Open Journal of Applied Sciences, 2013, 3, 8993 doi:10.4236/ojapps.2013.31B1018 Published Online April 2013 (http://www.scirp.org/journal/ojapps) 1A Cosmological Model with Varying G and in General Relativity Harpreet, R. K. Tiwari, H. S. Sahota Department of Applied Sciences , Sant Baba Bhag Singh Institute of Engineering & Technology, Khiala, Padhiana, Jalandhar 144030,Punjab, India Email: sahota.harpreet@rediffmail.com Received 2013 ABSTRACT In this paper homogeneous Bianchi type I spacetime with variable G and containing matter in the form of a perfect fluid assuming the cosmological term proportional to H2 (where H is Hubble Parameter). Initially the model has a point type singularity, gravitational constant G (t) is decreasing and cosmological constant is infinite at this time. When time increases, decrease. The model does not approach isotropy, if it is small. The model is quasiisotropic for large value of it. Keywords: Bianchi TypeI Universe; Varying G and ; Cosmology; Hubble Parameter 1. Introduction Cosmology is study of large scale structures of universe. The simplest homogeneous and anisotropic model is Bianchi type –I cosmological model which is gener alization of FriedmannRobertsonWalker (FRW) model. Cosmological model with cosmological constant are se rious participants to describe the dynamics of the uni verse. The origin of universe is greatest cosmological mystery even today. As we are aware that the expansion of the universe is undergoing time acceleration Perlmut ter et al.,(1997,1998,1999), Riess et al.,(1998,2004), Al len et al.,(2004), Peebles et al.,(2003), Padmanabhan. (2003) & Lima.(2004).To resolve the problem of a huge difference between the effective cosmological constant observed today and the vacuum energy density predicted by the quantum field theory, several mechanisms have been proposed by (Weinberg, 1989). A possible way is to consider a varying cosmological term due to the coupling of dynamic degree of freedom with the matter fields of the universe. The cosmological constant is small because the uni verse is old. Models with dynamically decaying cosmo logical term representing the energy density of vacuum have been studied by Vishwakarma, R. G. (2000, 2001,2005), Arbab, A. I.(1998), Berman, M. S.(1991a, 1991b). Cosmological scenarios with a time varying cosmological constant were proposed by several re searchers. A number of models with different decay laws for the variation of cosmological term were investigated during the last two decades Chen & Wu (1990); Pavan (1991); Carvalho et al.,(1992); Lima & Maia(1994); Lima & Trodden (1996); Arbab & AbdelRahman(1994); Cunha & Santos (2004); Carneiro & Lima(2005). A number of authers investigated Bianchies models, using the approach that there is link between variation of gravitational constant and cosmological constant [Abdel Rahman 1990;Berman 1991a ;Kalligas et al.1992; Ab dussattar and Vishkarma 2005;Pradhan et al 2006;Singh and Tiwari 2007 ]. Lot of work has been done by Saha (2005a, 2005b, 2006a, 2006b), in studying the anisot ropic Bianchi typeI Cosmological Model in general relativity with varying G and. In this paper we study homogeneous Bianchi type I spacetime with variable G and containing matter in the form of a perfect fluid. We obtain solution of the Einstein field equations assuming the cosmological term proportional to H2 (where H is Hubble Parameter). 1.1. The Metric and Field Equations We consider the Bianchi type  I metric in the orthogonal form (1) The nonzero components of the Ricci tensor Rij. We assume that cosmic matter is taken to be perfect fluid given by the energy momentum tensor (2) where p, are the isotropic pressure and energy density of the fluid. We take equation of state Copyright © 2013 SciRes. OJAppS
HARPREET ET AL. 90 (2 1) vi is four velocity vector of the fluid satisfying (3) Einstein's field equations with time dependent G and are (4) For the metric (1) and energymomentum tensor (2) in commoving system of coordinates, the field equation (4) yields. (5) (6) (7) (8) In view of vanishing of the divergence of Einstein tensor, we have (9) The usual energy conservation equation of general relativity quantities is (10) Equation (9) together with (10) puts G and in some sort of coupled field given by (11) implying that is a constant whenever G is constant. Using equation (21) in equation (10) and then integrating, we get k> 0 (111) We define, R as the average scale factor of Bianchi type I universe. (12) The Hubble parameter H, volume expansion , shear and deceleration parameter q are given by Einstein's field equations (5)(8) can be also written in terms of Hubble parameter H, shear and decelera tion parameter q as (13) (14) On integrating (5)  (8), we obtain (15) and (16) Where k1 and k2 are constants of integration. From (14), we obtain (17) Implying that 0 Thus the presence of positive lowers the upper limit of anisotropy whereas a negative contributes to the anisotropy. Equation (17) can also be written as (18) where 2 3 8 c G is the critical density and 8 vG is the vacuum density. From (13) and (14), we get, (19) Copyright © 2013 SciRes. OJAppS
HARPREET ET AL. 91 Thus the universe will be in decelerating phase for negative and for positive universe will slow down the rate of decrease showing that the rate of volume ex pansion decreases during time evolution and presence of positive , slows down the rate of this decrease whereas a negative would promote it. 1.2. Solution of the Field Equation The system of equations (5)(8) and (11) supply only five equations in seven unknown parameters (A, B, C,, p, , and G). Two extra equations are needed to solve the sys tem completely. For this purpose we take cosmological term is proportional to H2 . (20) This variation law was proposed by Olson, et al., (1987), Pavon(1991), Maia, et al., (1994); Silveira, et al., (1994, 1997), Torres, et al.,(1996).Because observations suggest that is very small in the present universe, a decreasing functional form permits to be large in early universe. Using equation (111) and equation (20) in equation (11) we get (21) From equations (13), (14), (20) and (201) we get (22) Find the time evolution of Hubble parameter, integrate (22), we get (23) where t0 is a constant of integration. The integration con stant is related to the choice of origin of time. From equation (23), we obtain the scale factor (24) By using equation (24) in (15) and (16), the metric (1), we get (25) where M = and m1,m2 and m3 are constants. For the model (25), the spatial V, density , gravitational con stant G and cosmological constant are (26) (27) (28) (29) Expansion scalar and shear are (30) (31) (32) (33) (34) (35) 2. Observations and Conclusions 1. Thus we observe that as spatial volume V0 at t = 0 and expansion scalar is infinite, which shows that universe starts evolving with zero volume at t = 0 infinite rate of expansion. Hence the model has a point type sin gularity at initial epoch. 2. Initially at t = 0 the energy density ‘ ’, pressures ‘p’, shear ,cosmological term tend all infinite. 3. As t increases the spatial volume increases but the expansion rate decreases. Thus the rate of expansion slows down with increase in time and tend to zero. 4. As t the spatial volume V becomes infinitely large. All parameters , , p, ,Ὠ, , 0 asymptotically but G is decreasing. Therefore at large value of t model gives empty universe. The cosmic scenario starts from a big bang at t = 0 and continues until t=. 5. The ratio / 0 as t∞. So the model approach isotropy for large value of t. 6. We also see for model (25), if t ∞ then q =2 im plies universe is decelerating as q is positive. 7. The possibility of G increasing with time, at least in some stages of the development of the universe, has been investigated by AbdelRahman (1990), Chow (1981), Levit (1980) and Milne (1935). 2 1 T Include Copyright © 2013 SciRes. OJAppS
HARPREET ET AL. 92 Berman (1990), Berman and Som (1990), Berman et al. (1989), and Bertolami (1986b, 1986a). This form of is physically reasonable as observations suggest that is very small in the present universe. A decreasing func tional form permits to be large in the early universe. 8. In summary, we observed on investigation that Bi anchi typeI cosmological model with variable G and in presence of perfect fluid with cosmological term is proportional to H2 (H is Hubble Parameter) suggested by Silveira et al., (1994,1997) and others. Initially the model has a point type singularity, gravitational constant G (t) is decreasing and cosmological constant is in finite at this time when time increases decrease.The model approach isotropy for large value of t ,the model is quasiisotropic i.e. 0. REFERENCES [1] A. Pradhan and P. Pandey, “Some Bianchi Type I Vis cous Fluid Cosmological Models with a Variable Cos mological Constant Astrophys,” 2006, Space Science, Vol. 301, p. 221. [2] B. 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