Convergence properties of halo merger trees; halo and substructure merger rates across cosmic history

We introduce GBPTREES: an algorithm for constructing merger trees from cosmological simulations, designed to identify and correct for pathological cases introduced by errors or ambiguities in the halo finding process. GBPTREES is built upon a halo matching method utilizing pseudo-radial moments constructed from radially sorted particle ID lists (no other information is required) and a scheme for classifying merger tree pathologies from networks of matches made to-and-from haloes across snapshots ranging forward-and-backward in time. Focusing on SUBFIND catalogues for this work, a sweep of parameters influencing our merger tree construction yields the optimal snapshot cadence and scanning range required for converged results. Pathologies proliferate when snapshots are spaced by less than or similar to 0.128 dynamical times; conveniently similar to that needed for convergence of semi-analytical modelling, as established by Benson et al. Total merger counts are converged at the level of similar to 5 per cent for friends-of-friends (FoF) haloes of size n(p) greater than or similar to 75 across a factor of 512 in mass resolution, but substructure rates converge more slowly with mass resolution, reaching convergence of similar to 10 per cent for n(p) greater than or similar to 100 and particle mass m(p) less than or similar to 10(9) M-circle dot. We present analytic fits to FoF and substructure merger rates across nearly all observed galactic history (z <= 8.5). While we find good agreement with the results presented by Fakhouri et al. for FoF haloes, a slightly flatter dependence on merger ratio and increased major merger rates are found, reducing previously reported discrepancies with extended Press-Schechter estimates. When appropriately defined, substructure merger rates show a similar mass ratio dependence as FoF rates, but with stronger mass and redshift dependencies for their normalization.