Core mass-halo mass relation of bosonic and fermionic dark matter halos harboring a supermassive black hole

We study the core mass-halo mass relation of bosonic dark matter halos, in the form of self-gravitating Bose-Einstein condensates, harboring a supermassive black hole. We use the ``velocity dispersion tracing'' relation according to which the velocity dispersion in the core v₂^2M₂/R₂ is of the same order as the velocity dispersion in the halo v₇^2M₇/r₇ (this relation can be justified from thermodynamical arguments) and the approximate analytical mass-radius relation of the quantum core in the presence of a central black hole obtained in our previous paper P. H. Chavanis, Eur. Phys. J. Plus 134, 352 (2019). For a given minimum halo mass (M₇) ₌₈₍10^8 M_ determined by the observations, the only free parameter of our model is the scattering length aₒ of the bosons (their mass m is then determined in order to match the characteristics of the minimum halo). For noninteracting bosons and for bosons with a repulsive self-interaction, we find that the core mass M₂ increases with the halo mass M₇ and achieves a maximum value (M₂) ₌₀ₗ at some halo mass (M₇) * before decreasing. The whole series of equilibria is stable. For bosons with an attractive self-interaction, we find that the core mass achieves a maximum value (M₂) ₌₀ₗ at some halo mass (M₇) * before decreasing. The series of equilibria becomes unstable above a maximum halo mass (M₇) ₌₀ₗ (M₇) *. In the absence of black hole (M₇) ₌₀ₗ= (M₇) *. At that point, the quantum core (similar to a dilute axion star) collapses. We perform a similar study for fermionic dark matter halos. We find that they behave similarly to bosonic dark matter halos with a repulsive self-interaction, the Pauli exclusion principle for fermions playing the role of the repulsive self-interaction for bosons.