Scaling Relations in the Phase-Space Structure of Dark Matter Haloes

We present new scaling relations for the isotropic phase-space distribution functions (DFs) and energy distributions of simulated dark matter haloes. These relations are inspired by those for the singular isothermal sphere with density profile $\rho(r)\propto r^{-2}$, for which the DF satisfies $f(E) \propto r_{\max}^{-2}(E)$ and the energy distribution satisfies $dM/dE \propto r_{\max}(E)$, with $r_{\max}(E)$ being the radius where the gravitational potential equals energy $E$. For the simulated haloes, we find $f(E)\propto r_{\max}^{-2.08}(E)$ and $dM/dE \propto r_{\max}(E)$ across broad energy ranges. In addition, the proportionality coefficients depend on the gravitational constant and the parameters of the best-fit Navarro-Frenk-White density profile. These scaling relations are satisfied by haloes over a wide mass range and provide an efficient method to approximate their DFs and energy distributions. Understanding the origin of these relations may shed more light on halo formation.