The virialization density of peaks with general density profiles under
  spherical collapse
  We calculate the non-linear virialization density, $\Delta_c$, of halos under
spherical collapse from peaks with an arbitrary initial and final density
profile. This is in contrast to the standard calculation of $\Delta_c$ which
assumes top-hat profiles. Given our formalism, the non-linear halo density can
be calculated once the shape of the initial peak's density profile and the
shape of the virialized halo's profile are provided. We solve for $\Delta_c$
for halos in an Einstein de-Sitter and $\Lambda$CDM universe. As examples, we
consider power-law initial profiles as well as spherically averaged peak
profiles calculated from the statistics of a Gaussian random field.
  We find that, depending on the profiles used, $\Delta_c$ is smaller by a
factor of a few to as much as a factor of 10 as compared to the density given
by the standard calculation ($\approx 200$). Using our results, we show that,
for halo finding algorithms that identify halos through an over-density
threshold, the halo mass function measured from cosmological simulations can be
enhanced at all halo masses by a factor of a few. This difference could be
important when using numerical simulations to assess the validity of analytic
models of the halo mass function.
