The importance of stepping up in the excursion set approach

Abstract

There is a simple analytic approximation for the first crossing distribution associated with random walks having correlated steps, which is very accurate in the limit of few steps. The approximation is accurate for the wide range of barrier shapes of interest in excursion set studies of cosmological structure formation. For example, it provides a useful fitting formula for the high-mass end of the dark halo mass function. The approximation 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. 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 is simple and fast, and yields excellent agreement, even at very low masses and for a wide variety of moving barriers, with considerably more computationally expensive Monte Carlo solution of the first crossing distribution. © 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.