  We explore observed dynamical trends in a wide range of dark matter dominated
systems (about seven orders of magnitude in mass) to constrain hypothetical
dark matter candidates and scenarios of structure formation. First, we argue
that neither generic warm dark matter (collisionless or collisional) nor
self-interacting dark matter can be responsible for the observed cores on all
scales. Both scenarios predict smaller cores for higher mass systems, in
conflict with observations; some cores must instead have a dynamical origin.
Second, we show that the core phase space densities of dwarf spheroidals,
rotating dwarf and low surface brightness galaxies, and clusters of galaxies
decrease with increasing velocity dispersion like Q ~ sigma^-3 ~ M^-1, as
predicted by a simple scaling argument based on merging equilibrium systems,
over a range of about eight orders of magnitude in Q. We discuss the processes
which set the overall normalization of the observed phase density hierarchy. As
an aside, we note that the observed phase-space scaling behavior and density
profiles of dark matter halos both resemble stellar components in elliptical
galaxies, likely reflecting a similar collisionless, hierarchical origin. Thus,
dark matter halos may suffer from the same systematic departures from homology
as seen in ellipticals, possibly explaining the shallower density profiles
observed in low mass halos. Finally, we use the maximum observed phase space
density in dwarf spheroidal galaxies to fix a minimum mass for relativistically
decoupled warm dark matter candidates of roughly 700 eV for thermal fermions,
and 300 eV for degenerate fermions.
