Ubiquity of density slope oscillations in the central regions of galaxy
  and cluster-sized systems
  One usually thinks of a radial density profile as having a monotonically
changing logarithmic slope, such as in NFW or Einasto profiles. However, in two
different classes of commonly used systems, this is often not the case. These
classes exhibit non-monotonic changes in their density profile slopes which we
call oscillations for short. We analyze these two unrelated classes separately.
Class 1 consists of systems that have density oscillations and that are defined
through their distribution function $f(E)$, or differential energy distribution
$N(E)$, such as isothermal spheres, King profiles, or DARKexp, a theoretically
derived model for relaxed collisionless systems. Systems defined through $f(E)$
or $N(E)$ generally have density slope oscillations. Class 1 system
oscillations can be found at small, intermediate, or large radii but we focus
on a limited set of Class 1 systems that have oscillations in the central
regions, usually at $\log(r/r_{-2})\lesssim -2$, where $r_{-2}$ is the largest
radius where $d\log(\rho)/d\log(r)=-2$. We show that the shape of their $N(E)$
can roughly predict the amplitude of oscillations. Class 2 systems which are a
product of dynamical evolution, consist of observed and simulated galaxies and
clusters, and pure dark matter halos. Oscillations in the density profile slope
seem pervasive in the central regions of Class 2 systems. We argue that in
these systems, slope oscillations are an indication that a system is not fully
relaxed. We show that these oscillations can be reproduced by small
modifications to $N(E)$ of DARKexp. These affect a small fraction of systems'
mass and are confined to $\log(r/r_{-2})\lesssim 0$. The size of these
modifications serves as a potential diagnostic for quantifying how far a system
is from being relaxed.
