The distribution function of dark matter in massive haloes

We study the distribution function (DF) of dark matter particles in haloes of mass range 10¹⁴-10¹⁵M_solar. In the numerical part of this work we measure the DF for a sample of relaxed haloes formed in the simulation of a standard Λ cold dark matter (ΛCDM) model. The DF is expressed as a function of energy E and the absolute value of the angular momentum L, a form suitable for comparison with theoretical models. By proper scaling we obtain the results that do not depend on the virial mass of the haloes. We demonstrate that the DF can be separated into energy and angular momentum components and propose a phenomenological model of the DF in the form . This formulation involves three parameters describing the anisotropy profile in terms of its asymptotic values (β₀ and β_∞) and the scale of transition between them (L₀). The energy part f_E(E) is obtained via inversion of the integral for spatial density. We provide a straightforward numerical scheme for this procedure as well as a simple analytical approximation for a typical halo formed in the simulation. The DF model is extensively compared with the simulations: using the model parameters obtained from fitting the anisotropy profile, we recover the DF from the simulation as well as the profiles of the dispersion and kurtosis of radial and tangential velocities. Finally, we show that our DF model reproduces the power-law behaviour of phase-space density Q = ρ(r)/σ³(r).