Diagnosing $\Lambda$HDE model with statefinder hierarchy and fractional growth parameter

Recently, a new dark energy model called $\Lambda$HDE was proposed. In this model, dark energy consists of two parts: cosmological constant $\Lambda$ and holographic dark energy (HDE). Two key parameters of this model are the fractional density of cosmological constant $\Omega_{\Lambda0}$, and the dimensionless HDE parameter $c$. Since these two parameters determine the dynamical properties of DE and the destiny of universe, it is important to study the impacts of different values of $\Omega_{\Lambda0}$ and $c$ on the $\Lambda$HDE model. In this paper, we apply various DE diagnostic tools to diagnose $\Lambda$HDE models with different values of $\Omega_{\Lambda0}$ and $c$; these tools include statefinder hierarchy \{$S_3^{(1)}, S_4^{(1)}$\}, fractional growth parameter $\epsilon$, and composite null diagnostic (CND), which is a combination of \{$S_3^{(1)}, S_4^{(1)}$\} and $\epsilon$. We find that: (1) adopting different values of $\Omega_{\Lambda0}$ only has quantitative impacts on the evolution of the $\Lambda$HDE model, while adopting different $c$ has qualitative impacts; (2) compared with $S_3^{(1)}$, $S_4^{(1)}$ can give larger differences among the cosmic evolutions of the $\Lambda$HDE model associated with different $\Omega_{\Lambda0}$ or different $c$; (3) compared with the case of using a single diagnostic, adopting a CND pair has much stronger ability to diagnose the $\Lambda$HDE model.