Fundamental physics with the Hubble Frontier Fields: constraining Dark Matter models with the abundance of extremely faint and distant galaxies

We show that the measured abundance of ultra-faint lensed galaxies at $z\approx 6$ in the Hubble Frontier Fields (HFF) provides stringent constraints on the parameter space of i) Dark Matter models based on keV sterile neutrinos; ii) the "fuzzy" wavelike Dark Matter models, based on Bose-Einstein condensate of ultra-light particles. For the case of the sterile neutrinos, we consider two production mechanisms: resonant production through the mixing with active neutrinos and the decay of scalar particles. For the former model, we derive constraints for the combination of sterile neutrino mass $m_{\nu}$ and mixing parameter $\sin^2(2\theta)$ which provide the tightest lower bounds on the mixing angle (and hence on the lepton asymmetry) derived so far by methods independent of baryonic physics. For the latter we compute the allowed combinations of the scalar mass, its coupling to the Higgs field, and the Yukawa coupling of the scalar to the sterile neutrinos. We compare our results to independent, existing astrophysical bounds on sterile neutrinos in the same mass range. For the case of "fuzzy" Dark Matter, we show that the observed number density $\approx 1/{\rm Mpc}^3$ of high-redshift galaxies in the HFF sets a lower limit $m_\psi\geq 8\cdot 10^{-22}$ eV (at 3-$\sigma$ confidence level) on the particle mass, a result that strongly disfavors wavelike bosonic Dark Matter as a viable model for structure formation. We discuss the impact on our results of uncertainties due to systematics in the selection of highly magnified, faint galaxies at high redshifts.