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 e ρ(r) ∝ r⁻², for which the DF satisfies f (E) ∝ r⁻²_max(E) and the energy distribution satisfies dM/dE ∝ r_max(E), with r_max(E) being the radius where the gravitational potential equals energy E. For the simulated haloes, we find (E) ∝ r^−2.08_max (E) and dM/dE ∝ r_max(E) across broad energy ranges. In addition, the proportionality coefficients depend on the gravitational constant and the parameters of the best-fitting 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.