Numerical convergence of simulations of galaxy formation: the abundance
  and internal structure of cold dark matter haloes
  We study the impact of numerical parameters on the properties of cold dark
matter haloes formed in collisionless cosmological simulations. We quantify
convergence in the median spherically-averaged circular velocity profiles for
haloes of widely varying particle number, as well as in the statistics of their
structural scaling relations and mass functions. In agreement with prior work
focused on single haloes, our results suggest that cosmological simulations
yield robust halo properties for a wide range of gravitational softening
parameters, $\epsilon$, provided: 1) $\epsilon$ is not larger than a
"convergence radius", $r_{\rm conv}$, which is dictated by 2-body relaxation
and determined by particle number, and 2) a sufficient number of timesteps are
taken to accurately resolve particle orbits with short dynamical times.
Provided these conditions are met, median circular velocity profiles converge
to within $\approx 10$ per cent for radii beyond which the local 2-body
relaxation timescale exceeds the Hubble time by a factor $\kappa\equiv t_{\rm
relax}/t_{\rm H}\gt 0.177$, with better convergence attained for higher
$\kappa$. We provide analytic estimates of $r_{\rm conv}$ that build on
previous attempts in two ways: first, by highlighting its explicit (but weak)
softening-dependence and, second, by providing a simpler criterion in which
$r_{\rm conv}$ is determined entirely by the mean inter-particle spacing, $l$;
for example, better than $10$ per cent convergence in circular velocity for
$r\gt 0.05\,l$. We show how these analytic criteria can be used to assess
convergence in structural scaling relations for dark matter haloes as a
function of their mass or maximum circular speed.
