Kantowski-Sachs String Cosmological Model with Bulk Viscosity in General Scalar Tensor Theory of Gravitation

Abstract

A new class of spatially homogeneous Kantowski-Sachs string cosmological models with bulk viscosity in Nordtvedt (1970) general scalar-tensor theory of gravitation with the help of a special case proposed by Schwinger (1970) is obtained. In this paper we have presented anisotropic as well as isotropic cosmological models. Some important features of the models, thus obtained, have been discussed. These exact models are new and more general and represent not only the early stages of evolution but also the present universe.

1. Introduction

Nordtvedt [1] proposed a general class of scalar-tensor gravitational theories in which the parameter 𝜔 of the BD theory is allowed to be an arbitrary (positive definite) function of the scalar field [𝜔→𝜔(𝜙)]. This general class of scalar-tensor gravitational theories includes the Jordan [2] and Brans-Dicke [3] theories as special cases.

Barker [4], Ruban and Finkelstein [5], Banerjee and Santos [6, 7], and Shanti and Rao [8, 9] are some of the authors who have investigated several aspects of Nordtvedt general scalar-tensor theory. Rao and Kumari [10] have discussed a cosmological model with negative constant deceleration parameter in this theory, and very recently Rao et al. [11] have obtained Kaluza-Klein radiating model in a general scalar-tensor theory of gravitation.

The field equations of general scalar-tensor theory proposed by Nordtvedt are 𝑅_𝑖𝑗−12𝑔_𝑖𝑗𝑅=−8𝜋𝜙⁻¹𝑇_𝑖𝑗−𝜔𝜙⁻²𝜙,_𝑖𝜙,_𝑗−12𝑔_𝑖𝑗𝜙,_𝑘𝜙^,𝑘−𝜙⁻¹𝜙_𝑖;𝑗−𝑔_𝑖𝑗𝜙^,𝑘_;𝑘,𝜙^,𝑘_;𝑘=8𝜋𝑇−13+2𝜔(3+2𝜔)𝑑𝜔𝑑𝜙𝜙,_𝑖𝜙^,𝑖, (1) where 𝑅_𝑖𝑗 is the Ricci tensor, 𝑅 is the scalar curvature, 𝑇_𝑖𝑗 is the stress energy tensor of the matter, and comma and semicolon denote partial and covariant differentiation, respectively.

Also, we have 𝑇^𝑖𝑗_;𝑗=0 (2) which is a consequence of the field equations (1).

Beside the Bianchi type metrics, the Kantowski-Sachs [12] models are also describing spatially homogeneous universes. For a review of Kantowski-Sachs of metrics one can refer to MacCallum [13]. These metrics represent homogeneous but anisotropically expanding (or contacting) cosmologies and provide models where the effects of anisotropic can be estimated and compared with all well-known Friedmann-Robertson-Walker class of cosmologies. Wang [14] has obtained Kantowski-Sachs string cosmological model with bulk viscosity in general relativity. Kandalkar et al. [15] have discussed Kantowski-Sachs viscous fluid cosmological model with a varying Λ. Kandalkar et al. [16] have obtained string cosmology in Kantowski-Sachs space-time with bulk viscosity and magnetic field. Rao et al. [17] have studied various Bianchi type string cosmological models in the presence of bulk viscosity. Recently Rao et al. [18] and Rao and Sireesha [19] have obtained Bianchi types II, VIII, and IX string cosmological models with bulk viscosity in Lyra [20] and Brans and Dicke [3] theory of gravitation, respectively.

Motivated by the above investigations, we study spatially homogeneous and anisotropic Kantowski-Sachs string cosmological model with bulk viscosity in Nordtvedt [1] general scalar-tensor theory with the help of a special case proposed by Schwinger [21], that is, 3+2𝜔(𝜙)=1/𝜆𝜙, where 𝜆 is a constant.

2. Metric and Energy Momentum Tensor

We consider a spatially homogeneous Kantowski-Sachs metric of the form 𝑑𝑠²=𝑑𝑡²−𝐴²𝑑𝑟²−𝐵²𝑑𝜃²+sin²𝜃𝑑𝜙², (3) where 𝐴and𝐵 are the functions of time 𝑡 only.

The energy momentum tensor for a bulk viscous fluid containing one-dimensional string is 𝑇_𝑖𝑗=𝜌+𝑝𝑢_𝑖𝑢_𝑗−𝑝𝑔_𝑖𝑗−𝜆_𝑠𝑥_𝑖𝑥_𝑗,𝑝=𝑝−3𝜉𝐻, (4)(5) where 𝑝=∈₀𝜌(0≤∈₀≤1). Here 𝑝 is the total pressure which includes the proper pressure 𝑝, 𝜌 is the rest energy density of the system, 𝜆_𝑠 is the tension in the string, 𝜉(𝑡)is the coefficient of bulk viscosity, 3𝜉𝐻 is usually known as bulk viscous pressure, 𝐻 is the Hubble parameter, 𝑢^𝑖 is the four velocity vector, and 𝑥^𝑖 is a space-like vector which represents the anisotropic directions of the string.

Here 𝑢^𝑖 and 𝑥^𝑖 satisfy the equations 𝑔_𝑖𝑗𝑢^𝑖𝑢^𝑗𝑔=1,_𝑖𝑗𝑥^𝑖𝑥^𝑗𝑢=−1,_𝑖𝑥_𝑖=0. (6) We assume that the string is to be lying along the 𝑥-axis. The one-dimensional strings are assumed to be loaded with particles, and the particle energy density is 𝜌_𝑝=𝜌−𝜆_𝑠. (7) In a comoving coordinate system, we get 𝑇¹₁=𝜆_𝑠−𝑝,𝑇²₂=𝑇³₃=−𝑝,𝑇⁴₄=𝜌, (8) where 𝜌,𝜆_𝑠,𝑝, and 𝜙 are functions of time 𝑡 only.

3. Solution of the Field Equations

Using comoving coordinates, the field equations (1) to (2) for the metric (3) with the help of (4) to (8) can be written as 2𝐵̈𝐵+𝐵̇²𝐵²+1𝐵²+𝜔𝜙̇²2𝜙²+𝜙̈𝜙𝜙̇𝐵̇+2=𝜆𝜙𝐵8𝜋_𝑠−𝑝𝜙,𝐴̈𝐴+𝐵̈𝐵+𝐴̇𝐵̇+𝜔𝜙̇𝐴𝐵2𝜙²²+𝜙̈𝜙+𝜙̇𝜙𝐴̇𝐴+𝐵̇𝐵=−8𝜋𝑝𝜙,2𝐴̇𝐵̇+𝐵̇𝐴𝐵𝐵²²+1𝐵²−𝜔𝜙̇²2𝜙²+𝜙̇𝜙𝐴̇𝐴𝐵̇+2𝐵=8𝜋𝜌𝜙,̈𝜙̇𝐴̇𝜙+𝐴𝐵̇+2𝐵=8𝜋(3+2𝜔)𝜌+𝜆_𝑠−3𝑝−1(3+2𝜔)𝑑𝜔̇𝜙𝑑𝜙²,𝜌̇+𝜌+𝑝𝐴̇𝐴𝐵̇+2𝐵−𝜆_𝑠𝐴̇𝐴=0. (9) Here the overhead dot denotes differentiation with respect to 𝑡.

By using the transformation 𝑑𝑡=𝐴𝐵²𝑑𝑇, the above field equations (9) can be written as 2𝐵^′′𝐵𝐵−3^′2𝐵²𝐴−2^′𝐵^′𝐴𝐵+𝐴²𝐵²+𝜔2𝜙^′2𝜙²+𝜙^′′𝜙−𝜙^′𝜙𝐴^′𝐴=8𝜋𝜙𝜆_𝑠−𝑝𝐴²𝐵⁴,𝐴^′′𝐴−𝐴^′2𝐴²+𝐵^′′𝐵𝐵−2^′2𝐵²𝐴−2^′𝐵^′+𝜔𝐴𝐵2𝜙^′2𝜙²+𝜙^′′𝜙−𝜙^′𝜙𝐵^′𝐵=−8𝜋𝑝𝜙𝐴²𝐵⁴,2𝐴^′𝐵^′+𝐵𝐴𝐵𝐵^′22+𝐴²𝐵²−𝜔2𝜙^′2𝜙²+𝜙^′𝜙𝐴^′𝐴𝐵+2^′𝐵=8𝜋𝜙𝜌𝐴²𝐵⁴,(3+2𝜔)𝜙^′′=8𝜋𝜌−3𝑝+𝜆_𝑠𝐴²𝐵⁴−𝜙^′2𝑑𝜔,𝜌𝑑𝜙_′+𝜌+𝑝𝐴^′𝐴𝐵+2^′𝐵−𝜆_𝑠𝐴^′𝐴=0. (10)(11)(12)(13)(14) From hereafter the overhead dash denotes differentiation with respect to 𝑇.

The field equations (10) to (13) are four independent equations with seven unknowns 𝐴,𝐵,𝜔,𝜙,𝜌,𝑝,and𝜆_𝑠.

From (10) to (13), we have (3+2𝜔)𝜙^′′+𝑑𝜔𝜙𝑑𝜙^′2𝐴=2𝜙^′′𝐴−𝐴^′2𝐴²𝐵+2^′′𝐵𝐵−3^′2𝐵²𝐴−2^′𝐵^′𝐴𝐵+𝐴²𝐵²𝜙+𝜔^′2𝜙+3𝜙^′′. (15) Here we obtain string cosmological model with bulk viscosity in Nordtvedt’s general scalar-tensor theory with the help of a special case proposed by Schwinger [21] in the form 𝜙=13+2𝜔𝜆𝜙,𝜆=constant. (16) From (15) and (16), we get 1𝜆𝜙^′′𝜙−𝜙^′2𝜙²+32𝜙^′2𝜙−3𝜙^′′𝐴=2𝜙^′′𝐴−𝐴^′2𝐴²𝐵+2^′′𝐵𝐵−3^′2𝐵²𝐴−2^′𝐵^′𝐴𝐵+𝐴²𝐵². (17) In order to solve the above equation completely, we assume that the expansion scalar is proportional to shear scalar. This condition leads to 𝐴=𝐵^𝑛,𝑛≠0. (18) From (17) and (18), we get 𝜙=𝑒^{(𝑘1𝑇+𝑘2)}, (19) where 𝑘₁and𝑘₂ are arbitrary constants and 𝐵^′′𝐵−3(𝑛+1)𝐵𝑛+2𝐵^′22=−1()𝐵𝑛+2^2𝑛+2−34()𝑘𝑛+2²¹. (20) From (20), we get 𝑘𝐵=⁴𝑘₃sech(𝑛+1)𝑘₄𝑇^1/(𝑛+1),𝑛≠−1, (21) where 𝑘²₃=1/(𝑛²+𝑛+1)  and  𝑘²₄=3𝑘²₁/4(2𝑛+1).

From (18) and (21), we get 𝑘𝐴=⁴𝑘₃sech(𝑛+1)𝑘₄𝑇^{𝑛/(𝑛+1)}. (22) From (12), we get the string energy density 1𝜌=8𝜋𝑘₅cosh(𝑛+1)𝑘₄𝑇^{2(𝑛+2)/(𝑛+1)}×𝑒^{(𝑘1𝑇+𝑘2)}(2𝑛+1)𝑘²₄tanh²(𝑛+1)𝑘₄𝑇+𝑘²₄𝑘²₃sech²(𝑛+1)𝑘₄𝑇−(𝑛+2)𝑘₁𝑘₄tanh(𝑛+1)𝑘₄3𝑇+4𝑘²₁−𝑘²₁,4𝜆 (23) where 𝑘₅=(𝑘₄/𝑘₃)^{2(𝑛+2)/(𝑛+1)}.

From (11), we get the total pressure 1𝑝=8𝜋𝑘₅cosh(𝑛+1)𝑘₄𝑇^{2(𝑛+2)/(𝑛+1)}×𝑒^{(𝑘1𝑇+𝑘2)}(2𝑛+1)𝑘²₄tanh²(𝑛+1)𝑘₄𝑇+(𝑛+1)²𝑘²₄sech²(𝑛+1)𝑘₄𝑇−𝑘₁𝑘₄tanh(𝑛+1)𝑘₄1𝑇−4𝑘²₁−𝑘²₁.4𝜆 (24) The proper pressure is given by 𝜀𝑝=⁰8𝜋𝑘₅cosh(𝑛+1)𝑘₄𝑇^{2(𝑛+2)/(𝑛+1)}×𝑒^{(𝑘1𝑇+𝑘2)}(2𝑛+1)𝑘²₄tanh²(𝑛+1)𝑘₄𝑇+𝑘²₄𝑘²₃sech²(𝑛+1)𝑘₄𝑇−(𝑛+2)𝑘₁𝑘₄tanh(𝑛+1)𝑘₄3𝑇+4𝑘²₁−𝑘²₁.4𝜆 (25) The coefficient of bulk viscosity is given by 1𝜉=8𝜋𝑘₅()()𝑘𝑛+2cosh𝑛+1₄𝑇^{2(𝑛+2)/(𝑛+1)}×𝑒^{(𝑘1𝑇+𝑘2)}𝑛²+21−𝜀₀+𝑛4−3𝜀₀×𝑘₄tanh(𝑛+1)𝑘₄𝑇−3𝜀₀+14𝑘²₁𝑘₄coth(𝑛+1)𝑘₄𝑇+𝜀₀(𝑛+2)−1𝑘₄+𝜀₀−1𝑘4𝜆𝑘²¹₄coth(𝑛+1)𝑘₄𝑇. (26) From (10) and (11), we get the string tension density 𝜆_𝑠=18𝜋𝑘₅cosh(𝑛+1)𝑘₄𝑇^{2(𝑛+2)/(𝑛+1)}×𝑒^{(𝑘1𝑇+𝑘2)}𝑛(2𝑛+1)𝑘²₄sech²(𝑛+1)𝑘₄𝑇+(𝑛−1)𝑘₁𝑘₄tanh(𝑛+1)𝑘₄𝑇. (27) The metric (3) can now be written as 𝑑𝑠²=1𝑘₅cosh(𝑛+1)𝑘₄𝑇^{2(𝑛+2)/(𝑛+1)}𝑑𝑇²−𝑘₄𝑘₃sech(𝑛+1)𝑘₄𝑇^{2𝑛/(𝑛+1)}𝑑𝑟²−𝑘₄𝑘₃sech(𝑛+1)𝑘₄𝑇^2/(𝑛+1)𝑑𝜃²+sin²𝜃𝑑𝜙². (28) Thus, for 𝑛≠1, the metric (28) together with (19) and (23) to (27) constitutes a Kantowski-Sachs anisotropic string cosmological model with bulk viscosity in Nordtvedt’s general scalar-tensor theory of gravitation with a special case proposed by Schwinger.

3.1. Isotropic Cosmological Model

For 𝑛=1, the metric (28) together with (19) and (23) to (27) constitutes isotropic string cosmological model with bulk viscosity in Nordtvedt’s general scalar-tensor theory of gravitation with a special case proposed by Schwinger.

4. Some Important Features of the Model

The volume element of the model (28) is given by 𝑉=−𝑔^1/2=𝑘₅sech(𝑛+1)𝑘₄𝑇^{(𝑛+2)/(𝑛+1)}sin𝜃. (29) The expression for the expansion scalar 𝜃 is given by 𝜃=𝑢^𝑖,_𝑖=−(𝑛+2)𝑘₄tanh(𝑛+1)𝑘₄𝑇, (30) and the shear 𝜎 is given by 𝜎²=12𝜎^𝑖𝑗𝜎_𝑖𝑗=7()18𝑛+2²𝑘²₄tanh²(𝑛+1)𝑘₄𝑇. (31) The deceleration parameter 𝑞 is given by 𝑞=−3𝜃⁻²𝜃_,𝑖𝑢^𝑖+13𝜃²,()𝑞=−−12𝑛+1(𝑛+2)𝑒^{2(𝑛+1)𝑘4𝑇}1−𝑒^{−2(𝑛+1)𝑘4𝑇2}+1. (32) The Hubble parameter 𝐻 is given by −()𝑘𝐻=𝑛+2₄tanh(𝑛+1)𝑘₄𝑇3. (33) The tensor of rotation 𝑤_𝑖𝑗=𝑢_𝑖,𝑗−𝑢_𝑗,𝑖 is identically zero, and hence this universe is nonrotational.

5. Conclusions

In this paper we have presented Kantowski-Sachs string cosmological models with bulk viscosity in Nordtvedt [1] general scalar-tensor theory with the help of a special case proposed by Schwinger [21]. The models presented here are free from singularities, and the spatial volume vanishes as 𝑇→∞. We observe that, as 𝑇 approaches to infinity, the expansion scalar 𝜃 leads to a constant value. This shows that the universe expands homogeneously. We also observe that the shear scalar 𝜎 and the Hubble parameter 𝐻 approach to constant value as 𝑇 tends to infinity. The energy density, total pressure, string tension density, and coefficient of bulk viscosity diverge with the increase of time. For this model, the deceleration parameter 𝑞 is negative for large values of 𝑇. Hence the model represents accelerating universe and conformed the well-known fact that the scalar field and the bulk viscosity will play a vital role in getting an accelerated universe. For 𝑛=1, the model represents isotropic universe. These anisotropic as well as isotropic exact models are new, more general, and free from singularities. Also the models presented here represent not only the early stages of evolution but also the present universe.

Acknowledgment

The second author (D. Neelima) is grateful to the Department of Science and Technology (DST), New Delhi, India, for providing INSPIRE fellowship.