Halo collapse: virialization by shear and rotation in dynamical
  dark-energy models. Effects on weak-lensing peaks
  The evolution of the virial overdensity $\Delta_{\rm vir}$ for $\Lambda$CDM
and seven dynamical dark-energy models is investigated in the extended
spherical collapse model (SCM). Here the virialization process is naturally
achieved by introducing shear and rotation instead of using the virial theorem.
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
features of the ordinary SCM, i.e., a constant $\Delta_{\rm vir}$ for the EdS
model and time-variation 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. We use the new recipes to predict the surface-mass-density profile of
dark matter haloes and the number of convergence density peaks for LSST- and
Euclid-like weak lensing surveys.
