Bianchi Type-I Universe with Cosmological Constant and Quadratic Equation of State in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.4924pt" id="M1" height="16.753pt" version="1.1" viewBox="-0.0657574 -12.2606 52.614 16.753" width="52.614pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-103" d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"/></g><g transform="matrix(.017,0,0,-0.017,11.059,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,17.202,0)"><path id="g50-83" d="M636 22C611 29 589 38 560 74C536 104 518 134 495 173L430 300C518 328 592 392 592 492C592 555 563 594 528 618C491 643 435 650 377 650H140L131 618C222 610 223 601 208 518L137 136C121 43 116 40 27 32L18 0H295L300 32C212 40 206 42 223 138L248 279H296C334 279 350 272 369 230C393 174 413 140 441 94C482 24 523 1 626 -8L636 22ZM497 480C497 422 468 374 417 343C387 324 353 315 304 315H255L298 556C303 583 309 598 319 603C331 610 348 613 367 613C448 613 497 573 497 480Z"/></g><g transform="matrix(.017,0,0,-0.017,28.116,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.017,0,0,-0.017,35.095,0)"><path id="g50-85" d="M655 676H628C608 656 600 650 571 650H156C118 650 114 651 102 676H77C63 615 45 548 24 485L60 486C81 531 97 562 114 580C134 603 148 611 252 611H308L217 127C201 41 196 38 105 32L98 0H384L393 32C300 38 288 41 304 127L393 611H466C553 611 571 604 582 580C592 559 597 530 595 485L630 489C636 548 645 624 655 676Z"/></g><g transform="matrix(.017,0,0,-0.017,46.168,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> Modified Gravity

Singh, G. P.; Bishi, Binaya K.

doi:https://doi.org/10.1155/2015/816826

Advances in High Energy Physics

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Research Article | Open Access

Volume 2015 | Article ID 816826 | https://doi.org/10.1155/2015/816826

Bianchi Type-I Universe with Cosmological Constant and Quadratic Equation of State in Modified Gravity

G. P. Singh¹and Binaya K. Bishi¹

Academic Editor: Enrico Lunghi

Received14 Aug 2015

Revised02 Nov 2015

Accepted08 Nov 2015

Published26 Nov 2015

Abstract

This paper deals with the study of Bianchi type-I universe in the context of gravity. Einstein’s field equations in gravity have been solved in the presence of cosmological constant and quadratic equation of state (EoS) , where is a constant. Here, we have discussed two classes of gravity; that is, and . A set of models has been taken into consideration based on the plausible relation. Also, we have studied some physical and kinematical properties of the models.

1. Introduction

It is known that [1–5] in the present scenario, our universe is accelerating. However, final satisfactory explanation about physical mechanism and driving force of accelerated expansion of the universe is yet to achieve as human mind has not achieved perfection. From the modern cosmology, it is known that a point of universe is filled with dark energy. It has been addressed by various slow rolling scalar fields. It is supposed that the dark energy is responsible for producing sufficient acceleration in the late time of evolution of the universe. Thus, it is much more essential to study the fundamental nature of the dark energy and several approaches have been made to understand it. The cosmological constant is assumed to be the simplest candidate of dark energy. It is the classical correction made to Einstein’s field equation by adding cosmological constant to the field equations. The introduction of cosmological constant to Einstein’s field equation is the most efficient way of generating accelerated expansion, but it faces serious problems like fine-tuning and cosmic coincidence problem in cosmology [6, 7]. Quintessence [8], phantom [9], k-essence [10], tachyons [11], and Chaplygin gas [12] are the other representative of dark energy. However, there is no direct detection of such exotic fluids. Researchers are taking an interest in exploring dark energy due to the lack of strong evidence of existence of dark energy. Several authors (Pimentel and Diaz-Rivera [13], Singh et al. [14], Singh et al. [15], and Jamil and Debnath [16]) have discussed cosmological model with cosmological constant in different contexts.

Dark energy can be explored in several ways, and modifying the geometric part of the Einstein-Hilbert action [17] is treated as the most efficient possible way. Based on its modifications, several alternative theories of gravity came into existence. Some of the modified theories of gravity are , , , and gravity. These models are proposed to explore the dark energy and other cosmological problems. Sharif and Azeem [18] discussed the Cosmological evolution for dark energy models in gravity. Jamil et al. [19] have studied the stability of the interactive models of the dark energy, matter, and radiation for a FRW model in gravity. Generalized second law of thermodynamics in gravity with entropy corrections has been studied by Bamba et al. [20]. In this work, they have used the power law and logarithmic corrected form of entropy for cosmological horizon and analysed the validity of the generalized second law of thermodynamics in specific scenarios of the quintessence and the phantom energy dominated eras. The modified theory produces both cosmic inflation and mimic behavior of dark energy, including present cosmic acceleration [21–23]. Amendola et al. [24] have discussed the cosmologically viable conditions in theory, which describe the dark energy models. Jamil et al. [25] have analysed the tachyon cosmology by the Noether symmetry approach. Azadi et al. [26] have discussed the static cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric theories of gravity. This article is devoted to construct the family of solutions with constant Ricci scalar explicitly and its possible relation to the Linet-Tian solution in general relativity. Momeni and Gholizade [27] discussed the constant curvature solutions in cylindrically symmetric metric gravity. In this paper, they have proved that, in gravity, the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in general relativity and further shown that constant curvature exact solution is applicable to the exterior of a string.

The basic paper on modified gravity was investigated by Harko et al. [28]. From the literature, it is found that Barrientos and Rubilar have pointed out that Harko et al. have missed an essential term, which has consequences in the equation of motion of test particles. Thus, the corrected derivation of this equation of motion is presented by Barrientos and Rubilar [29], who also discussed some of its consequences. Jamil et al. [30] have studied the reconstruction of some cosmological models in gravity, in which they have shown that dust fluid reproduces CDM, phantom-non-phantom era and the phantom cosmology. Jamil et al. [31] have proved that the first law of black hole thermodynamics is violated for gravity in general, but there might be some special case exit in which the first law of black hole thermodynamics is recovered. Momeni et al. [32] have investigated Noether symmetry issue for nonminimally model and mimetic . They have pointed out that Noether symmetry is able to provide a very excellent way to study cosmological implications of extended theories. We have observed from the literature that Bianchi type-I model is one of the important anisotropic cosmological models and hence it is widely studied in general relativity and alternative theories of gravitation. The Bianchi type-I model is discussed by Jamil et al. [33, 34] in different contexts. Recently, authors like Sahoo and Sivakumar [35], Ahmed and Pradhan [36], and Pradhan et al. [37] have investigated the cosmological models with cosmological constant in gravity for different Bianchi type space-time.

Quadratic equation of state is needed to explore in cosmological models due to its importance in brane world model and the study of dark energy and general relativistic dynamics for different models. The general form of the quadratic equation of state is given bywhere , , and are parameters. Equation (1) is nothing but the first term of Taylor expansion of any equation of state of the form about .

Nojiri and Odintsov [38] have studied the final state and thermodynamics of a dark energy universe, in which they discussed the model by considering the equation state of the form . Ananda and Bruni discussed the cosmological models by considering different form of nonlinear quadratic equation of state. Ananda and Bruni [39] have investigated the general relativistic dynamics of RW models with a nonlinear quadratic equation of state and analysed that the behaviour of the anisotropy at the singularity found in the brane scenario can be recreated in the general relativistic context by considering an equation of state of form (1). Also they have discussed the anisotropic homogeneous and inhomogeneous cosmological models in general relativity with the equation of state of the formand they tried to isotropize the universe at early times when the initial singularity is approached. Astashenok et al. [40] have analysed phantom cosmology without big rip singularity, in which they have considered the equation of state of the form . In our present study, we have considered the quadratic equation of state of the formwhere is a constant quantity and such type of consideration does not affect the quadratic nature of equation of state.

Nojiri and Odintsov [41] studied the effect of modification of general equation of state of dark energy ideal fluid by the insertion of inhomogeneous, Hubble parameter dependent term in the late-time universe. The quadratic equation of state may describe the dark energy or unified dark energy [41, 42]. Rahaman et al. [43] investigated the construction of an electromagnetic mass model using quadratic equation of state in the context of general theory of relativity. Feroze and Siddiqui [44] studied the general situation of a compact relativistic body by taking a quadratic equation of state for the matter distribution. Maharaj and Mafa Takisa [45] have investigated the regular models with quadratic equation of state. They have considered static and spherically symmetric space-time with a charged matter distribution and found new exact solutions to the Einstein-Maxwell system of equations which are physically reasonable.

A cosmological model based on a quadratic equation of state unifying vacuum energy, radiation, and dark energy has been discussed by Chavanis [46] and also a cosmological model describing the early inflation, the intermediate decelerating expansion, and the late accelerating expansion by a quadratic equation of state has been investigated by the same author [47]. Strange quark star model with quadratic equation of state has been investigated by Malaver [48] and they have obtained a class of models with quadratic equation of state for the radial pressure that correspond to anisotropic compact sphere, where the gravitational potential depends on an adjustable parameter . Recently, Reddy et al. [49] have studied the Bianchi type-I cosmological model with quadratic equation of state in the context of general theory of relativity.

Motivated by the aforesaid research, we have investigated the Bianchi type-I cosmological model in gravity with quadratic equation of state and cosmological constant. Here, we have discussed two classes of gravity.

2. Gravitational Field Equations of Modified Gravity Theory

Let us consider the action for the modified gravity aswhere is the arbitrary function of and . is the Ricci scalar and is the trace of the stress energy tensor of the matter . is the matter Lagrangian density. For the choice of , we will get the action for the different theories. If and , then (4) represents the action for gravity and general relativity, respectively. The stress energy tensor of matter is defined as and its stress by . If we consider that the matter Lagrangian density of matter depends only on and not on its derivatives, then it will lead us toBy varying action (4) with respect to the metric tensor component , we havewhere Here, , , is the De Alembert’s operator, and is the standard matter energy momentum tensor derived from the Lagrangian . By contracting (7), we obtained the relation between and aswhere . From (7) and (9), the gravitational field equations can be written asThe perfect fluid form of the stress energy tensor of the matter Lagrangian is given bywhere is the four-velocity vector and satisfies the relation and . and are the energy density and pressure of the fluid, respectively. From (8), we haveIt is to note that the functional depends on the physical nature of the matter field through tensor . Thus, each choice of leads us to different cosmological models. Harko et al. [28] presented three classes of as follows: In this present work, we have discussed two classes of ; that is, and .

For the choice of and with the help of (11) and (12), (7) takes the formwhich is the gravitational field equation in modified gravity for the class . For the choice of and with the help of (11) and (12), (7) takes the formwhich is regarded as the gravitational field equation in modified gravity for the class .

3. Field Equations and Cosmological Model for

In theory, the gravitational field equation (14) in the presence of cosmological constant is given as where prime denotes differentiation with respect to the argument. For the choice of , (16) takes the formLet us consider the Bianchi type-I space-time in the formwhere , , and are function of only. The field equation (17) for the line element (18) takes the form

4. Solution Procedure

Now, our problem is to solve Einstein’s modified field equations (19)–(22). Here, the system has four equations and six unknowns (, , , , , and ). To obtain the complete solution, we need two more physically plausible relations. The considered two physically plausible relations are(1)quadratic equation of state;(2)expansion law:(a)power law:(b)exponential law: where and are the positive constant quantity. According to the choice of expansion law, we have obtained two different models of the Bianchi type-I universe.

4.1. Power Law Model

With the help of (20)–(22), we have obtained the metric potentials as where and are constant of integration which satisfies the relation and . From (19)-(20) and along with (3), we have gotUsing (23) in (25), we have the metric potential asThe directional Hubble parameters are obtained as , . The Hubble parameter , deceleration parameter , expansion scalar , and Shear scalar are as follows: Using the observational value for [50], we have restricted as in case of power law model. Here, we noticed that , , and die out for larger values of . With the help of (27) from (26), the energy density is obtained as Using (29) in (3), we have the pressure as follows: With the help of (27)–(30) from (19), the cosmological constant is obtained as

Figures 1 and 2 show the variation of energy density and pressure against time for different values as in the figures. Here, we noticed that , when . In the increase of , energy density and pressure increase and decrease, respectively. Figure 3 represents the variation of cosmological constant against time for different values as in the figures. It is observed that cosmological constant is decreasing with the increase of and with the evolution of time it approaches towards zero. Variation of different energy conditions against time for different is presented in Figure 4. We observed from the figure that DEC (dominant energy condition, ) is satisfied, but NEC (null energy condition, ) and SEC (strong energy condition, ) are violated in this case. This violation may be responsible for the accelerated expansion of the universe.

(a)

(b)

(c)

4.2. Exponential Law Model

In this case, with the help of (24) in (25), we have found the metric potential asThe directional Hubble parameters are obtained as , . The Hubble parameter , deceleration parameter , expansion scalar , and Shear scalar are as follows: Here, we noticed that die out for larger values of . From (32) and (26), the energy density is expressed asUsing (34) in (3), the pressure is expressed asWith the help of (32)–(35) from (19), the cosmological constant is obtained as

Figures 5 and 7 show the variation of energy density and pressure against time for different values as in the figures. Here, we noticed that , when . In the increase of , energy density and pressure decrease, respectively. Variation of different energy conditions against time for different is presented in Figure 6. We observed from the figure that NEC and DEC are satisfied, but in this case SEC is violated. This violation may be responsible for the accelerated expansion of the universe. Figure 8 represents the variation of cosmological constant against time for different values as in the figures. It is observed that cosmological constant is not approaching towards zero with the evolution of time and also it takes negative values.

(a)

(b)

(c)

5. Field Equations and Cosmological Model for

In theory, the gravitational field equation (15) for the choice of and , along with cosmological constant , is given asIn this case, the field equations are given by

5.1. Power Law Model

Following the same procedure as in Section 4.1, we have obtained the same metric potential as in (27) and the other parameters like energy density , pressure , and cosmological constant are expressed as follows:

Here, also we have noticed similar qualitative results as in Section 4.1 (see Figures 9–12).

(a)

(b)

(c)

5.2. Exponential Law Model

Following the same procedure as in Section 4.2, we have obtained the same metric potential as in (34), and the other parameters like energy density , pressure , and cosmological constant are as follows:

Here, also we have noticed the similar qualitative results as in Section 4.2 (see Figures 13–16).

(a)

(b)

(c)

6. Concluding Remarks

In this paper, we have the Bianchi type-I cosmological model in modified gravity for two different classes of in the presence of cosmological constant and quadratic equation of state. Here, we have discussed two models based on the expansion law. From both models, case of , we have concluded the following points.(i)In both the models, energy density is decreasing function of and approaches towards zero with the evolution of time.(ii)In both the models, pressure is negative and approaches towards zero with the evolution of time.(iii)In both the models, cosmological constant is negative, but here we notice that in case of power law approaches towards zero with the evolution of time whereas it does not approach towards zero with the evolution of time in case of exponential law. Similar qualitative observations are also noticed for the case of . Here, all the observation are in fare agreement with the observational data.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2015 G. P. Singh and Binaya K. Bishi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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