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Bianchi Type III Anisotropic Cosmological Models in Gravity With Constant Deceleration Parameter

Pramod P. Khade, *Vinay M. Raut

Vidya Bharati Mahavidylaya, Amravati-444402.

E-mail: [email protected] *Shri Shivaji Science College, Amravati

[email protected] Abstract

The general class of anisotropic Bianchi cosmological models in modified theories of gravity with constant deceleration parameter have been considered. This paper deals with accelerating cosmological models in

a modified theory dubbed as gravity at the backdrop of an anisotropic Bianchi type III universe.

is a function of Ricci scalar R and the trace T of the energy momentum tensor and it replaces the Ricci scalar in the Einstein-Hilbert action of General Relativity. The exact solutions to the corresponding field equations are obtained in quadrature form. We have discussed two types of solutions of the average scale factor by using a special law for deceleration parameter. The solutions to the Einstein field equations are obtained for two different physical viable cosmologies. The physical aspects of the dark energy model is discussed.

Keywords: Dark energy; Anisotropic Bianchi Type III Space-time; gravity

1. Introduction Cosmology is a discipline to understand the nature of the origin, evolution, large scale structure, and ultimate fate of the universe. In the early 20th century the common world-view held that the Universe is static Å more or less the same throughout eternity. Even Albert Einstein supported this long-standing idea, and in order to get the steady state Universe he introduced cosmological constant in his famous system. So, when in 1922 the Russian meteorologist and mathematician Alexander Friedmann had published a set of possible mathematical solutions that gave a non static Universe [1, 2], Einstein rejected it noting that this model was indeed a mathematically possible solution to the field equations. This model has gained big popularity only after the works of Robertson and Walker and became known as FRW model. This model describes a homogeneous and isotropic Universe. By homogeneity we mean that space has the same metric properties (measures) in all points, whereas by isotropy we mean that the space has the same measures in all directions. This idea of expanding Universe suggested the presence of an initial singularity, which means the finiteness of time. Observation plays a major role in modern cosmology. The advent of new technologies in observations enforces the theorists to rethink on the formulation of the gravitational theories time to time. Einstein had to drop the cosmological constant from the field equations with the discovery of Hubble. The concept of decelerating expansion of the Universe had to drop by the theorists with the observation of type Ia supernovae in 1998. Since then CMB, BAO, SDSS and many more observations provide evidences in support of the accelerating expansion of the Universe. So, it is very important to take care of the observational results while building a theoretical model of the Universe. The accelerating expansion of the Universe is an important feature of present day cosmology. The Einstein field equations (EFEs) always lead to a decelerating expansion with the normal matter component in the Universe. The accelerating expansion can be described either by supplying some extra component in the energy momentum tensor part in the field equations or by doing some modifications in the geometrical part. With these principles, the past few years of research produced a plethora of cosmological models of the Universe explaining the accelerating expansion. Dark energy is a form of matter (energy) not observable in laboratory and it does not interact with electromagnetic radiation. These facts played decisive role in naming this object. In contrast to dark matter Å dark energy is uniformly distributed over the space; Å it does not intertwine under the influence of gravity in all scales; Å it has a strong negative pressure of the order of energy density. Based on these properties, cosmologists have suggested a number of dark energy models, those are able to explain the current accelerated phase of expansion of the Universe. What Dark Energy is? More is unknown than is known. We know how much dark energy there is because we know how it affects the Universe

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expansion. Other than that, it is a complete mystery. But it is an important mystery. The Wilkinson Microwave Anisotropy Probe (WMAP) measures the composition of the Universe. The top chart shows a pie chart of the relative constituents today. A similar chart (bottom) shows the composition at 380,000 years old (13.7 billion years ago) when the light WMAP observes emanated. The composition varies as the Universe expands: the dark matter and atoms become less dense as the Universe expands, like an ordinary gas, but the photon and neutrino particles also lose energy as the Universe expands, so their energy density decreases faster than the matter. They formed a larger fraction of the Universe 13.7 billion years ago. It appears that the dark energy density does not decrease at all, so it now dominates the Universe even though it was a tiny contributor 13.7 billion years ago. WMAP satellite experiment suggests 73% content of the Universe in the form of dark energy, 23% in the form of nonbaryonic dark matter and the rest 4% in the form of the usual baryonic matter as well as radiation. The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: empty space can possess its own energy. Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence, more of this energy-of-space would appear. As a result, this form of energy would cause the Universe to expand faster and faster. Unfortunately, no one understands why the cosmological constant should even be there, much less why it would have exactly the right value to cause the observed acceleration of the Universe.

EoS Parameter as Dark Energy. In addition to the models mentioned above recently attempt to describe dark energy by using a time-dependent parameter of equation of state [3-9] has been taken. In some of these models, it is assumed that the deceleration parameter is a constant. This yields two types of solutions, one of which is in the form of power function; while the second, exponential. These solutions describe the expanding nonsingular and singular Universes, respectively. The range of values for the equation of state w in both cases is in good agreement with recent observational data, namely: (i) SNe Ia data in [10], (ii) SNe Ia data collaborated with CMBR anisotropy and galaxy clustering statistics in [11], and (iii) a combination of cosmological datasets coming from CMB anisotropies, luminosity distances of high red-shift-type Ia supernovae and galaxy clustering [12,13]. Though the observations is in favor of a homogeneous and isotropic Universe, the possibility of anisotropic phase in the early Universe is also supported by some observations. Also the presence of anisotropy affect the evolution of energy density. This motivates the theorists to construct various models in different Bianchi space-times in different contexts [14-19]. Modified gravity models do not require dark energy to explain late time cosmic acceleration rather this aspect is taken care of by modifying the geometry. In this modified gravity framework, many authors have studied spatially homogeneous

Bianchi type cosmological models [20-24]. Irregularity energy density factor in theory that disturb the stability of homogeneous universe has been studied by Yousaf et al.[25] . Many authors have investigated the astrophysical and cosmological implications of the (�, �) gravity [26–29]. Jamil et al. [30] have reconstructed some cosmological models for some specific forms of (�, �) in this modified gravity. Shamir et al. [31] obtained exact solution of anisotropic Bianchi type I and type V cosmological models whereas Chaubey and Shukla [32] have obtained a newclass of Bianchi cosmological models using special law of variation of parameter. Using a decoupled form of (�, �), that is, (�, �) = (�) + (�) for Bianchi type V universe, Ahmed and Pradhan [33] have studied the energy conditions of perfect fluid cosmological models and Yadav [34] obtained some string solutions. Pawar and Solanke [35] have studied cosmological model filled with perfect fluid source in (�, �) gravity. Pawar and Agrawal [36] have obtained the solutions of dark energy cosmological model in the framework of the (�, �) theory of gravity. Recently Pawar et al. [37] have explored two fluid cosmological models in (�, �) theory. Mishra and Sahoo [38] solved the field equations of Bianchi type-VIh cosmological model in presence of perfect fluid in f (R,T) gravity. Sahoo and Mishra [39] studied Kaluza–Klein dark energy model in form of wet dark fluid in this theory. Reddy et al. [30] presented Kantowski-Sachs bulk viscous string model in (�, �) theory. Recently, Naidu et al. [41], Kiran and Reddy [42], and Reddy et al. [43] discussed the Bianchi type-V, Bianchi type-III, Kaluza-Klein space time with cosmic strings, and bulk viscosity in �(�, �) gravity, respectively. Caroll et al.[44], Nojiri and Odintsov [45-47] and Chiba et al.[48] are some of the authors who have investigated several aspects of f (R) gravity. Recently, Adhav [49] has obtained Bianchi type-I cosmological model in f (R,T ) gravity. Reddy et al.[50, 51] have discussed Bianchi type-

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III and Kaluza-Klein cosmological models in f (R,T ) gravity while Reddy and Shantikumar [52] studied some anisotropic cosmological models and Bianchi type-III dark energy model, respectively, in f (R,T ) gravity. Subsequently Kiran and Reddy [53] established the non-existence of Bianchi type-III bulk viscous string cosmological model in f (R,T ) gravity. Recently, Naidu et al. [54] presented Bianchi type-V bulk viscous string model in f (R,T ) gravity while Reddy et al. [55] have obtained the same in Saez-Ballester theory.

The above discussion and the investigations have inspired us to take up the investigation of dark

energy model in Bianchi type-III space time in gravity in the presence of anisotropic dark

energy. The paper is organized as follows. In section 2, A brief Review of gravity is described. The metric and field equations are presented in section 3. The exact solutions of the field equations are derived for an anisotropic Bianchi type III universe in section. The kinematic features of the models are discussed in section 4. In section 5, the conclusions of the present work are presented.

2. A Brief Review of gravity:

Harko et al. [56] developed another modified gravity known as gravity. In this theory, the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace of T of

the stress energy tensor. In this paper, we concentrate on gravity, with �being in this case a function of both � and �, manifesting a coupling between matter and geometry. Before going into the

details of (�, �) gravity, The field equations of gravity obtained from the action

(1)

where is an arbitrary function of the Ricci scalar is the stress energy tensor of matter

and is the matter Lagrangian density. In the present work, we have considered the matter Lagrangian

density to be , where is pressure of the cosmic fluid. We choose the system of units:

G is the Newtonian Gravitational constant and c is the speed of light in vacuum). The gravity field equations are obtained from the action S as

(2)

Where and and are the partial differentiation

of the respective functional with respect to their arguments. being the covariant

derivative. The stress energy tensor for a cosmic fluid with pressure p and energy density is given by

where is the four velocity vector in co-moving coordinate system. It is evident from equation (2) that the physical nature of the matter field decides the behavior of the field

equations of theory of gravity. Therefore different choice of the matter source will lead to

different cosmological models in gravity. In other words, one can construct viable cosmological

models with different choices of the functional . However, Harko et al. [56] have constructed

three possible models by considering the functional to be either of

or where

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are some arbitrary function of R and T. These functions may be chosen arbitrarily and the obtained results may then be matched with observations concerning late time acceleration or can be reconstructed from some plausible physical basis such as energy conditions and cosmic thermodynamics. Following four earlier work [57] we have considered here two

models: with and with linear functions

and . The field equations for the two specific choices reduces to

(3)

And (4) It is obvious that, because of the presence of linear functions of R and T, the above two specific choices

of the functional will behave a like and overlap for the specific choices of the model

parameters in view of this, we expect similar dynamics of the universe in both the models [57]. 3. Metric and field equations

The field equations in the modified gravity theory for the two specific choices of the functional in the backdrop of an anisotropic and spatially homogeneous universe modeled through a Bianchi type III metric

(5) Can be explicitly written as

(6)

(7)

(8)

(9)

(10)

where and are the metric functions of cosmic time only and is an arbitrary constant.

are constants and are decided from the choice of the functional In the above field equations, an overhead dot ( .) represents derivative with respect to time t . for the first case with

we have and whereas for the second case with the choice

we can have Here we have used the fact that trace of the

energy momentum tensor for our model is .

Integrating equation (10), we obtain where k is the positive constant of integration. Incorporation of equation (10) into the field equations (6)-(9) yields

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(11)

(12)

(13) Now we have three linearly independent equations (11)-(13) and five un known parameters. Two additional constraints relating to these parameters are required to obtain explicit solutions of the system. i) The law of variation of Hubble’s parameter that yields a constant value of deceleration parameter

Such type of relation have already been considered by Berman [58] for solving FRW models

ii) We assume that expansion in the model is proportional to the shear . This condition leads to

(14)

where n is proportionality constant. The average scale factor of Bianchi type III metric is given by

(15) The volume scale factor V is given by

(16)

is the expansion scalar defined by

(17) We define, generalized mean Hubble’s parameter H as

(18) where

are the directional Hubble’s parameter in the direction of x, y and z respectively. Equation (), may be reduces to

(19) Since the line element (5) is completely characterized by Hubble’s parameter H. Therefore, let us consider that mean Hubble parameter H is related to average scale factor R by following relation

(20)

where are constants. The deceleration parameter is defined as

(21) From equations (19) and (20), we get

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(22)

(23) Using (15) and (21), Equation (19) leads to

(24) The sign of q indicates whether the model inflates or not. The positive sign of q corresponds to standard decelerating model where as the negative sign of q indicates inflation. However the current observations

of SN Ia and CMBR favour accelerating models i.e. From equation (21), we obtain the law of average scale factor R as

(25)

where are the constant of integration.

From equation (24), for , it is clear that the condition for expansion of universe is

Therefore for expansion model of universe the deceleration parameter should not be greater than -1.

Case (i): when : Equation (10), (19) and (25) lead to

(26) Equation (14) and (26) lead to

(27) From equations (10) and (26), we obtain

(28)

where is the constant of integration and

Thus the Hubble’s parameter H, scalar of expansion and shear scalar are given by

(29)

(30)


(32)

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The average anisotropy parameter can be expressed as

(33) Using equations (11-13), (26) and (27), the energy density and pressure of the fluid is obtained as

(34)

(35)

Consequently, the effective dark energy equation of state (EoS) becomes

(36) The Trace T of the model is expressed as

(37)

Here we observe that energy density and pressure tends to infinity at . Cosmological

parameters tends to infinity at .Hence we can observe that the universe starts evolving

zero volume at and expands with cosmic time. We observe that , model does not approach isotropy at any time t. Scale factors and metric potentials are increasing function of time. Here we noticed that Hubble parameter is a decreasing function of time and it approaches towards zero with the evolution of time.

Case (ii) : When Equation (9), (14), (19) and (25) lead to


(39)

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From equations (10) and (38), we obtain

(40)

Where is the constant of integration and

Thus the Hubble’s parameter H, scalar of expansion and shear scalar are given by

(41)

(42)


(44)

The average anisotropy parameter can be expressed as

(45) Using equations (11-13), (39) and (40), the energy density and pressure of the fluid is obtained as

(46)

(47)

The effective dark energy equation of state becomes

(48) Trace energy is given by

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(49)

It is observed that energy density and pressure are decreasing function of time. The deceleration

parameter and , which gives the greatest value of H and the fastest rate of expansion. This type of solutions are consistent as per the recent observations for an accelerated expansion of the universe. From the anisotropy parameter it is observed that the model of the universe remains anisotropic throughout the evolution. 4. Conclusion: In this article, we have presented a new solutions to the field equations by using the law of variation of Hubbles parameter which yield constant deceleration parameter. An anisotropic Bianchi type III universe in the framework of modified gravity with constant deceleration parameter is studied, in two cases. where the geometry part of the Einstein-Hilbert action contains some stuff of the matter field. In this

context, we have considered two functional forms of . The choices of these models is based upon the idea that they should , more or less, behave like GR. Since

in both cases, , the model does not approach isotropy at any time. Therefore we cannot rule out the possibility of anisotropic nature of DE for Binachi Type III framework. To find the solution of the model we consider the law of variation of Hubble’s parameter that yields a constant value of deceleration parameter. We obtain two types of solutions, one of which is in the form of power function; while the second, exponential. References: [1] A.A.Friedmann: Z. Phys. 1922. V.10. P.377Ä386. [2] A.A. Friedmann: Z. Phys. 1924. V.21. P.326Ä332. [3] H. Amirhashchi, A. Pradhan,B. Saha : Astrophys. Space Sci. 2011. V.333. P.295Ä303. [4] H. Amirhashchi, A. Pradhan, B. Saha : Chin. Phys. Lett. 2011. V.3. P.039801. [5] A. Pradhan, H. Amirhashchi, B. Saha : Intern. J. Theor. Phys. 2011. V.50. P.2923Ä 2938. [6] A. Pradhan, H. Amirhashchi, B. Saha :Astroph. Space Sci. 2011. V.333. P.343Ä350. [7] B. Saha, H. Amirhashchi, A. Pradhan: Astrophys. Space Sci. 2012. V.342. P.257Ä267. [8] B. Saha, A. K. Yadav: Astrophys. Space Sci. 2012. V.341. P.651Ä656. [9] A. K. Yadav, B. Saha : Astrophys. Space Sci. 2012. V.337. P.759Ä765. [10] R.K. Knop et al.: Astrophys. J. 2003. V.598. P.102. [11] M. Tegmark et al.: Astrophys. J. 2004. V.606. P.702Ä740. [12] G. Hinshaw et al.: Astrophys. J. Suppl. Ser. 2009. V.180. P.225Ä245. [13] E. Komatsu et al.: Astrophys. J. Suppl. Ser. 2009. V.180. P.330Ä376. [14] D. R. K. Reddy, R. Santi Kumar: Astrophys. Space Sci. 344 (2013) 253. [15] P. K. Sahoo, M. Sivakumar, Astrophys Space Sci 357 (2015) 60. [16] G. P. Singh, B. K. Bishi: Astrophys. Space Sci. 360 (2015) 34. [17] D. Sofuoglu: Astrophys. Space Sci., 361 (2016) 12. [18] P. K. Sahoo, Parbati Sahoo, B. K. Bishi: Int. J. Geom. Methods Mod. Phys. 14 (2017) 1750097. [19] P. K. Sahoo, Parbati Sahoo, B. K. Bishi, S. Aygun: Mod. Phys. Lett. A 32 (2017) 1750105. [20] K. L. Mahanta: Astrophys. Space Sci. 353, 683 (2014) [21] B. Mishra and P. K. Sahoo: Astrophys. Space Sci.352, 331 (2014) [22] M. F. Shamir: Int. J. Theor. Phys.54, 1304 (2015). [23] N. Ahmed and A. Pradhan: Int. J. Theor. Phys. 53, 289 (2014). [24] B. Mishra, S. Tarai and S. K. Tripathi: Adv. High. Energy Phys. 2016, 8543560 (2016). [25] Z. Yousaf, K. Bamba, M. Z. H. Bhatti: Phys. Rev. D93 124048 (2016). [26] R. Myrzakulov: The European Physical Journal C, vol. 72, article 2203, 2012.

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