Axially Symmetric Bulk Viscous String Cosmological Models in GR and Brans-Dicke Theory of Gravitation

Abstract

Axially symmetric string cosmological models with bulk viscosity in Brans-Dicke (1961) and general relativity (GR) have been studied. The field equations have been solved by using the anisotropy feature of the universe in the axially symmetric space-time. Some important features of the models, thus obtained, have been discussed. We noticed that the presence of scalar field does not affect the geometry of the space-time but changes the matter distribution, and as a special case, it is always possible to obtain axially symmetric string cosmological model with bulk viscosity in general relativity.

1. Introduction

Brans and Dicke [1] theory of gravitation is a natural extension of general relativity which introduces an additional scalar field 𝜙 besides the metric tensor 𝑔_𝑖𝑗 and dimensionless coupling constant 𝜔. The Brans-Dicke field equations for combined scalar and tensor fields (using geometrized units with 𝑐=1, 𝐺=1) are given by 𝐺_𝑖𝑗=−8𝜋𝜙⁻¹𝑇_𝑖𝑗−𝜔𝜙⁻²𝜙_,𝑖𝜙_,𝑗−12𝑔_𝑖𝑗𝜙_,𝑘𝜙^,𝑘−𝜙⁻¹𝜙_𝑖;𝑗−𝑔_𝑖𝑗𝜙^,𝑘_;𝑘,𝜙^,𝑘_;𝑘=8𝜋(3+2𝜔)⁻¹𝑇, (1)(2) where 𝐺_𝑖𝑗=𝑅_𝑖𝑗−(1/2)𝑅𝑔_𝑖𝑗 is an Einstein tensor, 𝑅 is the scalar curvature,𝜔 is a dimension less constant, and 𝑇_𝑖𝑗 is the stress energy tensor of the matter. Also comma and semicolon denote partial and covariant differentiation, respectively.

Also, we have energy-conservation equation: 𝑇^𝑖𝑗_;𝑗=0. (3) Several aspects of Brans-Dicke cosmology have been extensively investigated by many authors. Reddy and Rao [2, 3] have obtained field of a charged particle and static plane symmetric solution in Brans-Dicke theory. The work of Singh and Rai [4] gives a detailed survey of Brans-Dicke cosmological models discussed by several authors. Rao et al. [5] have obtained exact Bianchi type-V perfect fluid cosmological models in Brans-Dicke theory of gravitation. Tiwari [6] has obtained Bianchi type-V cosmological models with constant deceleration parameter in a scalar-tensor theory. Rao and Vinutha [7] have obtained axially symmetric cosmological models in a scalar tensor theory based on Lyra [8] Manifold. Rao et al. [9] have discussed LRS Bianchi type-I dark energy cosmological model in Brans-Dicke theory of gravitation. Rao and Neelima [10] have discussed axially symmetric space-time with strange quark matter attached to the string cloud in self-creation theory and general relativity and established that the additional condition, special law of variation of Hubble parameter proposed by Berman [11], taken by Katore and Shaikh [12] in general relativity is superfluous. Rao and Sireesha [13, 14] have discussed axially symmetric, Bianchi type-II, -VIII, and -IX cosmological models with strange quark matter attached to string cloud in Brans-Dicke theory of gravitation, respectively.

Cosmic strings are important in the early stages of evolution of the universe before the particle creation. Cosmic strings are one-dimensional topological defects associated with spontaneous symmetry breaking whose plausible production site is cosmological phase transitions in the early universe. Letelier [15], Krori et al. [16], Mahanta and Mukherjee [17], and Bhattacharjee and Baruah [18] have studied several aspects of string cosmological models in general relativity. Rao et al. [19–21] have obtained axially symmetric string, perfect fluid cosmological models, and Bianchi type-II, -VIII, and -IX magnetized cosmological models in Brans-Dicke theory of gravitation.

In order to study the evolution of the universe, many authors constructed cosmological models containing a viscous fluid. The presence of viscosity in the fluid introduces many interesting features in the dynamics of homogeneous cosmological models. The possibility of bulk viscosity leading to inflationary like solutions in general relativistic FRW models has been discussed by several authors. Rao et al. [22] have discussed anisotropic universe with cosmic strings and bulk viscosity in a scalar-tensor theory of gravitation. Rao et al. [23] have obtained Bianchi type-II, -VIII, and -IX string cosmological models with bulk viscosity in a theory of gravitation. Rao and Sireesha [24, 25] have studied a higher-dimensional string, and Bianchi type-II, -VIII, and -IX string cosmological models with bulk viscosity, respectively, in Brans-Dicke theory of gravitation. Rao and Sireesha [26] have discussed axially symmetric string cosmological model with bulk viscosity in self-creation theory of gravitation.

In this paper, we will study axially symmetric string cosmological models with bulk viscosity in Brans-Dicke and general relativity.

2. Axially Symmetric Metric and Field Equations

We consider the axially symmetric space-time as 𝑑𝑠²=𝑑𝑡²−𝐴²(𝑡)𝑑𝜒²+𝑓²𝜒𝑑𝜑²−𝐵²(𝑡)𝑑𝑧², (4) where 𝐴 and 𝐵 are functions of time 𝑡 alone while 𝑓 is a function of coordinate 𝜒 alone.

The energy momentum tensor for a bulk viscous fluid containing one dimensional string is 𝑇_𝑖𝑗=𝜌+𝑝𝑢_𝑖𝑢_𝑗−𝑝𝑔_𝑖𝑗−𝜆𝑥_𝑖𝑥_𝑗,𝑝=𝑝−3𝜉𝐻, (5)(6) 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 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.

Also 𝑢^𝑖 and 𝑥^𝑖 satisfy the equations 𝑔_𝑖𝑗𝑢^𝑖𝑢^𝑗𝑔=1,_𝑖𝑗𝑥^𝑖𝑥^𝑗𝑢=−1,_𝑖𝑥_𝑖=0. (7)(8) Here we assume that the string is lying along the 𝑧-axis. The one-dimensional strings are assumed to be loaded with particles and the particle energy density is 𝜌_𝑝=𝜌−𝜆. (9) The components of energy momentum tensor are 𝑇¹₁=𝑇²₂=−𝑝,𝑇³₃=𝜆−𝑝,𝑇⁴₄=𝜌, (10) where 𝜌, 𝜆, and 𝑝 are functions of time 𝑡 only.

3. Cosmological Models

Now with the help of (5) to (10), the field equations (1) for the metric (4) can be written as 𝐴̈𝐴+𝐵̈𝐵+𝐴̇𝐵̇+𝜔𝜙̇𝐴𝐵2𝜙²²+𝜙̈𝜙+𝜙̇𝜙𝐴̇𝐴+𝐵̇𝐵=−8𝜋𝜙2𝐴̈𝑝,𝐴+𝐴̇²𝐴²−𝑓^′′𝐴²𝑓+𝜔𝜙̇²2𝜙²+𝜙̈𝜙𝜙̇+2𝜙𝐴̇𝐴=8𝜋𝜙𝜆−𝑝,𝐴̇²𝐴²𝐴̇𝐵̇+2−𝑓𝐴𝐵𝐴²𝑓^′′−𝜔𝜙̇²2𝜙²+𝜙̇𝜙2𝐴̇𝐴+𝐵̇𝐵=8𝜋𝜙̈𝜙̇2𝐴̇𝜌,𝜙+𝐴+𝐵̇𝐵=8𝜋(3+2𝜔)𝜌+𝜆−3𝑝̇,𝜌+𝜌+𝑝2𝐴̇𝐴+𝐵̇𝐵𝐵̇−𝜆𝐵=0. (11)(12)(13)(14)(15) Here the overhead dot and dash denote ordinary differentiation with respect to 𝑡 and 𝜒, respectively.

From (11) to (13), we can observe that it is possible to separate the terms of 𝑓(𝜒) to one side and the terms of 𝐴(𝑡),𝐵(𝑡),𝜙(𝑡),𝜌(𝑡),𝑝(𝑡), and 𝜆(𝑡) to another side. Hence, we can take that each part is equal to a constant. So, 𝑓^′′𝑓=𝑘²,𝑘²isaconstant. (16) If 𝑘=0, then 𝑓(𝜒)=𝑐₁𝜒+𝑐₂, 𝜒>0, where 𝑐₁ and 𝑐₂ are integrating constants.

Without loss of generality, by taking 𝑐₁=1 and 𝑐₂=0, we have 𝑓(𝜒)=𝜒.

Now the field equations (11) to (15) will reduce to 𝐴̈𝐴+𝐵̈𝐵+𝐴̇𝐵̇+𝜔𝜙̇𝐴𝐵2𝜙²²+𝜙̈𝜙+𝜙̇𝜙𝐴̇𝐴+𝐵̇𝐵=−8𝜋𝜙2𝐴̈𝑝,𝐴+𝐴̇²𝐴²+𝜔𝜙̇²2𝜙²+𝜙̈𝜙𝜙̇+2𝜙𝐴̇𝐴=8𝜋𝜙𝜆−𝑝,𝐴̇²𝐴²𝐴̇𝐵̇+2−𝜔𝜙̇𝐴𝐵2𝜙²²+𝜙̇𝜙2𝐴̇𝐴+𝐵̇𝐵=8𝜋𝜙̈𝜙̇2𝐴̇𝜌,𝜙+𝐴+𝐵̇𝐵=8𝜋(3+2𝜔)𝜌+𝜆−3𝑝̇,𝜌+𝜌+𝑝2𝐴̇𝐴+𝐵̇𝐵𝐵̇−𝜆𝐵=0. (17)(18)(19)(20)(21) The field equations (17) to (20) are four independent equations with six unknowns 𝐴,𝐵,𝜙,𝜌,𝑝, and𝜆.

From (17) to (20), we get 𝜔𝜙̈𝜙−12𝜙̇²𝜙²+𝜙̇𝜙2𝐴̇𝐴+𝐵̇𝐵𝐴̈=2𝐴𝐴̇𝐵̇+2+𝐵̈𝐴𝐵𝐵+𝐴̇²𝐴². (22) In order to get a deterministic solution, we take the following plausible physical condition: the shear scalar 𝜎 is proportional to scalar expansion 𝜃, which leads to the linear relationship between the metric potentials 𝐴 and 𝐵, that is, 𝐴=𝐵^𝑛,𝑛≠0, (23) where 𝑛 is an arbitrary constant.

From (22) and (23), we get 𝜔𝜙̈𝜙−12𝜙̇²𝜙²+()𝜙̇2𝑛+1𝜙𝐵̇𝐵=()𝐵̈2𝑛+1𝐵+3𝑛²𝐵̇²𝐵². (24) From (24) and (23), we get 𝐵=𝑘₃𝑘₁𝑡+𝑘₂^1/𝑘3,𝐴=𝑘₃𝑘₁𝑡+𝑘₂^𝑛/𝑘3,where𝑘₃=𝑛²+𝑛+1𝑛+1,𝑛≠−1, (25)𝑘₁≠0 and 𝑘₂ are arbitrary constants, and 𝜙=𝑘₁𝑡+𝑘₂^𝑟, (26) where 𝑟=𝑛(𝑛+2)/(𝑛²+𝑛+1)[−1±1+2(𝑛+1)/𝜔(𝑛+2)].

Since Brans-Dicke theory has to go to Einstein theory in all respects when 𝜔→∞, 𝜙= constant, then 𝑟 should be of the form 𝑛()𝑟=𝑛+2𝑛²+𝑛+1−1+2()1+𝑛+1𝜔().𝑛+2 (27) The metric (4) can now be written as 𝑑𝑠²=𝑑𝑡²−𝑘₃𝑘₁𝑡+𝑘₂^2𝑛/𝑘3𝑑𝜒²+𝜒²𝑑𝜑²−𝑘₃𝑘₁𝑡+𝑘₂^2/𝑘3𝑑𝑧². (28) From (19), we get the string energy density 𝑘𝜌=²¹8𝜋𝑘²₃𝑛(𝑛+2)+(2𝑛+1)𝑟𝑘₃−𝜔2𝑟²𝑘²₃𝑘₁𝑡+𝑘₂^𝑟−2. (29) From (17), we get the total pressure 𝑘𝑝=²¹8𝜋𝑘²₃(𝑛+1)𝑘₃−1−𝑛²+𝑟(1−𝑟)𝑘²₃−(𝑛+1)𝑟𝑘₃−𝜔2𝑟²𝑘²₃𝑘₁𝑡+𝑘₂^𝑟−2. (30) From (17) and (18), we get the string tension density 𝑘𝜆=²¹(𝑛−1)8𝜋𝑘²₃(2𝑛+1)+(𝑟−1)𝑘₃𝑘₁𝑡+𝑘₂^𝑟−2. (31) The proper pressure is given by 𝑝=𝜀₀𝜀𝜌=⁰𝑘²₁8𝜋𝑘²₃𝑛(𝑛+2)+(2𝑛+1)𝑟𝑘₃−𝜔2𝑟²𝑘²₃𝑘₁𝑡+𝑘₂^𝑟−2. (32)

The coefficient of bulk viscosity is given by 𝑘𝜉=¹8𝜋(2𝑛+1)𝑘₃×𝑛²1+𝜀₀+2𝑛𝜀₀+𝜔2𝑟²𝑘²₃1−𝜀₀+𝑟𝑘₃(2𝑛+1)𝜀₀+(𝑟−1)𝑘₃+(𝑛+1)]+(𝑛+1)1−𝑘₃×𝑘₁𝑡+𝑘₂^𝑟−1. (33) Thus, the metric (28) together with (26) and (29) to (33) constitutes axially symmetric string cosmological model with bulk viscosity in Brans-Dicke scalar-tensor theory gravitation where the constants are related by 3𝑛²+2𝑛+1+𝑘₃𝜔2𝑟𝑘₃(2−𝑟)−(2𝑛+1)(1+𝜔𝑟)=0. (34)

4. Axially Symmetric String Cosmological Models with Bulk Viscosity in General Relativity

It is interesting to note that when 𝜔→∞, 𝑟=0, and hence from (26), we get 𝜙= constant.

From (19), we get the string energy density 𝑘𝜌=²¹8𝜋𝑘²₃[𝑛(𝑛+2)]𝑘₁𝑡+𝑘₂⁻². (35) From (17), we get the total pressure 𝑘𝑝=²¹8𝜋𝑘²₃(𝑛+1)𝑘₃−1−𝑛²𝑘₁𝑡+𝑘₂⁻². (36) From (17) and (18), we get the string tension density 𝑘𝜆=²¹(𝑛−1)8𝜋𝑘²₃(2𝑛+1)−𝑘₃𝑘₁𝑡+𝑘₂⁻². (37) The proper pressure is given by 𝑝=𝜀₀𝜀𝜌=⁰𝑘²₁8𝜋𝑘²₃[𝑛(𝑛+2)]𝑘₁𝑡+𝑘₂⁻². (38) The coefficient of bulk viscosity is given by 𝑘𝜉=¹8𝜋(2𝑛+1)𝑘₃𝑛²1+𝜀₀+2𝑛𝜀₀+(𝑛+1)1−𝑘₃𝑘₁𝑡+𝑘₂⁻¹. (39) Then the present model (28) together with (35) to (39) represents axially symmetric string cosmological model with bulk viscosity in Einstein’s theory of gravitation. Also the solution identically satisfies the divergence equation (21).

5. Some Other Important Properties of the Models

The spatial volume of the model (28) is given by 𝑉=√−𝑔=𝜒𝑘₃𝑘₁𝑡+𝑘₂^{(2𝑛+1)/𝑘3},𝑘₃=𝑛²+𝑛+1.𝑛+1 (40) The expansion scalar 𝜃, calculated for the flow vector 𝑢^𝑖, is given by ()𝑘𝜃=2𝑛+1)(𝑛+1₁𝑛²+𝑛+1𝑘₁𝑡+𝑘₂. (41) The shear scalar 𝜎 is given by 𝜎²=12𝜎_𝑖𝑗𝜎^𝑖𝑗=7()182𝑛+1²(𝑛+1)²𝑘²₁𝑛²+𝑛+1²𝑘₁𝑡+𝑘₂². (42) The mean Hubble parameter 𝐻 is given by ()𝑘𝐻=2𝑛+1)(𝑛+1₁3𝑛²+𝑛+1𝑘₁𝑡+𝑘₂. (43) The mean anisotropy parameter 𝐴_𝑚 is given by 𝐴_𝑚=13³_𝑖=1𝐻_𝑖−𝐻𝐻²=2(𝑛−1)²(2𝑛+1)². (44) The density parameter Ω is given by 𝜌Ω=3𝐻²=3𝑛(𝑛+2)+(2𝑛+1)𝑟𝑘₃−(𝜔/2)𝑟²𝑘²₃16𝜋(𝑛−1)²×𝑘₁𝑡+𝑘₂^𝑟𝐴_𝑚. (45) The deceleration parameter is given by 𝑞=−3𝜃⁻²𝜃_,𝛼𝑢^𝛼+13𝜃²=𝑛²+2()12𝑛+1)(𝑛+1,𝑛≠−1,−2. (46) The tensor of rotation 𝑤_𝑖𝑗=𝑢_𝑖,𝑗−𝑢_𝑗,𝑖 is identically zero and hence, this universe is nonrotational.

6. Conclusions

In this paper, we have presented axially symmetric string cosmological models with bulk viscosity in Brans-Dicke theory of gravitation and general relativity. We observe that model (28) has no singularity at 𝑡=−𝑘₂/𝑘₁ for 𝑛>0 and at this point the proper volume will be zero, whereas when 𝑡→∞, the spatial volume becomes infinitely large. At 𝑡=−𝑘₂/𝑘₁, the expansion scalar 𝜃 and shear scalar 𝜎 tend to infinity whereas when 𝑡→∞, expansion scalar 𝜃 and shear scalar 𝜎 tend to zero. The Hubble parameter 𝐻 increases with time for 𝑛>0. Equation (45) shows that the density parameter Ω depends on the anisotropy parameter 𝐴_𝑚. Also, since the mean anisotropy parameter 𝐴_𝑚≠0, the model does not approach isotropy for 𝑛≠1. But for 𝑛=1, the mean anisotropy parameter is zero and hence, the model (28) will become isotropic. Also for 𝑛=1 the models presented here represent only bulk viscous isotropic cosmological models. It is interesting to note that the models represent accelerating universes for −1<𝑛<−1/2 and conformed to the well known fact that the scalar field and the bulk viscosity will play a vital role in getting an accelerated universe. Also recent observations of type Ia supernovae (SN Ia), Perlmutter et al. [27] and Riess et al. [28–30] proved that the decelerating parameter of the universe is in the range −1≤𝑞≤0, and the present day universe is undergoing accelerated expansion. Finally, we noticed that the presence of scalar field in Brans-Dicke theory does not affect the geometry of the space-time but changes the matter distribution in comparison with Einstein theory, and as a special case, it is always possible to obtain axially symmetric string cosmological model with bulk viscosity in general relativity.

Acknowledgment

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