Natural Science
Vol.07 No.04(2015), Article ID:55704,10 pages
10.4236/ns.2015.74021

Bulk Viscous Anisotropic Cosmological Models with Dynamical Cosmological Parameters G and L

S. Kotambkar1, G. P. Singh2, R. Kelkar3

1Department of Mathematics, Laxminarayan Institute of Technology, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur, India

2Department of Mathematics, Visvesvaraya, National Institute of Technology, Nagpur, India

3Department of Mathematics, S. B. Jain Institute of Technology Management and Research, Nagpur, India

Email: shubha.kotambkar@rediffmail.com, gpsingh@mth.vnit.ac.in, rupali.kelkar@yahoo.com

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 11 February 2015; accepted 14 April 2015; published 15 April 2015

ABSTRACT

This paper deals with the Bianchi type I anisotropic models of the universe, filled with a bulk viscous cosmic fluid, in the presence of variable gravitational and cosmological constants. Some exact solutions of Einstein’s gravitational field equations with bulk viscosity, gravitational and cosmological constants have been obtained. Several well known forms of cosmological terms have been considered to discuss the effect of cosmological variables. The new cosmological models presented in this paper approaches to isotropic model with evolution of universe. The physical and dynamical properties of the models have also been discussed.

Keywords:

Bianchi Type-I, Bulk Viscosity, Gravitational and Cosmological “Constant”

1. Introduction

It is well known that majority of exact solutions of Einstein’s field equations representing cosmological models of relativity for spatially homogeneous space times belongs to either Bianchi types or Kantowaski-Sachs models [1] . The isotropic Friedman-Robertson-Walker (FRW) models are special cases of anisotropic Bianchi type models. The present day cosmological observations provide strong evidence for homogeneous, isotropic and accelerating expansion of the universe. However, in its early stages of evolution, universe might have not so smoothed behavior. Experimental data and number of scientific arguments in the literature support the existence of anisotropic stage of universe which is supposed to be phased out during evolution of the universe. In order to study the early stages behavior of evolution of universe, several authors have studied different Bianchi types and Kantowski-Sachs cosmological models. Bianchi type I space-time become almost first choice of researchers for study of anisotropic expansion of the universe as it is the simplest generalization of the flat FRW space-time. A number of authors have studied various anisotropic cosmological models in different context [2] -[15] . Recently Pradhan et al. [16] have studied a new class of cosmological models in scalar tensor theory of gravitation. Bianchi type I cosmological models with viscous fluid in modified Brans-Dicke cosmology have been studied by Rai et al. [17] .

The cosmological “constant” L which was originally introduced by Einstein in 1917 in his field equations has fallen in and out of researchers interest several times. Its significance has been studied from time to time by various researchers. It is believed that the cosmological constant was very large in the early universe, relaxed to its present small value in the course of expansion of the universe by creating massive or mass less particles. Padmanabhan [18] pointed out that the observations, suggesting the existences of non zero cosmological constant, raise serious theoretical problems. The large discrepancy between the observational value of L and the theoretical value is termed as cosmological constant problem.

Many solutions convening to this problem were proposed by considering dynamical L [19] -[26] . It is predicted that the presence of a non zero cosmological constant is one of the most important reason for the acceleration [27] [28] . Discussion on cosmological constant problems and consequences on cosmology with a time varying cosmological constant is presented by several authors [29] [30] .

The Newtonian constant of gravitation G, plays a roll of coupling constant between geometry and matter in Einstein field equation. In order to achieve possible unification of gravitation and elementary particle physics or to incorporate Mach’s principle in general relativity, many other extension of Einstein theory with time dependent G have been proposed. A number of authors [31] -[40] have studied the variation of G and L , within the frame work of general relativity in different context.

It has been suggested in the literature that bulk viscosity is associated with grand unified theory, phase transition and string creation. Further bulk viscosity acts like a negative energy field in an expanding universe. Maartens [41] has presented exhaustive review of cosmological models with causal thermodynamics. Singh and Kale [42] have studied bulk viscous Bianchi Type V cosmological models with varying gravitational and cosmological constant. Recently Verma and Shriram [43] [44] have investigated Bianchi Types bulk viscous cosmological models with variable G and L . Singh [45] has discussed some cosmological models with time varying gravitational cosmological constants. Pradhan et al. [46] have investigated Bianchi type I transit cosmological models with time dependent gravitational and cosmological constants.

2. Field Equations

In this paper we have considered that the gravitational field of matter distribution is represented by a Bianchi type I metric space time.

(1)

Einstein field equations with cosmological “constant” for perfect fluid distribution are given by

(2)

where, the energy momentum tensor is given by

It is well known that in order to consider the effect of bulk viscosity, energy momentum tensor may be considered as

(3)

Here, is the energy density, p represent perfect fluid pressure and stands for bulk viscous stress.

The Einstein’s field Equation (2) for the space-time metric (1) yields following equations

(4)

(5)

(6)

(7)

By a combination of Equations (4)-(7), one can easily obtain

(8)

where .

The energy momentum conservation equation suggests

(9)

From Equations (8) and (9) we have

(10)

Further, Equations (4)-(6) yield the solutions

(11)

(12)

(13)

Here and are integration constants obeying the relation

The above results suggest that the metric function can be explicitly expresses in terms of volume V of the universe represented by Bianchi type I space-time. Now assuming a power law relation between the volume of the universe and it’s age

(14)

Equations (11)-(13) take the form

(15)

(16)

(17)

When n = 1, from Equations (11)-(13) and using (14), we have

(18)

(19)

(20)

With help of Equations (15)-(17), Hubble parameter H and deceleration parameter q can be obtained as

(21)

It can be easily seen that for accelerating model of the universe n > 3.

The shear scalar may be defined as [47]

which in this model takes the form

(22)

AS shear dies out.

The state finder parameter are given by

which is this model takes the form

(23)

3. Cosmological Models

In this section we shall consider different cases to discuss behavior of cosmological models.

3.1. Case I

In this case we consider ansatz given by Arbab [48]

(24)

Using Equations (15)-(17) and (24), Equation (7) takes the form

(25)

where

From Equations (10) and (24), we have

(26)

On dividing Equation (26) by Equation (25) and then integrating, one can obtain

(27)

From Figure 1 one can see that cosmological constant is increasing with evolution of the universe.

Using observational values we have. Here we consider n = 5.

By use of Equations (25) and (27), we can get the relation between energy density and cosmic time as

(28)

The condition ensures the positivity of energy density. It can be easily seen that the energy density is always decreasing during evolution of the universe for .

From Figure 2 one can see that energy density is decreasing with evolution of the universe.

Using observational values we have

Considering equation of state and Equation (24), the Equation (6) may be written as

, (29)

With the help of Equations (15) and (16), Equation (29) suggests

(30)

From Equations (25) and (30), we get

(31)

Equation (31) shows that bulk viscous pressure decreases with evolution of the universe.

The critical energy density and the critical vacuum energy density are respectively given by

For the anisotropic model can be expressed respectively as

(32)

(33)

Mass density parameter and the density parameter of the vacuum are given by

For the anisotropic Bianchi type I model can be expressed respectively as

(34)

(35)

3.2. Case II

Several authors have considered the Chen and Wu [49] ansatzs. Motivated by their consideration in this case we have assumed

Figure 1. Shows variation of G with cosmic time t. Here we consider n = 5 and.

Figure 2. Shows the variation of ρ with respect to cosmic time t. Here we consider.

(36)

Now with the help of Equations (15)-(17) and (36), Equation (7) takes the form

(37)

Differentiating Equation (36) and using Equation (10), we get

(38)

From Equations (37) and (38), one can easily obtain the solution for G in terms of cosmic time t.

(39)

where is the constant of integration.

By using Equations (37) and (39), we have

(40)

Here , ensures the positivity of energy density. It can be easily seen that is decreasing for all values of n > 1.

Figure 3 shows behavior of energy density against cosmic time t. One can see that energy density is decreasing with evolution of the universe.

With help of Equations (15)-(16) and (36), Equation (6) reduces to

(41)

Further, from Equations (37) and (41), we get

(42)

The critical energy density and the critical vacuum energy density for this model can be expressed as

(43)

(44)

Mass density parameter and the density parameter of the vacuum for this model are given by

(45)

(46)

3.3. Case III

Chen and Wu ansatz [49] was generalized by Carvalho et al. [50] by considering . On the similar line in this case we have modified assumption of case II by adding the term

(47)

Now, under the assumption (47), Equation (7) may be written as

(48)

Differentiating Equation (47) and using Equation (10), we get

(49)

From Equations (48) and (49) one can easily write expression for G in terms of t as

(50)

From Figure 4 one can see that cosmological constant is increasing with evolution of the universe.

Again, with help of Equation (50), Equation (48) gives energy density

(51)

Figure 3. Shows behavior of energy density against cosmic time t.

Figure 4. Shows variation of G with cosmic time t. Here we consider n = 5 and

where

The condition and ensure the positivity of . Further one can easily see that is decreasing with the evolution of universe.

From Figure 5 one can see that energy density is decreasing with evolution of the universe.

On the similar line of previous sections, we have obtained the expression for bulk viscous stress as

(52)

The critical energy density and the critical vacuum energy density for this model can be expressed as

(53)

Figure 5. Shows behavior of energy density against cosmic time t. Here we consider n = 5 and

(54)

Mass density parameter and the density parameter of the vacuum for this model are given by

(55)

(56)

4. Discussion

Presently it is well accepted fact that the expansion of the universe is accelerating at present epoch. For an accelerating model of the universe deceleration parameter must be negative (q < 0) and hence Equation (21) suggests n > 3. Considering the value n > 3 Equations (15)-(17) indicate that anisotropy will phase out with evolution of the universe and model approaches to isotropic cosmological model. For n = 1, q > 0 model has decelerated expansion. In case II, if we take n = 1 the cosmological constant similar to the results obtained Beesham [51] . This relation of and cosmic time t has been discussed by several authors [52] [53] . Further the model suggests the relation . This type of result has been reported in the references ([52] [53] and references their in). When the cosmic time t approaches infinity expansion stops and shear dies out. In order to have clear idea of variation in behavior of cosmological parameters several relevant graphs have been plotted. All graphs of cosmological parameters are in fair agreement with cosmological observations.

Acknowledgements

Authors would like to thank U.G.C. New Delhi for providing financial support under the scheme of Major research project (F. No. 41-765/2012(SR)).

References

  1. Kantowski, R. and Sachs, R.K. (1966) Some Spatially Homogeneous Anisotropic Relativistic Cosmological Models. Journal of Mathematical Physics, 7, 443. http://dx.doi.org/10.1063/1.1704952
  2. Kramer, D. and Stephani, H. (1980) Exact Solution of Einstein’s Field Equations. Proceedings General Relativity and Gravitation, Jena, 75-92.
  3. Lorenz, D. (1982) On the Solution for a Vacuum Bianchi Type-III Model with a Cosmological Constant. Journal of Physics A, 15, 2997. http://dx.doi.org/10.1088/0305-4470/15/9/045
  4. Maartens, R. and Maharaj, S.D. (1988) Anisotropic Spheres with Uniform density in General Relativity CNLS-88-7.
  5. Huang, W.-H. (1990) Anisotropic Cosmological Models with Energy Density Dependent Bulk Viscosity. Journal of Mathematical Physics, 31, 1456. http://dx.doi.org/10.1063/1.528736
  6. Singh, T. and Singh G.P. (1991) Bianchi Type I Cosmological Models in Lyra’s Geometry: Journal of Mathematical Physics, 32, 2456.
  7. Singh, T. and Singh G.P. (1992) Bianchi Type III and Kantowski-Schas Cosmological Models in Lyra Geometry: International Journal of Theoretical Physics, 31, 1433. http://dx.doi.org/10.1007/BF00673976
  8. Singh, T. and Agrawal, A.K. (1993) Homogeneous Anisotropic Cosmological Models with Variable Gravitational and Cosmological “Constants”. International Journal of Theoretical Physics, 32, 1041. http://dx.doi.org/10.1007/BF01215310
  9. Chimento, L.P. and Alejandro, S. (1997) Dissipative Cosmological Solutions. Classical and Quantum Gravity, 14, 1811. http://dx.doi.org/10.1088/0264-9381/14/7/016
  10. Pradhan, A. and Singh, S.K. (2004) Bianchi Type I Magnetofluid Cosmological Models with Variable Cosmological Cosntant Revisited. International Journal of Modern Physics D, 13, 503-516. http://dx.doi.org/10.1142/S0218271804004736
  11. Saha, B. (2005) Anisotropic Cosmological Models with Perfect Fluid and dark Energy. Chinese Journal of Physics, 43, 1035-1043.
  12. Reddy, D.R.K. and Naidu, R.L. (2006) On Kantowski―Sachs Cosmological Models in Bimetric Theory of Gravity. Astrophysics and Space Science, 301, 185-187. http://dx.doi.org/10.1007/s10509-006-1622-3
  13. Kumar, S. and Singh, C.P. (2007) Anisotropic Bianchi Type I Models with Constant Deceleration Parameter in General Relativity. Astrophysics and Space Science, 312, 57-62. http://dx.doi.org/10.1007/s10509-007-9623-4
  14. Singh, R.P. and Yadav, L. (2009) Some Bianchi Type I Cosmological Models of the Universe for Viscous Fluid Distribution in Lyra Geometry. Electronic Journal of Theoretical Physics, 6, 61-78.
  15. Pradhan, A., Amirhashchi, H. and Saha, B. (2011) Bianchi Type I Anisotropic Dark Energy Models with Constant Deceleration Parameter. International Journal of Theoretical Physics, 50, 2923-2938. http://dx.doi.org/10.1007/s10773-011-0793-z
  16. Pradhan, A., Singh, A.K. and Amirhashchi, H. (2012) A New Class of Bianchi Type I Cosmological Models in Scalar Tensor Theory of Gravitation and Late Time Acceleration. International Journal of Theoretical Physics, 51, 3769- 3786. http://dx.doi.org/10.1007/s10773-012-1262-z
  17. Rai, P., Rai, L.N. and Singh, V.K. (2012) Bianchi Type I Cosmological Model Filled with Viscous Fluid in a Modified Brans-Dicke Cosmology. International Journal of Theoretical Physics, 51, 2127-2134. http://dx.doi.org/10.1007/s10773-012-1092-z
  18. Padmanabhan, T. (2003) Cosmological Constant―The Weight of Vacuum. Physics Reports, 380, 235-320. http://dx.doi.org/10.1016/S0370-1573(03)00120-0
  19. Sahni, V. and Starobinsky, A. (2000) The Case of Positive Cosmological Lambda Term. International Journal of Modern Physics D, 9, 373-444. http://dx.doi.org/10.1142/S0218271800000542
  20. Vishwakarma, R.G. (2000) A Study of Angular Size Redshift Relation for Models in Which Lambda Decays as the Energy Density. Classical and Quantum Gravity, 17, 3833-3842.
  21. Vishwakarma, R.G. (2001) Consequences on Variable Lambda Models from Distant Type Ia Supernovae and Compact Radio Sources. Classical and Quantum Gravity, 18, 1159-1172.
  22. Vishwakarma, R.G. (2001) Study of the Magnitude Redshift Relation for Type Ia Supernovae in a Model Resulting from a Ricci Symmetry. General Relativity and Gravitation, 33, 1973-1984.
  23. Vishwakarma, R.G. (2002) A Machian Model of Dark Energy. Classical and Quantum Gravity, 19, 4747-4752. http://dx.doi.org/10.1088/0264-9381/19/18/309
  24. Peebles, P.J.E. and Ratra, B. (2003) The Cosmological Constant and Dark Energy. Reviews of Modern Physics, 75, 559-606. http://dx.doi.org/10.1103/RevModPhys.75.559
  25. Singh, G.P. and Kotambkar, S. (2001) Higher Dimensional Cosmological Model with Gravitational and Cosmological “Constant”. General Relativity and Gravitation, 33, 621-630. http://dx.doi.org/10.1023/A:1010278213135
  26. Padmanabhan, T. and Hamsa, P. (2013) CosMIn―The Solution to the Cosmological Constant Problem. International Journal of Modern Physics D, 22, Article ID: 1342001. http://dx.doi.org/10.1142/S0218271813420017
  27. Freese, K. and Kinny, W.H. (2003) The Ultimate Fate of Life in an Accelerating Universe. Physics Letters B, 558, 1-8. http://dx.doi.org/10.1016/S0370-2693(03)00239-9
  28. Carneiro, S., Pigozzo, C., Borges, H.A. and Alcaniz, J.S. (2006) Supernova Constraints on Decaying Vacuum Cosmology. Physical Review D, 74, Article ID: 023532. http://dx.doi.org/10.1103/PhysRevD.74.023532
  29. Narimani, A., Scott, D. and Niayesh, A. (2014) How Does Pressure Gravitate? Cosmological Constant Problem Confronts Observational Cosmology. Journal of Cosmology and Astroparticle Physics, 1408, 049.
  30. Miao, Y.G. and Zhao, Y.J. (2014) Thermodynamics Approach to Fluid Equation in Lovelock Gravity And f(R) Gravity Revisisted. International Journal of Modern Physics D, 23, Article ID: 1450062.
  31. Beesham, A. (1994) Bianchi Type I Cosmological Models with Variable G and Lambda. General Relativity and Gravitation, 26, 159-165. http://dx.doi.org/10.1007/BF02105151
  32. Abdel-Rehman, A.-M. M. (1990) A Note on Physical Interpretation of Weyl Gauging. General Relativity and Gravitation, 22, 665.
  33. Berman, M.S. (1991) Cosmological Models with Variable Cosmological Term. Physical Review, D 43, 1075-1078.
  34. Kalligas, D., Wesson, P. and Everitt, C.W.F. (1995) Bianchi Type I Cosmological Models with Variable G and Lambda: A Comment. General Relativity and Gravitation, 27, 645-650. http://dx.doi.org/10.1007/BF02108066
  35. Abdussattar and Vishwakarma, R.G. (1997) Some FRW Models with Variable G and Lambda. Classical and Quantum Gravity, 14, 945-953. http://dx.doi.org/10.1088/0264-9381/14/4/011
  36. Saha, B. (2001) Dirac Spinor in Bianchi Type I Universe with Time Dependent Gravitational and Cosmological Constant. Modern Physics Letters A, 16, 1287-1296. http://dx.doi.org/10.1142/S0217732301004546
  37. Singh, G.P. and Kotambkar, S. (2003) Higher Dimensional Dissipative Cosmology with Varying G and Lambda. Gravitation and Cosmology, 9, 206-210.
  38. Singh, J.P., Pradhan, A. and Singh, A.K. (2008) Bianchi Type I Cosmological Models with Variable G and Lambda Term in General Relativity. Astrophysics and Space Science, 314, 83-88. http://dx.doi.org/10.1007/s10509-008-9742-6
  39. Vishwakarma, R.G. (2005) A Model to Explain Varying Λ, G and σ2 Simulteneously. General Relativity and Gravitation, 37, 1305-1311. http://dx.doi.org/10.1007/s10714-005-0113-0
  40. Singh, C.P. and Kumar, S. (2009) Bianchi I Space-Time with Variable Gravitational and Cosmological Constants. International Journal of Theoretical Physics, 48, 2401-2411. http://dx.doi.org/10.1007/s10773-009-0030-1
  41. Maartens, R. (1995) Dissipative Cosmology. Classical and Quantum Gravity, 12, 1455-1465. http://dx.doi.org/10.1088/0264-9381/12/6/011
  42. Singh, G.P. and Kale, A.Y. (2009) Anisotropic Bulk Viscous Cosmological Models with Variable G and Lambda. International Journal of Theoretical Physics, 48, 1177-1185. http://dx.doi.org/10.1007/s10773-008-9891-y
  43. Verma, M.K. and Ram, S. (2010) Bulk Viscous Bianchi Type-III Cosmological Model with Time-Dependent G and Λ. International Journal of Theoretical Physics, 49, 693-700. http://dx.doi.org/10.1007/s10773-010-0248-y
  44. Verma, M.K. and Ram, S. (2011) Spatially Homogeneous Bulk Viscous Fluid Models with Time Dependent Gravitational Constant and Cosmological Term. Advanced Studies in Theoretical Physics, 5, 387-398.
  45. Singh, C.P. (2011) Cosmological Models with Time Varying Gravitational and Cosmological Constants. Astrophysics and Space Science, 331, 337-342. http://dx.doi.org/10.1007/s10509-010-0439-2
  46. Pradhan, A., Saha, B. and Rikhvitsky, V. (2014) Bianchi Type I Transit Cosmological Models with Time Dependent Gravitational and Cosmological Constants-Reexamined. Indian Journal of Physics.
  47. Raychaudhuri, A.K. (1955) Relativistic Cosmology. I. Physical Review, 98, 1123. http://dx.doi.org/10.1103/PhysRev.98.1123
  48. Arbab, I.A. (1997) Cosmological Models with Variable Cosmological and Gravitational “Constants” and Bulk Viscous Models. General Relativity and Gravitation, 29, 61-74. http://dx.doi.org/10.1023/A:1010252130608
  49. Chen, W. and Wu, Y.S. (1990) Implication of a Cosmological Constant Varying as R-2. Physical Review D, 41, 695- 698. http://dx.doi.org/10.1103/PhysRevD.41.695
  50. Carvalho, J.C., Lima, J.A.S. and Waga, I. (1992) On the Cosmological Consequences of a Time Dependent Lambda Term. Physical Review D, 46, 2404-2407.
  51. Beesham, A. (1993) Cosmological Models with a Variable Cosmological Term and Bulk Viscous Models. Physical Review D, 48, 3539-3543. http://dx.doi.org/10.1103/PhysRevD.48.3539
  52. Berman, M.S. (1991) Cosmological Models with Variable Cosmological Term. Physical Review D, 43, 1075-1078. http://dx.doi.org/10.1103/PhysRevD.43.1075
  53. Beesham, A. (1994) Bianchi Type I Cosmological Models with Variable G and Lambda. General Relativity and Gravitation, 26, 159-165. http://dx.doi.org/10.1007/BF02105151