Testing the conditional mass function of dark matter haloes against numerical N-body simulations

We compare the predicted conditional mass function (CMF) of dark matter haloes 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.