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 ≲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 ~5 per cent for friends-of-friends (FoF) haloes of size n_p ≳ 75 across a factor of 512 in mass resolution, but substructure rates converge more slowly with mass resolution, reaching convergence of ~10 per cent for n_p ≳ 100 and particle mass m_p ≲ 10⁹M_⊙. 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.