Fully Degenerate Self-Gravitating Fermionic Dark Matter: Implications to the Density Profile of the Cluster of Galaxies A1689, and the Mass Hierarchy of Black Holes

Equilibrium configurations of weakly interacting fully degenerate fermionic dark matter are considered at various scales in the Universe. We treat the general situations for the gravity from Newtonian to general relativity and the degeneracy from nonrelativistic to relativistic. A dimensionless equilibrium configuration is specified by a single parameter regardless of particle properties, the Fermi velocity at the center, and the scalings of mass and length are specified by the rest mass and statistical weight of the dark matter particle. We focus our attention to the flat-top nature of the mass column density profile of the cluster of galaxies, A1689, recently reported by Broadhurst et al. using gravitational lensing. We convert the column density profile to a volume density profile assuming spherical symmetry and derive a 3D encircled mass profile of A1689, which is compared with the model profiles of degenerate fermion structures. The flat-top profile is reproduced. The corresponding fermion mass ranges from 2 eV to 30 eV depending on the actual scale of the degenerate structure. If massive neutrinos are the dominant dark matter, the rest mass will be about 4.7 or 2.3 eV respectively for Majorana or Dirac neutrinos. The mass and size of the degenerate structure are $10^{14}M_\odot$ and 100 kpc for Majorana neutrinos, and 5$\times10^{14}M_\odot$ and 300 kpc for Dirac neutrinos. If we identify the fermions as heavier sterile neutrinos, they yield the characteristic mass hierarchy of black holes; giant black hole at the center of a galaxy and the intermediate mass black holes. Thus we propose the possibility that the mass hierarchy of fermions determines that of black holes in the Universe.