KANTOWSKI-SACHS INFLATIONARY UNIVERSE IN GENERAL RELATIVITY

Transcription

1 KNTOWSKI-SCHS INFLTIONY UNIVESE IN GENEL ELTIVITY This wor has been published in Internional J. of Theoretical Physics Volume 8, No.0, p (009.

2 [8.] INTODUCTION In recent years, there has been a lot of interest in cosmological models of the universe which are important in understanding the mysteries of the early stages of its evolution. In particular, inflionary model play an important role in solving a number of outstanding problems in cosmology lie the homogeneity, the isotropy and flness of the observed universe. The standard explanion for the flness of the universe is th it has undergone an early stage of the evolution a period of exponential expansion named as inflion. Inflion, the stage of accelered expansion of the universe, first proposed by Gliner (965, nowadays, receives a gre deal of tention. Guth (98, Linde (98 and La and Steinhardt (989 are some of the authors who have investiged several aspects of inflionary universe in general relivity. The role of self-interacting scalar fields in inflionary cosmology has been investiged by htacharjee and aruah (00, ali and Jain (00, ahaman et al. (00, eddy et al. (008, eddy and Naidu (008 and eddy (009. ecently Kore et al. (00 have investiged Einstein- osen inflionary universe in general relivity. In the last two chapters, we have studied various aspects of general relivity for different space-times. In this chapter we confine ourselves to study the Kantowsi-Sachs (966 inflionary 8

3 cosmological model in the presence of massless scalar field with a fl potential in Einstein s theory of general relivity. This chapter is organized as follows: In section [8.], Einstein s field euions are obtained with the aid of Kantowsi-Sachs space-time. The section [8.] deals with Kantowsi-Sachs inflionary model in Einstein s theory of general relivity. We have discussed some physical and geometrical properties of the model in section [8.]. The last section contains concluding remars. [8.] METIC ND FIELD EQUTIONS The homogeneous and anisotropic universe is described by Kantowsi-Sachs space-time in the form ds dt dr ( dθ sin θ d, (8.. where and are functions of cosmic time t only. Kantowsi-Sachs class of metric represents homogeneous but anisotropically expanding (contracting cosmologies and provides models where the effect of anisotropy can be estimed and compared with FW class of cosmologies (Thorne, 967. In the case of gravity minimally coupled to a scalar field V (, the Lagrangian L is L [ g dx, i, j V ( ] g, (8.. 9

4 which on variion of L with respect to dynamical fields lead to Einstein s field euions g T (8.. with energy momentum tensor T,, i, j, V ( g (8.. and dv (, (8..5 d i ; i where comma (, and semicolon (; indices ordinary and covariant differentiion respectively. The function depends on t only due to homogeneity. Units are taen such th 8π G c. Using co-moving coordine system from euion (8.., we have T T T V ( and T V (. (8..6 The Einstein s field euions (8.. with the help of euion (8..6 for the Kantowsi-Sachs metric (8.. are given by V (, (8..7 V (, (

5 V ( (8..9 and the euion (8..5 for the scalar field taes the form dv (. (8..0 d Here the subscript denotes differentiion with respect to t. [ For detailed calculions, please, refer to the ppendix [8..] ]. [8.] THE INFLTIONY MODEL We are interested, here, in inflionary solutions of the field euions (8..7-(8..0. Stein-Schabes (987 has shown th the scalar field will taen sufficient time to cross the fl region of the potential to allow the universe expands necessarily to become homogeneous and isotropic on the large scale of the order of the horizon size. The fl part of the potential is nurally associed with a vacuum energy th domines the dynamics for a period of time and identifies vacuum energy with an effective cosmological constant. Therefore we assumed the fl region when the potential is constant i.e. V ( = constant = V 0 (say. (8.. Since the field euions are highly non-linear, to get determine solution one can use a special law of variion for Hubble s parameter proposed by erman (98 which yields a constant negive decelerion parameter defined by 5

6 = constant, (8.. where ( t ( is the overall scale factor. Here the constant is taen as negive (i.e. it is an accelering model of the universe. Euion (8.. gives the solution ( t ( b, ( 0 (8.. where a ( 0 and b are constants of integrion. Thus field euions (8..7-(8..0 admit an exact solution given by exp [ ( ], (8.. b ( b exp [ ( b ], (8..5 ( b 0, (8..6 where V0,,. (8..7 a [ For detailed calculions, please, refer to the ppendix [8..] ]. fter a suitable transformion of coordine and constants of integrion, the Kantowsi-Sachs inflionary cosmological model corresponding to the solutions (8.. and (8..5 taes the form ds dt exp{ T } dr T exp{ T }( dθ sin θd. (8..8 5

7 [8.] SOME PHYSICL PMETES The model (8..8 represents Kantowsi-Sachs inflionary universe in general theory of relivity when the scalar field is minimally coupled to the gravitional field in which the fl region of potential is constant which is generally associed with vacuum energy. The investiged model has no initial singularity. The Physical and Kinemical parameters of the model (8..8 are given by the following expressions: Spial Volume V T, Expansion Scalar Shear Scalar θ, ( T σ V0 ( T (, T and Hubble Parameter H. ( T [ For detailed calculions, please, refer to the ppendix [8..] ]. We observed th the spial volume increases with time T, since ( 0, it becomes infinite for large values of T. Thus the Inflion is possible for large T in general theory of relivity for Kantowsi-Sachs space-time with a massless scalar field when the fl part of the potential is constant. The expansion scalar, shear scalar 5

8 and Hubble parameter become infinite T 0, but they vanish for large T. The Higg s field becomes constant for T = 0 or = and σ for large value of T it diverges. lso the rio ( θ become infinite for large T and hence the model does not approach isotropy. [8.5] CONCLUSION Kantowsi-Sachs cosmological model play an important role in understanding the early stages of evolution of the universe. In this chapter we have investiged Kantowsi-Sachs inflionary universe in Einstein s theory of general relivity. It is observed th the cosmological model is non-singular, expanding and does not approach anisotropy le times. This study of the inflionary model obtained here will throw some light on the structure formion of the universe which has astrophysical significance. For example, classical scalar fields are essential in the study of the present day cosmological models. We hope th our model will be useful for a better understanding of inflionary universe in Kantowsi-Sachs space-time. 5

9 PPENDIX [8..] Kantowsi-Sachs space-time is given by ds dt dr ( dθ sin θ d, where and are functions of cosmic time t only. The components of the metric tensor g are g, g, g sin θ, g and g 0 ( for i j g ( for i j g = 0 ( for i j. The non-vanishing Christoffel s symbols of second ind for the metric (8.. are: Γ g g x g x g x ([ g ] (. t t Similarly, Γ Γ, Γ Γ Γ Γ, Γ Γ cotθ, Γ sin θ cosθ, Γ, Γ sin θ. 55

10 icci tensor is defined as a a b a a b Γ Γ a i ia ΓiaΓbj Γba Γ. x x For i j Γ Γ Γ Γ] x [ Γ Γ = t = g =. Similarly, for i = j =,, g =. sin θ sin θ sin θ sin g. 56 θ

11 g. Now the icci invariant g = g g g g =. Einstein tensor is given by G g. For i = j = G G. Similarly, for i j,, G G, G. The components of j T i for euion (8.. are given by For i j T ( V ( 57

12 = V (. Similarly, for i j,, T T V (. T ( V ( = V (. For i j the field euions (8.. for the metric (8.. will be δ T V ( Similarly, for i j,, V (. V (, V ( and dv (. d 58

13 From euion (8.., we have On integring, we get a d adt. Integring yield 0. PPENDIX [8..] ( t ( ( b, where a and b are constants of integrion. Differentiing above euion, we obtain a. (8..a ( ( b Now we consider the fl region where the potential is constant i.e. V ( V (Say. 0 Euions (8..7-(8..9 with euion (8.. yield. (8..b V0 Elimining from euions (8..a and (8..b, with the help of, we get 59

14 60 0 ( ( V b a 0 ( ( V b a dt d. (8..c Euion (8..c yield b D b a V 0 ( (, where D is a constant of integrion. Without loss of generality, we tae D = 0 and integring, we get 0 ( exp. b a V m. (8..d Now using euion (8..d in euion (8..a and integre, we get 0 ( (.exp ( b a V b n, where m and n are constants of integrion, for m = n =, we have ] ( exp [ b, ] ( exp [ ( b b, 0 ( b, where,, 0 a V and 0 is a constant of integrion.

15 PPENDIX [8..] Expansion scalar θ u; i i = = i, i u u Γ i i V0 a V ( 0 θ b ( b a ( ( b a a θ ( ( b ut ( b T Therefore θ. ( T Shear Scalar σ ( g g = ( g g ( g g ( g g ( g g ( g g σ = V 0 T. ( T Hubble parameter = H a ( ( b. ( T 6

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