1 ISSN (Onlne): Index Coperncus Value (2013): 6.14 Impact Factor (2013): Banch Type-II Cosmologcal Model n Presence of Bulk Stress wth Varyng- n Gener...

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Bianchi Type-II Cosmological Model in Presence of Bulk Stress with Varying- in General Relativity V. G. Mete1, V.D.Elkar2 1

Department of Mathematics, R.D.I.K. & K.D. College, Badnera- Amravati, India.

2

Department of Mathematics, J.D.Patil Sangludkar Mahavidyalaya, Daryapur,Dist.Amravati, India.

Abstract: Bianchi Type-II space time is investigated in presence of bulk stress given by Landau and Lifshitz. To get a solution, a supplementary condition between metric potentials is used. The viscosity coefficient of bulk viscous fluid is assumed to be a simple power function of mass density where as the coefficient of shear viscosity is consider as proportional to the scale of expansion in the model. The cosmological term is found to be a decreasing function of time, which is supported by results found from recent type Ia supernovae observations. Also some physical and geometrical properties of the model are discussed.

Keywords: Bianchi Type-II space time, viscous fluid, variable cosmological constant. variations of cosmological term were investigated during the

1. Introduction The Einstein’s field equations has two parameters, the Newtonian gravitational constant G and the cosmological The Newtonian constant of gravitation G constant plays the role of a coupling constant between geometry of space and matter in Einstein’s field equation. In an evolving universe, it is natural to take this constant as a function of time. In the modern cosmological theories, the dynamic remains a focal point of interest as it cosmological term solves the cosmological constant problem in a natural way. There is significant observational evidence towards or a identifying Einstein’s cosmological constant component of material content of the universe that varies . slowly with time and space and so acts like Recent cosmological observations by the High-z Supernova term and the Supernova cosmological project suggest the existence of a positive cosmological constant

with

magnitude

These

observations on magnitude and red-shift of type Ia Supernova suggest that our universe may be accelerating with a large function of the cosmological density in the form - term. Earlier researchers on this

of the cosmological

topic, are contained in Zeldovich , Bertolami

, Weinberg

, Ratra & Peebles

,

Tsagas

and

, Carrol,

Maartens

,Vishwakarma , and Pradhan et. al. . This motivates us to study the cosmological models in which varies with time Cosmological scenarios with a time varying have been proposed by several researchers. A number of models with different decay laws for the

Paper ID: SUB15254

Carvalho et al.

, Pavon

Lima and Maia

Lima and Trodden

Arbab and Abdel-Rahman Carneiro and Lima

Cunha and Santos

Weinberger

, Heller and

Klimek , Misner Collins and Stewart have studied the effect of viscosity on the evolution of discussed cosmological models. Xing-Xiang Wang Kantowski-Sachs string cosmological model with bulk viscosity in general relativity. Also several aspects of viscous fluid cosmological model in early universe have been extensively investigated by many authors. Bali R. et. .Have studied Bianchi Type-III string cosmological al. models with time dependent bulk viscosity. Adhav et al. have studied Bianchi Type-V string cosmological model with bulk viscous fluid and Kantowski-Sachs cosmological model in general relativity. Recently Verma investigated spatially homogeneous bulk viscous et.al fluid models with time dependent gravitational constant and cosmological term. Accelerating Bianchi Type-I universe with time varying G and

-term in general relativity have

been investigated by Pradhan et al.

.

, Dolgov

Press and Turner . Some of the recent discussions on the cosmological constant and consequence on cosmology with a time varying cosmological constant have been discuss by Dolgov

last two decades .Chen and Wu

Recently Mete et.al.[51-53] have studied various aspects of cosmological models in general theory of gravitation. Katore et.al.[54] have investigated Bianchi–I inflationary universe in presence of massless scalar field with flat potential in general relativity. In this paper a new anisotropic L.R.S. Bianchi type-II stiff fluid cosmological model with variable

-term has been

investigated by assuming supplementary conditions where and are metric potential. The outline of this paper is as follows: Basic equation of model are given in Section.2, the solutions of the field equations is given in

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Section.3.In section 4 and 5, we have discuss the model.We conclude our result in Section 6.

3. Solution of the field Equations

2. The Metric and Field Equations

Equations (8) – (10) are three independent equations in seven unknowns

In an orthonormal frame, the metric for Bianchi Type-II space-time in the LRS case is given by ,

(1)

are given by

where the Cartan bases ,

,

,

. For the complete determinacy of the system, we assume that (12) and the coefficient of shear velocity is proportional to the scale expansion, i.e.

,

=

(2)

where n is real number. With this extra conditions, equations (8) and (9) leads to

where and are the time dependent metric functions. Assuming x,y,z) as local coordinates, the differential one forms

are given by

(3) The Energy momentum tensor for viscous fluid distribution in the presence of bulk stress given by Landau and Lifshitz as Ti = (ε + ρ )v i v j + pg ij

Condition (13) leads to

where is proportionality constant. Equation (14) together with (12) and (15) leads to

j

− η (v ij; + v;ij + v j v i v i;l + v i v l v;lj − (ξ − 23 η )θ ( g ij + v i v j ) ,

(4)

where which can be rewritten as

is the density,

where are two coefficients viscosity, and satisfying the relations

is the flow vector which on integration gives (5)

(19)

We choose the coordinates to be commoving, so that

The Einstein’s field equations with time-dependent gravitational units c=1, G=1) read as

(in

7) For the metric (1) and energy-momentum tensor (4) in commoving system of co-ordinates, the field equation (7) yields as

where is a constant of integration . After a suitable transformation of coordinates, the metric (1) reduces to the for

The pressure and density for model (20) are given by

and

where, where suffix 4 at the symbols A and B denotes ordinary differentiation with respect to t and give by

is the shear expansion

and

Paper ID: SUB15254

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 For the specification of ,now we assume that the fluid obeys an equation of state of the form where

is constant.

Thus, given we can solve the cosmological parameters. In most of the investigation involving bulk viscosity is assume to be a simple power function of the energy density ; Maartens,

(Pavon,

where

and

; Zimdahl,

are constant. If

Equation (24) may

correspond to a relative fluid (Weinberg realistic models (Santos,

)

). However, more

) are based on m lying in the

(29) quation

(27)

and

(29)

,

we

observe

that

positive cosmological constant is a decreasing function of time and approaches a small value in the present epoch.

4. Some Physical Aspects of the Model The straight forward calculation leads to the following

. regime Using (24) in (21), we obtain

expression for the scalar of expansion the fluid for the metric (20)

for the shear

+

of

(30)

We consider the two model corresponding to m=0 and m=1 3.1. Model- I : Equation (24) reduces to = When constant. Hence in this case Equation (25), with the use of (22) and (23), leads to

The expansion factor

decreases as a function of T and

asymptotically approaches zero with and p also approaches zero as Particular Model If we set form

Eliminating

If

the geometric space time (20) reduces to the

between (26) and (22), we have The pressure and density for model (32) are given by

3.2. Model-II Equation (24) reduces to . Hence When in this case Equation (25), with the use of (22) and (23), leads to

Eliminating

4.1. Model- I : Equation (24) reduces to = When constant. Hence in this case Equation (33), with the use of (23) and (34), leads to

between Equation (28) and (22) , we have Eliminating

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between Equation (34) and (35) , we have

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802

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Eliminating

between Equation (44) and (45) , we have

5.2. Model-II

4.2. Model-II : , Equation (24) reduces to . Hence When in this case Equation (33), with the use of (23) and (34), leads to

, Equation (24) reduces to . Hence When in this case Equation (43), with the use of (23) and (44), leads to

Eliminating between equation (37)and (34)

From the equation (46) and (48) we observe that the positive cosmological constant is a decreasing function of time and approaches small value. Some Physical aspect of the model The scalar of expansion is given by

Some Physical aspect of the model The scalar of expansion is given by

and the shear

and the shear

of the model (42)

of the model (32)

6. Conclusion 5. Special Model If we set

and

equation (19) leads to

Using the transformation the metric (1) takes the form

We have presented Bianchi type-II non static cosmological model in presence of bulk stress given by Landau L.D. and Lifshitz E.M. It is found that physically relevant solutions are possible for the Bianchi-II space time with bulk stress in term. For solving the field the presence of time varying equations we have assumed that the fluid obey an equation and bulk viscous fluid is of state of the form assumed to be the simple power function of mass density

The pressure and density for the model (42) is given by

5.1. Model- I : When Equation (24) reduces to = constant. Hence in this case Equation (43), with the use of (23) and (44), leads to

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Generally the models are expanding, given by shearing and non-rotating. The cosmological constant in all models given in section 3.1 and 3.2 are decreasing function of time and all approaches a small positive value at finite large times (i.e., the present epoch). These results are supported by the results from the supernova observations recently obtain by the High-z Supernova team and Supernova Cosmological project [1-7].Thus with our approach, we obtain a physically relevant decay law for the cosmological constant unlike other investigators where adhoc laws were used to arrive at a mathematical expressions for the decaying energy. Thus our models are more general than those studied earlier. In all models, the

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International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 physical parameters pressure and density are found to be decreasing function of time. Also we find that all the physical quantities the expansion scalar and the shear scalar.

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Author Profile Dr. V. G. Mete, Ph.D. is working as a Associate Professor and Head in P.G. Deptt. of Mathematics, R.D.I. K. & K.D. College, Badnera- Amravati. He has experience more than 24 years in the field of Relativity, cosmology and theories of gravitation. He has published more than 40 research papers in international journals and 8 research scholars are working under his guidance. He has completed two Minor research projects , funded by U.G.C. He received Best teacher award . V.D.Elkar, M.Phil. working as a Assistant Professor & Head in Deptt. of Mathematics, J.D.Patil Sangaludkar Mahavidyalaya, Daryapur-Amravati. He has more than 16 years teaching experience in U.G. level.Presently he is working as research fellow (F.I.P.) under the guidance of Dr.V.G.Mete.

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