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 Navarro-Frenk-White profile of cold dark matter. The quantum core is equivalent to a polytrope of index n=3/2 for fermions, n=2 for noninteracting bosons, and n=1 for bosons with a repulsive self-interaction in the Thomas-Fermi limit. We model the halo by an isothermal gas with an effective temperature T. We then derive the core mass--halo mass relation M₂ (Mₕ) of dark matter halos from an effective thermodynamical model by maximizing the entropy S (M₂) with respect to the core mass M₂ at fixed total mass and total energy. We obtain a general relation, valid for an arbitrary polytropic core, that is equivalent to 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 vₕ^2Mₕ/rₕ. We provide therefore a justification of this relation from thermodynamical arguments. In the case of fermions, we obtain a relation M₂Mₕ^1/2 that agrees with the relation found numerically by Ruffini et al. Mon. Not. R. Astron. Soc. 451, 622 (2015). In the case of noninteracting bosons, we obtain a relation M₂Mₕ^1/3 that agrees with the relation found numerically by Schive et al. Phys. Rev. Lett. 113, 261302 (2014). In the case of bosons with a repulsive self-interaction in the Thomas-Fermi limit, we predict a relation M₂Mₕ^2/3 that still has to be confirmed numerically. Using a Gaussian ansatz, we obtain a general approximate core mass--halo mass relation M₂ (Mₕ) that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self-interaction, we determine the maximum halo mass (Mₕ) ₌₀ₗ that can harbor a stable quantum core (dilute axion ``star''). Above that mass, the quantum core collapses. Finally, we argue that the fundamental mass scale of the bosonic dark matter particle is m_=/c^2=2. 0810^-33 eV/c^2 and that the fundamental mass scale of the fermionic dark matter particle is m_^*= (^3/Gc^3) ^1/4=m_{M₏}=5. 0410^-3 eV/c^2, where is the cosmological constant and M₏ is the Planck mass. Their ratio is m_^*/m_= (c^5/G) ^1/4=2. 4210^30 which explains the difference of mass between fermions and bosons in dark matter models. The actual value of the dark matter particle mass is equal to these mass scales multiplied by a large factor that we obtain from our model.