Astrophys Space Sci 323 (2009) 407-414

Viscous fluid cosmology with a cosmological constant C.P. Singh and Suresh Kumar* *E-mail: [email protected], Webpage: https://sites.google.com/site/sureshkumaryd/ Note: This version of the paper matches the version published in Astrophysics and Space Science. The definitive version is available at Astrophys Space Sci 323 (2009) 407-414. Abstract: A spatially homogeneous and anisotropic Bianchi type-I cosmological model filled dissipative fluid is studied in the presence of a cosmological constant. Exact solutions of the field equations are obtained with constant as well as time-varying cosmological constant by using a law of variation for mean Hubble parameter, which is related to average scale factor and yields a constant value for deceleration parameter. A barotropic equation of state (p = (γ − 1)ρ) together with a linear relation between shear viscosity and expansion scalar, is assumed. Graphical analysis of the anisotropy parameter reveals that the anisotropy of the universe damps out faster due to the viscous nature of cosmic fluid. However, the presence of viscous term does not change the fundamental nature of initial singularity. The thermodynamical aspects of the model are also discussed in detail. Key Words: Hubble parameter; Deceleration parameter; Cosmological constant; Viscous fluid . 1. Introduction Dissipative effects including both the bulk and shear viscosity, play a significant role in the early evolution of the universe. From a physical point of view, the inclusion of dissipative terms in the energy-momentum tensor of the cosmological fluid, seems appropriate generalization of the matter term in the gravitational field equations. A number of authors have investigated the role of viscosity in the evolution of universe. Misner [1] examined Bianchi type-I models in the presence of viscous term and concluded that viscosity of neutrinos can essentially reduce the initial anisotropy of the universe. A critical analysis of Misner’s work was carried out by Poroshkevich et al. [2], Stewart and Collins [3] and Stewart [4]. They deduced that the isotropization by decoupling neutrinos could have lead to the isotropy level observed today, only if at the end of the lepton era, the degree of anisotropy was already small. Grøn [5] has reviewed viscous cosmological models and deduced that viscosity plays an important role in the the process of isotropization of the universe. Apart from these qualitative discussions, suitable viscous fluid cosmological models have been considered in different contexts by several authors such as Pavon et al. [6], Pavon and Zimdahl [7], Burd and Coley [8], Maartens [9], Chimento et al. [10,11] and Fabris et al. [12]. The nature of cosmological solutions for homogeneous Bianchi type-I model was investigated by Belinski and Khalatnikov [13] by taking into account the dissipative process due to viscosity. They showed that viscosity can not remove the cosmological singularity but can cause a qualitatively new behavior of the solutions near the singularity. Bianchi type-I solutions in the case of stiff matter with shear viscosity being the power function of energy density, were obtained by Banerjee et al. [14,15] whereas models with bulk viscosity as a power function of energy density and stiff matter were studied by Huang [16]. Krori and Mukherjee [17] studied the effect of bulk viscosity in the evolution of anisotropic Bianchi-I models. Models with both the bulk and shear viscosity were obtained by Gavrilov et al. [18]. Although Murphy [19] claimed that the introduction of bulk viscosity can avoid the initial singularity at the finite past, the results obtained by Barrow [20] showed that it is, in general, not valid. Romomo and Pavon [21] have investigated the evolution of Bianchi type-I model with viscous fluid. Mak and Harko [22] studied the dynamics of a casual bulk viscous fluid model with constantly decelerating Bianchi type-I space-time. Saha in a series of works [23-25] discussed Bianchi type-I universe with viscous fluid. Again Saha and Rikhvitsky [26] studied the influence of viscous fluid and Λ-term in the evolution of the Bianchi type-I universe. Bali and Singh [27] have studied Bianchi-I space-time with bulk viscosity and variable cosmological constant. In a recent work, Singh and Kumar [28] have obtained viscous Bianchi type-I models with constant deceleration parameter. Here our intention is to see that how the constant deceleration parameter viscous models with Bianchi type-I metric behave in the presence of a cosmological constant. Therefore, in what follows, we consider a spatially homogeneous and anisotropic Bianchi-I space-time model filled with perfect fluid containing the bulk and shear viscosity content in the presence of a cosmological constant. This work is organized as follows: The model and field equations are given in Sect.2. The field equations are solved in Sect.3 by assuming some physically relevant 1

assumptions. Exact Bianchi type-I models are presented in subsections 3.1 and 3.2 with constant as well as timevarying Λ-term and physical behavior of the models is explained in each subsection. The concluding remarks are presented in Sect.4. 2. Model and field equations The spatially homogeneous and anisotropic Bianchi-I space-time is described by the line element ds2 = −dt2 + A2 (t)dx2 + B 2 (t)dy 2 + C 2 (t)dz 2 ,

(1)

where A(t) , B(t) and C(t) are the metric functions of cosmic time t . 1 We define a = (ABC) 3 as the average scale factor of Bianchi type-I universe so that the generalized Hubble parameter in anisotropic models may be defined as ! 1 A˙ B˙ C˙ a˙ , (2) + + H= = a 3 A B C where an over dot denotes derivative with respect to the cosmic time t . Also we have H= ˙

˙

1 (H1 + H2 + H3 ), 3

(3)

˙

C A , H2 = B where H1 = A B and H3 = C are directional Hubble factors in the directions of x , y and z axes respectively. By considering a time-dependent Λ in the general relativity, the usual energy momentum tensor is modified by the addition of a term, interpreted as a vacuum contribution (in units 8πG = 1) [29,30] (vac)

Tij

= −Λ(t) gij ,

(4)

which can be regarded as the energy-momentum tensor of a perfect fluid with its energy density ρv and pressure pv satisfying the equation of state pv = −ρv = −Λ . (5) The Einstein’s field equations are then given by 1 Rij − gij R = −Tij + Λ(t) gij , 2

(6)

where Tij is the stress energy tensor of matter which, in case of viscous fluid, has the form [31]

with

Tij = (ρ + p¯)ui uj + p¯gij − ηµij ,

(7)

2 p¯ = p − (ξ − η)ui;i = p − (3ξ − 2η)H 3

(8)

µij = ui;j + uj;i + ui uα uj;α + uj uα ui;α .

(9)

and In the above equations ξ and η stand for the bulk and shear viscosity coefficients, respectively; ρ is the matter density; p is the isotropic pressure and ui is the four-velocity vector satisfying ui ui = −1. In order to conveniently specify the source , we assume the perfect gas equation of state, which may be written as p = (γ − 1)ρ, 1 ≤ γ ≤ 2. (10) In a co-moving coordinate system, where ui = δ0i , the field equations (6), for the anisotropic Bianchi type-I space-time (1) and viscous fluid distribution (7), yield ¨ C¨ B˙ C˙ A˙ B + + = −¯ p + 2η + Λ, B C BC A

(11)

A¨ C˙ A˙ B˙ C¨ + + = −¯ p + 2η + Λ, C A CA B

(12)

2

¨ A˙ B˙ C˙ A¨ B + + = −¯ p + 2η + Λ, A B AB C

(13)

A˙ B˙ B˙ C˙ C˙ A˙ + + = ρ + Λ. (14) AB BC CA The usual definitions of the dynamical scalars such as the expansion scalar (θ) and the shear scalar (σ) are considered to be 3a˙ θ = ui;i = (15) a and  !2 !2 !2  ˙ ˙ ˙ 1 A 1 B C  − 1 θ2 , σ 2 = σij σ ij =  + + (16) 2 2 A B C 6 where

σij = ui;j +


1 1 ui;k uk uj + uj;k uk ui + θ (gij + ui uj ) . 2 3

(17)

¯ is defined as The anisotropy parameter (A)

3

1X A¯ = 3 i=1



Hi − H H

2

.

(18)

The covariant divergence of (6) leads to ˙ ρ˙ = −(ρ + p)θ + ξθ2 + 4ησ 2 − Λ.

(19)

1 θ˙ = −4πG(ρ + 3¯ p) − θ2 − 2σ 2 + Λ , 3

(20)

The Raychaudhary equation reads as

which shows that for ρ + 3¯ p = 0, acceleration is initiated by the Λ-term. The energy in a co-moving volume is U = ρV . The equation for production of entropy S in a co-moving volume due to the dissipative effects in a fluid with temperature T is given by
T S˙ = U˙ + pV˙ = 3V 3ξ + 2η A¯ H 2 . (21) In a cosmic fluid where the energy density and pressure of the cosmic fluid are functions of temperature only, ρ = ρ(T ), p = p(T ) and where the cosmic fluid has no net charge, we obtain easily [5] S=

V (ρ + p) . T

(22)

From (21) and (22), we get the following expression for the entropy production rate in viscous Bianchi type-I universe:
3 3ξ + 2η A¯ H 2 S˙ = . (23) S ρ+p For a fluid obeying the equation of state (10); (22) and (23) become S=

V γρ , T

(24)


3 3ξ + 2η A¯ H 2 S˙ = . S γρ

(25)

S˙ ξ + 4η(σ 2 /θ2 ) = , S γ(ρ/θ2 )

(26)

Equation (25) can be rewritten as

3

˙

which gives the rate of change of entropy with time. Clearly SS > 0 as long as viscosity coefficients are positive since σ 2 /θ2 > 0 and ρ/θ2 > 0. Hence S˙ > 0, which implies that the total entropy always increases with the ˙ change of proper time irrespective of an expanding or contracting model. Also we observe that SS → ∞ as θρ2 → 0, and the universe becomes homogeneous. Let the entropy density be s so that S γρ s= = . (27) V T It defines the entropy density in terms of the temperature. The first law of thermodynamics may be written as   ρV , (28) d (ρV ) + (γ − 1) ρdV = γ T d T which on integration, yields T = T0 ρ

γ−1 γ

,

(29)

where T0 is a constant. From (27) and (29), one can get 1

s = s0 ρ γ ,

(30)

where s0 is a constant. The entropy in a co-moving volume then varies according to 1

S = sV = s0 ρ γ V .

(31)

In the following section, we solve the field equations by assuming some suitable physical assumptions and derive the thermodynamical parameters as discussed above.

3. Solution of field equations Subtracting (11) from (12), (11) from (13), (12) from (13) and taking second integral of each, we get the following three relations respectively:  Z  R A −3 −2 ηdt = d1 exp x1 a e dt , (32) B  Z  R A −3 −2 ηdt = d2 exp x2 a e dt , (33) C  Z  R B = d3 exp x3 a−3 e−2 ηdt dt , (34) C where d1 , x1 , d2 , x2 , d3 and x3 are constants of integration. From (32)-(34), the metric functions can be explicitly written as  Z  R −3 −2 ηdt A(t) = a1 a exp b1 a e dt ,

where

(35)

 Z  R B(t) = a2 a exp b2 a−3 e−2 ηdt dt ,

(36)

 Z  R −3 −2 ηdt C(t) = a3 a exp b3 a e dt ,

(37)

a1 = b1 =

q p p 3 d3 , a3 = 3 (d2 d3 )−1 , d1 d2 , a2 = 3 d−1 1

x3 − x1 −(x2 + x3 ) x1 + x2 , b2 = , b3 = . 3 3 3 4

These constants satisfy the following two relations: a1 a2 a3 = 1,

b1 + b2 + b3 = 0.

(38)

Thus the metric functions are found explicitly in terms of the average scale factor a. Therefore an explicit form of the average scale factor a would be helpful in determining the scale factors. Berman [32] and Berman and Gomide [33] obtained some FRW cosmological models with constant deceleration parameter and showed that the constant deceleration parameter models stand adequately for our present view of different phases of the evolution of universe. In a series of works, Singh and Kumar [28,34-38] and Kumar and Singh [39-41] have extended the work to anisotropic Bianchi-I space-time in general relativity and some of its modified theories. According to the law, the variation of the average Hubble parameter is given by [39] H = Da−n ,

(39)

where D > 0 and n ≥ 0 are constants. The deceleration parameter (q), an important observational quantity, is defined by q=−

a¨ a . a˙ 2

(40)

From equations (2) and (39), we get a˙ = Da−n+1 .

(41)

Integration of (41) gives 1

a = (nDt) n ,

(n 6= 0)

(42)

where the constant of integration has been omitted by assuming that a = 0 at t = 0. For n = 0, we get a = C0 eDt ,

(43)

q = n − 1.

(44)

q = −1.

(45)

where C0 is a constant of integration. Substituting (42) into (40), we get Use of (43) into (40), yields This shows that the value of deceleration parameter is constant in both cases. It is interesting to note that the works of Perlmutter et al. [42-44], Riess et al. [45,46], Tonry et al. [47], Knop et al. [48] and John [49], on the basis of the recent observations of type Ia supernovae, reveal the approximate value of deceleration parameter in the range −1 < q < 0, which can be obtained from the relation (44) by restricting n in the range 0 < n < 1. Thus the relevance of the relation (39) is justified. Next, we assume that the coefficient of shear viscosity (η) is proportional to the expansion scalar (θ), i.e., η ∝ θ, which leads to η = η0 θ, (46) where η0 is proportionality constant. Such relation has already been proposed in the physical literature as a physically plausible relation [49-51]. In the following subsections we present exact Bianchi-I solutions for n 6= 0 and n = 0, respectively. 3.1 Cosmology for n 6= 0 Using (42) and (46) into (35)-(37), we get the following expressions for scale factors:   n−6η0 −3 1 b1 n n , (nDt) A(t) = a1 (nDt) exp D(n − 6η0 − 3)  n−6η0 −3 b2 B(t) = a2 (nDt) exp , (nDt) n D(n − 6η0 − 3)   n−6η0 −3 1 b3 C(t) = a3 (nDt) n exp . (nDt) n D(n − 6η0 − 3) 1 n



5

(47)

(48) (49)

provided n 6= 6η0 + 3. The physical parameters such as directional Hubble factors (Hi ), Hubble parameter (H), expansion scalar (θ), ¯ and shear scalar (σ) are given by spatial volume (V ), anisotropy parameter (A) Hi = (nt)−1 + bi (nDt)

−6η0 −3 n

,

(50)

H = (nt)−1 ,

(51)

θ = 3(nt)−1 ,

(52)

3

V = (nDt) n , A¯ =

2n−12η0 −6 1 n , (b21 + b22 + b23 )(nDt) 2 3D

σ 2 = β0 (nDt) b21

(53)

−12η0 −6 n

(54)

.

(55)

b22

where β0 = + + b1 b2 . Shear viscosity of the model reads as η = 3η0 (nt)−1 ,

(56)

Substituting (47)-(49) into (13) and (14), we get p¯ − Λ = (2n − 6η0 − 3)(nt)−2 − β(nDt) ρ + Λ = 3(nt)−2 − β(nDt)

−12η0 −6 n

−12η0 −6 n

,

.

(57) (58)

Now we solve (57) and (58) with (10) in the following cases:

(i) Model with constant Λ-term In this case, the expressions for energy density, isotropic pressure and bulk viscosity are respectively given by ρ = 3(nt)−2 − β(nDt)

−12η0 −6 n

− Λ.

i h −12η0 −6 −Λ , p = (γ − 1) 3(nt)−2 − β(nDt) n n−12η0 −6 γΛ (2 − γ)β D (3γ − 2n + 6η0 ) n − (nDt)−1 + (nDt) (nt), 3 3D 3 From (29)-(31), the thermodynamical parameters of the model read as

ξ=

(59) (60) (61)

h i γ−1 −12η0 −6 γ T = T0 3(nt)−2 − β(nDt) n −Λ ,

(62)

i γ1 h −12η0 −6 −Λ s = s0 3(nt)−2 − β(nDt) n ,

(63)

h i γ1 −12η0 −6 3 . −Λ S = s0 (nDt) n 3(nt)−2 − β(nDt) n

(64)

The entropy production rate is given by

−12η0 −6 3(3γ − 2n)(nt)−3 + 3D(2 − γ + 4η0 )β(nDt) n −1 − 3γΛ(nt)−1 S˙ i h = , −12η0 −6 S −Λ γ 3(nt)−2 − β(nDt) n

6

(65)

(ii) Model with variable Λ and constant ξ Assuming ξ = constant = ξ0 (say), equations (57) and (58) can be solved with (10) to yield the following expressions for energy density, pressure and cosmological constant: i −12η0 −6 1h (2n − 6η0 )(nt)−2 − 2β0 (nDt) n + 3ξ0 (nt)−1 , γ

ρ=

i −12η0 −6 (γ − 1) h (2n − 6η0 )(nt)−2 − 2β0 (nDt) n + 3ξ0 (nt)−1 , γ     −12η0 −6 3ξ0 2 2n − 6η0 − (nt)−2 − − 1 β0 (nDt) n (nt)−1 , Λ= 3− γ γ γ p=

(66)

(67) (68)

The thermodynamical parameters and entropy production rate of the model are given by  n o γ−1 γ −12η0 −6 1 −1 −2 T = T0 + 3ξ0 (nt) , (2n − 6η0 )(nt) − 2β0 (nDt) n γ

(69)

 n o γ1 −12η0 −6 1 −1 −2 , s = s0 + 3ξ0 (nt) (2n − 6η0 )(nt) − 2β0 (nDt) n γ

(70)

3

S = s0 (nDt) n

 n o γ1 −12η0 −6 1 + 3ξ0 (nt)−1 (2n − 6η0 )(nt)−2 − 2β0 (nDt) n . γ

−12η0 −6 S˙ −4n(n − 3η0 )(nt)−3 − 3nξ0 (nt)−2 + 12β0 D(2η0 − 1)(nDt) n −1 h i , = 3(nt)−1 + −12η0 −6 S γ (2n − 6η0 )(nt)−2 − 2β0 (nDt) n + 3ξ0 (nt)−1

(71)

(72)

(iii) Model with variable Λ and ξ ∝ ρ

Assuming ξ = ξ0 ρ, equations (57) and (58) can be solved with (10) to yield the expressions for energy density, pressure, bulk viscosity and cosmological constant, respectively, as follows: ρ=

h i −12η0 −6 1 −2 n (2n − 6η )(nt) − 2β (nDt) , 0 0 γ − ξ0 (nt)−1

i −12η0 −6 (γ − 1) h −2 n , (2n − 6η )(nt) − 2β (nDt) 0 0 γ − ξ0 (nt)−1 h i −12η0 −6 ξ0 −2 n (2n − 6η )(nt) − 2β (nDt) ξ= , 0 0 γ − ξ0 (nt)−1     −12η0 −6 2 2n − 6η0 −2 (nt) − − 1 β0 (nDt) n , Λ= 3− γ − ξ0 (nt)−1 γ − ξ0 (nt)−1 p=

(73)

(74) (75) (76)

The thermodynamical parameters and entropy production rate of the model are given by T = T0



s = s0

n o γ−1 γ −12η0 −6 1 −2 (2n − 6η0 )(nt) − 2β0 (nDt) n , −1 γ − ξ0 (nt)

(77)

n o γ1 −12η0 −6 1 −2 n (2n − 6η )(nt) − 2β (nDt) , 0 0 γ − ξ0 (nt)−1

(78)



S = s0 (nDt)

3 n



n o γ1 −12η0 −6 1 −2 n . (2n − 6η0 )(nt) − 2β0 (nDt) γ − ξ0 (nt)−1 S˙ ρ˙ = 3(nt)−1 + , S γρ 7

(79) (80)

where ρ is given by (73). In each case, the above solutions satisfy equations (19) and (20) identically, and therefore represent exact solutions of the Einstein’s field equations (11)-(14). It is observed that the spatial volume is zero at t = 0 and expansion scalar is infinite, which shows that the universe starts evolving with zero volume at t = 0 with a big bang. The scale factors also vanish at t = 0 and hence the model has a point singularity at the initial epoch. The pressure, energy density, shear viscosity, Hubble factors and shear scalar diverge at t = 0. The bulk viscosity is infinite at the initial epoch except case (ii), where it is constant throughout the evolution. The anisotropy parameter also tend to infinity at t = 0 provided n < 6η0 + 3. The universe exhibits the power-law expansion after the big bang impulse. As t increases, the scale factors and spatial volume increase but the expansion scalar decreases. The shear viscosity decreases with time drops to zero for large times. In case (i), the solutions are physically valid provided Λ < 0. At late times ρ, p, T and s converge to non-zero constants. In case (ii) and (iii), all time varying parameters except S decrease with time and vanish at late times. Also the magnitude of shear as well as the expansion rate diminish in the course of expansion. As t → ∞, scale factors and volume become infinite whereas H1 , H2 , H3 , θ, A¯ and σ 2 tend to zero. Therefore the model would essentially give an empty universe for large times t.

0.7 0.6 0.5 A¯

0.4

η0 = 0 η0 = 0.5

0.3 0.2 0.1 0 2

2.5

t3

3.5

4

¯ versus time t with n = 0.5 , β0 = C0 = D = 1 for different values of η0 . Fig.1: Plot of anisotropy parameter (A) It is interesting to note that the anisotropy parameter decreases faster with time due to the presence of viscosity in the interval 1/nD < t < ∞ as may be observed from Fig.1. Therefore the solutions reveal that viscosity has played an important role in the process of isotropization of the large scale structure of the universe. From (52) and (55), we get −12η0 −6+2n σ2 β0 n . (81) = (nDt) 2 2 θ 9D This shows that the ratio of shear and expansion scalars decays to zero as t → ∞ provided n < 6η0 + 3. So the model approaches to isotropy for large values of t. Also we observe that the rate of decay falls in the absence of shear viscosity. Finally, we conclude that the model represents shearing, non-rotating and expanding model of the universe, which starts with a big bang and approaches to isotropy at late times. 3.2 Cosmology for n = 0 Using (43) and (46) into (35)-(37), we get the following expressions for scale factors:   b1 −3(2η0 +1)Dt , e A(t) = a1 C0 exp Dt − 3D(2η0 + 1)C03   b2 −3(2η0 +1)Dt B(t) = a2 C0 exp Dt − , e 3D(2η0 + 1)C03 8

(82) (83)

 C(t) = a3 C0 exp Dt −

 b3 −3(2η0 +1)Dt . e 3D(2η0 + 1)C03

(84)

The other physical parameters of the model have the following expressions: Hi = D + bi C0−3 e−3(2η0 +1)Dt ,

(i = 1, 2, 3)

(85)

H=D,

(86)

θ = 3D ,

(87)

V = C03 e3Dt ,

(88)

2β0 C0−6 −6(2η0 +1)Dt A¯ = e , 3D2

(89)

σ 2 = β0 C0−6 e−6(2η0 +1)Dt .

(90)

η = 3η0 D,

(91)

p¯ − Λ = (6η0 − 3)D2 − β0 C0−6 e−6(2η0 +1)Dt ,

(92)

ρ + Λ = 3D2 − β0 C0−6 e−6(2η0 +1)Dt .

(93)

Shear viscosity of the model reads as Substituting (82)-(84) into (13) and (14), we get

Now we solve (92) and (93) with (10) in the following cases: (i) Model with constant Λ-term In this case, the expressions for energy density, isotropic pressure and bulk viscosity are respectively given by ρ = 3D2 − β0 C0−6 e−6(2η0 +1)Dt − Λ ,

(94)

h i p = (γ − 1) 3D2 − β0 C0−6 e−6(2η0 +1)Dt − Λ ,

(95)

(2 − γ)β0 C0−6 −6(2η0 +1)Dt γΛ e − , 3D 3D The thermodynamical parameters of the model read as ξ = D(γ − 2η0 ) +

(96)

h i γ−1 γ , T = T0 3D2 − β0 C0−6 e−6(2η0 +1)Dt − Λ

(97)

h i γ1 s = s0 3D2 − β0 C0−6 e−6(2η0 +1)Dt − Λ ,

(98)

h i γ1 S = s0 C03 e3Dt 3D2 − β0 C0−6 e−6(2η0 +1)Dt − Λ .

(99)

The entropy production rate is given by

9γD3 + 3D(2 − γ + 4η0 )βe−6(2η0 +1)Dt − 3DγΛ S˙   . = S γ 3D2 − βC0−6 e−6(2η0 +1)Dt − Λ 9

(100)

(ii) Model with variable Λ and constant ξ Assuming ξ = constant = ξ0 (say), equations (92) and (93) can be solved with (10) to yield the following expressions for energy density, pressure and cosmological constant: i 1h 6η0 D2 + 3Dξ0 − 2β0 C0−6 e−6(2η0 +1)Dt , γ

(101)

i (γ − 1) h 6η0 D2 + 3Dξ0 − 2β0 C0−6 e−6(2η0 +1)Dt , γ

(102)

ρ=

p=

3Dξ0 + 3D2 (2η0 − γ) − β0 C0−6 e−6(2η0 +1)Dt , 2−γ

Λ=

(103)

The thermodynamical parameters and entropy production rate of the model are given by  n o γ−1 γ 1 −6 −6(2η0 +1)Dt 2 , T = T0 6η0 D + 3Dξ0 − 2β0 C0 e γ

(104)

 n o γ1 1 −6 −6(2η0 +1)Dt 2 s = s0 , 6η0 D + 3Dξ0 − 2β0 C0 e γ

(105)

S = s0 C03 e3Dt

 n o γ1 1 . 6η0 D2 + 3Dξ0 − 2β0 C0−6 e−6(2η0 +1)Dt γ

S˙ 12Dβ0 (2η0 + 1)C0−6 e−6(2η0 +1)Dt . = 3D +  S γ 6η0 D2 + 3Dξ0 − 2β0 C0−6 e−6(2η0 +1)Dt

(106) (107)

(iii) Model with variable Λ and ξ ∝ ρ

Assuming ξ = ξ0 ρ, equations (92) and (93) can be solved with (10) to yield the expressions for energy density, pressure, bulk viscosity and cosmological constant, respectively, as follows: ρ=

i h 2 3η0 D2 − β0 C0−6 e−6(2η0 +1)Dt , γ − 3Dξ0

i 2(γ − 1) h 3η0 D2 − β0 C0−6 e−6(2η0 +1)Dt , γ − 3Dξ0 i h 2ξ0 3η0 D2 − β0 C0−6 e−6(2η0 +1)Dt , ξ= γ − 3Dξ0     2 2η0 2 + − 1 β0 C0−6 e−6(2η0 +1)Dt , Λ = 3D 1 − γ − 3Dξ0 γ − 3Dξ0 p=

(108)

(109) (110) (111)

The thermodynamical parameters and entropy production rate of the model are given by T = T0



s = s0

S=

o γ−1 n γ 2 , 3η0 D2 − β0 C0−6 e−6(2η0 +1)Dt γ − 3Dξ0

(112)

n o γ1 2 −6 −6(2η0 +1)Dt 2 , 3η0 D − β0 C0 e γ − 3Dξ0

(113)



s0 C03 e3Dt



o γ1 n 2 −6 −6(2η0 +1)Dt 2 3η0 D − β0 C0 e . γ − 3Dξ0

6Dβ0 (2η0 + 1)C0−6 e−6(2η0 +1)Dt S˙  . = 3D +  S γ 3η0 D2 − β0 C0−6 e−6(2η0 +1)Dt 10

(114)

(115)

We find that, in each case, the above solutions satisfy equations (19) and (20) identically. The model has no initial singularity. The spatial volume, scale factors, pressure, energy density, shear viscosity, bulk viscosity and the other cosmological parameters are constants at t = 0. Thus the universe starts with a non-singular state at t = 0. As t increases, the scale factors and the spatial volume increase exponentially. The anisotropy parameter starts with maximum value 2β0 C0−6 /3D2 at t = 0 and decreases faster due to the viscosity effects as may be observed from Fig.2. Therefore viscosity has played a key role in the process of isotropization of universe. The universe exhibits uniform exponential expansion in this model, expansion scalar being a constant. As t → ∞, the scale factors and volume of the universe become infinitely large whereas the anisotropy parameter and shear scalar tend to zero. The pressure, energy density, cosmological constant, shear viscosity, bulk viscosity and Hubble factors converge to constants.

0.8 η0 = 0 η0 = 0.5

A¯

0.6

0.4

0.2

0 0

0.2

0.4

t

0.6

0.8

1

¯ versus time t with n = 0.5 and β0 = C0 = D = 1 for different values of Fig.2: Plot of anisotropy parameter (A) η0 . The ratio of shear and expansion scalars is given by β0 C0−6 −6(2η0 +1)Dt σ2 = e , θ2 9D2

(116)

which decays exponentially and the rate of decay falls in the absence of shear viscosity. It tends to zero as t → ∞, which implies that the model approaches to isotropy at late times. Hence the model represents a shearing, non-rotating and expanding universe with a finite start approaching to isotropy at late times. 4. Conclusion In this paper we have studied a spatially homogeneous and anisotropic Bianchi-I space-time, in general relativity, with bulk and shear viscosity in the presence of a cosmological constant. Exact Bianchi type-I models have been obtained in subsections 3.1 and 3.2. The physical behavior of the models is discussed in detail. The basic equations of thermodynamics for Bianchi-I universe have been deduced and thermodynamic aspects of the models have been discussed. It is found that physically relevant solutions are possible for the Bianchi-I space-time with bulk and shear viscosity in the presence of a constant as well as time-varying Λ-term. The shear viscosity is found to be responsible for the faster removal of initial anisotropies in the universe, which can be observed from Fig.1 and Fig.2. Therefore the isotropy, observed in the present universe, is a possible consequence of the viscous effects in the cosmic fluid. Further if we take Λ = 0, we retrieve the solutions obtained in our earlier work [28]. Hence we have extended our previous work [28] by taking into account the vacuum energy due to a cosmological constant.

11

References 1. Misner, C.W.: Astrophys. J. 151, 431 (1968) 2. Poroshkevich, A.G., Zeldovich, Y.B., Nivokov, I.D.: Zh. Eksp. Theor. Fiz. 53, 644 (1967) 3. Stewart, J.M., Collins : 4. Stewart, J.M.: Mon. Notice Roy. Soc. 145, 347 (1969) 5. Grøn, Ø.: Astrophys. Space Sci. 173, 191 (1990) 6. Pavon, D., Bafaluy, J., Jou, D.: Class. Quantum Grav. 8, 347 (1991) 7. Pavon, D., Zimdahl, W.: Phys. Lett. A 179, 261 (1993) 8. Burd, A., Coley, A.: Class. Quantum Grav. 11, 83 (1994) 9. Maartens, R.: Class. Quantum Grav. 12, 1455 (1995) 10. Chimento, L.P., Jakubi, A.S., Pavon, D.: Phys. Rev. D 62, 063508 (2000) 11. Chimento, L.P., Jakubi, A.S., Pavon, D.: Phys. Rev. D 67, 087302 (2003) 12. Fabris, J.C., Goncalves, S.V.B., Rebeiro, R.S.: Gen. Relativ. Gravit. 38, 495 (2006) 13. Belinskii, V.A., Khalatnikov, I.M.: Sov. Phys. JETP 42, 205 (1976) 14. Banerjee, A., Duttachoudhury, S.B., Sanyal, A.K.: Gen. Relativ. Grav. 18, 461 (1986) 15. Banerjee, A., Sanyal, A.K.: Gen. Relativ. Grav. 20, 103 (1988) 16. Huang, W.: J. Math. Phys. 31, 1456 (1990) 17. Krori, K.D., Mukherjee, A.: Gen. Relativ. Grav. 32, 1429 (2000) 18. Gavrilov, V.R., Melnikov, V.N., Triay, R.: Class. Quantum Grav. 14, 2203 (1997) 19. Murphy, J.L.: Phys. Rev. D 8, 4231 (1973) 20. Barrow, J.D.: Nucl. Phys. B 310, 743 (1988) 21. Romano, V., Pavon, D.: In Proceedings of 10th Italian Conference on General Relativity and Gravitational Physics 543, (1988) 22. Mak, M.K., Harko, T.: Int. J. Mod. Phys. D 11, 893 (2002) 23. Saha, B.: Mod. Phys. Lett. A 20, 2127 (2005) 24. Saha, B.:Physica D 219, 168 (2006) 25. Saha, B.: Astrophys. Space Sci. 312, 3 (2007) 26. Saha, B., Rikhvitsky, V.: J. Math. Phys. 49, 112502 (2008) 27. Bali, R., Singh, J.P.: Int. J. Theor. Phys. 47, 3288 (2008) 28. Singh, C.P., Kumar, S., Int. J. Theor. Phys. 48, 925 (2009). ¨ 29. Ozer, M., Taha, M.O.: Phys. Lett. B 171, 363 (1986) ¨ 30. Ozer, M., Taha, M.O.: Nucl. Phys. B 287, 776 (1987) 31. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, Pergamon, New York, pp. 47 (1959) 32. Berman, M.S.: Nuovo Cimento B 74, 182 (1983)

12

33. Berman, M.S., Gomide: Gen. Relativ. Grav. 20, 191 (1988) 34. Singh, C.P., Kumar, S.: Int. J. Mod. Phys. D 15, 419 (2006) 35. Singh, C.P., Kumar, S.: Pramana J. Phys. 68, 707 (2007) 36. Singh, C.P., Kumar, S.: Astrophys. Space Sci. 310, 31 (2007) 37. Singh, C.P., Kumar, S.: Int. J. Theor. Phys. 47, 3171 (2008). 38. Singh, C.P., Kumar, S.: DOI 10.1007/s10773-009-0030-1 (2009) 39. Kumar, S., Singh, C.P.: Astrophys. Space Sci. 312, 57 (2007) 40. Kumar, S., Singh, C.P.: Int. J. Theor. Phys. 47, 1722 (2008) 41. Kumar, S., Singh, C.P.: Int. J. Mod. Phys. A 23, 813 (2008) 42. Perlmutter , S., et al.: Astrophys. J. 483, 565 (1997) 43. Perlmutter , S., et al.: Nature 391, 51 (1998) 44. Perlmutter , S., et al.: Astrophys. J. 517, 565 (1999) 45. Riess, A.G., et al.: Astron. J. 116, 1009 (1998) 46. Riess, A.G., et al.: Astron. J. 607, 665 (2004) 47. Tonry, J.L., et al.: Astrophys. J. 594, 1 (2003) 48. Knop, R.A., et al.: Astrophys. J. 598, 102 (2003) 49. John, M.V.: The Astrophys. J. 614, 1 (2004) 50. Pradhan, A., Srivastav, S.K., Yadav, M.K.: Astrophys. Space Sci. 298, 419 (2005) 51. Pradhan, A., Pandey, P.: Astrophys. Space Sci. 301, 127 (2006) 52. Singh, T., Chaubey, R.: Pramana J. Phys. 68, 721 (2007) 53. Vishwakarma, R.G.: Mon. Notice Roy. Astron. Soc. 345, 545 (2003) 54. Arbab, A.I.: Astrophys. Space Sci. 259, 371 (1998) 55. Beesham, A., Ghosh, S.G., Lombard, R.G.: Gen. Relativ. Grav. 32, 471 (2000) 56. Nightingale, J.P.: Astrophys. J. 185, 105 (1973) 57. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, Pergamon, New York, pp. 47 (1959) 58. Raychaudhuri, A.K.: Phys. Rev. 98, 1123 (1955) 59. Heller, M., Klimek, Z.: Astrophys. Space Sci. 33, 37 (1975)

13