The nearly universal merger rate of dark matter haloes in ΛCDM cosmology

We construct merger trees from the largest data base of dark matter haloes to date provided by the Millennium Simulation to quantify the merger rates of haloes over a broad range of descendant halo mass (10 12 ≤ M 0 ≤ 1015 M ⊙ ), progenitor mass ratio (10 -3 ≤ ξ ≤ 1), and redshift (0 ≤ z ≤ 6). We find the mean merger rate per halo, Bin, to have very simple dependence on M 0 , ξ, and z, and propose a universal fitting form for B/n that is accurate to 10-20 per cent. Overall, B/n depends very weakly on the halo mass (oc M 0 0 .08) and scales as a power law in the progenitor mass ratio (oc ξ -2 ) for minor mergers (ξ ≤ 0.1) with a mild upturn for major mergers. As a function of time, we find the merger rate per Gyr to evolve roughly as (1 + z) nm with r m = 2-2.3, while the rate per unit redshift is nearly independent of z. Several tests are performed to assess how our merger rates are affected by e.g. the time interval between Millennium outputs, binary versus multiple progenitor mergers, and mass conservation and diffuse accretion during mergers. In particular, we find halo fragmentations to be a general issue in merger tree construction from N-body simulations and compare two methods for handling these events. We compare our results with predictions of two analytical models for halo mergers based on the extended Press-Schechter (EPS) model and the coagulation theory. We find that the EPS model overpredicts the major merger rates and underpredicts the minor merger rates by up to a factor of a few.