The locations of halo formation and the peaks formalism

We investigate the problem of predicting the halo mass function from the properties of the Lagrangian density field. We focus on a perturbation spectrum with a small-scale cut-off (as in warm dark matter cosmologies). This cut-off results in a strong suppression of low mass objects, providing additional leverage to rigorously test which perturbations collapse and to what mass. We find that all haloes are consistent with forming near peaks of the initial density field, with a strong correlation between proto-halo density and ellipticity. We demonstrate that, while standard excursion set theory with correlated steps completely fails to reproduce the mass function, the inclusion of the peaks constraint leads to the correct number of haloes but significantly underpredicts the masses of low-mass objects (with the predicted halo mass function at low masses behaving like dn/dln m ~ m^2/3). This prediction is very robust and cannot be easily altered within the framework of a single collapse barrier. The nature of collapse in the presence of a small-scale cut-off thus reveals that excursion set calculations require a more detailed understanding of the collapse-time of a general ellipsoidal perturbation to predict the ultimate collapsed mass of a peak -- a problem that has been hidden in the large abundance of small-scale structure in CDM. We demonstrate how this problem can be resolved within the excursion set framework.