Are the most super-massive dark compact objects harbored at the center of dark matter halos?

Essay selected for Honorable mention 2014 by the Gravity Research Foundation. We study an isothermal system of semi-degenerate self-gravitating fermions in General Relativity (GR). The most general solutions present mass density profiles with a central degenerate compact core governed by quantum statistics followed by an extended plateau, and ending in a power law behaviour $r^{-2}$. By fixing the fermion mass $m$ in the keV regime, the different solutions depending on the free parameters of the model: the degeneracy and temperature parameters at the center, are systematically constructed along the one-parameter sequences of equilibrium configurations up to the critical point, which is represented by the maximum in a central density (${\rho }_0$) Vs. core mass ($M_c$) diagram. We show that for fully degenerate cores, the Oppenheimer-Volkoff (OV) mass limit $M_c^{cr}\propto M_{pl}^3\mathrm{/}{m}^2$ is obtained, while instead for low degenerate cores, the critical core mass increases showing the temperature effects in a non linear way. The main result of this work is that when applying this theory to model the distribution of dark matter in big elliptical galaxies from miliparsec distance-scales up to ${10}^2$ Kpc, we do not find any critical core-halo configuration of self-gravitating fermions, able to explain both the most super-massive dark object at their center together with the DM halo simultaneously.