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

Good and Proper: Self-similarity of N-body Simulations with Proper Force Softening