The importance of stepping up in the excursion set approach

TL;DR

This paper introduces an improved, simple, and fast approximation method for the first crossing distribution in the excursion set approach, accurately modeling halo mass functions across a wide mass range.

Contribution

It presents a new formulation that corrects divergence issues at low masses, requiring minimal variables and matching Monte-Carlo results closely.

Findings

Accurate approximation for first crossing distribution across various barriers

Effective correction for low-mass divergence

Excellent agreement with Monte-Carlo simulations

Abstract

Recently, we provided a simple but accurate formula which closely approximates the first crossing distribution associated with random walks having correlated steps. The approximation is accurate for the wide range of barrier shapes of current interest and is based on the requirement that, in addition to having the right height, the walk must cross the barrier going upwards. Therefore, it only requires knowledge of the bivariate distribution of the walk height and slope, and is particularly useful for excursion set models of the massive end of the halo mass function. However, it diverges at lower masses. We show how to cure this divergence by using a formulation which requires knowledge of just one other variable. While our analysis is general, we use examples based on Gaussian initial conditions to illustrate our results. Our formulation, which is simple and fast, yields excellent agreement with the considerably more computationally expensive Monte-Carlo solution of the first crossing distribution, for a wide variety of moving barriers, even at very low masses.