Good and proper: self-similarity of <i>N</i>-body simulations with proper force softening

Analysis of self-similarity in scale-free $N$-body simulations reveals the spatial and temporal scales for which statistics measured cosmological are converged to physical continuum limit. We examine how range two-point correlation function is depends on force softening length whether it held constant comoving or proper coordinates. find that a reaches roughly 1/30th inter-particle spacing by end simulation resolves same as while using third fewer time steps, scale factors typical $\Lambda$CDM simulations. additionally infer an inherent resolution limit, set particle mass scaling $a^{-1/2}$, beyond reducing does not improve resolution. postulate mapping these results with spectral index $n=-2$