The mass function of dark matter haloes

We combine data from a number of N-body simulations to predict the abundance of dark haloes in cold dark matter (CDM) universes over more than four orders of magnitude in mass. A comparison of different simulations suggests that the dominant uncertainty in our results is systematic and is smaller than 1030 per cent at all masses, depending on the halo definition used. In particular, our `Hubble volume' simulations of tCDM and LCDM cosmologies allow the abundance of massive clusters to be predicted with uncertainties well below those expected in all currently planned observational surveys. We show that for a range of CDM cosmologies and for a suitable halo definition, the simulated mass function is almost independent of epoch, of cosmological parameters and of the initial power spectrum when expressed in appropriate variables. This universality is of exactly the kind predicted by the familiar PressSchechter model, although this model predicts a mass function shape that differs from our numerical results, overestimating the abundance of `typical' haloes and underestimating that of massive systems.