Anisotropic Bianchi Type-III Bulk Viscous Fluid Universe in Lyra Geometry

Abstract

An anisotropic Bianchi type-III cosmological model is investigated in the presence of a bulk viscous fluid within the framework of Lyra geometry with time-dependent displacement vector. It is shown that the field equations are solvable for any arbitrary function of a scale factor. To get the deterministic model of the universe, we have assumed that (i) a simple power-law form of a scale factor and (ii) the bulk viscosity coefficient are proportional to the energy density of the matter. The exact solutions of the Einstein’s field equations are obtained which represent an expanding, shearing, and decelerating model of the universe. Some physical and kinematical behaviors of the cosmological model are briefly discussed.

1. Introduction

After Einstein (1916) proposed his theory of general relativity which provided a geometrical description of gravitation, many physicists attempted to generalize the idea of geometrizing the gravitation to include a geometrical description of electromagnetism. One of the first attempts was made by Weyl [1] who proposed a more general theory by formulating a new kind of gauge theory involving metric tensor to geometrize gravitation and electromagnetism. But Weyl theory was criticized due to the nonintegrability of length of vector under parallel displacement. Later, Lyra [2] suggested a modification of Riemannian geometry by introducing a gauge function into the structureless manifold which removed the nonintegrability condition. This modified geometry is known as Lyra geometry. Subsequently, Sen [3] formulated a new scalar-tensor theory of gravitation and constructed an analogue of the Einstein’s field equations based on Lyra geometry. He investigated that the static model with finite density in Lyra manifold is similar to the static model in Einstein’s general relativity. Halford [4] has shown that the constant displacement vector field in Lyra geometry plays the role of cosmological constant in general relativity. He has also shown that the scalar-tensor treatment based in Lyra geometry predicts the same effects, within observational limits, as in Einstein’s theory (Halford, [5]).

Soleng [6] has investigated cosmological models based on Lyra geometry and has shown that the constant gauge vector field either includes a creation field and be identical to Hoyle’s creation cosmology (Hoyle, [7], Hoyle, and Narlikar [8, 9]) or contains a special vacuum field which together with the gauge vector term may be considered as a cosmological term. In the latter case, solutions are identical to the general relativistic cosmologies with a cosmological term.

The cosmological models based on Lyra geometry with constant and time-dependent displacement vector fields have been investigated by a number of authors, namely, Beesham [10], T. Singh and G. P. Singh [11], Chakraborty and Ghosh [12], Rahaman and Bera [13], Rahaman et al. [14, 15], Pradhan and Vishwakarma [16], Ram and Singh [17], Pradhan et al. [18, 19], Ram et al. [20], Pradhan [21, 22], Mohanty et al. [23], Bali and Chandnani [24], and so forth.

Bali and Chandnani [25] have investigated a Bianchi type-III bulk viscous dust filled universe in Lyra geometry under certain physical assumptions. Recently, V. K. Yadav and L. Yadav [26] have presented Bianchi type-III bulk viscous and barotropic perfect fluid cosmological models in Lyra’s geometry with the assumption that the coefficient of viscosity of dissipative fluid is a power function of the energy density. In this paper, we investigate a Bianchi type-III universe filled with a bulk viscous fluid within the framework of Lyra geometry with time-dependent displacement vector field without assuming the barotropic equation of state for the matter field. The organization of the paper is as follows. In Section 2, we present the metric and Einstein’s field equations. In Section 3, we deal with the solution of the field equations. We first show that the field equations are solvable for any arbitrary scale function. There after, we obtain exact solutions of the field equation by assuming (i) a power-law form of a scale factor and (ii) the bulk viscosity coefficient is directly proportional to the energy density of the matter. In Section 4, we discuss the physical and dynamical behaviors of the universe. Section 5 summarizes the main results presented in the paper.

2. Metric and Field Equations

The diagonal form of the Bianchi type-III space-time is considered in the form 𝑑𝑠²=𝑑𝑡²−𝐴²𝑑𝑥²−𝐵²𝑒^−2𝛼𝑥𝑑𝑦²−𝐶²𝑑𝑧², (1) where 𝛼 is a constant and 𝐴, 𝐵, and 𝐶 are functions of cosmic time 𝑡.

Einstein’s field equations in normal gauge for Lyra’s geometry given by Sen [3] are 𝑅_𝜇𝜈−12𝑅𝑔_𝜇𝜈+32𝜙_𝜇𝜙_𝜈−12𝑔_𝜇𝜈𝜙_𝛼𝜙^𝛼=−𝑇_𝜇𝜈, (2) where 𝜙_𝜇 is the displacement vector field defined as 𝜙_𝜇=(0,0,0,𝛽(𝑡)). The energy-momentum tensor 𝑇_𝜇𝜈 for bulk viscous fluid is given by 𝑇_𝜇𝜈=𝜌+𝑝𝑢_𝜇𝑢_𝜈−𝑝𝑔_𝜇𝜈, (3) where 𝜌 is the energy density and 𝑢^𝜈 is the four-velocity vector of the fluid satisfying 𝑢_𝜇𝑢^𝜈=1. The effective pressure 𝑝 is given by 𝑝=𝑝−𝜉𝑢^𝜇_;𝜇, (4) where 𝑝 is the isotropic pressure, 𝛽 is the gauge function, and 𝜉 is the bulk viscosity coefficient.

For the metric (1), the field equations (2) together with (3) and (4) lead to 𝐵̈𝐵+𝐶̈𝐶+𝐵̇𝐶̇𝐵𝐶=−3𝑝−4𝛽²,𝐶̈𝐶+𝐴̈𝐴+𝐶̇𝐴̇𝐶𝐴=−3𝑝−4𝛽²,𝐴̈𝐴+𝐵̈𝐵+𝐴̇𝐵̇−𝛼𝐴𝐵𝐴²²=−3𝑝−4𝛽²,𝐴̇𝐵̇+𝐵̇𝐶̇𝐴𝐵+𝐶̇𝐴̇𝐵𝐶−𝛼𝐶𝐴𝐴²²3=𝜌+4𝛽²,𝛼𝐴̇𝐴−𝐵̇𝐵=0, (5)(6)(7)(8)(9) where an overhead dot denotes differentiation with respect to time 𝑡. Here, we have used the geometrized unit in which 8𝜋𝐺=1and𝑐=1.

The energy conservation equation 𝑇^𝜈_𝜇;𝜈=0 leads to 𝜌̇+𝜌+𝑝𝐴̇𝐴+𝐵̇𝐵+𝐶̇𝐶=0. (10) The conservation of left-hand side of (2) yields (Bali and Chandnani [25]) 𝛽𝛽̇+𝛽²𝐴̇𝐴+𝐵̇𝐵+𝐶̇𝐶=0. (11) The physical quantities that are important in cosmology are proper volume 𝑉, average scale factor 𝑅, expansion scalar 𝜃, shear scalar 𝜎, and Hubble parameter 𝐻. For the metric (1), they have the following form: 𝑉=𝑅³=𝐴𝐵𝐶,𝜃=𝑢^𝜇_;𝜇=𝐴̇𝐴+𝐵̇𝐵+𝐶̇𝐶,𝜎²=12𝜎_𝜇𝜈𝜎^𝜇𝜈=12𝐴̇𝐴²+𝐵̇𝐵²+𝐶̇𝐶²−16𝜃²,𝑅̇𝐻=𝑅. (12)(13)(14)(15) An important observational quantity in cosmology is the deceleration parameter 𝑞 which is defined as 𝑅𝑅̈𝑞=−𝑅̇². (16) The sign of 𝑞 indicates whether the model inflates or not. The positive sign of 𝑞 corresponds to a standard decelerating model, whereas negative sign indicates inflation.

3. Exact Solutions

We now solve the field equations (5)–(11) by using the method developed by Mazumdar [27] and further used by Verma and Ram [28].

From (9), we obtain 𝐴=𝑘𝐵, (17) where 𝑘 is an integration constant. Without loss of generality, we take 𝑘=1. Using (17), (5)–(11) reduce to 𝐵̈𝐵+𝐶̈𝐶+𝐵̇𝐶̇𝐵𝐶=−3𝑝−4𝛽²,2𝐵̈𝐵+𝐵̇²𝐵²−𝛼²𝐵²=−3𝑝−4𝛽²,𝐵̇²𝐵²+2𝐵̇𝐶̇−𝛼𝐵𝐶𝐵²²3=𝜌+4𝛽²,𝜌̇+𝜌+𝑝2𝐵̇𝐵+𝐶̇𝐶𝛽̇=0,𝛽+2𝐵̇𝐵+𝐶̇𝐶=0. (18)(19)(20)(21)(22) From (18) and (19), we get 𝐵̈𝐵−𝐶̈𝐶+𝐵̇²𝐵²−𝐵̇𝐶̇−𝛼𝐵𝐶𝐵²²=0. (23) This is an equation involving two unknown functions 𝐵 and 𝐶 which will admit solution if one of them is a known function of 𝑡. To get a physically realistic model, Bali and Chandanani [25] and V. K. Yadav and L. Yadav [26] have assumed a supplementary condition 𝐵=𝐶^𝑛 between the metric potentials 𝐵 and 𝐶. This condition is based on the physical assumption that the shear scalar 𝜎 is proportional to the expansion scalar 𝜃. In order to obtain a simple but physically realistic solution, we make the mathematical assumption that 𝐶=𝑡^𝑛, (24) where 𝑛 is positive real number. For this, we first show that (23) is solvable for an arbitrary choice of 𝐶. Multiplying (23) by 𝐵²𝐶, we get 𝐵𝐶𝐵̈−𝐵𝐵̇𝐶̇+𝐶𝐵̇²−𝐵²𝐶̈=𝛼²𝐶. (25) This can be written in the following form: 𝑑𝑑𝑡−𝐵²𝐶̇+𝐵𝐶𝐵̇=𝛼²𝐶. (26) The first integral of (26) is −𝐵²𝐶̇+𝐵𝐶𝐵̇=𝛼²𝐶𝑑𝑡+𝑘₁, (27) where 𝑘₁ is a constant of integration. We can write (27) in the following form: 𝑑𝐵𝑑𝑡²−2𝐶̇𝐶𝐵²=𝐹(𝑡), (28) where 𝐹(𝑡)=2𝛼²𝐶𝐶𝑑𝑡+𝑘₁. (29) The general solution of (28) is given by 𝐵²=𝐶²𝐹(𝑡)𝐶²𝑑𝑡+𝑘₂, (30) where 𝑘₂ is a constant of integration. Note that in this case the solution of Einstein’s field equations reduces to the integration of (30) if 𝐶 is explicitly known function of 𝑡. Making use of (24) in (29) and (30), we obtain the general solution 𝐵² of the form 𝐵²=𝛼²𝑡²1−𝑛²+2𝑘₁𝛼²𝑡^1−𝑛1−3𝑛+𝑘₂𝑡^2𝑛. (31) For further discussion of the solutions, we take 𝑘₁=𝑘₂=0. Then, 𝐵²=𝛼²𝑡²1−𝑛²,𝑛≠1. (32) Hence, the metric of our solutions can be written in the form 𝑑𝑠²=𝑑𝑡²−𝛼²𝑡²1−𝑛²𝑑𝑥²+𝑒^−2𝛼𝑥𝑑𝑦²−𝑡^2𝑛𝑑𝑧², (33) where 0<𝑛<1.

4. Some Physical and Kinematical Properties of the Model

The model (33) represents an anisotropic Bianchi type-III cosmological universe filled with a bulk viscous fluid in the framework of Lyra geometry.

From (22), (24), and (32), the gauge function 𝛽 has the expression given by 𝑐𝛽=¹1−𝑛²𝛼²𝑡^−(𝑛+2), (34) where 𝑐₁ is an integration constant. Using (24), (32), and (34) into (18) and (20), we obtain 𝑛𝑝=−²𝑡²+3𝑐²₁1−𝑛²²4𝛼⁴𝑡^2𝑛+4,𝑛()𝜌=𝑛+2𝑡²−3𝑐²₁1−𝑛²²4𝛼⁴𝑡^2𝑛+4. (35)(36) It is clear that, given 𝜉(𝑡), we can determine isotropic pressure 𝑝. In most of the investigations involving bulk viscosity, 𝜉(𝑡) is assumed to be simple power function of energy density (Weinberg, [29]): 𝜉(𝑡)=𝜉₀𝜌^𝑚, (37) where 𝜉₀ and 𝑚>0 are constants. The case 𝑚=1 has been considered by Murphy [30] which corresponds to a radiating fluid. We take 𝑚=1 so that 𝜉=𝜉₀𝜌. (38) From(4), (13), (36), (37), and (38), we obtain 𝜉𝑝=⁰𝑛(𝑛+2)²𝑡³−𝑛²𝑡²−3𝑐²₁1−𝑛²²4𝛼⁴𝑡^2𝑛+4𝜉1+⁰(𝑛+2)𝑡. (39) Clearly, the viscosity contributes significantly to the isotropic pressure of the fluid.

The physical and kinematical parameters of the model (33) have the following expressions: 𝑉=𝑅³=𝛼²1−𝑛²𝑡^𝑛+2,𝜃=𝑛+2𝑡,𝜎=1−𝑛√,3𝑡𝐻=𝑛+2,3𝑡𝑞=1−𝑛.2+𝑛 (40) The deceleration parameter is positive which shows the decelerating behavior of the cosmological model. It is worthwhile to mention the work of Vishwakarma [31], where he has shown that the decelerating model is also consistent with recent CMB observations model by WNAP, as well as with the high-redshift supernovae Ia data including 1997ff at 𝑍=1.755.

We observe that the spatial volume 𝑉 is zero at 𝑡=0, and it increases with the cosmic time. This means that the model starts expanding with a big bang at 𝑡=0. All the physical and kinematical parameters 𝜌, 𝑝, 𝜃, and 𝜎 diverge at this initial singularity. The physical and kinematical parameters are well defined and are decreasing functions for 0<𝑡<∞, and ultimately tend to zero for large time. The gauge function 𝛽(𝑡) and bulk viscosity coefficient 𝜉(𝑡) are infinite at the beginning and gradually decrease as time increases and ultimately tend to zero at late times. Since 𝜎/𝜃=(1−𝑛)/√3(𝑛+2) = const, the anisotropy in the universe is maintained throughout the passage of time.

5. Conclusions

We have presented an anisotropic Bianchi type-III cosmological model in the presence of a bulk viscous fluid within the framework of Lyra’s geometry with time-dependent displacement vector. The model describes an expanding, shearing, and decelerating universe with a big-bang singularity at 𝑡=0. All the physical and kinematical parameters start off with extremely large values, which continue to decrease with the expansion of the universe and ultimately tend to zero for large time. As 𝜌 tends to zero as 𝑡 tends to infinity, the model would essentially give an empty space-time for large time.

Acknowledgment

The authors are extremely grateful to the referee for his valuable suggestions.