# Ghost Dark Energy in the DGP Braneworld

Authors: M. Abdollahi Zadeh; A. Sheykhi
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We investigate the ghost model of dark energy in the framework of DGP braneworld. We explore the cosmological consequences of this model by determining the equation of state parameter,ωD, the deceleration, and the density parameters. We also examine the stability of this model by studying the squared of the sound speed in the presence/absence of interaction term between dark energy and dark matter. We find out that in the absence of interaction between two dark sectors of the universe we have ωD→-1 in the late time, while in the presence of interaction ωD can cross the phantom line -1. In both cases the squared of sound speed vs2 does not show any signal of stability. We also determine the statefinder diagnosis of this model as well as the ωD-ωD′ plane and compare the results with the ΛCDM model. We find that ωD-ωD′ plane meets the freezing region in the absence of interaction between two dark sectors, while it meets both the thawing and the freezing regions in the interacting case.


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## Body

## 1. Introduction

The current acceleration of the universe expansion is strongly confirmed by the type Ia supernova observations [1] and also supported by the astrophysical data from WMAP [2]. A component which is responsible for this acceleration is called dark energy (DE) with negative pressure, which can overcome the gravity force between the galaxies and push them to accelerate, though its nature and origin is still an open question in the modern cosmology. There are two approaches for explanation of the cosmic acceleration. The first one is the modified gravity models such as f(R) gravity [3] and scalar-tensor theories [4], and the second approach is the idea of the existence of a strange type of energy whose gravity is repulsive such as the cosmological constant Λ [5] and the dynamical DE models [6, 7]. Against the cosmological constant Λ which has constant equation of state (EoS) parameter, ωD=-1, further observations detect a small variation in the EoS parameter of DE in favor of a dynamical DE with ωD>-1 in the past and even ωD<-1 in the late time [8].An interesting model for probing the dynamical DE model is the ghost dark energy (GDE) model proposed in [9]. The advantages of this model are that it does not introduce any new degree of freedom in contrast to the most DE models which explain the accelerated expansion by introducing new degree(s) of freedom or by modifying the underlying theory of gravity. This is important because, with introducing new degrees of freedom, one needs to investigate the nature and new consequences in the universe so it seems to be impressive and economic if we can explain DE puzzle by using currently known fluids and fields of nature. Actually, GDE model which is based on the Veneziano ghost in Quantum Chromodynamics (QCD) can act as the source of DE [10] and its existence are required for resolution of the U(1) problem in QCD [11]. Indeed, the ghosts are decoupled from the physical states and make no contribution in flat Minkowski space, but it produces a small vacuum energy density in a dynamic background or a curved spacetime proportional to ΛQCD3H, where H is the Hubble parameter and ΛQCD is QCD mass scale of order a 100MeV [12]. Different features of GDE have been studied in ample details [13].Independent of the DE puzzle, for explanation of the cosmic acceleration, special attention is also paid to extra dimensional theories, in which our universe is realized as a3-brane embedded in a higher dimensional spacetime. Based on the braneworld model, all the particle fields in the standard model are confined to a four-dimensional brane, while gravity is free to propagate in all dimension. One of the original models of braneworld is introduced by Dvali-Gabadadze-Porrati (DGP) [14], which describes our universe as a 4D brane embedded in a 5D Minkowskian bulk with infinite size. In this model the recovery of the usual gravitational laws on the brane is obtained by adding an Einstein-Hilbert term to the action of the brane computed with the brane intrinsic curvature. It is a well-known that the DGP model has two branches of solutions. The self-accelerating branch of DGP model can explain the late time cosmic speed-up without recourse to DE or other components of energy [15, 16]. However, the self-accelerating DGP branch has ghost instabilities and it cannot realize phantom divide crossing by itself. To realize phantom divide crossing it is necessary to add at least a component of energy on the brane. On the other hand, the normal DGP branch cannot explain acceleration but it has the potential to realize a phantom-like phase by dynamical screening on the brane. Adding a DE component to the normal branch solution brings new facilities to explain late time acceleration and also better matching with observations. These are the motivations to add DE to this braneworld setup [17, 18]. In this work we would like to investigate the GDE model in the framework of the DGP braneworld. This study is of great importance, since we can incorporate and disclose the effects of the extra dimension on the evolution of the cosmological parameters on the brane when the DE source is in the form of GDE.This paper is organized as follows. In Section2 we formulate the GDE model in the context of the DGP braneworld. We also consider both interacting and noninteracting cases and explore various cosmological parameters as well as cosmological planes. Besides the discussion of instability analysis, we study the ωD-ωD′ plane and properties of statefinder parameters. We finish with closing remarks in Section 3.


## 2. The GDE in the DGP Model

In the DGP cosmology, a homogeneous, spatially flat, and isotropic3-dimensional brane which is embedded in a 5-dimensional Minkowskian bulk can be described by the following Friedmann equation [19](1)H2=ρm+ρD3mp2+14rc2+ϵ2rc2,or equivalently(2)H2-ϵrcH=13mp2ρm+ρD,where H=a˙/a is the Hubble parameter, rc=mpl2/(2m53) [20] is the crossover length scale reflecting the competition between 4D and 5D effects of gravity, and ϵ=±1 corresponds to the two branches of solutions of the DGP model. Before going any further, it is worth noting that if H-1≪rc (early times) the 4D general relativity is recovered; otherwise the 5D effect becomes significant. Also ϵ=+1 corresponds to the self-accelerating solution where the universe may accelerate in the late time purely due to modification of gravity [15, 16], while ϵ=-1 can produce the acceleration only if a DE component is included on the brane. Here, to accommodate GDE into the formalism we take ϵ=-1.The fractional energy density parameters are defined as(3)Ωm=ρm3mp2H2,ΩD=ρD3mp2H2,Ωrc=14rc2H02,where H0 is the Hubble parameter at redshift z=0. The Friedmann equation (2) can be rewritten in terms (3) as(4)Ωm+ΩD+2ϵH0HΩrc=1.We introduce ΩDGP=2ϵΩrcH0/H, which comes from the extra dimension. Thus the Friedmann equation (4) can be reexpressed as(5)Ωm+ΩD+ΩDGP=1.For the GDE density we have(6)ρD=αH, where α is a constant of order ΛQCD3 and ΛQCD is the QCD mass scale [12]. Taking the time derivative of the energy density ρD and using (6) we obtain(7)ρ˙D=ρDH˙H. For the FRW universe filled with DE and DM, with mutual interaction, the energy-momentum conservation law can be written as(8)ρ˙m+3Hρm=Q,(9)ρ˙D+3H1+ωDρD=-Q,where Q=3b2H(ρD+ρm) is considered as the interaction term between DE and DM also b2 is the coupling constant of interaction Q.We know that (i) our universe is in a DE dominated phase and (ii) our universe that is our habitat is stable. These imply that any variable DE model should result in a stable DE dominated universe. So it is worth investigating the stability of the GDE in DGP braneworld against perturbation. The intended indicator for checking the stability of a proposed DE model is to study the behavior of the squared sound speed (vs2=dP/dρ) [21]. If vs2<0 we have the classical instability of a given perturbation because the perturbation of the background energy density is an oscillatory function and may grow or decay with time. When vs2>0, we expect a stable universe against perturbations because the perturbation in the energy density propagates in the environment. We continue discussion of stability in the linear perturbation regime where the perturbed energy density of the background can be written as(10)ρt,x=ρt+δρt,x, where ρ(t) is unperturbed background energy density. For the energy conservation equation (∇μTμν=0) which yields [21](11)δρ¨=vs2∇2δρt,x, we encounter two cases. In the first case where vs2>0, we observe an ordinary wave equation which has a wave solution in the form δρ=δρ0e-iωt+ik→.x→ (stable universe). In the second case where vs2<0, the frequency of the oscillations becomes pure imaginary and the density perturbations will grow with time as δρ=δρ0eωt+ik→.x→ (unstable universe). Since vs2 plays a crucial role in determining the stability of DE model, we rewrite it in terms of EoS parameter as(12)vs2=P˙ρ˙=ρ˙DwD+ρDw˙Dρ˙D1+u+ρDu˙, where P=PD is the pressure of DE, ρ=ρm+ρD is the total energy density of DE and DM, and u=Ωm/ΩD is the energy density ration.On the other sides, Sahni et al. [22] proposed new geometrical diagnostic pair parameter {r,s}, known as statefinder parameter, for checking the viability of newly introduced DE models. Unlike the physical variables which depend on the properties of physical fields describing DE models, the statefinder pair primarily depends on the scale factor and hence it depends on the metric of the spacetime. The r and s parameters are defined as [22](13)r=a⃛aH3,s=r-13q-1/2, where r can rewrite as(14)r=1+3H˙H2+H¨H3. and then(15)r=2q2+q-q˙H. Let us note that in the {r,s} plane, s>0 corresponds to a quintessence-like model of DE and s<0 corresponds to a phantom-like model of DE. Also the studies on a flat ΛCDM model and matter dominated universe (SCDM) show that for these models {r,s}={1,0} and {r,s}={1,1}, respectively. In above equations q is the deceleration parameter which is given by(16)q=-1-H˙H2. In what follows we discuss the ωD-ωD′ plane which introduced by Caldwell and Linder [23] for analyzing the dynamical property of various DE models and distinguish these models (ωD′ represents the evolution of ωD). The models can be categorized into two different classes: (i) ωD′>0 and ωD<0 which present the thawing region. (ii) ωD′<0 and ωD<0 which present the freezing region. It should be noted that the ΛCDM model corresponds to a fixed point {ωD=-1,ωD′=0} in the ωD-ωD′ plane. We shall consider the noninteracting and interacting cases, separatively.It deserves to mention here that in all figures we selected values for the fixed parameters, which is custom and the observational evidence confirms it, asΩrc=0.0003[24] orΩD(z=0)that can be between 0.7 and  .75 andH(z=0)that can be between 60 and 100. It should be noted that the results and conclusions are not sensitive to these choices and also do not do any observational work on this title; thus we by helping papers and experience selected these values for the fixed parameters.


### 2.1. Noninteracting Case

We start to obtain the cosmological parameters for GDE in the DGP braneworld by ignoring the interaction term(Q=0). The deceleration parameter q can be obtained by taking the time derivative of (2), which lead to(17)H˙H2=-31-ΩDGP+3ΩD2-ΩDGP-ΩD. Using relation (16), we find(18)q=-1--31-ΩDGP+3ΩD2-ΩDGP-ΩD. Inserting (7) in (9) we have(19)ωD=-1-13H˙H2, where by replacing (17) in it, we get(20)ωD=-12-ΩDGP-ΩD. Also we obtain ωD′ from the above equation as(21)ωD′=-3-1+ΩDGP+ΩDΩDGP+ΩD-2+ΩDGP+ΩD3. Note that, in order to find the evolution of density parameter ΩD, we take the time derivative of relation ΩD=ρD/(3mp2H2), after combining the result with (7) and (17), which yields(22)ΩD′=ΩD1+q.In Figure 1, we plot the evolution of ΩD versus redshift parameter z. It is obvious that ΩD tends to 0 in the early universe where 1+z→∞, while at the late time where 1+z→0, we have ΩD→1. Clearly, (20) for the EoS parameter shows that, at the late time where ΩD→1, the EoS parameter mimics the cosmological constant, namely, ωD→-1,which is showed inFigure 2. In Figure 3, the behavior of the deceleration parameter q is plotted and indicates that indeed there is a decelerated expansion at the early stage of the universe followed by an accelerated expansion. The energy density ratio is defined as u=Ωm/ΩD, which by using (5) can be written(23)u=-1+1ΩD1-ΩDGP.Differentiating (23) and (20) and then substituting the results in (12) we get the squared of sound speed as(24)vs2=-2ΩD-1+ΩDGP+ΩD-2+ΩDGP-2+ΩDGP+ΩD2.Figure 1
The evolution ofΩD versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 2
The evolution ofωD versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 3
The evolution of the deceleration parameterq versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.The evolution ofvs2 against z for the noninteracting GDE in the framework of DGP braneworld is plotted in Figure 4. From graphical analysis of vs2 one concludes that this model does not indicate any signal of stability, that is, vs2<0 during the history of the universe. We can also find the statefinder parameters r and s by taking derivative of (18) and using (13) and (15). The results are(25)r=10+18-2+ΩDGP+ΩD3+45-2+ΩDGP+ΩD2+36-2+ΩDGP+ΩD,(26)s=2-1+ΩDGP+ΩD2-2+ΩDGP+ΩD2.Figure 4
The evolution of the squared of sound speedvs2 versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.The graphical behavior of the statefinder parameters{r,s} given in (25) and (26) is plotted in Figures 5 and 6, showing that, at late time where ΩD→1, we have {r,s}={1,0} which implies that GDE mimics the cosmological constant at the late time, as expected.Figure 5
The evolution of the statefinder parameterr versus the redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 6
The evolution of the statefinder parameters versus the redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Let us study the trajectory in the statefinder plane and analyze this model from the statefinder viewpoint. For this purpose, we plot the statefinder diagram in ther-s in Figure 7, which shows the cure gets to the point {r,s}={1,0} in the end, which implies that the model corresponds to the ΛCDM model at the late time. For complementarity of the diagnostic, we also plot the trajectories of statefinder pair r-q in Figure 8 which ends in the future to r=1, q=-1 corresponding to the de Sitter expansion.Figure 7
The evolution of the statefinder parameterr versus s for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 8
The evolution of the statefinder parameterr versus the deceleration parameter q for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.TheωD-ωD′ plane for the noninteracting GDE in the DGP scenario is shown in Figure 9. Again, we see that this plane corresponds to ΛCDM model, i.e., (ωD=-1,ωD′=0), and the trajectory meets the freezing region as well.Figure 9
TheωD-ωD′ diagram for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.


### 2.2. Interacting Case

Differentiating the modified Friedmann equation (2) and using (7) and (8) we reach(27)H˙H2=3b2-11-ΩDGP+3ΩD2-ΩDGP-ΩD,(28)q=-1-3b2-11-ΩDGP+3ΩD2-ΩDGP-ΩD. Next, the EoS parameter can be determined by substituting (7) into the semiconservation law equation (9) and using (27). We find(29)ωD=b2ΩDGP-2ΩDGP-1+ΩDΩD-2+ΩDGP+ΩD.Taking differentiation with respect to x=ln⁡a from above equation we get(30)ωD′=-3-1+b2-1+ΩDGP-ΩDb2-2+ΩDGP2+-ΩDGP+b2-4+3ΩDGP-ΩDΩDΩD-2+ΩDGP+ΩD3, where the prime indicates derivative with respect to x=ln⁡a. We can obtain the equation of motion for ΩD as(31)ΩD′=3ΩD-1+ΩD+ΩDGP+b21-ΩDGP-2+ΩDGP+ΩD,To illustrate the cosmological consequences of the interacting GDE in the DGP braneworld, we plot their evolution in terms of redshift parameterz. In Figure 10, we present the graphical of ΩD versus z for the different values of the coupling constant b2. As expected, we see both ΩD→1 and ΩD→0 for late time and early time, respectively. The graphical behavior of the EoS parameter for the different values of b2 shows crossing of phantom line as plotted in Figure 11. The stability of interacting GDE in DGP model can be obtained by differentiating with respect to time of (23) and (29)(32)vs2=-b2-2+ΩDGP3+ΩD6+-6+ΩDGPΩDGP+ΩD-2+2ΩDGP+2ΩD-2+ΩDGP-2+ΩDGP+ΩD2.Figure 10
The evolution ofΩD versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 11
The evolution ofωD versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.The evolution of the deceleration parameterq and the squared of sound speed vs2 versus redshift parameter z are plotted in Figures 12 and 13, respectively. In Figure 12, we see for different values of b2 with the interacting GDE in DGP model, our universe has a phase transition from deceleration to an acceleration, while by keeping the same situation in Figure 13, this universe cannot be stable. As the value of b2 decreases the severity of instability also decreases. Like previous section, the statefinder parameters are obtained by taking derivative of (28) and using (13) and (15)(33)r=10+181+b2-b2ΩDGP-2+ΩDGP+ΩD3+9-1+b2-1+ΩDGP-5+b2-3+2ΩDGP-2+ΩDGP+ΩD2+94+b24-3ΩDGP-2+ΩDGP+ΩD,(34)s=2+2-2b2-1+ΩDGP-2+ΩDGP+ΩD2+4+b23-2ΩDGP-2+ΩDGP+ΩD+b22b21-ΩDGp+ΩDGP+ΩD.Figure 12
The evolution of the deceleration parameterq versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 13
The evolution of the squared of sound speedvs2 versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.We obtain{r,s}={1,0} for ΛCDM model from (33) and (34) in the limiting case where b2=0, ΩDGP=0, and ΩD→1 (in the late time). Also, Figures 14 and 15 show that r and s are positive through the entire life of the universe and turn to 1 and 0 at the late time, respectively. The {r,s} evolutionary trajectories for the interacting GDE in the framework of the DGP braneworld for different values of b2 are shown in Figure 16. From Figure 16, we can see that at the late time all curves tend to the ΛCDM fixed point {r=1,s=0}; also different b2 results in different evolution trajectories of statefinder which states that r is smaller when b2 is larger. The r-q diagrams are plotted for different values of b2 in Figure 17 which mimics the de Sitter expansion, namely, r=1, q=-1 in the far future where z→0. In Figure 18, we plot the ωD-ωD′ plane for different values of b2 which show that the trajectories meet both the thawing and the freezing regions as well.Figure 14
The evolution of the statefinder parameterr versus the redshift parameter z for interacting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 15
The evolution of the statefinder parameters versus the redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 16
The evolution of the statefinder parameterr versus s for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 17
The evolution of the statefinder parameterr versus the deceleration parameter q for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 18
TheωD-ωD′ diagram for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.


## 2.1. Noninteracting Case

We start to obtain the cosmological parameters for GDE in the DGP braneworld by ignoring the interaction term(Q=0). The deceleration parameter q can be obtained by taking the time derivative of (2), which lead to(17)H˙H2=-31-ΩDGP+3ΩD2-ΩDGP-ΩD. Using relation (16), we find(18)q=-1--31-ΩDGP+3ΩD2-ΩDGP-ΩD. Inserting (7) in (9) we have(19)ωD=-1-13H˙H2, where by replacing (17) in it, we get(20)ωD=-12-ΩDGP-ΩD. Also we obtain ωD′ from the above equation as(21)ωD′=-3-1+ΩDGP+ΩDΩDGP+ΩD-2+ΩDGP+ΩD3. Note that, in order to find the evolution of density parameter ΩD, we take the time derivative of relation ΩD=ρD/(3mp2H2), after combining the result with (7) and (17), which yields(22)ΩD′=ΩD1+q.In Figure 1, we plot the evolution of ΩD versus redshift parameter z. It is obvious that ΩD tends to 0 in the early universe where 1+z→∞, while at the late time where 1+z→0, we have ΩD→1. Clearly, (20) for the EoS parameter shows that, at the late time where ΩD→1, the EoS parameter mimics the cosmological constant, namely, ωD→-1,which is showed inFigure 2. In Figure 3, the behavior of the deceleration parameter q is plotted and indicates that indeed there is a decelerated expansion at the early stage of the universe followed by an accelerated expansion. The energy density ratio is defined as u=Ωm/ΩD, which by using (5) can be written(23)u=-1+1ΩD1-ΩDGP.Differentiating (23) and (20) and then substituting the results in (12) we get the squared of sound speed as(24)vs2=-2ΩD-1+ΩDGP+ΩD-2+ΩDGP-2+ΩDGP+ΩD2.Figure 1
The evolution ofΩD versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 2
The evolution ofωD versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 3
The evolution of the deceleration parameterq versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.The evolution ofvs2 against z for the noninteracting GDE in the framework of DGP braneworld is plotted in Figure 4. From graphical analysis of vs2 one concludes that this model does not indicate any signal of stability, that is, vs2<0 during the history of the universe. We can also find the statefinder parameters r and s by taking derivative of (18) and using (13) and (15). The results are(25)r=10+18-2+ΩDGP+ΩD3+45-2+ΩDGP+ΩD2+36-2+ΩDGP+ΩD,(26)s=2-1+ΩDGP+ΩD2-2+ΩDGP+ΩD2.Figure 4
The evolution of the squared of sound speedvs2 versus redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.The graphical behavior of the statefinder parameters{r,s} given in (25) and (26) is plotted in Figures 5 and 6, showing that, at late time where ΩD→1, we have {r,s}={1,0} which implies that GDE mimics the cosmological constant at the late time, as expected.Figure 5
The evolution of the statefinder parameterr versus the redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 6
The evolution of the statefinder parameters versus the redshift parameter z for noninteracting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Let us study the trajectory in the statefinder plane and analyze this model from the statefinder viewpoint. For this purpose, we plot the statefinder diagram in ther-s in Figure 7, which shows the cure gets to the point {r,s}={1,0} in the end, which implies that the model corresponds to the ΛCDM model at the late time. For complementarity of the diagnostic, we also plot the trajectories of statefinder pair r-q in Figure 8 which ends in the future to r=1, q=-1 corresponding to the de Sitter expansion.Figure 7
The evolution of the statefinder parameterr versus s for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 8
The evolution of the statefinder parameterr versus the deceleration parameter q for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.TheωD-ωD′ plane for the noninteracting GDE in the DGP scenario is shown in Figure 9. Again, we see that this plane corresponds to ΛCDM model, i.e., (ωD=-1,ωD′=0), and the trajectory meets the freezing region as well.Figure 9
TheωD-ωD′ diagram for noninteracting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.


## 2.2. Interacting Case

Differentiating the modified Friedmann equation (2) and using (7) and (8) we reach(27)H˙H2=3b2-11-ΩDGP+3ΩD2-ΩDGP-ΩD,(28)q=-1-3b2-11-ΩDGP+3ΩD2-ΩDGP-ΩD. Next, the EoS parameter can be determined by substituting (7) into the semiconservation law equation (9) and using (27). We find(29)ωD=b2ΩDGP-2ΩDGP-1+ΩDΩD-2+ΩDGP+ΩD.Taking differentiation with respect to x=ln⁡a from above equation we get(30)ωD′=-3-1+b2-1+ΩDGP-ΩDb2-2+ΩDGP2+-ΩDGP+b2-4+3ΩDGP-ΩDΩDΩD-2+ΩDGP+ΩD3, where the prime indicates derivative with respect to x=ln⁡a. We can obtain the equation of motion for ΩD as(31)ΩD′=3ΩD-1+ΩD+ΩDGP+b21-ΩDGP-2+ΩDGP+ΩD,To illustrate the cosmological consequences of the interacting GDE in the DGP braneworld, we plot their evolution in terms of redshift parameterz. In Figure 10, we present the graphical of ΩD versus z for the different values of the coupling constant b2. As expected, we see both ΩD→1 and ΩD→0 for late time and early time, respectively. The graphical behavior of the EoS parameter for the different values of b2 shows crossing of phantom line as plotted in Figure 11. The stability of interacting GDE in DGP model can be obtained by differentiating with respect to time of (23) and (29)(32)vs2=-b2-2+ΩDGP3+ΩD6+-6+ΩDGPΩDGP+ΩD-2+2ΩDGP+2ΩD-2+ΩDGP-2+ΩDGP+ΩD2.Figure 10
The evolution ofΩD versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 11
The evolution ofωD versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.The evolution of the deceleration parameterq and the squared of sound speed vs2 versus redshift parameter z are plotted in Figures 12 and 13, respectively. In Figure 12, we see for different values of b2 with the interacting GDE in DGP model, our universe has a phase transition from deceleration to an acceleration, while by keeping the same situation in Figure 13, this universe cannot be stable. As the value of b2 decreases the severity of instability also decreases. Like previous section, the statefinder parameters are obtained by taking derivative of (28) and using (13) and (15)(33)r=10+181+b2-b2ΩDGP-2+ΩDGP+ΩD3+9-1+b2-1+ΩDGP-5+b2-3+2ΩDGP-2+ΩDGP+ΩD2+94+b24-3ΩDGP-2+ΩDGP+ΩD,(34)s=2+2-2b2-1+ΩDGP-2+ΩDGP+ΩD2+4+b23-2ΩDGP-2+ΩDGP+ΩD+b22b21-ΩDGp+ΩDGP+ΩD.Figure 12
The evolution of the deceleration parameterq versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.Figure 13
The evolution of the squared of sound speedvs2 versus redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=0.0003.We obtain{r,s}={1,0} for ΛCDM model from (33) and (34) in the limiting case where b2=0, ΩDGP=0, and ΩD→1 (in the late time). Also, Figures 14 and 15 show that r and s are positive through the entire life of the universe and turn to 1 and 0 at the late time, respectively. The {r,s} evolutionary trajectories for the interacting GDE in the framework of the DGP braneworld for different values of b2 are shown in Figure 16. From Figure 16, we can see that at the late time all curves tend to the ΛCDM fixed point {r=1,s=0}; also different b2 results in different evolution trajectories of statefinder which states that r is smaller when b2 is larger. The r-q diagrams are plotted for different values of b2 in Figure 17 which mimics the de Sitter expansion, namely, r=1, q=-1 in the far future where z→0. In Figure 18, we plot the ωD-ωD′ plane for different values of b2 which show that the trajectories meet both the thawing and the freezing regions as well.Figure 14
The evolution of the statefinder parameterr versus the redshift parameter z for interacting GDE in DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 15
The evolution of the statefinder parameters versus the redshift parameter z for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 16
The evolution of the statefinder parameterr versus s for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 17
The evolution of the statefinder parameterr versus the deceleration parameter q for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.Figure 18
TheωD-ωD′ diagram for interacting GDE in the DGP model. Here, we have taken ΩD(z=0)=0.73, H(z=0)=67, and Ωrc=.0003.


## 3. Closing Remarks

We have made a versatile study on both noninteracting and interacting GDE in the framework DGP model through well-known cosmological parameters as well as planes. We summarize our results as follows. For noninteracting case, we have found that the density parameter tends to zero at the early universe while at the late time we haveΩD→1. Meanwhile the EoS parameter cannot cross the phantom line and mimics the cosmological constant at the late time (Figure 2). We have shown that our universe has a phase transition from deceleration to an acceleration, though we do not receive any signal of stability. The statefinder plane shows that the trajectory corresponds to quintessence model (s>0 and r<1) while at late time we have {r,s}={1,0} for ΛCDM model as expected. The ωD-ωD′ plane in Figure 9 meets the freezing region as well.For interacting case, we find that the density and the deceleration parameters as well as the EoS parameter are consistent with observational data. We have seem that as the value ofb2 decreases the severity of instability decreases. From r-s plane, we can see that at the late time all cures tend to the ΛCDM fixed point {r=1,s=0}. Besides, for different values of b2, the different evolution trajectories of statefinder are shown which indicates that r is smaller when b2 is larger. The r-q plane is plotted in Figure 17 which mimics the de Sitter expansion, namely, r=1, q=-1 in the far future where z→0. In the end, the ωD-ωD′ plane exhibits both freezing and thawing regions of the universe for all values of b2. Again, in this case vs2<0 which implies that interacting GDE in the DGP braneworld is not stable against perturbation.



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*Source: 2514927-2018-10-14.xml*