  We propose a novel use of high-redshift galaxies, discovered in deep Hubble
Space Telescope (HST) fields around strong lensing clusters. These fields probe
small comoving volumes (about 1000 cubic Mpc) at high magnification ({\mu} >
10), and can detect otherwise inaccessible ultra-faint galaxies. Even a few
galaxies found in such small volumes require a very high number density of
collapsed dark matter (DM) halos. This implies significant primordial power on
small scales, allowing these observations to rule out popular alternatives to
standard cold dark matter (CDM) models, such as warm dark matter (WDM). In this
work, we analytically compute WDM halo mass functions at z = 10, including the
effects of both particle free-streaming and residual velocity dispersion. The
preliminary number density corresponding to the two galaxies at z about 10
already detected by the Cluster Lensing And Supernova survey with Hubble
(CLASH) constrains the WDM particle mass to m_x > 1 (0.9) keV at 84% (99.9%)
confidence limit (for a thermal relic relativistic at decoupling). This limit
depends only on the WDM halo mass function and, unlike previous constraints on
m_x, is independent of any astrophysical modeling. The forthcoming HST Frontier
Fields can significantly tighten these constraints.
