A COSMOLOGICAL MODEL WITH VARYING G AND IN GENERAL RELATIVITY

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1 A COSMOLOGICAL MODEL WITH VARYING G AND IN GENERAL RELATIVITY Harpreet 1, R.K. Tiwari and *H.S. Sahota 1, Sant Baba Bhag Singh Institute of Engineering and Technology, Department of Applied Sciences, Khiala, Padhiana, Jalandhar Government Engineering College, Reva, M.P. *Author for Correspondence ABSTRACT Spatially homogeneous and anisotropic Cosmological models play significant role in the description of the early stages of evolution of the universe. The problem of the cosmological constant is still unsettled. The authors recently considered time dependent G and with Bianchi type I Cosmological model.in this paper homogeneous Bianchi type -I space-time with variable G and containing matter in the form of a perfect fluid assuming the cosmological term proportional to R - (where R is scale factor). 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, unlike in some earlier works we have neither assumed equation of state nor particular form of G. The model does not approach isotropy. If k is small the model is quasi-isotropic. Key Words: Bianchi Type-I Universe, Varying G and, Cosmology INTRODUCTION Cosmological models which are spatially homogeneous and anisotropic play significant roles in the description of the universe in its early stages of evolution. Bianchi type- I space times is the simplest generalization of Friedmann - Robertson - Walker (FRW) flat models. There is significant observational evidence that the expansion of the universe is undergoing time acceleration Perlmutter et al., (1997, 1998, 1999); Riess et al., (1998, 004); Allen et al., (004); Peebles et al., (00); Padmanabhan (00) and Lima (004). Now days, the problem of cosmological constant is one of the most salient and unsettled problems in cosmology. A number of models with different decay laws for the variation of cosmological term were investigated during the last two decades Chen and Wu (1990); Pavan (1991); Carvalho et al., (199); Lima and Maia (1994); Lima and Trodden (1996); Arbaband Abdel-Rahman (1994); Cunha and Santos (004) and Carneiro and Lima(005). To resolve the problem of a huge difference between the effective cosmological constant observed today and the vacuum energy density predicted by the quantum filed 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. Models with dynamically decaying cosmological term representing the energy density of vacuum have been studied by Vishwakarma (000, 001 and 005); Arbab (1998) and Berman (1991, 1991b). On the other hand numerous modifications of general relativity to allow for a variable G based on different arguments have been proposed by Wesson (1980); Kalligas et al., (199); Abdel Rahman (1990); Pradhan et al., (00, 004, 006 and 007) and Tiwari et al., (009 and 010). A lot of work has been done by Saha (005a, 005b, 006a and 006b), in studying the anisotropic Bianchi type-i Cosmological Model in general relativity with varying G and. Recently we have studied Bianchi type I Cosmological model with time dependent G and (008). In this paper we study homogeneous Bianchi type-i space-time with variable G and containing matter in the form of a perfect fluid. We obtain solution of the Einstein field equations by assuming the cosmological term proportional to R - (where R is scale factor). 4

2 The Metric and Field Equations We consider the Bianchi type - I metric in the orthogonal form ds = -dt + A (t) dx + B (t) dy + C (t) dz (1) The non-vanishing Christoffel symbols of the second kind are 4 AA A A 1 11, 14, 1 41, 4 BB A A B, 4, 4 CC B C, 4 C The non-zero components of the Ricci tensor R ij A C R AA AA A C B C 11 R BB BB A C, A B R CC CC A B C A B R 44, A B C The Ricci scalar AA BC CA A B C R AB BC CA A B C Matter content is taken to be perfect fluid given by the energy- momentum tensor T ij = ( + p) v i v j + pg ij () Where v i is four velocity vector of the fluid satisfying g ij v i v j = -1 () And p, are respectively the isotropic pressure and energy density of the fluid. The Einstein's field equations with time dependent G and are R ij - ½ R l lg ij = -8G(t) T ij + (t) g ij (4) For the metric (1) and energy-momentum tensor () in commoving system of coordinates, the field equation (4) yields B C BC 8Gp B C BC A C AC 8Gp A B AC (6) A B AB 8Gp A B AB (7) AB BC AC 8G AB BC AC (8) In view of vanishing of the divergence of Einstein tensor, we get A B C 8 G p 8G 0 A B C (9) j The usual energy conservation equation T 0yields.. i; j A B C p 0 (10) A B C Equation (9) together with (10) puts G and in some sort of coupled field given by 8 G 0 (11) (5) 44

3 Implying that is a constant whenever G is constant. Here and elsewhere a dot stands for ordinary differentiation with respect to time t. We define, R = (ABC) 1/ (1) As the average scale factor of Bianchi type- I universe. The Hubble parameter H, volume expansion, shear and deceleration parameter q are given by R H k, k 0 (Constant) R R H RR q 1 H R Einstein's field equations (5)-(8) can be also written in terms of Hubble parameter H, shear and deceleration parameter q as H (q 1) 8Gp (1) H 8G (14) On integrating (5) - (8), we obtain A B k1 A B R (15) And B C k B C R (16) Where k 1 and K are constants of integration. From (14), we obtain 4G 1 (17) Implying that 0 1 8G 1 0,0 Thus the presence of positive lowers the upper limit of anisotropy whereas a negative gives more room for anisotropy. Equation (17) can also be written as 8G v 1 1 H H H (18) H Where c is the critical density and v 8G 8G is the vacuum density. From (1) and (14), we get, 1Gp 1G( p) (19) d dt Showing that the rate of volume expansion decreases during time evolution and presence of positive, slows down the rate of this decrease whereas a negative would promote it. Solution of the Field Equations The system of equations (5)-(8) and (11) supply only five equations in seven unknown (A, B, C,, p,, and G). Two extra equations are needed to solve the system completely. For this purpose (i) we take cosmological term is proportional to R -, where a is a positive constant. c c 45

4 a i.e. (0) R This variation law was proposed by Olson et al., (1987); Pavon (1991); Maia et al., (1994); Silveira et al., (1994 and 1997) and 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. Next condition we assume that the volume expansion is proportional to Eigen values of shear tensor ij. It is believed that evolution of one parameter should also be responsible for the evolution of the others (005). Following Collins et al., (1980) that the volume expansion is having a constant ratio to the anisotropy which gives, 1 n A B n C (1) Where n1 and n are constants. Without loss of generality we take n 1 = n = 1. i.e. A = B C () Using condition () in (15) and (16), we obtain the metric (1) takes the following form, after suitable transformation: ds ( K1 K ) K1 4 K1 K 4K1K dt Tdx T dy DISCUSSION For the model () average scale factor R is given by R = T 1/ (4) Volume expansion, Hubble parameter H and shear for the model are H 1 (5) T k (6) T k Clearly,.Therefore the model does not approach isotropy. If k is small, the model is quasiisotropic i.e. 0. Using (0) in (5), (8) and (11) the pressure p, density, gravitational constant G and cosmological constant are: 5k 5k k k p / 1 1 at 1 (4k1 k ) T 8k 4 5k1 5k1k k at (4k1 k ) 5k 5k k m m T (7) / 1 1 at 1 (4k1 k ) T (8) 1 8k 4 5k1 5k 1k k at (4k1 k ) k dz () 46

5 1 4 5k1 5k1k k G k at (9) (4k1 k ) And = a T -/ Where k is a positive integration constant. In the model we observe that the spatial volume V is zero at T=0 and expansion scalar is infinite at T=0 which shows that universe starts evolving with zero volume and infinite rate of expansion at T=0. The scale factors also vanish at T=0 and hence the model has a point type singularity at initial epoch. Initially at T=0 the energy density, pressures p, shear, cosmological term tend all infinite but G is finite. As t increases the spatial volume increases but the expansion rate decreases as time increases. As T the spatial volume V becomes infinitely large. All parameters,, p,,, tend to zero asymptotically but G is decreasing. A partial list of cosmological models in which the gravitational constant G is decreasing function of time are contained in Grn (1986); Helling et al., (198) and Rowan- Robinson (1981). The possibility of G increasing with time, at least in some stages of the development of the universe, has been investigated by Abdel-Rahman (1990); Chow (1981); Levit (1980) and Milne (195). 1 Include Berman (1990); Berman and Som (1990); Berman et al., (1989) and Bertolami (1986b T and 1986a). This form of is physically reasonable as observations suggest that is very small in the present universe. A decreasing functional form permits to be large in the early universe. Recent cosmological observations Perlmutter et al., (1997, 1998 and 1999); Riess et al., (1998 and 004); Garnavich et al., (1998a) and Schmidt et al., (1998) suggest the positive cosmological constant with the magnitude (Gh/c ) The density parameter 4 5k1 5k1k k at c (4k1 k ) And the ratio between vacuum and critical density is given by 4 V at c It is interesting that the above results reduce to the result of Hoyle et al., (1997). CONCLUSION We have investigated the Bianchi type-i cosmological model with variable G and in presence of perfect fluid with cosmological term is proportional to R - (R is scale factor) suggested by Silveira et al., (1994 and 1997) and others. 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. It is interesting that Beesham (1994); Lima and Carvalho (1994); Kalligas et al., (1995) and Lima (1996) have also derived the Bianchi type I cosmological models with variable G and assuming a particular from for G and by taking equation of state whereas we have neither assumed equation of state nor particular form of G. The model does not approach isotropy if k is small the model is quasi-isotropic i.e

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