# Dust Fluid Cosmological Model

Authors: Anita Bagora (Menaria); Rakeshwar Purohit
Journal: ISRN Astronomy and Astrophysics
 (2013)

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## Abstract

Bianchi type I tilted bulk viscous fluid cosmological model filled with dust fluid is investigated. We assume thatζθ=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.


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## Body

## 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
(1)Tij=(∈+p)vivj+pgij,vivj=-1∈>0,p>0,
where vi 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 statestatep=(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 thatζθ=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(2)ds2=-dt2+dx2+B2dy2+C2dz2,
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
(3)Tij=(ρ+p)vivj+pgij+qivj+viqj-ζθ(gij+vivj),
together with
(4)gijvivj=-1,(5)qiqj>0,qivi=0,
In the above, p is the isotropic pressure, ρ the matter density, and qi the heat conduction vector orthogonal to vi. The fluid flow vector vi has the components (sinhλ,0,0,coshλ) satisfying (4), with λ being the tilt angle.The Einstein’s field equation is(6)Rij-12Rgij=-8πTij,(c=G=1).
The field equation for the line element (2) leads to
(7)B44B+C44C+B4C4BC=-8π[(ρ+p)sinh2λB44B+C44C+B4C4BC=-8π+p+2q1sinhλ-Kcosh2λ],(8)C44C=-8π(p-K),(9)B44B=-8π(p-K),(10)B4C4BC=-8π[-(ρ+p)cosh2λ+pB4C4BC=-8π-2q1sinhλ-Ksinh2λ],(11)(ρ+p)sinhλcoshλ+q1coshλ+q1sinh2λcoshλ-Ksinhλcoshλ=0,
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
(12)p=0.Also, we assume that(13)ζθ=K.The conditionζθ=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
(14)B44B+C44C+2B4C4BC=8π[ρ-p+K].
Using the condition of dust fluid from (12) in (14), we have
(15)B44B+C44C+2B4C4BC=8πρ+8πK.
Equations (8) and (9) lead to
(16)B44B-C44C=0.Let us assume that(17)BC=μ,BC=ν.
With the help of (17), (16) leads to
(18)ν4ν=aμ,
where “a” is a constant of integration.Again (8) and (9) lead to
(19)B44B+C44C=-16πp+16πK.
Using condition (12) in (19), we have
(20)B44B+C44C=16πK.
Using (17) and (18) in (20), we have
(21)2μ44-μ42μ=-a2μ+32πKμ.
This leads to
(22)μ42=a2+32πKμ2+bμ,
where “b” is a constant of integration and μ4=f(μ).Equation (18) leads to
(23)logν=∫adμμa2+32πKμ2+bμ.
Hence, the metric (2) reduces to the form
(24)ds2=-dμ2f2+dx2+μνdy2+μνdz2.
This leads to
(25)ds2=-dT2a2+32πKT2+bT+dX2+TνdY2+TνdZ2,
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
(26)ds2=-dT2a2+bT+dX2+TνdY2+TνdZ2,
where ν is determined by (23) with μ=T and K=0.


## 4. Some Physical and Geometrical Features

The matter densityρ for the model (25) is given by
(27)8πρ=12T(b+48πKT).The tilt angleλ is given by
(28)coshλ=48πKT+b32πKT,sinhλ=16πKT+b32πKT.
The scalar of expansion θ calculated for the flow vector vi for the model (25) is given by
(29)θ=(96πKT+b)8T2(a2+32πKT2+bT)2πK(48πKT+b).
The components of flow vector vi and heat conduction vector qi for the model (25) are given by
(30)ν1=16πKT+b32πKT,v4=48πKT+b32πKT,q1=-(b+48πKT)128π3/2K3/2b+16πKT2T,q4=b+16πKT128π3/2K3/2b+48πKT2T.
The nonvanishing components of shear tensor (σij) and rotation tensor (ωij) are given by
(31)σ11=-(b+24πKT)192(b+48πKT)(a2+32πKT2+bT)π3K3T5,σ14=-(b+24πKT)192(b+16πKT)(a2+32πKT2+bT)π3K3T5,ω14=18T3/2a2+32πKT2+bT2πK(b+16πKT).
Thus,
(32)σ11ν1+σ14ν4=0.
Similarly,
(33)ω11ν1+ω14ν4=0.
The physical significance of conditions (32) and (33) is explained by Ellis [24]. The shear tensor (σij) 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 σijνj=0, which leads to
(34)σ11ν1+σ14ν4=0(∵ν1≠0,ν4≠0).
Shear (σ) is given by
(35)σ2=12σijσij.
Thus,  σ2≥0 and σ=0⇔σij=0.The vorticity tensor (ωij) determines a rigid rotation of cluster of galaxies with respect to a local inertial rest frame. Thus, we have
(36)ωij=ηijkℓωkωℓ,
where ηijkℓ is pseudotensor and ωi=(1/2)ηijkℓνjωkℓ.Thus,ωijνj=0.This leads to(37)ω11ν1+ω14ν4=0(∵ν1≠0,ν4≠0).
The magnitude of ωij is ω and is defined as
(38)ω2=12ωijωij.
Also ω=0⇔ωij=0.


### 4.1. Special Model I

When the bulk viscosity is absent, that is,K=0, (22) reduces to
(39)μ42=a2+bμ.
Putting a2+bμ=ξ in (39), we have
(40)μ=b(t2-M2)4,
where   M=2a/b.Again by (23), we have
(41)ν=N(tb+2atb-2a),
where “N” is a constant of integration.The metric (2) reduces to
(42)ds2=-dT2a2+bT+dX2+Nb(t2-M2)(tb+2a)4(tb-2a)dY2+b(t2-M2)(tb-2a)4N(tb+2a)dZ2.
For the model (42), density ρ and tilt angle λ are given by
(43)ρ=28π(t2-M2),coshλ=1bb2-4a22(1-t2),sinhλ=1b2b2t2-b2-4a22(1-t2).
The components of flow vector vi and heat conduction vector qi for the model (42) are given by
(44)ν1=16πKT+b32πKT,v4=48πKT+b32πKT,q1=-(b+48πKT)128π3/2K3/2b+16πKT2T,q4=b+16πKT128π3/2K3/2b+48πKT2T.
The scalar of expansion θ and the nonvanishing components of (σij) and (ωij) are given by
(45)θ=-t(4a2-2b2+b2t2)b(1-t2)(b2t2-4a2)b2-4a22(1-t2),σ11=t(b2-4a2)(2b2t2-b2-4a2)3b3(1-t2)2(b2t2-4a2)b2-4a22(1-t2),σ14=-t(b2-4a2)(2b2t2-b2-4a2)3b3(1-t2)2(b2t2-4a2)2b2t2-b2-4a22(1-t2),ω14=(b2-4a2)(3b2-2b2t2-4a2)4b3(1-t2)22(2b2t2-b2-4a2)(1-t2).


### 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(46)μ=k1e(32πK)t,ν=constant.
The metric (2) becomes
(47)ds2=-dt2+dx2+k1k2e(32πK)tdy2+k1e(32πK)tk2dz2.
The expansion θ is given by
(48)θ=32πK2(32πKe(32πK)t+1)3/212.
The tilt angle λ is given by
(49)coshλ=12(32πKe(32πK)t+1).
Here, at t=0, the expansion is constant and at t=∞ it vanishes, that is, zero.


## 4.1. Special Model I

When the bulk viscosity is absent, that is,K=0, (22) reduces to
(39)μ42=a2+bμ.
Putting a2+bμ=ξ in (39), we have
(40)μ=b(t2-M2)4,
where   M=2a/b.Again by (23), we have
(41)ν=N(tb+2atb-2a),
where “N” is a constant of integration.The metric (2) reduces to
(42)ds2=-dT2a2+bT+dX2+Nb(t2-M2)(tb+2a)4(tb-2a)dY2+b(t2-M2)(tb-2a)4N(tb+2a)dZ2.
For the model (42), density ρ and tilt angle λ are given by
(43)ρ=28π(t2-M2),coshλ=1bb2-4a22(1-t2),sinhλ=1b2b2t2-b2-4a22(1-t2).
The components of flow vector vi and heat conduction vector qi for the model (42) are given by
(44)ν1=16πKT+b32πKT,v4=48πKT+b32πKT,q1=-(b+48πKT)128π3/2K3/2b+16πKT2T,q4=b+16πKT128π3/2K3/2b+48πKT2T.
The scalar of expansion θ and the nonvanishing components of (σij) and (ωij) are given by
(45)θ=-t(4a2-2b2+b2t2)b(1-t2)(b2t2-4a2)b2-4a22(1-t2),σ11=t(b2-4a2)(2b2t2-b2-4a2)3b3(1-t2)2(b2t2-4a2)b2-4a22(1-t2),σ14=-t(b2-4a2)(2b2t2-b2-4a2)3b3(1-t2)2(b2t2-4a2)2b2t2-b2-4a22(1-t2),ω14=(b2-4a2)(3b2-2b2t2-4a2)4b3(1-t2)22(2b2t2-b2-4a2)(1-t2).


## 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(46)μ=k1e(32πK)t,ν=constant.
The metric (2) becomes
(47)ds2=-dt2+dx2+k1k2e(32πK)tdy2+k1e(32πK)tk2dz2.
The expansion θ is given by
(48)θ=32πK2(32πKe(32πK)t+1)3/212.
The tilt angle λ is given by
(49)coshλ=12(32πKe(32πK)t+1).
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 atT=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 LimT→∞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.



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*Source: 293298-2013-02-17.xml*