Dark matter concentrations in galactic nuclei according to polytropic models

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 (χ ≡ 2GM/Rc2), 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 6), 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