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