Fermionic Dark Matter spikes: origin and growth of Black Hole seeds

We characterize the overdensity (spike) of fermionic dark matter (DM) particles around a supermassive black hole (SMBH) within a general relativistic analysis. The initial DM halo distribution is obtained by solving the equilibrium equations of a self-gravitating system of massive fermions at a finite temperature, according to the Ruffini-Arg\"uelles-Rueda (RAR) model. The final fermionic DM spike is calculated around a Schwarzschild SMBH. We explore two possible interpretations for the origin and evolution of the SMBH seed. One corresponds to the traditional scenario proposed by Gondolo & Silk (1999), where a small BH mass of baryonic origin sits at the halo's center and grows adiabatically. The other one, from DM origin, where the dense and degenerate fermion core predicted by the RAR model grows adiabatically by capturing baryons until its gravitational collapse, providing a heavy SMBH seed, whose specific value depends on the fermion mass. We study different initial fermionic DM profiles that the theory allows. We show that overall dilute (i.e., Boltzmannian) fermionic DM develops the well-known spike with density profile $\rho \sim r^{-3/2}$. Instead, for semi-degenerate fermions with a dense and compact core surrounded by a diluted halo, we find novel spike profiles that depend on the particle mass and nature. In the more general case, fermionic spikes do not develop a simple power-law profile. Furthermore, the SMBH does not always imply an enhancement of the surrounding DM density; it might also deplete it. Thus, the self-consistent inclusion of the DM candidate nature and mass in determining the structure and distribution of DM in galaxies, including the DM spikes around SMBHs, is essential for the specification of DM astrophysical probes such as BH mergers, gravitational waves, or stellar orbits.