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 ΛCDM 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 in the pre-assembled halo develops an r^-γ cusp (r is the halocentric radius) in the density profile of the accreted material. Accretion and relaxation in this r^-γ inner cusp develops an r^-β outer fall-off with β ≈ 5 - 2γ in the quasi-steady state. Spherical collapse theory dictates that a quasi-steady outer halo must settle to β ≈ 3 since then the mass enclosed within a radially moving shell barely changes with time. This implies that the quasi-steady γ must be approximately 1, which is possible in the quasilinear framework only if (i) the pre-assembled halo harbors an r^-γ_P profile with γ_P ≥ 0.5, (ii) its fluctuations are sufficiently correlated in time (red noise), and (iii) the initial value of γ is smaller than 1, implying that the r^-1 cusp is a neutral equilibrium. Self-consistent quasilinear relaxation therefore establishes the Navarro-Frenk-White (NFW) profile. We demonstrate for the first time how this profile emerges as a quasi-steady state of collisionless relaxation.