Merger rates of dark matter haloes

We derive analytic merger rates for dark matter haloes within the framework of the extended Press–Schechter (EPS) formalism. These rates become self-consistent within EPS once we realize that the typical merger in the limit of a small time-step involves more than two progenitors, contrary to the assumption of binary mergers adopted in earlier studies. We present a general method for computing merger rates that span the range of solutions permitted by the EPS conditional mass function, and focus on a specific solution that attempts to match the merger rates in N-body simulations. The corrected EPS merger rates are more accurate than the earlier estimates of Lacey & Cole by ∼20 per cent for major mergers and by up to a factor of ∼3 for minor mergers of mass ratio 1:104. Based on the revised merger rates, we provide a new algorithm for constructing Monte Carlo EPS merger trees, which could be useful in semi-analytic modelling. We provide analytic expressions and plot numerical results for several quantities that are very useful in studies of galaxy formation. This includes (i) the rate of mergers of a given mass ratio per given final halo, (ii) the fraction of mass added by mergers to a halo and (iii) the rate of mergers per given main progenitor. The creation and destruction rates of haloes serve for a self-consistency check. Our method for computing merger rates can be applied to conditional mass functions beyond EPS, such as those obtained by the ellipsoidal collapse model or extracted from N-body simulations.