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 data set 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 (10¹⁰ <~ M₀ <~ 10¹⁵M_solar), progenitor mass ratio (10^-5 <~ ξ <= 1) and redshift (0 <= z <~ 15). We update our earlier fitting form for the mean merger rate per halo as a function of M₀, ξ and z. The overall behaviour 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 (~ M^0.13₀); and it is nearly a power law in the progenitor mass ratio (~ ξ^-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 46M_solar yr^-1 for 10¹²M_solar 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.