Table of Contents
ISRN Astronomy and Astrophysics
VolumeΒ 2011, Article IDΒ 565842, 3 pages
http://dx.doi.org/10.5402/2011/565842
Research Article

LRS Bianchi Type II Inflationary Universe with Massless Scalar Field

Department of Mathematics, University of Rajasthan, Jaipur 302004, India

Received 5 September 2011; Accepted 16 October 2011

Academic Editor: P. A.Β Hughes

Copyright Β© 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.

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 𝑑, we have assumed that the scale factor βˆΌπ‘’3𝐻𝑑, that is, 𝑅2π‘†βˆΌπ‘’3𝐻𝑑 and 𝑉(πœ™) = constant where 𝑉 is effective potential and πœ™ is Higg's field. We find that spatial volume increases with time and the model isotropizes for large value of 𝑑 under special condition. The Higg's field decreases slowly and tends to a constant value when π‘‘β†’βˆž. 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 Μ‡πœ™πœ™=0=; πœ™=0 being a local minimum of 𝑉(πœ™) where 𝑉 is effective potential and πœ™ is Higg’s field which breaks the symmetry. In this case, the energy-momentum tensor of particles βˆΌπ‘‡4 (𝑇 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, 𝑇𝑖𝑗=𝑔𝑖𝑗𝑉(0) where 𝑉(0) is effective potential at vanishing temperature (Zel’dovich & Khlokov [1978]). This leads toπ‘Ž3=𝑒3𝐻𝑑,(1) where π‘Ž is scale factor and 𝐻 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 𝑉(πœ™) has flat region while Higg’s field (πœ™) 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 = 𝑒3𝐻𝑑 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 π‘ŽβˆΌπ‘’π»π‘‘, 𝐻 is Hubble constant as introduced by Kirzhnits and Linde [12]. The model represents an anisotropic universe which isotropizes for large value of 𝑑 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 form𝑑𝑠2=πœ‚π‘Žπ‘πœƒπ‘Žπœƒπ‘(2) withπœƒ2=𝑆(π‘‘π‘¦βˆ’π‘₯𝑑𝑧).(3) Therefore, the metric (2) leads to the form 𝑑𝑠2=βˆ’π‘‘π‘‘2+𝑅2𝑑π‘₯2+𝑆2(π‘‘π‘¦βˆ’π‘₯𝑑𝑧)2+𝑅2𝑑𝑧2,(4) where 𝑅 and 𝑆 are functions of 𝑑-alone.

The Lagrangian is that of gravity minimally coupled to be a scalar field 𝑉(πœ™) given byξ€œβˆšπΏ=ξ‚€1βˆ’π‘”π‘…βˆ’2π‘”π‘–π‘—πœ•π‘–πœ™πœ•π‘—ξ‚π‘‘πœ™βˆ’π‘‰(πœ™)4π‘₯(5) (Notations have their usual meaning and in geometrized unit 𝐺=𝑐=1). Now from the variation of 𝐿 with respect to the dynamical fields, we obtain Einstein field equation for massless scalar field 𝑉(πœ™) asπ‘…π‘–π‘—βˆ’12𝑅𝑔𝑖𝑗=βˆ’8πœ‹π‘‡π‘–π‘—,(6) where𝑇𝑖𝑗=πœ•π‘–πœ™πœ•π‘—ξ‚ƒ1πœ™βˆ’2πœ•πœŒπœ™πœ•πœŒξ‚„π‘”πœ™+𝑉(πœ™)𝑖𝑗(7) with1βˆšπœ•βˆ’π‘”π‘–ξ‚ƒβˆšβˆ’π‘”πœ•π‘–πœ™ξ‚„=βˆ’π‘‘π‘‰π‘‘πœ™,(8) where 𝑣𝑖 the flow vector, πœ™ the Higg’s field, 𝑉the potential, and 𝑔𝑖𝑗 the metric tensor. Here πœ•π‘–πœ™=πœ•πœ™πœ•π‘₯𝑖,πœ•πœŒπœ™=π‘”πœŒβ„“πœ•πœ™πœ•π‘₯β„“.(9) The Einstein’s field equation (6) for the metric (4) leads to𝑅44𝑅+𝑆44𝑆+𝑅4𝑆4+𝑆𝑅𝑆24𝑅41=βˆ’8πœ‹2Μ‡πœ™2ξ‚„,βˆ’π‘‰(πœ™)2𝑅44𝑅+𝑅24𝑅2βˆ’3𝑆24𝑅41=βˆ’8πœ‹2Μ‡πœ™2ξ‚„,π‘…βˆ’π‘‰(πœ™)24𝑅2+2𝑅4𝑆4βˆ’1𝑅𝑆4𝑆2𝑅41=8πœ‹2Μ‡πœ™2ξ‚„.+𝑉(πœ™)(10) Equations (8) for scalar field leads to πœ™44+ξ‚΅2𝑅4𝑅+𝑆4𝑆,πœ™4=βˆ’π‘‘π‘‰π‘‘πœ™.(11)

3. Solution of Field Equations

To get deterministic solution in terms of cosmic time 𝑑, we assume that scale factor βˆΌπ‘’3𝐻𝑑, that is, 𝑅2π‘†βˆΌπ‘’3𝐻𝑑,𝐻isHubbleconstant(12) as considered by Bali [11]. We also assume that effective potential 𝑉(πœ™)= constant. Thus (11) leads toπœ™44+ξ‚΅2𝑅4𝑅+𝑆4π‘†ξ‚Άπœ™4=0.(13) Equation (10) leads to𝑅44𝑅+𝑆44𝑆+3𝑅4𝑆4+𝑅𝑅𝑆24𝑅2=π‘˜,(14) where16πœ‹π‘‰(πœ™)=π‘˜(constant).(15) From (12), we have2𝑅4𝑅+𝑆4𝑆=3𝐻.(16) Using (16) in (14), we get𝑅44π‘…βˆ’π‘…24𝑅2+3𝐻𝑅4𝑅=𝛽,(17) where𝛽=9𝐻2βˆ’π‘˜.(18) From (17), we have𝑅4𝑅=𝛽3𝐻+π›Ύπ‘’βˆ’3𝐻𝑑,(19)𝛾 being constant of integration. Equation (19) leads to𝑅=ℓ𝑒𝛽𝑑/3π»ξ‚€βˆ’π›Ύexp𝑒3π»βˆ’3𝐻𝑑.(20) Equations (19) and (16) lead to1𝑆=β„“2𝑒(3π»βˆ’(2𝛽/3𝐻))𝑑exp2𝛾𝑒3π»βˆ’3𝐻𝑑.(21) Hence the metric (4) reduces to the form𝑑𝑠2=βˆ’π‘‘π‘‘2+β„“2𝑒2𝛽𝑑/3𝐻2ξ‚€βˆ’π›Ύexp𝑒3π»βˆ’3𝐻𝑑𝑑π‘₯2+1β„“4𝑒(6π»βˆ’(4𝛽/3𝐻))𝑑2ξ‚΅βˆ’exp2𝛾𝑒3π»βˆ’3𝐻𝑑(π‘‘π‘¦βˆ’π‘₯𝑑𝑧)2+β„“2𝑒2𝛽𝑑/3𝐻2ξ‚€βˆ’π›Ύexp𝑒3π»βˆ’3𝐻𝑑𝑑𝑧2.(22)

To determine Higg’s field πœ™
Using the assumed condition 𝑉(πœ™)=constant in (11), we have πœ™44+ξ‚΅2𝑅4𝑅+𝑆4π‘†ξ‚Άπœ™4=0.(23) Equation (23) leads to πœ™=βˆ’π‘Žπ‘’βˆ’3𝐻𝑑3𝐻+𝑏,(24) where π‘Ž and 𝑏 are constants.

4. Some Physical and Geometrical Aspects

The spatial volume (𝑉3) for the model (22) is given by𝑉3=𝑅2𝑆=𝑒3𝐻𝑑.(25) The expansion (πœƒ) is given byξ‚΅πœƒ=2𝑅4𝑅+𝑆4𝑆=3𝐻.(26) The shear (𝜎) is given by1𝜎=√3𝑅4π‘…βˆ’π‘†4𝑆,(27) which leads to1𝜎=√3ξ€·βˆ’3𝐻+3π›Ύπ‘’βˆ’3𝐻𝑑+𝛽𝐻.(28) The deceleration parameter (π‘ž) is given byΜˆπ‘ž=βˆ’π‘‰/𝑉̇𝑉2/𝑉2=βˆ’1,(29) wherė𝑉𝑉=𝑉4𝑉;Μˆπ‘‰π‘‰=𝑉44𝑉.(30)

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 𝑑 and 𝛽=3𝐻2. The Higg’s field decreases slowly, and it tends to a constant value when π‘‘β†’βˆž. There is uniform expansion and deceleration parameter π‘ž=βˆ’1. Hence the model leads to de-Sitter space-time, and the model represents accelerating universe.

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