Analysis of generalized ghost pilgrim dark energy in nonflat FRW universe
Abstract
This work is based on pilgrim dark energy conjecture which states that phantomlike dark energy possesses the enough resistive force to preclude the formation of black hole. The nonflat geometry is considered which contains the interacting generalized ghost pilgrim dark energy with cold dark matter. Some wellknown cosmological parameters (evolution parameter (\(\omega _{\Lambda }\)) and squared speed of sound) and planes (\(\omega _{\Lambda }\)–\(\omega _{\Lambda }'\) and statefinder) are constructed in this scenario. The discussion of these parameters is totally done through pilgrim dark energy parameter (\(u\)) and interacting parameter (\(d^2\)). It is interesting to mention here that the analysis of evolution parameter supports the conjecture of pilgrim dark energy. Also, this model remains stable against small perturbation in most of the cases of \(u\) and \(d^2\). Further, the cosmological planes correspond to \(\Lambda \)CDM limit as well as different wellknown dark energy models.
1 Introduction
The accelerated expansion of the universe is one of the active topic in cosmology since its prediction [1]. It is suggested through different cosmological and astrological data arisen from wellknown observational schemes [2, 3, 4, 5, 6] that this rapid expansion is due to an unknown force termed as dark energy (DE). Despite of many efforts from different observational and theoretical ways, the problem of DE is still not well settled due to its unknown nature. In order to justify the source of accelerating expansion (i.e., the nature of DE) of the universe, two different approaches have been adopted. One way is to modify the geometric part of EinsteinHilbert action (termed as modified theories of gravity) for the discussion of expansion phenomenon [7, 8, 9, 10, 11]. The second approach is to propose the different forms of DE called dynamical DE models.
According to Wei, the formation of BH can be avoided through appropriate resistive force which is capable to prevent the matter collapse. In this phenomenon, phantomlike DE can play important role which possesses strong repulsive force as compare to quintessence DE. The effective role of phantomlike DE onto the mass of the BH in the universe has also been observed in many different ways. The accretion phenomenon is one of them which favor the possibility of avoidance of BH formation due to presence of phantomlike DE in the universe. It has been suggested that accretion of phantom DE (which is attained through family of Chaplygin gas models [16, 17, 18, 19, 20, 21]) reduces the mass of BH. On the other hand, there also exists a possibility of increasing of BH mass due to phantom energy accretion process which leads to the violation of cosmic censorship hypothesis [22]. Hence, this phenomenon is still unresolved.
It is strongly believed that the presence of phantom DE in the universe will force it towards big rip singularity. This represents that the phantomlike universe possesses ability to prevent the BH formation. The proposal of PDE model [15] also works on this phenomenon which states that phantom DE contains enough repulsive force which can resist against the BH formation. Wei [15] developed cosmological parameters for PDE model with Hubble horizon and provided different possibilities for avoiding the BH formation through PDE parameter. He adopted different possible theoretical and observational ways to make the BH free phantom universe. Also, PDE via reconstruction scheme is discussed in modified theory of gravity such as \(f(T)\) gravity [23]. The behavior of cosmological parameters along with validity of generalized second law of thermodynamics are explored as well.
In addition, we worked on PDE models interacting with cold dark matter (CDM) and pointed different ways in order to meet the PDE phenomenon [24, 25, 26]. In this work, the generalized ghost version of PDE model so called GGPDE interacting with CDM is considered in nonflat universe. In this context, different cosmological parameters (EoS parameter and squared speed of sound) and planes (\(\omega _{\Lambda }\)–\(\omega _{\Lambda }'\) and statefinder) are developed. The format of the paper is as follows. Section 2 contains the basic cosmological scenario, whereas Sect. 3 explores above mentioned cosmological parameters and planes. The concluding remarks of the results are given in the last section.
2 Nonflat FRW universe and basic equations
3 Cosmological parameters
Here, we discuss the evolution of the Hubble parameter, the universe and stability of the interacting model GGPDE. For this purpose, we extract EoS parameter and squared speed of sound.
3.1 Hubble parameter
3.2 The equation of state parameter
3.3 Stability analysis
3.4 \(\omega _{\Lambda }{}\omega '_{\Lambda }\) analysis
3.5 Statefinder parameters
4 Results and discussions
It is wellknown that total energy density of the universe contains contribution of its different constituents in ratio of \(\Omega _k<\Omega _m<\Omega _\Lambda \) (the cosmic curvature density \(\Omega _k\) is found to be fractional). The early inflation era indicates that the universe is nonflat if the number of efolding are small. It is predicted through many inflationary models that the order of spatial curvature (\(\Omega _k\)) in the universe should be less than \(10^{5}\) (but there are also exist some models which allow larger curvature) [55, 56]. Also, the bound on EoS parameter of different DE models was established in the nonflat scenario of the universe by using observations of SNe Ia, BAO and CMBR [57]. The range \((0.2851,0.0099)\) of \(\Omega _k\) at \(95~\%\) confidence level was obtained with the help of WMAP 5 year data [56] which was improved upto \((0.0181,0.0071)\) by using the data of BAO and SNe Ia. The range \(0.0133<\Omega _k<0.0084\) was obtained by using latest WMAP \(7\)years [58].
 Two recent analysis have greatly improved the precision of the cosmic distance scale. Riess et al. [61] use HST observations of Cepheid variables in the host galaxies of eight SNe Ia to calibrate the supernova magnituderedshift relation. Their “best estimate” of the Hubble constant, from fitting the calibrated SNe magnituderedshift relation, iswhere the error is \(1\sigma \) level and includes known sources of systematic errors. Freedman et al. [62], as part of the Carnegie Hubble Program, use Spitzer Space Telescope midinfrared observations to recalibrate secondary distance methods used in the HST Key Project. These authors find$$\begin{aligned} H_0&= 73.8\pm 2.4~\hbox {km}~\hbox {s}^{1}\hbox { Mpc}^{1} \qquad \text {(Cepheids+SNe Ia)}, \end{aligned}$$It can be observed through all plots (Figs. 1, 2, 3) for all values of interacting parameter \(d^2\) that \(H(a)\) shows increasing behavior which is consistent with the above observations.$$\begin{aligned} H_0&= [74.3 \pm 1.5\text { (statistical)} \pm 2.1 \text { (systematic)}]\hbox { km}~\hbox {s}^{1}~\\&\qquad \hbox {Mpc}^{1}\text { (Carnegie HP)}, \end{aligned}$$
 In Fig. 4 (\(u=0.5\)), it can be observed that the EoS parameter starts from phantom region (with comparatively large negative value) and goes towards lower negative value of phantom region for all cases of interacting parameter. For \(u=0.5\) (Fig. 5), it starts from quintessence phase and turns towards phantom region by crossing vacuum dominated era of the universe for the cases (\(d^2=0.02,~0.03\)). However, it remains in the phantom region for \(d^2=0.04\). Also, Fig. 6 provided that EoS parameter starts comparatively high value of phantom region and always remains in that region for all values of interacting parameter. It can also be observed that EoS parameter attains high phantom region with the increase of interacting parameter. Moreover, Ade et al. [47] (Planck data) have put the following constraints on the EoS parameterby implying different combination of observational schemes at \(95~\%\) confidence level. It can be seen from Figs. 4, 5 and 6 that the EoS parameter also meets the above mentioned values for all cases of interacting parameter which shows consistency of our results. The above discussion shows that all the models provides fully support the PDE phenomenon.$$\begin{aligned} \omega _{\Lambda }&= 1.13^{+0.24}_{0.25} \qquad \text {(Planck+WP+BAO)},\\ \omega _{\Lambda }&= 1.09\pm 0.17, \qquad \text {(Planck+WP+Union 2.1)}\\ \omega _{\Lambda }&= 1.13^{+0.13}_{0.14},\qquad \text {(Planck+WP+SNLS)},\\ \omega _{\Lambda }&= 1.24^{+0.18}_{0.19},\qquad (\text {Planck+WP}+H_0). \end{aligned}$$

In Fig. 7, it is observed that GGPDE remains stable against small perturbation at the present epoch as well as recent present epoch. It can be viewed from Fig. 8 (\(u=0.5\)) that the GGPDE model exhibits stability for all values of interacting parameter in this scenario due to positive behavior of squared speed of sound. In case of \(u=1\) (Fig. 9), the squared speed of sound also exhibits stability of the model for all cases of \(d^2\).
 The \(\omega _{\Lambda }{}\omega '_{\Lambda }\) plane for the current DE model is constructed by plotting the \(\omega '_{\Lambda }\) versus \(\omega _{\Lambda }\) for three different values of \(u\) as shown in Figs. 10, 11 and 12. The specific values of other constant are the same as above plots. Figures 10 and 12 provide thawing region while Fig. 11 exhibits freezing region. The \(\Lambda \)CDM limit, i.e., \((\omega _{\Lambda },\omega '_{\Lambda })=(1,0)\) only achieved for \(u=0.5\) with (\(d^2=0\)) as shown in Fig. 10. Also, Ade et al. [47] have obtained the following constraints on \(w_{\Lambda }\) and \(w'_{\Lambda }\):at \(95~\%\) confidence level. Also, other data with different combinations of observational schemes such as (Planck+WP+Union 2.1) and (Planck+WP+SNLS) favor the above constraints. In the present case, the trajectories of \(\omega '_{\Lambda }\) against \(\omega _{\Lambda }\) also meet the above mentioned values for all cases of interacting parameter which shows consistency of our results as shown in Figs. 10, 11 and 12. Hence, \(\omega _{\Lambda }{}\omega '_{\Lambda }\) plane provides consistent behavior with the present day observations in all cases of \(u\).$$\begin{aligned} \omega _{\Lambda }&= 1.13^{+0.24}_{0.25} \qquad \text {(Planck+WP+BAO)},\\ \omega '_{\Lambda }&< 1.32,\qquad \text {(Planck+WP+BAO)} \end{aligned}$$

The \(rs\) plane corresponding to this scenario is shown in Figs. 13, 14 and 15. It is observed that the trajectories of \(rs\) plane for all cases of interacting parameter corresponds to \(\Lambda \)CDM model for \(u=0.5\) as shown in Fig. 13. However, the trajectories of \(rs\) meet \(\Lambda \)CDM limit for only \(d^2=0\) (in case of \(u=0.5\)) and \(d^2=0.02,~0.03\) (in case of \(u=0.5\)) as shown in Figs. 14 and 15, respectively. Also, the trajectories coincide with the Chaplygin gas model in all cases of \(u\).
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