# On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes

## TL;DR

This paper derives an exact analytical solution for first crossing distributions in fractional Brownian motion, compares it with simulations and approximations, and applies it to dark matter halo mass functions, highlighting discrepancies with N-body simulations.

## Contribution

It provides the first exact analytical solution for fractional Brownian motion crossing distributions and applies it to dark matter halo modeling, improving understanding of structure formation.

## Key findings

- Analytical solutions agree well with simulations
- Approximate solutions differ significantly from exact results
- Dark matter multiplicity functions from fractional Brownian motion poorly fit N-body data

## Abstract

We construct an integral equation for the first crossing distributions for fractional Brownian motion in the case of a constant barrier and we present an exact analytical solution. Additionally we present first crossing distributions derived by simulating paths from fractional Brownian motion. We compare the results of the analytical solutions with both those of simulations and those of some approximated solutions which have been used in the literature. Finally, we present multiplicity functions for dark matter structures resulting from our analytical approach and we compare with those resulting from N-body simulations. We show that the results of analytical solutions are in good agreement with those of path simulations but differ significantly from those derived from approximated solutions. Additionally, multiplicity functions derived from fractional Brownian motion are poor fits of the those which result from N-body simulations. We also present comparisons with other models which exist in the literature and we discuss different ways of improving the agreement between analytical results and N-body simulations.