View of HIGHER DIMENSIONAL COSMOLOGICAL STRING MODELS WITH SCALAR MESON FIELD

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Peer Reviewed and Refereed Journal, ISSN NO. 2456-1037 Available Online: www.ajeee.co.in/index.php/AJEEE

Vol. 06, Issue 04, April 2021 IMPACT FACTOR: 7.98 (INTERNATIONAL JOURNAL) 47 HIGHER DIMENSIONAL COSMOLOGICAL STRING MODELS WITH SCALAR MESON

FIELD

Dr. Bhumika Panigrahi

Associate Professor, Department of Mathematics, Gopal Krishna College of Engineering & Technology, Gourahari Vihar, Raniput, Jeypore - 764 001, Orissa, INDIA

Abstract- In this paper we have discussed some five dimensional string cosmological models in presence of cosmic string with mass less scalar meson field in Lyra Manifold are constructed.

Keywords: Five Dimensions Cosmological Models, Lyra Manifold, Mass Less Scalar and Cosmic Strings.

1 INTRODUCTION

Now a day, there has been a lot of interest in the study of large scale structure of the universe recently cosmic strings have attracted many astrophysicists to try to achieve a plausible description of the early stage of the universe. The study of cosmic strings in elementary particle physics arise from the gauge theories with spontaneous broken symmetry, after the big bang, it is believed that the universe might have experienced a no. of phase transition by producing vacuum domain structures such as domain walls strings.

In this paper we have considered five dimensional cosmological models in presence of cosmic string with scalar meson fluid based on Lyra geometry.

The analog of Einstein’s field equation based an lyra geometry in normal gaugeas obtained by Sen [16] and Sen and Dunn [17] are

 

ij m

m ij j i ij

ij

 g   g   χ 

4 3 2

R 3 2

R 1    

(1) Where



_i is the displacement vector and other symbols have their usual meanings as in the Riemannian geometry.

2. THE METRIC AND THE FIELD EQUATION

Here we consider the five dimensional spherically metric in the form given by:



² ² ² ² ² ²



²

2

dt e dr r d r sin d e dy

ds  

^

     

^

(2) 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

  , 0 , 0 , 0 , 0 



_i



(4) Where β is constant.

The energy momentum tensor Tij for Cosmic string with mass less scalar meson is

, ) , 2 ( , 1 ,

T

_ij

  u

_i

u

_j

 

_s

x

_i

x

_j

 u

_i

u

_j

 g

_ij

u

_k

u

^k

 M

²

u

²

(5) where



is the particle density,

λ

_sis the string tension density,

u

ⁱis the five velocity vector,

g

_ij is the covariant fundamental tensor,

x

ⁱis the direction of anisotropy of cosmic string satisfying.

1 u

u

_i ⁱ

  x

_i

x

ⁱ



(6) and

u

_i

x

ⁱ

 0

(7) Further the expansion scalar is given by

.

;  u



 

(8) Using equation (4), (5) and (6) the explicit form of field equation (1) for the line element (6.2) are obtained as

2 , 4

3 4

3 4 λ

3

₂ ² ²

2

u

 M







   



^ ^



(9)

2 4

3 4 2 2 4

λ

3

₂ ² ²

2

u

 M

 



      









 

(10) And

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Vol. 06, Issue 04, April 2021 IMPACT FACTOR: 7.98 (INTERNATIONAL JOURNAL) 48

    4 2

3 1

1 1 2

3 1

1 1 2

3 ² ² ² ²

1 1 2 2

1 1

2 M u

e C

e C a

a e

C e C

a ^Att

Att Att

Att



 



 









 



 









 











 

2 λ χ

4 χ 3 2 λ 3 2

λ

3

² ²

s 2

2

u

 M









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

3. COSMOLOGICAL SOLUTIONS

Here there are five unknowns viz.,

λ

,



,



,



^and



_sinvolved in three field equations (9)–(11). In order to avoid the in sufficiency of field equations for solving five unknowns through three field equations,

we consider

  a 

(12) Where a ≠ 0 is a parameter.

Case I: (

  

_s

 0

)

The equation of state is given by

 0

 

_s



(13) solving the equation (9) (10) (11) and (12) we get

exp

 

 





 





 



 

  

  C e

^At

t A

a 2 log 1

¹

1 exp  2



(14) exp

  



 





 



 

  

  C e

^At

t A

a a 2 log 1

₁

1 exp  2



(15)

where

1 2

  A  a

(16)

e

AC

C

₁



exp

  



 





 



 

  

  C e

^At

t A

a a 2 log 1

₁

1 exp  2



   

At 1

C C

2 At 2

2 2 2 2 2

1 At 1

1 e 1 C e

3 3a 3 M u

2 a 1 1 e 2 a 1 1 C e 4 2

  

   

     

               

(17)

   

At 1

C C

2 At 2

2 2 2 2 2

1

s At

1

1 e 1 C e

3 3a 3 M u

2 a 1 1 e 2 a 1 1 C e 4 2

  

   

     

               

(18) And

 



 







 



 



  

 

 



  

^ _At^At

e C

e C a

a a

1 1

1 1 1 2 2

3 2

3  



(19) if a=-3,



=0

Case II: Geometric String

]) 21 [ (Letlier



s

 

In this case, due to paucity of one equation, an additional constraint relating these parameters is required to obtain explicit exact solution of the system of field equation. Therefore we take

λ,  0

 a a



is a parameter and the equation of state i.e.

  λ

_s Solving the equation (6.9) (6.10) (6.11) & (6.12), we get

 

 



 

  

  C e

^At

t A

a 2 log 1

¹

1  2



(20)

 

 



 

  

  C e

^At

t A

a a 2 log 1

₁

1  2



(21) where

1 2

  A  a

(22)

e

AC

C

₁



(23)

 

¹ ¹¹ ²³

 

¹ ¹¹ ³⁴ ²

2

3 ² ² ² ²

1 1 2 2

1 1 2

2

u M e

C e C a

a e

C e C

a ^Att

Att Att

Att   

 









 



 









 











 

(24)



 









 



 



 



 



  ^ _At^At

e C

e C a

a a

1 1

1 1 1 2 2

3 2

and 3  

(25)

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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Vol. 06, Issue 04, April 2021 IMPACT FACTOR: 7.98 (INTERNATIONAL JOURNAL) 49 a = -3 then



^{= 0}

4. DISCUSSION For



^=constant

Some Physical and geometrical properties of the models:-

(a)

 

 



 

 



^ _At

1 At 1

C - 1

C 1 1 λ 2

σ e

e

 a

(26) (b) Expansion Scalar



 









 



 



 



 



 

 _i ^ _At^At

e C

e C a

a u a

1 1

1 1 1 2 2

3 2

3

 



^

(27) when t



0,

  cons tan t

and t



-



,

  

therefore when t



0,

  cons tan t

and t



-



,

  

, i.e the universe reduces to black holes.

(c) Since

0

lim

t 

 a



 

 



  



3 2 2 λ

a 3

λ θ

σ

=

Constant (28) when

  0 





undefined

(d) The spatial volume of the universe is



³^λ ⁴ ²



¹²

12

θ e sin (-g)

V    e r

^

(29)

where

 

 



 

  

  2 log 1 C

₁ ^At

1 2 e

t A

 a



when

  0

, V= constant and

  0

^{, V=}



The result shows the universe reduces to infinity, i.e. the universe reduces to black holes.

5. CONCLUSION

In this paper it is shown that the cosmological model for Takabayasi String does exist in five dimensional string cosmological model with mass less scalar meson field.

The expansion in the models stops at infinite time. Thus there is finite expansion in the model.

The result shows the universe reduces to infinity i.e. the universe reduces to black holes.

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2 2 2 4 λ 3

θ e sin



r

 e

