A dark energy model with higher order derivatives of H in the f ( R , T ) modified gravity model

In this paper, 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 f (R, T ) = μR + νT modified gravity model, with μ and ν being two constant parameters. R and T represent the curvature and torsion scalars, respectively. We found that the Hubble parameter exhibits a decaying behavior until redshifts z ≈ −0.5 (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 ω < −1 to ω > −1 (showing a quintomlike behavior). We also found that the model considered can attain the late time accelerated phase of the universe. Using the statefinder parameters r and s, we derived that the studied model can attain the ΛCDM phase of the universe and can interpolate between dust and ΛCDM phase of the universe. Finally, studying the squared speed of sound v s , we found that the considered model is classically stable in the earlier stage of the universe, but classically unstable in the current stage.


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 Λ or it is originated from other sources dynamically changing with time [3].For good reviews on DE see [4][5][6].
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][22][23][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 ISRN High Energy Physics 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  = 0, we obtain the energy density of DE with Granda-Oliveros cut-off, and in the limiting case corresponding to  = 0,  = 1, and  = 2, 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][36][37][38][39][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.

The 𝑓(𝑅,𝑇) = 𝜇𝑅 + ]𝑇 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 8 =  = 1) 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 (  = 0).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]:  ∝  DE ,  ∝   , and  ∝ (  +  DE ).
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  37 -model, whose action  is given by [29] where  is defined as  = det(   ) = √− (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  = (, ȧ ) and V = V(, ȧ ), where ȧ 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  37 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 V.
We now assume the following expressions for  and, V: where , , , and  are real constants.We also have that  and V 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

Interacting DE in 𝑓(𝑅,𝑇) Gravity
Solving the differential equation for   given in (8), we derive the following expression for   : where  0 indicates the present day value of   .Using (1) and ( 25) in the right-hand side of (21), we obtain the following expression of  2 as function of the scale factor: where  1 and  2 are two constants of integration.In order to have a real and definite expression of  2 given in (26), the following conditions must be satisfied: We can now derive the expressions of the first and the second time derivative of the Hubble parameter , that is, Ḣ and Ḧ, 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  DE : Taking into account the expression of  DE given in ( 29), we derive that the expression of the pressure  DE of DE is given by Using the expressions of the energy density  DE and the pressure  DE of DE given, respectively, in ( 29) and ( 30), and the expression of   given in (25), we get the EoS parameter  DE for DE and the total EoS parameter  tot as follows: We must remember here that we are considering the case of pressureless DM, so that   = 0.
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  2 and Ḣ are given, respectively, in ( 26) and ( 27).The deceleration parameter, the Hubble parameter , and the dimensionless energy density parameters Ω DE , Ω  , and Ω  (which will be considered and studied in the following Sections) are a set of useful parameters if it is needed to describe cosmological observations.

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 ΛCDM 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 ISRN High Energy Physics
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  ≈ −0.5.In fact, in a very late stage  > −0.5, it shows an increasing pattern.
In Figure 2, we have plotted the time derivative of Hubble parameter Ḣ against the redshift .We have observed that for  = 3, Ḣ transits from negative to positive side at  ≈ −0.5.However, for  = 2 and 4 this transition occurs at lower redshift  ≈ −0.1.In Figures 3 and 4  we have observed that for  = 2,  DE crosses the phantom divide −1 at  ≈ 0. For  = 3 the phantom divide is crossed at  ≈ −0.2.However, for  = 4, the equation of state (EoS) parameter for DE stays below −1.Thus, for  = 2 and 3,  DE transits from quintessence to phantom, that is, has a quintom-like behavior.Instead, for  = 4, the EoS parameter has a phantom-like behavior.In Figure 4, we have plotted the effective EoS parameter  eff .In this case, for all values of  considered, there is a crossing of phantom divide.Moreover, for  = 4,  eff crosses the phantom divide earlier with respect to the other cases, in particular for  ≈ 0.2.The deceleration parameter  has been plotted as a function of  in Figure 5.For  = 2, and 3, there is a transition from positive to negative , that is, transition from decelerated to accelerated expansion.For  = 3, the deceleration parameter changes sign at  = 0, and for  = 2, it changes sign at  ≈ 0.1.However, for  = 4, the deceleration parameter always stays at negative level.Thus, for  = 4, we are getting ever-accelerating universe.
Next, we have plotted in Figure 6 the fractional density of DE, given by Ω DE =  DE /3 H2 (), and the fractional density of matter, given by Ω  =   /3 H2 (), against the redshift .H2 is defined as H2 () = ( + ])   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,  > 0 corresponds to a quintessence-like model of DE and  < 0 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 ( = 1,  = 0) in {, } plane [45], which corresponds to ΛCDM 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 { = 1,  = 0}| ΛCDM .Thus, the (, ) model is capable of attaining the ΛCDM phase of the universe.Furthermore, for finite ,    → −∞.Thus, the model can interpolate between dust and ΛCDM phase of the universe.Finally, in Figure 8, we plotted the squared speed of the sound, defined as V 2  = ṗ / ρ , 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 V 2  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
2 + (1/6)[(1 + ) − + ](1 + ) − ].The solid lines correspond to Ω DE and the dashed lines correspond to Ω DM .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  > 1, the dark energy density is largely dominated by dark matter density.We denote the cross-over point by  cross and it comes out to be  cross ≈ 0.5, that is, where Ω DE = Ω DM for all values of  considered in this work.Hence, the (, ) model, based on which we have reconstructed DE density, is capable of achieving

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

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

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

Figure 6 :
Figure 6: The fractional densities Ω DE (smooth lines) and Ω DM (dashed lines) as function of redshift .

Figure 7 :
Figure 7: Statefinder trajectories for various choices of parameters.