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 $f(E)\propto r_{\max }^{}(E)$ and the energy distribution satisfies $dM\mathrm{/}dE\propto r_{\max }(E)$, with $r_{\max }(E)$ being the radius where the gravitational potential equals energy $E$. For the simulated haloes, we find $f(E)\propto r_{\max }^{}(E)$ and $dM\mathrm{/}dE\propto r_{\max }(E)$ 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.