Merger rates in hierarchical models of galaxy formation – II. Comparison with N-body simulations

We have made a detailed comparison of the results of N-body simulations with the analytical description of the merging histories of dark matter haloes presented in Lacey & Cole, which is based on an extension of the Press-Schechter method. We find the analytical predictions for the halo mass function, merger rates and formation times to be remarkably accurate. The N-body simulations used 1283 particles and were of self- similar clustering, with Ω=1 and initial power spectra P(k)∝kn with spectral indices n =−2, −1,0. The analytical model is, however, expected to apply for arbitrary Ω and more general power spectra. Dark matter haloes were identified in the simulations using two different methods and at a range of overdensities. For haloes selected at mean overdensities ∼100-200, the analytical mass function was found to provide a good fit to the simulations with a collapse threshold close to that predicted by the spherical collapse model, with a typical error of ≲30 per cent over a range of 103 in mass, which is the full dynamical range of our N-body simulation. This was insensitive to the type of filtering used. Over a range of 102- 103 in mass, there was also good agreement with the analytical predictions for merger rates including their dependence on the masses of the two haloes involved and the time interval being considered, and for formation times, including the dependence on halo mass and formation epoch. The analytical Press-Schechter mass function and its extension to halo lifetimes and merger rates thus provide a very useful description of the growth of dark matter haloes through hierarchical clustering, and should provide a valuable tool in studies of the formation and evolution of galaxies and galaxy clusters.