A Dark Energy Model with Higher Order Derivatives of  <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06579971pt" id="M1" height="11.4135pt" version="1.1" viewBox="-0.0657574 -11.3477 14.7205 11.4135" width="14.7205pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-73" d="M870 650H600L593 618C681 609 684 605 669 516L641 367H261L286 516C302 603 313 610 391 618L398 650H132L123 618C214 610 218 606 200 516L127 130C113 46 107 40 25 32L18 0H290L298 32C204 40 200 42 214 130L250 321H631L596 131C580 47 573 39 484 32L475 0H753L762 32C673 39 668 46 685 130L757 516C772 606 778 610 861 618L870 650Z"/></g></svg> in the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.8905pt" id="M2" height="15.1511pt" version="1.1" viewBox="-0.0657574 -12.2606 48.0324 15.1511" width="48.0324pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g58-100" d="M56 439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z"/></g><g transform="matrix(.017,0,0,-0.017,6.732,0)"><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,-0.017,12.875,0)"><path id="g58-80" d="M639 21C616 28 593 39 566 68C542 92 523 117 486 170C457 215 415 282 397 314C472 346 524 398 524 485C524 545 497 590 456 617C417 641 369 650 296 650H43V617C122 612 130 608 130 523V129C130 45 122 38 38 33V0H306V33C225 38 216 45 216 129V282H249C302 282 319 270 336 243C365 195 401 132 436 84C483 18 525 -5 597 -8C610 -9 622 -9 635 -9L639 21ZM216 319V559C216 586 219 596 227 603S250 613 281 613C356 613 429 576 429 469C429 416 413 379 374 348C349 327 316 319 265 319H216Z"/></g><g transform="matrix(.017,0,0,-0.017,23.842,0)"><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,-0.017,30.819,0)"><path id="g58-82" d="M601 496C596 555 591 627 591 675H567C551 656 544 650 516 650H120C91 650 79 651 64 675H41C38 619 34 555 29 492H63C75 537 83 566 96 582C110 604 128 612 207 612H272V126C272 46 264 38 170 33V0H464V33C368 38 359 48 359 126V612H430C502 612 520 605 535 582C547 566 556 541 567 492L601 496Z"/></g><g transform="matrix(.017,0,0,-0.017,41.596,0)"><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 Model

Pasqua, Antonio; Chattopadhyay, Surajit; Myrzakulov, Ratbay

doi:https://doi.org/10.1155/2014/535010

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Research Article | Open Access

Volume 2014 | Article ID 535010 | https://doi.org/10.1155/2014/535010

A Dark Energy Model with Higher Order Derivatives of in the Modified Gravity Model

Antonio Pasqua,¹Surajit Chattopadhyay,²and Ratbay Myrzakulov³

Academic Editor: C. A. D. S. Pires, B. Bilki

Received20 Sept 2013

Accepted23 Oct 2013

Published29 Jan 2014

Abstract

We consider a model of dark energy (DE) which contains three terms (one proportional to the squared Hubble parameter, one to the first derivative, and one to the second derivative with respect to the cosmic time of the Hubble parameter) in the light of the modified gravity model, with and being two constant parameters. and represent the curvature and torsion scalars, respectively. We found that the Hubble parameter exhibits a decaying behavior until redshifts (when it starts to increase) and the time derivative of the Hubble parameter goes from negative to positive values for different redshifts. The equation of state (EoS) parameter of DE and the effective EoS parameter exhibit a transition from to (showing a quintom-like behavior). We also found that the model considered can attain the late-time accelerated phase of the universe. Using the statefinder parameters and , we derived that the studied model can attain the phase of the universe and can interpolate between dust and phase of the universe. Finally, studying the squared speed of sound , we found that the considered model is classically stable in the earlier stage of the universe but classically unstable in the current stage.

1. Introduction

The late-time accelerated expansion of the universe (which is well-established from different cosmological observations) [1, 2] is a major challenge for cosmologists. The universe underwent two phases of accelerated expansion: the inflationary stage in the very early universe and a late-time acceleration in which our universe entered only recently. Models trying to explain this late-time acceleration are dubbed as dark energy (DE) models. An important step toward the comprehension of the nature of DE is to understand whether it is produced by a cosmological constant or it is originated from other sources dynamically changing with time [3]. For good reviews on DE see [4–6].

In a recent paper, Nojiri and Odintsov [7] described the reasons why modified gravity approach is extremely attractive in the applications for late accelerating universe and DE. Another good review on modified gravity was made by Clifton et al. [8]. Many different theories of modified gravity have been recently proposed: some of them are (with being the Ricci scalar curvature) [9, 10], (with being the torsion scalar) [11–14], Hořava-Lifshitz [15, 16], and Gauss-Bonnet [17–20] theories.

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, we describe some important features of the gravity. The recent motivation for studying gravity came from the necessity to explain the apparent late-time accelerating expansion of the universe. Detailed reviews on gravity can be found in [21–24]. Thermodynamic aspects of gravity have been investigated in the works of [25, 26]. A generalization of the modified theory of gravity that includes an explicit coupling of an arbitrary function of with the matter Lagrangian density leads to a non-geodesic motion of massive particles and an extra force, orthogonal to the four-velocity, arises. [27]. Harko et al. [28] recently suggested an extension of standard general relativity, where the gravitational Lagrangian is given by an arbitrary function of and and called this model . The model depends on a source term, representing the variation of the matter stress-energy tensor with respect to the metric. A general expression for this source term can be obtained as a function of the matter Lagrangian . In a recent paper, Myrzakulov [29] proposed gravity model and studied its main properties of FRW cosmology. Moreover, Myrzakulov [30] recently derived exact solutions for a specific model which is a linear combination of and , that is, , where and are two free constant parameters. Moreover, it was demonstrated that, for some specific values of and , the expansion of universe results to be accelerated without the necessity to introduce extra dark components. Recently, Chattopadhyay [31] studied the properties of interacting Ricci DE considering the model . Pasqua et al. [32] recently considered the modified holographic Ricci dark energy (MHRDE) model in the context of the specific model we are considering in this work. Moreover, Alvarenga et al. [33] studied the evolution of scalar cosmological perturbations in the metric formalism in the framework of modified theory of gravity.

In this work, we consider a DE model proposed in the recent paper of Chen and Jing [34]. The DE model considered contains three different terms, one proportional to the squared Hubble parameter, one to the first derivative with respect to the cosmic time of the Hubble parameter, and one proportional to the second derivative with respect to the cosmic time of the Hubble parameter: where , , and are three positive constant parameters. The first term is divided by the Hubble parameter in order that all the three terms have the same dimensions. The energy density given in (1) can be considered as an extension and generalization of other two DE models widely studied in recent time, that is, the Ricci DE (RDE) model and the DE energy density with Granda-Oliveros cut-off. In fact, in the limiting case corresponding to , we obtain the energy density of DE with Granda-Oliveros cut-off, and in the limiting case corresponding to , , and , we recover the RDE model for flat universe (i.e., with curvature parameter equal to zero).

In this work we are considering DE interacting with pressureless DM which has energy density . Various forms of interacting DE models have been constructed in order to fulfil the observational requirements. Many different works are presently available where the interacting DE have been discussed in detail. Some examples of interacting DE are presented in [35–40].

This work aims to reconstruct the DE model considered under gravity and it is organized as follows. In Section 2, we describe the main features of the model. In Section 3, we consider the energy density of DE given in (1) in the context of gravity considering the particular model considered. In Section 4, we study the statefinder parameters and for the energy density model we are considering in this work. In Section 5, we write a detailed discussion about the results found in this work. Finally, in Section 6, we write the Conclusions of this work.

2. The Model

The metric of a spatially flat, homogeneous, and isotropic universe in Friedmann-Lemaitre-Robertson-Walker (FLRW) model is given by where represents a dimensionless scale factor (which gives information about the expansion of the universe), indicates the cosmic time, represents the radial component, and and are the two angular coordinates.

We also know that the tetrad orthonormal components are related to the metric through the following relation:

The Einstein field equations are given by where and indicate (choosing units of ) the total energy density and the total pressure, respectively. The conservation equation is given by where

We must emphasize here that we are considering pressureless DM (). Since the components do not satisfy the conservation equation separately in presence of interaction, we reconstruct the conservation equation by introducing an interaction term which can be expressed in any of the following forms [41]: , , and .

In this paper, we consider as interaction term the second of the three forms mentioned above. Accordingly, the conservation equation is reconstructed as where indicates an interaction constant parameter which gives information about the strength of the interaction between DE and DM. The present day value of is still not known exactly and it is under debate.

One of the most interesting models of gravity is the so-called -model, whose action is given by [29] where is defined as (with being the determinant of the metric tensor ), is the matter Lagrangian, is the curvature scalar, and is the torsion scalar.

In this paper, we consider the following expressions for the curvature scalar and for the torsion scalar given, respectively, by

We now consider the particular case corresponding to and , where is the derivative of the scale factor with respect to the cosmic time . Moreover, the scale factor , the torsion scalar , and the curvature scalar are considered as independent dynamical variables. Then, after some algebraic calculations, the action given in (9) can be rewritten as where the Lagrangian is given by

The quantities , , , and are, respectively, the first derivative of with respect to , the first derivative of with respect to , the second derivative of with respect to , and the second derivative of with respect to and .

The equations of gravity are usually more complicated with respect to the equations of Einstein’s theory of general relativity even if the FLRW metric is considered. For this reason, as stated before, we consider the following simple particular model of gravity: with and being two constant parameters.

The equations system of this model of gravity is given by where We get from (14)

Then (16) can be rewritten as follows: or equivalently

The above system has two equations and five unknown functions, which are , , , , and .

We now assume the following expressions for and, : where , , , and are real constants. We also have that and can be expressed as

Then, the system made by (18) leads to

Finally, we have that the EoS parameter for this model is given by the relation

3. Interacting DE in Gravity

Solving the differential equation for given in (8), we derive the following expression for : where indicates the present day value of .

Using (1) and (25) in the right-hand side of (21), we obtain the following expression of as function of the scale factor: where and are two constants of integration.

In order to have a real and definite expression of given in (26), the following conditions must be satisfied:(1),(2),(3),(4),(5).

We can now derive the expressions of the first and the second time derivative of the Hubble parameter , that is, and , as functions of the scale factor differentiating (26) with respect to the cosmic time : Using (26), (27), and (28) in (1), we obtain the following expression of the energy density :

Taking into account the expression of given in (29), we derive that the expression of the pressure of DE is given by

Using the expressions of the energy density and the pressure of DE given, respectively, in (29) and (30), and the expression of given in (25), we get the EoS parameter for DE and the total EoS parameter as follows:

We must remember here that we are considering the case of pressureless DM, so that .

We now want to consider the properties of the deceleration parameter for the model we are considering. The deceleration parameter is generally defined as follows: where the expressions of and are given, respectively, in (26) and (27). The deceleration parameter, the Hubble parameter , and the dimensionless energy density parameters , , and (which will be considered and studied in the following Sections) are a set of useful parameters if it is needed to describe cosmological observations.

4. The Statefinder Parameters

In order to have a better comprehension of the properties of the DE model taken into account, we can compare it with a model independent diagnostics which is able to differentiate between a wide variety of dynamical DE models, including the model. We consider here the diagnostic, also known as statefinder diagnostic, which introduces a pair of parameters defined, respectively, as follows:

Using (26), (27), and (29), we get the statefinder parameters as with

5. Discussion

In this Section, we discuss the behavior of the physical quantities derived in the previous sections. We have considered the following values for the parameters involved: , , , , , , , , , , , and . We considered three different cases corresponding to three different values of the parameter , that is, , and .

In Figure 1, we plotted the expression of the Hubble parameter , obtained from (26), as function of the redshift . It is evident that the Hubble parameter has a decaying behavior with varying values of the parameter and the redshift going from higher to lower redshifts. However, this decaying pattern is apparent till . In fact, in a very late stage , it shows an increasing pattern.

In Figure 2, we have plotted the time derivative of Hubble parameter against the redshift . We have observed that for , transits from negative to positive side at . However, for and this transition occurs at lower redshift . In Figures 3 and 4 we have plotted, respectively, the equation of state (EoS) parameter for DE, defined as , and the effective EoS parameter, defined as , for the three different values of considered in this work. In Figure 3, we have observed that for , crosses the phantom divide at . For the phantom divide is crossed at . However, for , the equation of state (EoS) parameter for DE stays below . Thus, for and , transits from quintessence to phantom, that is, has a quintom-like behavior. Instead, for , the EoS parameter has a phantom-like behavior. In Figure 4, we have plotted the effective EoS parameter . In this case, for all values of considered, there is a crossing of phantom divide. Moreover, for , crosses the phantom divide earlier with respect to the other cases, in particular for .

The deceleration parameter has been plotted as a function of in Figure 5. For , and , there is a transition from positive to negative , that is, transition from decelerated to accelerated expansion. For , the deceleration parameter changes sign at , and for , it changes sign at . However, for , the deceleration parameter always stays at negative level. Thus, for , we are getting ever-accelerating universe.

Next, we have plotted in Figure 6 the fractional density of DE, given by , and the fractional density of matter, given by , against the redshift . is defined as . The solid lines correspond to and the dashed lines correspond to . In this Figure, there is a clear indication of transition of the universe from dark matter dominated phase to the dark energy dominated phase. At very early stage of the universe , the dark energy density is largely dominated by dark matter density. We denote the cross-over point by and it comes out to be , that is, where for all values of considered in this work. Hence, the model, based on which we have reconstructed DE density, is capable of achieving the present DE dominated universe from the earlier dark matter dominated universe.

Sahni et al. [42] recently demonstrated that the statefinder diagnostic is effectively able to discriminate between different models of DE. Chaplygin gas, braneworld, quintessence, and cosmological constant models were investigated by Alam et al. [43] using the statefinder diagnostics; they observed that the statefinder pair could differentiate between these different models. An investigation on statefinder parameters for differentiating between DE and modified gravity was carried out in [44]. Statefinder diagnostics for gravity has been well studied in Wu and Yu [45]. In the plane, corresponds to a quintessence-like model of DE and corresponds to a phantom-like model of DE. Moreover, an evolution from phantom to quintessence or inverse is given by crossing of the fixed point in plane [45], which corresponds to scenario. The statefinder parameters have been plotted in Figure 7 for different values of the parameter . It is clearly visible that the trajectories are converging towards the fixed point . Thus, the model is capable of attaining the phase of the universe. Furthermore, for finite , . Thus, the model can interpolate between dust and phase of the universe.

Finally, in Figure 8, we plotted the squared speed of the sound, defined as , where the upper dot indicates derivative with respect to the cosmic time for the model we are considering as a function of . The sign of the squared speed of sound is fundamental in order to study the stability of a background evolution. A negative value of implies a classical instability of a given perturbation in general relativity [46, 47]. Myung [47] recently observed that the squared speed of sound for HDE stays always negative if the future event horizon is considered as IR cutoff, while for Chaplygin gas and tachyon is observed to benonnegative. Kim et al. [46] found that for Agegraphic DE (ADE) stays always negative, which leads to the instability of the perfect fluid for the model. Moreover, it was found that the ghost QCD [48] DE model is unstable. In a recent work, Sharif and Jawad [49] have shown that interacting new HDE is characterized by negative squared speed of the sound.

In the recent work of Pasqua et al. [32], authors observed that the interacting modified holographic Ricci DE (MHRDE) model in gravity is classically stable.

Jawad et al. [50] showed that model in HDE scenario with power-law scale factor is classically unstable.

Pasqua et al. [51] showed that the DE model based on generalized uncertainty principle (GUP) with power-law form of the scale factor is instable.

We have observed that, for all values of , is positive till the redshift . However, after this stage it enters the negative region. Thus, although the model is classically stable in the early universe, for present universe it is classically unstable.

6. Concluding Remarks

In this work, we have considered a recently proposed model of energy density of DE which depends on three terms, one proportional to the squared Hubble parameter , one proportional to the time derivative of , and one proportional to the second time derivative of interacting with pressureless DM in the framework of the modified gravity theory for the special model given by , where and represent two constants. The DE model considered here reduces to other two well-studied DE model (the Ricci DE model and the DE energy density model with Granda-Oliveros cut-off) for some particular values of the three parameters involved, that is, , , and . We have derived the expressions and studied the behavior of some important physical quantities which gave useful hints about the model studied. The Hubble parameter exhibits a decaying behavior going from higher to lower redshifts until the redshift of about , when it starts to increase. The time derivative of the Hubble parameter, that is, , shows a transition from negative to positive values for different values of the redshift according to the value of considered. We have observed that the equation of state (EoS) parameter of DE exhibits a transition from to , that is, transition from quintessence to phantom (i.e., quintom) with the evolution of the universe for and , instead it is always negative for . Moreover, the effective equation of state (EoS) parameter always shows a transition from quintessence to phantom. Hence, we can conclude that the reconstructed model based on the gravity model considered leads to an equation of state (EoS) parameter that has a quintom-like behavior. We have further observed that the said model is capable of attaining the dark energy dominated accelerated phase of the universe from dark matter dominated decelerated phase of the universe. Through statefinder trajectories we have shown that the is capable of attaining the phase of the universe and can interpolate between dust and phase of the universe. Through squared speed of sound we have seen that the model under consideration is classically stable in the earlier stage of the universe but classically unstable in the current stage.

Conflict of Interests

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

Acknowledgment

The second author acknowledges financial support from the Department of Science and Technology, Govtvernment of India, under project Grant no. SR/FTP/PS-167/2011.

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Copyright © 2014 Antonio Pasqua et al. 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.

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