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

Pierre-Henri Chavanis (1, 2)

1 LPT - Laboratoire de Physique Théorique
2 PhyStat - Physique Statistique des Systèmes Complexes (LPT)

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 ${\mathrm{\Sigma }}_0={\rho }_0{r}_h=141M_{\odot }/{\mathrm{pc}}^2$ obtained from the observations. For a fermion mass $m=165\mathrm{eV}/c^2$, the “minimum halo” has a mass $(M_h{)}_{\min }={10}^8{M}_{\odot }$ and a radius $(r_h{)}_{\min }=597\mathrm{pc}$ 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 $(M_h{)}_{\min }<M_h<(M_h{)}_{\mathrm{CCP}}=6.73\times {10}^8{M}_{\odot }$ (canonical critical point), the quantum core is surrounded by a tenuous classical isothermal atmosphere. Dark matter halos with a mass $M_h>(M_h{)}_{\mathrm{CCP}}$ 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 $(M_h{)}_{\mathrm{CCP}}<M_h<(M_h{)}_{\mathrm{MCP}}=1.08\times {10}^{10}{M}_{\odot }$ (microcanonical critical point), it is microcanonically stable. By maximizing the entropy at fixed mass and energy we find that the mass of the quantum core scales with the halo mass as $M_c/(M_h{)}_{\min }=1.47[M_h/(M_h{)}_{\min }{]}^{3/8}$. This relation is equivalent to the “velocity dispersion tracing” relation according to which the velocity dispersion in the core $v_c^2\sim G{M}_c/R_c$ is of the same order as the velocity dispersion in the halo $v_h^2\sim G{M}_h/r_h$. We provide therefore a justification of this relation from thermodynamical arguments. The fermion ball represents a large quantum bulge which is either present now or may have, in the past, triggered the collapse of the surrounding gas, leading to a supermassive black hole and a quasar. When $M_h>(M_h{)}_{\mathrm{MCP}}$, 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 $M_h>M_{\mathrm{OV}}$, 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 $m=165\mathrm{eV}/c^2$ 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 $M_c=9.45\times {10}^9{M}_{\odot }$ and radius $R_c=240\mathrm{pc}$ in possible agreement with the observations. We also consider another model involving a larger fermion mass $m=54.6\mathrm{keV}/c^2$. In this model, a fermion ball of mass $M_c=4.2\times {10}^6{M}_{\odot }$ and radius $R_c=6\times {10}^{-4}\mathrm{pc}$ could mimic the effect of a supermassive black hole at the center of the Milky Way (Sagittarius ${\mathrm{A}}^{*}$). In bigger galaxies, the fermion ball should be replaced by a supermassive black hole of mass $M_{\mathrm{BH}}=2.10\times {10}^8{M}_{\odot }$ which could account for active galactic nuclei. For an even larger fermion mass $m=386\mathrm{keV}/c^2$, a supermassive black hole of mass $M_{\mathrm{BH}}=4.2\times {10}^6{M}_{\odot }$ should be formed in the Milky Way instead of a fermion ball. However, models with a fermion mass $m=54.6\mathrm{keV}/c^2$ predict that ultracompact dark matter halos of mass $\sim {10}^8{M}_{\odot }$ should contain a fermionic core of mass $M_c\sim {10}^4{M}_{\odot }$ and radius $R_c\sim 5\mathrm{mpc}$ similar to intermediate mass black holes, a prediction which may be challenged by observations.

Mots clés: density; solution; stability; gravitation; collapse; atmosphere; critical phenomena; statistical; galaxy; entropy; rotation; AGN; specific heat; quasar; ground state; Fermi gas; gas; equation of state; pressure; finite temperature; Pauli; formation; Navarro-Frenk-White profile; structure; catastrophe theory; dark matter; halo; fermion; mass; velocity; dispersion; black hole; model; thermodynamical; surface

Domaines: Physique des Hautes Energies - Phénoménologie [hep-ph]; Astrophysique [astro-ph]

Pierre-Henri Chavanis. Predictive model of fermionic dark matter halos with a quantum core and an isothermal atmosphere. Physical Review D, 2022, 106 (4), pp.043538. ⟨10.1103/PhysRevD.106.043538⟩. ⟨hal-03770723⟩