Collisionless relaxation to equilibrium distributions in cold dark matter halos: Origin of the Navarro-Frenk-White profile

Collisionless self-gravitating systems such as cold dark matter halos are known to harbor universal density profiles despite the intricate nonlinear physics of hierarchical structure formation in the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) 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 ${f}_{0}$ 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 preassembled halo develops an ${r}^{\ensuremath{-}\ensuremath{\gamma}}$ cusp ($r$ is the halocentric radius) in the density profile of the accreted material. Accretion and relaxation in this ${r}^{\ensuremath{-}\ensuremath{\gamma}}$ inner cusp develops an ${r}^{\ensuremath{-}\ensuremath{\beta}}$ outer fall-off with $\ensuremath{\beta}\ensuremath{\approx}5\ensuremath{-}2\ensuremath{\gamma}$ in the quasisteady state. Spherical collapse theory dictates that a quasi-steady outer halo must settle to $\ensuremath{\beta}\ensuremath{\approx}3$ since then the mass enclosed within a radially moving shell barely changes with time. This implies that the quasisteady $\ensuremath{\gamma}$ must be approximately 1, which is possible in the quasilinear framework only if (i) the preassembled halo harbors an ${r}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{P}}}$ profile with ${\ensuremath{\gamma}}_{\mathrm{P}}\ensuremath{\gtrsim}0.5$, (ii) its fluctuations are sufficiently correlated in time (red noise), and (iii) the initial value of $\ensuremath{\gamma}$ is smaller than 1, implying that the ${r}^{\ensuremath{-}1}$ 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 quasisteady state of collisionless relaxation.