A prescription for the conditional mass function of dark matter haloes

The unconditional mass function (UMF) of dark matter haloes has been determined accurately in the literature, showing excellent agreement with high-resolution numerical simulations. However, this is not the case for the conditional mass function (CMF). Here, we propose a simple analytical procedure to derive the CMF by rescaling the UMF to the constrained environment using the appropriate mean and variance of the density field at the constrained point. This method introduces two major modifications with respect to the standard rescaling procedure. First of all, rather than using in the scaling procedure the properties of the environment averaged over all the conditioning regions, we implement the rescaling locally. We show that for high masses this modification may lead to substantially different results. Secondly, we modify the (local) standard rescaling procedure in such a manner as to force normalization, in the sense that when one integrates the CMF over all possible values of the constraint multiplied by their corresponding probability distribution, the UMF is recovered. In practice, we do this by replacing in the standard procedure the value δc (the linear density contrast for collapse) by certain adjustable effective parameter δeff. In order to test the method, we compare our prescription with the results obtained from numerical simulations in voids. We find that when our modified rescaling is applied locally to any existing numerical fit of the UMF, and the appropriate value for δeff is chosen, the resulting CMF is, in all cases, in very good agreement with the numerical results. Based on these results, we finally present a very accurate analytical fit to the (accumulated) CMF obtained with our procedure, as a function of the parameters that describe the conditioning region (size and mean linear density contrast), the redshift and the relevant cosmological parameters (σ8 and Γ). This analytical fit may be useful for any theoretical treatment of the large-scale structure, and has been already used successfully in regard with the statistic of voids.