ISSN: 2455-7064

Contents lists available at http://www.albertscience.com

ASIO Journal of Chemistry, Physics, Mathematics & Applied Sciences (ASIO-JCPMAS)

Volume 1, Issue 2, 2016, 14-16

dids no.: 03.2016-26169319, dids Link: http://dids.info/didslink/06.2016-55614657/

Pag

e1

4

BIANCHI TYPE-III BULK VISCOUS STRING COSMOLOGICAL MODEL

Sapna Shrimali1†, Teena Joshi2

1Department of Mathematics, Pacific Academy of Higher Education & Research University, Udaipur(Rajasthan)-313024,India.

2Department of Mathematics, Pacific Academy of Higher Education & Research University, Udaipur(Rajasthan)-313024,India.

ARTICLE INFO ABSTRACT

Research Article History

Received: 06 June, 2016 Accepted: 23 June, 2016

Corresponding Author:

†Sapna Shrimali

Department of Mathematics, Pacific Academy of Higher Education & Research University, Udaipur(Rajasthan)-313024,India.

Mail ID: shrimalisapna@gmail.com

Bianchi type-III bulk viscous stringcosmological model was

investigated. To obtain the determinate model it was assumed that

Bulk viscosity ξ is inversely proportional to scalar expansion θ. Some

physical and geometrical features of the model are also discussed.

Keywords: Bianchi type-III, cosmology, bulk viscosity, space-times.

© www.albertscience.com, All Right Reserved.

INTRODUCTION

In the recent years the study of cosmology has been

considerable. Cosmology deals with large scale

structure of the universe extending to distance of

billions of light years. Due to phase transition

symmetry of the universe was broken. This results

into number of stable topological defects; for example,

there are issues associated with domain walls and

monopoles. The Bianchi models are cosmological

model that have spatially homogeneous sections,

invariant under the action of three dimensional Lie

group. The classification of the three dimensional Lie

algebra is called Bianchi classification. This is labeled

by number I-IX. Bianchi [1] has shown nine distinct set

of structure constant of groups. The cosmic string

refers to one dimensional topological formed in

symmetric breaking phase transition in early universe.

The idea first proposed by Tom Kibble [2] in 1976.

The general relativistic treatment of cosmic string has

been given by Letelier [3, 4], Vilenkin [5] and Satchel

[6]. Singh [7], Adhav [8], Pradhan [9, 10] & Soni [11]

investigated Bianchi type III cosmological model with

bulk viscosity. Matter was highly ionized so there is

possibility of presence of magnetic field and It was

discussed by Wang [12,13]. Tyagi and Singh [14]

discussed LRS Bianchi type III universe and barotropic

perfect fluid. Recently Dubey and Saini [15] discussed

Bianchi type III Cosmological model filled with an

electro magnetized with Nambu strings in general

relativity. Deo, Punwatkar and Patil [16] investigated

LRS Bianchi type –III cosmological model in General

Theory of Relativity with the matter magnetized wet

dark energy. In this paper we have investigated

Bianchi type-III bulk viscous string cosmological

model. To obtain the determinate model it was

assumed that Bulk viscosity ξ is inversely proportional

to scalar expansion θ.

FIELD EQUATIONS

We consider Bianchi type III space time in the general form

Bianchi III space-time given by metric

2 2 2 2 2 2 2 2( )

xds dt A dx B e dy dz

(1)

W here A and B are function of t and is constant.

Sapna Shrimali / ASIO Journal of Chemistry, Physics, Mathematics & Applied Sciences (ASIO-JCPMAS), 2016, 1(2): 14-16

doi no.: 05.2016-11672519, dids Link: http://doi-ds.org/doilink/06.2016-73817465/

Pag

e1

5

The energy momentum tensor for a cloud o f string dust w ith a bulk

viscous fluid of string given by

; ( )l v

u u x x u l g u uT

(2)

W here

1, 0u u x x u x

(3)

is proper energy density for a cloud string w ith particle attached

to them , is string tensor density, is velocity of particle, is

unit space like vector representing the direction of string

u x

. If particle

density of configration is denoted by th en w e havep

p (4)

The Einstein's field equation without time dependent cosmological

constant

18

2G R R GT

(5)

W here R is Ricci tensor and R=g is the R icci scalar.R

For the line element (1) and the field equation (5) can be written as

2

22 8

B BG

B B (6)

8A B AB

GA B AB

(7)

2

28 ( )

A B ABG

A B AB A

(8)

2 2

2 22 8

AB BG

AB B A

(9)

0A B

A B (10)

Where dot represents ordinary diferentiation with respect to .t

The particle density is given byp

2

28 2

p

B AB

ABB (11)

The scalar expansion and shear is given by

2A B

A B (12)

2 2

2

2 2

12

3

A B AB

ABA B

(13)

From equation (10), we have

A= B

W here is constant of integration.

(14)

From equation (14), w ithout loss of grav ity we have to take

=1 so that,

A B (15)

In order to obtain the more general solution, we assume

k (16)

aGH

(17)

W here a and k are the positive constant and H is Hubble parameter,

define by

3H (18)

Substituing equation (15) in (12), we have

3B

B (19)

By using equation (16), (17), (18) and(19) in equation(6), we got

2

22 8 8

B BG Gk

B B (20)

2

22

B Bl

B B (21)

W here

8l Gk (22)

M ultipling in equation (21), we haveB

B

1

2

B B Bl

BB B

(23)

Integrating equation (23), we got

1

2

1

l

B B m

(24)

Again integrating , we obtained

2

3

1 2

3

2

llB m t m

(25)

1 2Where m and m are constant of integration.

T hus

2

3

1 2

3

2

llA m t m

(26)

Therefore equation (1) reduced to

Sapna Shrimali / ASIO Journal of Chemistry, Physics, Mathematics & Applied Sciences (ASIO-JCPMAS), 2016, 1(2): 14-16

doi no.: 05.2016-11672519, dids Link: http://doi-ds.org/doilink/06.2016-73817465/

Pag

e1

6

4

32 2 2 2 2 2

1 2

3( )

2

lxl

ds dt m t m dx e dy dz

(27)

1

1 2

2 1

3

Bm

B l m t m

(28)

For the model of equation (27), the othe r physical and geometrical

parameter.The expressions for the energy density , the string

tension density , the cofficient of bulk viscosity , the scalar

expan

2sion and the shear scalar are

42

32

1 1 2

1 2

1 2 1 33

8 3 2

llm m t m

G l m t m

(29)

42

3

1 2

3

8 2

llm t m

G

(30)

1

1

1 2

2 1

3 3

km

l m t m

(31)

1

1 2

2 13

3m

l m t m

(32)

20

0

CONCLUSION

It is seen that w hen t , but scalar expasion tends

to finite and tends to finite w hen t 0 d ue to presence of bulk

viscosity. H ence m odel represets the sh earing and non rotating

expanding unive

rse w ith the big bang start. S ince , the

m odel approches isotropy for large v

li

alue of t

m 0

.

t

REFERENCES

1. L. Bianchi, Memori di Mathematical Fisca della

Societa Italiana della Science, 1898, 11, 276.

2. T. W. B. Kibble, Journal of Physics A: Mathematics

and General, 1976, 9, 1387.

3. P. S. Letelier, Phys. Rev., 1979, D 20, 1294.

4. P. S. Letelier, Phys. Rev., 1983 D 28(10), 2414.

5. A. Velenkin, Phys. Rev., 1981, D 23, 853.

6. J. Stachel. Phys. Rev., 1980, D 21, 2171.

7. V. Singh, Int. J. Math. Sci., 2012, Vol 2,No 2, 547-

556.

8. K. S. Adhav, M.R. Ugale, C.B. Kale, M.P. Bhande,

Bulg. J. Phys. 2007, 34, 260-272.

9. Pradhan, V. K. Yadav and L Chakrabarty, Int. J.

Mod. Phys., 2001, D 10, 339.

10. Pradhan, V. K. Yadav and N. N. Saste, Int. J. Mod.

Phys., 2002, D 11, 893.

11. P. Soni, S. Shrimali, ARPN J. Sci, 2014, Vol 4, No 4,

249-253.

12. X.X. Wang, Chin. Phys Lette, 2003,20,1205.

13. X.X. Wang, Chin. Phys Lette, 2003, 20, 615.

14. Tyagi, G.P. Singh, Prespacetime journal, 2015, Vol

6, Issue 2, PP 143-150.

15. Dubey R. K., Saini A., Yadav N., IJAUR, 2016, vol 3,

issue 2, 1-6.

16. Deo S., Puwatkar G., Patil U., IJMA, 2016, 7(3),

113-118.