A dynamics-based density profile for dark haloes -- III. Parameter space

In the previous paper of this series, we proposed a new function to fit halo density profiles out to large radii. This truncated Einasto profile models the inner, orbiting matter as $\rho_{\rm orb} \propto \exp \left[-2/\alpha\ (r / r_{\rm s})^\alpha - 1/\beta\ (r / r_{\rm t})^\beta \right]$ and the outer, infalling term as a power-law overdensity. In this paper, we analyse the resulting parameter space of scale radius $r_{\rm s}$, truncation radius $r_{\rm t}$, steepening $\alpha$, truncation sharpness $\beta$, infalling normalisation $\delta_1$, and infalling slope $s$. We show that these parameters are non-degenerate in averaged profiles, and that fits to the total profiles generally recover the underlying properties of the orbiting and infalling terms. We study the connection between profile parameters and halo properties such as mass (or peak height) and accretion rate. We find that the commonly cited dependence of $\alpha$ on peak height is an artefact of fitting Einasto profiles to the actual, truncated profiles. In our fits, $\alpha$ is independent of mass but dependent on accretion rate. When fitting individual halo profiles, the parameters exhibit significant scatter but otherwise follow the same trends. We confirm that the entire profiles are sensitive to the accretion history of haloes, and that the two radial scales $r_{\rm s}$ and $r_{\rm t}$ particularly respond to the formation time and recent accretion rate. As a result, $r_{\rm t}$ is a more accurate measure of the accretion rate than the commonly used radius where the density slope is steepest.