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 utilising 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 catalogs 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 $\lesssim{0.128}$ dynamical times; conveniently similar to that needed for convergence of semi-analytical modelling, as established by Benson etal Total merger counts are converged at the level of $\sim{5}$% for friends-of-friends (FoF) haloes of size $n_{\rm p}\gtrsim{75}$ across a factor of 512 in mass resolution, but substructure rates converge more slowly with mass resolution, reaching convergence of $\sim{10}$% for $n_{\rm p}\gtrsim{100}$ and particle mass $m_{\rm p}{\lesssim}10^{9}M_\odot$. We present analytic fits to FoF and substructure merger rates across nearly all observed galactic history ($z{\le}8.5$). While we find good agreement with the results presented by Fakhouri etal for FoF haloes, a slightly flatter dependance 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 dependance as FoF rates, but with stronger mass and redshift dependencies for their normalisation.