Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 STIN OSMOLOIL MODELS IN BINHI TYPE-III SPE-TIME WITH BULK VISOSITY ND Λ TEM PEETI SONI SPN SHIMLI search Scholar Department of Mathematics Pacific cademy of Higher Education & esearch University Udaipur (ajasthan)india ssoc. Prof. Department of Mathematics Pacific cademy of Higher Education & esearch University Udaipur (ajasthan)india Email: preeti00ni@gmail.com shrimalisapna@gmail.com BSTT The present study deals with exact solution of Einstein field equations in Bianchi type-iii string cosmological models with bulk viscosity and variable cosmological term Λ. Exact solutions of the field equations are obtained by assuming the conditions: the coefficient of the viscosity is proportional to the expansion scalar expansion scalar is proportional to shear scalar and Λ is proportional to the Hubble parameter. The physical implications of the models are also discussed in detail. Keywords: Bianchi-III space-time Bulk Viscosity Variable Λ expansion scalar shear Hubble parameter. INODUTION osmic string in recent years have drawn considerable attention among researchers for various aspects such that the study of the early universe. It is well known that in an earlier stage of the universe when the radiation in the form of photons as well as neutrinos decoupled from matter it behaved like a viscous fluid. Misner [ ] have studied the effect of viscosity on the evolution of cosmological models. Nightingale [] has investigated the role of viscosity in cosmology. Thus we should consider the presence of a material distribution other than a perfect fluid to have realistic cosmological models. Heller and Klimek [] emphasized that the introduction of bulk viscosity effectively removes the initial singularities with in a certain class of cosmological models. The effect of bulk viscosity on the cosmological evolution has been studied by a number of authors in the fram work of general theory of relativity viz. Pavon et al. [5] Maartens [6] Kalyani and Singh [7] Singh et al. [8]. 7 On other hand one outstanding problem in cosmology is the cosmological constant problem [9 0]. Since its introduction and its significance has been studied from time to time by various workers [-]. In modern cosmological theories the cosmological constant remains a focal point of interest. wide range of observations now suggests compellingly that the universe possesses a non-zero cosmological constant []. ecently Singh et al. [5] investigated Bianchi type- III cosmological models with gravitational constant and the cosmological constant Λ. In this chapter we have investigated Bianchi type-iii string cosmological models with bulk viscosity and cosmological term Λ(t). To obtain an explicit solution we assume that the coefficient of the viscosity is proportional to the expansion scalar and the expansion scalar is proportional to the shear scalar and for the cosmological term Λ we assume that it is proportional the Hubble parameter Λ H.
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 B B (a). DEIVITON OF FIELD EQUTION. THE VITTIONL FIELD: In our case we consider the Bianchi type- III metric in the form ds dt dx B e dy dz x () where B and are functions of time alone and a is a constant. The metric () has the following non-zero christoffel symbols. B x e B B B B B B B B B B B B (b) (c) (d) B B (e) From () one finds the following icci scalar for the Bianchi type-iiii universe B B B B B B () The non-vanishing components of Einstein j j tensor i i are given by B B B B (a) BBe x. (b) Where dots on B represents the ordinary differentiation with respect to t. The surviving components of the mixed icci tensor j i are as follows: B B B B (c) 8
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 B B B B (d) T (8d) B B (e) T 0. (8e) B. VISOUS FLUID: The energy momentum tensor for a cloud of string dust with a bulk viscous fluid of string is given by T = u u x x u ; l g u u j j j l j j i i i i i Where u i and x i satisfy the condition. i i i uu = x x = u x = 0 (6) i i i In equation (5) is the proper energy density for a cloud string with particle attached them is the coefficient of bulk viscosity is the string tension density of particles u i is the cloud four-velocity vector and x i is a unit space-like vector representing the direction of string. If the particle density of the configuration is denoted by p then we have ρ = ρ + p (5) (7) In a co-moving system of references such that ui = (0 0 0 ) we have T (8a). FIELD EQUTION: Einstein s field equations with 8 = and variable cosmological term Λ (t) in suitable units are ij gij Tij t g (9) The field equation (9) and (8) subsequently lead to the following system of equations: B B B B (0a) (0b) B B B B B B B B (0c) (0d) T (8b) B 0 B (0e) T (8c) 9 The scalar expansion and components of shear ij are given by
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 B B () scalar is proportional to the shear scalar which leads to the B = n (7) B B () x B e B B () Where n is a constant and the second condition is Λ = ah (8) Where H is Hubble parameter and a is a positive constant. From equation (0e) we have = B (9) where B B. () = 0 (5) Therefore B B B B B B (6) From equation (9) without any loss of generality we can take = B (0) In order to obtain the more general solution we assume that the coefficient of bulk viscosity is inversely proportional to the expansion therefore we have = K () Substituting equation (7) into equation () we have = n () By the use of the equation (7) (9) (0) and () the field equation (0a) reduces to. SOLUTIONS OF THE FIELD EQUTION: s the field equations (0) are a system of five equations with seven unknown parameters B and Λ. Thus two more relations are needed to solve the system completely. We assume that one condition is that the expansion a nn K n Equation () can be rewritten as () / n n n K n a n () On integrating equation () we obtain 0
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 Kn an t n n e (5) Where is constant of integration. gain integrating equation () we get n K n n n e an n K n n n a n t K n n n (6) Where is constant of integration Therefore from equation (7) (9) and (5) we obtain ds K n n dt n n e an nn K n n nn an t K n n n x K n n dx e dy n n K n n an n t K n n n e dz an n (9) For the model (8) the expressions for the energy density the string tension density the particle density p the expansion scalar the shear scalar and the cosmological term are respectively given by B n K n n n e an K n n n n e an n n K n n n n a n t K n n n n n K n n (7) n n a n t K n n n (8) K n n n n n a n a n t t n n n e e an n n K n n K n n n n e an an t n a n K n n n n n K n n a n a n t t n n n e e an (0) Hence the metric () reduces to form
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 n a 5na a K n n n n n e an an t n an t n n n n K n n n n n e an n n K n n K n n 8na a K n n n n an n t n n e K an n eferences [] Misner.W. (967) : Nature. 0. () [] Misner.W. (968) : J. strophys. 5. [] Nightingale J.P. (97) : J. strophys. K n n 85 05. n n [] Heller M. and Klimek M. (975) : strophys. Space Sci. 6. a n a n [5] Pavon D. Bafaluy J.and Jou D. (99) : n t t n n e e lass Quantum ravity. 8 57. an [6] Maartens. (995) : lass Quantum ravity. 55. () [7] Kalyani D. and Singh.P. (997) : In new direction in elativity and osmology. a n K n N n n e e an a n a n t t n n K n n n n () a n a n t t n n n e e an n n (5) n an K n n n t n n e K a n n In summery we have presented a new class of Bianchi type-iiii string cosmological () models with bulk viscosity and cosmological term Λ(t). To obtain an explicit solution we adopt the conditions and Λ H. Thus the cosmological model for a string n n n K n n p cosmology with bulk viscosity and n n cosmological term is obtained. The model describes a shearing non-rotating continuously a n a n n t t expanding universe with a big-bang start. The n n e e physical and geometrical aspects of the model an are also discussed. In the absences of cosmological term Λ the model reduces to the string model with bulk viscosity investigated by Bali and Pradhan (007).
Jan. 05. Vol. 6. No. 0 0-05 ES & F. ll rights reserved ISSN05-869 [8] Singh T. Beesham. and Mbokazi W.S. (998) : en. ela. ravit. 0 57. [9] Ng Y. J. (99) : Int. J. Mod. Phys. D 5. [0] Weinberg S. (989) : ev. Mod. Phys. 6. [] Zeldovich Ya. B. (968) :Sov. Phys. Usp. 8. [] Linde. D. (97) : ZETP. Lett. 9 8. [] Krauss L. M. and Turner M. S. (995) : en. el. rav. 7 7. [].. eiss et al. (00) : stron. J. 607 665. [5] Singh J. P. Tiwari. K.and Pratibha Shukla (007) : hin. Phys. Lett. ().