Research ArticleA Dark Energy Model with Higher Order Derivatives of𝐻 in the 𝑓(𝑅, 𝑇)Modified Gravity Model

Antonio Pasqua,1 Surajit Chattopadhyay,2 and Ratbay Myrzakulov3

1 Department of Physics, University of Trieste, Via Valerio 2, 34127 Trieste, Italy2 Pailan College of Management and Technology, Bengal Pailan Park, Kolkata 700 104, India3 Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan

Correspondence should be addressed to Surajit Chattopadhyay; surajit 2008@yahoo.co.in

Received 20 September 2013; Accepted 23 October 2013; Published 29 January 2014

Academic Editors: B. Bilki and C. A. D. S. Pires

Copyright © 2014 Antonio Pasqua et al.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a model of dark energy (DE) which contains three terms (one proportional to the squared Hubble parameter, one tothe first derivative, and one to the second derivative with respect to the cosmic time of the Hubble parameter) in the light of the𝑓(𝑅, 𝑇) = 𝜇𝑅+]𝑇modified gravitymodel, with 𝜇 and ] being two constant parameters.𝑅 and𝑇 represent the curvature and torsionscalars, respectively. We found that the Hubble parameter exhibits a decaying behavior until redshifts 𝑧 ≈ −0.5 (when it starts toincrease) and the time derivative of the Hubble parameter goes from negative to positive values for different redshifts.The equationof state (EoS) parameter of DE and the effective EoS parameter exhibit a transition from𝜔 < −1 to 𝜔 > −1 (showing a quintom-likebehavior). We also found that the model considered can attain the late-time accelerated phase of the universe. Using the statefinderparameters 𝑟 and 𝑠, we derived that the studied model can attain theΛCDMphase of the universe and can interpolate between dustandΛCDMphase of the universe. Finally, studying the squared speed of sound V2

𝑠, we found that the consideredmodel 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 (whichis well-established from different cosmological observations)[1, 2] is a major challenge for cosmologists. The universeunderwent two phases of accelerated expansion: the infla-tionary stage in the very early universe and a late-time accel-eration in which our universe entered only recently. Modelstrying to explain this late-time acceleration are dubbed asdark energy (DE) models. An important step toward thecomprehension of the nature of DE is to understand whetherit is produced by a cosmological constantΛ or it is originatedfrom other sources dynamically changing with time [3]. Forgood reviews on DE see [4–6].

In a recent paper, Nojiri and Odintsov [7] described thereasonswhymodified gravity approach is extremely attractivein the applications for late accelerating universe and DE.Another good review on modified gravity was made byClifton et al. [8]. Many different theories of modified gravity

have been recently proposed: some of them are 𝑓(𝑅) (with𝑅 being the Ricci scalar curvature) [9, 10], 𝑓(𝑇) (with 𝑇being the torsion scalar) [11–14], Horava-Lifshitz [15, 16], andGauss-Bonnet [17–20] theories.

In this paper, we concentrate on 𝑓(𝑅, 𝑇) gravity, with 𝑓being in this case a function of both 𝑅 and 𝑇, manifesting acoupling betweenmatter and geometry. Before going into thedetails of 𝑓(𝑅, 𝑇) gravity, we describe some important fea-tures of the 𝑓(𝑅) gravity. The recent motivation for studying𝑓(𝑅) gravity came from the necessity to explain the apparentlate-time accelerating expansion of the universe. Detailedreviews on 𝑓(𝑅) gravity can be found in [21–24]. Thermody-namic aspects of 𝑓(𝑅) gravity have been investigated in theworks of [25, 26]. A generalization of the 𝑓(𝑅)modified the-ory of gravity that includes an explicit coupling of an arbitraryfunction of 𝑅 with the matter Lagrangian density 𝐿

𝑚leads

to a non-geodesic motion of massive particles and an extraforce, orthogonal to the four-velocity, arises. [27]. Harko et al.[28] recently suggested an extension of standard general

Hindawi Publishing CorporationISRN High Energy PhysicsVolume 2014, Article ID 535010, 11 pageshttp://dx.doi.org/10.1155/2014/535010

2 ISRN High Energy Physics

relativity, where the gravitational Lagrangian is given by anarbitrary function of 𝑅 and 𝑇 and called this model 𝑓(𝑅, 𝑇).The 𝑓(𝑅, 𝑇) model depends on a source term, representingthe variation of the matter stress-energy tensor with respectto the metric. A general expression for this source term canbe obtained as a function of the matter Lagrangian 𝐿

𝑚. In

a recent paper, Myrzakulov [29] proposed 𝑓(𝑅, 𝑇) gravitymodel and studied its main properties of FRW cosmology.Moreover, Myrzakulov [30] recently derived exact solutionsfor a specific 𝑓(𝑅, 𝑇)model which is a linear combination of𝑅 and 𝑇, that is, 𝑓(𝑅, 𝑇) = 𝜇𝑅 + ]𝑇, where 𝜇 and ] are twofree constant parameters.Moreover, it was demonstrated that,for some specific values of 𝜇 and ], the expansion of universeresults to be accelerated without the necessity to introduceextra dark components. Recently, Chattopadhyay [31] studiedthe properties of interacting Ricci DE considering the model𝑓(𝑅, 𝑇) = 𝜇𝑅 + ]𝑇. Pasqua et al. [32] recently considered themodified holographic Ricci dark energy (MHRDE) model inthe context of the specific 𝑓(𝑅, 𝑇) model we are consideringin this work. Moreover, Alvarenga et al. [33] studied theevolution of scalar cosmological perturbations in the metricformalism in the framework of 𝑓(𝑅, 𝑇) modified theory ofgravity.

In this work, we consider a DE model proposed in therecent paper of Chen and Jing [34]. The DE model consid-ered contains three different terms, one proportional to thesquared Hubble parameter, one to the first derivative withrespect to the cosmic time of the Hubble parameter, andone proportional to the second derivative with respect to thecosmic time of the Hubble parameter:

𝜌DE = 𝜀��

𝐻+ 𝜆�� + 𝜃𝐻2, (1)

where 𝜀, 𝜆, and 𝜃 are three positive constant parameters.The first term is divided by the Hubble parameter 𝐻 inorder that all the three terms have the same dimensions. Theenergy density given in (1) can be considered as an extensionand generalization of other two DE models widely studiedin recent time, that is, the Ricci DE (RDE) model and theDE energy density with Granda-Oliveros cut-off. In fact, inthe limiting case corresponding to 𝜀 = 0, we obtain theenergy density of DE with Granda-Oliveros cut-off, and inthe limiting case corresponding to 𝜀 = 0, 𝜆 = 1, and 𝜃 = 2, werecover the RDE model for flat universe (i.e., with curvatureparameter 𝑘 equal to zero).

In this work we are considering DE interacting withpressureless DMwhich has energy density 𝜌

𝑚. Various forms

of interacting DE models have been constructed in order tofulfil the observational requirements. Many different worksare presently available where the interacting DE have beendiscussed in detail. Some examples of interacting DE arepresented in [35–40].

This work aims to reconstruct the DE model consideredunder 𝑓(𝑅, 𝑇) gravity and it is organized as follows. InSection 2, we describe the main features of the 𝑓(𝑅, 𝑇) =𝜇𝑅+]𝑇model. In Section 3, we consider the energy density ofDE given in (1) in the context of 𝑓(𝑅, 𝑇) gravity consideringthe particular model considered. In Section 4, we study thestatefinder parameters 𝑟 and 𝑠 for the energy density model

we are considering in this work. In Section 5, we write adetailed discussion about the results found in this work.Finally, in Section 6, we write the Conclusions of this work.

2. The 𝑓(𝑅,𝑇) = 𝜇𝑅 + ]𝑇 Model

The metric of a spatially flat, homogeneous, and isotropicuniverse in Friedmann-Lemaitre-Robertson-Walker (FLRW)model is given by

𝑑𝑠2 = 𝑑𝑡2 − 𝑎2 (𝑡) [𝑑𝑟2 + 𝑟2 (𝑑𝜃2 + sin2𝜃𝑑𝜑2)] , (2)

where 𝑎(𝑡) represents a dimensionless scale factor (whichgives information about the expansion of the universe), 𝑡indicates the cosmic time, 𝑟 represents the radial component,and 𝜃 and 𝜑 are the two angular coordinates.

We also know that the tetrad orthonormal components𝑒𝑖(𝑥𝜇) are related to themetric through the following relation:

𝑔𝜇] = 𝜂

𝑖𝑗𝑒𝑖𝜇𝑒𝑗]. (3)

The Einstein field equations are given by

𝐻2 =1

3𝜌,

�� = −1

2(𝜌 + 𝑝) ,

(4)

where 𝜌 and 𝑝 indicate (choosing units of 8𝜋𝐺 = 𝑐 = 1) thetotal energy density and the total pressure, respectively. Theconservation equation is given by

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0, (5)

where

𝜌 = 𝜌DE + 𝜌𝑚,

𝑝 = 𝑝DE.(6)

We must emphasize here that we are considering pres-sureless DM (𝑝

𝑚= 0). Since the components do not

satisfy the conservation equation separately in presence ofinteraction, we reconstruct the conservation equation byintroducing an interaction term 𝑄 which can be expressedin any of the following forms [41]:𝑄 ∝ 𝐻𝜌DE,𝑄 ∝ 𝐻𝜌

𝑚, and

𝑄 ∝ 𝐻(𝜌𝑚+ 𝜌DE).

In this paper, we consider as interaction term the secondof the three formsmentioned above. Accordingly, the conser-vation equation is reconstructed as

𝜌DE + 3𝐻 (𝜌DE + 𝑝DE) = 3𝐻𝛿𝜌𝑚, (7)

𝜌𝑚+ 3𝐻𝜌

𝑚= −3𝐻𝛿𝜌

𝑚, (8)

where 𝛿 indicates an interaction constant parameter whichgives information about the strength of the interactionbetween DE and DM. The present day value of 𝛿 is still notknown exactly and it is under debate.

ISRN High Energy Physics 3

One of the most interesting models of 𝑓(𝑅, 𝑇) gravity isthe so-called𝑀

37-model, whose action 𝑆 is given by [29]

𝑆 = ∫𝑓 (𝑅, 𝑇) 𝑒𝑑4𝑥 + ∫𝐿

𝑚𝑒𝑑4𝑥, (9)

where 𝑒 is defined as 𝑒 = det(𝑒𝑖𝜇) = √−𝑔 (with 𝑔 being the

determinant of the metric tensor 𝑔𝜇]), 𝐿𝑚 is the matter

Lagrangian, 𝑅 is the curvature scalar, and 𝑇 is the torsionscalar.

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

𝑅 = 𝑢 + 6 (�� + 2𝐻2) ,

𝑇 = V − 6𝐻2.(10)

We now consider the particular case corresponding to𝑢 = 𝑢(𝑎, 𝑎) and V = V(𝑎, 𝑎), where 𝑎 is the derivative of thescale factor with respect to the cosmic time 𝑡. Moreover, thescale factor 𝑎(𝑡), the torsion scalar𝑇, and the curvature scalar𝑅 are considered as independent dynamical variables. Then,after some algebraic calculations, the action given in (9) canbe rewritten as

𝑆37

= ∫𝑑𝑡𝐿37, (11)

where the Lagrangian 𝐿37is given by

𝐿37

= 𝑎3 (𝑓 − 𝑇𝑓𝑇− 𝑅𝑓𝑅+ V𝑓𝑇+ 𝑢𝑓𝑅) ,

−6 (𝑓𝑅+ 𝑓𝑇) 𝑎 𝑎2 − 6 (𝑓

𝑅𝑅�� + 𝑓𝑅𝑇��) 𝑎2 𝑎 − 𝑎3𝐿

𝑚.

(12)

The quantities 𝑓𝑅, 𝑓𝑇, 𝑓𝑅𝑅, and 𝑓

𝑅𝑇are, respectively, the

first derivative of 𝑓 with respect to 𝑅, the first derivative of 𝑓with respect to 𝑇, the second derivative of 𝑓 with respect to𝑅, and the second derivative of 𝑓 with respect to 𝑅 and 𝑇.

The equations of 𝑓(𝑅, 𝑇) gravity are usually more com-plicated with respect to the equations of Einstein’s theory ofgeneral relativity even if the FLRWmetric is considered. Forthis reason, as stated before, we consider the following simpleparticular model of 𝑓(𝑅, 𝑇) gravity:

𝑓 (𝑅, 𝑇) = 𝜇𝑅 + ]𝑇, (13)

with 𝜇 and ] being two constant parameters.The equations system of this model of 𝑓(𝑅, 𝑇) gravity is

given by

𝜇𝐷1+ ]𝐸1+ 𝐾 (𝜇𝑅 + ]𝑇) = −2𝑎3𝜌,

𝜇𝐴1+ ]𝐵1+𝑀(𝜇𝑅 + ]𝑇) = 6𝑎2𝑝,

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0,

(14)

where

𝐷1= −6𝑎 𝑎2 + 𝑎3𝑢

𝑎𝑎 − 𝑎3 (𝑢 − 𝑅)

= 6𝑎2 𝑎 + 𝑎3 𝑎𝑢𝑎= 𝑎3 (6

𝑎

𝑎+ 𝑎𝑢𝑎) ,

𝐸1= −6𝑎 𝑎2 + 𝑎3 𝑎V

𝑎− 𝑎3 (V − 𝑇)

= −12𝑎 𝑎2 + 𝑎3 𝑎V𝑎= 𝑎3 (−12

𝑎2

𝑎2+ 𝑎V𝑎) ,

𝐾 = −𝑎3,

𝐴1= 12 𝑎2 + 6𝑎 𝑎 + 3𝑎2 𝑎𝑢

𝑎+ 𝑎3𝑢

𝑎− 𝑎3𝑢𝑎,

𝐵1= −24 𝑎2 − 12𝑎 𝑎 + 3𝑎2 𝑎V

𝑎+ 𝑎3V

𝑎− 𝑎3V𝑎,

𝑀 = −3𝑎2.

(15)

We get from (14)

−6 (𝜇 + ])𝑎2

𝑎2+ 𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢 − ]V = −2𝜌,

− 2 (𝜇 + ]) (𝑎2

𝑎2+ 2

𝑎

𝑎) + 𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢

− ]V +𝜇

3𝑎 (��𝑎− 𝑢𝑎) +

]3𝑎 (V𝑎− V𝑎) = 2𝑝,

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0.

(16)

Then (16) can be rewritten as follows:

3 (𝜇 + ])𝑎2

𝑎2−1

2(𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢 − ]V) = 𝜌,

(𝜇 + ]) (𝑎2

𝑎2+ 2

𝑎

𝑎) −

1

2(𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢 − ]V)

−𝜇

6𝑎 (��𝑎− 𝑢𝑎) −

]6𝑎 (V𝑎− V𝑎) = −𝑝,

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0,

(17)

or equivalently

3 (𝜇 + ])𝐻2 −1

2(𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢 − ]V) = 𝜌,

(𝜇 + ]) (2�� + 3𝐻2) −1

2(𝜇 𝑎𝑢

𝑎+ ] 𝑎V

𝑎− 𝜇𝑢 − ]V)

−𝜇

6𝑎 (��𝑎− 𝑢𝑎) −

]6𝑎 (V𝑎− V𝑎) = −𝑝,

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0.

(18)

The above system has two equations and five unknownfunctions, which are 𝑎, 𝜌, 𝑝, 𝑢, and V.

We now assume the following expressions for 𝑢 and, V:

𝑢 = 𝛼𝑎𝑛,

V = 𝛽𝑎𝑚,(19)

4 ISRN High Energy Physics

where 𝑚, 𝑛, 𝛼, and 𝛽 are real constants. We also have that 𝑢and V can be expressed as

𝑢 = 𝛼(V𝛽)𝑛/𝑚

,

V = 𝛽(𝑢

𝛼)𝑚/𝑛

.

(20)

Then, the system made by (18) leads to

3 (𝜇 + ])𝐻2 +1

2(𝜇𝛼𝑎𝑛 + ]𝛽𝑎𝑚) = 𝜌, (21)

(𝜇 + ]) (2�� + 3𝐻2) +𝜇𝛼 (𝑛 + 3)

6𝑎𝑛 +

]𝛽 (𝑚 + 3)

6𝑎𝑚 = −𝑝,

(22)

𝜌 + 3𝐻 (𝜌 + 𝑝) = 0. (23)

Finally, we have that the EoS parameter 𝜔 for this modelis given by the relation

𝜔=𝑝

𝜌= −1−

2 (𝜇+]) ��−(𝜇/6) 𝑎 (��𝑎−𝑢𝑎)−(]/6) 𝑎 (V 𝑎−V𝑎)

3 (𝜇+])𝐻2−(1/2) (𝜇 𝑎𝑢𝑎+] 𝑎V

𝑎−𝜇𝑢−]V)

.

(24)

3. Interacting DE in 𝑓(𝑅,𝑇) Gravity

Solving the differential equation for 𝜌𝑚given in (8), we derive

the following expression for 𝜌𝑚:

𝜌𝑚= 𝜌𝑚0𝑎−3(1+𝛿), (25)

where 𝜌𝑚0

indicates the present day value of 𝜌𝑚.

Using (1) and (25) in the right-hand side of (21), we obtainthe following expression of𝐻2 as function of the scale factor:

𝐻2 = 𝐶1𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 𝐶2𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝛼𝜇𝑎𝑛

𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]

+𝛽]𝑎𝑚

𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]

− (2𝑎−3(1+𝛿)𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

−3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

,

(26)

where 𝐶1and 𝐶

2are two constants of integration.

In order to have a real and definite expression of𝐻2 givenin (26), the following conditions must be satisfied:

(1) 𝜀 = 0,(2) 𝜆2 − 8𝜀[𝜃 − 3(𝜇 + ])] ≥ 0,

(3) 𝑛2𝜀 + 𝑛𝜆 + 2[𝜃 − 3(𝜇 + ])] = 0,(4) 𝑚2𝜀 + 𝑚𝜆 + 2[𝜃 − 3(𝜇 + ])] = 0,(5) 9(1 + 𝛿)2 + 2𝜃 − 3[𝜆(1 + 𝛿) + 2(𝜇 + ])] = 0.

We can now derive the expressions of the first and thesecond time derivative of the Hubble parameter𝐻, that is, ��and ��, as functions of the scale factor 𝑎 differentiating (26)with respect to the cosmic time 𝑡:

�� = −𝐶1

2(

𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝐶2

2(

−𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝑛𝛼𝜇𝑎𝑛

2 {𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+𝑚𝛽]𝑎𝑚

2 {𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+ (3 (1 + 𝛿) 𝑎−3(1+𝛿)𝜌

𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

− 3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])])−1

,

(27)

�� = 𝐻{{{{{

𝐶1

2(

𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

2

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝐶2

2(

−𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

2

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝑛2𝛼𝜇𝑎𝑛

2 {𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+𝑚2𝛽]𝑎𝑚

2 {𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

− (9(1 + 𝛿)2𝑎−3(1+𝛿)𝜌

𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

− 3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1}}}}}

.

(28)

ISRN High Energy Physics 5

Using (26), (27), and (28) in (1), we obtain the followingexpression of the energy density 𝜌DE:

𝜌DE =1

2[

(𝑛2𝜀 + 2𝜃 + 𝑛𝜆) 𝛼𝜇𝑎𝑛

{𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+(𝑚2𝜀 + 2𝜃 + 2𝑚𝜆) 𝛽]𝑎𝑚

{𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+ 6𝐶1(𝜇 + ]) 𝑎−(𝜆+√𝜆

2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 6𝐶2(𝜇 + ]) 𝑎(−𝜆+√𝜆

2−8𝜀[𝜃−3(𝜇+])])/2𝜀

− (2 [9(1 + 𝛿)2𝜀 + 2𝜃 − 3 (1 + 𝛿) 𝜆] 𝑎

−3(1+𝛿)𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

−3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

] .

(29)

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

𝑝DE = 𝐶1(𝜇 + ])(

−6𝜀 + 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

× 𝐶2(𝜇 + ])(

6𝜀 − 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

− ([2𝜃 + 𝑛 (𝑛𝜀 + 𝜆)] (𝑛 + 3) 𝛼𝜇𝑎𝑛)

× (6 {𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]})−1

− ([2𝜃 + 𝑛 (𝑛𝜀 + 𝜆)] (𝑚 + 3) 𝛽]𝑎𝑚)

× (6 {𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]})−1

− ([9(1 + 𝛿)2𝜀 + 2𝜃 − 3 (1 + 𝛿) 𝜆] 𝑎

−3(1+𝛿)𝛿𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃 − 3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])])

−1

.

(30)

Using the expressions of the energy density 𝜌DE and thepressure 𝑝DE of DE given, respectively, in (29) and (30), and

the expression of 𝜌𝑚given in (25), we get the EoS parameter

𝜔DE for DE and the total EoS parameter 𝜔tot as follows:

𝜔DE =𝑝DE𝜌DE

,

𝜔tot =𝑝DE

𝜌DE + 𝜌𝑚

.

(31)

We must remember here that we are considering the caseof pressureless DM, so that 𝑝

𝑚= 0.

We now want to consider the properties of the decel-eration parameter 𝑞 for the model we are considering. Thedeceleration parameter 𝑞 is generally defined as follows:

𝑞 = −1 −𝑎 𝑎

𝑎2= −1 −

��

𝐻2, (32)

where the expressions of 𝐻2 and �� are given, respectively,in (26) and (27). The deceleration parameter, the Hubbleparameter𝐻, and the dimensionless energy density parame-ters ΩDE, Ω𝑚, and Ω

𝑘(which will be considered and studied

in the following Sections) are a set of useful parameters if it isneeded to describe cosmological observations.

4. The Statefinder Parameters

In order to have a better comprehension of the properties ofthe DE model taken into account, we can compare it with amodel independent diagnostics which is able to differentiatebetween a wide variety of dynamical DE models, includingthe ΛCDM model. We consider here the diagnostic, alsoknown as statefinder diagnostic, which introduces a pair ofparameters {𝑟, 𝑠} defined, respectively, as follows:

𝑟 = 1 + 3��

𝐻2+

��

𝐻3= 1 +

3�� + ��/𝐻

𝐻2,

𝑠 = −3𝐻�� + ��

3𝐻 (2�� + 3𝐻2)= −

3�� + ��/𝐻

3 (2�� + 3𝐻2).

(33)

Using (26), (27), and (29), we get the statefinder parame-ters as

𝑟 = 1 +𝜌1

𝜌2

,

𝑠 =𝜁1

𝜁2

,

(34)

with

𝜌1= 𝐶1(𝜇 + ]) (𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])])

×(−6𝜀 + 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

8𝜀2)

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 𝐶2(𝜇 + ]) (−𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])])

6 ISRN High Energy Physics

×(6𝜀 − 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

8𝜀2)

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+(𝑛 + 3) 𝑛𝛼𝜇𝑎𝑛

2 {𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+(𝑚 + 3)𝑚𝛽]𝑎𝑚

2 {𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

− (9𝑎−3(1+𝛿)𝛿 (1 + 𝛿) 𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

−3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

,

𝜌2= 𝐶1𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 𝐶2𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+𝛼𝜇𝑎𝑛

𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]

+𝛽]𝑎𝑚

𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]

− (2𝑎−3(1+𝛿)𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

− 3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

,

𝜁1= −𝐶1(𝜇 + ]) (𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])])

×(−6𝜀 + 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

8𝜀2)

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

− 𝐶2(𝜇 + ]) (−𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])])

×(6𝜀 − 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

8𝜀2)

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

−(𝑛 + 3) 𝑛𝛼𝜇𝑎𝑛

2 {𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

−(𝑚 + 3)𝑚𝛽]𝑎𝑚

2 {𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+ (9𝑎−3(1+𝛿)𝛿 (1 + 𝛿) 𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

− 3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

,

𝜁2= −3𝐶

1(

−6𝜀 + 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎−(𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 3𝐶2(

6𝜀 − 𝜆 + √𝜆2 − 8𝜀 [𝜃 − 3 (𝜇 + ])]

2𝜀)

× 𝑎(−𝜆+√𝜆2−8𝜀[𝜃−3(𝜇+])])/2𝜀

+ 3(𝑛 + 3) 𝛼𝜇𝑎𝑛

{𝑛2𝜀 + 𝑛𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+ 3(𝑚 + 3) 𝛽]𝑎𝑚

{𝑚2𝜀 + 𝑚𝜆 + 2 [𝜃 − 3 (𝜇 + ])]}

+ (18𝑎−3(1+𝛿)𝛿𝜌𝑚0)

× (9(1 + 𝛿)2𝜀 + 2𝜃

−3 [𝜆 (1 + 𝛿) + 2 (𝜇 + ])] )−1

.

(35)

5. Discussion

In this Section, we discuss the behavior of the physical quan-tities derived in the previous sections. We have consideredthe following values for the parameters involved: 𝐶

1= 0.2,

𝐶2= 1.2,𝑚 = 1.2, 𝑛 = 1.4, 𝛽 = 1.2, 𝛼 = 1.5, 𝜃 = 0.002, 𝜆 = 2,

] = 0.5, 𝜇 = 0.9, 𝛿 = 0.05, and 𝜌𝑚0

= 0.23. We consideredthree different cases corresponding to three different valuesof the parameter 𝜀, that is, 𝜀 = 2, 3, and 4.

In Figure 1, we plotted the expression of the Hubbleparameter𝐻, obtained from (26), as function of the redshift𝑧. It is evident that the Hubble parameter 𝐻 has a decayingbehavior with varying values of the parameter 𝜀 and theredshift 𝑧 going from higher to lower redshifts. However, thisdecaying pattern is apparent till 𝑧 ≈ −0.5. In fact, in a verylate stage 𝑧 > −0.5, it shows an increasing pattern.

In Figure 2, we have plotted the time derivative of Hubbleparameter �� against the redshift 𝑧. We have observed thatfor 𝜀 = 3, �� transits from negative to positive side at𝑧 ≈ −0.5. However, for 𝜀 = 2 and 4 this transition occursat lower redshift 𝑧 ≈ −0.1. In Figures 3 and 4 we haveplotted, respectively, the equation of state (EoS) parameterfor DE, defined as 𝜔DE = 𝑝DE/𝜌DE, and the effective EoSparameter, defined as 𝜔eff = 𝑝DE/(𝜌DE + 𝜌DM), for the threedifferent values of 𝜀 considered in this work. In Figure 3,

ISRN High Energy Physics 7

−1.0 −0.5 0.0

0.0

1.0

1.0

1.5

1.5

2.0

2.0

0.5

0.5

z

H(z)

Figure 1: The Hubble parameter 𝐻, obtained from (26), as a func-tion of redshift 𝑧. The red, green, and blue lines correspond to 𝜀 =2, 3, and 4, respectively.

we have observed that for 𝜀 = 2, 𝜔DE crosses the phantomdivide −1 at 𝑧 ≈ 0. For 𝜀 = 3 the phantom divide is crossed at𝑧 ≈ −0.2. However, for 𝜀 = 4, the equation of state (EoS)parameter for DE stays below −1. Thus, for 𝜀 = 2 and 3,𝜔DE transits from quintessence to phantom, that is, has aquintom-like behavior. Instead, for 𝜀 = 4, the EoS parameterhas a phantom-like behavior. In Figure 4, we have plotted theeffective EoS parameter 𝜔eff. In this case, for all values of 𝜀considered, there is a crossing of phantom divide. Moreover,for 𝜀 = 4, 𝜔eff crosses the phantom divide earlier with respectto the other cases, in particular for 𝑧 ≈ 0.2.

The deceleration parameter 𝑞 has been plotted as a func-tion of 𝑧 in Figure 5. For 𝜀 = 2, and 3, there is a transition frompositive to negative 𝑞, that is, transition from decelerated toaccelerated expansion. For 𝜀 = 3, the deceleration parameterchanges sign at 𝑧 = 0, and for 𝜀 = 2, it changes sign at 𝑧 ≈ 0.1.However, for 𝜀 = 4, the deceleration parameter always stays atnegative level.Thus, for 𝜀 = 4, we are getting ever-acceleratinguniverse.

Next, we have plotted in Figure 6 the fractional density ofDE, given by ΩDE = 𝜌DE/3��

2(𝑧), and the fractional densityof matter, given by Ω

𝑚= 𝜌𝑚/3��2(𝑧), against the redshift 𝑧.

��2 is defined as ��2(𝑧) = (𝜇 + ])𝐻2 + (1/6)[𝛼𝜇(1 + 𝑧)−𝑛 +𝛽](1 + 𝑧)−𝑚]. The solid lines correspond to ΩDE and thedashed lines correspond to ΩDM. In this Figure, there is aclear indication of transition of the universe fromdarkmatterdominated phase to the dark energy dominated phase. Atvery early stage of the universe 𝑧 > 1, the dark energy densityis largely dominated by dark matter density. We denote thecross-over point by 𝑧cross and it comes out to be 𝑧cross ≈ 0.5,that is, where ΩDE = ΩDM for all values of 𝜀 consideredin this work. Hence, the 𝑓(𝑅, 𝑇) model, based on whichwe have reconstructed DE density, is capable of achieving

−1.0 −0.5 0.0 1.0 1.5 2.00.5

z

2

1

0

−1

−2

H(z)

Figure 2:The time derivative of reconstructed Hubble parameter asa function of redshift 𝑧. The red, green and blue lines correspond to𝜀 = 2, 3, and 4, respectively.

−2.0

−1.5

−1.0

−1.0

−0.5

−0.5

0.0

0.0 1.00.5

z

𝜔DE(z)

Figure 3: The EoS parameter 𝜔DE for the reconstructed DE.

the present DE dominated universe from the earlier darkmatter dominated universe.

Sahni et al. [42] recently demonstrated that the statefinderdiagnostic is effectively able to discriminate between differentmodels of DE. Chaplygin gas, braneworld, quintessence,and cosmological constant models were investigated byAlam et al. [43] using the statefinder diagnostics; theyobserved that the statefinder pair could differentiate betweenthese different models. An investigation on statefinderparameters for differentiating between DE and modified

8 ISRN High Energy Physics

−1.5

−1.0

−1.0

−0.5

−0.5

0.0

0.0 1.00.5

z

𝜔eff(z)

Figure 4: The effective EoS parameter 𝜔eff(𝑧) = 𝑝DE/(𝜌DE + 𝜌DM).

2

1

0

−1

−2

−1.0 −0.5 0.0 1.0 1.50.5 2.0

z

q(z)

Figure 5: The deceleration parameter 𝑞 as a function of 𝑧.

gravity was carried out in [44]. Statefinder diagnostics for𝑓(𝑇) gravity has been well studied in Wu and Yu [45]. In the{𝑟, 𝑠} plane, 𝑠 > 0 corresponds to a quintessence-like modelof DE and 𝑠 < 0 corresponds to a phantom-like model ofDE.Moreover, an evolution fromphantom to quintessence orinverse is given by crossing of the fixed point (𝑟 = 1, 𝑠 = 0) in{𝑟, 𝑠} plane [45], which corresponds to ΛCDM scenario. Thestatefinder parameters {𝑟, 𝑠} have been plotted in Figure 7 fordifferent values of the parameter 𝜀. It is clearly visible that the{𝑟−𝑠} trajectories are converging towards the fixed point {𝑟 =1, 𝑠 = 0}|

ΛCDM.Thus, the𝑓(𝑅, 𝑇)model is capable of attainingthe ΛCDM phase of the universe. Furthermore, for finite 𝑟,

−1.0 −0.5 0.0 1.00.5

z

1.0

0.8

0.6

0.4

0.2

0.0

,ΩDM

ΩDE

Figure 6: The fractional densities ΩDE (smooth lines) and ΩDM(dashed lines) as function of redshift 𝑧.

s

r

−2 −1 0 1 2

0.10

0.05

0.00

−0.05

−0.10

−0.15

−0.20

Figure 7: Statefinder trajectories for various choices of parameters.

𝑠 → −∞. Thus, the model can interpolate between dust andΛCDM phase of the universe.

Finally, in Figure 8, we plotted the squared speed of thesound, defined as V2

𝑠= ��/ 𝜌, where the upper dot indicates

derivative with respect to the cosmic time 𝑡 for the modelwe are considering as a function of 𝑧. The sign of thesquared speed of sound is fundamental in order to studythe stability of a background evolution. A negative value ofV2𝑠implies a classical instability of a given perturbation in

general relativity [46, 47]. Myung [47] recently observed thatthe squared speed of sound for HDE stays always negative if

ISRN High Energy Physics 9

−1 10 2 3 4 5 6

0.0

−0.5

−1.0

−1.5

z

�2 s

Figure 8: Squared speed of sound V2𝑠= ��/ 𝜌 as a function of redshift

𝑧.

the future event horizon is considered as IR cutoff, while forChaplygin gas and tachyon V2

𝑠is observed to benonnegative.

Kim et al. [46] found that V2𝑠for Agegraphic DE (ADE) stays

always negative, which leads to the instability of the perfectfluid for the model. Moreover, it was found that the ghostQCD [48] DE model is unstable. In a recent work, Sharifand Jawad [49] have shown that interacting new HDE ischaracterized by negative squared speed of the sound.

In the recent work of Pasqua et al. [32], authorsobserved that the interacting modified holographic Ricci DE(MHRDE) model in 𝑓(𝑅, 𝑇) = 𝜇𝑅 + ]𝑇 gravity is classicallystable.

Jawad et al. [50] showed that𝑓(𝐺)model inHDE scenariowith power-law scale factor is classically unstable.

Pasqua et al. [51] showed that the DE model based ongeneralized uncertainty principle (GUP) with power-lawform of the scale factor 𝑎(𝑡) is instable.

We have observed that, for all values of 𝜀, V2𝑠is positive

till the redshift 𝑧 ≈ 0.5. However, after this stage it entersthe negative region. Thus, although the model is classicallystable in the early universe, for present universe it is classicallyunstable.

6. Concluding Remarks

In this work, we have considered a recently proposed modelof energy density of DE which depends on three terms,one proportional to the squared Hubble parameter 𝐻, oneproportional to the time derivative of 𝐻, and one propor-tional to the second time derivative of 𝐻 interacting withpressureless DM in the framework of the 𝑓(𝑅, 𝑇) modifiedgravity theory for the special model given by 𝑓(𝑅, 𝑇) =𝜇𝑅 + ]𝑇, where 𝜇 and ] represent two constants. The DE

model considered here reduces to other two well-studiedDE model (the Ricci DE model and the DE energy densitymodel with Granda-Oliveros cut-off) for some particularvalues of the three parameters involved, that is, 𝜀, 𝜆, and 𝜃.We have derived the expressions and studied the behaviorof some important physical quantities which gave usefulhints about the model studied. The Hubble parameter 𝐻exhibits a decaying behavior going from higher to lowerredshifts until the redshift of about 𝑧 ≈ −0.5, when it startsto increase.The time derivative of the Hubble parameter, thatis, ��, shows a transition from negative to positive values fordifferent values of the redshift 𝑧 according to the value of𝜀 considered. We have observed that the equation of state(EoS) parameter of DE exhibits a transition from 𝜔 < −1to 𝜔 > −1, that is, transition from quintessence to phantom(i.e., quintom) with the evolution of the universe for 𝜀 = 2and 𝜀 = 3, instead it is always negative for 𝜀 = 4. Moreover,the effective equation of state (EoS) parameter 𝜔eff alwaysshows a transition from quintessence to phantom. Hence,we can conclude that the reconstructed DE model based onthe 𝑓(𝑅, 𝑇) gravity model considered leads to an equationof state (EoS) parameter that has a quintom-like behavior.We have further observed that the said model is capable ofattaining the dark energy dominated accelerated phase of theuniverse from dark matter dominated decelerated phase ofthe universe. Through statefinder trajectories we have shownthat the𝑓(𝑅, 𝑇) is capable of attaining theΛCDMphase of theuniverse and can interpolate between dust and ΛCDM phaseof the universe. Through squared speed of sound we haveseen that the model under consideration is classically stablein the earlier stage of the universe but classically unstable inthe current stage.

Conflict of Interests

The authors declare that there is no conflict of inter-ests regarding the publication of this paper.

Acknowledgment

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

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ISRN High Energy Physics 11

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