We calculate the radial profiles of galaxies where the nuclear region is self-gravitating, consisting of self-interacting dark matter (SIDM) with $F$ degrees of freedom. For sufficiently high density this dark matter becomes collisional, regardless of its behaviour on galaxy scales. Our calculations show a spike in the central density profile, with properties determined by the dark matter microphysics, and the densities can reach the `mean density of a black hole (from dividing the black-hole mass by the volume enclosed by the Schwarzschild radius). For a galaxy halo of given compactness ($chi=2GM/Rc^2$), certain values for the dark matter entropy yield a dense central object lacking an event horizon. For some soft equations of state of the SIDM (e.g. $Fge6$), there are multiple horizonless solutions at given compactness. Although light propagates around and through a sphere composed of dark matter, it is gravitationally lensed and redshifted. While some calculations give non-singular solutions, others yield solutions with a central singularity. In all cases the density transitions smoothly from the central body to the dark-matter envelope around it, and to the galaxys dark matter halo. We propose that pulsar timing observations will be able to distinguish between systems with a centrally dense dark matter sphere (for different equations of state) and conventional galactic nuclei that harbour a supermassive black hole.
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