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.
