Bianchi Type IX Magnetized Bulk Viscous String Cosmological Model in General Relativity

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1 4 Theoretical Physics, Vol., No., March 07 htts://dx.doi.org/0.606/t ianchi Tye IX Magnetized ulk Viscous String Cosmological Model in General Relativity V. G. Mete, V. D. Elkar, V.S. Deshmukh 3 Deartment of Mathematics, R.D.I.K. & K.D. College, adnera Amravati (M.S.), India Deartment of Mathematics, J.D.Patil Sangludkar Mahavidyalaya, Daryaur, Dist. Amravati, India 3 Deartment of Mathematics, P.R.M.I.T.&R., adnera Amravati (M.S.), India vmete56@gmail.com Abstract. In this aer we resent a bulk viscous magnetized ianchi tye IX string cosmological model in general relativity. In order to find determinate solution of the investigated model, we assume the conditions :(i) Scalar exansion is roortional to shear scalar and (ii) ξθ = constant, where ξ is bulk viscosity and θ is scalar exansion. Some hysical and geometrical roerties of the investigated model are also discussed. Keywords: ianchi tye IX sace time, bulk viscous, magnetized cosmic string. Introduction It is well known that Einstein s general theory of relativity has been successful in finding different models for the universe. Einstein s theory of general relativity is considered as the only successful theory of gravitation in terms of geometry which has the most beautiful structure of the theoretical hysics. In recent years, ianchi tye-ix cosmological models are very oular for relativistic studies. These models are also used to examine the role of certain anisotroic sources during the formation of large scale structure as we see the universe today. Many researchers have taken keen interest in studying ianchi Tye IX universes, because familiar solutions like Robertson Walker universe with ositive curvature, the de-sitter universe, the Taub-Nat solutions etc. are articular case of ianchi Tye IX sace time. Number of authors Chakraborty [], Raj ali and Dave[], Raj ali and Yadav [3] have studied ianchi tye-ix string as well as viscous fluid models in general relativity. Pradhan [4] have studied some homogeneous ianchi tye-ix viscous fluid cosmological model with varying cosmological constant. On the other hand, ulk viscosity lays a significant role in the early evolution of the universe and contributes to the accelerated exansion hase of the universe oularly known as the inflationary hase. The effect of viscosity on the evolution of cosmological models and the role of viscosity in avoiding the initial big bang singularity have been studied by several authors [5-6]. In late nineties, two teams studying distant tye Ia-suernovae (SNe-Ia) indeendently resented evidence of exansion [7-8] and confirmed later by cross checks from the cosmic microwave background radiation (CMR)[9] and large scale structure (LSS) [0-3]. In addition, the magnetic field has imortant role at the cosmological scale and is resent in galactic and intergalactic sace. It lays a vital role in the descrition of the energy distribution in the universe as it contains highly ionized matter. Strong magnetic fields can be created due to adiabatic comression in cluster of galaxies. Large scale magnetic fields give rise to anisotroies in the universe. Primordial magnetic fields of cosmological origin have been discussed by Asseo et.al [4] and Madsen [5]. Melvin [6] suggested, in the cosmological solution for dust and electromagnetic fields, that during the evolution of the universe, the matter was in highly ionized state and smoothly couled with electromagnetic and consequently formed a neutral matter as a result of exansion of the universe. Hence, in a string dust universe the resence of magnetic fields is not unrealistic. The magnetic field has the significant role in the dynamics of the universe deending on the direction of the field lines [7-8]. Tyagi et.al.[9] has studied homogeneous anisotroic ianchi tye-ix cosmological models for erfect fluid distribution with electromagnetic field. Recently Tyagi and Singh.[0], Ghate and Sontakke [-] have investigated ianchi tye-ix cosmological models in different context. Mete and Elkar [3] have studied ianchi Coyright 07 Isaac Scientific Publishing

2 Theoretical Physics, Vol., No., March 07 5 tye-ix cosmological models for viscous fluid with electromagnetic field and time deendent Λ -term. Recently ianchi tye-v magnetized cosmological model with wet dark fluid in general relativity has been studied by Mete et.al [4]. Motivated by the above discussions, in this aer, we have focused uon the roblem of establishing a formalism for studying a new integrability of magnetized ianchi tye IX bulk viscous string cosmological models in general relativity. The behaviors of the models in the resence magnetic fields are also discussed. Metric and Field Equations We consider ianchi tye IX sace time in the form ds = dt + A dx + dy + ( sin y + A cos y) dz A cosydxdz () where A and are scale factors and are functions of cosmic time t. The energy momentum tensor for a cloud string with a magnetic field in a co-moving coordinate system is T = ρuu λxx ξθ( uu + g ) + E () ij i j i j i j ij ij where the vector u i describes the cloud 4-velocity and x i reresents a direction of anisotroy, i.e. the string satisfies the relations uu i = xx i =, ux i = 0 (3) i i i Here ρ is the rest energy of the cloud strings with massive articles attached to them. It is given by ρ = ρ + λ, where ρ being the rest energy density of articles attached to the strings and λ is the density of tension that characterizes the strings. The energy momentum for the magnetic field is j lm j lm E = FF g + gf F i il jm i lm (4) 4π 4 j where E is the electromagnetic field tensor satisfying Maxwell's equations i ij F + F + F = 0 or ij; k jk; l ki; j ( F g) = 0 (5) j x In comoving coordinates, the incident magnetic field is taken along x-axis. With the hel of Maxwell's equations (5), the only nonvanishing comonent of F is F = H siny = constant (6) 3 In order to determine the system comletely, we consider Takabayasi's equation of state [5], ρ = l λ (7) where l is constant. The Einstein's field equations (in gravitational unit c =, G = ) read as R gr= 8πT (8) ij ij ij For the metric(), the above field equations with the hel of equations ()- (7) takes the form A H + + = λ + ξθ (9) π A A A H = ξθ + (0) 4 4 A A 4 8π A A H + + = ρ () 4 4 A 4 8π 3 Solution in Presence of ulk Viscosity and Magnetic Field The field equations (9) to () are a system of three highly non-linear differential equations with five ij Coyright 07 Isaac Scientific Publishing

3 6 Theoretical Physics, Vol., No., March 07 unknown arameters A,, ξλρ,,. The system is thus initially undetermined. To obtain a deterministic solution, the following hysical conditions are used. Firstly, we assume that the coefficient of bulk viscosity ξ is inversely roortional to the exansion. This condition leads to ξθ = k (constant) () The motive behind assuming this condition is exlained in the literature [6-8]. Secondly, we consider the exansion scalar θ is roortional to the shear σ, i.e. θ σ [9]. This condition leads to A = n (3) where n 0 a constant. From equations (9)-(), with the hel of () and (3), eliminating λ, we have 4 β n 3 δ k + α = + γ + + (4) 44 3 where l (n+ ) l 3l ( ll ) α =, β =, γ =, δ =. l l 4 4 l l df ( ) To solve equation (4), we denote = f ( ) then = f 4 44 ( ) and equation (4) can be d reduced to the linear differential equation in the form d ( ) f ( ) + α f = Q (5) d β n 3 δ k where Q = + γ Integrating equation (5), we obtain ( n ) β k C f ( ) = + γ + δ + + (6) α α n + α α α + Thus we get, ( ) n β k C dt = + γ + δ + + α n + α α α + where C is constant of integration. Hence line element () becomes ( n ) β γ δ k C ds = α d α n + α α α + n n n + dx + dy + ( sin y + cos y) dz cosydxdz Using the suitable transformation of coordinates equation (8) reduces to ( n) β γ δ T T k T C ds = dt α n + α α α + α T + T n dx + T dy + ( T sin y + T n cos y) dz T n cosydxdz where =T can be determined by equation (9). 4 Some Physical Proerties of the Model For the model (9), the hysical arameters: the energy density( ρ), the string tension density( λ), the article energy density( ρ ) scalar ( θ ) and shear scalar σ α d (7) (8) (9), the coefficient of bulk viscosity ( ξ), the satial volume (V), exansion ( ), the Hubble arameter ( H ) and the deceleration arameter ( ) q are Coyright 07 Isaac Scientific Publishing

4 Theoretical Physics, Vol., No., March 07 7 obtained as follows: ρ ( n + ) β γ(n + ) (n + ) δ T LT C(n ) T T M(0) α n + α 4 α ( n ) α = H k(n + ) where L =, M = 8π (α + ) λ = ρ l (0) θ ξ ρ ρ λ = = ρ l k k β γ δ k T T T CT θ n + α n + α α (α + ) ( n ) 4 α = = () () n + V = T siny (3) A β γ δ k ( n ) T T T CT A α n + α α ( α + ) 4 4 ( n ) 4 α = + = A β γ δ α 4 4 ( n ) 4 k σ = = ( n ) T + T + T + CT + 3 A 3 α n + α α (α + ) (5) Hence σ ( n ) lim = T θ 3( n + ) = constant (6) Here we observe that when T 0, σ and σ = constant, as T. θ The Hubble arameter: θ ( n + ) β γ ( n ) δ 4 α k H = = T + T + T + CT α n + α α (α + ) and the deceleration arameter ( q ) 5 Conclusion β γ( n ) δ T + T + C( α ) T 3 α n + α α q = + ( n + ) β γ δ k T + T + T + + CT α n + α α (α + ) 3 n 5 α 3 ( n ) 4 α In this aer, we have studied magnetized ianchi tye-ix bulk viscous string cosmological model in general relativity. To get the deterministic model we have assumed two conditions. Firstly coefficient of bulk viscosity is inversely roortional to exansion scalar. i.e. ξθ = constant and secondly exansion scalar is roortional to the shear scalar, i.e. θ σ. We observe that the satial volume V 0 when T 0 and V when T, thus the model is exanding. For the anisotroy when T =, σ 0, this imlies that the model does not aroach isotroy at late time when < n <. Also, for θ σ n = the shear scalar σ is zero and as θ 0 T. Thus the model becomes isotroic at late time when < n <. The shear scalar does not tend to zero faster than exansion at late time. 3 (4) (7) (8) Coyright 07 Isaac Scientific Publishing

5 8 Theoretical Physics, Vol., No., March 07 Acknowledgement. The author thanks an unknown reviewer for helful advices which ermitted to imrove this work. References. S. Chakraborty, A study on ianchi IX cosmological model, Astrohysics and Sace Science,80 (), ,99. Raj ali and S. Dave, ianchi Tye-IX string cosmological model in general relativity, Pramana J. Phy., 56 (4),53 58, Raj ali and M.K.Yadav, ianchi Tye-IX viscous fluid cosmological model in general Relativity, Pramana J. Phy., 64 (), 87 96, A.Pradhan, S.K. Srivastav and M.K.Yadav, Some homogeneous ianchi tye- IX viscous fluid cosmological models with a varying Λ, Astrohysics and Sace Science;98,49-43, Shri Ram and C.P.Singh,, Anisotroic ianchi Tye -II Cosmological Models in Self Creation Cosmology, Astrohy. Sa. Sci.,57,87, A. Pradhan and H.R. Pandey, ulk Viscous Cosmological Models in arber s Second Self Creation Theory, Int. J. Mod. Phy., A.G. Riess et al., Tye Ia Suernova Discoveries at z? from the Hubble Sace Telescoe: Evidence for the Past Deceleration and Constraints on Dark Energy Evolution, The Astrohys. J., 607(), , D.N. Sergel et al., Tye Ia Suernova Discoveries at z? from the Hubble Sace Telescoe: Evidence for the Past Deceleration and Constraints on Dark Energy Evolution, Astrohys. J. Sul. Ser., 70(), , D.N. Sergel et al., First-YearWilkinson Microwave Anisotroy Probe (WMAP)Observations: Determinations of cosmological arameters, Astrohys. J. Sul. Ser., 48(),75-8, E.Hawkins et al., The df galaxy redshift survey: correlation functions, eculiar velocities and the matter density of the universe, Mon. Notices of the Royal Astrono.Soc., 346(), 78 96,003. K. Abazajian et. al., The First Data Release of the Sloan Digital Sky Survey, Astronom. J., 8, 50-5, 004. K. Abazajian et.al., The Second Data Release of the Sloan Digital Sky Survey, Phy. Rev. D, 69, 03-, M.Tegmark et.al., Cosmological arameters from SDSS and WMAP, Phys. Rev. D, 69(0),03-50, E. Asseoet.al. Extra galactic magnetic fields. Phys. Re. Sol, H. Phys. Re. 48, , M.S.Madsen, Magnetic field in cosmology, Not. R. Astron. 37, 09-7, M. A. Melvin, Homogeneous axial cosmologies with electromagnetic field and dust,ann. New York Acad. Sci. 6, 53-74, K.L. Mahanta and A.K. iswal, Rom. Journ. Phys., 58 (3-4), 39-46,03 8. E.J. King and P Coles, Dynamics of a magnetized ianchi-i Universe with Vacuum Energy, Classical and Quantum Gravity, 4, 06-07, A.Tyagi and D. Chhajed, Homogeneous anisotroic ianchi tye -IX cosmological model for erfect fluid distribution with electromagnetic field, American J. Math. and Statis., (3),9-, 0 0. A. Tyagi, G.P. Singh, Magnetized bulk viscous ianchi tye -IX cosmological model with varying Λ, Ultra Scientist,(), ,00.. H.R. Ghate and A.S. Sontakke, ianchi tye IX cosmological models with a anisotroic dark energy, International Journal of Scientific and Engineering Research, 4(6), , 03. H.R. Ghate and A.S. Sontakke, inary mixture of Anisotroic dark energy and Perfectfluid in ianchi Tye- IX sace-time, JPMS, 3(),-3, V.G.Mete and V.D.Elkar, ianchi tye-ix cosmological model for viscous fluid with electromagnetic field and time deendent Λ term, Presacetime Journal 6 (), , V.G.Mete, K.R.Mule and V.D.Elkar, ianchi tye-v magnetized cosmological model with wet dark fluid in general relativity, Int. J. of current research, 8(), , T. Takabayasi Quantum Mechanics, Determination, Causality and Particles, Reidal: Dordrecht, the Netherlands, 79, Rajali and A. Pradhan, ianchi tye III string cosmological models with time deendent bulk viscosity, Chin. Phys. Lett.4, , 007 Coyright 07 Isaac Scientific Publishing

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