A self-consistent quasilinear theory for collisionless relaxation to universal quasi-steady state attractors in cold dark matter halos

Collisionless self-gravitating systems, e.g., cold dark matter halos, harbor universal density profiles despite the intricate non-linear physics of hierarchical structure formation, the origin of which has been a persistent mystery. To solve this problem, we develop a self-consistent quasilinear theory (QLT) in action-angle space for the collisionless relaxation of driven, inhomogeneous, self-gravitating systems by perturbing the governing Vlasov-Poisson equations. We obtain a quasilinear diffusion equation (QLDE) for the secular evolution of the mean distribution function $f_0$ of a halo due to linear fluctuations (induced by random perturbations in the force field) that are collectively dressed by self-gravity, a phenomenon described by the response matrix. Unlike previous studies, we treat collective dressing up to all orders. Well-known halo density profiles $\rho(r)$ commonly observed in $N$-body simulations, including the $r^{-1}$ NFW cusp, an Einasto central core, and the $r^{-1.5}$ prompt cusp, emerge as quasi-steady state attractor solutions of the QLDE. The $r^{-1}$ cusp is a constant flux steady-state solution for a constantly accreting massive halo perturbed by small-scale white noise fluctuations induced by substructure. It is an outcome of the universal nature of collisionless relaxation: lower energy particles attract more particles, gain higher effective mass and get less accelerated by the fluctuating force field. The zero-flux steady state solution for an isolated halo is an $f_0$ that is flat in energy, and the corresponding $\rho(r)$ can either be cored or an $r^{-1.5}$ cusp depending on the inner boundary condition. The latter forms around a central dense object, e.g., a compact subhalo or a black hole. We demonstrate for the first time that these halo profiles emerge as quasi-steady state attractors of collisionless relaxation described by a self-consistent QLT.