Derivation of the core mass -- halo mass relation of fermionic and bosonic dark matter halos from an effective thermodynamical model

We consider the possibility that dark matter halos are made of quantum particles such as fermions or bosons in the form of Bose-Einstein condensates. In that case, they generically have a "core-halo" structure with a quantum core that depends on the type of particle considered and a halo that is relatively independent of the dark matter particle and that is similar to the NFW profile of cold dark matter. We model the halo by an isothermal gas with an effective temperature $T$. We then derive the core mass -- halo mass relation $M_c(M_v)$ of dark matter halos from an effective thermodynamical model by extremizing the free energy $F(M_c)$ with respect to the core mass $M_c$. We obtain a general relation that is equivalent to the "velocity dispersion tracing" relation according to which the velocity dispersion in the core $v_c^2\sim GM_c/R_c$ is of the same order as the velocity dispersion in the halo $v_v^2\sim GM_v/r_v$. We provide therefore a justification of this relation from thermodynamical arguments. In the case of fermions, we obtain a relation $M_c\propto M_v^{1/2}$ that agrees with the relation found numerically by Ruffini {\it et al.}. In the case of noninteracting bosons, we obtain a relation $M_c\propto M_v^{1/3}$ that agrees with the relation found numerically by Schive {\it et al.}. In the case of bosons with a repulsive self-interaction in the Thomas-Fermi limit, we predict a relation $M_c\propto M_v^{2/3}$ that still has to be confirmed numerically. We also obtain a general approximate core mass -- halo mass relation that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass that can harbor a stable quantum core (dilute axion "star").