The Merger Rates and Mass Assembly Histories of Dark Matter Haloes in
  the Two Millennium Simulations
  We construct merger trees of dark matter haloes and quantify their merger
rates and mass growth rates using the joint dataset from the Millennium and
Millennium-II simulations. The finer resolution of the Millennium-II Simulation
has allowed us to extend our earlier analysis of halo merger statistics to an
unprecedentedly wide range of descendant halo mass (10^10 < M0 < 10^15 Msun),
progenitor mass ratio (10^-5 < xi < 1), and redshift (0 < z < 15). We update
our earlier fitting form for the mean merger rate per halo as a function of
M_0, xi, and z. The overall behavior of this quantity is unchanged: the rate
per unit redshift is nearly independent of z out to z~15; the dependence on
halo mass is weak (M0^0.13); and it is nearly a power law in the progenitor
mass ratio (xi^-2). We also present a simple and accurate fitting formula for
the mean mass growth rate of haloes as a function of mass and redshift. This
mean rate is 46 Msun/yr for 10^12 Msun haloes at z=0, and it increases with
mass as M^{1.1} and with redshift as (1+z)^2.5 (for z > 1). When the fit for
the mean mass growth rate is integrated over a halo's history, we find
excellent match to the mean mass assembly histories of the simulated haloes. By
combining merger rates and mass assembly histories, we present results for the
number of mergers over a halo's history and the statistics of the redshift of
the last major merger.
