Dust Fluid Cosmological Model

Abstract

Bianchi type I tilted bulk viscous fluid cosmological model filled with dust fluid is investigated. We assume that 𝜁𝜃=𝐾 (constant), where 𝜁 is the coefficient of bulk viscosity and 𝜃 is the expansion in the model. It has been assumed that the expansion in the model is only in two directions; that is, one of the components of Hubble parameters (𝐻₁=𝐴₄/𝐴) is zero. The physical and geometrical aspects of the model in the presence and absence of bulk viscosity are also discussed. Also, we have discussed two special models and their physical properties. From this, we present a particular example based on dust fluid.

1. Introduction

Homogeneous and anisotropic cosmological models have been studied widely in the framework of general relativity. These models are more restricted than the inhomogeneous models. But in spite of this, they explain a number of observed phenomena quite satisfactorily. In recent years, there has been a considerable interest in investigating spatially homogeneous and anisotropic cosmological models in which matter does not move orthogonal to the hypersurface of homogeneity. Such types of models are called tilted cosmological models.

The general dynamics of these cosmological models have been studied in detail by King and Ellis [1], Ellis and King [2], and Collins and Ellis [3]. Ellis and Baldwin [4] have investigated that we are likely to be living in a tilted universe and they have indicated how we may detect it. King and Ellis [1] have found that there is no Bianchi type I tilted model if it has been obtained under the assumption that the matter takes the perfect fluid form 𝑇_𝑖𝑗=∈+𝑝𝑣_𝑖𝑣_𝑗+𝑝𝑔^𝑗_𝑖,𝑣_𝑖𝑣^𝑗=−1∈>0,𝑝>0, (1) where 𝑣^𝑖 is the velocity flow vector and ∈, 𝑝 are the density and pressure of the fluid, respectively.

A realistic treatment of the problem requires the consideration of material distribution other than the perfect fluid. It is well known that in early stages of the universe when radiation in the form of photon as well as neutrino decoupled from matter, it behaved like a viscous fluid. The possibilities of bulk viscosity-driven inflationary solution of full Israel-Stewart theory in different cases are discussed by Zimdahl [5]. Murphy [6] has considered a Robertson-Walker universe with cosmological fluid possessing bulk viscosity, and the coefficient of bulk viscosity has been assumed to be proportional to the density. He has found a nonsingular solution of the Einstein’s field equation claiming that the initial singularity can be avoidable by introducing the concept of bulk viscosity. Although Belinskii and Khalatnikov [7] later criticized Murphy’s model for corresponding to very peculiar parameter choices, the model has nevertheless continued to attract interest during the years.

After the introduction by Guth [8] of the inflationary cosmological model, it has been pointed out that sources for viscosity are present in the cosmic continuum as the inflation inducing phase transition starts. Bulk viscosity associated with the grand unified theory (GUT) phase transition can lead to sufficient inflation (exponential or generalized), independent of the details of the phase transition, provided it is small [9]. A great amount of work has been done in order to study Bianchi type I cosmological models with both linear and nonlinear viscosity. An extensive review of the subject can be found in the paper of Gron [10], who has studied and carried out further the research on viscous cosmological models. Mainly he has studied inflationary cosmological models of Bianchi type I with shear, bulk, and nonlinear viscosity to a great extent.

Bali [11] discussed expanding and rotating magneto viscous fluid cosmological model in general relativity. He has obtained a cosmological model of Bianchi type I in which the distribution consists of an electrically neutral viscous fluid with an infinite electrical conductivity in the pressure of magnetic field. Patel and Koppar [12] derived four cosmological models having nonzero expansion and shear. One of them has nonzero constant shear viscosity coefficient.

Bianchi type I models with bulk and shear viscosity and an equation of state state𝑝=(𝛾−1)𝜌, with 1≤𝛾≤2 were investigated by Belinskii and Khalatnikov [13]. Models with the coefficient of shear viscosity proportional to the density have been analyzed by Banerjee and Santos [14]. Tilted Bianchi type I cosmological models in the presence a perfect and bulk viscous fluid with heat flow are derived by Pradhan and Pandey [15] and Pradhan and Rai [16], respectively.

Beside this, 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. Bali and Sharma [17] have investigated tilted Bianchi type I dust fluid and shown that model has cigar-type singularity when 𝑇=0. Concerning the tilted perfect fluid models, Bradley [18] have stated that there does not exist tilted dust self-similar model. Bagora et al. [19, 20] have discussed tilted Bianchi type I cosmological models in different context with dust fluid.

Motivated by these researches, we have studied tilted homogeneous cosmological model for dust fluid in the presence and absence of bulk viscosity. It has been shown that the tilted nature of the model is preserved due to bulk viscosity. We assume that 𝜁𝜃=𝐾 (constant), where 𝜁 is the coefficient of bulk viscosity and 𝜃 is the expansion in the model. It has been assumed that the expansion in the model is only in two directions; that is, one of the components of Hubble parameters (𝐻₁=𝐴₄/𝐴) is zero. The physical and geometrical aspects of the model in the presence and absence of bulk viscosity are also discussed.

2. The Metric and Field Equations

We consider the Bianchi type I metric in the form 𝑑𝑠²=−𝑑𝑡²+𝑑𝑥²+𝐵²𝑑𝑦²+𝐶²𝑑𝑧², (2) where 𝐵 and 𝐶 are functions of “𝑡” alone.

The energy-momentum tensor for perfect fluid distribution with heat conduction given by Ellis [21] and for bulk viscosity given by Landau and Lifshitz [22] is given by 𝑇^𝑗_𝑖=𝜌+𝑝𝑣_𝑖𝑣^𝑗+𝑝𝑔^𝑗_𝑖+𝑞_𝑖𝑣^𝑗+𝑣_𝑖𝑞^𝑗−𝜁𝜃𝑔^𝑗_𝑖+𝑣_𝑖𝑣^𝑗, (3) together with 𝑔_𝑖𝑗𝑣^𝑖𝑣^𝑗𝑞=−1,_𝑖𝑞^𝑗𝑞>0,_𝑖𝑣^𝑖=0, (4)(5) In the above, 𝑝 is the isotropic pressure, 𝜌 the matter density, and 𝑞_𝑖 the heat conduction vector orthogonal to 𝑣^𝑖. The fluid flow vector 𝑣^𝑖 has the components (sinh𝜆,0,0,cosh𝜆) satisfying (4), with 𝜆 being the tilt angle.

The Einstein’s field equation is𝑅^𝑗_𝑖−12𝑅𝑔^𝑗_𝑖=−8𝜋𝑇^𝑗_𝑖,(𝑐=𝐺=1). (6) The field equation for the line element (2) leads to 𝐵₄₄𝐵+𝐶₄₄𝐶+𝐵₄𝐶₄𝐵𝐶=−8𝜋𝜌+𝑝sinh²𝜆+𝑝+2𝑞₁sinh𝜆−𝐾cosh²𝜆,𝐶₄₄𝐶,𝐵=−8𝜋𝑝−𝐾₄₄𝐵,𝐵=−8𝜋𝑝−𝐾₄𝐶₄−𝐵𝐶=−8𝜋𝜌+𝑝cosh²𝜆+𝑝−2𝑞₁sinh𝜆−𝐾sinh²𝜆,𝜌+𝑝sinh𝜆cosh𝜆+𝑞₁cosh𝜆+𝑞₁sinh²𝜆cosh𝜆−𝐾sinh𝜆cosh𝜆=0, (7)(8)(9)(10)(11) where the suffix “4” stands for ordinary differentiation with respect to the cosmic time “𝑡” alone.

3. Solutions of Field Equations

Equations from (7) to (11) are five equations in six unknown 𝐵, 𝐶, 𝜌, 𝑝, 𝑞₁, and 𝜆. For the complete determination of these quantities, we assume that the model is filled with dust fluid which leads to 𝑝=0. (12)

Also, we assume that 𝜁𝜃=𝐾. (13)

The condition 𝜁𝜃=𝐾 is due to the peculiar characteristic of the bulk viscosity. It acts like a negative energy field in an expanding universe (Johri and Sudarshan [23]); that is, 𝜁𝜃=𝐾. According to that, expansion is inversely proportional to bulk viscosity.

From (7) and (10), we have 𝐵₄₄𝐵+𝐶₄₄𝐶+2𝐵₄𝐶₄.𝐵𝐶=8𝜋𝜌−𝑝+𝐾 (14) Using the condition of dust fluid from (12) in (14), we have 𝐵₄₄𝐵+𝐶₄₄𝐶+2𝐵₄𝐶₄𝐵𝐶=8𝜋𝜌+8𝜋𝐾. (15) Equations (8) and (9) lead to 𝐵₄₄𝐵−𝐶₄₄𝐶=0. (16)

Let us assume that 𝐵𝐵𝐶=𝜇,𝐶=𝜈. (17) With the help of (17), (16) leads to 𝜈₄𝜈=𝑎𝜇, (18) where “𝑎” is a constant of integration.

Again (8) and (9) lead to 𝐵₄₄𝐵+𝐶₄₄𝐶=−16𝜋𝑝+16𝜋𝐾. (19) Using condition (12) in (19), we have 𝐵₄₄𝐵+𝐶₄₄𝐶=16𝜋𝐾. (20) Using (17) and (18) in (20), we have 2𝜇₄₄−𝜇²₄𝜇=−𝑎²𝜇+32𝜋𝐾𝜇. (21) This leads to 𝜇²₄=𝑎²+32𝜋𝐾𝜇²+𝑏𝜇, (22) where “𝑏” is a constant of integration and 𝜇₄=𝑓(𝜇).

Equation (18) leads to log𝜈=𝑎𝑑𝜇𝜇𝑎²+32𝜋𝐾𝜇².+𝑏𝜇 (23) Hence, the metric (2) reduces to the form 𝑑𝑠²=−𝑑𝜇²𝑓²+𝑑𝑥²+𝜇𝜈𝑑𝑦²+𝜇𝜈𝑑𝑧². (24) This leads to 𝑑𝑠²=−𝑑𝑇²𝑎²+32𝜋𝐾𝑇²+𝑏𝑇+𝑑𝑋²+𝑇𝜈𝑑𝑌²+𝑇𝜈𝑑𝑍², (25) where 𝑥=𝑋, 𝑦=𝑌, 𝑧=𝑍, and 𝜈 is determined by (23) with 𝜇=𝑇.

In the absence of bulk viscosity the metric (25) reduces to 𝑑𝑠²=−𝑑𝑇²𝑎²+𝑏𝑇+𝑑𝑋²+𝑇𝜈𝑑𝑌²+𝑇𝜈𝑑𝑍², (26) where 𝜈 is determined by (23) with 𝜇=𝑇 and 𝐾=0.

4. Some Physical and Geometrical Features

The matter density 𝜌 for the model (25) is given by 18𝜋𝜌=().2𝑇𝑏+48𝜋𝐾𝑇 (27)

The tilt angle 𝜆 is given by cosh𝜆=48𝜋𝐾𝑇+𝑏,32𝜋𝐾𝑇sinh𝜆=16𝜋𝐾𝑇+𝑏.32𝜋𝐾𝑇 (28) The scalar of expansion 𝜃 calculated for the flow vector 𝑣^𝑖 for the model (25) is given by ()𝜃=96𝜋𝐾𝑇+𝑏8𝑇²𝑎²+32𝜋𝐾𝑇²+𝑏𝑇().2𝜋𝐾48𝜋𝐾𝑇+𝑏 (29) The components of flow vector 𝑣^𝑖 and heat conduction vector 𝑞_𝑖 for the model (25) are given by 𝜈¹=16𝜋𝐾𝑇+𝑏,𝑣32𝜋𝐾𝑇₄=48𝜋𝐾𝑇+𝑏,𝑞32𝜋𝐾𝑇₁=−(𝑏+48𝜋𝐾𝑇)128𝜋^3/2𝐾^3/2𝑏+16𝜋𝐾𝑇,𝑞2𝑇₄=𝑏+16𝜋𝐾𝑇128𝜋^3/2𝐾^3/2𝑏+48𝜋𝐾𝑇.2𝑇 (30) The nonvanishing components of shear tensor (𝜎_𝑖𝑗) and rotation tensor (𝜔_𝑖𝑗) are given by 𝜎₁₁=−(𝑏+24𝜋𝐾𝑇)192(𝑏+48𝜋𝐾𝑇)𝑎²+32𝜋𝐾𝑇²+𝑏𝑇𝜋³𝐾³𝑇⁵,𝜎₁₄=−(𝑏+24𝜋𝐾𝑇)192(𝑏+16𝜋𝐾𝑇)𝑎²+32𝜋𝐾𝑇²+𝑏𝑇𝜋³𝐾³𝑇⁵,𝜔₁₄=18𝑇^3/2𝑎²+32𝜋𝐾𝑇²+𝑏𝑇().2𝜋𝐾𝑏+16𝜋𝐾𝑇 (31) Thus, 𝜎₁₁𝜈¹+𝜎₁₄𝜈⁴=0. (32) Similarly, 𝜔₁₁𝜈¹+𝜔₁₄𝜈⁴=0. (33) The physical significance of conditions (32) and (33) is explained by Ellis [24]. The shear tensor (𝜎_𝑖𝑗) determines the distortion arising in the fluid flow, leaving the volume invariant. The direction of principal axis is unchanged by the distortion, but all other directions are changed. Thus, we have 𝜎_𝑖𝑗𝜈^𝑗=0, which leads to 𝜎₁₁𝜈¹+𝜎₁₄𝜈⁴=0∵𝜈₁≠0,𝜈₄≠0. (34) Shear (𝜎) is given by 𝜎²=12𝜎_𝑖𝑗𝜎^𝑖𝑗. (35) Thus, 𝜎²≥0 and 𝜎=0⇔𝜎_𝑖𝑗=0.

The vorticity tensor (𝜔_𝑖𝑗) determines a rigid rotation of cluster of galaxies with respect to a local inertial rest frame. Thus, we have 𝜔_𝑖𝑗=𝜂_{𝑖𝑗𝑘ℓ}𝜔^𝑘𝜔^ℓ, (36) where 𝜂_{𝑖𝑗𝑘ℓ} is pseudotensor and 𝜔^𝑖=(1/2)𝜂_{𝑖𝑗𝑘ℓ}𝜈_𝑗𝜔_𝑘ℓ.

Thus, 𝜔_𝑖𝑗𝜈^𝑗=0.

This leads to 𝜔₁₁𝜈¹+𝜔₁₄𝜈⁴=0∵𝜈₁≠0,𝜈₄≠0. (37) The magnitude of 𝜔_𝑖𝑗 is 𝜔 and is defined as 𝜔²=12𝜔_𝑖𝑗𝜔^𝑖𝑗. (38) Also 𝜔=0⇔𝜔_𝑖𝑗=0.

4.1. Special Model I

When the bulk viscosity is absent, that is, 𝐾=0, (22) reduces to 𝜇²₄=𝑎²+𝑏𝜇. (39) Putting 𝑎²+𝑏𝜇=𝜉 in (39), we have 𝑏𝑡𝜇=²−𝑀²4, (40) where 𝑀=2𝑎/𝑏.

Again by (23), we have 𝜈=𝑁𝑡𝑏+2𝑎𝑡𝑏−2𝑎, (41) where “𝑁” is a constant of integration.

The metric (2) reduces to 𝑑𝑠²=−𝑑𝑇²𝑎²+𝑏𝑇+𝑑𝑋²+𝑁𝑏𝑡²−𝑀²(𝑡𝑏+2𝑎)4(𝑡𝑏−2𝑎)𝑑𝑌²+𝑏𝑡²−𝑀²(𝑡𝑏−2𝑎)4𝑁(𝑡𝑏+2𝑎)𝑑𝑍². (42) For the model (42), density 𝜌 and tilt angle 𝜆 are given by 2𝜌=8𝜋𝑡²−𝑀²,1cosh𝜆=𝑏𝑏²−4𝑎²21−𝑡²,1sinh𝜆=𝑏2𝑏²𝑡²−𝑏²−4𝑎²21−𝑡². (43) The components of flow vector 𝑣^𝑖 and heat conduction vector 𝑞_𝑖 for the model (42) are given by 𝜈¹=16𝜋𝐾𝑇+𝑏,𝑣32𝜋𝐾𝑇₄=48𝜋𝐾𝑇+𝑏,𝑞32𝜋𝐾𝑇₁=−(𝑏+48𝜋𝐾𝑇)128𝜋^3/2𝐾^3/2𝑏+16𝜋𝐾𝑇,𝑞2𝑇₄=𝑏+16𝜋𝐾𝑇128𝜋^3/2𝐾^3/2𝑏+48𝜋𝐾𝑇.2𝑇 (44) The scalar of expansion 𝜃 and the nonvanishing components of (𝜎_𝑖𝑗) and (𝜔_𝑖𝑗) are given by 𝜃=−𝑡4𝑎²−2𝑏²+𝑏²𝑡²𝑏1−𝑡²𝑏²𝑡²−4𝑎²𝑏²−4𝑎²21−𝑡²,𝜎₁₁=𝑡𝑏²−4𝑎²2𝑏²𝑡²−𝑏²−4𝑎²3𝑏³1−𝑡²²𝑏²𝑡²−4𝑎²𝑏²−4𝑎²21−𝑡²,𝜎₁₄=−𝑡𝑏²−4𝑎²2𝑏²𝑡²−𝑏²−4𝑎²3𝑏³1−𝑡²²𝑏²𝑡²−4𝑎²2𝑏²𝑡²−𝑏²−4𝑎²21−𝑡²,𝜔₁₄=𝑏²−4𝑎²3𝑏²−2𝑏²𝑡²−4𝑎²4𝑏³1−𝑡²²22𝑏²𝑡²−𝑏²−4𝑎²1−𝑡². (45)

4.2. Special Model II

When bulk viscosity is present then we find the model in terms of “𝑡”; for this we assume that 𝑏=0, 𝑎=0.

Then we have 𝜇=𝑘₁𝑒^{(√32𝜋𝐾)𝑡},𝜈=constant. (46) The metric (2) becomes 𝑑𝑠²=−𝑑𝑡²+𝑑𝑥²+𝑘₁𝑘₂𝑒^{(√32𝜋𝐾)𝑡}𝑑𝑦²+𝑘₁𝑒^{(√32𝜋𝐾)𝑡}𝑘₂𝑑𝑧². (47) The expansion 𝜃 is given by √𝜃=32𝜋𝐾232𝜋𝐾𝑒^{(√32𝜋𝐾)𝑡}+1^3/212. (48) The tilt angle 𝜆 is given by cosh𝜆=1232𝜋𝐾𝑒^{(√32𝜋𝐾)𝑡}.+1 (49) Here, at 𝑡=0, the expansion is constant and at 𝑡=∞ it vanishes, that is, zero.

5. Discussion

The model starts with a big bang from its initial stage at 𝑇=0 and continues to expand till 𝑇=∞. The model has point-type singularity at 𝑇=0 (MacCallum [25]). The model represents shearing and rotating universe in general and rotation goes on decreasing as time increases. Since Lim_𝑇→∞𝜎/𝜃≠0, then the model does not approach isotropy for large values of 𝑇. Density 𝜌→0 as 𝑇→∞ and 𝜌→∞ as 𝑇→0. When 𝑇→0, 𝑞¹→∞ and 𝑞⁴→∞. Also 𝑞¹ and 𝑞⁴ tend to zero as 𝑇→0. At 𝑇=0, the Hubble parameters tend to infinite at the time of initial singularity of vanishing as 𝑇→∞.

The effect of bulk viscosity is to produce change in perfect fluid and exhibit essential influence on the character of the solution. All the physical characters of the model (42) remain finite and regular for the entire range of variable −∞<𝑡<∞. Therefore, this model is free of singularity and in this case the model has singularity-free solution. Also this model is throughout a rotating model.