Bianchi Type IX Two Fluids Cosmological Models in General Relativity

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 ISSN 2229-5518

689

Bianchi Type IX Two Fluids Cosmological Models in General Relativity D. D. Pawar1 , V. J. Dagwal2 Abstract— In this paper we have studied Bianchi Type IX two fluids cosmological models with matter and radiating source. In the model one of the fluids represents the matter content of the universe and another fluid is the CMB radiation. To get the deterministic model, we have asn sumed a supplementary condition a = b where a & b are metric potentials and n is the constant. We have also investigated the behaviors of some physical parameters. Keywords: Bianchi Type-IX Space Time, CMB radiation, Two Fluids.

——————————  ——————————

1 INTRODUCTION

T

sent through out the evolution of the universe. HE cosmic microwave background (CMB) is one of the

2 FIELD EQUATIONS

cornerstones of the homogeneous, isotropic model. Anisotropies in the CMB are related to small perturbation, superimposed on the perfectly smooth background, which are believed to seed formation of galaxies and large-scale structure in the universe. Coley and Dunn1 (1990), have studied Bianchi

We consider the metric in the form

ds 2   dt 2  a 2 t  dx 2  b 2 t  dy 2 

b

2

sin 2 y  a 2 cos 2 y  dz 2  2a 2 cos ydxdz (1)

where a & b are functions of t alone. The Einstein’s field equation is

type VI0 model with two fluid source. Pant and Oli2 (2002) has been investigated two fluid cosmological models using Bianchi type II space-time. Oli (2008) presented anisotropic, ho3

R ij 

1 i g j R  T ji . 2

(2)

mogeneous two fluid cosmological models in a Bianchi type I

The energy momentum tensor for two fluids sources is given

space time with a variable gravitational constant G and cos-

by

mological constant. Many researchers were investigated the

Ti j  Ti (jm )  Ti (jr ) ,

Friedmann-Robertson-Walker models with two fluids source [Davidson4 (1962), McIntosh5 (1968), Coley and Tupper6 (1986), Coley7 (1988), Verma8 (2009) etc.]

where and

Recently Adhav9 et al. (2011) examined two fluid cosmological models in Bianchi type V space-time. Here we have studied two-fluid models in Kantowski-Suchs space time. In these models we assume that both the fluids are pre-

School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Vishnupuri Nanded-431 606 (India) E-mail: [email protected]

) Ti (m j is the energy momentum tensor for matter field

Ti (rj ) is the energy momentum tensor for radiation field

which are given by

Ti (jm )   pm   m  uim u mj  pm g i j ,

Ti (jr )  with

————————————————

(3)

4 1  r uir u rj   r g i j , 3 3 g i j uim u mj  1 , g i j uir u rj  1 .

(4) (5) (6)

1

Govt.College of Engineering, Amravati-444 604 (India) E-mail: [email protected] 2

The off diagonal equations of (2) together with energy conditions imply that the matter and radiation are both comoving. We get,

IJSER © 2013 http://www.ijser.org


ui( m )  0,0,0,1 ,

ui( r )  0,0,0,1

When

(7)

f1 

The Einstein’s field equation (2) reduces to –



3a 2 2b 1 b 2    2  2  Pm  r 4 4b b b b 3

a b ab a 2     2  Pm  r a b ab 4b 3

,

df 2 f 2 2b 2 n 3 2  2 1  n    . (19) db b 1  n  b 1  n 

(9)

Equation (19) leads to

2

b 2n1 1   b 21n  . 2 2n1  n  1  n

f2

2     2 Pm   m   r  , 3   Pm 

df . db

Equating (16), (17) and (18) we have

(8)

ab b 1 a  2 2 4  ab b b 4b 2

2

,

690





where  is integration constant.

(10)

Equating (14) and (15) we get

r  0. 3

2n b     10 Pm   m .  n  1 2n  n  1 2 1 n

(11)

b

2 n 1



3 SOLUTION OF THE FIELD EQUATION

(21)

Equations (8) to (11) are four independent equations in five unknowns a, b,  r ,

m

and

pm . For the complete de-

terminacy of the system, we need extra conditions. We assume a relation in metric potentials as

a  bn

(12)

(13)

Pm    1 m ,

(23)

total energy density as (14)

(15)

 2 n 1 1 2n  T 21 n   m    T  , (24) 10  9    n  1 2n  n  1 

r 

Equating (8), (9) and (12) we get

b 2 2b 2n3 2   2b  21  n    b 1  n  b1  n 

1   2

We get energy density matter, energy density of radiation and

Equating (13) and (14) we get .

(22)

For the equation of state of matter we shall assume the -law


5b 1 b 2 3a 3ab      8Pm . b b2 b2 a ab

1

 T 2 n 1  1 21 n   dT 2  T 2 n dX 2 ds      T 2  2n 1  n  1  n   T 2 dY 2   T 2 sin 2 Y  T 2 n cos 2 Y  dZ 2  2T 2 n cos YdXdZ 2

where b = T, x = X, y = Y and z = Z


a b 3ab b 2 1      2 Pm   m a b ab b 2 b 2

The metric (1) reduces to the form

.

when n is the constant.

 a 2b   5 Pm  m a b 2

(20)

3    1  2 n 1 2n T 21 n  T     , (25) 10  9    n  1 2n  n  1 

   r  m , (16)

We assume that

(26)

3  2   2 n 1  2n  T 21 n      T . 10  9    n  1 2n  n  1 

(27)

b  f b  ,

(17)

Case I: Dust model when

b  f f 1 .

(18)

The scalar of expansion, shear scalar and deceleration parameIJSER © 2013 http://www.ijser.org

 1


691

ter are given by

(34) 1

2  T 2 n 4 1 2 n  2  ,   3H   n  2    2  T 2   2n 1  n  T 1  n 

 2  n 2  1  T 2 n  4 1 2 n  4 ,      T  2 2 3   2n  n  1 T 1  n  2

(35)

(28)

 T 4 n 8 2 q  1   n  2 n  1  2  4n 1  n  T 8 T 2 n  6 T 2 n 6 ,(36)   1  n  1  n  n1  n  1  n 2

  T 2 n 4 2 2 1 2 2 n  4 ,    n  1   2   T 2 3   2n  n  1 T 1  n 





(29)

 T 4 n 8  T 8 q  1  6  n  2   n  1   2  4n 1  n  1  n  T 2 n 6    nT 2 n 6 2 n 1  n  1  n  2



1 T 1  n 2  4

2

   nT 4 n 8   

 nT 2 n6 

m 

1



T 4 1 n



2 2





  nT  4n8  

 2 n 1 2n T 21 n     T  n  1 2n  n  1  

1 2

2n T 21 n    n  1 2nn  1

m 



2 n 4

 T  1 3  2  T  2 n  2   2  2n1  n  T 1  n 





2n T 21 n    n  1 2nn  1

 T 2 n 4  1 3  2  T  2  n  2   2  2n1  n  T 1  n 



1 2

13  n  2   T 2 n  4  1 2  n  2   2   T   2  2n 1  n  T 1  n  

,

(38)

(32)



, (31)



r  0 ,

n  21 T 2 n1  

r 

2

2

 2 n 1 2n T 21 n   T    .   2n  n  1   n  1 

The density parameters are



,

(37)

(30)

n  21 T 2n1 

2

13  n  2   T 2 n  4  1 2  n  2   2   T   2  2n 1  n  T 1  n  

2

 . (33)



 2 n 1 2n T 21 n   T    .   2n  n  1   n  1 

2 2

13  n  2   T 2 n  4  1 2  n  2   2   T   2  2n 1  n  T 1  n  

2

.

(39)

Case II: Radiation universe: Case III: Hard Universe:

4 When   3 The scalar of expansion, shear scalar and deceleration parame-

Let

 

ter

4     ,2  3 

5 3

1

2  T 2 n 4 1 2 n  2    3H   n  2    2  T 2   2n 1  n  T 1  n 

The scalar of expansion, shear scalar and deceleration parameter in hard universe are



692

(46)

1

2  T 2 n 4 1   3H   n  2    2 1 n 2   T 2 n  2   ,  2n 1  n  T   

2 

(40)

(47)

  T 2 n 4 2 1 2 n  4 ,    n 2  1   2   T 2 3 2 n n  1 T 1  n       2

q  1  6n  2 n  1 2

 T 4 n 8  T 8 T 2 n  6     2 1  n  1  n   4n 1  n , T 2 n 6  nT 2 n6 2 n1  n  1  n

(41)



4 n 8

 T 2 q  1  6n  2 n  1  2  4n 1  n  T 8 T 2 n 6 ,(42)  1  n  n1  n  1  n 2







  T 2 n 4 2 2 1 2 n  4 , n  1   2   T  2 3   2n  n  1 T 1  n 

1



T 4 1  n2



2











  nT 4 n8  

T 4 1 n



2 2

r  m 

 2 n 1 2n  T 21 n   T     n  1 2n  n  1  

1 m  2 23  n  2    T 2 n 4 1 2  n  2   2  T   2  2n 1  n  T 1  n  

  nT  4n8  

1 2 33n  2 

 2n1 2n T 21 n   T    n  1 2nn  1 

,

,(49)

 T 2 n 4  1  2  T  2  n  2    2  2n1  n  T 1  n 



2



  m  r ,

(43)

r 



1



(48)

 2 n 1 2n T 21 n     . T  2n  n  1   n  1 



3 2 23  n  2    T 2 n 4 1  2   T 2  n  2    2 2 n 1  n  T 1  n    

,

2 2 33n  2 

 2n1 2n T 21n   T    .   n  1 2nn  1  

.(50)

 T 2 n 4  1  2  T  2 n  2    2  2n1  n  T 1  n 



(44)

2



21 n 



 2 n 1  2n T   . T  2n  n  1   n  1 

2 2 23  n  2    T 2 n 4 1 2  n  2   2  T   2  2n 1  n  T 1  n  

4 CONCLUSION

.

Here we observe that when n = –2 then expansion

 0

(45)

q  1  0

indicates the accelerating model of

the universe. Also the energy density of matter

Case IV: Zeldovich Universe: When

and

total energy

 2

r  0

  T 2 n 4 1 2 n  2    3H   n  2    2   T 2   2n 1  n  T 1  n 

1 2

for dust model only.





lim    2 n2  1 In all cases 0    T    n  12


and

   but energy density of radiation

2

,

m  


Therefore these models do not approach isotropy for large value of T. These models are expanding, the expansion in the model starts with big bang at T = 0.

Acknowledgments: The author D D Pawar especially thanks to University Grant Commission, Pune for provides the grant under MRP scheme.

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