Testing the conditional mass function of dark matter halos against
  numerical N-body simulations
  We compare the predicted conditional mass function (CMF) of dark matter halos
from two theoretical prescriptions against numerical N-body simulations, both
in overdense and underdense regions and at different Eulerian scales ranging
from $5$ to $30\,h^{-1}\,$Mpc. In particular, we consider in detail a
locally-implemented rescaling of the unconditional mass function (UMF) already
discussed in the literature, and also a generalization of the standard
rescaling method described in the extended Press-Schechter formalism. First, we
test the consistency of these two rescalings by verifying the normalization of
the CMF at different scales, and showing that none of the proposed cases
provides a normalized CMF. In order to satisfy the normalization condition, we
include a modification in the rescaling procedure. After this modification, the
resulting CMF generally provides a better description of numerical results. We
finally present an analytical fit to the ratio between the CMF and the UMF
(also known as the matter-to-halo bias function) in underdense regions, which
could be of special interest to speed-up the computation of the halo abundance
when studying void statistics. In this case, the CMF prescription based on the
locally-implemented rescaling provides a slightly better description of the
numerical results when compared to the standard rescaling.
