How general is the global density slope-anisotropy inequality?

TL;DR

This paper investigates the universality of the density slope-anisotropy inequality in spherical systems, deriving a unifying criterion and providing evidence that it may be a fundamental property of such systems.

Contribution

The authors derive a new criterion for the density slope-anisotropy inequality applicable to all spherical systems with separable augmented density, unifying previous results.

Findings

The inequality holds at all radii in many spherical systems.

It is satisfied by multiple multi-component stellar systems.

The results suggest the inequality might be universally valid for spherical systems.

Abstract

Following the seminal result of An & Evans, known as the central density slope-anisotropy theorem, successive investigations unexpectedly revealed that the density slope-anisotropy inequality holds not only at the center, but at all radii in a very large class of spherical systems whenever the phase-space distribution function is positive. In this paper we derive a criterion that holds for all spherical systems in which the augmented density is a separable function of radius and potential: this new finding allows to unify all the previous results in a very elegant way, and opens the way for more general investigations. As a first application, we prove that the global density slope-anisotropy inequality is also satisfied by all the explored additional families of multi-component stellar systems. The present results, and the absence of known counter-examples, lead us to conjecture that the global density slope-anisotropy inequality could actually be a universal property of spherical systems with positive distribution function.