ž+π›½ξ…žξ…žβˆ’π›Όξ…ž2βˆ’2π›Όξ…žπ›½ξ…ž+𝑒4𝛽4+πœ”πœ™ξ…ž22πœ™2βˆ’π›Όξ…žπœ™ξ…žπœ™+πœ™ξ…žξ…žπœ™=0,(3.2)2π›Όξ…žξ…žβˆ’π›Όξ…ž2βˆ’2π›Όξ…žπ›½ξ…ž+𝛿𝑒(2𝛼+2𝛽)βˆ’34𝑒4𝛽+πœ”πœ™ξ…ž22πœ™2βˆ’π›½ξ…žπœ™ξ…žπœ™+πœ™ξ…žξ…žπœ™=8πœ‹πœ†πœ™π‘’(4𝛼+2𝛽),(3.3)2π›Όξ…žπ›½ξ…ž+π›Όξ…ž2+𝛿𝑒(2𝛼+2𝛽)βˆ’14𝑒4π›½βˆ’πœ”πœ™ξ…ž22πœ™2+π›½ξ…žπœ™ξ…žπœ™+2π›Όξ…žπœ™ξ…žπœ™=8πœ‹πœŒπœ™π‘’(4𝛼+2𝛽)𝛼,(3.4)ξ…žβˆ’π›½ξ…žξ€Έβ„Ž(πœƒ)πœ™ξ…žπœ™πœ™=0,(3.5)ξ…žξ…ž=8πœ‹3+2πœ”(πœ†+𝜌)𝑒(4𝛼+2𝛽)𝜌,(3.6)ξ…ž+2πœŒπ›Όξ…ž+(πœŒβˆ’πœ†)π›½ξ…ž=0,(3.7) where β€œξ…žβ€ denotes differentiation with respect to β€œπ‘‡β€.

Since we are considering the Bianchi types II, VIII, and IX metrics, we have β„Ž(πœƒ)=πœƒ, β„Ž(πœƒ)=sinhπœƒ, and β„Ž(πœƒ)=cosπœƒ for Bianchi types II, VIII, and IX metrics, respectively. Therefore, from (3.5), we will consider the following possible cases with β„Ž(πœƒ)β‰ 0: (1)π›Όξ…žβˆ’π›½ξ…ž=0,πœ™ξ…žβ‰ 0,(2)π›Όξ…žβˆ’π›½ξ…žβ‰ 0,πœ™β€²=0,(3)π›Όξ…žβˆ’π›½ξ…ž=0,πœ™β€²=0.(3.8)

Case 1 (for π›Όξ…žβˆ’π›½ξ…ž=0 and πœ™ξ…žβ‰ 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 2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝑒4𝛽4+πœ”πœ™ξ…ž22πœ™2βˆ’π›½ξ…žπœ™ξ…žπœ™+πœ™ξ…žξ…žπœ™=0,(3.10)2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝛿𝑒4π›½βˆ’34𝑒4𝛽+πœ”πœ™ξ…ž22πœ™2βˆ’π›½ξ…žπœ™ξ…žπœ™+πœ™ξ…žξ…žπœ™=8πœ‹πœ†πœ™π‘’6𝛽,(3.11)3π›½ξ…ž2+𝛿𝑒4π›½βˆ’14𝑒4π›½βˆ’πœ”πœ™ξ…ž22πœ™2+3π›½ξ…žπœ™ξ…žπœ™=8πœ‹πœŒπœ™π‘’6π›½πœ™,(3.12)ξ…žξ…ž=8πœ‹3+2πœ”(πœ†+𝜌)𝑒6π›½πœŒ,(3.13)ξ…ž+(3πœŒβˆ’πœ†)π›½ξ…ž=0,(3.14) where β€œξ…žβ€ denotes differentiation with respect to β€œπ‘‡β€.

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

Case 2 (for πœ†+𝜌=0 (Reddy string)). From (3.13), we get πœ™ξ…žξ…ž=0,(3.17) then πœ™=π‘Žπ‘‡+𝑏.(3.18) Without loss of generality by taking the constants of integration π‘Ž=1 and 𝑏=0, we get πœ™=𝑇.(3.19) Now from (3.11), (3.12), and (3.19), we get 2π›½ξ…žξ…ž+2𝛿𝑒4π›½βˆ’π‘’4𝛽+2π›½ξ…žπ‘‡=0.(3.20)
For Bianchi Type II Metric (𝛿=0)
From (3.20), we get π‘‡π›½ξ…žξ…ž+π›½ξ…ž=12𝑇𝑒4𝛽.(3.21) From (3.21), we get 𝑒4𝛽=𝑐12𝑇2cosech2𝑐1log𝑇+𝑐2ξ€Έ,(3.22) where 𝑐1 and 𝑐2 are integration constants.
Using (3.22) in (3.11) and (3.12), we have 𝑇8πœ‹πœŒ=βˆ’8πœ‹πœ†=22𝑐1𝑐sinh1log𝑇+𝑐2ξ€Έ.(3.23) The corresponding metric can be written in the form 𝑑𝑠2=𝑐13𝑇3cosπ‘’π‘β„Ž3𝑐1log𝑇+𝑐2𝑑𝑇2βˆ’π‘1𝑇𝑐cosπ‘’π‘β„Ž1log𝑇+𝑐2ξ€Έξ€Ίπ‘‘πœƒ2+π‘‘πœ™2ξ€»βˆ’π‘1𝑇𝑐cosπ‘’π‘β„Ž1log𝑇+𝑐2ξ€Έ[]π‘‘πœ“+π‘‘πœ™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=βˆ’1)
From (3.20), we get π‘‡π›½ξ…žξ…ž+π›½ξ…ž=32𝑇𝑒4𝛽.(3.25) From (3.25), we get 𝑒4𝛽=𝑐323𝑇2cosech2𝑐3log𝑇+𝑐4ξ€Έ,(3.26) where 𝑐3 and 𝑐4 are integration constants.
Using (3.26) in (3.11) and (3.12), we have √8πœ‹πœŒ=βˆ’8πœ‹πœ†=3𝑇2𝑐3𝑐sinh3log𝑇+𝑐4ξ€Έ.(3.27) The corresponding metric can be written in the form 𝑑𝑠2=𝑐339𝑇3cosπ‘’π‘β„Ž3𝑐3log𝑇+𝑐4𝑑𝑇2βˆ’π‘3βˆšξ€·π‘3𝑇cosπ‘’π‘β„Ž3log𝑇+𝑐4ξ€Έξ€Ίπ‘‘πœƒ2+cosh2π‘‘πœ™2ξ€»βˆ’π‘1βˆšξ€·π‘3𝑇cosπ‘’π‘β„Ž3log𝑇+𝑐4ξ€Έ[]π‘‘πœ“+sinhπœƒπ‘‘πœ™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 π‘‡π›½ξ…žξ…ž+π›½ξ…ž1=βˆ’2𝑇𝑒4𝛽.(3.29) From (3.29), we get 𝑒4𝛽=𝑐52𝑇2secβ„Ž2𝑐5log𝑇+𝑐6ξ€Έ,(3.30) where 𝑐5 and 𝑐6 are integration constants.
Using (3.30) in (3.11) and (3.12), we have 𝑇8πœ‹πœŒ=βˆ’8πœ‹πœ†=22𝑐5𝑐cosh5log𝑇+𝑐6ξ€Έ.(3.31) The corresponding metric can be written in the form 𝑑𝑠2=𝑐53𝑇3secβ„Ž3𝑐5log𝑇+𝑐6𝑑𝑇2βˆ’π‘5𝑇𝑐secβ„Ž5log𝑇+𝑐6ξ€Έξ€Ίπ‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€»βˆ’π‘5𝑇𝑐secβ„Ž5log𝑇+𝑐6ξ€Έ[]π‘‘πœ“+cosπœƒπ‘‘πœ™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 πœƒ, and shear 𝜎 for the models (3.24), (3.28), and (3.32) are given by 𝑉=(βˆ’π‘”)1/2=𝑐13/2𝑇3/2cosπ‘’π‘β„Ž3/2𝑐1log𝑇+𝑐2ξ€Έ,πœƒ=𝑒;𝑖𝑖=3𝑐1𝑐2𝑇coth1log𝑇+𝑐2ξ€Έ,𝜎2=3𝑐128𝑇2coth2𝑐1log𝑇+𝑐2ξ€Έ(3.33) for the Bianchi type II model, 𝑉=(βˆ’π‘”)1/2=𝑐33/2𝑇3/2cosπ‘’π‘β„Ž3/2𝑐3log𝑇+𝑐4ξ€Έcoshπœƒ,πœƒ=𝑒;𝑖𝑖=3𝑐3𝑐2𝑇coth3log𝑇+𝑐4ξ€Έ,𝜎2=3𝑐328𝑇2coth2𝑐3log𝑇+𝑐4ξ€Έ(3.34) for the Bianchi type VIII model, and 𝑉=(βˆ’π‘”)1/2=𝑐53/2𝑇3/2secβ„Ž3/2𝑐5log𝑇+𝑐6ξ€Έsinπœƒ,πœƒ=𝑒;𝑖𝑖=3𝑐5𝑐2𝑇tanh5log𝑇+𝑐6ξ€Έ,𝜎2=3𝑐528𝑇2tanh2𝑐5log𝑇+𝑐6ξ€Έ(3.35) for the Bianchi type IX model.

Case 3 (for πœ†=𝜌 (geometric string)). From (3.14), we get 𝜌=𝑐1π‘’βˆ’2𝛽.(3.36) From (3.11) and (3.12), we get (π›Ώβˆ’1)=8πœ‹πœŒπœ™π‘’2𝛽.(3.37) From (3.37) and (3.36), we get (π›Ώβˆ’1)=8πœ‹πœ™π‘1.(3.38)
For Bianchi Types II and VIII Metrics (𝛿=0andβˆ’1)
From (3.38), we have ξ€·πœ™=constantsay𝑐2ξ€Έ.(3.39) Using (3.39), the field equations (3.10) to (3.13) reduce to 2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝑒4𝛽4=0,(3.40)2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝛿𝑒4π›½βˆ’34𝑒4𝛽=8πœ‹πœŒπ‘2𝑒6𝛽,(3.41)3π›½ξ…ž2+𝛿𝑒4π›½βˆ’14𝑒4𝛽=8πœ‹πœŒπ‘2𝑒6𝛽,(3.42)0=8πœ‹3+2πœ”(2𝜌)𝑒6𝛽.(3.43) From (3.43), we get 𝜌=0,(3.44) and, since πœ†=𝜌, we will get πœ†=0.
From (3.40) to (3.42), we get 2π›½ξ…žξ…ž+𝑒4𝛽=0.(3.45) From (3.45), we get 𝑆2=𝑒2𝛽=2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έ.(3.46) The corresponding metrics can be written in the form 𝑑𝑠2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1𝑑𝑇2βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έξ€Ίπ‘‘πœƒ2+π‘‘πœ™2ξ€»βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έ[]π‘‘πœ“+πœƒπ‘‘πœ™2,𝑑𝑠2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1𝑑𝑇2βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έξ€Ίπ‘‘πœƒ2+cosh2πœƒπ‘‘πœ™2ξ€»βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έ[]π‘‘πœ“+sinhπœƒπ‘‘πœ™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 2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝑒4𝛽4+πœ”πœ™ξ…ž22πœ™2βˆ’π›½ξ…žπœ™ξ…žπœ™+πœ™ξ…žξ…žπœ™=0,(3.50)3π›½ξ…ž2+𝛿𝑒4π›½βˆ’14𝑒4π›½βˆ’πœ”πœ™ξ…ž22πœ™2+3π›½ξ…žπœ™ξ…žπœ™πœ™=0,(3.51)ξ…žξ…ž=0.(3.52) From (3.52), we get πœ™=𝑐3𝑇+𝑐4.(3.53) Without loss of generality by taking the constants of integration 𝑐3=1 and 𝑐4=0, we get πœ™=𝑇.(3.54) From (3.50), (3.51), and (3.54), we get 2π›½ξ…žξ…ž+𝑒4𝛽+2π›½ξ…žπ‘‡=0,(3.55) that is, π‘‡π›½ξ…žξ…ž+π›½ξ…ž1=βˆ’2𝑇𝑒4𝛽.(3.56) From (3.56), we get 𝑒4𝛽=𝑐12𝑇2secβ„Ž2𝑐1log𝑇+𝑐2ξ€Έ.(3.57) The corresponding metric can be written in the form 𝑑𝑠2=𝑐13𝑇3secβ„Ž3𝑐1log𝑇+𝑐2𝑑𝑇2βˆ’π‘1𝑇𝑐secβ„Ž1log𝑇+𝑐2ξ€Έξ€Ίπ‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€»βˆ’π‘1𝑇𝑐secβ„Ž1log𝑇+𝑐2ξ€Έ[]π‘‘πœ“+cosπœƒπ‘‘πœ™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 πœƒ, and the shear 𝜎 for the models (3.47) and (3.58) are given by 𝑉=(βˆ’π‘”)1/2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1ξ€Έ,πœƒ=6π‘š1tanh2π‘š1𝑇+𝑛1ξ€Έ,𝜎2=6π‘š21secβ„Ž22π‘š1𝑇+𝑛1ξ€Έ(3.59) for the Bianchi type II cosmological model (𝛿=0), 𝑉=(βˆ’π‘”)1/2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1ξ€Έξ€Ίcoshπœƒ,πœƒ=2π‘š1tanh2π‘š1𝑇+𝑛1ξ€Έξ€»,𝜎coshπœƒ+sinhπœƒ2=ξ€Ί2π‘š21tanh22π‘š1𝑇+𝑛1ξ€Έcosh2πœƒ+sinh2πœƒ+4π‘š1tanh2π‘š1𝑇+𝑛1ξ€Έξ€»coshπœƒsinhπœƒ(3.60) for the Bianchi type VIII cosmological model (𝛿=βˆ’1), and 𝑉=(βˆ’π‘”)1/2=𝑐13/2𝑇3/2secβ„Ž3/2𝑐1log𝑇+𝑐2ξ€Έ,πœƒ=𝑒;𝑖𝑖=3𝑐1𝑐2𝑇tanh1log𝑇+𝑐2ξ€Έ,𝜎2=3𝑐128𝑇2tanh2𝑐1log𝑇+𝑐2ξ€Έ(3.61) for the Bianchi type IX cosmological model (𝛿=1).

Case 4 (for π›Όξ…žβˆ’π›½ξ…žβ‰ 0 and πœ™β€²=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 π›Όξ…žβˆ’π›½ξ…ž=0 and πœ™ξ…ž=0). Here, we get 𝛼=𝛽+𝑐.
Without loss of generality by taking the constant of integration 𝑐=0, we get 𝛼=𝛽,πœ™=constantsay𝑐1ξ€Έ.(3.62) Using (3.62), the field equations (3.2) to (3.7) will reduce to 2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝑒4𝛽4=0,(3.63)2π›½ξ…žξ…žβˆ’3π›½ξ…ž2+𝛿𝑒4π›½βˆ’34𝑒4𝛽=8πœ‹πœ†π‘1𝑒6𝛽,(3.64)3π›½ξ…ž2+𝛿𝑒4π›½βˆ’14𝑒4𝛽=8πœ‹πœŒπ‘1𝑒6𝛽,(3.65)0=8πœ‹3+2πœ”(𝜌+πœ†)𝑒6π›½πœŒ,(3.66)ξ…ž+(3πœŒβˆ’πœ†)π›½ξ…ž=0.(3.67) From (3.66), we get 𝜌+πœ†=0.(3.68) From (3.63) to (3.65) and (3.68), we have 2π›½ξ…žξ…ž+2𝛿𝑒4π›½βˆ’π‘’4𝛽=0.(3.69)For Bianchi Type II Metric (𝛿=0)
From (3.69), we get 𝑒𝛽=(π‘Žπ‘‡+𝑏)βˆ’1/2,whereπ‘Ž2=1.(3.70) From (3.64), (3.65), and (3.70), we have 8πœ‹πœ†=βˆ’π‘1(π‘Žπ‘‡+𝑏)2𝑐,8πœ‹πœŒ=1(π‘Žπ‘‡+𝑏)2.(3.71) From (3.71) we get πœ†+𝜌=0.
The corresponding metric can be written in the form 𝑑𝑠2=(π‘Žπ‘‡+𝑏)βˆ’3𝑑𝑇2βˆ’(π‘Žπ‘‡+𝑏)βˆ’1ξ€Ίπ‘‘πœƒ2+π‘‘πœ™2ξ€»βˆ’(π‘Žπ‘‡+𝑏)βˆ’1[]π‘‘πœ“+πœƒπ‘‘πœ™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=βˆ’1)
From (3.69), we get 𝑒𝛽=(π‘Žπ‘‡+𝑏)βˆ’1/2,whereπ‘Ž2=3.(3.73) From (3.64) and (3.65), we have 8πœ‹πœ†=βˆ’π‘1(π‘Žπ‘‡+𝑏),8πœ‹πœŒ=𝑐1(π‘Žπ‘‡+𝑏).(3.74) Therefore, from (3.74), we have πœ†+𝜌=0.(3.75) The corresponding metric can be written in the form 𝑑𝑠2=(π‘Žπ‘‡+𝑏)βˆ’3𝑑𝑇2βˆ’(π‘Žπ‘‡+𝑏)βˆ’1ξ€Ίπ‘‘πœƒ2+cosh2πœƒπ‘‘πœ™2ξ€»βˆ’(π‘Žπ‘‡+𝑏)βˆ’1[]π‘‘πœ“+sinhπœƒπ‘‘πœ™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 2π›½ξ…žξ…ž+𝑒4𝛽=0.(3.77) From (3.77), we get 𝑆2=𝑒2𝛽=2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έ.(3.78) From (3.63) and (3.64), we get (π›Ώβˆ’1)=8πœ‹πœ†π‘1𝑒2𝛽.(3.79) From (3.66) & (3.79), we get πœ†=𝜌=0.(3.80) The corresponding metric can be written in the form 𝑑𝑠2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1𝑑𝑇2βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έξ€Ίπ‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€»βˆ’2π‘š1secβ„Ž2π‘š1𝑇+𝑛1ξ€Έ[]π‘‘πœ“+cosπœƒπ‘‘πœ™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 πœƒ, and the shear 𝜎 for the models (3.72), (3.76), and (3.81) are given by 𝑉=(βˆ’π‘”)1/2=(π‘Žπ‘‡+𝑏)βˆ’3/2,πœƒ=βˆ’3π‘Ž(π‘Žπ‘‡+𝑏),𝜎2=3π‘Ž22(π‘Žπ‘‡+𝑏)2(3.82) for the Bianchi type II model, 𝑉=(βˆ’π‘”)1/2=(π‘Žπ‘‡+𝑏)βˆ’3/2coshπœƒ,πœƒ=tanhπœƒ(π‘Žπ‘‡+𝑏)βˆ’3/2βˆ’3π‘Ž(π‘Žπ‘‡+𝑏)βˆ’1/2,𝜎2=tanh2πœƒ(π‘Žπ‘‡+𝑏)βˆ’3+3π‘Ž2(π‘Žπ‘‡+𝑏)βˆ’π‘Žtanhπœƒ(π‘Žπ‘‡+𝑏)(3.83) for the Bianchi type VIII model, and 𝑉=(βˆ’π‘”)1/2=8π‘š13secβ„Ž32π‘š1𝑇+𝑛1ξ€Έsinπœƒ,πœƒ=𝑒;𝑖𝑖=ξ€Ί2π‘š1tanh2π‘š1𝑇+𝑛1ξ€Έξ€»,𝜎sinπœƒ+sin2πœƒ2=ξ€Ί2π‘š12tanh22π‘š1β„Œ+2cos2πœƒ+4π‘š1tanh2π‘š1ξ€»β„Œcosπœƒ3,(3.84) where β„Œ=(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 πœ†+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 π‘‡β†’βˆž; 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 πœ†=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 π‘‡β†’βˆž, 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 πœ†+𝜌=0 and also got Bianchi type IX vacuum cosmological model of general relativity, since the scalar field πœ™ is constant. The spatial volume of the models (3.72), (3.76), and (3.81) are decreasing as π‘‡β†’βˆž; that is, the models are contracting with the increase of time. Also the models (3.72) and (3.76) have initial singularity at 𝑇=βˆ’π‘/π‘Ž, π‘Žβ‰ 0, and the model (3.81) has no initial singularity.

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