Excursion set peaks: a self-consistent model of dark halo abundances and clustering

We describe how to extend the excursion set peaks framework so that its predictions of dark halo abundances and clustering can be compared directly with simulations. These extensions include: a halo mass definition which uses the TopHat filter in real space; the mean dependence of the critical density for collapse δc on halo mass m; and the scatter around this mean value. All three of these are motivated by the physics of triaxial rather than spherical collapse. A comparison of the resulting mass function with N-body results shows that, if one uses δc(m) and its scatter as determined from simulations, then all three are necessary ingredients for obtaining ∼10 per cent accuracy. For example, assuming a constant value of δc with no scatter, as motivated by the physics of spherical collapse, leads to many more massive haloes than seen in simulations. The same model is also in excellent agreement with N-body results for the linear halo bias, especially at the high mass end where the traditional peak-background split argument applied to the mass function fit is known to underpredict the measured bias by ∼10 per cent. In the excursion set language, our model is about walks centred on special positions (peaks) in the initial conditions – we discuss what it implies for the usual calculation in which all walks contribute to the statistics.