We analyze the behavior of pilgrim dark energy with Hubble horizon in f(G) gravity. We reconstruct the f(G) models through correspondence phenomenon by assuming two values of pilgrim dark energy parameter (u=2,-2). We evaluate the equation of state parameter which shows evolution of the universe in the quintessence, vacuum, and phantom phase for both cases of u and give favor the pilgrim dark energy phenomenon. Also, squared speed of sound exhibits the stability of f(G) model for both cases of u. The wPDE-wPDE′ also provides freezing and thawing regions in this scenario. In this framework, the r-s plane also corresponds to different dark energy scenarios.

1. Introduction

It is confirmed through different observational schemes that our universe undergoes accelerated expansion. The pioneer observations about this accelerated expansion phenomenon was made by a variety of astronomers through supernova type Ia [1, 2]. The subsequent attempts in this regard also favor this phenomenon [3–6]. The background force which acts upon behind this phenomenon is an exotic type called dark energy (DE) whose nature is still under consideration. In order to understand its nature, various approaches have been adopted, the proposal of different dynamical DE models [7–14] and modified theories of gravity [15–19] to peruse the DE problem.

Moreover, the accelerated expansion is also discussed widely in the literature through correspondence phenomenon of dynamical DE models with modified gravity. Many works have been done in this direction by adopting different scenarios in modified gravity theories [20–25]. Further, Setare and Saridakis [26] considered the holographic and Gauss-Bonnet DE models separately to investigate the conditions under which these models can be simultaneously valid. They found that this correspondence phenomenon leads to accelerated expansion of the universe. Also, they investigated the correspondence of HDE model with canonical, phantom, and quintom models minimally coupled to gravity and pointed out consistent results about acceleration of the universe [27]. Further, Setare [28–32] has studied this correspondence scenario incorporating different dynamical DE models as well as modified theories of gravity in this respect. We have also investigated the correspondence phenomenon of various DE models with modified gravities and found that these scenarios favor the accelerated expansion phenomenon of the universe [33–39].

The bound of energy density as suggested by Cohen et al. [40] predicts the formation of black hole (BH) in quantum gravity. However, it is suggested that formation of BH can be avoided through appropriate repulsive force which resists the matter collapse phenomenon. This force can only provide phantom DE in spite of other phases of DE like vacuum and quintessence DE. By keeping in mind this phenomenon, Wei [41] has suggested the DE model called pilgrim DE (PDE) on the speculation in which phantom DE possesses the large negative pressure as compared to the quintessence DE which helps in violating the null energy condition and possibly prevent the formation of BH. In the past, many applications of phantom DE exist in the literature. For instance, phantom DE also plays an important role on the reduction of masses of BHs [42–45] and in the wormhole physics where the event horizon can be avoided due to its presence [46–50].

In the present work, we present the reconstruction scenario of newly proposed DE model called pilgrim DE (PDE) with the f(G) gravity. We will construct f(G) models and analyze it through different cosmological parameters as well as cosmological planes. The paper is organized as follows: next section contains the discussion of f(G) models. In Section 3, we provide the analysis of EoS parameter, squared speed of sound, wPDE-wPDE′, and r-s planes. The last section contains the summary of the outcomes.

2. Pilgrim Dark Energy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mi>f</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>G</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula> Models

It was demonstrated by Sahni and Starobinsky [51] that there is also a possibility to write the field equations in modified gravities in terms of usual Einstein field equations by transferring all additional terms from the left-hand side into the right-hand side of the Einstein equations and called them as an effective energy momentum tensor of DE. Here, we consider f(G) gravity where the modified field equations are [51, 52](1)3H2=ρG+ρm,2H˙+3H2=pG+pm.The subscripts m indicate the matter contribution of energy density and pressure and H=a˙/a is the Hubble parameter with dot representing the time derivative. The Gauss-Bonnet invariant G for FRW metric becomes G=24H2(H˙+H2). Also,(2)ρG=ρm+GfG-f-24H3f˙G,pG=-8H2f¨G-16H(H2+H˙)f˙G-f+GfG.Here, we will incorporate with the pressureless matter, that is, pm=0, while the energy density of matter ρm satisfies the following energy conservation equation:(3)ρ˙m+3Hρm=0.This equation has a solution which is given by the following expression:(4)ρm=ρm0a-3,where ρm0 is an arbitrary constant.

It is predicted that phantom DE with strong negative pressure can push the universe towards the big rip singularity where all the physical objects lose the gravitational bounds and finally dispersed. This prediction supports the phenomenon of the avoidance of BH formation and motivated Wei [41] in constructing PDE model. The PDE model is defined as(5)ρPDE=3n2mp4-sL-u,where L is the IR cutoff and size of the system, u is the PDE parameter, n is conventional constant, and mp is reduced Planck mass. Wei analyzed the PDE model through different possible theoretical and observational ways to make the BH-free phantom universe with Hubble horizon (L=H-1) through PDE parameter (u). Further, Sharif and Jawad [43–45] have analyzed this proposal in detail by choosing different IR cutoffs through well-known cosmological parameters in flat and nonflat universes. This model has also been in different modified gravities [53, 54].

Here, we make correspondence between PDE and f(G) model by equating their energy densities; that is, ρG=ρPDE. It yields(6)f-24H2H2+H˙fG+242H4·(2H˙2+HH¨+4H2H˙)fGG=-3n2mp4-sHu.To find out an analytic solution of (6), we assume a power-law form of the scale factor as follows:(7)at=a0tm,where the constant a0 represents the present day value of the scale factor. By using (7) in the reconstructed equation, we obtain(8)G2d2f(G)dG2+m-124Gdf(G)dG-m-124f(G)=n232m2um-1u-8/4Gu.Equation (8) is a second-order linear differential equation whose solution is given by(9)f(G)=Gum-10.25(8-u)m-2un232(l+-2u)(l--2u)+AGl-+BGl+,where A and B are integration constants and(10)l±=121±1+12m-12+116m-14-14m-12.Equation (9) represents the reconstructed PDE f(G) model which we have plotted for three different values of m= 2, 2.2, and 2.4 as shown in Figures 1 and 2. The assumptions of other constants parameters are A=0.5,B=0.2,n=0.91. Figure 1 shows that the f(G) model versus G exhibits decreasing behavior initially and then approaches G=0 after some interval. After that, it shows increasing behavior forever for all values of m. Figure 2 represents that the reconstructed function f(G) shows rapid decrease initially and then approaches positive value forever for all values of m.

Plot of f(G) versus G for PDE parameter u=2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Plot of f(G) versus G for PDE parameter u=-2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

In this section, we discuss the cosmological parameters such as EoS parameter and squared speed of sound. Also, we develop cosmological planes such as wPDE-wPDE′ and r-s.

3.1. Equation of State Parameter

Through reconstruction scenario, that is, ρG=ρPDE and pG=pPDE, we can get EoS as follows:(11)wPDE=pPDEρPDE.We plot the above EoS parameter versus redshift parameter z for PDE parameter u=2,-2 as shown in Figures 3 and 4, respectively, while keeping the other constants same as in previous section. We use the expression of scale factor in terms of redshift parameter a=a0(1+z)-1. It can be observed from Figure 3 (u=2) that the EoS parameter starts from quintessence phase (at the present time), crosses the phantom divide line, and then goes towards phantom phase of the universe. After achieving the extreme value of phantom phase, the EoS parameter turns back and approaches dust like matter in the end. This type of behavior of EoS parameter is termed as quintom. Figure 4 (u=-2) shows similar behavior of EoS parameter as in case of u=2. It is concluded that both PDE f(G) models favor the PDE phenomenon.

Plot of wPDE versus z for PDE parameter u=2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Plot of wPDE versus z for PDE parameter u=-2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

3.2. Squared Speed of Sound

Here, we analyze the squared speed of sound for the stability analysis of PDE f(G) model. In the present case, the squared speed of sound takes the form(12)vs2=p˙PDEρ˙PDE.The sign of vs2 is very important to see the stability of background evolution of the model. A positive value indicates a stable model whereas instability of a given perturbation corresponds to the negative value of vs2. The plot of squared speed of sound with respect to redshift parameter for two values of PDE parameter is as shown in Figures 5 and 6. It can be observed from Figures 5 and 6 that the PDE f(G) model for u=2,-2 shows stability at the present as well as at latter epoch because vs2≥0 for all the time.

Plot of vs2 versus z for PDE parameter u=2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Plot of vs2 versus z for PDE parameter u=-2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

This plane is firstly proposed by Caldwell and Linder [55] and tested the behavior of quintessence scalar field DE model through this plane. Also, this can be divided into two regions such as thawing and freezing. The thawing region is described as (ωΛ′>0,ωΛ<0) while freezing region as (ωΛ′<0,ωΛ<0). Here, we develop the ωPDE-ωPDE′ plane for reconstructed PDE f(G) models corresponding to u=2,-2 as shown in Figures 7 and 8, respectively, for three different values of m= 2, 2.2, and 2.4. It can be observed from Figure 7 that ωPDE-ωPDE′ plane corresponds to freezing region only for m=2,2.4 while it lies in both regions (freezing and thawing) for m=2.2. However, ωPDE-ωPDE′ plane corresponds to thawing region only for all cases of m. Hence, ωPDE-ωPDE′ planes corresponding to m=2,2.2,2.4 show consistency with the accelerated expansion of the universe.

Trajectories of wPDE-wPDE′ for reconstructed PDE f(G) model for u=2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Trajectories of wPDE-wPDE′ for reconstructed PDE f(G) model for u=-2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Up to now, a variety of DE models have been developed for explaining the phenomenon of DE in the accelerated expansion of the universe. It is necessary to differentiate these models so that one can decide which one provides better explanation for the current status of the universe. Many DE models give the same present value of the deceleration and Hubble parameter, so these parameters could not be able to discriminate the DE models. For this purpose, Sahni et al. [56] introduced two new dimensionless parameters by combining the Hubble and deceleration parameters which are expressed as(13)r=a⃛aH3,s=r-13(q-1/2),where(14)q=-a¨aH2=-1+H˙H2.The statefinders are useful in the sense that we can find the distance of a given DE model from ΛCDM limit. The well-known regions described by these cosmological parameters are as follows: (r,s)=(1,0) indicates ΛCDM limit and (r,s)=(1,1) shows CDM limit, while s>0 and r<1 represent the region of phantom and quintessence DE eras.

The r-s planes develop corresponding to reconstructed PDE f(G) models with u=2,-2 for three different values of m= 2, 2.2, and 2.4 as shown in Figures 9 and 10. Figure 9 shows that r-s plane (PDE f(G) models with u=2) corresponds to Chaplygin gas model and DE regions (quintessence and phantom) for all three cases of m. Also, r-s plane (PDE f(G) models with u=-2) corresponds to only Chaplygin gas model as shown in Figure 10 in all three cases of m. However, ΛCDM limit can not be achieved in both cases of PDE parameter of u.

Trajectories of r-s for reconstructed PDE f(G) model for u=2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

Trajectories of r-s for reconstructed PDE f(G) model for u=-2 with m=2 (red), m=2.2 (green), and m=2.4 (blue).

4. Concluding Remarks

In this paper, we have considered the reconstruction phenomenon of a well-known PDE model with f(G) gravity in the presence of power law scale factor. We have reconstructed f(G) models with respect to two values of PDE parameter; that is, u=2,-2. To check the significant cosmological aspects of these reconstructed models, we have presented the analysis of different cosmological parameters as well as cosmological planes. We summarized the results of them as follows.

The f(G) model versus G exhibits decreasing behavior initially and then approaches G=0 after some interval as shown in Figure 1. After that, it shows increasing behavior forever for all values of m. Figure 2 represents that the reconstructed function f(G) shows rapid decrease initially and then approaches positive value forever for all values of m.

It can be observed from Figure 3 (u=2) that the EoS parameter starts from quintessence phase (at the present time), crosses the phantom divide line, and then goes towards phantom phase of the universe. After achieving the extreme value of phantom phase, the EoS parameter turns back and approaches dust-like matter in the end. This type of behavior of EoS parameter is termed as quintom. Figure 4 (u=-2) shows similar behavior of EoS parameter as in case of u=2. It is concluded that both PDE f(G) models favor the PDE phenomenon.

The plot of squared speed of sound with respect to redshift parameter for two values of PDE parameter is as shown in Figures 5 and 6. It can be observed from Figures 5 and 6 that the PDE f(G) model for u=2,-2 shows stability at the present as well as at latter epoch because vs2≥0 for all the time.

It can be observed from Figure 7 that ωPDE-ωPDE′ plane corresponds to freezing region only for m=2,2.4 while it lies in both regions (freezing and thawing) for m=2.2. However, ωPDE-ωPDE′ plane corresponds to thawing region only for all cases of m. Hence, ωPDE-ωPDE′ planes corresponding to m= 2, 2.2, and 2.4 show consistency with the accelerated expansion of the universe.

Figure 9 shows that r-s plane (PDE f(G) models with u=2) corresponds to Chaplygin gas model and DE regions (quintessence and phantom) for all three cases of m. Also, r-s plane (PDE f(G) models with u=-2) corresponds to only Chaplygin gas model as shown in Figure 10 in all three cases of m. However, ΛCDM limit can not be achieved in both cases of PDE parameter of u.

Conflict of Interests

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

RiessA. G.FilippenkoA. V.ChallisP.Observational evidence from supernovae for an accelerating universe and a cosmological constantPerlmutterS.AlderingG.GoldhaberG.Measurements of omega and lambda from 42 high-redshift supernovaeCaldwellR. R.DoranM.Cosmic microwave background and supernova constraints on quintessence: concordance regions and target modelsKoivistoT.MotaD. F.Dark energy anisotropic stress and large scale structure formationDanielS. F.CaldwellR. R.CoorayA.MelchiorriA.Large scale structure as a probe of gravitational slipHoekstraH.JainB.Weak gravitational lensing and its cosmological applicationsArmendariz-PiconC.DamourT.MukhanovV.k-InflationCaldwellR. R.A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of stateBaglaJ. S.JassalH. K.PadmanabhanT.Cosmology with tachyon field as dark energyHsuS. D. H.Entropy bounds and dark energyLiM.A model of holographic dark energyFengB.WangX. L.ZhangX. M.Dark energy constraints from the cosmic age and supernovaZhangX.WuF. Q.ZhangJ.New generalized Chaplygin gas as a scheme for unification of dark energy and dark matterCaiR.-G.A dark energy model characterized by the age of the universeBransC. H.DickeR. H.Mach's principle and a relativistic theory of gravitationLinderE. V.Einstein's other gravity and the acceleration of the UniverseDuttaS.SaridakisE. N.Observational constraints on Hořava-Lifshitz cosmologySharifM.RaniS.Generalized teleparallel gravity via some scalar field dark energy modelsSharifM.RaniS.Nonlinear electrodynamics in f(T) gravity and generalized second law of thermodynamicsNojiriS.OdintsovS. D.Modified Gauss-Bonnet theory as gravitational alternative for dark energyNojiriS.OdintsovS. D.Modified f(R) gravity consistent with realistic cosmology: from a matter dominated epoch to a dark energy universeNojiriS.OdintsovS. D.ŠtefančićH.Transition from a matter-dominated era to a dark energy universeNojiriS.OdintsovS. D.Modified gravity and its reconstruction from the universe expansion historyNojiriS.OdintsovS. D.Diego Sez-GmezD.Cosmological reconstruction of realistic modified F(R) gravitiesNojiriS.OdintsovS. D.Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant modelsSetareM. R.SaridakisE. N.Correspondence between holographic and Gauss-Bonnet dark energy modelsSetareM. R.SaridakisE. N.Coupled oscillators as models of quintom dark energySetareM. R.Bulk-brane interaction and holographic dark energySetareM. R.The holographic dark energy in non-flat Brans-Dicke cosmologySetareM. R.Interacting holographic phantomSetareM. R.Holographic modified gravitySetareM. R.MomeniD.MoayediS. K.Interacting dark energy in Hořava-Lifshitz cosmologyJawadA.PasquaA.ChattopadhyayS.Correspondence between f(G) gravity and holographic dark energy via power-law solutionJawadA.ChattopadhyayS.PasquaA.Reconstruction of f(G) gravity with the new agegraphic dark-energy modelJawadA.PasquaA.ChattopadhyayS.Holographic reconstruction of f(G) gravity for scale factors pertaining to emergent, logamediate and intermediate scenariosJawadA.ChattopadhyayS.PasquaA.A holographic reconstruction of the modified f(R) Horava-Lifshitz gravity with scale factor in power-law formJawadA.ChattopadhyayS.PasquaA.Power-law solution of new agegraphic modified f(R) Horava-Lifshitz gravityJawadA.Reconstruction of TeX models via well-known scale factorsJawadA.Analysis of QCD ghost F(R~) gravityCohenA. G.KaplanD. B.NelsonA. E.Effective field theory, black holes, and the cosmological constantWeiH.Pilgrim dark energySharifM.JawadA.Phantom-like generalized cosmic chaplygin gas and traversable wormhole solutionsLoboF. S. N.Phantom energy traversable wormholesLoboF. S. N.Phantom energy traversable wormholesSushkovS.Wormholes supported by a phantom energySharifM.JawadA.Thermodynamics in closed universe with entropy correctionsMartin-MorunoP.On the formalism of dark energy accretion onto black- and worm-holesJamilM.RashidM. A.QadirA.Charged black holes in phantom cosmologyBabichevE.ChernovS.DokuchaevV.EroshenkoY.Ultrahard fluid and scalar field in the Kerr-Newman metricBhadraJ.DebnathU.Accretion of new variable modified Chaplygin gas and generalized cosmic Chaplygin gas onto Schwarzschild and Kerr-Newman black holesSahniV.StarobinskyA.Reconstructing dark energySadjadiH. M.On the second law of thermodynamics in modified Gauss-Bonnet gravitySharifM.RaniS.Pilgrim dark energy in f(T) gravityChattopadhyayS.JawadA.MomeniD.MyrzakulovR.Pilgrim dark energy in f(T,T_{G}) cosmologyCaldwellR. R.LinderE. V.Limits of quintessenceSahniV.SainiT. D.StarobinskyA. A.AlamU.Statefinder—a new geometrical diagnostic of dark energy