Bianchi Type-III and Kantowski-Sachs Universes with Wet Dark Fluid

The Bianchi type-III and Kantowski-Sachs (KS) Universes filled with dark energy from a wet dark fluid has been considered. A new Equation of state for the dark energy component of the universe has been used. It is modeled on the Equation of state = ( ) p     which can describe a liquid, for example water. The exact solutions to the corresponding field Equations are obtained in quadrature form. The solution for constant deceleration parameter have been studied in detail for power-law and exponential forms both. The case = 0  , = 1  and 1 = 3  have been also analysed.


Introduction
The nature of the dark energy component of the universe [1][2][3] remains one of the deepest mysteries of cosmology.There is certainly no lack of candidates: cosmological constant, quintessence [4][5][6], k-essence [7][8][9], phantom energy [10][11][12].Modifications of the Friedmann Equation such as Cardassian expansion [13,14] as well as what might be derived from brane cosmology [15][16][17] have also been used to explain the acceleration of the universe.A particular case of the linear Equation of state has used in the cosmological context by Xanthopuolos [18], he considered space-times with two hypersueface orthogonal, spacelike, commuting killing fields.
In this work, we use Wet Dark Fluid (WDF) as a model for dark energy.This model is in the spirit of the generalized Chaplygin gas (GCG) [19], where a physically motivated Equation of state is offered with properties relevant for the dark energy problem.Here the motivation stems from an empirical Equation of state proposed by Tait [20] and Hayword [21] to treat water and aqueous solution.The Equation of state for WDF is very simple, and is motivated by the fact that it is a good approximation for many fluids, including water, in which the internal attraction of the molecules makes negative pressures possible.One of the virtues of this model is that the square of the sound speed, 2 s c , which depends on p    , can be positive (as opposed to the case of phantom energy, say), while still giving rise to cosmic acceleration in the current epoch.
We treat Equation (1) as a phemenological Equation [22].Holman et al. [23] have shown that this model can be made consistent with the most recent SNIa data [24], the WMAP results [25,26] as well as constraints coming from measurements of the matter power spectrum [27].
The parameters  and   are taken to be positive and we restrict ourselves to 0 1    .Note that if s c 2 denotes the adiabatic sound speed in WDF, then = s c  .(refer Babichev et al. [28]).
To find the WDF energy density, we use the energy conservation Equation From Equation of state (1) and using 3 = H V V  in above Equation, we have where C is a constant of integration.Here V is volume expansion.
WDF naturally includes two components: a piece that behaves as a cosmological constant as well as a standard fluid with an Equation of state = p  0 .We can show that if we take , this fluid will not violate the strong energy condition Chaubey and Chaubey et al. ([29,30]) have studied some anisotropic cosmological universes with wet dark fluid.In this paper we study the Bianchi type-III and Kantowski-Sachs Universes with matter term with dark energy treated as a Dark Fluid satisfying the Equation of state (1).The solution has been obtained in the quadrature form.The models with constant deceleration parameter have been studied in detail.

Bianchi Type -III Universe
We take Bianchi type-III metric in form is the gravitational constant and overhead dot denotes differentiation with respect to .


t The energy-momentum tensor of the source is given by   = .
where is the flow vector satisfying In a co-moving system of coordinates, from Equation (9) we find ) Now using Equation (11) in Equations ( 6)-( 8) we obtain Let be a function of defined by V t Now adding three times Equation ( 14), two times Equatin (13) in Equation ( 12), we get From Equations ( 15) and ( 16) we have The conservational law for the energy-momentum tensor gives From Equations ( 18) and ( 19) we have with being an integration constant. 1 Rewriting (18) in the form and taking into account that the pressure and the energy density obeying an equation of state of type , we conclude that WDF  and WDF , hence the right hand side of the Equation ( 17) is a function of only.
From the mechanical point of view Equation ( 22) can be interpreted as Equation of motion of a single particle with unit mass under the force Here  can be viewed as energy and   U V as the potential of the force F .Compairing the Equations (20) and (23) we find (13) 4 2  Finally, we write the solution to the Equation (20) in quadrature form where the integration constant 0 t can be taken to be zero, since it only gives a shift in time.

Kantowski-Sachs Universe
We take Kantowski-Sachs metric in form Here is the gravitational constant and overhead dot denotes differentiation with respect to . t Now using Equations ( 9)- (11) in Equations ( 30)-( 32) we obtain From Equations ( 36) and (37) we have After simplification, we get After simlification, we get 0 2 ( 1 )

Bianchi Type -III Universe
Case 1: We consider these subcases separately.
Case I (a) when  15) and (45), we get From Equation ( 3) and ( 45) we have and from Equation ( 1) and ( 49) we get The physical quantities of observational interest in cosmology are the expansion scalar  , the mean anisotropy parameter A , the shear scalar 2   and the deceleration parameter .They are defined as [31,32], With the use of Equations ( 51)-( 54) we can express the physical quantities as For large , the shear dies out.t Case I (b) when From Equations ( 15) and (60), we get From Equations ( 3) and (60), we have and from Equations ( 1) and ( 63) we get With the use of Equations ( 51)-( 54) we can express the physical quantities as For large , the shear dies out.t Case I (c) when Then Equation (46) reduces to From Equations ( 15) and (70), we get Copyright © 2011 SciRes.
From Equations ( 3) and (70) we have and from Equations ( 1) and ( 73) we get = 0 With the use of Equations ( 51)-( 54) we can express the physical quantities as For large , the shear dies out.
which gives Then for small (i.e.near singularity Then Equation (80) reduces to From Equations ( 15) and (82), we get From Equations ( 3) and (82) we have   and from Equations ( 1) and ( 85) we get With the use of Equations ( 51)-( 54) we can express the physical quantities as For large cosmic time, the shear dies out and ,


and the model reduces to vacuum.
which gives 3/ 2 3 16 12 = s i n h 3 16 3 Then for small (i.e.near singularity ), Then Equation (92) reduces to From Equations ( 15) and (94), we get From Equations ( 3) and (96) we have and from Equations ( 1) and (97) we get  28) reduces to which gives From Equations ( 15) and (104), we get From Equations ( 3) and (104) we have and from Equations ( 1) and (107) we get With the use of Equations ( 51)-(54) we can express the physical quantities as For large cosmic time, the shear dies out.
With the use of Equations ( 51)-(54) we can express the physical quantities as The model has no singularity.

Case II (b)
  Then for small (i.e.near singularity ), Then Equation (114) reduces to From Equations ( 3) and (126), we have and from Equations ( 1) and (129), we get With the use of Equations ( 51)-(54) we can express the physical quantities as The model has no singularity.

IJAA
With the use of Equations ( 51)-(54) we can express the physical quantities as 2 ) 83 The model has no singularity.

Kantowski-Sachs Universe
Case 1: From Equations ( 36) and ( 146), we get From Equations ( 3) and (146) we have and from Equations ( 1) and ( 149) we get With the use of Equations ( 51)-( 54) we can express the physical quantities as c o s2 4 = 9 3 sin 2 16 8 which gives Then for small (i.e.near singularity ), t = 0 t Then Equation (156) reduces to From Equations ( 36) and (158), we get From Equations ( 3) and ( 158) we have   and from Equations ( 1) and ( 161) we get = 0 With the use of Equations ( 51)-( 54) we can express the physical quantities as Copyright © 2011 SciRes.IJAA which gives Then for small (i.e.near singularity ), Then Equation (168) reduces to From Equations (36) and (170), we get From Equations ( 3) and (170) we have and from Equations ( 1) and (173) we get = 0 With the use of Equations ( 51)-( 54) we can express the physical quantities as For large cosmic time, the shear dies out and ,p 0   and the model reduces to vacuum.
From Equations ( 36) and (180), we get From Equations ( 3) and (180) we have and from Equations ( 1) and ( 183) we get = 0 WDF p (184) With the use of Equations ( 51)-(54) we can express the physical quantities as For large cosmic time, the shear dies out.
Case II (a)   From Equations ( 3) and (191), we have and from Equations ( 1) and (195), we get With the use of Equations ( 51)-( 54) we can express the physical quantities as The model has no singularity.

IJAA
Then Equation (192) reduces to The model has no singularity.

Case II (c)
  Then for small (i.e.near singularity ), From Equations ( 3) and (212), we have and from Equations ( 1) and (215), we get With the use of Equations ( 51) -( 54) we can express the physical quantities as ) The model has no singularity.

Models with Constant Deceleration Parameter
and from ( 1) and (224), we get Case I Power-Law From ( 3) and ( 221), we have and from ( 1) and (232), we get With the use of Equations ( 51)-( 54) we can express the physical quantities as Case I (c) When From ( 3) and (221), we have and from ( 1) and (240), we get and from ( 1) and (249), we get With the use of Equations ( 51)-(54) we ca the physical quantities as = 1 q  The model has no singularity.

Conclusions
The Bianchi type-II s have been consi rse e for the Dark Energy component of the universe (known as dark wet fluid).The solution has been obtained in quadrature form.The models with constant deceleration parameter have been discussed in detail.The behaviour of the models for large time have been analyzed.
SciRes.IJAAWith the use of Equations (51)-(54) we can express the physical quantities as 3 For large cosmic time, the shear dies out and ,  and the model reduces to vacuum.
225) Here we take = b V at , (221) where and b are constants, a With the use of Equations (51) -(54) we can express the physical quantities as Here we discuss three interesing cases = -Sachs (KS) unive dered for a new Equation of stat