Spherical collapse in $\nu \Lambda CDM$
  The abundance of massive dark matter halos hosting galaxy clusters provides
an important test of the masses of relic neutrino species. The dominant effect
of neutrino mass is to lower the typical amplitude of density perturbations
that eventually form halos, but for neutrino masses $> 0.4eV$ the threshold for
halo formation can be changed significantly as well. We study the spherical
collapse model for halo formation in cosmologies with neutrino masses in the
range $m_{\nu i} =0.05eV$- $1eV$ and find that halo formation is differently
sensitive to $\Omega_\nu$ and $m_\nu$. That is, different neutrino hierarchies
with common $\Omega_\nu$ are in principle distinguishable. The added
sensitivity to $m_{\nu}$ is small but potentially important for scenarios with
heavier sterile neutrinos. Massive neutrinos cause the evolution of density
perturbations to be scale-dependent at high redshift which complicates the
usual mapping between the collapse threshold and halo abundance. We propose one
way of handling this and compute the correction to the halo mass function
within this framework. For $\sum m_{\nu i} < 0.3eV$, our prescription for the
halo abundance is only $\sim< 15\%$ different than the standard calculation.
However for larger neutrino masses the differences approach $50-10\%$ which, if
verified by simulations, could alter neutrino mass constraints from cluster
abundance.
