Predictive model of fermionic dark matter halos with a quantum core and an isothermal atmosphere

We develop a thermodynamical model of fermionic dark matter halos at finite temperature. Statistical equilibrium states may be justified by a process of violent collisionless relaxation in the sense of Lynden-Bell or from a collisional relaxation of nongravitational origin if the fermions are self-interacting. The most probable state (maximum entropy state) generically has a ``core-halo'' structure with a quantum core (fermion ball) surrounded by an isothermal atmosphere. The quantum core is equivalent to a polytrope of index n=3/2. The Pauli exclusion principle creates a quantum pressure that prevents gravitational collapse and solves the core-cusp problem of the cold dark matter model. The isothermal atmosphere (which is similar to the Navarro-Frenk-White profile of cold dark matter) accounts for the flat rotation curves of the galaxies at large distances. We numerically solve the equation of hydrostatic equilibrium with the Fermi-Dirac equation of state and determine the density profiles and rotation curves of fermionic dark matter halos. We impose that the surface density of the dark matter halos has the universal value ₀=₀r₇=141 M_/pc^2 obtained from the observations. For a fermion mass m=165 eV/c^2, the ``minimum halo'' has a mass (M₇) ₌₈₍=10^8 M_ and a radius (r₇) ₌₈₍=597 pc similar to dwarf spheroidals like Fornax. This ultracompact halo corresponds to a completely degenerate fermion ball at T=0. This is the ground state of the self-gravitating Fermi gas. For ultracompact dark matter halos with a mass (M₇) ₌₈₍ (M₇) ₂₂₏ are dominated by the classical isothermal atmosphere. They may be purely gaseous (similar to the Burkert profile) or harbor a fermion ball. The gaseous solution is stable in all statistical ensembles. The core-halo solution is canonically unstable (having a negative specific heat) but, for small dark matter halos with a mass (M₇) ₂₂₏ (M₇) ₌₂₏, the quantum core-halo solution is microcanonically unstable. Large dark matter halos may undergo a gravothermal catastrophe leading ultimately to the formation of a small out-of-equilibrium condensed core or, in the case of very large dark matter halos with M₇>M₎ₕ, to a supermassive black hole when the core mass overcomes the Oppenheimer-Volkoff (OV) limit. The isothermal halo is left undisturbed and is in agreement with the Burkert profile. Our model has no free parameter (the mass m=165 eV/c^2 of the fermionic particle is determined by the minimum halo) so it is completely predictive. It predicts that the Milky Way should harbor a fermionic dark matter bulge of mass M₂=9. 4510^9 M_ and radius R₂=240 pc in possible agreement with the observations. We also consider another model involving a larger fermion mass m=54. 6 keV/c^2. In this model, a fermion ball of mass M₂=4. 210^6 M_ and radius R₂=610^-4 pc could mimic the effect of a supermassive black hole at the center of the Milky Way (Sagittarius A^*). In bigger galaxies, the fermion ball should be replaced by a supermassive black hole of mass M₁₇=2. 1010^8 M_ which could account for active galactic nuclei. For an even larger fermion mass m=386 keV/c^2, a supermassive black hole of mass M₁₇=4. 210^6 M_ should be formed in the Milky Way instead of a fermion ball. However, models with a fermion mass m=54. 6 keV/c^2 predict that ultracompact dark matter halos of mass 10^8 M_ should contain a fermionic core of mass M₂10^4 M_ and radius R₂5 mpc similar to intermediate mass black holes, a prediction which may be challenged by observations.