Cmjv03i01p0109

J. Comp. & Math. Sci. Vol.3 (1), 109-113 (2012)

LRS Bianchi Type-I Cosmological Model Universe with Bulk Viscosity R. K. TIWARI and DIVYA SINGH Department of Mathematics, Govt. Model Science College Rewa - 486001 M. P., India.

(Received on: 11th February, 2012) ABSTRACT Exact solutions for an anisotropic Bianchi type-I model with bulk viscosity and variable G and Λ are obtained we have found an expanding universe and isotropy at late times. The gravitational constant is found to increase with time and the cosmological constant decreases with time as Λα t

−2

.

Keywords : Cosmology variable G, Λ Bianchi Models.

1. INTRODUCTION In recent papers Kalligas, Wesson, and Everitt1, Beesham2, Arbab and Beesham[3] have investigated FRW model as well as anisotropic Bianchi type-I cosmological models with variable gravitational constant G and cosmological constant Λ . Along the same time. Arbab4,5 has considered a viscous cosmological model with variable G and Λ and shown that the universe isotropic is the course of expansion. We wish here to derive some results in Bianchi type-I cosmology with viscous fluid where G and Λ vary with time using a slightly different method from that of Arbab6 .

In order to solve Einstein’s field equations, we normally assume a form for the matter content or suppose that the space

time admits killing vector symmetries.

Solutions to the field equations may also be generated by a law of variation of cosmological term Λ . The phenomenological Λ - decay scenarios have been considered by a number of authors Freese7, Pradhan8, Singh C.P.9, Chen and Wu10 −2 considered Λ varying as R , Carvalho and Lima11 generalized it by taking

Λ = α R −2 + β H 2 where R is the scale factor of Robertson-Walker metric, H is the

Hubble parameter and α , β are adjustable dimensionless parameter on the basis of quantum field estimation in the curved expanding background. Schützhold[12,13] recently proposed a vacuum density proportional to the Hubble parameter this leads to a vacuum energy density decaying

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R. K. Tiwari, et al., J. Comp. & Math. Sci. Vol.3 (1), 109-113 (2012)

as Λ ≈ m 3 H where m ≈ 150 MeV is the energy scale of the chiral phase transmission of QCD. In a recent year Borges and Carnerio14 have considered an isotropic and homogeneous flat space filled with matter and cosmological term proportional to Hubble parameter H. Very recently we have presented perfect fluid cosmological models with variable G and Λ in FRW model and Bianchi type-I model15,16.

In the paper we study Bianchi type-I cosmological model with bulk viscosity and time dependent gravitational constant G and the cosmological constant Λ . We obtain solution of the Einstein equation assuming that cosmological term proportional to the Hubble parameter. The paper is organized as follows : Basic equations of the model is in Sec. 2 and solution is given in Sec. 3. We discuss the model and conclude our results in Sec. 4.

2. MODEL AND FIELD EQUATIONS The spatially homogeneous and anisotropic Bianchi type-I space time is described by the line element. ds2 = dt 2 - A2(t) dx2 - B2(t) [dy2 +dz2]

(1)

The spatial volume of this model is given by

V = AB

2

(2)

(

)

1

We define R = AB 2 3 as the average scale factor so that the Hubble parameters of anisotropic models may be defined as

H=

R& 1  A& 2 B&   =  + R 3  A B 

(3)

Here and elsewhere a dot stands for ordinary time derivative of the concerned quantity.

H=

1 ( H1 + H 2 + H 3 ) 3

Where H1 =

(4)

A& B& C& , H2 = and H3 = are A B C

directional Hubble’s factors in the directions of x,y, and z respectively.

We assume that the cosmic matter is represented by the energy momentum tensor of an imperfect fluid.

Tij = ( ρ + p *) ui ui – p*gij

(5)

Where ρ and p are energy density, effective pressure and ui is the four velocity vector of the fluid satisfying the relation uiui = -1 Here p * = p - 3ηH.

(6)

η is the coefficient of viscosity The bulk viscosity takes the from η = ηo ρn ηo ≥ 0 ,

(7) n is a const.

We assume that the matter content obeys on equation of state. p = (γ-1) ρ,

0≤ γ≤2

(8)

The Einstein’s filed equation with time dependent G and Λ are Rij -

1 Rg ij = 8πG (t )Tij − Λ (t ) g ij 2

(9)

For the metric (1) and energy momentum tensor (5) in co-moving system of coordinates the fields equation (9) yields.

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R. K. Tiwari, et al., J. Comp. & Math. Sci. Vol.3 (1), 109-113 (2012)

A& B& B& 2 2 + = − 8πGρ − Λ AB B 2

(10)

&& B && A& B& A + + = 8πGp * − Λ A B AB

(11)

&& B& 2 2B + 2 = 8πGp * − Λ B B

(12)

In view of the vanishing of divergence of Einstein tensor, we have 8 π G&

ρ + Λ& + 8πG

 &   &  ρ& + (ρ + p *) A + 2 B  A   B   

= 0 (13)

The conventional energy conservation splits equation (13) leads

& A B&  ρ& + ( ρ + p )  + 2  = 0 B A

(14)

and

& + 8πG& ρ = 8πG (9ηH 2 ) Λ

(15)

Using equation (3) in equation (15) we obtain

ρ& + 3γρH = 0

(16)

3. SOLUTION OF THE FIELD EQUATIONS The system of equations (7), (8) (10) – (12) and (15) supply only seven equation in eight unknowns (A, B,C, ρ , p ,η , G and Λ) one extra equation is needed to solve the system completely. For this purpose we take cosmological term is proportional to the Hubble parameter. This variation law for vacuum density has initially proposed.

Recently Borges and Carnerio[14] have considered a cosmological term proportional to H. Thus we take the decaying vacuum energy density.

Λ = aH

(17)

For constant G and ρ(γ = 0) equation (17) gives H = k1ebt ,

(18)

Where k1 and b are constant And hence R= k2 exp [k1 exp (bt)] t, constant

where k2 is (19)

This extra or “hyper inflationary solution has been obtained by Barrow.

During this epoch Λ grows enormously.

This solution does not hold for the isotropic case it is shown however by Beesham the de-Sitter type solutions do not exits for either constant or non-constant energy density .

Using equation (7), (16) , (17) in equation (15), one gets 8πG& ρ

+

a 3γ

 2   ρ& ρ&&   2 −  ρ ρ  

=

2

8πη 0  ρ&  n   ρ G   γ2 ρ

(20)

If we assume the density to have the form

ρ = c1t κ Where c1 , κ are constants

(21)

The equation (21) becomes

− akt − k − 2 κ 2η G& − 1− n 0 2 t kn − k − 2G = 24πc1γ c1 γ

(22)

This equation admits a power law solution of the form

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R. K. Tiwari, et al., J. Comp. & Math. Sci. Vol.3 (1), 109-113 (2012)

G = c2t l , Where c2 ,ι are constants

(23)

8πGσ = c3t −2 + c4t −1

(30)

It follows from equation (22) that

Where c3 and c4 are constant

1 n and l = k =− l − n, 1− n

c3 =

This result has already been obtained for the isotropic case. Hence eq. (21), (23) and (16) become

ρ = c1t G = c2 t H=

The Vacuum energy ρ v and critical density

ρ c are given by.

−1 1− n

(24)

n 1− n

(25)

1 3γ (1 − n)t

(26)

ρv =

ρc =

and

a Λ= t −1 3(1 − n)γ

(27)

2  A& B&  8πGσ = 2  −   A B

(28)

Using equation, (14) and equation (10) the above equation becomes

a 24πγc2 (1 − n)t

(31)

1 1− n

3H 2 1 = 2−n 8πG 24πγ 2 c2 (1 − n) 2 t 1−n

(32)

ρ = 24πγ 2 c1 c2 (1 − n) 2 t ρc

The deceleration parameter model is q

= −1 −

q for the

H& = 3γ (1 − n) − 1 H2

(33)

CONCLUSION We conclude that the form of

ρ , G and Λ for the anisotropic universe are

2

2  ρ&    + 48πGρ + 6Λ γ 2  ρ 

Λ = 8πG

and the ratio between matter densities and critical density is given by.

Ω=

These parameters are the same as the ones for the isotropic case we now consider the anisotropy energy ( σ ) defined by

8πGσ =

2 , c4 = 8πG γ (1 − n) 2

(29)

From equation (27), we see that the anisotropy energy ( σ ) becomes

similar to those of the isotropic one. Except for the inflationary period the anisotropic energy (σ ) varies as 8πGσ ~ t −2 . This results has been obtained by Kalligas et al.1.

During the desittter expansion σ decreases

Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)

R. K. Tiwari, et al., J. Comp. & Math. Sci. Vol.3 (1), 109-113 (2012)

exponentially with time. This results is equivalent to the asymptotic solution obtained by Kalligas. A hyperinflation is obtained for the period during which the density and the gravitational ‘constant’ were constant. The Λ retains its variation, Λ ∝ t −2 , which is thought to be fundamental.

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Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)