We explore observed trends in the cores of a wide range of relaxed dark matter-dominated halos (about 7 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 mean core phase-space densities Q of dwarf spheroidal galaxies, rotating dwarf and low surface brightness galaxies, and clusters of galaxies decrease with increasing velocity dispersion like Q ∝ σ-3 ∝ M-1, as predicted by a simple scaling argument based on quietly merging equilibrium systems over a range of about 8 orders of magnitude in Q. We discuss the processes that set the overall normalization of the observed phase density hierarchy. We note the resemblance between the observed phase-space scaling behavior and density profiles of dark matter halos and stellar components in elliptical galaxies and conjecture that dark matter halos may suffer from the same systematic departures from homology as seen in elliptical galaxies, thus 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.
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