Abstract

We have obtained and presented spatially homogeneous Bianchi types II, VIII, and IX string cosmological models with bulk viscosity in a theory of gravitation proposed by Sen (1957) based on Lyra (1951) geometry. It is observed that only vacuum cosmological model exists in case of Bianchi type IX universe. Some physical and geometrical properties of the models are also discussed.

1. Introduction

Lyra [1] proposed a modification to Riemannian geometry by introducing an additional gauge function into the structure less manifold, as a result of which the cosmological constant arises naturally from the geometry. This bears a remarkable resemblance to Weyl’s [2] geometry. In subsequent investigations Sen [3] and Sen and Dunn [4] formulated a new scalar-tensor theory of gravitation and constructed an analog of Einstein’s field equations based on Lyra’s geometry. Halford [5] has shown that the scalar-tensor treatment based on Lyra’s geometry predicts the same effects as in general relativity.

The field equations in normal gauge in Lyra’s manifold as obtained by Sen [3] are𝑅𝑖𝑗−12𝑅𝑔𝑖𝑗+32𝜙𝑖𝜙𝑗−34𝑔𝑖𝑗𝜙𝑘𝜙𝑘=𝑇𝑖𝑗,(1.1) where 𝑇𝑖𝑗 is the stress energy tensor of the matter, 𝜙𝑖 is the displacement field, and other symbols have their usual meaning as in Riemannian geometry. The displacement field 𝜙𝑖 can be written as 𝜙𝑖=(0,0,0,𝛽(𝑡)).

The study of string theory has received considerable attention in cosmology. 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 [6], Krori et al. [7], Mahanta and Mukheriee [8], and Bhattacharjee and Baruah [9] have studied several aspects of string cosmological models in general relativity. Reddy and Rao [10] have studied axially symmetric cosmic strings and domain walls in a scalar tensor theory proposed by sen [3] based on Lyra [1] geometry. Mohanty and Mahanta [11] have studied five-dimensional axially symmetric string cosmological model in Lyra [1] manifold. Rao and Vinutha [12] have studied axially symmetric cosmological models in a scalar tensor theory of gravitation based on Lyra [1] geometry.

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 [13–17]. Roy and Tiwari [18], Mohanty and Pattanaik [19], Mohanty and Pradhan [20], Singh and Shriram [21], and Sing [22] are some of the authors who have investigated cosmological models with bulk viscosity in general relativity. Wang [23–25], Bali and Dave [26], Bali and Pradhan [27], Tripathy et al. [28], Tripathy et al. [29], and recently Rao et al. [30] have studied various Bianchi type cosmological models in the presence of cosmic strings and bulk viscosity.

Bianchi type spacetimes play a vital role in understanding and description of the early stages of evolution of the universe. In particular, the study of Bianchi types II, VIII, and IX universes is important because familiar solutions like FRW universe with positive curvature, the de Sitter universe, the Taub-Nut solutions, and so forth, correspond to Bianchi types II, VIII, and IX spacetimes. Bali and Dave [31], and Bali and Yadav [32] studied Bianchi type IX string as well as viscous fluid models in general relativity. Reddy et al. [33] studied Bianchi types II, VIII, and IX models in scale covariant theory of gravitation. Shanthi and Rao [34] studied Bianchi types VIII and IX models in Lyttleton-Bondi Universe. Also Rao and Sanyasi Raju [35], and Sanyasi Raju and Rao [36] have studied Bianchi types VIII, and IX models in Zero mass scalar fields and self-creation cosmology. Rahaman et al. [37] have investigated Bianchi type IX string cosmological model in a theory of gravitation formulated by Sen [3] based on Lyra [1] manifold. Rao et al. [38–40] have obtained Bianchi types II, VIII, and IX string, perfect fluid cosmological models in Saez-Ballester theory of gravitation, and string cosmological models in general relativity as well as self-creation theory of gravitation, respectively.

In this paper, we will discuss Bianchi types II, VIII, and IX string cosmological models with bulk viscosity in a theory of gravitation proposed by Sen [3] based on Lyra [1] geometry.

2. Metric and Energy Momentum Tensor

We consider a spatially homogeneous Bianchi types II, VIII, and IX metrics of the following form:𝑑𝑠2=−𝑑𝑡2+𝑅2𝑑𝜃2+𝑓2(𝜃)𝑑𝜙2+𝑆2[]𝑑𝜑+ℎ(𝜃)𝑑𝜙2,(2.1) where (𝜃,𝜙,𝜑) are the Eulerian angles, and 𝑅 and 𝑆 are functions of 𝑡 only.

It represents, Bianchi type II if 𝑓(𝜃)=1 and ℎ(𝜃)=𝜃, Bianchi type VIII if 𝑓(𝜃)=cosh𝜃 and ℎ(𝜃)=sinh𝜃, and Bianchi type IX if 𝑓(𝜃)=sin𝜃 and ℎ(𝜃)=cos𝜃.

The energy momentum tensor for a bulk viscous fluid containing one-dimensional string as𝑇𝑖𝑗=𝜌+𝑝𝑢𝑖𝑢𝑗+𝑝𝑔𝑖𝑗−𝜆𝑥𝑖𝑥𝑗,(2.2)𝑝=𝑝−3𝜉𝐻,(2.3) is the total pressure which includes the proper pressure, 𝜌 is the rest energy density of the system, 𝜆 is 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, 𝛽 the gauge function, 𝑢𝑖=𝛿𝑖4 is the four velocity vector and 𝑥𝑖 is a space-like vector which represents the anisotropic directions of the string.

Here 𝑢𝑖 and 𝑥𝑖 satisfy the 𝑔𝑖𝑗𝑢𝑖𝑢𝑗𝑔=−1,(2.4)𝑖𝑗𝑥𝑖𝑥𝑗𝑢=1,(2.5)𝑖𝑥𝑖=0.(2.6) We assume the string to be lying along the 𝑧-axis. The one-dimensional strings are assumed to be loaded with particles and the particle energy density is 𝜌𝑝=𝜌−𝜆.

In a commoving coordinate system, we get𝑇11=𝑇22=𝑝,𝑇33=𝑝−𝜆,𝑇44=−𝜌,(2.7) where 𝜌,𝜆,𝑝, and 𝜙 are functions of time “𝑡” only.

3. Solutions of Field Equations

Now with the help of (2.2) to (2.7), the field (1.1) for the metric (2.1) can be written as ̈𝑅𝑅+̈𝑆𝑆+̇𝑅̇𝑆+𝑆𝑅𝑆24𝑅4+34𝛽2=𝑝,(3.1)2̈𝑅𝑅+̇𝑅2+𝛿𝑅2−3𝑆24𝑅4+34𝛽2=𝑝−𝜆,(3.2)2̇𝑅̇𝑆+̇𝑅𝑅𝑆2+𝛿𝑅2−𝑆24𝑅4−34𝛽2=−𝜌.(3.3) Here the over head dot denotes differentiation with respect to “𝑡”.

When 𝛿=0,−1,+1, the fields, (3.1) to (3.3) correspond to the Bianchi types II, VIII, and IX universes, respectively.

By taking the transformation 𝑑𝑡=𝑅2𝑆𝑑𝑇, the previous fields (3.1) to (3.3) can be written as𝑅′𝑅+𝑆′𝑆−𝑅′𝑅2−2𝑅′𝑆′+𝑆𝑅𝑆44+34𝛽2𝑅4𝑆2=𝑝𝑅4𝑆2,(3.4)2𝑅′𝑅−𝑅′𝑅2−2𝑅′𝑆′−𝑅𝑆3𝑆44+𝛿𝑅2𝑆2+34𝛽2𝑅4𝑆2=𝑅𝑝−𝜆4𝑆2,(3.5)𝑅′𝑅2+2𝑅′𝑆′−𝑆𝑅𝑆44+𝛿𝑅2𝑆2−34𝛽2𝑅4𝑆2=−𝜌𝑅4𝑆2.(3.6) Here the over head dash denotes differentiation with respect to “𝑇”.

The fields (3.4) to (3.6) are only three independent equations with seven unknowns 𝑅, 𝑆, 𝜌, 𝜆, 𝑝, 𝛽, and 𝜉, which are functions of “𝑇”. Since these equations are highly nonlinear in nature, in order to get a deterministic solution, we take the following plausible physical conditions.(1) The shear scalar 𝜎 is proportional to scalar expansion 𝜃, so that we can take a linear relationship between the metric potentials 𝑅 and 𝑆, that is, 𝑅=𝑆𝑛,(3.7)where 𝑛 is an arbitrary constant.(2) A more general relationship between the proper rest energy density 𝜌 and string tension density 𝜆 is taken to be 𝜌=𝑟𝜆,(3.8) where 𝑟 is an arbitrary constant which can take both positive and negative values. The negative value of 𝑟 leads to the absence of strings in the universe and the positive value shows the presence of one dimensional string in the cosmic fluid. The energy density of the particles attached to the strings is 𝜌𝑝=𝜌−𝜆=(𝑟−1)𝜆.(3.9)(3)For a barotropic fluid, the combined effect of the proper pressure and the barotropic bulk viscous pressure can be expressed as 𝑝=𝑝−3𝜉𝐻=(∈𝜌),(3.10) where ∈=∈0−𝜍 and 𝑝=∈0𝜌(0≤∈0≤1).

Using (3.7), the fields (3.4), (3.5), and (3.6) can be written as 𝑆𝑆(𝑆𝑛+1)−2𝑆2𝑛2+𝑆+3𝑛+144+34𝛽2𝑆4𝑛+2=𝑝𝑆4𝑛+2,(3.11)𝑆𝑆(𝑆2𝑛)−2𝑆2𝑛2−+4𝑛3𝑆44+𝛿𝑆2𝑛+2+34𝛽2𝑆4𝑛+2=𝑆𝑝−𝜆4𝑛+2,(3.12)𝑆2𝑆2𝑛2−𝑆+2𝑛44+𝛿𝑆2𝑛+2−34𝛽2𝑆4𝑛+2=−𝜌𝑆4𝑛+2.(3.13) From (3.11) and (3.12), we get 𝑆𝑆(𝑆1−𝑛)−2𝑆2(1−𝑛)+𝑆4−𝛿𝑆2𝑛+2=𝜆𝑆4𝑛+2.(3.14) From (3.11), (3.13), (3.8), and (3.10), we obtain 𝑆𝑆(1+𝑛)+𝛿𝑆2𝑛+2=𝑟𝜆(∈−1)𝑆4𝑛+2.(3.15) From (3.14) and (3.15), we get 𝐶1𝑆′𝑆+𝐶2𝑆4−𝐶3𝛿𝑆2𝑛+2=0,(3.16) where 𝐶1=𝑟(∈−1)(1−𝑛)−(1+𝑛),𝐶2=𝑟(∈−1),𝐶3=𝑟(∈−1)+1.

3.1. Bianchi Type II (𝛿=0) Cosmological Model

If 𝛿=0, (3.16) can be written as𝐶1𝑆′𝑆+𝐶2𝑆4=0,(3.17) where 𝐶1=𝑟(∈−1)(1−𝑛)−(1+𝑛),𝐶2=𝑟(∈−1).

From (3.17), with suitable substitution, we get𝑆′2=𝑊2𝑆6+𝛾2𝑆2,(3.18) where 𝛾2 is an integrating constant and𝑊2=−𝐶22𝐶1=−𝑟(∈−1)2(𝑟(∈−1)(1−𝑛)−(1+𝑛)).(3.19)

From (3.18), we get𝑆2=𝐶32𝑊coth2𝐶3𝑇−11/2,(3.20) where 𝐶3=−2𝛾.

From (3.20) and (3.7), we get𝑅2=𝐶32𝑊𝑛coth2𝐶3𝑇−1𝑛/2.(3.21) From (3.11), (3.13), and (3.10), we obtain𝑆𝑆(1+𝑛)=𝜌(∈−1)𝑆4𝑛+2.(3.22) From (3.22), we get the energy density:𝜌=(1+𝑛)(2𝑊)1+2𝑛2𝐶(∈−1)32𝑛−1coth2𝐶3𝑇−1(1−2𝑛)/2.(3.23) The total pressure is given by𝑝=∈𝜌=∈(1+𝑛)(2𝑊)1+2𝑛2𝐶(∈−1)32𝑛−1coth2𝐶3𝑇−1(1−2𝑛)/2.(3.24) The proper pressure is given by𝑝=∈0∈𝜌=0(1+𝑛)(2𝑊)1+2𝑛𝐶2(∈−1)32𝑛−1coth2𝐶3𝑇−1(1−2𝑛)/2.(3.25) From (3.11), (3.13), and (3.10), we obtain𝑆2𝑆22𝑛2−𝑆+4𝑛𝑆(1+𝑛)−2𝑆44−64𝛽2𝑆4𝑛+2=−𝜌(1+∈)𝑆4𝑛+2.(3.26) From (3.26), we get the displacement vector:𝛽2=⎡⎢⎢⎢⎢⎣𝐶31−2𝑛3(2𝑊)1−2𝑛4𝑛2𝑊2−1+8𝑊2𝑛∈+1∈−1coth2𝐶3𝑇−1(1−2𝑛)/2+(2𝑊)1+2𝑛3𝐶32𝑛−1𝑛2+2𝑛coth2𝐶3𝑇−1−(2𝑛+1)/2⎤⎥⎥⎥⎥⎦.(3.27) From (3.12), (3.20), (3.24), and (3.27), we get the string tension density:𝐶𝜆=31−2𝑛(2𝑊)1−2𝑛2𝑊2(1−𝑛)+1coth2𝐶3𝑇−1(1−2𝑛)/2.(3.28) The particle energy density is given by𝜌𝑝=𝜌−𝜆=(𝑟−1)𝜆=(𝑟−1)𝐶31−2𝑛(2𝑊)1−2𝑛2𝑊2(1−𝑛)+1coth2𝐶3𝑇−1(1−2𝑛)/2.(3.29) The coefficient of bulk viscosity is given by 𝜉=𝜁(1+𝑛)(2w)1+2𝑛(𝜖−1)(2𝑛+1)𝐶32𝑛coth2𝐶3𝑡−1(1−2𝑛)/2coth−1𝐶3𝑡.(3.30) The components of Hubble parameter 𝐻1, 𝐻2, and 𝐻3 are given by 𝐻1=𝑅𝑅=𝑛𝐶32𝐶coth3𝑇,𝐻2=𝑆′𝑆=𝐶32𝐶coth3𝑇,𝐻3=𝑅′𝑅=𝑛𝐶32𝐶coth3𝑇.(3.31) Therefore the generalized mean Hubble parameter (𝐻) is1𝐻=3𝐻1+𝐻2+𝐻3=(2𝑛+1)𝐶36𝐶coth3𝑇.(3.32) The metric (2.1), in this case, can be written as𝑑𝑠2𝐶=−32𝑊(2𝑛+1)coth2𝐶3𝑇−1(2𝑛+1)/2𝑑𝑇2+𝐶32𝑊𝑛coth2𝐶3𝑇−1𝑛/2×𝑑𝜃2+𝑑𝜙2+𝐶32𝑊coth2𝐶3𝑇−11/2(𝑑𝜑+𝜃𝑑𝜙)2.(3.33) Thus (3.33) together with (3.23), (3.24), and (3.28) constitutes a Bianchi type-II string cosmological model with bulk viscosity in Sen [3] theory of gravitation.

3.2. Bianchi Type VIII (𝛿=−1) Cosmological Model

If 𝛿=−1, (3.16) can be written as𝐶1𝑆′𝑆+𝐶2𝑆4+𝐶3𝑆2𝑛+2=0,(3.34) where 𝐶1=𝑟(∈−1)(1−𝑛)−(1+𝑛),𝐶2=𝑟(∈−1),𝐶3=𝑟(∈−1)+1.

From (3.34), with suitable substitution and for 𝑛=1, we get𝑆′2=𝑊2𝑆6+𝛾2𝑆2,(3.35) where 𝛾2 is an integrating constant and𝑊2=−𝐶2+𝐶32𝐶1=(2𝑟(∈−1)+1)4.(3.36)

From (3.35), we get𝑆2=𝑅2=𝐶42𝑊coth2𝐶4𝑇−11/2,where𝐶4=−2𝛾.(3.37) From (3.11), (3.13), and (3.10), we obtain2𝑆𝑆−𝑆4=𝜌(∈−1)𝑆6.(3.38) From (3.38), we get the energy density:𝜌=2𝑊𝐶4(∈−1)4𝑊2−1coth2𝐶4𝑇−1−1/2.(3.39) The total pressure is given by𝑝=∈𝜌=2∈𝑊𝐶4(∈−1)4𝑊2−1coth2𝐶4𝑇−1−1/2.(3.40) The proper pressure is given by𝑝=∈0𝜌=2∈0𝑊𝐶4(∈−1)4𝑊2−1coth2𝐶4𝑇−1−1/2.(3.41) From (3.11), (3.13), and (3.10), we obtain6𝑆′2𝑆2−2𝑆′𝑆−6𝑆44−64𝛽2𝑆6=−𝜌(1+∈)𝑆6.(3.42) From (3.42), we get the displacement vector𝛽2=2𝑊3𝐶44𝑊2−3+2∈+1∈−14𝑊2−1coth2𝐶4𝑇−1−1/2+8𝑊3𝐶4coth2𝐶4𝑇−1−3/2.(3.43) From (3.12), (3.37), (3.40), and (3.43), we get the string tension density:𝜆=4𝑊𝐶4coth2𝐶4𝑇−1−1/2.(3.44) The particle energy density is given by 𝜌𝑝=𝜌−𝜆=(𝑟−1)𝜆=(𝑟−1)4𝑊𝐶4coth2𝐶4𝑇−1−1/2.(3.45) The coefficient of bulk viscosity is given by𝜉=4𝜁𝑊3𝐶42(∈−1)4𝑊2−1coth2𝐶4𝑇−1−1/2coth−1𝐶4𝑇.(3.46) The components of Hubble parameter 𝐻1, 𝐻2, and 𝐻3 are given by𝐻1=𝐻2=𝐻3=𝑅′𝑅=𝑆′𝑆=𝐶42𝐶coth4𝑇.(3.47) Therefore the generalized mean Hubble parameter (𝐻) is1𝐻=3𝐻1+𝐻2+𝐻3=c42𝐶coth4𝑇.(3.48) The metric (2.1), in this case, can be written as𝑑𝑠2𝐶=−42𝑊3coth2𝐶4𝑇−13/2𝑑𝑇2+𝐶42𝑊coth2𝐶4𝑇−11/2×𝑑𝜃2+cosh2𝜃𝑑𝜙2+𝐶42𝑊coth2𝐶4𝑇−11/2(𝑑𝜑+sinh𝜃𝑑𝜙)2.(3.49) Thus (3.49) together with (3.39), (3.40), and (3.44) constitutes a Bianchi type VIII string cosmological model with bulk viscosity in Sen [3] theory of gravitation.

3.3. Bianchi Type IX (𝛿=1) Cosmological Model

If 𝛿=1, (3.16) can be written as𝐶1𝑆𝑆+𝐶2𝑆4−𝐶3𝑆2𝑛+2=0,(3.50) where 𝐶1𝐶=𝑟(∈−1)(1−𝑛)−(1+𝑛),(3.51)2𝐶=𝑟(∈−1),(3.52)3=𝑟(∈−1)+1.(3.53)

From (3.50), with suitable substitution and for 𝑛=1, we get𝑆′2=𝑊2𝑆6+𝛾2𝑆2,(3.54) where 𝛾2 is an integrating constant and𝑊2=𝐶3−𝐶22𝐶1=−14.(3.55) Integrating (3.54), we get𝑆2=𝑅2=𝐶42𝑊coth2𝐶4𝑇−11/2,(3.56) where 𝐶4=−2𝛾.(3.57) From (3.11), (3.13), and (3.10), we obtain2𝑆′𝑆+𝑆4=𝜌(∈−1)𝑆6.(3.58) From (3.58), we get the energy density:𝜌=2𝑊𝐶4(∈−1)4𝑊2+1coth2𝐶4𝑇−1−1/2.(3.59) The total pressure is given by𝑝=∈𝜌=2∈𝑊𝐶4(∈−1)4𝑊2+1coth2𝐶4𝑇−1−1/2.(3.60) The proper pressure is given by𝑝=∈0𝜌=2∈0𝑊𝐶4(∈−1)4𝑊2+1coth2𝐶4𝑇−1−1/2.(3.61) Since 𝑊2=(𝐶3−𝐶2)/2𝐶1=(−1/4), from (3.59) to (3.61) and (3.12), we can observe that the energy density 𝜌, total pressure 𝑝, proper pressure 𝑝, and string tension density 𝜆 will vanish.

From (3.11), (3.13), and (3.10), we obtain6𝑆′2𝑆2−2𝑆′𝑆+2𝑆44−64𝛽2𝑆6=−𝜌(1+∈)𝑆6.(3.62) From (3.62), we get the displacement vector:𝛽2=−2𝑊𝐶4coth2𝐶4𝑇−1−3/2.(3.63) The coefficient of bulk viscosity is given by 𝜉=4𝜁𝑊3𝐶42(∈−1)4𝑊2+1coth2𝐶4𝑇−1−1/2coth−1𝐶4𝑇.(3.64) Since 𝑊2=(𝐶3−𝐶2)/2𝐶1=(−1/4), the coefficient of bulk viscosity 𝜉 will vanish.

The components of Hubble parameter 𝐻1, 𝐻2, and 𝐻3 are given by𝐻1=𝐻2=𝐻3=𝑅′𝑅=𝑆′𝑆=𝐶42𝐶coth4𝑇.(3.65)

Therefore the generalized mean Hubble parameter (𝐻) is1𝐻=3𝐻1+𝐻2+𝐻3=𝐶42𝐶coth4𝑇.(3.66) The metric (2.1), in this case, can be written as𝑑𝑠2𝐶=−42𝑊3coth2𝐶4𝑇−13/2𝑑𝑇2+𝐶42𝑊coth2𝐶4𝑇−11/2×𝑑𝜃2+sin2𝜃𝑑𝜙2+𝐶42𝑊coth2𝐶4𝑇−11/2(𝑑𝜑+cos𝜃𝑑𝜙)2.(3.67) Thus (3.67) together with (3.59) and (3.60) constitutes a Bianchi type IX vacuum cosmological model in Sen [3] theory of gravitation.

3.4. The Cosmological Models in the Absence of Bulk Viscosity

It is interesting to note that in the absence of bulk viscosity, by taking 𝜁=0 in (3.30), (3.46), and (3.64), we get the Bianchi types II, VIII, and IX perfect fluid string cosmological models, respectively, and if we assign the value zero to ∈ and ∈0, the present models reduce to string cosmological models in Sen [3] theory of gravitation.

4. Physical and Geometrical Properties

4.1. Bianchi Type II Cosmological Model (𝛿=0)

The spatial volume for the model is𝑉=(−𝑔)1/2𝐶32𝑊2𝑛+1/2coth2𝐶3𝑇−12𝑛+1/4.(4.1) The expression for expansion scalar 𝜃 calculated for the flow vector 𝑢𝑖 is given by𝜃=𝑢𝑖;𝑖=(2𝑛+1)(2𝑊)2𝑛+1/22𝐶32𝑛−1/2coth2𝐶3𝑇−1−(2𝑛+1)/4𝐶coth3𝑇,(4.2) and the shear 𝜎 is given by𝜎2=12𝜎𝑖𝑗𝜎𝑖𝑗=5(2𝑛+1)2(2𝑊)2𝑛+172𝐶32𝑛−1coth2𝐶3𝑇−1−(2𝑛+1)/2coth2𝐶3𝑇.(4.3) The deceleration parameter 𝑞 is given by𝑞=−3𝜃−2𝜃,𝑖𝑢𝑖+13𝜃2=6𝐶2𝑛+132𝑊2n+1/2coth2𝐶3𝑇−1(2𝑛+1)/4coth−2𝐶3𝑇×2𝑛+12coth2𝐶3𝑇+cosech2𝐶3𝑇−1.(4.4)

4.2. Bianchi Type VIII Cosmological Model (𝛿=−1)

The spatial volume for the model is𝑉=(−𝑔)1/2=𝐶42𝑊3/2coth2𝐶4𝑇−13/4cosh𝜃.(4.5) The expression for expansion scalar 𝜃 calculated for the flow vector 𝑢𝑖 is given by𝜃=𝑢𝑖;𝑖=3(2𝑊)3/22𝐶41/2coth2𝐶4𝑇−1−3/4𝐶coth4𝑇,(4.6) and the shear 𝜎 is given by𝜎2=12𝜎𝑖𝑗𝜎𝑖𝑗=5𝑊3𝐶4coth2𝐶4𝑇−1−3/2coth2𝐶4𝑇.(4.7) The deceleration parameter 𝑞 is given by𝑞=−3𝜃−2𝜃,𝑖𝑢𝑖+13𝜃2𝐶=242𝑊3/2coth2𝐶4𝑇−13/4coth−2𝐶4𝑇32coth2𝐶4𝑇+cosech2𝐶4𝑇−1.(4.8)

4.3. Bianchi Type IX Cosmological Model (𝛿=1)

The spatial volume for the model is𝑉=(−𝑔)1/2=𝐶42𝑊3/2coth2𝐶4𝑇−13/4sin𝜃.(4.9) The expression for expansion scalar 𝜃 calculated for the flow vector 𝑢𝑖 is given by 𝜃=𝑢𝑖;𝑖=3(2𝑊)3/22𝐶41/2coth2𝐶4𝑇−1−3/4𝐶coth4𝑇,(4.10) and the shear 𝜎 is given by𝜎2=12𝜎𝑖𝑗𝜎𝑖𝑗=5𝑊3𝐶4coth2𝐶4𝑇−1−3/2coth2𝐶4𝑇.(4.11) The deceleration parameter 𝑞 is given by𝑞=−3𝜃−2𝜃,𝑖𝑢𝑖+13𝜃2𝐶=242𝑊3/2coth2𝐶4𝑇−13/4coth−2𝐶4𝑇32coth2𝐶4𝑇+cosech2𝐶4𝑇−1.(4.12)

5. Conclusion

In this paper we have presented Bianchi types II, VIII, and IX string cosmological models with bulk viscosity in a theory of gravitation proposed by Sen [3] based on Lyra [1] geometry. It is observed that in case of Bianchi type IX only vacuum cosmological model exists. The models have no initial singularity at 𝑇=0 and the spatial volume is decreasing as time 𝑇 increases; that is, all the three models are contracting. For Bianchi type II cosmological model, the energy density 𝜌 and the total pressure 𝑝 will tend to infinity as 𝑇 approaches to zero, if 2𝑛−1<0. Also the expansion scalar 𝜃 and the shear scalar 𝜎 for this model will tend to infinity as 𝑇 approaches to zero, if 2𝑛+1<0. But the energy density 𝜌, the total pressure 𝑝, the expansion scalar 𝜃, and the shear scalar 𝜎 will tend to zero as 𝑇 approaches to zero for Bianchi types VIII and IX cosmological models. Since the deceleration parameter 𝑞 is greater than zero for all the models, they represent decelerating universes. Since lim𝑇→∞𝜎2/𝜃2=5/18≠0, the models do not approach isotropy for large values of 𝑇.

Acknowledgment

K. V. S. Sireesha is grateful to the Department of Science and Technology (DST), New Delhi, India for providing INSPIRE fellowship.