View of FIVE- DIMENSIONAL COSMOLOGICAL MODELS WITH SCALAR MESON FIELD & BULK VISCOSITY

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FIVE- DIMENSIONAL COSMOLOGICAL MODELS WITH SCALAR MESON FIELD & BULK VISCOSITY

Dr. Bhumika Panigrahi

Associate Professor, Department of Mathematics, Gopal Krishna College of Engineering & Technology, Gourahari Vihar, Raniput, Jeypore - 764001, Orissa, INDIA

Abstract - In this paper we have investigated some five dimensional bulk viscous cosmological models in Lyra geometry in presence of scalar meson field. Some physical features of the models are also discussed.

Keywords: Five dimensions, Lyra Manifold, Bulk viscous coefficient, Scalar Meson Field.

1. INTRODUCTION

The higher dimensional cosmological models play an important role in description of the universe in its early stages of evolution. In recent years, many cosmologists have investigated physics of the early phase of universe in the contest of higher dimensional space times. (The most famous five dimensional theory proposed by Kaluza [1] and Klein [2], was the 1^st theory in which gravitation could be unified in a single geometrical structure). A number of authors [3,4,5,6,7,8] have studied the physics of the universe in higher dimensional space- time as a part of bulk viscous cosmology investigation. In General Theory of Relativity, a scalar field solution is an exact solution of the Einstein field equation. Here the gravitational field is due to entirely the field energy and momentum of a scalar field. This kind of field may or may not be massless. It may be taken to have minimal curvature coupling or some other choice, such as conformal coupling.

The unification of gravitational forces with other forces in nature is not possible in the usual four dimensional space time. So in higher dimensional quautum field theory this may be possible (Appelquist et. al., [9]; Weinberg, [10];

Chodos and Detweler,[11] ). This idea is important in the field of cosmology, as it is known that the universe was much smaller in the early stage of the evolution than today. So we predict that the present four dimensional spacetime of the universe could have proceded by a higher dimensional space-time. The extra dimensional reduced to a volume of the order of the pauk length with the passage of time, which are not observable at the present stage of the universe.

Freund [12], Appelquist and Chodos [13], Randjbar Daemiet.al, [14], Rahaman et. al., [15] and Singh et al., [16] claimed through the solutions of the field equations that there is an expansion of four dimensional space time while fifth dimensions contracts or remain constant.

Further Guth [17] and Alvarez and Gavela [18] observe that large amount of entropy is produced during contraction process, which provides an alternatives resolution to the flatness and horizon problem, as compared to usual inflationary scenario.

2. THE METRIC AND THE FIELD EQUATION

The Einstein’s field equation based an Lyra manifold as proposed by Sen [19]

and Sen and Dunn [20] in normal gauge may be written as

 

 

   









 χ '

4 3 2 R 3 2

R

_ij

1 g

_ij



_i



_j

g

_ij



_m



^m _ij

T

_ij

(1) where

T

_ijis energy momentum tensor for scalar meson field and

T

_ij

'

is energy momentum tensor of bulk viscosity

 , ^,

^{2 2}



2 u, 1

u,

_{i j}

g

_ij

u

_k

u

^k

M u

ij

  



and

^T

_ij^'

^{ }  ^u

_i

^u

_j

^{ g}

_ij



where



is the displacement vector and other symbols have their usual meanings as in the Riemannian geometry.The Klein Gorden takes the form

g u

^ij

,

_ij

 M u

²

 0

. Here we consider the five dimensional spherically symmetric metric in the form.



^{2 2 2 2 2 2}



²

2

2

dt e dr r d r sin d e dy

ds  

^

     

^

(2)

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where



and



are the function of time coordinate only.

It is assumed here that the coordinates to be commoving

i.e., u⁰=1 and u¹= u² = u³ = u⁴= 0 (3) Further, we consider the displacement vector



_i in the form



_i

   , 0 , 0 , 0 , 0 

(4) where β is constant.

The energy momentum tensor Tij for viscous fluid with massive scalar field is



^{' 2}

  

T

_ij

, ,

_{i j}

1

_ij

,

_k ^k

M

² _{i j ij}

u u g u u u u u g

2 

    

(5) where



is the bulk viscous coefficient, uⁱ is the five velocity vector, gij is the covariant fundamental tensor.

Under the commoving coordinate system we have,

1 u

u

_i ⁱ



_. (6)

Further the expansion scalar is given by

; . u



 

(7)

Using eqⁿ (4), (5) and (6) the explicit form of field equation (1) for the line element (2) are obtained as

2 , 4

3 4

3 4 λ

3

₂ ^{2 2}

2

u

 M





^{ }









(8)

2 4

3 4 2 2 4

λ

3

₂ ^{2 2}

2

2

u

 M





















 

 



 





 

(9) and

2 4

3 2 λ 3 2

λ

3

₂ ^{2 2}

2

u

 M

















(10) where over head dot denotes differentiation w.r.t. ‘t’. In the following section we intend to derive the exact solution of the field equation using β (constant) and



(constant) in order to overcome the difficulties due to non linear nature of the field equations.

3. COSMOLOGICAL SOLUTIONS

Here there are three unknowns viz.,

λ

,



, and



involved in three field equations (8) – (10).In order to find the solution of the three field equation, we consider

  a 

(11)where a (≠ 0 ) is a parameter.

Case I: β = Constant

In this case there are three unknowns

λ

,



^and



involved in three field equations (8) – (10). In order to derive explicit solution we consider

  a 

and solving equation (8) – (10), we obtain.

  _ ^

   

 C e t

D A 2 log 1

₁ ^At



^,

(12)

  _ ^

   

 C e t

aD A 2 log 1

₁ ^At



(13)

and

  

 





 



 



  

 

 



  

^

1 1 2

3 2

3

1 1

At At

e C

e D C a a 



(14) where

  ^{3 3}   ¹ 

 a a

D 

,

 ³     ^{3 3 1}

2   

 a a a

A 

and

C

₁

= e

^AC. in this case the metric (2) takes the form

   ^{dr r dθ r sin θ d}  ^{ }

e dt ds

At 1 At

1 log1Ce

A aD 2 2 2 2 2 2 e 2

C 1 Alog D 2 2

2 

  



  











^t

 e

^t

(15) Case II: β=β(t)

In this case, there are three unknowns

λ

,



, and



involved in three field equation (8) – (10). In order to find the solution of the field equations we take i.e., µ = a𝜆 where a is an arbitrary constant.

Substituting this equation in the field equation (8), (9) and (10), we obtain

  ,

2 4

1 3 4

3

₂ ^{2 2}

2

u a   M





 

(16)

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 

2 4

3 3 4 2

1 2

2 2 2

2 2

M u

a

a   a     



 

  

^





   

(17)

and

4 2

3 2 3 2

3

₂ ^{2 2}

2

u

 M















 



, (18) Further, substituting the value of



²

from (10) in (16), (17) and (18) we get.

2 0 3

²

 



 



  

^





 ^a 

.

(19) On integration yields

  , 3

log

₂



₂ ^{2 3}

 

 a t b

^a

a



(20) for

  a 

, we obtain

  , 3

log

₂



₂ ^{2 3}

 

 a t b

^aa

a



(21) In this case line element (2) can be written as

(22) 4. DISCUSSION

Case – I :



^=constant

Some Physical and geometrical properties of the models:-

(a) The anisotropy

σ

is defined as (Raychaudhuri, [21])

 

 







 



^

1 C

1 λ C

σ

At

1 At 1

e

D e

.

(b) Spatial Volume



^{3λ 4 2}



¹²

12

θ e sin (-g)

V    e r

^

3 λ 2

2

sin θ

2

e r e



 

where

^{ D} _ ^ log  1 ^ C

1

^e

^At

 ^{ t} _ ^

A λ 2

and also

^{ aD} _ ^ log  1 ^ C

1

^e

^At

 ^{ t} _ ^

A

 2

(c) Expansion Scalar

At 1

; At

1

C 1

3 3

θ u λ

2 2 C 1

e

a a

D e



  

 

   

               





.

(d)

  a

 

 



  

_



3 2 2 λ

a 3

λ θ

σ

. For a=-3,

θ

σ

tends to infinity. Alsoffor

infinity large value of a,

σ

θ

tends to zero, hence model tends to isotropy.

Case-II:

     t

(a) The anisotropy

σ

is defined as (Raychaudhuri, [21] )

 ³  ²

₂² ₂



λ

σ a a t b

a



 



^ .

(b) Spatial Volume



^{3λ 4 2}



¹²

12

θ e sin (-g)

V    e r

^

3 λ 2

2

sin θ

2

e r e



 

where

λ  log  a

2

t  b

2



^2a3_.

It is observed that when ²

2

t b

  a

_the

spatial volume V is zero,hence the universe starts expanding with zero volume.

(c) Expansion Scalar

 

²

 

²



;

2 2 2 2

2

3 3

θ u λ

2 2 3

a a

a a

a a t b a t b







   

              





It is observed that when ²

,

2

t b

a 

 

tends to infinite.

(d)

  a

 

 



  

_



3 2 2 λ

a 3

λ θ

σ

.

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The model does not approaches to isotropy as

σ  0

and



θ

σ

_constant

   0 ,

for all

t

Thus in this case the anisotropy exist throughout the evolution.

when

t  0 , σ  constant

when

t   , σ  constant

So the shape of the universe does not change during evolution.

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