Abstract

We examine homogeneous and anisotropic Bianchi type I cosmological model with viscous stiff matter and time-varying cosmological term Λ which scales with Hubble parameter H. The resulting model approaches isotropy. The cosmological term Λ relaxes to a genuine cosmological constant, and the model in the absence of bulk viscosity tends to a deSitter universe asymptotically. Our scenario presents an initial epoch with decelerating expansion followed by late-time acceleration consistent with observations. Bulk viscosity advances the accelerating phase in the model and prevents the matter density to vanish for large times.

1. Introduction

Stiff fluid cosmological models create more interest in the study because, for these models, the speed of light equals the speed of sound, and its governing equations have the same characteristics as those of gravitational field [1]. Barrow [2] in his investigation has also pointed out that the entropy level of the universe makes it likely that its initial state was isotropic and quiescent (𝑝=𝜔𝜌, 𝜔∈(−1,0)) rather than chaotic only if the equation of state for high density of matter tends to stiff 𝜌=𝑝 (𝜌 being matter density and 𝑝, the isotropic pressure). Keeping in view the importance of stiff fluid models, Bali and Sharma [3], Bali et al. [4], and Mak and Harko [5, 6] have investigated cosmological models for stiff fluid distribution in different contexts.

It has been argued for a long time that the dissipative process in early stages of cosmic expansion may well account for the high degree of isotropy we observe today. Dissipative effects involving both the bulk and shear viscosity play a significant role in the early evolution of the universe. The inclusion of dissipative terms in the energy-momentum tensor of cosmic fluid seems to be the best motivated generalization of the matter term of the gravitational field equations. Eckart [7] developed the first relativistic theory of nonequilibrium thermodynamics to the effect of bulk viscosity. Padmanabhan and Chitre [8] pointed out that the presence of bulk viscosity leads to inflationary-like situations in general relativity. Johri and Sudarshan [9] have shown that bulk viscosity acts like a negative energy field in an expanding universe. Romano and Pavón [10] have investigated the evolution of Bianchi type I universe with viscous fluid. The effect of bulk viscosity on the cosmological evolution has been investigated by a number of authors namely, Saha [11–13], Singh et al. [14], Sahni and Starobinsky [15], Peebles and Ratra [16], Bali and Pradhan [17], Bali and Kumawat [18], and Bali [19].

A wide range of observations suggest that universe possesses a non-zero cosmological constant. The cosmological constant Λ is the most favoured candidate of dark energy representing energy density of vacuum. Zel’dovich [20], Dreitlein [21], and Krauss and Turner [22] have studied its significance. Recently Barrow and Shaw [23] suggested that cosmological term corresponds to a very small value of the order 10−122 when applied to Friedmann universe. Linde [24] has investigated that Λ is a function of temperature and is related to the spontaneous symmetry-breaking process. A number of cosmological models in which Λ decays with time have been investigated by Sahni and Starobinsky [15], Bertolami [25], Beesham [26], Berman [27], Singh and Desikan [28], Abdussattar and Vishwakarma [29], Bronnikov et al. [30], Bali and Singh [31], and Ram and Verma [32].

In this paper we investigate viscous Bianchi type I cosmological model containing stiff matter and a cosmological term Λ scaling with Hubble parameter 𝐻. We find that the model evolves with decelerating expansion in the initial epoch followed by late-time accelerating phase. The model approaches isotropy in the limit of large times, and shear viscosity accelerates the isotropization process. The presence of bulk viscosity prevents the matter density to vanish asymptotically. In the absence of viscosity, the model tends to a deSitter universe for large time [33].

2. Basic Equations

The line element for homogeneous, anisotropic Bianchi type I space-time with flat space sections in synchronous coordinate frame is given by 𝑑𝑠2=−𝑑𝑡2+𝐴2(𝑡)𝑑𝑥2+𝐵2(𝑡)𝑑𝑦2+𝐶2(𝑡)𝑑𝑧2.(2.1) We consider the matter component of source field to be viscous fluid described by the energy-momentum tensor 𝑇𝑀𝑖𝑗=𝜌+−𝑝𝑣𝑖𝑣𝑗+−𝑝𝑔𝑖𝑗−2𝜂𝜎𝑖𝑗,(2.2) where −𝑝 is the effective pressure related to the equilibrium pressure 𝑝 by −𝑝=𝑝−𝜁𝑣𝑖;𝑖.(2.3) Here 𝜌 is the energy density of matter, 𝜂≥0 and 𝜁≥0 are the coefficients of shear and bulk viscosity respectively, 𝑣𝑖, the four-velocity vector of the fluid satisfying, 𝑣𝑖𝑣𝑖=−1, 𝜃 is the expansion scalar and 𝜎𝑖𝑗 is the shear tensor defined by 𝜃=𝑣𝑖;𝑖,𝜎𝑖𝑗=12𝑣𝑖;𝑘ℎ𝑘𝑗+𝑣𝑗;𝑘ℎ𝑘𝑖−13𝜃ℎ𝑖𝑗,(2.4) where ℎ𝑖𝑗=𝑔𝑖𝑗+𝑣𝑖𝑣𝑗 is the projection tensor.

The shear viscosity characterizes a change in shape of a fixed volume of the fluid, whereas the bulk viscosity characterizes a change in volume of the fluid of a fixed shape.

We choose gravitational units such that 8𝜋𝐺=𝑐=1. Since the vacuum has the symmetry of the background, its energy-momentum tensor has the form 𝑇Λ𝑖𝑗=Λ𝑔𝑖𝑗, where Λ is a function of time in a homogeneous space. In comoving coordinates system (𝑣𝑖=𝛿𝑖4), it corresponds to a perfect fluid with energy density 𝜌Λ=Λ and pressure 𝑝Λ=−Λ. The Einstein’s field equations with viscous matter and vacuum energy are given by 𝑅𝑖𝑗−12𝑅𝑘𝑘𝑔𝑖𝑗=−𝑇total𝑖𝑗,(2.5) where 𝑇total𝑖𝑗=𝑇𝑀𝑖𝑗+𝑇Λ𝑖𝑗=(𝜌𝑡+𝑝𝑡)𝑣𝑖𝑣𝑗+𝑝𝑡𝑔𝑖𝑗 with the understanding that 𝜌𝑡=𝜌+Λ and 𝑝𝑡=𝑝−Λ are total energy density and total pressure, respectively. The Bianchi identities require that 𝑇total𝑖𝑗 has a vanishing divergence. The surviving components of the field equations (2.5) for the Bianchi type I metric are 2𝑝−𝜁−3𝜂̈𝐵𝜃−Λ=−𝐵−̈𝐶𝐶−̇𝐵̇𝐶̇𝐴𝐵𝐶+2𝜂𝐴,2(2.6)𝑝−𝜁−3𝜂̈𝐶𝜃−Λ=−𝐶−̈𝐴𝐴−̇𝐶̇𝐴̇𝐵𝐶𝐴+2𝜂𝐵,2(2.7)𝑝−𝜁−3𝜂̈𝐴𝜃−Λ=−𝐴−̈𝐵𝐵−̇𝐴̇𝐵̇𝐶𝐴𝐵+2𝜂𝐶,̇𝐴̇𝐵(2.8)𝜌+Λ=+̇𝐵̇𝐶𝐴𝐵+̇𝐴̇𝐶𝐵𝐶.𝐴𝐶(2.9) Here and henceforth, an overhead dot (·) denotes ordinary derivative with respect to cosmic time 𝑡, and semicolon (;) stands for covariant derivative. We define average scale factor 𝑅 for Bianchi type I space-time as 𝑅3=𝐴𝐵𝐶.(2.10) Generalized Hubble parameter 𝐻 and generalized deceleration parameter 𝑞 are defined as ̇𝑅𝐻=𝑅=13𝐻1+𝐻2+𝐻3,𝑅̈𝑅𝑞=−̇𝑅2̇𝐻=−𝐻2−1,(2.11) where 𝐻1=̇𝐴/𝐴, 𝐻2=̇𝐵/𝐵, and 𝐻3=̇𝐶/𝐶 are directional Hubble's factors along 𝑥, 𝑦, and 𝑧 directions, respectively. For Bianchi type I metric expressions for volume expansion scalar 𝜃 and shear tensor 𝜎𝑖𝑗 come out to be 𝜎𝜃=3𝐻,11=𝐻1−𝐻,𝜎22=𝐻2−𝐻,𝜎33=𝐻3−𝐻,𝜎44=0.(2.12) Magnitude 𝜎 of the shear tensor 𝜎𝑖𝑗 is given by 𝜎2=12𝜎𝑖𝑗𝜎𝑖𝑗=16𝐻1−𝐻22+𝐻2−𝐻32+𝐻3−𝐻12.(2.13) From (2.6) to (2.8), we obtain the following two equations after integration: ̇𝐴𝐴−̇𝐵𝐵=𝑘1𝑒𝐴𝐵𝐶∫−2𝜂𝑑𝑡,̇𝐵𝐵−̇𝐶𝐶=𝑘2𝑒𝐴𝐵𝐶∫−2𝜂𝑑𝑡.(2.14) From (2.10) and (2.14), we obtain that ̇𝐴𝐴=̇𝑅𝑅+2𝑘1+𝑘23𝑅3𝑒∫−2𝜂𝑑𝑡,̇𝐵𝐵=̇𝑅𝑅+𝑘2−𝑘13𝑅3𝑒∫−2𝜂𝑑𝑡,̇𝐶𝐶=̇𝑅𝑅−𝑘1+2𝑘23𝑅3𝑒∫−2𝜂𝑑𝑡.(2.15) We consider the shear viscosity 𝜂 scaling with the expansion scalar 𝜃 [11], that is, 𝜂=3𝜂0̇𝑅𝑅,(2.16)𝜂0 being constant and bulk viscosity 𝜁 of the form given by Meng and Ren [34] 𝜁=𝜁0+𝜁1̇𝑅𝑅,(2.17) where 𝜁0 and 𝜁1 are constants. Using (2.16) in (2.15), we obtain that ̇𝐴𝐴=̇𝑅𝑅+2𝑘1+𝑘23𝑅3+6𝜂0,̇𝐵𝐵=̇𝑅𝑅+𝑘2−𝑘13𝑅3+6𝜂0,̇𝐶𝐶=̇𝑅𝑅−𝑘1+2𝑘23𝑅3+6𝜂0.(2.18) Shear scalar 𝜎 in this case turns out to be 𝑘𝜎=√3𝑅3+6𝜂0;𝑘2=𝑘21+𝑘22+𝑘1𝑘2(2.19) implying that the shear viscosity accelerates the isotropization. Field equations (2.6)–(2.9) can be expressed in terms of 𝐻, and 𝑞 as −𝑝−Λ=(2𝑞−1)𝐻2−𝜎2,(2.20)𝜌+Λ=3𝐻2−𝜎2.(2.21) From (2.20) and (2.21), we get the following: ̈𝑅𝑅1=−6(𝜌+3𝑝)−2𝜎23+Λ3+𝜁𝜃2,𝑑𝜃𝑑𝑡=Λ−2𝜎2−12𝜃(𝜌+3𝑝)−23+3𝜁𝜃2.(2.22) We observe that bulk viscosity and positive Λ drive the acceleration of the universe whereas active gravitational mass density and anisotropy are responsible for deceleration. Also, presence of bulk viscosity slows down the rate of decrease of volume expansion 𝜃.

Again from (2.20) and (2.21), we obtain that 𝑞=2−𝜌−𝑝+2Λ+𝜁𝜃2𝐻2(2.23) implying that the presence of bulk viscosity lowers the value of deceleration parameter.

3. Solutions of the Field Equations and Discussion

We observe that (2.6)–(2.9), (2.16), and (2.17) are six equations in eight unknowns 𝐴, 𝐵, 𝐶, 𝜌, 𝑝, 𝜂, 𝜁, and Λ. We require two more equations to obtain a determinate solution. Following Schützhold [35, 36], we consider the decaying vacuum energy density Λ=𝑎𝐻,(3.1) where 𝑎 is a positive constant and non-vacuum component of matter to be stiff fluid 𝜌=𝑝.(3.2) Let ΛΩ=𝜌(3.3) be the ratio between the vacuum and matter densities. From (2.21) and (3.1), we get the following: 𝑎=3𝐻ΩΩ+11−3𝜎2𝜃2,(3.4) which determines the precise value of 𝑎 from the observed current values of 𝐻, Ω, and 𝜎/𝜃.

Equations (2.20) and (2.21) together with (3.1) and (3.2) give rise to ̇𝐻+3𝐻23−𝑎𝐻−2𝜁𝐻=0(3.5) determining the time evolution of Hubble parameter. Equation (3.5) with the use of (2.17) assumes the form ̇3𝐻+3−2𝜁1𝐻2−3𝑎+2𝜁0𝐻=0.(3.6) We integrate (3.6) to obtain that 𝑏𝐻=𝑑1−𝑒−𝑏𝑡,(3.7) where the integration constants are related to the choice of origin and 𝑏=𝑎+(3/2)𝜁0, 𝑑=3−(3/2)𝜁1.

On integration, (3.7) yields 𝑅𝑑𝑒=𝑚𝑏𝑡,−1(3.8) where 𝑚 is constant of integration. For the model, matter density 𝜌, vacuum density Λ, and cosmological parameters 𝜃, 𝜎, and 𝑞 have the following expressions: 𝜌=𝑏(3𝑏−𝑎𝑑)+𝑎𝑏𝑑𝑒−𝑏𝑡𝑑21−𝑒−𝑏𝑡2−𝑘23𝑚𝑒𝑏𝑡−1(6+12𝜂0)/𝑑,(3.9)Λ=𝑎𝑏𝑑1−𝑒−𝑏𝑡,(3.10)𝜃=3𝑏𝑑1−𝑒−𝑏𝑡,𝑘(3.11)𝜎=√3𝑚𝑒𝑏𝑡−1(3+6𝜂0)/𝑑,(3.12)𝑞=𝑑𝑒−𝑏𝑡−1.(3.13) We observe that the model has singularity at 𝑡=0. It evolves with a big bang at 𝑡=0, where 𝜌, Λ, 𝜃, and 𝜎 all diverge. Matter density 𝜌 varies as 𝑅−2𝑑 whereas vacuum energy density Λ as 𝑅−𝑑. Thus, the matter density decays faster than the vacuum energy density. Bulk viscosity component 𝜁1 arrests the rate of decay. Energy density associated with anisotropy decays as 𝑅−6−12𝜂0 contributing to matter production. Therefore, the presence of shear viscosity accelerates the decay of anisotropy energy. At early times Ω≈0 and 𝑞≈2−(3/2)𝜁1 concluding that matter density is dominant in the beginning of the universe and deceleration of the expanding universe is reduced by the presence of bulk viscous component 𝜁1. In the limit of large times, we obtain that 𝜌≈𝑏(3𝑏−𝑎𝑑)𝑑2,𝜃≈3𝑏𝑑,Λ≈𝑎𝑏𝑑,𝑞≈−1,Ω≈𝑎𝑑,𝜎3𝑏−𝑎𝑑𝜃≈0.(3.14)

We observe that the presence of bulk viscosity does not allow the matter density to become zero in infinitely far future. Also, at late times cosmological term Λ tends to a genuine cosmological constant. Anisotropy 𝜎/𝜃 in the model vanishes asymptotically. The model evolves with decelerating expansion followed by accelerating one at late times. Initial anisotropy of the model dies out asymptotically. We also observe that our solution does not exactly tend to a deSitter universe due to bulk viscosity.

As particular cases of the above solution, when only one component 𝜁0 or 𝜁1 of the bulk viscosity is present, we can obtain the physical quantities of the resulting models by replacing 𝑏 by 𝑎 when 𝜁0=0, 𝜁1≠0 and 𝑑 by 3 when 𝜁0≠0, 𝜁1=0 in (3.8) and (3.13). In case only 𝜁0 is nonzero, we see that matter density varies as 𝑅−6 in accordance with the standard model whereas vacuum energy density decays as 𝑅−3. In this case 𝑞≈2 at the initial epoch. When only 𝜁1 is nonvanishing component of bulk viscosity 𝜁, matter density 𝜌 decays as 𝑅−2𝑑 and vacuum density Λ as 𝑅−𝑑. Since 𝑑=3−(3/2)𝜁1<3, the presence of bulk viscosity component 𝜁1 slows down the rate of decay. In this case 𝑞≈2−(3/2)𝜁1 at early times implying that deceleration of the expanding universe goes down by 𝜁1.

We observe that initial epoch of the universe is dominated by matter density, and it decays faster than vacuum energy density. Time 𝑡Λ for which the cosmological term Λ turns out to be of the order of matter energy density is given by the condition Λ=𝜌. For this condition, we obtain from (3.4) and (3.7) 𝑡Λ=1(3/2)𝜁0+3𝐻0/21−𝜎20/3𝐻20ln2𝑑2𝑑−3+𝜎20/𝐻20−3𝜁0/𝐻.(3.15) In the presence of bulk viscosity, the ratio Ω=Λ/𝜌 tends to a constant in the limit of large times. We also observe that the model starts with decelerating expansion, and expansion in the model changes from decelerating phase to accelerating one. Equating 𝑞=0, we can calculate the time 𝑡𝑞 when expansion changes from decelerating to accelerating phase. We use (3.4) and (3.13) to get that 𝑡𝑞=ln𝑑(3/2)𝜁0+3𝐻0Ω0/Ω0+11−𝜎20/3𝐻20.(3.16) Therefore, due to bulk viscosity, accelerating phase occurs earlier than in the case of perfect fluid. Time evolution of some cosmological parameters is shown graphically in Figures 1, 2, 3, and 4.


Figure 1 
Variation of vacuum energy Λ with cosmic time 𝑡 .

Figure 2 
Variation of matter energy density 𝜌 with cosmic time 𝑡 .

Figure 3 
Variation of deceleration parameter 𝑞 with cosmic time 𝑡 .

Figure 4 
Variation of expansion anisotropy 𝜎 / 𝜃 with cosmic time 𝑡 .

4. Conclusion

We have examined the possibility of viscous stiff matter distribution in the background of homogeneous anisotropic Bianchi type I space-time with a cosmological term Λ which scales with Hubble parameter 𝐻. Coefficient of shear viscosity 𝜂 is assumed to vary as expansion scalar 𝜃 whereas bulk viscosity 𝜁 is taken of the form 𝜁=𝜁0+𝜁1(̇𝑅/𝑅), 𝜁0, 𝜁1 being constants. Exact solutions of Einstein's field equations have been obtained. The resulting model evolves with decelerating expansion in the initial epoch followed by a late time accelerated expansion presenting an appropriate expansion history of the universe consistent with observations [37–39]. We observe that the model approaches isotropy for large values of 𝑡, and presence of shear viscosity accelerates the isotropization process. Cosmological term Λ being very large at initial times relaxes to a genuine cosmological constant asymptotically. In the absence of bulk viscosity, the model results in a deSitter universe for large values of 𝑡 with √𝐻=Λ/3=𝑎/3. We find that the initial phase is dominated by matter density which tends asymptotically to a constant. The ratio Ω between the vacuum and matter densities tends asymptotically to a constant. Presence of bulk viscosity prevents matter density to vanish for large 𝑡. We also observe that bulk viscous component 𝜁1 slows down the rate of decrease of volume expansion. Due to bulk viscosity, accelerating phase occurs earlier than the perfect fluid case.

Acknowledgment

Authors are thankful to Inter University Centre for Astronomy and Astrophysics, Pune, India for providing facility where the part of this work was completed.