Magnetized bulk viscous Bianchi type IX cosmological models with ...

Ultra Scientist Vol. 22(2)M, 658-664 (2010).

Magnetized bulk viscous Bianchi type IX cosmological models with variable Λ ATUL TYAGI1 and GAJENDRA PAL SINGH2 1

Department of Mathematics & Statistics, University College of Science Mohanlal Sukhadia University, Udaipur – 313001 (INDIA) e-mail: [email protected] 2 Department of Mathematics Maharaja College of Engineering Udaipur : 313024 (INDIA) e-mail : [email protected] (Acceptance Date 14th June, 2010)

Abstract Some Bianchi type IX cosmological models with electromagnetic field are investigated in presence of bulk viscous fluid. For the complete determination of the model we assume that expansion (θ) in the model is proportional to the eigen value σ11 of the shear tensor σij. This condition n

leads to A  B , where  and n are constants. Also the fluid obey the equation of state p =  where 0    1. Some physical and geometrical features of the models are also discussed. Key words : Bianchi IX, Electromagnetic field, Bulk Viscous fluid, Variable Λ

1. Introduction

Models with a dynamic cosmological term Λ(t) are becoming popular as they solve the cosmological constant problem in a natural way. The cosmological constant problem is one of the outstanding problems in cosmology1. A wide range of observations suggest that the universe possess a non-zero cosmological constant2. A cosmological term corresponding to the energy-density of vacuum in the context of quantum field theory. The cosmological term, which is a measure of the energy of empty

space, provides a repulsive force opposing the gravitational pull between the galaxies. The energy it represents, counts as mass by Einstein’s mass-energy equivalent formula. But recent researches suggest that the cosmological term has a very small value of the order of 10-56 cm-2 3. Linde4 has suggested that Λ is a function of temperature and is related to the spontaneous symmetry breaking process. Therefore, it could be a function of time in a spatially homogenous expanding universe5. The latest measurements of the Hubble

Atul Tyagi, et al. parameter6 point to an intrinsic fragility of standard FRW cosmology in such a way that models without a cosmological constant seem to be effectively ruled out7. The large scale distribution of galaxies in our universe shows that the matter distribution can satisfactorily be described by perfect fluid. However, a realistic treatment of the problem requires the consideration of matter distribution other than the perfect fluid. When neutrino decoupling occurs, the matter behaves like a viscous fluid in an early stage of universe. Israel and Vardabs 8 believed that viscous effects did play a role at least at the early stage of the universe. Misner9 has studied the effect of viscosity on the evolution of cosmological model. Belinskii and Khalatinkov 10 have investigated the effect of viscosity in isotropic Friedmann model where coefficient of viscosity is function of energy density. Bali and Jain11 have obtained some expanding and shearing viscous fluid cosmological models. Singh and Yadav12 have investigated Bianchi type-I viscous fluid cosmological model in which coefficient of shear is proportional to the expansion of model. Landau & Lifshitz13 proved that the existence of bulk viscosity is equivalent to restoring equilibrium states. In general theory of relativity, the effect of bulk viscosity on the cosmological evolution has been studied by several researchers Banerjee and Sanyal14, Pradhan et.al.15,16, Singh et.al.17 Bali & Pradhan18, Bali and Singh19. The occurrence of magnetic field on galactic scale is well-established fact today and their importance for a variety of astrophysical phenomena is generally acknowledged, as

659 pointed out by Zeldovich et al.20. Also Harrison21 has suggested that magnetic field could have a cosmological origin. As a natural consequence we should include magnetic field in energymomentum tensor of early universe. Bali & Yadav22 have investigated some Bianchi type IX cosmological models in general relativity, Bali and Dave23 have investigated Bianchi type IX string cosmological model in general relatively. Pradhan et al. 24 have investigated some homogenous Bianchi type IX viscous fluid cosmological models with a varying Λ. Vaidya and Patel25 have studied spatially homogeneous space time of Bianchi type IX cosmological model. In this paper, we study some homogeneous Bianchi type IX cosmological models for bulk viscous fluid in presence of electromagnetic field. The cosmological constant is supposed to be time dependent. We assume that current is flowing along x-axis so the magnetic field is in yz-plane. Thus F23 is the only nonvanishing component of electromagnetic field tensor. Physical & geometrical features in presence & absence of magnetic field are discussed. 2. The metric and field equation : We consider homogenous anisotropic Bianchi type-IX metric in the form of ds2 = –dt2 + A2dx2 + B2dy2 + (B2Sin2y+A2 Cos2y) dz2 – 2A2 cosy dx dz Where A & B are functions of t alone.

(1)

We assume that the cosmic matter consist of

Magnetized bulk viscous Bianchi type IX cosmological models with variable Λ.

660

magnetic bulk viscous fluid given by





Ti j  p  ρ vi v j  pg ij  Ei j

matter distribution (2) yield. (2)

Where ρ is energy density,, p is effective pressure and v j the four velocity of fluid satisfying

v i v j  1 Ei j 

 B44 B42 1 3 A2 H2    2  2    8π p  4 B B B 4B 8πB 4  

(9)  A44 A4 B4 B44 A2 H2         8 π p   A AB B 4B 4 8πB 4 

  

(3)

(10)

The electromagnetic field is given by

 2 A4 B4 B42 1 A2 H2    2  2     8π  ρ  4 AB B B 4B 8 πB 4  

1  1   Fi F j  g ij Fm F m  (4)  4π  4  The Maxwell’s equation is

 F ij  g  0 j x





The equation (5) leads us to F23 = H Sin y, where H is constant.

(5)

p  p  v; ij

(7)

Where  stand for co-efficient of bulk viscosity that determines the magnitude of the viscous stress relative to expansion. The co-ordinate are considered to be comoving so that v1  v 2  v 3  0 & v 4  1 The Einstein field equation ( G  C  1 ) with time dependent cosmological term.

1 Rg ij  g ij  8πTi j 2

(11) The Suffix 4 by the symbol A and B denote the differentiation with respect to t. 3. Solution of field equation : Equation (9)-(11) are three equation

(6)

The effective pressure p is related to the equilibrium pressure p given by:

Ri j 

2

(8)

In the co-moving system of co-ordinate the field equation (8) for the metric (1) & the

in five unknown A, B, Λ, p ,  for the complete determinacy of the system, we need two extra condition. We assume first that the expansion (θ) in the model is proportional to the eigen value σ11 of the shear tensor σij. This condition leads to A =  Bn (12) From (8) & (9) we get

A44 B44 A4 B4 B42 A 2 1 k    2  4  2  4 A B AB B B B B (13) 2

Where k  2H Equation (12) & (13) together lead us

BB44  n  1B42 

1  k 2  1   n  1  B 2 B 2  2n  (14)

Atul Tyagi, et al.

661 Where B  T , x  X , y  Y , z  Z

Now putting B4  f ( B ) in (14) we get

Bff ' ( n  1 ) f

2

1  k  2  4. Some physical and geometrical features:  1  n  1  B 2 B 2 2n  (15)

Equations (15) lead us 2

2

df 2 2 1 k    n  1 f 2    3  3 2 n  dB B n  1  B B B 

(16) Equations (16) lead us

f2

1  1 k 2    n  1  n  1 nB 2 2nB 22 n  

The pressure and density for the model (19) are given by



4n( n  1 )T 4 2 n

8πρ 

(17)





3n  2n  1 2  2n  1L  



L B 2n  2



n2 k n2  n  2 8π p   1  n 2 T 2 2n1  n T 4



Where L is constant of integration

T 2n4

(20)

n( n  2 ) n  2 ( n  1 )k  n2 1 T 2 2nn  1T 4





 2 n  2 n  1 2 n  1L    (21) 4 n( 1  n )T 4 2 n T 2 n4

The reality conditions given by Ellis Equations (17) lead us dB



t  M

1  1 k 2  L    2 2n  2 22 n  n 1  n  1 nB 2nB  B

(18)

Where M is constant of integration. Value of B can be determine by Eq. (18) After a suitable transformation of co-ordinate metric (1) reduce in form of  dT 2

2

ds 

1  1 k 2  L    2 n2  2 22n  n  1  n  1 nT 2nT  T

  2 T 2 n dX 2  T 2 dY 2  ( T 2 Sin 2Y   2 T 2 n cos 2 Y ) dZ 2  2 2T 2 n Cos Y dXdZ

(1) ρ  p >0 (2) ρ  3 p >0 Condition (1) leads to

(22)

n n  1k  n  2 n  1 2  n 2  1 T 2 nn  1T 4 4n (n 1)T 42 n







2n  1L  4 πξθ > 0 T 2 n 4

(23)

The condition (2) leads to

  2n n  1n  4k 2 n 1 2   n  1T 2 n1  nT 4 nT 4 2 n 

42n  1L  24 πξθ  2 > 0 T 2n  4

(24)

The scalar of expansion  calculated for flow (19)

vector

vi

is given by

Magnetized bulk viscous Bianchi type IX cosmological models with variable Λ.

662

θ  n  2 

(25) Also we assume the fluid obey an equation of state of the form p  ργ , Where 0    1 (26) Equation (20), (21) and (26) lead us

1 γ

n1k n  n   n  2  γ   1 n2 T 2 2nn 1T 4





2 2 2n  T n  1 3

σ 11 

1 σ 22  T 2 1  n  3 σ 33 

21γ n1γ

1  1 k 2  L  4    2 n  1  T n  1 nT 2nT 42n  T 2 n 4



2 3n2  n  2  n2 3n 2 γ 42n 4nn 1T



 



 2Tn 1L 1 γ  8πξθ

(27)

2n4

The non vanishing component of shear σ ij are given by

1  1 k 2  L      n  1  n  1T 2 nT 4 2nT 4 2 n  T 2 n  4

1  1 k 2  L    2 n 4  2 4 4 2 n  n  1  n  1T nT 2nT  T

(28)

(29)

1 1  1 k 2  L  T 2 sin 2 Y  2 2T 2 n cos 2 Y n  1    2n 4  2 4 4 2 n  3 n  1  n  1T nT 2nT  T





(30)

σ13  σ 31

2 2 2n 1  1 k 2  L    T cos Y n  1    2n  4  2 4 4 2 n  3 n  1  n  1T nT 2nT  T

(31)

Hence σ2 

2   1 n  1 1 2  k 4   4 2n   2Ln 4 3  n  1  T nT 2 nT   T

  2 T 2 n dX 2  T 2 dY 2  ( T 2 Sin 2Y   2T 2n cos 2 Y ) dZ 2  2 2T 2 n Cos Y dXdZ (33)

(32) 5. Solution in absence of magnetic field :

The pressure and density of the model (33) are given by

In the absence of magnetic filed the geometry of space time is given by

8π p 

 dT 2

2

ds 

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





3n  2n  1 2 n2  1  n 2 T 2 4n( n  1 )T 4 2 n







2n  1L   T 2 n 4

(34)

Atul Tyagi, et al.

663 g220, g330 therefore there is a point type singularity.

n( n  2 )  2 n  1n  2 8πρ  2  n  1 T 2 4n( 1  n )T 4 2 n





Acknowledgement



2n  1L   T 2 n 4

(35)

Using (26) in (34), (35) we get

1 γ  

n n  n  2γ 1n2 T2

 

References

2 3n2  n  2  n2 3n 2 γ  42n 4nn 1T





 



2n 1L 1 γ  8πξθ

(36)

T 2n4

6. Conclusion We have obtained a new class of anisotropic cosmological models including an electromagnetic bulk viscous fluid as the source. In general the models represent expanding, shearing and non rotating universe. When -2