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We study an isothermal system of semi-degenerate self-gravitating fermions in general relativity (GR). The most general solutions present is mass density profiles with a central degenerate compact core governed by quantum statistics followed by an extended plateau, and ending in a power law behavior 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 (ρ 0 ) versus core mass (M c ) diagram. We show that for fully degenerate cores, the Oppenheimer–Volkoff (OV) mass limit [Formula: see text] is obtained, while instead for low degenerate cores, the critical core mass increases showing the temperature effects in a nonlinear way. The main result of this work is that when applying this theory to model the distribution of dark matter (DM) 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.

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