LRS Bianchi Type II Inflationary Universe with Massless Scalar Field

Abstract

Inflationary scenario in locally rotationally symmetric (LRS) Bianchi Type II space-time with massless scalar field with flat potential is discussed. To get the deterministic solution in terms of cosmic time , we have assumed that the scale factor , that is, and  = constant where is effective potential and is Higg's field. We find that spatial volume increases with time and the model isotropizes for large value of under special condition. The Higg's field decreases slowly and tends to a constant value when . The model represents uniform expansion but accelerating universe and leads to de-Sitter type metric.

1. Introduction

Inflationary universes provide a potential solution to the formation of structure problem in Big-Bang cosmology like Horizon problem, Flatness problem, and magnetic monopole problem. Guth [1] introduced the concept of inflation while investigating the problem of why we see no magnetic monopole today. He found that a positive-energy false vacuum generates an exponential expansion of space according to general relativity. In Guth inflationary universe, the scalar field is assumed to start at ; being a local minimum of () where is effective potential and is Higg’s field which breaks the symmetry. In this case, the energy-momentum tensor of particles ( being absolute temperature) almost vanishes in the course of expansion of the universe and the total energy-momentum tensor reduces to vacuum energy, that is, where is effective potential at vanishing temperature (Zel’dovich & Khlokov [1978]). This leads to where is scale factor and is Hubble constant. Rothman and Ellis [2] have pointed out that we can have a solution of isotropic problem if we work with anisotropic metric and these metrics can be isotropized under various general circumstances. Stein-Schabes [3] has shown that inflation will take place if effective potential has flat region while Higg’s field evolves slowly but the universe expands in an exponential way due to vacuum field energy. Burd [4] has discussed inflationary scenario in FRW (Friedmann-Robertson-Walker) model. Anninos et al. [5] discussed the significance of inflation for isotropization of universe. In modern cosmology, inflation is an essential ingredient. During the inflationary epoch, the scale factor of the universe grew exponentially allowing a small causally coherent region to grow enough to be identified with the present observable universe. Linde [6] proposed a chaotic model with an assumption that the present universe is originated from chaotic distribution of initial scalar field when potential energy of the field dominates over that of kinetic energy. Later on, it has been shown by Bunn et al. [7] that chaotic scenario can be realized even when scalar field is kinetic energy dominated. Paul et al. [8] have shown that Linde’s chaotic scenario is fairly general and can be accommodated even if universe is anisotropic. Bali and Jain [9] has discussed inflationary scenario in LRS Bianchi Type I space-time in the presence of massless scalar field with flat potential. Reddy et al. [10] have investigated inflationary scenario in Kantowski-Sachs space-time. Recently Bali [11] investigated inflationary scenario in anisotropic Bianchi Type I space-time with flat potential considering the scale factor = as introduced by Kirzhnits and Linde [12].

Motivated by the above-mentioned research works, we have investigated inflationary scenario in LRS Bianchi Type II space-time with flat potential, and assuming the condition scale factor , is Hubble constant as introduced by Kirzhnits and Linde [12]. The model represents an anisotropic universe which isotropizes for large value of under special condition as shown by Rothman and Ellis [2]. The model represents uniform expansion but accelerating universe. The model leads to de-Sitter space-time.

2. Metric and Field Equations

We consider Bianchi Type II metric in the form with Therefore, the metric (2) leads to the form where and are functions of -alone.

The Lagrangian is that of gravity minimally coupled to be a scalar field given by (Notations have their usual meaning and in geometrized unit ). Now from the variation of with respect to the dynamical fields, we obtain Einstein field equation for massless scalar field as where with where the flow vector, the Higg’s field, the potential, and the metric tensor. Here The Einstein’s field equation (6) for the metric (4) leads to Equations (8) for scalar field leads to

3. Solution of Field Equations

To get deterministic solution in terms of cosmic time , we assume that scale factor , that is, as considered by Bali [11]. We also assume that effective potential = constant. Thus (11) leads to Equation (10) leads to where From (12), we have Using (16) in (14), we get where From (17), we have being constant of integration. Equation (19) leads to Equations (19) and (16) lead to Hence the metric (4) reduces to the form

To determine Higg’s field
Using the assumed condition in (11), we have Equation (23) leads to where and are constants.

4. Some Physical and Geometrical Aspects

The spatial volume () for the model (22) is given by The expansion () is given by The shear () is given by which leads to The deceleration parameter () is given by where

5. Conclusions

The spatial volume increases with time. Hence inflationary scenario exists in Bianchi Type II space-time. The model (22) in general represents an anisotropic universe. However the model isotropizes for large values of and . The Higg’s field decreases slowly, and it tends to a constant value when . There is uniform expansion and deceleration parameter . Hence the model leads to de-Sitter space-time, and the model represents accelerating universe.

References

1. A. H. Guth, “Inflationary universe: a possible solution to the horizon and flatness problems,” Physical Review D, vol. 23, no. 2, pp. 347–356, 1981. View at: Publisher Site | Google Scholar
2. T. Rothman and G. F. R. Ellis, “Can inflation occur in anisotropic cosmologies?” Physics Letters B, vol. 180, no. 1-2, pp. 19–24, 1986. View at: Google Scholar
3. J. A. Stein-Schabes, “Inflation in spherically symmetric inhomogeneous models,” Physical Review D, vol. 35, no. 8, pp. 2345–2351, 1987. View at: Publisher Site | Google Scholar
4. A. Burd, “Inflation in open FLRW universes,” Classical and Quantum Gravity, vol. 10, no. 8, pp. 1495–1505, 1993. View at: Publisher Site | Google Scholar
5. P. Anninos, R. A. Matzner, T. Rothman, and M. P. Ryan Jr., “How does inflation isotropize the universe?” Physical Review D, vol. 43, no. 12, pp. 3821–3832, 1991. View at: Publisher Site | Google Scholar
6. A. D. Linde, “Chaotic inflation with constrained fields,” Physics Letters B, vol. 202, no. 2, pp. 194–197, 1988. View at: Google Scholar
7. E. F. Bunn, A. R. Liddle, and M. White, “Four-year COBE normalization of inflationary cosmologies,” Physical Review D, vol. 54, no. 10, pp. R5917–R5921, 1996. View at: Google Scholar
8. B. C. Paul, D. P. Datta, and S. Mukherjee, “Chaotic inflationary scenario in an anisotropic universe,” Modern Physics Letters, vol. 1, p. 149, 1986. View at: Google Scholar
9. R. Bali and V. C. Jain, “Bianchi type I inflationary universe in general relativity,” Pramana, vol. 59, no. 1, pp. 1–7, 2002. View at: Google Scholar
10. D. R. K. Reddy, K. S. Adhav, S. D. Katore, and K. S. Wankhade, “Kantowaski-Sachs inflationary universe in general relativity,” International Journal of Theoretical Physics, vol. 48, no. 10, pp. 2884–2888, 2009. View at: Publisher Site | Google Scholar
11. R. Bali, “Inflationary scenario in bianchi type I space-time,” International Journal of Theoretical Physics, vol. 50, no. 10, pp. 3043–3048, 2011. View at: Publisher Site | Google Scholar
12. D. A. Kirzhnits and A. D. Linde, “Symmetry behavior in gauge theories,” Annals of Physics, vol. 101, no. 1, pp. 195–238, 1976. View at: Google Scholar

Copyright

Copyright © 2011 Raj Bali and Laxmi Poonia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.