^{3}.

The purpose of this paper is to study the cosmological evolution of the universe in the framework of dynamical Chern-Simons modified gravity. We take pilgrim dark energy model with Hubble and event horizons in interacting scenario with cold dark matter. For this scenario, we discuss cosmological parameters such as Hubble and equation of state and cosmological plane like

The astronomers have put up a struggle in the subject of cosmology and pointed out that the rapid expansion of the universe is because of an unknown force dubbed as dark energy (DE) with the help of different cosmological and astronomical data (arising from well-known observational schemes) [

Among all dynamical models, holographic DE (HDE) model has been constructed in the framework of quantum gravity on the basis of holographic principle [

Moreover, different modified theories of gravity have also been developed for explaining the accelerated expansion. The Chern-Simons modified gravity is one of them which is recently developed [

In this paper, we study some DE models such as pilgrim DE (PDE) with Hubble and event horizons in the framework of dynamical Chern-Simons modified gravity. We discuss equation of state (EoS) and Hubble parameters, cosmological plane using EoS parameter, and stability of the models in interacting scenario. We examine these scenarios with respect to redshift function, interacting parameter, and PDE parameter. We check the compatibility of these results by comparing with the data from recent observations. The paper is organized as follows. In the next section, we briefly discuss the dynamics of Chern-Simons modified gravity. Section

The action which describes the Chern-Simons theory is defined as follows [

Field equation (

The idea of Cohen et al. [

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 [

In the present work, we choose Hubble as well as event horizons for the cosmological analysis taking PDE model in the underlying gravity.

In this section, we study the cosmological parameters such as EoS and Hubble parameters and plane which involves

Here, we address the discussion of the basic cosmological parameters such as Hubble and EoS for PDE models. Firstly we require the rate of change of PDE with Hubble horizon which is given by

Plots of

The EoS parameter can be obtained with the help of (

Plots of

The time derivative of event horizon PDE takes the form

Plots of

For the event horizon, the EoS parameter turns out to be

We analyze the behavior of above EoS parameter versus redshift parameter through PDE as well as interacting parameters (Figure

Plots of

Taking the evolutionary form of EoS parameter as

Thawing region is the region where EoS parameter nearly evolves from

In freezing region, the evolution parameter for EoS parameter remains negative; that is,

Taking derivative of (

For event horizon PDE,

The

Plots of

Plots of

The stability analysis of PDE models in the present framework is being discussed in this section. For this purpose, we extract the following squared speed of sound expression:

Plots of

For event horizon PDE, the squared speed of sound becomes

Plot of

Riess et al. [

It is interesting to mention here that the ranges of EoS parameter for both PDE models lie within the following observational constraints. Ade et al. [

The constructed planes in both PDE models (Figures

Sharif and Jawad [

In another paper [

Up to now, most of the works on this gravity have been restricted to cosmological solutions of scale factor in flat FRW universe in the presence of family of well-known HDE models [

Firstly, we have extracted the Hubble parameter for both PDE models. We have observed for Hubble horizon PDE that the evolution of Hubble parameter lies within the range

For Hubble horizon PDE, we have observed that the EoS parameter (Figure

We have also constructed the

The graphical behavior of squared speed of sound is displayed against redshift parameter in Figure

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

^{*}observations: determination of cosmological parameters