  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. $F\ge6$), 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 galaxy's 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.
