A closer look at Viaggiu Holographic Dark Energy with Statefinder Hierarchy and Fractional Growth Diagnostics

We employ the statefinder hierarchy along with the fractional growth parameter to investigate the recently proposed noninteracting Viaggiu holographic dark energy (VHDE) model with the future event horizon as the infrared cut-off. By adjusting different values of the VHDE model parameter $\delta$, we illustrate the evolution trajectories of several important parameters, prticularly, the statefinder hierarchy $S_3^{(1)}$, $S_3^{(2)}$, $S_4^{(1)}$, $S_4^{(2)}$ and the fractional growth parameter $\epsilon$ against the redshift $z$ and the fractional energy density of matter, ${\mathrm{\Omega }}_m$. Together with the growth rate of matter perturbations, we present the statefinder hierarchy that defines a composite null diagnostic (CND), which can differentiate dynamical dark energy from $\mathrm{\Lambda }$CDM. The joint use of the statefinder hierarchy and the fractional growth parameter offers a powerful diagnostic tool for probing VHDE, effectively resolving the degeneracy between different parameter choices within the model. We find that employing a CND pair, rather than a single diagnostic, significantly enhances the diagnostic power for the VHDE model. Finally, we perform an ${\omega }_d-{\omega }_d^{\mathrm{\prime }}$ analysis and, in addition, assess the squared sound speed $v_s^2$ to determine the model’s dynamical stability under small perturbations.