Dark matter constitutes the great majority of the matter content in the Universe, but its microscopic nature remains an intriguing mystery, with profound implications for particle physics, astrophysics and cosmology. Here we shed light on the longstanding issue of whether the dark matter is warm or cold by combining the measurements of the galaxy luminosity functions out to high redshifts z~10 from the Hubble Space Telescope with the recent cosmological data on the reionization history of the Universe from the Planck mission. We derive robust and tight bounds on the mass of warm dark matter particle, finding that the current data require it to be in the narrow range between 2 and 3 keV. In addition, we show that a mass not exceeding 3 keV is also concurrently indicated by astrophysical constraints related to the local number of satellites in Milky Way-sized galaxies, though it is in marginal tension with analysis of the Lyman-alpha forest. For warm dark matter masses above 3 keV as well as for cold dark matter, to satisfy the Planck constraints on the optical depth and not to run into the satellite problem would require invoking astrophysical processes that inhibit galaxy formation in halos with mass M_H< a few 10^8 M_sun, corresponding to a limiting UV magnitude M_UV~-11. Anyway, we predict a downturn of the galaxy luminosity function at z~8 faintward of M_UV~-12, and stress that its detailed shape is extremely informative both on particle physics and on the astrophysics of galaxy formation in small halos. These expectations will be tested via the Hubble Frontier Fields and with the advent of the James Webb Space Telescope, which will enable probing the very faint end of the galaxy luminosity function out to z~8-10.
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