LAUSR

We study the evolution of the virial overdensity $\Delta_{\rm vir}$ for the $\Lambda$CDM and seven dynamical dark energy models by means of the extended spherical collapse model (SCM), in which the virialization process is naturally achieved by introducing shear and rotation. We generalise two approaches proposed in the literature and show that, regardless of the dark-energy model, the new virialization term can be calibrated on the peculiar velocity of the shell as measured from Einstein-de Sitter simulations. The two virialization recipes qualitatively reproduce the same features of the ordinary SCM, with a constant $\Delta_{\rm vir}$ for the EdS model and time-varying for dark-energy models, but without any mass dependence. Depending on the actual description of virialization and on the dark energy model, the value of $\Delta_{\rm vir}$ varies between 10 and 40 percent. As pivotal quantity in spherical overdensity algorithms, $\Delta_{\rm vir}$ might fairly affect the halo mass function computed from $N$-body simulations and observed density fields.