Exploring the Sharma-Mittal HDE models with different diagnostic tools

In this paper, we have examined the Sharma-Mittal holographic dark energy model (SMHDE) in the framework of an isotropic and spatially homogeneous flat Friedmann-Robertson-Walker(FRW) Universe by considering different values of parameter \(\delta\) and \(R\), where the infrared cut-off is taken care by the Hubble horizon. We examined the SMHDE model through the analysis of Statefinder hierarchy and the growth rate of perturbation. The evolutionary trajectories of the statefinder hierarchy \(S_3^1\), \(S_3^2\) \(S_4^1\), \(S_4^2\) versus redshift z, show satisfactory behavior throughout the Universe evaluation. One promising tool for investigating the dark energy models is the composite null diagnostic(CND) \(\{ S_3^1 - \epsilon\}\), where the evolutionary trajectories of the \(S_3^1 - \epsilon\) pair present different property and the departure from \(\Lambda\)CDM could be well evaluated. Additionally, we investigated the dynamical analysis of the model by \(\omega_{D}-\omega^{'}_{D}\) pair analysis.