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

Abstract

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 $\Lambda$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 pre-assembled halo develops an $r^{-\gamma}$ inner density cusp, accretion and relaxation in which develops an $r^{-\beta}$ outer fall-off with $\beta \approx 5 - 2\gamma$ in the quasi-steady state. Spherical collapse theory dictates that a quasi-steady outer halo must settle to $\beta \approx 3$, for which the mass enclosed within a shell barely changes with time. This implies that $\gamma\approx 1$, which is possible in the quasilinear framework only if (i) the pre-assembled halo harbors an $r^{-\gamma_{\mathrm{P}}}$ profile with $\gamma_{\mathrm{P}} \gtrsim 0.5$, (ii) its fluctuations are correlated in time (red noise), and (iii) the initial value of $\gamma$ is smaller than $1$, implying that the $r^{-1}$ cusp is a neutral equilibrium. We demonstrate for the first time how the Navarro-Frenk-White (NFW) profile emerges as a quasi-steady state of collisionless relaxation.