📝 SummarXiv

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 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 $r^{-\gamma}$ profiles with $0\leq \gamma \leq 2$. 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.