Statistical ensembles of virialized halo matter density profiles

TL;DR

This paper develops a statistical framework for modeling spherically symmetric, virialized dark matter haloes, deriving mean density profiles that align with observed NFW profiles, especially in massive systems.

Contribution

It introduces an exact analytical solution for ensembles of halo density profiles, predicting universal features like the NFW inner slope from statistical principles.

Findings

Mean profiles have an infinitely deep potential well.

NFW profiles emerge as the most probable configurations.

Agreement with NFW profiles improves for more massive haloes.

Abstract

We define and study statistical ensembles of matter density profiles describing spherically symmetric, virialized dark matter haloes of finite extent with a given mass and total gravitational potential energy. Our ensembles include spatial degrees of freedom only, a microstate being a spherically symmetric matter density function. We provide an exact solution for the grand canonical partition functional, and show its equivalence to that of the microcanonical ensemble. We obtain analytically the mean profiles that correspond to an overwhelming majority of microstates. All such profiles have an infinitely deep potential well, with the singular isothermal sphere arising in the infinite temperature limit. Systems with virial radius larger than gravitational radius exhibit a localization of a finite fraction of the energy in the very center. The universal logarithmic inner slope of unity of the NFW haloes is predicted at any mass and energy if an upper bound is set to the maximal depth of the potential well. In this case, the statistically favored mean profiles compare well to the NFW profiles. For very massive haloes the agreement becomes exact.