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

Authors: V. U. M. Rao; M. Vijaya Santhi
Journal: ISRN Mathematical Physics
 (2012)

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## 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) forλ+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.


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## Body

## 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(1.1)Gij=-8πϕ-1Tij-ωϕ-2(ϕ,iϕ,j-12gijϕ,kϕ,k)-ϕ-1(ϕi;j-gijϕ;k,k),ϕ;k,k=8π(3+2ω)-1T,
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(1.2)Tij,j=0
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(2.1)ds2=dt2-R2[dθ2+f2(θ)dϕ2]-S2[dψ+h(θ)dϕ]2,
where (θ,ϕ,ψ) are the Eulerian angles, R and S are functions of t only. It represents Bianchi type II iff(θ)=1 and h(θ)=θ,Bianchi type VIII iff(θ)=coshθ and h(θ)=sinhθ,Bianchi type IX iff(θ)=sinθ and h(θ)=cosθ.The energy momentum tensor for cosmic strings [17] is
(2.2)Tij=ρuiuj-λxixj,
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
(2.3)uiui=-xixi=1,uixi=0.
In the commoving coordinate system, we have from (2.2) and (2.3)
(2.4)T11=T22=0,T33=λ,T44=ρ,Tji=0fori≠j.
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
(3.1)R̈R+S̈S+ṘṠRS+S24R4+ωϕ̇22ϕ2+ϕ̈ϕ+Ṙϕ̇Rϕ+Ṡϕ̇Sϕ=0,2R̈R+Ṙ2+δR2-3S24R4+ωϕ̇22ϕ2+ϕ̈ϕ+2Ṙϕ̇Rϕ=8πλϕ,2ṘṠRS-S24R4+Ṙ2+δR2-ωϕ̇22ϕ2+2Ṙϕ̇Rϕ+Ṡϕ̇Sϕ=8πρϕ,(ṠS-ṘR)h(θ)ϕ̇ϕ=0,ϕ̈+ϕ̇(2ṘR+ṠS)=8π3+2ω(λ+ρ),ρ̇+2ρṘR+(ρ-λ)ṠS=0,
where “.” denotes differentiation with respect to “t”.Whenδ=0,-1 & +1, the field equation (3.1) correspond to the Bianchi types II, VIII, and IX universes, respectively.Using the transformationR=eα, S=eβ, dt=R2SdT, (3.1) reduce to
(3.2)α′′+β′′-α′2-2α′β′+e4β4+ωϕ′22ϕ2-α′ϕ′ϕ+ϕ′′ϕ=0,(3.3)2α′′-α′2-2α′β′+δe(2α+2β)-34e4β+ωϕ′22ϕ2-β′ϕ′ϕ+ϕ′′ϕ=8πλϕe(4α+2β),(3.4)2α′β′+α′2+δe(2α+2β)-14e4β-ωϕ′22ϕ2+β′ϕ′ϕ+2α′ϕ′ϕ=8πρϕe(4α+2β),(3.5)(α′-β′)h(θ)ϕ′ϕ=0,(3.6)ϕ′′=8π3+2ω(λ+ρ)e(4α+2β),(3.7)ρ′+2ρα′+(ρ-λ)β′=0,
where “′” denotes differentiation with respect to “T”.Since we are considering the Bianchi types II, VIII, and IX metrics, we haveh(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:
(3.8)(1)α′-β′=0,ϕ′≠0,(2)α′-β′≠0,ϕ′=0,(3)α′-β′=0,ϕ′=0.Case 1 (forα′-β′=0 and ϕ′≠0).
Here, we getα=β+c.
Without loss of generality by taking the constant of integrationc=0, we get
(3.9)α=β.
By using (3.9), (3.2) to (3.7) will reduce to
(3.10)2β′′-3β′2+e4β4+ωϕ′22ϕ2-β′ϕ′ϕ+ϕ′′ϕ=0,(3.11)2β′′-3β′2+δe4β-34e4β+ωϕ′22ϕ2-β′ϕ′ϕ+ϕ′′ϕ=8πλϕe6β,(3.12)3β′2+δe4β-14e4β-ωϕ′22ϕ2+3β′ϕ′ϕ=8πρϕe6β,(3.13)ϕ′′=8π3+2ω(λ+ρ)e6β,(3.14)ρ′+(3ρ-λ)β′=0,
where “′” denotes differentiation with respect to “T”.Latelier [17] discussed, in general, the following equations of state:
(3.15)ρ=λ(geometricstring),ρ=(1+ω)λ(p-string),
and Reddy [10] obtained inflationary string cosmological models in Brans-Dicke scalar-tensor theory of gravitation for
(3.16)ρ+λ=0(Reddystring).
Here we will present string cosmological models corresponding to ρ+λ=0 and ρ=λ.Case 2 (forλ+ρ=0 (Reddy string)).
From (3.13), we get
(3.17)ϕ′′=0,
then
(3.18)ϕ=aT+b.
Without loss of generality by taking the constants of integration a=1 and b=0, we get
(3.19)ϕ=T.
Now from (3.11), (3.12), and (3.19), we get
(3.20)2β′′+2δe4β-e4β+2β′T=0.
For Bianchi Type II Metric(δ=0)
From (3.20), we get
(3.21)Tβ′′+β′=12Te4β.
From (3.21), we get
(3.22)e4β=c12T2cosech2(c1log⁡⁡T+c2),
where c1 and c2 are integration constants.
Using (3.22) in (3.11) and (3.12), we have
(3.23)8πρ=-8πλ=T22c1sinh⁡⁡(c1log⁡⁡T+c2).
The corresponding metric can be written in the form
(3.24)ds2=c13T3cos⁡ech3(c1log⁡⁡T+c2)dT2-c1Tcos⁡ech(c1log⁡⁡T+c2)[dθ2+dϕ2]-c1Tcos⁡ech(c1log⁡⁡T+c2)[dψ+dϕ]2.
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
(3.25)Tβ′′+β′=32Te4β.
From (3.25), we get
(3.26)e4β=c323T2cos⁡⁡ech2(c3log⁡⁡T+c4),
where c3 and c4 are integration constants.
Using (3.26) in (3.11) and (3.12), we have
(3.27)8πρ=-8πλ=3T2c3sinh⁡⁡(c3log⁡⁡T+c4).
The corresponding metric can be written in the form
(3.28)ds2=c339T3cos⁡ech3(c3log⁡⁡T+c4)dT2-c33Tcos⁡ech(c3log⁡⁡T+c4)[dθ2+cosh⁡⁡2dϕ2]-c13Tcos⁡ech(c3log⁡⁡T+c4)[dψ+sinh⁡⁡θdϕ]2.
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
(3.29)Tβ′′+β′=-12Te4β.
From (3.29), we get
(3.30)e4β=c52T2sec⁡h2(c5log⁡⁡T+c6),
where c5 and c6 are integration constants.
Using (3.30) in (3.11) and (3.12), we have
(3.31)8πρ=-8πλ=T22c5cosh⁡⁡(c5log⁡⁡T+c6).
The corresponding metric can be written in the form
(3.32)ds2=c53T3sec⁡h3(c5log⁡⁡T+c6)dT2-c5Tsec⁡h(c5log⁡⁡T+c6)[dθ2+sin⁡⁡2θdϕ2]-c5Tsec⁡h(c5log⁡⁡T+c6)[dψ+cos⁡⁡θdϕ]2.
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 elementV, expansion θ, and shear σ for the models (3.24), (3.28), and (3.32) are given by
(3.33)V=(-g)1/2=c13/2T3/2cos⁡ech3/2(c1log⁡⁡T+c2),θ=u;ii=3c12Tcoth⁡⁡(c1log⁡⁡T+c2),σ2=3c128T2coth⁡⁡2(c1log⁡⁡T+c2)
for the Bianchi type II model,
(3.34)V=(-g)1/2=c33/2T3/2cos⁡ech3/2(c3log⁡⁡T+c4)cosh⁡⁡θ,θ=u;ii=3c32Tcoth⁡⁡(c3log⁡⁡T+c4),σ2=3c328T2coth⁡⁡2(c3log⁡⁡T+c4)
for the Bianchi type VIII model, and
(3.35)V=(-g)1/2=c53/2T3/2sec⁡h3/2(c5log⁡⁡T+c6)sin⁡θ,θ=u;ii=3c52Ttanh⁡(c5log⁡⁡T+c6),σ2=3c528T2tanh⁡2(c5log⁡⁡T+c6)
for the Bianchi type IX model.Case 3 (forλ=ρ (geometric string)).
From (3.14), we get
(3.36)ρ=c1e-2β.
From (3.11) and (3.12), we get
(3.37)(δ-1)=8πρϕe2β.
From (3.37) and (3.36), we get
(3.38)(δ-1)=8πϕc1.
For Bianchi Types II and VIII Metrics(δ=0and-1)
From (3.38), we have
(3.39)ϕ=constant(sayc2).
Using (3.39), the field equations (3.10) to (3.13) reduce to
(3.40)2β′′-3β′2+e4β4=0,(3.41)2β′′-3β′2+δe4β-34e4β=8πρc2e6β,(3.42)3β′2+δe4β-14e4β=8πρc2e6β,(3.43)0=8π3+2ω(2ρ)e6β.
From (3.43), we get
(3.44)ρ=0,
and, since λ=ρ, we will get λ=0.
From (3.40) to (3.42), we get
(3.45)2β′′+e4β=0.
From (3.45), we get
(3.46)S2=e2β=2m1sec⁡h2m1(T+n1).
The corresponding metrics can be written in the form
(3.47)ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+dϕ2]-2m1sec⁡h2m1(T+n1)[dψ+θdϕ]2,ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+cosh⁡⁡2θdϕ2]-2m1sec⁡h2m1(T+n1)[dψ+sinh⁡⁡θdϕ]2.
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
(3.48)c1=0.
Using (3.48) in (3.36), we get
(3.49)ρ=0.
Using (3.49) the field equations (3.10) to (3.14) reduce to
(3.50)2β′′-3β′2+e4β4+ωϕ′22ϕ2-β′ϕ′ϕ+ϕ′′ϕ=0,(3.51)3β′2+δe4β-14e4β-ωϕ′22ϕ2+3β′ϕ′ϕ=0,(3.52)ϕ′′=0.
From (3.52), we get
(3.53)ϕ=c3T+c4.
Without loss of generality by taking the constants of integration c3=1 and c4=0, we get
(3.54)ϕ=T.
From (3.50), (3.51), and (3.54), we get
(3.55)2β′′+e4β+2β′T=0,
that is,
(3.56)Tβ′′+β′=-12Te4β.
From (3.56), we get
(3.57)e4β=c12T2sec⁡h2(c1log⁡⁡T+c2).
The corresponding metric can be written in the form
(3.58)ds2=c13T3sec⁡h3(c1log⁡⁡T+c2)dT2-c1Tsec⁡h(c1log⁡⁡T+c2)[dθ2+sin⁡⁡2θdϕ2]-c1Tsec⁡h(c1log⁡⁡T+c2)[dψ+cos⁡⁡θdϕ]2.
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 volumeV, expansion θ, and the shear σ for the models (3.47) and (3.58) are given by
(3.59)V=(-g)1/2=8m13sec⁡h32m1(T+n1),θ=6m1tanh⁡2m1(T+n1),σ2=6m12sec⁡h22m1(T+n1)
for the Bianchi type II cosmological model (δ=0),
(3.60)V=(-g)1/2=8m13sec⁡h32m1(T+n1)cosh⁡⁡θ,θ=[2m1tanh⁡2m1(T+n1)cosh⁡⁡θ+sinh⁡⁡θ],σ2=[2m12tanh⁡22m1(T+n1)cosh⁡⁡2θ+sinh⁡⁡2θ+4m1tanh⁡2m1(T+n1)cosh⁡⁡θsinh⁡⁡θcosh⁡⁡2]
for the Bianchi type VIII cosmological model (δ=-1), and
(3.61)V=(-g)1/2=c13/2T3/2sec⁡h3/2(c1log⁡⁡T+c2),θ=u;ii=3c12Ttanh⁡(c1log⁡⁡T+c2),σ2=3c128T2tanh⁡2(c1log⁡⁡T+c2)
for the Bianchi type IX cosmological model (δ=1).Case 4 (forα′-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 (forα′-β′=0 and ϕ′=0).
Here, we getα=β+c.
Without loss of generality by taking the constant of integrationc=0, we get
(3.62)α=β,ϕ=constant(sayc1).
Using (3.62), the field equations (3.2) to (3.7) will reduce to
(3.63)2β′′-3β′2+e4β4=0,(3.64)2β′′-3β′2+δe4β-34e4β=8πλc1e6β,(3.65)3β′2+δe4β-14e4β=8πρc1e6β,(3.66)0=8π3+2ω(ρ+λ)e6β,(3.67)ρ′+(3ρ-λ)β′=0.
From (3.66), we get
(3.68)ρ+λ=0.
From (3.63) to (3.65) and (3.68), we have
(3.69)2β′′+2δe4β-e4β=0.For Bianchi Type II Metric(δ=0)
From (3.69), we get
(3.70)eβ=(aT+b)-1/2,wherea2=1.
From (3.64), (3.65), and (3.70), we have
(3.71)8πλ=-c1(aT+b)2,8πρ=c1(aT+b)2.
From (3.71) we get λ+ρ=0.
The corresponding metric can be written in the form(3.72)ds2=(aT+b)-3dT2-(aT+b)-1[dθ2+dϕ2]-(aT+b)-1[dψ+θdϕ]2.
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
(3.73)eβ=(aT+b)-1/2,wherea2=3.
From (3.64) and (3.65), we have
(3.74)8πλ=-c1(aT+b),8πρ=c1(aT+b).
Therefore, from (3.74), we have
(3.75)λ+ρ=0.
The corresponding metric can be written in the form
(3.76)ds2=(aT+b)-3dT2-(aT+b)-1[dθ2+cosh⁡⁡2θdϕ2]-(aT+b)-1[dψ+sinh⁡⁡θdϕ]2.
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
(3.77)2β′′+e4β=0.
From (3.77), we get
(3.78)S2=e2β=2m1sec⁡h2m1(T+n1).
From (3.63) and (3.64), we get
(3.79)(δ-1)=8πλc1e2β.
From (3.66) & (3.79), we get
(3.80)λ=ρ=0.
The corresponding metric can be written in the form
(3.81)ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+sin⁡⁡2θdϕ2]-2m1sec⁡h2m1(T+n1)[dψ+cos⁡⁡θdϕ]2.
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 volumeV, expansion θ, and the shear σ for the models (3.72), (3.76), and (3.81) are given by
(3.82)V=(-g)1/2=(aT+b)-3/2,θ=-3a(aT+b),σ2=3a22(aT+b)2
for the Bianchi type II model,
(3.83)V=(-g)1/2=(aT+b)-3/2cosh⁡⁡θ,θ=tanh⁡θ(aT+b)-3/2-3a(aT+b)-1/2,σ2=tanh⁡2θ(aT+b)-3+3a2(aT+b)-atanh⁡θ(aT+b)
for the Bianchi type VIII model, and
(3.84)V=(-g)1/2=8m13sec⁡h32m1(T+n1)sin⁡⁡θ,θ=u;ii=[2m1tanh⁡2m1(T+n1)sin⁡⁡θ+sin⁡⁡2θ],σ2=[2m12tanh⁡22m1H+2cos⁡⁡2θ+4m1tanh⁡2m1Hcos⁡⁡θ]3,
where ℌ=(T+n1) for the Bianchi type IX model.


## 3.1. Physical and Geometrical Properties

The volume elementV, expansion θ, and shear σ for the models (3.24), (3.28), and (3.32) are given by
(3.33)V=(-g)1/2=c13/2T3/2cos⁡ech3/2(c1log⁡⁡T+c2),θ=u;ii=3c12Tcoth⁡⁡(c1log⁡⁡T+c2),σ2=3c128T2coth⁡⁡2(c1log⁡⁡T+c2)
for the Bianchi type II model,
(3.34)V=(-g)1/2=c33/2T3/2cos⁡ech3/2(c3log⁡⁡T+c4)cosh⁡⁡θ,θ=u;ii=3c32Tcoth⁡⁡(c3log⁡⁡T+c4),σ2=3c328T2coth⁡⁡2(c3log⁡⁡T+c4)
for the Bianchi type VIII model, and
(3.35)V=(-g)1/2=c53/2T3/2sec⁡h3/2(c5log⁡⁡T+c6)sin⁡θ,θ=u;ii=3c52Ttanh⁡(c5log⁡⁡T+c6),σ2=3c528T2tanh⁡2(c5log⁡⁡T+c6)
for the Bianchi type IX model.Case 3 (forλ=ρ (geometric string)).
From (3.14), we get
(3.36)ρ=c1e-2β.
From (3.11) and (3.12), we get
(3.37)(δ-1)=8πρϕe2β.
From (3.37) and (3.36), we get
(3.38)(δ-1)=8πϕc1.
For Bianchi Types II and VIII Metrics(δ=0and-1)
From (3.38), we have
(3.39)ϕ=constant(sayc2).
Using (3.39), the field equations (3.10) to (3.13) reduce to
(3.40)2β′′-3β′2+e4β4=0,(3.41)2β′′-3β′2+δe4β-34e4β=8πρc2e6β,(3.42)3β′2+δe4β-14e4β=8πρc2e6β,(3.43)0=8π3+2ω(2ρ)e6β.
From (3.43), we get
(3.44)ρ=0,
and, since λ=ρ, we will get λ=0.
From (3.40) to (3.42), we get
(3.45)2β′′+e4β=0.
From (3.45), we get
(3.46)S2=e2β=2m1sec⁡h2m1(T+n1).
The corresponding metrics can be written in the form
(3.47)ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+dϕ2]-2m1sec⁡h2m1(T+n1)[dψ+θdϕ]2,ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+cosh⁡⁡2θdϕ2]-2m1sec⁡h2m1(T+n1)[dψ+sinh⁡⁡θdϕ]2.
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
(3.48)c1=0.
Using (3.48) in (3.36), we get
(3.49)ρ=0.
Using (3.49) the field equations (3.10) to (3.14) reduce to
(3.50)2β′′-3β′2+e4β4+ωϕ′22ϕ2-β′ϕ′ϕ+ϕ′′ϕ=0,(3.51)3β′2+δe4β-14e4β-ωϕ′22ϕ2+3β′ϕ′ϕ=0,(3.52)ϕ′′=0.
From (3.52), we get
(3.53)ϕ=c3T+c4.
Without loss of generality by taking the constants of integration c3=1 and c4=0, we get
(3.54)ϕ=T.
From (3.50), (3.51), and (3.54), we get
(3.55)2β′′+e4β+2β′T=0,
that is,
(3.56)Tβ′′+β′=-12Te4β.
From (3.56), we get
(3.57)e4β=c12T2sec⁡h2(c1log⁡⁡T+c2).
The corresponding metric can be written in the form
(3.58)ds2=c13T3sec⁡h3(c1log⁡⁡T+c2)dT2-c1Tsec⁡h(c1log⁡⁡T+c2)[dθ2+sin⁡⁡2θdϕ2]-c1Tsec⁡h(c1log⁡⁡T+c2)[dψ+cos⁡⁡θdϕ]2.
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 volumeV, expansion θ, and the shear σ for the models (3.47) and (3.58) are given by
(3.59)V=(-g)1/2=8m13sec⁡h32m1(T+n1),θ=6m1tanh⁡2m1(T+n1),σ2=6m12sec⁡h22m1(T+n1)
for the Bianchi type II cosmological model (δ=0),
(3.60)V=(-g)1/2=8m13sec⁡h32m1(T+n1)cosh⁡⁡θ,θ=[2m1tanh⁡2m1(T+n1)cosh⁡⁡θ+sinh⁡⁡θ],σ2=[2m12tanh⁡22m1(T+n1)cosh⁡⁡2θ+sinh⁡⁡2θ+4m1tanh⁡2m1(T+n1)cosh⁡⁡θsinh⁡⁡θcosh⁡⁡2]
for the Bianchi type VIII cosmological model (δ=-1), and
(3.61)V=(-g)1/2=c13/2T3/2sec⁡h3/2(c1log⁡⁡T+c2),θ=u;ii=3c12Ttanh⁡(c1log⁡⁡T+c2),σ2=3c128T2tanh⁡2(c1log⁡⁡T+c2)
for the Bianchi type IX cosmological model (δ=1).Case 4 (forα′-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 (forα′-β′=0 and ϕ′=0).
Here, we getα=β+c.
Without loss of generality by taking the constant of integrationc=0, we get
(3.62)α=β,ϕ=constant(sayc1).
Using (3.62), the field equations (3.2) to (3.7) will reduce to
(3.63)2β′′-3β′2+e4β4=0,(3.64)2β′′-3β′2+δe4β-34e4β=8πλc1e6β,(3.65)3β′2+δe4β-14e4β=8πρc1e6β,(3.66)0=8π3+2ω(ρ+λ)e6β,(3.67)ρ′+(3ρ-λ)β′=0.
From (3.66), we get
(3.68)ρ+λ=0.
From (3.63) to (3.65) and (3.68), we have
(3.69)2β′′+2δe4β-e4β=0.For Bianchi Type II Metric(δ=0)
From (3.69), we get
(3.70)eβ=(aT+b)-1/2,wherea2=1.
From (3.64), (3.65), and (3.70), we have
(3.71)8πλ=-c1(aT+b)2,8πρ=c1(aT+b)2.
From (3.71) we get λ+ρ=0.
The corresponding metric can be written in the form(3.72)ds2=(aT+b)-3dT2-(aT+b)-1[dθ2+dϕ2]-(aT+b)-1[dψ+θdϕ]2.
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
(3.73)eβ=(aT+b)-1/2,wherea2=3.
From (3.64) and (3.65), we have
(3.74)8πλ=-c1(aT+b),8πρ=c1(aT+b).
Therefore, from (3.74), we have
(3.75)λ+ρ=0.
The corresponding metric can be written in the form
(3.76)ds2=(aT+b)-3dT2-(aT+b)-1[dθ2+cosh⁡⁡2θdϕ2]-(aT+b)-1[dψ+sinh⁡⁡θdϕ]2.
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
(3.77)2β′′+e4β=0.
From (3.77), we get
(3.78)S2=e2β=2m1sec⁡h2m1(T+n1).
From (3.63) and (3.64), we get
(3.79)(δ-1)=8πλc1e2β.
From (3.66) & (3.79), we get
(3.80)λ=ρ=0.
The corresponding metric can be written in the form
(3.81)ds2=8m13sec⁡h32m1(T+n1)dT2-2m1sec⁡h2m1(T+n1)[dθ2+sin⁡⁡2θdϕ2]-2m1sec⁡h2m1(T+n1)[dψ+cos⁡⁡θdϕ]2.
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 volumeV, expansion θ, and the shear σ for the models (3.72), (3.76), and (3.81) are given by
(3.82)V=(-g)1/2=(aT+b)-3/2,θ=-3a(aT+b),σ2=3a22(aT+b)2
for the Bianchi type II model,
(3.83)V=(-g)1/2=(aT+b)-3/2cosh⁡⁡θ,θ=tanh⁡θ(aT+b)-3/2-3a(aT+b)-1/2,σ2=tanh⁡2θ(aT+b)-3+3a2(aT+b)-atanh⁡θ(aT+b)
for the Bianchi type VIII model, and
(3.84)V=(-g)1/2=8m13sec⁡h32m1(T+n1)sin⁡⁡θ,θ=u;ii=[2m1tanh⁡2m1(T+n1)sin⁡⁡θ+sin⁡⁡2θ],σ2=[2m12tanh⁡22m1H+2cos⁡⁡2θ+4m1tanh⁡2m1Hcos⁡⁡θ]3,
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 of3, 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 Case5, 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.



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*Source: 573967-2011-11-28.xml*