An excursion set model of hierarchical clustering: ellipsoidal collapse and the moving barrier

The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the non-linear processes of collapse and virialization are modelled. We present simple analytic approximations that allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function, which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low-density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high-density regions are collapsed haloes viewed at small lookback times. Although we have applied our analytic formulae only to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.