Tilted Bianchi Type IX Dust Fluid Cosmological Model in General Relativity

Abstract

Some tilted Bianchi type IX dust fluid cosmological model is investigated. To get a deterministic model we assume that 𝑎=𝑏^1/2, where 𝑎 and 𝑏 are metric potentials and the fluid is pressureless, that is, 𝑝=0. Some physical and geometric properties of the model are discussed. Also, we have discussed a special model in terms of 𝑡 only.

1. Introduction

Bianchi type IX cosmological models are interesting because these models allow not only expansion but also rotation and shear, and in general these are anisotropic. Many relativists have taken keen interest in studying Bianchi type IX universe because familiar solutions like Robertson-Walker universe, the de Sitter universe, the Taub-Nut solutions, and so forth are of Bianchi type IX space-times. Bianchi type IX universe includes closed FRW models. The homogeneous and isotropic FRW cosmological models, which are used to describe standard cosmological models, are particular cases of Bianchi type I, V and IX space-times according to the constant curvature of the physical three-space, 𝑡 = constant, that is zero, negative or positive. In these models, neutrino viscosity explains the large radiation entropy in the universe and the degree of isotropy of the cosmic background radiation. The standard cosmological models are too restrictive because of the insistence on the isotropy of the physical three spaces; several attempts have been made to study nonstandard cosmological models [1–3]. It is therefore interesting to carry out the detailed studies of gravitational fields, which are described by space-time of various Bianchi types. Vaidya and Patel [4] have studied spatially homogeneous space time of Bianchi type IX, and they have outlined a general scheme for the derivation of exact solutions of Einstein’s field equations in the presence of perfect fluid and pure radiation fields. Krori et al. [5] and Chakraborty and Nandy [6] have derived cosmological models of Bianchi type II, VIII, and IX. These and many other researchers namely Uggla and Zur-Muhlen [7], Burd et al. [8], King [9], and Paternoga and Graham et al. [10] have studied Bianchi type IX space-time in different context. Bali and Yadav [11] have investigated Bianchi type IX viscous fluid cosmological models. Pradhan et al. [12] derived Bianchi type IX viscous fluid cosmological models with a varying cosmological constant.

There has been a considerable interest in spatially homogeneous and anisotropic cosmological models in which the fluid flow is not normal to the hypersurface of homogeneity. These are called tilted universes. The general dynamics of tilted cosmological models have been studied by King and Ellis [13], Ellis and King [14]. Bradley and Sviestins [15] have shown that heat flow is expected for tilted universes. Mukherjee [16] has investigated Bianchi type I cosmological model with heat flux in general relativity. The tilted cosmological models with heat flux have been investigated by a number of authors namely Banerjee and Sanyal [17], Coley [18], Roy and Prasad [19], Roy and Banerjee [20]. Bagora [21, 22] have also investigated Bianchi type I and III tilted cosmological model in a different context.

In general relativity, a dust solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid which has positive mass energy density but vanishing pressure. Concerning the tilted perfect fluid models, Bradley [23] obtained all tilted and expanding dust self-similar cosmologies that are hypersurface homogeneous. Carr [24] classified spherically symmetric self similar dust models. Bagora et al. [25, 26] have investigated tilted dust magnetic cosmological models.

Motivated by the situations discussed above, in this paper we shall focus on the problem tilted Bianchi type IX cosmological models with dust fluid, and we investigated dust cosmological model. To get the deterministic model, we have assumed supplementary condition 𝑎=𝑏^1/2, where 𝑎 and 𝑏 are metric potentials. The all physical aspects of the model are discussed. Also, we discussed special model in terms of “𝑡”.

2. Field Equations and Solutions

We consider the homogeneous anisotropic Bianchi type IX metric in the following form: 𝑑𝑠²=−𝑑𝑡²+𝑎²𝑑𝑥²+𝑏²𝑑𝑦²+𝑏²sin²𝑦+𝑎²cos²𝑦𝑑𝑧²−2𝑎²cos𝑦𝑑𝑥𝑑𝑧, (1) where 𝑎 and 𝑏 are functions of cosmic time “𝑡” alone.

The energy-momentum tensor for perfect fluid distribution with heat conduction is given by Ellis [27] as 𝑇_𝑖𝑗=∈+𝑝𝜈_𝑖𝜈^𝑗+𝑝𝑔^𝑗_𝑖+𝑞_𝑖𝜈^𝑗+𝜈_𝑖𝑞^𝑗, (2) together with 𝑔_𝑖𝑗𝜈^𝑖𝜈^𝑗𝑞=−1,_𝑖𝑞^𝑗𝑞>0,_𝑖𝜈^𝑖=0. (3)(4)(5)

Here 𝑝 is the pressure and ∈ the density and 𝑞_𝑖 the heat conduction vector orthogonal to 𝑣^𝑖.

The fluid flow vector 𝑣^𝑖 has the components (sinh𝜆/𝑎,0,0,cosh𝜆) satisfying the condition (3) and 𝜆 is the tilt angle.

The Einstein’s field equation 𝑅^𝑗_𝑖−12𝑅𝑔^𝑗_𝑖=−8𝜋𝑇^𝑗_𝑖,(Λ=0,𝑐=𝐺=1). (6) The field equation for the line element (1) leads to 2𝑏̈𝑏+𝑏̇²𝑏²−34𝑎²𝑏⁴+1𝑏²=−8𝜋∈+𝑝sinh²𝜆+𝑝+2𝑞₁sinh𝜆𝑎,𝑎̈𝑎+𝑏̈𝑏+𝑎𝑏̇̇+𝑎𝑎𝑏4𝑏²⁴2𝑎𝑏̇̇=−8𝜋𝑝,+𝑏̇𝑎𝑏𝑏²²+1𝑏²−𝑎²4𝑏⁴=8𝜋∈+𝑝cosh²𝜆−𝑝+2𝑞₁sinh𝜆𝑎,∈+𝑝𝑎sinh𝜆cosh𝜆+𝑞₁cosh𝜆+𝑞₁sinh²𝜆cosh𝜆=0, (7)(8)(9)(10) where (⋅) denotes the ordinary differentiation with respect to cosmic time “𝑡”.

3. Solutions of Field Equations

Equations from (7) to (10) are four equations in six unknown 𝑎, 𝑏, ∈, 𝑝, 𝜆, and 𝑞₁. Thus we require two more conditions to determine complete solution.

Firstly we assume that the model is filled with dust of perfect fluid which leads to 𝑃=0. (11)

Secondly, we assumed relation between metric potentials as 𝑎=𝑏^1/2. (12) The motive behind assuming the conditions 𝑎=𝑏^1/2 is explained as follows. Referring to Throne [28], the observations of velocity-redshift relation for extra-galactic sources suggest that the Hubble expansion of the universe is isotropic to within 30% [29, 30]. More precisely, the redshift studies place the limit 𝜎/𝐻≤0.30, where 𝜎 is the shear and 𝐻 the Hubble constant. Collins et al. [31] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous hypersurface satisfies the condition 𝜎/𝜃= constant. The condition 𝜎¹₁/𝜃 = constant leads to 𝑎=𝑏^1/2 when the tilt angle 𝜆=0. Here 𝜎¹₁ is the eigenvalue of shear tensor 𝜎^𝑗_𝑖.

Equations (7) and (9) lead to 2𝑏̈𝑏+2𝑏̇²𝑏²+2𝑎𝑏̇̇+2𝑎𝑏𝑏²−𝑎²𝑏⁴=8𝜋∈−𝑝. (13) Using (11) in (13), we have 2𝑏̈𝑏+2𝑏̇²𝑏²+2𝑎𝑏̇̇+2𝑎𝑏𝑏²−𝑎²𝑏⁴=8𝜋∈. (14) Again from equations (7) and (9), we have 𝑎̈𝑎+𝑏̈𝑏+𝑎𝑏̇̇+𝑎𝑎𝑏2𝑏²⁴=0. (15) Using condition (12) in (15), we have 3𝑏̈+12𝑏4𝑏̇𝑏²=−14𝑏³. (16) This leads to 2𝑓𝑓^′+𝑓²13𝑏=−3𝑏², (17) where 𝑏̇=𝑓(𝑏) and 𝑏̈=𝑓𝑓^′.

Equation (17) leads to 𝑏̇=2𝐾𝑏^2/3+1,̈−2𝑏𝑏=2𝐾𝑏^2/3+112𝑏², (18) where “𝐾” is constant of integration.

The metric (1) reduces to the following form: 𝑑𝑆²=−2𝑇𝑑𝑇²2𝑇^2/3𝐾+1+𝑇𝑑𝑋²+𝑇²𝑑𝑌²+𝑇²sin²𝑌+𝑇cos²𝑌𝑑𝑍²−2𝑇cos𝑌𝑑𝑍𝑑𝑋, (19) where 𝑏=𝑇, 𝑥=𝑋, 𝑦=𝑌, 𝑧=𝑍.

4. Some Physical and Geometrical Properties

The density for the model (19) is given by 8𝜋∈=8𝐾𝑇^2/3+6𝑇3𝑇³. (20) The tilt angle 𝜆 is given by cosh𝜆=38𝐾𝑇^2/3+4𝑇+328𝐾𝑇^2/3,+9sinh𝜆=8𝐾𝑇^2/3+12𝑇−928𝐾𝑇^2/3.+9 (21) The scalar expansion 𝜃 calculated for the flow vector 𝑣^𝑖 is given by 𝜓𝜃=¹2𝑇2𝐾𝑇^2/3+138𝐾𝑇^2/3+4𝑇+38𝐾𝑇^2/3+9³. (22) The nonvanishing components of shear tensor (𝜎_𝑖𝑗) and rotation tensor (𝜔_𝑖𝑗) are given by 𝜎₁₁=𝜓₂42𝐾𝑇^2/3+18𝐾𝑇^2/3+4𝑇+33𝑇8𝐾𝑇^2/3+9⁵,𝜎₂₂=𝜓₃12𝑇2𝐾𝑇^2/3+138𝐾𝑇^2/3+4𝑇+38𝐾𝑇^2/3+9³,𝜎₃₃=𝜓₄cos²𝑦+𝑇𝜓₅sin²𝑦×122𝐾𝑇^2/3+13𝑇8𝐾𝑇^2/3+4𝑇+38𝐾𝑇^2/3+9³,𝜎₄₄=8𝐾𝑇^2/3+12𝑇−9𝜓₆×12𝑇2𝐾𝑇^2/3+13𝑇8𝐾𝑇^2/3+4𝑇+38𝐾𝑇^2/3+9⁵,𝜎₁₄=−𝜓₂12𝑇2𝐾𝑇^2/3+18𝐾𝑇^2/3+12𝑇−98𝐾𝑇^2/3+9⁵,𝜔₁₄=12𝑇2𝐾𝑇^2/3+18𝐾𝑇^2/3+12𝑇−928𝐾𝑇^2/3.+9 (23)

Here 𝜓₁=384𝐾²𝑇^4/3+200𝐾𝑇^5/3+243𝑇+600𝐾𝑇^2/3𝜓+162,₂=192𝐾²𝑇^4/3+64𝐾𝑇^5/3+192𝐾𝑇^2/3𝜓+81,₃=192𝐾²𝑇^4/3+64𝐾𝑇^5/3+108𝑇+192𝐾𝑇^2/3𝜓−27,₄=384𝐾²𝑇^4/3+160𝐾𝑇^5/3+108𝑇+480𝐾𝑇^2/3,𝜓₅=960𝐾²𝑇^4/3+448𝐾𝑇^5/3+432𝑇+1344𝐾𝑇^2/3𝜓+405,₆=192𝐾²𝑇^4/3−64𝐾𝑇^5/3+192𝐾𝑇^2/3+81. (24) The flow vectors 𝑣^𝑖 and heat conduction vectors 𝑞_𝑖 for the model (19) are given by 𝜈¹=8𝐾𝑇^2/3+12𝑇−92𝑇8𝐾𝑇^2/3,𝑣+9₄=38𝐾𝑇^2/3+4𝑇+328𝐾𝑇^2/3,𝑞+9₁=−8𝐾𝑇^2/3+4𝑇+32𝜋𝑇³8𝐾𝑇^2/3+12𝑇+928𝐾𝑇^2/3,𝑞+9₄=8𝐾𝑇^2/3+12𝑇−96𝜋𝑇^7/238𝐾𝑇^2/3+4𝑇+328𝐾𝑇^2/3.+9 (25)

The rates of expansion 𝐻_𝑖 in the direction of 𝑥, 𝑦, and 𝑧 axes are given by 𝐻₁=14𝑇^3/24𝐾𝑇^2/3𝐻+2,₂=12𝑇^3/24𝐾𝑇^2/3𝐻+2,₃=12𝑇^3/24𝐾𝑇^2/3+2. (26)

4.1. Special Model I

To find model in terms of 𝑡, we assume the constant 𝐾=0. Then, we have 3𝑏=𝑡+𝑘₁^2/32, (27) where “𝑘₁” is constant of integration.

Then metric (1) reduces in the following form: 𝑑𝑠²=−𝑑𝜏²+34𝜏^2/3𝑑𝑥²+34𝜏^2/3𝑑𝑦²+3−2cos𝑦𝑑𝑥𝑑𝑧4𝜏^2/334𝜏^2/3sin²𝑦+cos𝑦𝑑𝑧². (28)

The density for the model (28) is given by 28𝜋∈=9𝜏^4/3. (29)

The tilt angle 𝜆 is given by cosh𝜆=𝜏^2/3+12,sinh𝜆=𝜏^2/3−12. (30) The scalar expansion 𝜃 calculated for the flow vector 𝑣^𝑖 is given by 2𝜃=162𝜏^2/3+13581𝜏3𝜏^2/3.+1 (31) The flow vectors 𝑣^𝑖 and heat conduction vectors 𝑞_𝑖 for the model (28) are given by 𝜈¹=𝜏^2/3−1𝜏^1/3,𝑣⁴=𝜏^2/3+12,𝑞₁=−𝜏^2/3+1𝜏^2/3−1144𝜋𝑇^5/3,𝑞₄=𝜏^2/3−12𝜏^2/3+1144𝜋𝜏². (32)

The nonvanishing components of shear tensor (𝜎_𝑖𝑗) and rotation tensor (𝜔_𝑖𝑗) are given by 𝜎₁₁=3𝜏^2/3+14𝜏^1/3𝜏^2/3+13,𝜎₂₂=3𝜏^2/3−124𝜏^1/313𝜏^2/3,𝜎+1₄₄=𝜏^2/3−19𝜏13𝜏^2/3,𝜎+1₁₄=2𝜏^2/3+39𝜏^2/33𝜏^2/3+13,𝜔₁₄=16𝜏^1/33𝜏^2/3−12. (33)

The rates of expansion 𝐻_𝑖 in the direction of 𝑥, 𝑦, and 𝑧 axes are given by 𝐻₁=1,𝐻3𝜏₂=2,𝐻3𝜏₃=2.3𝜏 (34)

5. Conclusions

The model (19) has point-type singularity at 𝑇=0. The model starts to expand with the big bang at 𝑇=0 and stops at 𝑇=∞. The model is expanding, shearing, rotating and anisotropic in general. Since ^𝐿𝑡_𝑇→∞(𝜎/𝜃)≠0 then the model does not approach isotropy for large values of 𝑇. The heat conduction vectors 𝑞¹→∞, and 𝑞⁴→∞ at initial stage whereas as 𝑇=∞, 𝑞¹→0, and 𝑞⁴→0. The 𝑥, 𝑦 and 𝑧 components of Hubble parameters 𝐻_𝑖→∞, 𝑇→0 and 𝐻_𝑖→0 as 𝑇→∞. Therefore, these are decreasing function of time.

In the model (28), initially the rate of expansion 𝜃 is infinite but for the large time as 𝜏→∞ the expansion will be completely vanished. The components of Hubble parameters are well behaved at 𝜏>1 and will become zero for large time.

For particularly at 𝜏=𝜏₀, 𝐻₁, 𝐻₂, 𝐻₃, 𝜃, 𝜎² and ∈ are all infinite and the volume is zero. This shows that model has finite singularity at 𝜏=𝜏₀.Therefore, this model starts expanding with power law expansion, and it has finite physical singularity.