1. Introduction

It is well known that Einstein’s general theory of relativity has been successful in developing various cosmological models to study a structure formation and early stages of evolution of the universe. Einstein pointed out that general relativity does not account satisfactorily for inertial properties of matter i.e. Mach’s principle is not substantiated by general relativity. Several attempts have been made to generalize the general theory of gravitation by incorporating Mach’s principle and other desired features which were lacking in the original theory. Alternative theories of gravitation have been proposed to Einstein’s theory to incorporate certain desirable features in the general theory. In the last decades, as an alternative to general relativity, scalar tensor theories and modified theories of gravitation have been proposed. Brans and Dicke [1], Saez and Ballester [2] have much importance amongst the scalar-tensor theories of gravitation. Recently gravity and gravity [3, 4] theories have much importance amongst the modified theories of gravity because these theories are supposed to provide natural gravitation alternatives to dark energy. From the cosmological observations, it is known that the energy composition of the universe has 76% dark energy, 20% dark matter and 4% Baryon matter. This is confirmed by the high red shift supernovae experiments [5, 6] and cosmic microwave background radiation observations [7, 8]. Coperland et al. [9] have given a comprehensive review of gravity and Harko et al. [4] proposed theory of gravity as an alternative amongst the modified theory of gravity of Einstein’s theory of gravitation. Several authors have studied theory of gravity in different contexts [10-15]. Recently Chirde and Kadam [16] have studied A new class of Bianchi type bulk viscous and string cosmological model in theory of gravitation.

In theory of gravity, the field equations are obtained from the Hilbert-Einstein type variation principle.

(1)

where is an arbitrary function of the Ricci scalar R and the trace T of the stress-energy tensor of the matter is the matter Lagrangian density.

The stress-energy tensor of matter is defined as [17]

(2)

and its trace by respectively. By assuming that the Lagrangian density of matter depends only on the metric tensor components and not on its derivatives, we obtain

In the present paper, we use the natural system of units with so that the Einstein gravitational constant is defined as .

The corresponding field equations of the gravity are found by varying the action with respect to the metric [4]:

(3)

where ,

is the covariant derivative and is the standard matter energy-momentum tensor derived from the Lagrangian .

Contracting (3) gives the relation between the Ricci scalar and the trace of the stress-energy tensor,

(4)

where .

Generally the field equations also depend [through the tensor ] on the physical nature of the matter field. Hence several theoretical models corresponding to different matter sources in gravity can be obtained.

By considering some particular class of modified gravity model obtained by explicitly specifying the functional form of . Assuming that the function given by

(5)

where is an arbitrary function of the trace of the stress-energy tensor of matter.

From equation (3), the gravitational field equations are given by

(6)

where the prime denotes a derivative with respect to the argument.

If the matter source is a perfect fluid,

then the above equation (6) reduces to

(7)

where the prime denotes derivative with respect to the argument.

The Deceleration parameters (q) are useful in studying expansion of the universe. We know that the universe has

(i) decelerating expansion if ,

(ii) an expansion with constant rate if ,

(iii) accelerating power law expansion if ,

(iv) exponential expansion (de-Sitter expansion) if

(v) super-exponential expansion if .

Many relativists have studied DE cosmological models with different form of deceleration parameters. Singha and Debnath [18], Adhav et al. [19, 20] have obtained DE models with special form of deceleration parameter.

The study of Bianchi type cosmological models are important in achieving better understanding of anisotropy in the universe. Moreover the anisotropic universes have greater generality than FRW isotropic models. The simplicity of the field equations made Bianchi type space-times useful. Bianchi type I-IX cosmological models are homogeneous and anisotropic. Bianchi type-IX universe are studied by the number of cosmologists because of familiar solutions like Robertson–Walker Universe, the de-sitter universe, the Taub-Nut solutions etc. Chakraborty [21], Bali and Dave [22], Bali and Yadav [23], Pradhan et al. [24], Tyagi et al. [25], Ghate and Sontakke [26, 27] have obtained Bianchi type-IX cosmological models in different contexts.

Bulk viscosity plays a very important role in getting accelerated expansion of the universe. The matter behaves like viscous fluid in the early stages of the universe when neutrinos decoupling occurs. The effect of viscosity on the evolution of the universe had been studied by Misner [28, 29]. Several authors have studied bulk viscous cosmological models in general relativity [30-37] Johri and Sudharsan [38], Pimental [39], Banerjee and Beesham [40], Singh et al. [41], Rao et al. [42], Naidu et al. [43], Reddy et al. [44] have studied bulk viscous string cosmological models in scalar-tensor theories of gravitation. Recently Naidu et al. [45], Reddy et al. [46] have studied Bianchi type-V and Kaluza-Klein cosmological models with bulk viscosity and cosmic strings in theory of gravitation. Motivated by the above discussion and investigations in modified theory of gravity, in this paper we have investigated Bianchi type-IX bulk viscous string cosmological model in theory of gravity with special form of deceleration parameter.

Bianchi type-IX space-time has considered when universe is filled with cosmic strings and bulk viscosity in theory of gravity with special form of deceleration parameter. This work is organized as follows: In Section 2, the model and field equations have been presented. The field equations have been solved in Section 3 by applying special form of deceleration parameter. The physical and geometrical behavior of the model have been discussed in Section 4. In the last Section 5, concluding remarks have been expressed.

2. Metric and Field Equations

Bianchi type-IX metric is considered in the form,

(8)

where a, b are scale factors and are functions of cosmic time t.

(9)

and

(10)

where is the rest energy density of the system, is the coefficient of bulk viscosity, is bulk viscous pressure, H is Hubble’s parameter, is the direction of the string and is the string tension density. Also is a four-velocity vector which satisfies . Here we consider, and as function of time t only.

We assume the string to be lying along the x-axis. One-dimensional strings are assumed to be loaded with particles and the particle energy density is .

In co-moving coordinate system, we get

(11)

Now we choose the function of the trace of the stress-energy tensor of the matter so that

(12)

where is constant.

In the co-moving coordinate system, the field equations (7) for the metric (8) and with the help of (9)-(12) can be written as

(13)

(14)

(15)

where the overdot denotes the differentiation with respect to t.

3. Solutions of Field Equations

The field equations (13)–(15) are a system three independent equations in six unknowns . Three additional conditions relating these unknowns may be used to obtain explicit solutions of the systems.

(i) Firstly, we assume that the expansion in the model is proportional to the shear . This condition leads to

(16)

where m is proportionality constant.

The motive behind assuming condition is explained with reference to Thorne [47], the observations of the velocity red-shift relation for extragalactic sources suggest that Hubble expansion of the universe is isotropic today within percent [48, 49]. Collin et al. [50] have pointed out that for spatially homogeneous metric, the normal congruence to the homogeneous expansion satisfies that the condition is constant.

(ii) Secondly for a barotropic fluid, the combined effect of the proper pressure and the bulk viscous pressure can be expressed as

(17)

where

Here are constants.

(iii) Lastly following Singha and Debnath [17], we use a special form of deceleration parameter as:

(18)

where R is the average scale factor, is constant.

Solving equation (18), the average scale factor R is given by

(19)

where and are constants of integration.

For the metric (8), the scale factor R is given by

(20)

Solving field equations (13)-(15), with the help of equations (19) and (20), we obtain

(21)

(22)

Using equations (21) and (22), the metric (8) takes the form

(23)

Equation (23) represents Bianchi type-IX bulk viscous strings cosmological model in gravity with special form of deceleration parameter.

4. Some Physical Properties of the Model

For the cosmological model (23), the physical quantities Spatial volume , Hubble parameter , Expansion scalar , Mean Anisotropy parameter , Shear scalar , Energy density , String tension density , Bulk viscosity pressure , Coefficient of bulk viscosity and Particle energy density are obtained as follows :

Spatial volume,

(24)

Hubble parameter,

(25)

Expansion scalar,

(26)

Mean Anisotropy Parameter,

(27)

Shear scalar,

(28)

where

(29)

Energy density,

(30)

String tension density,

(31)

Particle energy density,

(32)

Bulk viscous pressure,

(33)

Coefficient of bulk viscosity,

(34)

4.1. Physical Behavior of the Model

For the cosmological model (23), we observed that, the spatial scale factors has finite value at the initial epoch and increases for large values of t. In particular for and , the model has a big bang type singularity. It can also be observed that, at , the universe starts evolving with finite volume and expands infinitely with the increase in cosmic time t (Figure 1). The values of expansion and shear tend to infinity for large values of showing that the universe is expanding with increase of time (Figure 3, 4).

From equations (27) and (29), the mean anisotropy parameter and indicates that the model is anisotropic throughout the evolution of the universe except at (i.e. the model does not approach isotropy).

In figure 5, the plot of deceleration parameter verses time is given from which we conclude that the model is decelerating at an initial phase and changes the expansion from decelerating to accelerating. Hence the model is consistent with the recent cosmological observations [5-6, 51-55].

Bulk viscosity in the model decreases with increase in time which is in accordance with the well-known fact that bulk viscosity decreases with time and leads to inflationary model [56].

For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically (Figure 1-5).

Figure 1. The Plot of Volume verses time

Figure 2. The Plot of Hubble parameter verses time

Figure 3. The Plot of Expansion scalar verses time

Figure 4. The Plot of Shear scalar verses time

Figure 5. The Plot of deceleration parameter verses time

5. Conclusions

Bianchi type-IX bulk viscous strings cosmological model has been discussed in the frame work of gravity proposed by Harko et al. (Phys. Rev. D 84:024020, 2011) when the source for energy momentum tensor is bulk viscous fluid containing one dimensional cosmic strings. The solution of the field equations has obtained by choosing the special form of deceleration parameter . It is observed that in early phase of universe, the value of deceleration parameter is positive while as , the value of . Hence the universe had a decelerated expansion in the past and has accelerated expansion at present which is in good agreement with the recent observations of SN Ia. It is worth to mention that, the model obtained is point type singular, expanding, shearing, non-rotating and do not approach isotropy for large t. Further the models are anisotropic throughout the evolution. This shows that the present model not only represent accelerating universe but also confirms the well-known fact that the bulk viscosity will play a vital role in getting the accelerated expansion. We hope that our model will be useful in the discussion of structure formation in the early universe and an accelerating expansion of the universe at present.