Collisionless relaxation to equilibrium distributions in cold dark matter halos: origin of the NFW profile

Collisionless self-gravitating systems such as cold dark matter halos are known to harbor universal density profiles despite the intricate non-linear physics of hierarchical structure formation in the $Λ$CDM paradigm. The origin of such states has been a persistent mystery, particularly because the physics of collisionless relaxation has remained poorly understood. To solve this long-standing problem, we develop a self-consistent quasilinear theory in action-angle space for the collisionless relaxation of inhomogeneous, self-gravitating systems by perturbing the governing Vlasov-Poisson equations. We obtain a quasilinear diffusion equation that describes the secular evolution of the mean coarse-grained distribution function f₀ of accreted matter in the fluctuating force field of a spherical isotropic halo. The diffusion coefficient not only depends on the fluctuation power spectrum but also on the evolving potential of the system, which reflects the self-consistency of the problem. Diffusive heating by an initially cored halo develops an r⁻¹ cusp in the density profile of the accreted material, with r the halocentric radius, if it is initially shallower than r⁻¹. This is fundamentally a consequence of the virial theorem: self-gravitating systems have a negative specific heat and want to cool down when energized. The inner halo relaxes to an r⁻¹ cusp because its central region is the coldest among all r-γ profiles with 0≤ γ≤ 2. Accretion and relaxation in the r⁻¹ cusp develops an r⁻³ outer fall-off, thereby establishing the Navarro-Frenk-White (NFW) density profile. We demonstrate for the first time that this profile emerges as a steady state solution to the problem of self-consistent collisionless relaxation.