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

## TL;DR

This paper evaluates theoretical models of the conditional mass function of dark matter halos against N-body simulations, identifying modifications needed for normalization and providing an analytical bias function for underdense regions.

## Contribution

It compares two rescaling methods for the CMF, introduces a normalization modification, and offers an analytical bias function for void studies.

## Key findings

- Neither rescaling method alone provides a normalized CMF.
- A modified rescaling improves the match with simulations.
- The analytical bias function aids in void statistics computations.

## Abstract

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.