Conditional mass functions and merger rates of dark matter ellipsoidal collapse model

Analytic models based on spherical and ellipsoidal gravitational collapse have been used to derive the mass functions of dark matter haloes and their progenitors (the conditional mass function). The ellipsoidal model generally provides a better match to simulation results, but there has been no simple analytic expression in this model for the conditional mass function that is accurate for small time-steps, a limit that is important for generating halo merger trees and computing halo merger rates. We remedy the situation by deriving accurate analytic formulae for the first-crossing distribution, the conditional mass function and the halo merger rate in the ellipsoidal collapse model in the limit of small look-back times. We show that our formulae provide a closer match to the Millennium simulation results than those in the spherical collapse model and the ellipsoidal model of Sheth and Tormen.