Cosmic structures from a mathematical perspective 1. Dark matter halo
  mass density profiles
  The shapes of individual self-gravitating structures of an ensemble of
identical, collisionless particles have remained elusive for decades. In
particular, a reason why mass density profiles like the Navarro-Frenk-White or
the Einasto profile are good fits to simulation- and observation-based dark
matter halos has not been found. Given the class of three dimensional,
spherically symmetric power-law probability density distributions to locate
individual particles in the ensemble mentioned above, we derive the
constraining equation for the power-law index for the most and least likely
joint ensemble configurations. We find that any dark matter halo can be
partitioned into three regions: a core, an intermediate part, and an outskirts
part up to boundary radius $r_\mathrm{max}$. The power-law index of the core is
determined by the mean radius of the particle distribution within the core. The
intermediate region becomes isothermal in the limit of infinitely many
particles. The slope of the mass density profile far from the centre is
determined by the choice of $r_\mathrm{max}$ with respect to the outmost halo
particle, such that two typical limiting cases arise that explain the
$r^{-3}$-slope for galaxy-cluster outskirts and the $r^{-4}$-slope for galactic
outskirts. Hence, we succeed in deriving the mass density profiles of common
fitting functions from a general viewpoint. These results also allow to find a
simple explanation for the cusp-core-problem and to separate the halo
description from its dynamics.
