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

Understand

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 $M_c$.