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

摘要

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 M_c(M_v) of dark matter halos from an effective thermodynamical model by maximizing the entropy S(M_c) with respect to the core mass M_c 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 v_c~2 ~ GM_c/R_c is of the same order as the velocity dispersion in the halo v_v~2 ~ 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 ∝ M_v~(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 M_c ∝ M_v~(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 M_c ∝ M_v~(2/3) that still has to be confirmed numerically. Using a Gaussian ansatz, we obtain a general approximate core mass-halo mass relation M_c(M_v) that is valid for bosons with arbitrary repulsive or attractive self-interaction. For an attractive self