Ellipsoidal collapse and an improved model for the number and spatial distribution of dark matter haloes

The Press-Schechter, excursion set approach allows one to make predictions about the shape and evolution of the mass function of bound objects. The ap proach combines the assumption that objects collapse spherically with the a ssumption that the initial density fluctuations were Gaussian and small. Th e predicted mass function is reasonably accurate, although it has fewer hig h-mass and more low-mass objects than are seen in simulations of hierarchic al clustering. We show that the discrepancy between theory and simulation c an be reduced substantially if bound structures are assumed to form from an ellipsoidal, rather than a spherical, collapse. In the original, standard, spherical model, a region collapses if the initial density within it excee ds a threshold value, delta (sc). This value is independent of the initial size of the region, and since the mass of the collapsed object is related t o its initial size, this means that delta (sc) is independent of final mass . In the ellipsoidal model, the collapse of a region depends on the surroun ding shear field, as well as on its initial overdensity. In Gaussian random fields, the distribution of these quantities depends on the size of the re gion considered. Since the mass of a region is related to its initial size, there is a relation between the density threshold value required for colla pse and the mass of the final object. We provide a fitting function to this delta (ec)(m) relation which simplifies the inclusion of ellipsoidal dynam ics in the excursion set approach. We discuss the relation between the excu rsion set predictions and the halo distribution in high-resolution N-body s imulations, and use our new formulation of the approach to show that our si mple parametrization of the ellipsoidal collapse model represents an improv ement on the spherical model on an object-by-object basis. Finally, we show that the associated statistical predictions, the mass function and the lar ge-scale halo-to-mass bias relation, are also more accurate than the standa rd predictions.