# Correspondence off(R,∇R) Modified Gravity with Scalar Field Models

Authors: Abdul Jawad; Ujjal Debnath
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

This paper is devoted to study the scalar field dark energy models by taking its different aspects in the framework off(R,∇R) gravity. We consider flat FRW universe to construct the equation of state parameter governed by f(R,∇R) gravity. The stability of the model is discussed with the help of squared speed of sound parameter. It is found that models show quintessence behavior of the universe in stable as well as unstable modes. We also develop the correspondence of f(R,∇R) model with some scalar field dark energy models like quintessence, tachyonic field, k-essence, dilaton, hessence, and DBI-essence. The nature of scalar fields and corresponding scalar potentials is being analyzed in f(R,∇R) gravity graphically which show consistency with the present day observations about accelerated phenomenon.


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## Body

## 1. Introduction

It is strongly believed that the universe is experiencing an accelerated expansion nowadays. The type Ia Supernovae [1, 2], large scale structure [3], and cosmic microwave background [4] observations have provided main evidence for this cosmic acceleration. This acceleration is caused by some unknown matter having positive energy density and negative pressure is dubbed as “dark energy” (DE). The combined analysis of cosmological observations suggests that the universe is spatially flat and consists of about 70% DE, 30% dust matter, and negligible radiations. Recent WMAP data analysis [5] also has given the confirmation of this acceleration.The most appealing and simplest candidate for DE is the cosmological constantΛ [6–8] which has the equation of state (EoS) parameter ω=-1. Over the past decade, there have been many theoretical models for mimicking DE behavior, such as Λ CDM, containing a mixture of cosmological constant Λ and cold dark matter (CDM). However, two problems arise from this scenario, namely, the “fine-tuning” and “cosmic coincidence” problems [9]. In order to solve these two problems, many dynamical DE theoretical models have been proposed [10–14].The dynamical DE scenario is often realized by some scalar field mechanism which suggests that the energy formed with negative pressure is provided by a scalar field evolving down a proper potential. The scalar field or quintessence (a scalar field slowly evolving down its potential) [15, 16] is one of the most favored candidate of DE. Provided that the evolution of the field is slow enough, the kinetic energy density is less than the potential energy density, giving rise to the negative pressure responsible for the cosmic acceleration. A lot of scalar field DE models have been studied [17], includingk-essence [10], dilaton [18], DBI-essence [19, 20], hessence [21], tachyon [11], phantom [22], ghost condensate [23, 24], and quintom [12, 25, 26]. In addition, other proposals on DE include scenarios of interacting DE models [27, 28], brane-world models [29], and Chaplygin gas models [13].Another approach to explore the accelerated expansion of the universe is the modified theories of gravity. In this scenario, cosmic acceleration of the universe would not arise from mysterious DE as a substance but rather from the dynamics of modified gravity. There are several models of modified gravity which include DGP brane,f(R) gravity, f(T) gravity, f(G) gravity, Gauss-Bonnet gravity, Horava-Lifshitz gravity, and Brans-Dicke gravity [30–41]. The f(R,∇R) gravity is a theory based on an action integral whose kernel is a function of the Ricci scalar R and terms involving its derivatives. Cuzinatto et al. [42] have explored the cosmological consequences of a particular model in the context of this gravity theory. They tested the model against some of the observational data available. A more recent example is the (nonlocal) higher derivative cosmology presented in [43–46].Recently the reconstruction procedure or correspondences between various DE models become very challenging subject in cosmological phenomena. The correspondence between different DE models, reconstruction of DE/gravity, and their cosmological implications have been discussed by several authors [47–60]. In the present work, we assume f(R,∇R) gravity model in FRW universe. We discuss the correspondence of f(R,∇R) gravity with some scalar field dark energy models like quintessence, tachyon, dilaton,k-essence, hessence, and DBI-essence models. The nature of scalar fields and the potentials is analyzed by graphical behavior of their scalar field and potential functions. The scheme of the paper is as follows. In Section 2, we briefly provide a review of f(R,∇R) gravity and discuss behavior of EoS and stability parameters. Section 3 devotes the analysis of correspondence between this gravity model and some scalar field models. The last section summarizes the obtained results.


## 2. Brief Review off(R,∇R) Gravity

The action off(R,∇R) gravity is given by [42]
(1)S=∫‍d4x-gR2κ2+β8κ2∇μR∇μR-Lm,whereκ2=8πG=mp-2, β is the coupling constant, ∇μ is the covariant derivative, and Lm is the matter Lagrangian density. An important feature of S is the presence of ghosts [46]. However, here this is not a serious issue because (1) should be considered as an effective action of a fundamental theory of gravity, which make it ghost-free theory. The extended theory of gravity such as String-inspired nonlocal higher derivative gravitational model draws the attention of the researchers. The ghost-free action of this model is given by (the derivation has been discussed in detail by Biswas et al. [45])
(2)S=∫‍d4x-gmp22R+12RF□M*2R,
where M* denotes the mass scale and term
(3)F□M*2=∑n=1∞‍fn□n.
In order to maintain consistency with the action as low energy limiting case, the only suitable contribution of (3) is the first term of the series. That is, the corresponding term is f1≫fn, ∀n≥2. This leads to the equivalence of both actions (1) and (2) up to surface term and yields a relation; that is,
(4)f1=β4κ2.
Taking the variation of action (1) with respect to gμν, the field equations turn out to be [42]
(5)Rμν-12gμνR+βHμν=1mp2Tμν,
where
(6)Hμν=∇μ∇ν[□R]+12∇μR∇νR-Rμν□R-gμν□[□R]-14gμν∇ρR∇ρR.
Here □=∇μ∇μ and Rμν is the Ricci tensor.For FRW universe, the Friedmann equations in the scenario off(R,∇R) model turn out to be
(7)3H2+βF1=1mp2ρm,3H2+2H˙+βF2=-1mp2pm.
Here F1 and F2 are defined in terms of Hubble parameter as follows:
(8)F1=18HH(4)+108H2H⃛-18H˙H⃛+9H¨2+90H3H¨+216HH˙H¨-72H˙3+288HH˙2-216H4H˙,F2=6H(5)+54HH(4)+138H2H⃛+126H˙H⃛+81H¨2+18H3H¨+498HH˙H¨+120H˙3-216H4H˙,
where H(5) denotes the fifth order time derivative of Hubble parameter. The equation of continuity of CDM takes the following form:
(9)ρ˙m+3Hρm=0,which givesρm=ρm0a-3.
The term ρm0 represents the present value of cold matter density. We can extract the EoS parameter; that is, ωR=pR/ρm+ρR, where
(10)ρR=mp2βF1,pR=mp2βF2.
It follows that
(11)ωR=mp2βF2mp2βF1+ρm0a-3.
For the reconstructed scenario, we consider the solution of the field equation in exact power law form given by
(12)a(t)=a0ts-tn,ts>t,
where a0 is the present value of scale factor and n is a constant. The term ts denotes the finite future singularity time and the scale factor (12) is used to check type II (sudden singularity) or type IV (corresponds to H˙) for positive values of n. We plot effective EoS parameter ωR versus time as shown in Figure 1(a) for different values of n such that n=1,2,3. We choose some typical values of constants in this respect; that is, β=2.02, mp2=1, and ts=0.6. This parameter shows increasing behavior and transition from phantom to quintessence phase. For n=1, ωR becomes nearly constant at ωR=-0.4 which represents the quintessence era. As we increase the value of n, this parameter again expresses transition from phantom to quintessence with the passage of time but converges to a value that nearly corresponds to vacuum era of the universe.Plots of (a) effective EoS parameter and (b) squared speed of sound versus cosmic timet, for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)To check the stability of the model, we plot squared speed of sound(vs2=p˙/ρ˙) versus time as shown in Figure 1(b). The sign of squared speed of sound has a significant role in this regard. A positive value indicates the stability of model whereas instability stands for the negative value of vs2. In the plot, we obtain two regions corresponding to the sign as t<0.6 and t>0.6. For the region t<0.6, it is found that vs2 represents positively decreasing behavior which gives the stability of the model. In the second region, squared speed of sound represents the instability of the model by experiencing negative behavior for all values of n.


## 3. Correspondence betweenf(R,∇R) and Scalar Field Models

Here, we investigate the correspondence off(R,∇R) gravity with some of the scalar field models such as quintessence, tachyon,k-essence, dilaton, hessence, and DBI-essence models. The scalar field models have been developed through particle physics as well as string theory and nowadays these are used as a DE candidate.


### 3.1. Quintessence

The quintessence scalar field model has been proposed to resolve the fine-tuning problem taking EoS as time dependent instead of constant. This model can explain current cosmic acceleration by giving negative pressure when potential dominates the kinetic term. It is minimally coupled with gravity and used as a useful tool for discussing early inflation as well as the late time acceleration. The action of this model is described as [14–16]
(13)SQ=∫‍d4x-g-12∇ϕ2-V(ϕ),
where (∇ϕ)2=gμν∂μϕ∂νϕ and V(ϕ) denotes the potential of the field. Varying this action with respect to metric tensor gμν, we obtain the corresponding energy-momentum tensor as follows:
(14)Tμν=∂μϕ∂νϕ-gμν12gαβ∂αϕ∂βϕ+V(ϕ).
This gives the energy density and pressure for FRW universe as follows:
(15)ρQ=-T00=12ϕ˙2+V(ϕ),pQ=Tii=12ϕ˙2-V(ϕ).
These expressions yield
(16)ϕ˙2=ρQ+pQ,V(ϕ)=12(ρQ-pQ).
Replacing ρQ and pQ by the energy density and pressure of f(R,∇R) model, we study the behavior of the resulting scalar field and potential function.The evolution trajectories of scalar field and potential function versus time forf(R,∇R) quintessence model are shown in Figure 2 taking the same values of constants as for effective EoS parameter. In left plot (a), ϕ(t) represents increasing behavior and becomes more steeper as time goes on, particularly for greater value of n. The increasing value of scalar field indicates that kinetic energy is decreasing. The potential function versus time in plot (b) expresses decreasing behavior. The plot of quintessence potential in terms of scalar field is shown in Figure 3 representing increasing behavior. The gradually decreasing kinetic energy while potential remains positive for f(R,∇R) quintessence model represents accelerated expansion of the universe.Plots off(R,∇R) quintessence scalar field model: (a) scalar field ϕ and (b) potential V versus cosmic time t.
(a)
Scalar fieldϕ(t)
(b)
Potential functionV(t)Figure 3
Plot off(R,∇R) quintessence potential V versus ϕ(t) for n=1 (red), n=2 (green), and n=3 (blue).


### 3.2. Tachyon

The tachyon model is motivated from string theory and used as a possible candidate to explain DE scenario. It is argued that rolling tachyon condensates possess useful cosmological consequences. The EoS of this rolling tachyon is smoothly interpolated between-1 and 0 which is an interesting feature of this model. This has led a flurry of attempts to construct more viable cosmological models for tachyon which serves as a suitable candidate of inflation at high energy. It is realized that tachyon can be used as a source of DE depending on different forms of its potential. The effective Lagrangian for tachyon is [11, 14]
(17)ST=-∫‍d4xV(ϕ)-det⁡(gμν+∂μϕ∂νϕ).
Varying this action with respect to the metric tensor, we have
(18)Tμν=V(ϕ)∂μϕ∂νϕ1+gαβ∂αϕ∂βϕ-gμνV(ϕ)1+gαβ∂αϕ∂βϕ,
leading to the energy density and pressure as
(19)ρT=V(ϕ)1-ϕ˙2,pT=-V(ϕ)1-ϕ˙2,
and EoS parameter becomes
(20)ωT=ϕ˙2-1.
Taking into account (19) and (20), we have
(21)ϕ˙2=1+ωT,V(ϕ)=ρTωT,
which can be replaced by ρR and ωR in order to check the behavior of scalar field and potential function of f(R,∇R) tachyon model.We plot scalar field and potential function off(R,∇R) tachyon model as shown in Figure 4. The evolution of this model is much similar to the quintessence model. The scalar field represents increasing behavior versus time and indicates more steeper behavior for n=1. This leads to the decreasing kinetic energy, approximately zero value. This represents the vacuum behavior of the universe depicted by (20). The corresponding potential function expresses decreasing but positive behavior with respect to t. Its decreasing behavior from maxima gives inverse proportionality to the scalar field for the later times. This type of behavior corresponds to scaling solutions in the brane-world cosmology [61, 62].Plots off(R,∇R) tachyon model: (a) scalar field ϕ versus time t and (b) potential V versus time t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
Scalar fieldϕ(t)
(b)
Potential functionV(t)


### 3.3.k-Essence

Armendáriz-Picón et al. [63] introduced the concept inflation driven by kinetic energy (termed ask-inflation) for discussing the inflation in the early universe at high energies. This scenario was used for DE purpose by Chiba et al. [64]. Moreover, this analysis was extended by Armendáriz-Picón et al. [65] to a more generalized form of Lagrangian, named as “k-essence.” The quintessence scalar field relies on potential energy in order to explain late time cosmic acceleration of the universe. However, this model takes the modifications of kinetic energy in the scalar fields to address the cosmic expansion. The generalized form of scalar field action is
(22)SK=∫‍d4x-gp(ϕ,χ),
where p(ϕ,χ) shows the pressure density as a function of potential ϕ and χ=(1/2)ϕ˙2. The energy density and pressure are given by
(23)ρK=V(ϕ)(-χ+3χ2),pK=V(ϕ)(-χ+χ2).
The corresponding EoS parameter yields
(24)ωK=1-χ1-3χ,
which represents the fact that the accelerated expansion of the universe is experienced for the interval (1/3,2/3) for χ. The potential function takes the form
(25)V(ϕ)=1-3ωK22(1-ωK)ρK.EquatingωK=ωR and ρK=ρR in (24) and (25), the evolution trajectories of ϕ and χ versus t for f(R,∇R)k-essence scalar field model are given in Figure 5. The plot (a) represents increasing behavior of scalar field with respect to time for all values of n. The plot (b) of χ versus t shows compatible behavior of the model for all n from early era to later times. This indicates that for t≥0.6,χ(t) approaches to the required interval for accelerated expansion of the universe. Also, the potential function versus t and ϕ is shown in Figures 6(a)-6(b), expressing positively decreasing behavior.Plots off(R,∇R)k-essence scalar field: (a) ϕ versus cosmic time t and (b) kinetic energy term χ versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)Plots off(R,∇R)k-essence potential (a) V versus cosmic time t and (b) V versus ϕ for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


### 3.4. Dilaton

It is shown [14] that phantom field with negative kinetic energy plagued a problem of quantum instabilities. To avoid this problem, dilaton model has been originated which is later used to explain DE puzzle. Also, it is realized that this model avoids some quantum instabilities with respect to the phantom field models of DE. This model is found in low energy effective string theory, where dilatonic higher-order corrections were made to the tree-level action. The Lagrangian of dilaton field can be expressed in terms of pressure of scalar field as [14, 18]
(26)pD=-χ+c1ebϕχ2,
where c1 and c2 are taken to be positive constants. This Lagrangian (pressure) produces the following energy density:
(27)ρD=-χ+3c1ebϕχ2.
The EoS parameter takes the form
(28)ωD=1-c1ebϕχ1-3c1ebϕχ.By keepingρD=ρR and pD=pR in above three equations, we can get kintetic energy corresponding scalar field numerically by keeping same values for other constants while c1=1.2. The behavior of this model is approximately the same as of thek-essence model. Figure 7(a) represents the increasing behavior of scalar field versus time. In plot (b), the kinetic energy term χebϕ versus time shows increasing behavior with the passage of time. The EoS parameter given in (28) indicates the bound of kinetic energy term such as (5/18,5/9) for accelerated expansion of the universe. In this regard, we observe that the value n=1 is inconsistent with this interval while the remaining values of n represent compatible results.Plots off(R,∇R) dilaton scalar field model (a) ϕ versus cosmic time t and (b) kinetic energy term χebϕ versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


### 3.5. Hessence

The noncanonical complex scalar field,Φ=ϕ1+ιϕ2 named as hessence, has the Lagrangian density as [21]
(29)LH=12∂μϕ2-ϕ2∂μϕ2-V(ϕ),
where ϕ=ϕ12+ϕ22 with ϕ1=ϕcosh⁡θ and ϕ2=ϕsinh⁡θ which lead to coth⁡θ=ϕ1/ϕ2. These two variables (ϕ,θ) are introduced to describe the hessence scalar field. The hessence comprises the special case of quintom model in terms of ϕ1 and ϕ2. For FRW spacetime, the energy density and pressure take the form
(30)pH=12ϕ˙2-Q2a6ϕ2-V(ϕ),ρH=12ϕ˙2-Q2a6ϕ2+V(ϕ),
where Q represents the total conserved charge within the physical volume due to the internal symmetry. Using (30), we obtain the scalar field and potential of hessence scalar filed model as
(31)ϕ˙2-Q2a-6ϕ-2=ρH-pH,V(ϕ)=12(ρH-pH).
Replacing energy density and potential of hessence scalar field with ρR and pR in the above equations, we plot scalar field and potential of f(R,∇R) hessence scalar field model as shown in Figure 8. The scalar field ϕ expresses increasing behavior with respect to t as shown in plot (a). In plot (b), the potential function represents the same behavior as the remaining scalar field models. Initially, this indicates decreasing behavior with the passage of time.Plots off(R,∇R) hessence scalar field model: (a) ϕ versus cosmic time t and (b) potential V versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


### 3.6. DBI-Essence

Here we consider Dirac-Born-Infeld- (DBI-) essence scalar field model whose action is given by [19, 20]
(32)Sdbi=∫‍d4xa3(t)T(ϕ)1-ϕ˙2T(ϕ)+V(ϕ)-T(ϕ),
where T(ϕ) and V(ϕ) are the warped brane tension and the DBI potential, respectively. The corresponding energy density and pressure yield
(33)ρdbi=(γ-1)T(ϕ)+V(ϕ),pdbi=1-1γT(ϕ)-V(ϕ),
where γ=1-ϕ˙2/Tϕ-1/2 is reminiscent from the usual relativistic Lorentz factor. Now, we consider here two particular cases which involve γ as a constant and vice versa.Case 1 (γ=constant).
In this case, we assumeγ as constant and T(ϕ)=mϕ˙-2 where m>1 for the sake of simplicity. The γ becomes
(34)γ=mm-1.
Using (33), we obtain the following expressions for DBI-essence for constant γ:
(35)ϕ˙2=γm(γ2-1)(ρdbi+pdbi),(36)V(ϕ)=12[(ρdbi-pdbi)-(γ-2+1γ)T(ϕ)],
where T(ϕ)=γ/(γ2-1)(ρdbi+pdbi). Inserting energy density and pressure of f(R,∇R) model with those of ρdbi and pdbi in (35), we plot ϕ and V versus t as shown in Figure 9. This represents approximately the same behavior as for f(R,∇R) quintessence scalar field model. The scalar field shows increasing behavior while potential represents decreasing but positive behavior.Plots off(R,∇R) DBI-essence scalar field (with constant γ): (a) ϕ versus cosmic time t and (b) potential versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)Case 2 (γ≠constant).
Here we takeγ not as a constant. For this purpose, we assume γ=ϕ˙s where s is an arbitrary constant. This case leads to the following expressions:
(37)ϕ˙=ρdbi+pdbi1/(s+2),V(ϕ)=12[(ρdbi-pdbi)-(γ-2+1γ)T(ϕ)],
where T(ϕ)=ϕ˙2s+2/(ϕ˙2s-1). In order to plot scalar field and potential of f(R,∇R) DBI-essence scalar field (with γ≠ constant), we substitute ρdbi=ρR and pdbi=pR in (37). Figure 10 represents the same behavior as for constant γ; however, the values of scalar field and potential are quite different in both cases.Plots off(R,∇R) DBI-essence scalar field (with γ≠ constant): (a) ϕ versus cosmic time t and (b) potential versus t (right panel) for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


## 3.1. Quintessence

The quintessence scalar field model has been proposed to resolve the fine-tuning problem taking EoS as time dependent instead of constant. This model can explain current cosmic acceleration by giving negative pressure when potential dominates the kinetic term. It is minimally coupled with gravity and used as a useful tool for discussing early inflation as well as the late time acceleration. The action of this model is described as [14–16]
(13)SQ=∫‍d4x-g-12∇ϕ2-V(ϕ),
where (∇ϕ)2=gμν∂μϕ∂νϕ and V(ϕ) denotes the potential of the field. Varying this action with respect to metric tensor gμν, we obtain the corresponding energy-momentum tensor as follows:
(14)Tμν=∂μϕ∂νϕ-gμν12gαβ∂αϕ∂βϕ+V(ϕ).
This gives the energy density and pressure for FRW universe as follows:
(15)ρQ=-T00=12ϕ˙2+V(ϕ),pQ=Tii=12ϕ˙2-V(ϕ).
These expressions yield
(16)ϕ˙2=ρQ+pQ,V(ϕ)=12(ρQ-pQ).
Replacing ρQ and pQ by the energy density and pressure of f(R,∇R) model, we study the behavior of the resulting scalar field and potential function.The evolution trajectories of scalar field and potential function versus time forf(R,∇R) quintessence model are shown in Figure 2 taking the same values of constants as for effective EoS parameter. In left plot (a), ϕ(t) represents increasing behavior and becomes more steeper as time goes on, particularly for greater value of n. The increasing value of scalar field indicates that kinetic energy is decreasing. The potential function versus time in plot (b) expresses decreasing behavior. The plot of quintessence potential in terms of scalar field is shown in Figure 3 representing increasing behavior. The gradually decreasing kinetic energy while potential remains positive for f(R,∇R) quintessence model represents accelerated expansion of the universe.Plots off(R,∇R) quintessence scalar field model: (a) scalar field ϕ and (b) potential V versus cosmic time t.
(a)
Scalar fieldϕ(t)
(b)
Potential functionV(t)Figure 3
Plot off(R,∇R) quintessence potential V versus ϕ(t) for n=1 (red), n=2 (green), and n=3 (blue).


## 3.2. Tachyon

The tachyon model is motivated from string theory and used as a possible candidate to explain DE scenario. It is argued that rolling tachyon condensates possess useful cosmological consequences. The EoS of this rolling tachyon is smoothly interpolated between-1 and 0 which is an interesting feature of this model. This has led a flurry of attempts to construct more viable cosmological models for tachyon which serves as a suitable candidate of inflation at high energy. It is realized that tachyon can be used as a source of DE depending on different forms of its potential. The effective Lagrangian for tachyon is [11, 14]
(17)ST=-∫‍d4xV(ϕ)-det⁡(gμν+∂μϕ∂νϕ).
Varying this action with respect to the metric tensor, we have
(18)Tμν=V(ϕ)∂μϕ∂νϕ1+gαβ∂αϕ∂βϕ-gμνV(ϕ)1+gαβ∂αϕ∂βϕ,
leading to the energy density and pressure as
(19)ρT=V(ϕ)1-ϕ˙2,pT=-V(ϕ)1-ϕ˙2,
and EoS parameter becomes
(20)ωT=ϕ˙2-1.
Taking into account (19) and (20), we have
(21)ϕ˙2=1+ωT,V(ϕ)=ρTωT,
which can be replaced by ρR and ωR in order to check the behavior of scalar field and potential function of f(R,∇R) tachyon model.We plot scalar field and potential function off(R,∇R) tachyon model as shown in Figure 4. The evolution of this model is much similar to the quintessence model. The scalar field represents increasing behavior versus time and indicates more steeper behavior for n=1. This leads to the decreasing kinetic energy, approximately zero value. This represents the vacuum behavior of the universe depicted by (20). The corresponding potential function expresses decreasing but positive behavior with respect to t. Its decreasing behavior from maxima gives inverse proportionality to the scalar field for the later times. This type of behavior corresponds to scaling solutions in the brane-world cosmology [61, 62].Plots off(R,∇R) tachyon model: (a) scalar field ϕ versus time t and (b) potential V versus time t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
Scalar fieldϕ(t)
(b)
Potential functionV(t)


## 3.3.k-Essence

Armendáriz-Picón et al. [63] introduced the concept inflation driven by kinetic energy (termed ask-inflation) for discussing the inflation in the early universe at high energies. This scenario was used for DE purpose by Chiba et al. [64]. Moreover, this analysis was extended by Armendáriz-Picón et al. [65] to a more generalized form of Lagrangian, named as “k-essence.” The quintessence scalar field relies on potential energy in order to explain late time cosmic acceleration of the universe. However, this model takes the modifications of kinetic energy in the scalar fields to address the cosmic expansion. The generalized form of scalar field action is
(22)SK=∫‍d4x-gp(ϕ,χ),
where p(ϕ,χ) shows the pressure density as a function of potential ϕ and χ=(1/2)ϕ˙2. The energy density and pressure are given by
(23)ρK=V(ϕ)(-χ+3χ2),pK=V(ϕ)(-χ+χ2).
The corresponding EoS parameter yields
(24)ωK=1-χ1-3χ,
which represents the fact that the accelerated expansion of the universe is experienced for the interval (1/3,2/3) for χ. The potential function takes the form
(25)V(ϕ)=1-3ωK22(1-ωK)ρK.EquatingωK=ωR and ρK=ρR in (24) and (25), the evolution trajectories of ϕ and χ versus t for f(R,∇R)k-essence scalar field model are given in Figure 5. The plot (a) represents increasing behavior of scalar field with respect to time for all values of n. The plot (b) of χ versus t shows compatible behavior of the model for all n from early era to later times. This indicates that for t≥0.6,χ(t) approaches to the required interval for accelerated expansion of the universe. Also, the potential function versus t and ϕ is shown in Figures 6(a)-6(b), expressing positively decreasing behavior.Plots off(R,∇R)k-essence scalar field: (a) ϕ versus cosmic time t and (b) kinetic energy term χ versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)Plots off(R,∇R)k-essence potential (a) V versus cosmic time t and (b) V versus ϕ for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


## 3.4. Dilaton

It is shown [14] that phantom field with negative kinetic energy plagued a problem of quantum instabilities. To avoid this problem, dilaton model has been originated which is later used to explain DE puzzle. Also, it is realized that this model avoids some quantum instabilities with respect to the phantom field models of DE. This model is found in low energy effective string theory, where dilatonic higher-order corrections were made to the tree-level action. The Lagrangian of dilaton field can be expressed in terms of pressure of scalar field as [14, 18]
(26)pD=-χ+c1ebϕχ2,
where c1 and c2 are taken to be positive constants. This Lagrangian (pressure) produces the following energy density:
(27)ρD=-χ+3c1ebϕχ2.
The EoS parameter takes the form
(28)ωD=1-c1ebϕχ1-3c1ebϕχ.By keepingρD=ρR and pD=pR in above three equations, we can get kintetic energy corresponding scalar field numerically by keeping same values for other constants while c1=1.2. The behavior of this model is approximately the same as of thek-essence model. Figure 7(a) represents the increasing behavior of scalar field versus time. In plot (b), the kinetic energy term χebϕ versus time shows increasing behavior with the passage of time. The EoS parameter given in (28) indicates the bound of kinetic energy term such as (5/18,5/9) for accelerated expansion of the universe. In this regard, we observe that the value n=1 is inconsistent with this interval while the remaining values of n represent compatible results.Plots off(R,∇R) dilaton scalar field model (a) ϕ versus cosmic time t and (b) kinetic energy term χebϕ versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


## 3.5. Hessence

The noncanonical complex scalar field,Φ=ϕ1+ιϕ2 named as hessence, has the Lagrangian density as [21]
(29)LH=12∂μϕ2-ϕ2∂μϕ2-V(ϕ),
where ϕ=ϕ12+ϕ22 with ϕ1=ϕcosh⁡θ and ϕ2=ϕsinh⁡θ which lead to coth⁡θ=ϕ1/ϕ2. These two variables (ϕ,θ) are introduced to describe the hessence scalar field. The hessence comprises the special case of quintom model in terms of ϕ1 and ϕ2. For FRW spacetime, the energy density and pressure take the form
(30)pH=12ϕ˙2-Q2a6ϕ2-V(ϕ),ρH=12ϕ˙2-Q2a6ϕ2+V(ϕ),
where Q represents the total conserved charge within the physical volume due to the internal symmetry. Using (30), we obtain the scalar field and potential of hessence scalar filed model as
(31)ϕ˙2-Q2a-6ϕ-2=ρH-pH,V(ϕ)=12(ρH-pH).
Replacing energy density and potential of hessence scalar field with ρR and pR in the above equations, we plot scalar field and potential of f(R,∇R) hessence scalar field model as shown in Figure 8. The scalar field ϕ expresses increasing behavior with respect to t as shown in plot (a). In plot (b), the potential function represents the same behavior as the remaining scalar field models. Initially, this indicates decreasing behavior with the passage of time.Plots off(R,∇R) hessence scalar field model: (a) ϕ versus cosmic time t and (b) potential V versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


## 3.6. DBI-Essence

Here we consider Dirac-Born-Infeld- (DBI-) essence scalar field model whose action is given by [19, 20]
(32)Sdbi=∫‍d4xa3(t)T(ϕ)1-ϕ˙2T(ϕ)+V(ϕ)-T(ϕ),
where T(ϕ) and V(ϕ) are the warped brane tension and the DBI potential, respectively. The corresponding energy density and pressure yield
(33)ρdbi=(γ-1)T(ϕ)+V(ϕ),pdbi=1-1γT(ϕ)-V(ϕ),
where γ=1-ϕ˙2/Tϕ-1/2 is reminiscent from the usual relativistic Lorentz factor. Now, we consider here two particular cases which involve γ as a constant and vice versa.Case 1 (γ=constant).
In this case, we assumeγ as constant and T(ϕ)=mϕ˙-2 where m>1 for the sake of simplicity. The γ becomes
(34)γ=mm-1.
Using (33), we obtain the following expressions for DBI-essence for constant γ:
(35)ϕ˙2=γm(γ2-1)(ρdbi+pdbi),(36)V(ϕ)=12[(ρdbi-pdbi)-(γ-2+1γ)T(ϕ)],
where T(ϕ)=γ/(γ2-1)(ρdbi+pdbi). Inserting energy density and pressure of f(R,∇R) model with those of ρdbi and pdbi in (35), we plot ϕ and V versus t as shown in Figure 9. This represents approximately the same behavior as for f(R,∇R) quintessence scalar field model. The scalar field shows increasing behavior while potential represents decreasing but positive behavior.Plots off(R,∇R) DBI-essence scalar field (with constant γ): (a) ϕ versus cosmic time t and (b) potential versus t for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)Case 2 (γ≠constant).
Here we takeγ not as a constant. For this purpose, we assume γ=ϕ˙s where s is an arbitrary constant. This case leads to the following expressions:
(37)ϕ˙=ρdbi+pdbi1/(s+2),V(ϕ)=12[(ρdbi-pdbi)-(γ-2+1γ)T(ϕ)],
where T(ϕ)=ϕ˙2s+2/(ϕ˙2s-1). In order to plot scalar field and potential of f(R,∇R) DBI-essence scalar field (with γ≠ constant), we substitute ρdbi=ρR and pdbi=pR in (37). Figure 10 represents the same behavior as for constant γ; however, the values of scalar field and potential are quite different in both cases.Plots off(R,∇R) DBI-essence scalar field (with γ≠ constant): (a) ϕ versus cosmic time t and (b) potential versus t (right panel) for n=1 (red), n=2 (green), and n=3 (blue).
(a)
(b)


## 4. Concluding Remarks

In this work, we have discussed the flat FRW universe model in the framework off(R,∇R) gravity. We have obtained the modified Friedmann equations and the effective energy density as well as pressure coming from extra terms of the f(R,∇R) gravity sector. These can be treated as DE provided that the strong energy condition violates. We have assumed the power law solution of scale factor a(t)=a0(ts-t)n, ts>t. The term ts denotes the finite future singularity time and the scale factor is used to check type II (sudden singularity) or type IV (corresponds to H˙) for positive values of n. We have plotted the effective EoS parameter ωR versus time in Figure 1(a) for different values of n=1,2,3 and β=2.02, mp2=1, and ts=0.6. The EoS parameter shows increasing nature and transition from phantom to quintessence phase. For n=1, the ωR becomes nearly constant at ωR=-0.4 which represents the quintessence era. For increasing values of n, the EoS parameter again expresses transition from phantom to quintessence in the course of time, but this converges to vacuum era of the universe. In Figure 1(b), we have plotted squared speed of sound vs2 versus time. A positive value indicates the stability of model whereas instability stands for the negative value of vs2. We obtain two regions corresponding to the sign as t<0.6 and t>0.6. For the region t<0.6, it is found that squared speed of sound represents positively decreasing behavior which gives the stability of the model for all values of n. In the second region, squared speed of sound represents negative behavior which gives the instability (classically unstable) of the model for all values of n.Since scalar field models of DE are effective theories of an underlying theory of DE, we have studied the correspondence between the effective DE coming fromf(R,∇R) gravity with other scalar field DE models like quintessence, tachyon,k-essence, dilaton, hessence, and DBI-essence and constructed the scalar field as well as corresponding scalar potentials which describe the dynamics of the scalar fields graphically. We have taken the same values of the parameters in all cases taking different values of power of scale factor. The physical interpretations of all the DE models are summarized as follows.The evolution trajectories of scalar field and potential function versus time for the correspondence off(R,∇R) withquintessence dark energy model are shown in Figure 2 by taking the same values of constants as for effective EoS parameter ωR. From Figure 2(a), we have observed that the quintessence scalar field ϕ(t) represents increasing behavior and becomes more steeper as time goes on, particularly for greater value of n. The increasing value of scalar field indicates that kinetic energy is decreasing. The potential function versus time in plot 2(b) expresses decreasing nature. The plot of quintessence potential in terms of scalar field is shown in Figure 3 representing increasing behavior. The gradually decreasing kinetic energy while potential remains positive for quintessence model represents accelerated expansion of the universe.We have plotted the scalar field and potential function oftachyon model in Figure 4 for the correspondence scenario. The scalar field (Figure 4(a)) represents increasing behavior versus time and indicates more steeper behavior for n=1. This leads to the positively decreasing kinetic energy, which approximately tends to zero (i.e., the vacuum). The corresponding potential function (Figure 4(b)) expresses decreasing but positive behavior with respect to t. Its decreasing behavior from maxima gives inverse proportionality to the scalar field for the later times. This type of behavior corresponds to scaling solutions in the brane-world cosmology.Thek-essence model expresses the evolution of scalar field and kinetic energy term versus t. The evolution trajectories of ϕ and χ versus t for f(R,∇R)k-essence scalar field model are given in Figure 5. The plot 5(a) represents increasing behavior of scalar field with respect to time for all values of n. The plot 5(b) of χ versus t shows compatible behavior of the model for all n from early era to later times. This indicates that, for t≥0.6, χ(t) approaches to the required interval for accelerated expansion of the universe. Also, the potential function versus t and ϕ is shown in Figures 6(a)-6(b), expressing positively decreasing nature.The scalar field and kinetic energy term ofdilaton model are drawn in Figure 7, keeping the same values for other constants while c1=1.2. Figure 7(a) represents the increasing behavior of scalar field versus time. In the plot 7(b), the kinetic energy term χebϕ versus time shows increasing behavior in the course of time. The EoS parameter indicates the bound of kinetic energy term such as (5/18,5/9) for accelerated expansion of the universe. In this regard, we observe that the value n=1 is inconsistent with this interval while the remaining values of n represent compatible results.Forhessence model, scalar field and potential are shown in Figure 8. The scalar field expresses increasing behavior with respect to t, which is shown in Figure 8(a). In plot 8(b), the potential function represents the same behavior as the remaining scalar field models. Initially, this indicates decreasing behavior with the passage of time. It is noted that if Q=0, the hessence model may generate the scalar field model for quintessence. So quintessence model is the special case of hessence model.We have studied scalar field and potential functions ofDBI-essence model taking constant and variable DBI-parameter, which have been drawn in Figures 9 and 10. The scalar field shows increasing behavior while potential represents decreasing but positive behavior in both cases.



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