  Cosmological surveys aim to use the evolution of the abundance of galaxy
clusters to accurately constrain the cosmological model. In the context of
LCDM, we show that it is possible to achieve the required percent level
accuracy in the halo mass function with gravity-only cosmological simulations,
and we provide simulation start and run parameter guidelines for doing so. Some
previous works have had sufficient statistical precision, but lacked robust
verification of absolute accuracy. Convergence tests of the mass function with,
for example, simulation start redshift can exhibit false convergence of the
mass function due to counteracting errors, potentially misleading one to infer
overly optimistic estimations of simulation accuracy. Percent level accuracy is
possible if initial condition particle mapping uses second order Lagrangian
Perturbation Theory, and if the start epoch is between 10 and 50 expansion
factors before the epoch of halo formation of interest. The mass function for
halos with fewer than ~1000 particles is highly sensitive to simulation
parameters and start redshift, implying a practical minimum mass resolution
limit due to mass discreteness. The narrow range in converged start redshift
suggests that it is not presently possible for a single simulation to capture
accurately the cluster mass function while also starting early enough to model
accurately the numbers of reionisation era galaxies, whose baryon feedback
processes may affect later cluster properties. Ultimately, to fully exploit
current and future cosmological surveys will require accurate modeling of
baryon physics and observable properties, a formidable challenge for which
accurate gravity-only simulations are just an initial step.
