On the virialization threshold for halo mass functions

In a recent study by Euclid collaboration, the halo mass function (HMF) has been fitted with accuracy better than $1\%$ for the $\Lambda$CDM model. Several parameters were introduced and fitted against N-body simulations, assuming the usual linearly extrapolated matter density contrast at the collapse time, $\delta_c$, as a basic threshold for halo formation. As a result, a new function that multiplies $\delta_c$ was introduced, producing an effective threshold that varies both with redshift and mass scale. We show that the redshift evolution of this effective threshold is similar to the one of the linear extrapolated matter density contrast at the virialization time, $\delta_{\rm v}$. Assuming the Euclid HMF as a fiducial model, we refit the Sheth-Tormen (ST) HMF using $\delta_{\rm v}$ as a threshold. This new fit improves the agreement between ST-HMF and the Euclid one with respect to Despali et al. (2016) fit, specially at high masses. Interestingly, the parameters $a$ and $p$ in this refit have values closer to the Press-Schechter limit of the ST-HMF, showing that the use of $\delta_{\rm v}$ can provide semi-analytical HMF less dependent on extra parameters. Moreover, we analyze the consistency of the ST-HMF fitted with $\delta_{\rm v}$ in smooth dark energy models with time-varying equation of state, finding an overall good agreement with the evolution of halo abundances expected from the linear evolution of perturbations and the Euclid HMF extrapolated to these scenarios. These findings suggest that the use $\delta_{\rm v}$ as a basic function to describe the threshold for halo formation can be a good guide when considering extrapolations for models beyond $\Lambda$CDM, which are typically harder to study in simulations.