INFLATIONARY SOLUTIONS OF BIANCHI TYPE I STRING MODELS WITH BULK VISCOUS FLUID IN BIMETRIC THEORY OF GRAVITATION

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1 International Journal of Researches In iosciences, griculture & Technology May 06 ISSN No. Online X INFLTIONRY SOLUTIONS OF INCHI TYPE I STRING MODELS WITH ULK VISCOUS FLUID IN IMETRIC THEORY OF GRVITTION Viay Lepse Department of Mathematics, Science College Pauni-4490 bstract In this paper, we presented the solutions of inflationary solutions of bianchi type i string models with bulk viscous fluid in bimetric theory of gravitation. The bulk viscosity leads inflationary like solutions in the mode with cosmetic string and dominated the nature of Hubbell parameter, the expansion and shar in the evaluation of the universe Keywords: Cosmology, Viscosity Introduction ulk viscous fluid has maor contribution in the evaluation of the universe at early stage as well as at later stage. On large scale distribution of galaxies in the universe shows that the matter distribution is satisfactorily described by the perfect fluid. However, when neutrino decoupling occurs, then the matter behaves like viscous fluid in early stage of the universe. The researchers like Misner 967, 968 suggested that strong dissipation due to the neutrino viscosity may considerably reduce the anisotropy of the black-body radiation. Weinberg 97 has described that the viscosity mechanism in cosmology that can explain the anomalously high entropy per baryon in the present universe. elinskii and Khalatnikow 975 have suggested the nature of cosmological solutions for the homogeneous ianchi type I models by taking into account the dissipative process due to viscosity. Chimento et al. 997 have investigated the cosmological models with bulk viscosity and studied their nature. lso there are many researchers like Santos et al. 985, aneree et al. 985, Huang 990, Desikan 997, Krori et al. 000,ali et al. 00, Singh et al. 008, Xu Dou et al. 0, Tiwari et al. 0,, Verma 0, Gilberto et al. 0, Sharma 0, Zeyauddian et al. 03, Singh et al. 03, Rao et al. 03 and enaum 04 who have evaluated the models of the universe with bulk viscous fluid and discussed the effect of bulk viscosity on the evaluation of universe. Metric and Rosen s Field Equations We consider the ianchi Type I metric ds. where dt dx dy C, and C are the functions of t only. The flat metric corresponding to metric 5.. is. d dt dx dy dz The energy momentum tensor T i of the source of bulk viscous fluid with string is given by Ti viv xi x l v l gi viv.3 with v iv xi x.4 and v i x i 0.5 Here is the rest energy density and p where p particles loaded on the string and the string tension density respectively, dz and denote the particle density i x is the direction of string, is the four velocity vector, is the coefficient of bulk viscosity and is the scalar expansion which is given by i v

2 International Journal of Researches In iosciences, griculture & Technology May 06 ISSN No. Online X l v l.6 Thus, the Rosen s field equations are, C C 6 C C C C C C C 6 C C C C C 6 C C C C C 6 C From differential equations,we write d dt d C dt C which on integrating, we get a C. where a is constant of integration and choosing the constant of integration a C. Using equation., the differential equations can be written as, so that On subtracting equations.4 and.5, we get 6 dding equations.5 and.6, we have 4 6 The conservation law for the energy-momentum tensor, yield We are going to solve these two differential equations.7 and.8 in Five unknowns,,, and. Here is a known quantity with the function of and. In order to have solution, we assume

3 International Journal of Researches In iosciences, griculture & Technology May 06 ISSN No. Online X three extra conditions. First we assume the relation between and the scalar expansion is proportional to shear assumed by Saha 005 as On differentiating the above equation, we get fter solving the above equations,we have 4 8 ssuming the second condition.9 constant, i.e., the coefficient of bulk viscosity varies with the scalar expansion, since the viscosity, being dissipative phenomenon, is believed to affect the expansion of the universe. Originally, this condition is assumed by Santos et al. 985 and then many other researchers like Saha 005 and ali et al. 007, etc. The rest energy density and the string tension density for the composite matter of viscous fluid and dark energy can be taken together as,.0 DE DE and the energy momentum tensor equation 5..3 can be decomposed as Ti DE viv DE xi x gi viv. in which DE is the component of energy density corresponding to dark energy and DE component of string tension density corresponding to dark energy. The term density and is the string tension density corresponding to the viscous fluid. 3. Model with Viscous Fluid The viscous fluid obeys the equation of state, 0 3. is the is the rest energy Here 0 Dust Universe, Radiation Universe,, Hard Universe and 3 3 Zel dovich Universe or stiff matter. In view of this energy density differential equation becomes and string tension density, from the equation 3., the DE DE DE 3. In the evaluation of energy density and string tension density, in case of bulk viscous fluid and dark energy, we bifurcate the conservation law equation 5.3. as 3

4 International Journal of Researches In iosciences, griculture & Technology May 06 ISSN No. Online X 3.3 With the help of equation of state 3., we write equation 3.3 as, On integrating, we get where is an integrating constant. Using the above value of energy density Thus, the energy density, we get string tension density and string tension density corresponding to the viscous fluid model are given by and 3.4 OR 8 we are going to solve the integral equation 3.5 and by straight forward calculations, we arrived at the solution bsinh c c t / in which c and c are constants of integration and Using the equation3.6, we write the values of and C as bsinh c / c t C bsinh c / c t The volume V is b 8 c 4 / as = constant / V bsinh c c t 3.8 Here it is to be noted that the equation of state parameter is non-zero i.e., 0, since the scale factors,, C and the volume V do not exist at 0. For 0, the required metric corresponding to viscous fluid is

5 International Journal of Researches In iosciences, griculture & Technology May 06 ISSN No. Online X ds dt 4 / b sinh c c t dx / b sinh c c t dy dz 3.9 In this viscous fluid model for 0, the physical parameters like scalar expansion, the coefficient of bulk viscosity, anisotropic parameter, the shear scalar, and the deceleration parameter q are t 4 coth,, t 4 coth 3.0 coth 3 t, q 3 sech 8 The nature of scalar expansion and shear have been shown in similar in nature to that of volume V and Hubble parameter H and they do not contribute any new behavior geometrically and physically. The bulk viscosity coefficient has the direct connection with the Hubble parameter H which has the scaling relation with the red shift, the expansion and the shear in the evaluation of the universe and it dominate H, and. For small effect of, the rate of expansion and shear both are very high and they slowing down as increases, and the rate of expansion stopped and shear vanish, for high range of. 4. Conclusion ianchi type I models with the assumptions of bulk viscous fluid and dark energy have been deduced by solving the Rosen s field equations in imetric theory of gravitation and observed that the bulk viscosity leads inflationary like solutions in the evaluation of the universe. The string viscous fluid models for 0, starts evolving with non-zero volume, non-zero energy density, string tension density and particle density and goes over to vacuum matter died at t and further the matter suddenly appeared in the model whose energy density, particle density and string phases goes on increasing. The bulk viscosity coefficient has the direct connection with the Hubble parameter H which has the scaling relation with red shift, with the expansion and with the shear in the evaluation of the universe and it dominates H, and. For small effect of, the rate of expansion and shear both are very high and they slowing down as increases and the rate of expansion stopped and shear vanishes, for high range of. The deceleration parameter q is negative shows the model is acceleratingly expanding. It is anisotropize and has shear. We have deduced the solutions of Rosen s field equations for ianchi type I space-time in the presence of string bulk viscous fluid and dark energy quintessence and Chaplygin gas. The bulk viscosity leads inflationary like solutions in the models with cosmic strings and dominated the nature of Hubble parameter H, the expansion and the shear in the evaluation of the universe. Five models have been investigated namely, Non-dust string viscous fluid model, Dust string viscous fluid model, Quintessence dark energy model with viscosity for q and for q and chaplygin gas dark energy. It is noticed that the geometry of quintessence dark energy model, for 0 is coincide with the geometry of non-dust string fluid model for 0. The dust q string viscous fluid model and Chaplygin gas dark energy model yield same geometry except some physical behavior. Other geometrical and physical aspects of the models are studied. t 8 5

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