View of Inflationary Scenario in Bianchi Type II Space with Bulk Viscosity in General Relativity

Inflationary Scenario in Bianchi Type II Space with Bulk Viscosity in General Relativity

Laxmi Poonia¹, Sanjay Sharma^2*

*Correspondence: [email protected]

1-2Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan, India ORCIDs:

Laxmi POONIA: https://orcid.org/0000-0001- Sanjay SHARMA:https://orcid.org/0000-0001-9049-1349

Abstract:This study is about bulk viscous inflationary model with flat potential under framework of LRS Bianchi type II metric. To drive relativistic solution of the field equations we choose theproportionally condition between coefficient of shear and expansion scalar which leads to asuitable relation 𝑎 = 𝑏^𝑛 between the metric functions where n is the constant other than one.

Some dynamical features of the universe are also discussed.

Keywords:Bianchi Type II, Inflationary Model, Bulk Viscosity, Flat Potential, General Relativity

1.Introduction

In recent scenario, inflationary cosmology has become curious subject of study to explaining the evolution of universe and formation of galactic structures, it widely acceptable by many cosmologist that major cosmological problem like isotropy, homogeneity, flatness are successfully explain by the theory of inflation. The choice of anisotropic metric with system of field’s equations allows us to construct mathematical model of cosmos to understanding the accelerated fate of current physical universe. The system of fields equations are basically set of non-linear differential equations we required solutions of its in various applications in astrophysics and cosmology.The analysis of microwave background has also provided some physical evidence regarding the cosmos inflation. Staronbinsky [1] explained the model of initial universe after epoch. Guth[2] studies the various aspects of cosmos inflation by proposing the fact that false vacuum energy are responsible for this phenomenon. The role of higg’s fields with potential V are significantly are used in various research.Many cosmologists [3-11] have derived different model to understanding the theory of inflation and scalar field 𝜑 in various manner.

LRS Bianchi- II metric has significant role in developing models which helps to understanding the early evolution of cosmos and inflationary nature in more details. Inflationary model within framework of bulk viscosity is very helpful to illustrating many physical and structural features in dynamics of current universe. Mishner [12] studied inflationary model under effect of bulk viscosity in different manner. Heller and Klimek [13] has constructed viscous fluid model without initial singulaties. Gron [14-15] has discussed Bianchi - I space with bulk, nonlinear viscosity and in shearing mode.. The significant role of bulk viscosity in cosmos inflation is studied by many researchers [16-18] and Bali et al.[19-21] in different aspects. Agrawal [22] investigate LRS

1224 Bianchi II model inflationary model. Reddy [23] investigated Bianchi Type V space time for massless scalar field under flat potential. Sharma [24] derived Bianchi II inflationary model in general relativity. Bali and Poonia [25] constructed inflationary cosmological model under framework of Bianchi type III with bulk viscosity.

Motivated by this discussion, we have investigated Bulk viscous inflationary cosmological model with flat potential for LRS Bianchi type II metric. To get inflationary solution we suppose 𝜉𝜃 = 𝛼 (constant) as proposed by Brevik et al.[26]. The paper work is classified as following sections given as: Section-2 concerned with metric and system of nonlinear fields equations.

Section-3 contains solutions of fields equations in inflationary context.Section-4 are concerned with dynamical aspects of constructed model.Section-5 deals with conclusion and discussion.

2. Fields Equations with metric

LRS Bianchi-II space in orthonormal frame can be obtained by line element 𝑑𝑠² = 𝑔_𝜇𝛾𝜃^𝜇𝜃^𝛾 , 𝑔_𝜇𝛾= diag (1, 1, 1, -1) (1) Here 𝑎 𝑡 and 𝑏(𝑡) are the metric function

where𝜃¹ = 𝑎(𝑡)𝜔¹, 𝜃² = 𝑏(𝑡)𝜔², 𝜃³ = 𝑎(𝑡)𝜔³ , 𝜃⁴ = 𝑑𝑡

with𝜔¹ = 𝑑𝑥 , 𝜔² = 𝑑𝑦 − 𝑥𝑑𝑧 , 𝜔³ = 𝑑𝑧 (2)

from equation (1) and (2) metric can be written as

𝑑𝑠² = −𝑑𝑡² + 𝑎² 𝑡 𝑑𝑥²+ 𝑏² 𝑡 𝑑𝑦 − 𝑥𝑑𝑧 ²+ 𝑎² 𝑡 𝑑𝑧² (3) The field of gravity minimallyto scalar region with potential 𝑉 𝜑 is given by 𝑆 = 𝑅 −¹₂𝜑_,𝑖𝜑_,𝑗𝑔^𝑖𝑗 − 𝑉 𝜑 −𝑔 𝑑𝑥⁴ (4) Einstein’s field equation for the model is given by

𝑅_𝑖^𝑗 −¹

2𝑅𝑔_𝑖^𝑗 = −𝑇_𝑖^𝑗 (5)

(In geometrical unit 8πG = c = 1)

where energy momentum tensor for scalar field under consideration of bulk viscosity is given by 𝑇_𝑖𝑗 = 𝜑_,𝑖𝜑_,𝑗 − ¹

2𝜑_,𝑙𝜑^,𝑙+ 𝑉 𝜑 𝑔_𝑖𝑗 − 𝜉𝜃 𝑔_𝑖𝑗 + 𝑢_𝑖𝑢_𝑗 (6) with ¹

−𝑔𝜕_,𝑖 −𝑔𝜑_,𝑖 = −^{𝑑𝑉 𝜑}

𝑑𝜑 (7)

where𝜉 and 𝜃 be the bulk viscosity coefficient and scalar expansion respectively we assume comoving coordinate system as 𝑢^𝑖 = 0,0,0,1

For LRS Bianchi type II metric (1), system of field equation (2) can be obtained as 2^𝑎44

𝑎 +^𝑎42

𝑎² −³

4 𝑏²

𝑎⁴ = − ¹

2𝜑₄²− 𝑉 𝜑 − 𝜉𝜃 (8)

𝑎₄₄ 𝑎 +^𝑏44

𝑏 +^𝑎4𝑏4

𝑎𝑏 +¹

4 𝑏²

𝑎⁴ = − ¹

2𝜑₄²− 𝑉 𝜑 − 𝜉𝜃 (9)

𝑎42

𝑎² + 2^𝑎4𝑏4

𝑎𝑏 −¹

4 𝑏²

𝑎⁴ = ¹

2𝜑₄²+ 𝑉 𝜑 (10)

where indices 4 in metric coefficient shows the ordinary differentiation with respect to cosmic time

Equation (7) provides 𝜑₄₄+ 2^𝑎4

𝑎 +^𝑏4

𝑏 𝜑₄ = −^𝑑𝑉

𝑑𝜑 (11)

The physical parameter of developed model which are significantly used to find solution of field equations and discussing geometrical features are given by

Proper volume of developed model (1) is given by

𝑉 = −𝑔 = 𝑎²𝑏 (12)

Expansion scalar 𝜃 for model is given by 𝜃 = 𝑢_,𝑖^𝑖 = 2^𝑎4

𝑎 +^𝑏4

𝑏 (13)

Shear scalar for model is given by 𝜎² = ¹

2𝜎_𝑖𝑗𝜎^𝑖𝑗 (14) where𝜎_𝑖𝑗 =¹

2 𝑢_{𝑖 ,𝑘}𝐷_𝑗^𝑘 + 𝑢_{𝑗 ,𝑘}𝐷_𝑖^𝑘 −

1

3𝜃 𝑔_𝑖𝑗 + 𝑢_𝑖𝑢_𝑗 (15)

and𝐷_𝑖^𝑗 = 𝛿_𝑖^𝑗 − 𝑢^𝑖𝑢_𝑗 which leads to 𝜎² = ¹

3 𝑎4

𝑎 −^𝑏4

𝑏 2

(16) Formula for Hubble parameter is given by

𝐻 =¹

3 2^𝑎4

𝑎 +^𝑏4

𝑏 (17)

we obtained deceleration parameter by given relation 𝑞 = −

𝑅44 𝑅 𝑅₄²

𝑅²

(18)

3. Fields Equations with solutions

Equation (8-10) is independent equations with five unknown 𝑎, 𝑏, 𝜑, 𝜃 𝑎𝑛𝑑 𝜉 . For this purpose we required extra condition

𝜉𝜃 = 𝛼 (Constant) (19)

i.e. coefficient of bulk viscosityis inversely proportional to expansion scalarand shear scalar (σ) is directly proportional to expansion (θ) which leads to

𝑎 = 𝑏^𝑛𝑛 > 1 (20)

1226 wehave also assumed constant potential V with flat region

i.e. 𝑉 𝜑 = 𝜍 (Constant)

Equation (11) leads to 𝜑₄₄+ 2^𝑎4

𝑎 +^𝑏4

𝑏 𝜑₄ = 0 on integrating we have 𝜑₄ = ^𝜑0

𝑎²𝑏 (21)

From the equations (8) and (9), we obtained

𝑎44 𝑎 −^𝑏44

𝑏 +^𝑎42

𝑎² −^𝑎4𝑏4

𝑎𝑏 −^𝑏2

𝑎⁴ = 0 (22)

Equations (20) and (22) give 2𝑏₄₄ + 4𝑛^𝑏4

2 𝑏 = ²

𝑛−1𝑏^3−4𝑛 (23)

we consider 𝑏₄ = 𝑓 𝑏 (24)

which provide 𝑏₄₄ = 𝑓𝑓^′ (25)

where𝑓^′ = ^𝑑𝑓

𝑑𝑏

from equations (23),(24) and (25) we have

𝑑𝑓²

𝑑𝑏 + 4𝑛^𝑓2

𝑏 = ²

𝑛−1𝑏^3−4𝑛 (26)

which leads to 𝑓² = ¹

2 𝑛−1 𝑏^4−4𝑛 + 𝐷𝑏^−4𝑛 (27)

where D is the integrating constant From equation (17) and (18) we have 𝑏₄ = ¹

2 𝑛−1 𝑏^4−4𝑛 + 𝐷𝑏^−4𝑛

1

2 (28)

Equation (28) leads to _{2 𝑛−1} ¹ 𝑏^4−4𝑛 + 𝐷𝑏^{−4𝑛 −}

1

2𝑑𝑏 = ± 𝑡 − 𝑡₀

where𝑡₀ is the integration constant The metric (1) can be reduces in to form 𝑑𝑠² = − ¹

2 𝑛−1 𝑇^{4 1−𝑛} + 𝐷𝑇^{−4𝑛 −1}𝑑𝑇²+ 𝑇^2𝑛 𝑑𝑋²+ 𝑑𝑍² + 𝑇² 𝑑𝑌 − 𝑋𝑑𝑍^{2 2}(29) using the transformation 𝑏 = 𝑇 , 𝑥 = 𝑋, 𝑦 = 𝑌, 𝑎𝑛𝑑 𝑧 = 𝑍

4. Structural and dynamical features of the model The proper volume is given by

𝑉 = 𝑇^2𝑛+1 (30)

The shear scalar 𝜎² of model is obtained by 𝜎² = ¹

3 𝑛 − 1 ^{2 1}

2 𝑛−1 𝑇^{2 2𝑛 −1} + 𝐷 𝑇^{−2 2𝑛+1} (31)

Fig. 1 volume (V) versus time (T) Fig. 2 coefficient of shear ( σ)verses time(T)

The expansion (θ) for constructed model is given by θ = 2n + 1 ¹

2 n−1 T^{2 2n −1} + D T^{−2 2n+1}

1

2 (32)

The result for hubble parameter (H) is given as H = ¹

3 2n + 1 ¹

2 n−1 T^{2 2n −1} + D T^{−2 2n+1}

1

2 (33)

Fig. 3 coefficient of expansion (θ) versus time(T) Fig. 4 Hubble parameter (H)verses time (T)

The deceleration parameter 𝑞 for model is obtained by 𝑞 = −1 + ³

2𝑛 +1 (33)

Coefficient of bulk viscosity (ξ) is given by

𝜉 = ^𝛼

2𝑛+1 ¹

2 𝑛 −1 𝑇2 2𝑛 −1 +𝐷 𝑇^{−2 2𝑛 +1} 1 2

(34)

Scalar field is given by

𝜑 = 𝜑₀

𝑇^2𝑛+1𝑑𝑇 + 𝐶

1228 5. Conclusion and discussion

The proper volume growths with time and become infinite at late time which indicates that cosmic inflation is possible in developed model. Since ^𝜎

𝜃 = constant i.e. model maintained the anisotropy at late time, but the model become shear free and isotropic at n=1. The negative deceleration parameter shows accelerated phase of universe. The scalar of expansion and Hubble parameter become divergent at initial epoch, t=0 and tends to zero for infinite T and at 𝑛 = −¹

2

i.e. universe start with infinite expansion. The shear scalar is decreasing function of time and tends to zero for infinite large T. Higgs field decrease slowly and become finite for large time.

The bulk viscosity coefficient leads to cosmic inflation in present scenario.

References

[1] Starobinsky, A.A. 1980 A new type of isotropic cosmological models without singularity Phys.

Lett. B91 pp.99-102

[2] Guth, A.H.1980 Inflationary Universe: A Possible solution of the horizon and flatness problem Phys. Lett. B 91pp. 99-102

[3] Albrecht, A. and Steinhardt P. J. 1982 Cosmology for grand unified theories with radiatively induced symmetry breaking Phys. Rev. Lett.48 pp. 1220-1223.

[4] La, D. and Steinhardt, P.J. 1989 Extended Inflationary Cosmology Physical Review Letters 62, pp.

376-378.

[5] Wald, R.M. 1983 Asymptotic behavior of homogeneous cosmological models in presence of a positive cosmological constant Phys. Rev. D 28 pp. 2118

[6] Ellis G.F.R. and Madsen, M.S. 1991 Exact scalar field cosmologies Class. Quant. Gravit.8 pp.667-676

[7] Sharma, S. and Poonia, L. 2021 Cosmic acceleration in LRS Bianchi Type I space-time with bulk viscosity in general relativity IOP Conf. Ser.: Mater. Sci. Eng.1045 012025

[8] Sharma, S. and Poonia, L. 2021 Cosmic inflation in Bianchi Type IX space with bulk viscosity Advances in Mathematics: Scientific Journal10(1) pp.527-534

[9] Poonia, L. and Sharma, S. 2020 Inhomogeneous cylindrically Bianchi Type I space-time with flat potential Journal of Physics: Conference Series Journal 1706,012042

[10] Bhattacharya, R. and Baruah, K.K. 2001 String cosmologies with a scalar field Ind. J. Pure. Appl.

Math.32 pp. 47

[11] Bali, R. and Jain, V.C. 2002 Bianchi Type I Inflationary Universe in General Relativity Pramana59 pp.1-7

[12] Mishner, C.W. 1968 The Isotropy of Universe Astrophys. J. 151 pp. 431-458

[13] Heller, N. and Klimek, Z.1975 Viscous universes without initial singularity Astrophys. Space Sci.33 pp. L37-L 39

[14] Gron, O. 1985 Expansion isotropization during the inflationary era Phys. Rev. D32 pp.2522 [15] Gron, O. 1990 Viscous inflationary universe models Astrophys. Space Sci.173 pp.191-225 [16] Saha, B.2005 Bianchi Type I Universe with Viscous Fluid Mod. Phys. Lett. A20 pp. 2127-2143 [17] Brevik, I.K. and Gron, O. 2013 Relativistic Viscous Universe Models Rec. Adv. in Cosmo. Nova

Sci. Pub. New York chapter 4 pp.97-127

[18] Sahni, V. and Staronbinsky, A. 2000 The Case for a Positive Cosmological Lambda-term Int. J.

Mod. Phys. D 9 pp. 373-444.

[19] Bali, R., Singh, P. and Singh, J. 2012 Bianchi Type V Viscous Fluid Cosmological Models in Presence of Decaying Vacuum Energy Astrophys. Space Sci. 341 pp.701-706

[20] Bali, R. and Sharma, K. 2003 Tilted Bianchi Type I stiff fluid magnetized cosmological model in general relativity Astrophys. Space Sci.283 pp.11-22

[21] Bali, R. and Kumawat, P. 2008 Bulk viscous LRS Bianchi type V tilted stiff fluid cosmological

model in general relativity Phys. Lett. B665 pp. 332-337

[22] Agrawal, S. 2017 LRS Bianchi Type II Cosmological Model With Stiff Fluid and Varying Λ term JIAR1pp.46-55

[23] Reddy, D.R.K. 2009 Bianchi Type V Inflationary Universe in General Relativity Int.J. Theor.

Phys.48 pp.2036-2040

[24] Sharma, S. 2012 Bianchi Type-II Bulk Viscous Cosmological Model in General Relativity Int. J.

Theor. Appl. Sci.4 pp.36 -41

[25] Bali,R. and Poonia, L. 2014 Bianchi Type-III Inflationary Cosmological Model with Bulk Viscosity in General Relativity Prespacetime J.5 pp. 981-986

[26] Brevik, E., Elizalde, S. and Odintsov, S.D. 2011Viscous Little Rip Cosmology Phys.Rev. D84 103508

[27] Lorentz, D.1980 An exact Bianchi-type II cosmological model with matter and an electromagnetic fieldPhys. Lett. A79 pp. 19-20

[28] Stein- Shabes, J. A. 1987 Inflation in spherically symmetric inhomogeneous models Phys. Rev. D 35 pp.2345-2351