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 t, we have assumed that the scale factor ~e3Ht, that is, R2S~e3Ht and V(ϕ) = constant where V is effective potential and ϕ is Higg's field. We find that spatial volume increases with time and the model isotropizes for large value of t under special condition. The Higg's field decreases slowly and tends to a constant value when t→∞. 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 ϕ=0=ϕ̇; ϕ=0 being a local minimum of V(ϕ) where V is effective potential and ϕ is Higg’s field which breaks the symmetry. In this case, the energy-momentum tensor of particles ~T4 (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 toa3=e3Ht,
where a 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(ϕ) has flat region while Higg’s field (ϕ) 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 = e3Ht 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 a~eHt, H is Hubble constant as introduced by Kirzhnits and Linde [12]. The model represents an anisotropic universe which isotropizes for large value of t 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=ηabθaθb
withθ2=S(dy-xdz).
Therefore, the metric (2) leads to the form
ds2=-dt2+R2dx2+S2(dy-xdz)2+R2dz2,
where R and S are functions of t-alone.

The Lagrangian is that of gravity minimally coupled to be a scalar field V(ϕ) given byL=∫-g(R-12gij∂iϕ∂jϕ-V(ϕ))d4x
(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(ϕ) asRij-12Rgij=-8πTij,
whereTij=∂iϕ∂jϕ-[12∂ρϕ∂ρϕ+V(ϕ)]gij
with1-g∂i[-g∂iϕ]=-dVdϕ,
where vi the flow vector, ϕ the Higg’s field, Vthe potential, and gij the metric tensor. Here
∂iϕ=∂ϕ∂xi,∂ρϕ=gρl∂ϕ∂xl.
The Einstein’s field equation (6) for the metric (4) leads toR44R+S44S+R4S4RS+S24R4=-8π[12ϕ̇2-V(ϕ)],2R44R+R42R2-3S24R4=-8π[12ϕ̇2-V(ϕ)],R42R2+2R4S4RS-14S2R4=8π[12ϕ̇2+V(ϕ)].
Equations (8) for scalar field leads to
ϕ44+(2R4R+S4S),ϕ4=-dVdϕ.

3. Solution of Field Equations

To get deterministic solution in terms of cosmic time t, we assume that scale factor ~e3Ht, that is, R2S~e3Ht,His Hubble constant
as considered by Bali [11]. We also assume that effective potential V(ϕ)= constant. Thus (11) leads toϕ44+(2R4R+S4S)ϕ4=0.
Equation (10) leads toR44R+S44S+3R4S4RS+R42R2=k,
where16πV(ϕ)=k(constant).
From (12), we have2R4R+S4S=3H.
Using (16) in (14), we getR44R-R42R2+3HR4R=β,
whereβ=9H2-k.
From (17), we haveR4R=β3H+γe-3Ht,γ being constant of integration. Equation (19) leads toR=leβt/3Hexp(-γ3He-3Ht).
Equations (19) and (16) lead toS=1l2e(3H-(2β/3H))texp(2γ3He-3Ht).
Hence the metric (4) reduces to the formds2=-dt2+l2e2βt/3Hexp{2(-γ3He-3Ht)}dx2+1l4e(6H-(4β/3H))texp{2(-2γ3He-3Ht)}(dy-xdz)2+l2e2βt/3Hexp{2(-γ3He-3Ht)}dz2.

To determine Higg’s field <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M64"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>

Using the assumed condition V(ϕ)=constant in (11), we have
ϕ44+(2R4R+S4S)ϕ4=0.
Equation (23) leads to
ϕ=-ae-3Ht3H+b,
where a and b are constants.

4. Some Physical and Geometrical Aspects

The spatial volume (V3) for the model (22) is given byV3=R2S=e3Ht.
The expansion (θ) is given byθ=(2R4R+S4S)=3H.
The shear (σ) is given byσ=13(R4R-S4S),
which leads toσ=13(-3H+3γe-3Ht)+βH.
The deceleration parameter (q) is given byq=-V̈/VV̇2/V2=-1,
whereV̇V=V4V;V̈V=V44V.

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 t and β=3H2. The Higg’s field decreases slowly, and it tends to a constant value when t→∞. There is uniform expansion and deceleration parameter q=-1. Hence the model leads to de-Sitter space-time, and the model represents accelerating universe.

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