Cosmological mass functions and moving barrier models

In the ellipsoidal collapse model, the critical density for the collapse of a gravitationally bound object is a function of its mass. In the excursion set formalism, this translates into a moving barrier problem such that the mass function of dark matter haloes is given by the first crossing probability of a random walker across the barrier. In this paper, we study this first crossing probability analytically. Complete solutions are obtained for barriers that vary as square root of time and square of time. Large and small time asymptotic behaviour is derived. For arbitrary power-law barriers, the large time behaviour is determined. The solutions allow us to derive useful inferences about the scaling of the conditional mass function in terms of present day halo masses and look back redshifts. As an application of our results, we compare the estimates of major merger rates of haloes in constant and moving barrier models and find that for massive haloes ($10^{12-13} M_\odot$) the latter predicts significantly higher merger rates towards high redshifts ($z \gtrsim 4$)