Self-Similar Mass Accretion History in Scale-Free Simulations

Using a scale-free $N$-body simulation generated with the ABACUS $N$-body code, we test the robustness of halo mass accretion histories via their convergence to self-similarity. We compare two halo finders, ROCKSTAR and COMPASO. We find superior self-similarity in halo mass accretion histories determined using ROCKSTAR, with convergence to 5% or better between $\sim10^2$ to $10^5$ particles. For COMPASO we find weaker convergence over a similar region, with at least 10% between $\sim10^2$ to $10^4$ particles. Furthermore, we find the convergence to self-similarity improves as the simulation evolves, with the largest and deepest regions of convergence appearing after the scale factor quadrupled from the time at which non-linear structures begin to form. With sufficient time evolution, halo mass accretion histories are converged to self-similarity within 5% with as few as $\sim70$ particles for COMPASO and within 2% for as few as $\sim30$ particles for ROCKSTAR.