Testing dark matter halo properties using self-similarity

We use self-similarity in N-body simulations of scale-free models to test for resolution dependence in the mass function and two-point correlation functions of dark matter haloes. We use 1024³ particle simulations performed with ABACUS , and compare results obtained with two halo finders: friends-of-friends (FOF ), and ROCKSTAR . The FOF mass functions show a systematic deviation from self-similarity which is explained by resolution dependence of the FOF mass assignment previously reported in the literature. Weak evidence for convergence is observed only starting from haloes of several thousand particles, and mass functions are overestimated by at least as much as $20-25{{\ \rm per\ cent}}$ for haloes of 50 particles. The mass function of the default ROCKSTAR halo catalogue (with bound virial spherical overdensity mass), on the other hand, shows good convergence of the order of 50 to 100 particles per halo, with no detectable evidence at the few percent level of any systematic dependence for larger particle number. Tests show that the mass unbinding procedure in ROCKSTAR is the key factor in obtaining this much improved resolution. Applying the same analysis to the halo-halo two point correlation function, we find again strong evidence for convergence only for ROCKSTAR haloes, at separations sufficiently large so that haloes do not overlap. At these separations, we can exclude dependence on resolution at the $5-10{{\ \rm per\ cent}}$ level once haloes have of the order of 50 to 100 particles. At smaller separations results are not converged even at significantly larger particle number, and bigger simulations would be required to establish the resolution required for convergence.