A dynamics-based density profile for dark haloes -- II. Fitting function

The density profiles of dark matter haloes are commonly described by fitting functions such as the NFW or Einasto models, but these approximations break down in the transition region where halos become dominated by newly accreting matter. In Paper I we dynamically split simulation particles into orbiting and infalling components and analysed their separate profiles. Here we propose simple, accurate fitting functions designed to capture the asymptotic shapes of the two terms at large and small radii. The orbiting term is described as a truncated Einasto profile, $\rho_{\rm orb} \propto \exp \left[-2/\alpha\ (r / r_{\rm s})^\alpha - 1/\beta\ (r / r_{\rm t})^\beta \right]$, with a five-parameter space of normalization, physically distinct scale and truncation radii, and $\alpha$ and $\beta$, which control how rapidly the profiles steepen. The infalling profile is modelled as a power law in overdensity that smoothly transitions to a constant at the halo centre. We show that these formulae fit the averaged, total profiles in simulations to about 5% accuracy across almost all of an expansive parameter space in halo mass, redshift, cosmology, and accretion rate. When fixing $\alpha = 0.18$ and $\beta = 3$, the formula becomes a three-parameter model for the orbiting term that fits individual halos better than the Einasto profile on average.