1. INTRODUCTION:

Tikekar and Patel [12] and Chackraborty and Chackraborty [6] have presented the exact solutions of Bioanchi type-III and spherically symmetric cosmology respectively for a cloud string. It is well known that in an early stage of universe when neutrino decoupling occurs, the matter behaves like a viscous fluid. The cosmological models of a fluid with viscosity play a significant role in study the evolution of the universe. Recently, string cosmological models of Bianchi type I, II, III with bulk viscosity have been discussed by several authors [3, 13-17]. The magnetic field has an important role at the cosmological scale and is present in galactic and intergalactic spaces. The importance of the magnetic field for various astrophysical phenomena has been studied in many papers. Melvin [9] has pointed out that during the evolution of the universe, the matter was in a highly ionized state and its smoothly coupled with the field, subsequently forming neutral matter as a result of universe expansion. Therefore the possibility of the presence of magnetic field in the cloud string universe is not unrealistic and has been investigated by many authors [7, 10, 15]. Some other workers in this field are Bahera [2], Bali and Dave [4]. Ball et al. [5], Kibble [8], Takabayski [11], Yadav at. al [18], Zimadahl [22], Yadav and Kumar [19], Zeldovich et. al. [20, 21]. Tyagi and Sharma [23], Sharma et al. [24] and Roy et. al. [25]. In this paper we have studied Bianchi type-III string cosmological model in the presence of bulk viscosity with and without magnetic field. To obtain an exact solution (we have used different craditionsive and equation of state and an assumption that the scalar of expansion is proportional to the shear scalar , which leads to the relation between metric potential . The physical and geometric aspects of the models absence of magnetic field are also discussed in presence and absence of magnetic field.

The Bianchi type-III space-time metric we considered here is [16]. (2.1) 222222x222dsdtdxedydz where , , and are the functions of time t only. The energy-momentum tensor for a cloud of string with bulk viscosity and magnetic field [15]. (2.2) ijijijijjiijJuu(uug)E where p, is the rest energy density of the cloud of strings with particles attached to them, p is the rest energy density of particle is the tension density of the cloud of string, i;iu, is the scalar of expansion, and  is the coefficient of bulk viscosity. According to Letelier [8a] the energy density for the coupled system and p is is restricted to be positive, while the tension density may be positive or negative. The vector ui describes the cloud four-velocity and i represents a direction of anisotropy, i.e., the direction of string. They satisfy the standard relation [8(a)].

Eij is the energy-momentum tensor for the magnetic field.

(2.4)

hkhkijihjkijhk

11EgFFgFF44 Where Fij is the electromagnetic field tensor, which satisfies the Maxwell equations.

(2.5) ij[ij;h];jF0,Fg0

Einstein‘s equation we consider here is

(2.6) ijijij

1RRgT2

Where we have choose the units such that c = 1 and 8G = 1. In the co-moving coordinates iii00iuandu, and the incident magnetic field taken along the z-axis, with the help of Maxwell equation [2.5], the only non-vanishing component of Fij is [15]. (2.7) F12 = constant = H The Einstein equation (2.6) for the metric (2.1) can be written as following system of equations [15, 16]

(2.8)

2 222x

H 8e

 (2.9)

2 222x

H 8e

 (2.10)

2

2222x1H 8e

 (2.11)

2

2222x1H 8e

 (2.12) 0

Where the dot denotes the differentiation with respect to time t. From Eq. (2.12), we have

(2.13)  (2.14)

nAB where A, B and n are +ve constants The expression for scalar of expansion and shear scalar are

(2.15)

i;iu

(2.16)

2ijij

1

2

222

2221

3



We see that the five independent equation (2.9). (2.12) and (2.14) connecting six unknown variable (,,,,,). Thus, one more relation connecting these variables is needed to solve these equations. In order to obtain explicit solutions, one additional relation is needed and we adopt an assumption that the shear scalar of expansion is proportional to the shear scalar of expansion , which leads to (3.1) NCDr where a, b and is a constant. Now we consider B = C and D = 1 = n, the equation (2.14) and (3.1) reduces to

(3.2) A (3.3) N

From equation (2.9), (2.10), (2.11), with the help of eq. (3.2) eliminating , and , we obtain

(3.4) AA(A1)(A1)(A1)

2A

2222x1H(2K1) 8e



(3.5)

2(2N1)2N2A(N1)(N2)(A1) A(N1)A(N1)



2(4N1)22xH(2A1) 8A(N1)e



To solve Eq. (3.5), we denote , then

d d

and the eq. (3.5) can be reduced to the following form

(3.6)

2(N1)2

d(A1) dA(N1)



(4N1)22x(2A1)M A(NDe



where

(3.7)

N[2A(N1)(N2)]

a(N1)

 (3.8) 2HM8

Equation (3.6) can be written as

(3.9)

222(2N1)2

d2(A1)()dA(N1)



2(4N1)22x

2(2a1)M A(N1)e



Therefore, we get

(3.10) 222(A1)dtA(N1)(N1)1





12(4N2)2222x(2A1)MKdA(N1)(N1)1e



where K is the constant of integration. For this solution, the geometry of the universe is described by the metric.

(3.11) 

22(N1)222(A1)dtA(N1)(N1)1  22N22N2x222dxedydz Under suitable transformation of coordinates, Eq. (3.11) reduces to

(3.12) 

22(N1)222(A1)dtTA(N1)(N1)1





1(4N2)22222x(2A1)MTKTdTA(N1)(N1)1e



22N22N2x222TdxTedyTdz The expressions for the energy density , the string tension density , the particle density p the coefficient of bulk viscosity , the scalar of expression and the shear scalar 2 for (3.12) are given by

(3.13) 

2N222

A(2N1)TA(N1)(N1)1



 

24N222x

2AN3A(N1)3A3MTA(N1)(N1)1e

 2(1)N(N2)KT

(3.14) A



(3.15) p11A



(3.16) 222

(A1) A(N1)(NH)1





2 222x

(NN1)(12A)

A(N1)(N1)1e



(22)N(N1)AN(N1)(N2)KTA(N1)





2N

222(2N1).T A(N1)(NH)1

 

222x(2A1)M A(N1)(N1)1e



(3.18) 

222N222(N1)(A1).T3A(N1)(NH)1

222x

(2A1)M

A(N1)(N1)1e



In this paper, we have studied Bianchi type-III string cosmological model in the presence of bulk viscosity and magnetic field. To obtain an exact solution, an equation of state = AE + B and an assumption that the scalar of expansion is proportional to the shear scalar , which leads to the relation between metric potential . Then the cosmological model for a cosmic string with bulk viscosity and magnetic field is obtained. The physical and geometric aspects of the model in the presence and absence of magnetic field are also discussed. Out model describes a shearing non-rotating continuously expanding universe with a big-bang start. In the absence of magnetic field it reduces to the string model with bulk viscosity. From Equations (3.13) and (3.15) it is observed that the energy condition > 0 and > 0 are fulfilled, provided. K0,N1andAN(N2)/(N1) Or K0,N1andA1/(N(2N3) When K > 0, N > 1 and A > N(N+2)/(N–1), the string tension density 0; however, 0when K > 0, N > 1 and A < –1/N(2N+3) The above expression (3.13) – (3.16) indicate that the magnetic field is related with p,,and. Here a term of M is involved in the expression for 2p,,,,andrespectively, and it represents the effect of magnetic field on the model. It is seen that in the case N > 1, whether A > N(N+2)/(N–1) or A < –1/(N(2N+3), we have + 1 > 0. Hence equation (3.17) shows that the scalar of expansion tends to infinitely large with J0, but when , therefore the model describes a shearing non rotating expanding universe with the big-bang start. We can see from the above discussion that the bulk viscosity plays a significant role in the evolution of universe [1, 8(b)]. Furthermore, since Jlim0, the model does not approach isotropy for large values of T. In the special case A = 1, the model represents a geometric string model [17]. In the absence of magnetic field M = 0, the metric (3.12) reduces to the string model with bulk viscosity i.e.,

(4.1)



21

22(N1)KJ2222(A1)dsTdTA(N1)(N1)1





22N22N2x222TdxTedyTdz

1. Amirhaschi, H. et. al. (2012) : Rom. Journ. Phys., Vol. 57, No. 3-4, 748-760. 2. Bahera, D., Tripathi, S.K. and Routry, J. R. (2010) : Int. J. Theor. Phys., 49, 2569. 3. Bali, R. and Dave, S. (2003): Astrophys. Space. Sci., 288, 503. 4. Bali, R and Dave, S. (2002) : Astrophys. Space. Sci., 282, 461 5. Bali, R., Tinker, S. and Singh, P. (2010) : Int. J. Theor. Phys., 49, 1431. 6. Chakraborty, S. and Chakraborty, A.K. (1992): Math. Phys., 33, 2336. 7. Chakraborty, N.C. and Chakraborty, S. (2001) : Int. J. Mod. Phys. D. 10, 723. 8. Kibble, T.W.B. (1976) : J. Phys., A. 9, 1387. 8.(a) Leteller, P.S. (1983) : Phys. Rev., D. 28, 2414. 8.(b) Marteens. R. (1995) : Class Quantum Gav., 12, 1455. 9. Melvin, M.A. (1975) : Ann. New York, Acad. Sci., 262, 253. 11. Takabaysi, T. (1976): Quantum Mechanics Determinism, Casuality and Particles (Dordrecht : Reidel) P. 179. 12. Tiekekar, R. and Patel, L.K. (1992): Gen. Relativ. Gravit., 24, 394. 13. Wang, X.X. (2003): Chin. Phys., Lett., 20, 1674. 14. Wang, X.X. (2004) : Comm. Theor. Phys., 41, 726 15. Wang. X.X. (2004) : Astrophys. Space. Sci., 293, 433. 16. Wang, X.X. (2005) : Chin. Phys. Let., 22, 29. 17. Wang, X.X. (2006) Chin. Phys. Lett., 23, 1702. 18. Yadav, A.K. Pradhan, A. and Singh, A.K. (2008) : Astrophys. Space. Sci., 3, 4, 84 19. Yadav, R.B.S. and Kumar, M. (2020), Int. J. Cont. Res. Vol. 7, p. 59-64. 20. Zlldovich, Ya. B. (1980) : Mon. Not. R.Astron. Soc., 192, 663. 21. Zimdahl et. al. (1975), Sov. Phys. JETP, 40, 1. 22. Zimdahi, W. (1996) : Phys. Rev. D., 5483, 53. 23. Tyagi, A and Sharma, K. (2010) : Ind. Acad. Of Math, 32,2. 24. Sharma, A et. al. (2016); Prespect-Time Jour, 7, 615. 25. Roy, V, Vomroy, Neelima, D. (2013) : Astronomy and Astrophys. Jour., 3, 342.

Research Scholar, Department of Mathematics, Magadh University, Bodh Gaya, Bihar