ANISOTROPIC BIANCHI TYPE-II COSMOLOGICAL MODELS INSELF-CREATION COSMOLOGY

SHRI RAM and C.P. SINGHDepartment of Applied Mathematics, Institute of Technology, Banaras Hindu University,

Varanasi-221005, India

(Received 1 August, 1997; accepted 21 April, 1998)

Abstract. Totally anisotropic Bianchi type-II cosmological solutions are presented in Barber’ssecond self-creation theory of gravitation both in vacuum and in the presence of stiff-matter. Thecorresponding cosmological models have no finite singularity. The stiff-matter model gives essen-tially an empty universe for large time. Some physical and kinematical properties of the models arediscussed.

1. Introduction

The consideration of cosmological models which are spatially homogeneous butanisotropic have significant roles in the description of the universe at its earlystages of evolution. Bianchi spaces I–IX are useful tools in constructing modelsof spatially homogeneous cosmologies. Several modification of Einstein’s generalrelativity have been proposed and extensively studied sofar by many cosmologistsin an attempt to unify gravitation, electromagnetic fields and many other effects inthe universe. In an attempt to produce a continuous creation theory, Barber (1982)has proposed two self-creation cosmologies by modifying Brans and Dicke (1961)theory and general relativity. First one is the modified Brans-Dicke theory whichis unsatisfactory since the equivalence principle is violated (Brans, 1987). Thesecond theory is a modification in general relativity to a variable G-theory. In thistheory the scalar fieldφ does not directly gravitate, but simply divides the mattertensor, acting as a reciprocal of gravitational constant. The scalar field couplesto the trace of the energy-momentum tensor. Barber (1982), Pimentol (1985) andSoleng (1987) have discussed the Friedmann-Robertson-Walker (FRW) modelsin Barber’s second theory of gravitation. Reddy and Venkateswarlu (1989) havestudied the Bianchi-type-VI0 cosmological model in this theory. Shanthi and Rao(1991) have studied the Bianchi type II and III cosmological models in Barbersecond theory of gravitation both in vacuum and in the presence of stiff-matter.

In this paper, we present homogeneous and totally anisotropic Bianchi type IIcosmological solutions in Barber’s second theory of gravitation both in vacuumand in the presence of stiff-matter. We also discuss the physical and kinematicalproperties of the solutions.

Astrophysics and Space Science257: 287–292, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

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288 SHRI RAM AND C.P. SINGH

2. Field Equations

We consider homogeneous and totally anisotropic Bianchi type II line-element inthe form (cf. Guzman, 1991)

ds2 = −dt2 +A2(dx − z dy)2 + B2dy2 + c2 dz2 (1)

whereA, B andC are functions of timet only. The field equations in Barber’ssecond self-creation theory of gravitation are given by (Barber, 1982)

Rij − 1

2gijR = −8π

φTij (2)

and

�φ = 8π3 λT (3)

whereλ is a coupling constant to be determined from experiments (|λ| ≤ 0.1,λ = 0 corresponds to general relativity) andG = 1/φ. Tij is the energy-momentumtensor andT the trace of the energy-momentum tensorT ik . Other symbols havetheir usual meanings.

The energy-momentum tensorTij for perfect fluid distribution is given by

Tij = (ρ + p)uiuj + pgij , uiui = −1 (4)

whereui is the four-velocity vector of the fluid andp andρ are pressure and matterenergy-density, respectively. In comoving coordinates, the field equations (2) and(3) for the metric (1) give

B44

B+ C44

C+ B4C4

BC− 3

4

A2

B2C2= −8π

φp, (5)

C44

C+ A44

A+ C4A4

CA+ 1

4

A2

B2C2= −8π

φp, (6)

A44

A+ B44

B+ A4B4

AB+ 1

4

A2

B2C2= −8π

φp, (7)

A4B4

AB+ B4C4

BC+ C4A4

CA− 1

4

A2

B2C2= 8π

φρ, (8)

φ44+ φ4

(A4

A+ B4

B+ C4

C

)= 8π

3λ(ρ − 3p). (9)

Again using correspondence to general relativity we define equivalent densities andpressures as

ρeq = ρ/φ (10)

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ANISOTROPIC BIANCHI TYPE-II COSMOLOGICAL MODELS 289

peq = p/φ. (11)

The energy-conservation equation of general relativity (cf. Soleng, 1987) is givenby (

ρ

φ

)4

+(ρ

φ+ p

φ

) (A4A+ B4

B+ C4

C

) = 0. (12)

A subscript 4 and 44 after an unknown function denote first and second derivativeswith respect to time.

3. Solutions of Field Equations

We introduce the transformation

A = exp(α), B = exp(β), C = exp(γ ), dt = AB C dT . (13)

By use of (13), the field equations (5)–(9) and (12) reduce to

γ ′′ + β ′′ − α′β ′ − β ′γ ′ − γ ′α′ − 3

4exp(4α) = −8π

φp exp(2α + 2β + 2γ ),

(14)

γ ′′ + α′′ − α′β ′ − β ′γ ′ − γ ′α′ + 1

4exp(4α) = −8π

φp exp(2α + 2β + 2γ ),

(15)

α′′ + β ′′ − α′β ′ − β ′γ ′ − γ ′α′ + 1

4exp(4α) = −8π

φp exp(2α + 2β + 2γ ),

(16)

α′β ′ + β ′γ ′ + γ ′α′ − 1

4exp(4α) = 8π

φρ exp(2α + 2β + 2γ ), (17)

φ′′ = 8π

3λ(ρ − 3p)exp(2α + 2β + 2γ ), (18)

and(ρ

φ

)′+(ρ

φ+ p

φ

)(α′ + β ′ + γ ′) = 0; (19)

where a dash denotes ordinary differentiation with respect toT . Adding Equations(14), (15), (16) and (17), we obtain

α′′ + β ′′ + γ ′′ − α′β ′ − β ′γ ′ − γ ′α′ − 1

4exp(4α)

= 8π

φ· 1

2(ρ − 3p)exp(2α + 2β + 2γ ). (20)

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290 SHRI RAM AND C.P. SINGH

A linear combination of Equations (14)–(17) and (20) provides

α′′ + 1

2exp(4α) = 8π

φ

1

2(ρ − p)exp(2α + 2β + 2γ ) (21)

β ′′ − 1

2exp(4α) = 8π

φ

1

2(ρ − p)exp(2α + 2β + 2γ ) (22)

γ ′′ − 1

2exp(4α) = 8π

φ

1

2(ρ − p)exp(2α + 2β + 2γ ) (23)

and

α′′ + β ′′ + γ ′′ − 2α′β ′ − 2β ′γ ′ − 2γ ′α′

= −8π

φ· 1

2(ρ + 3p)exp(2α + 2β + 2γ ). (24)

The field equations (21)–(24) are highly non-linear havingα, β, γ, ρ, p andφ asunknowns, the general solutions of these equations are extremely difficult. So, weconsider the following physically important cases and obtain the correspondingmodels of the Universe.

CASE I: VACUUM MODEL (ρ = p = 0)

In vacuum equations (19) and (21)–(23) give the solutions

A2 = exp(2α) = a sech[a(T +m)] (25)

B2 = exp(2β) = k1 exp(bT ) cosh[a(T +m)] (26)

C2 = exp(2γ ) = k2 exp(cT ) cosh[a(T +m)] (27)

wherea, b, c,m, k1 andk2 are constants of integration. Without loss of any gener-ality we can takek1 = k2 = 1, the equation (18) gives

φ = m1T +m2 (28)

wherem1 andm2 are constants of integration. The equation (24) gives a relationbetween the constants, related by

a2− bc = 0. (29)

The corresponding vacuum model can now be written in the form

ds2 = −K exp(b+ c)T cosh[a(T +m)]dT 2+ a sech[a(T +m)]

(dx − z dy)2 + k1 exp(bT ) cosh[a(T +m)]dy2

+k2 exp(cT ) cosh[a(T +m)] dz2 (30)

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ANISOTROPIC BIANCHI TYPE-II COSMOLOGICAL MODELS 291

whereK is another constant.Equation (30) represents anisotropic Bianchi type-II vacuum universe in Bar-

ber’s second theory of gravitation. It is clear from Equation (28) that the scalarfield φ becomes constant form1 = 0 and hence, the model (30) reduces to generalrelativistic case. The model has no singularity atT = 0.

CASE II. Z EL’ DOVICH UNIVERSE (ρ = p)Whenρ = p, Equations (21)–(23) have the same solutions as given by Equations(25), (26) and (27). Equation (18) gives

φ = cos(√

16πλc1

3T + C2

)(31)

whereC1 andC2 are constants of integration, Equation (24) gives the constantsrelation

a2− bc + 16πC1 = 0. (32)

Now the metric (1), in view of (31), takes the form

ds2 = cosh[a(T +m)][−s exp{(b+ c)T }dT 2+ k1 exp(bT )dy2

+k2 exp(cT )dz2] + a sech[a(T +m)] (dx − zdy)2 (33)

wheres is another constant.The pressurep and energy-densityρ in the model (33) are given by

ρ = p = C1C3

acos

(√16πλC1

3T + C2

)sech{a(T +m)} exp{−(b+ c)T }

(34)

whereC3 is another constant. For pressure and energy-density to be positive, it isnecessary thatC1, C3 > 0. The physical and kinematical parameters for model(33) are given by

Spatial volume:V 3 = (−g) 12 = [a3 cosh[a(T +m)]exp(b+ c)T ] 12 (35)

Expansion scalar:θ = 1

2[a tanh{a(T +m)} + (b + c)] (36)

Shear scalar:σ 2 = 1

3[a2 tanh2{a(T +m)} + 1

2a(b + c) tanh{a(T +m)}

+1

4(b2+ c2 − bc)] (37)

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292 SHRI RAM AND C.P. SINGH

Hubble parameter:H = 1

6[tanh{a(T +m)} + (b + c)] (38)

Deceleration parameter:q = −6a2 sech2{a(T +m)}[tanh{a(T +m)} + (b+ c)]2 − 1, (39)

wherea3 is constant.The solution (25), (26), (27), (31) and (34) satisfy the energy-conservation

equation given in (19). The spatial volume in the model (33), given by Equation(35) shows the expansion of the universe with time. The pressure, energy-densityand scalar field are not singular atT = 0. The model has no finite singularity.As T tends to infinity, pressure and energy-density tend to zero and the scalarfield becomes constant. Thus, Equation (33) represents an expanding cosmologicalmodel which would essentially gives an empty universe for large timeT . As Ttends to infinity, the kinematical parametersθ , σ 2 andH tend to constant limitsand since limT→∞ σ/θ = constant, the model is not isotropic for largeT . Thedeceleration parameterq is negative forT <∞ and tends to−1 asT →∞. Thismodel does not admit rotation and acceleration.

Whenλ → 0, the scalar field(φ) from Equation (21) becomes constant andhence model (33) represents general relativistic anisotropic Bianchi type II uni-verse in the presence of stiff-matter. In the case, whenb = c. Equation (33)reduces to corresponding LRS (locally rotationally symmetric) Bianchi type-IImodel obtained by Shanthi and Rao (1991).

4. Concluding Remarks

We have presented totally anisotropic Bianchi type-II cosmological models both invacuum and in the presence of stiff-matter in Barber’s second self-creation theory.We have studied the physical and kinematical behaviors at infinity of the modelfilled with stiff-matter. This model, free from finite singularity would essentiallygives an empty space for infinitely large time.

References

Barber, G.A.: 1982,Gen. Relativ. Gravitation14, 117.Brans, C.: 1987,Gen. Relativ. Gravitation19, 949.Brans, C. and Dicke, R.H.: 1961,Phys. Rev.124, 925.Guzman, E.: 1991,Astrophys. Space Sci.179, 331.Pimentel, L.O.: 1985,Astrophys. Space Sci.116, 395.Reddy, D.R.K. and Venkateswarlu, R.: 1989,Astrophys. Space Sci.155, 135.Shanthi, K. and Rao, V.V.M.: 1991,Astrophys. Space Sci.179, 147.Soleng, H.H.: 1987,Astrophys. Space Sci.139, 13.

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