Constraining the nature of dark matter with the star formation history
  of the faintest Local Group dwarf galaxy satellites
  $\Lambda$-Warm Dark Matter (WDM) has been proposed as alternative scenario to
$\Lambda$ cold dark matter (CDM), motivated by discrepancies at the scale of
dwarf galaxies, with less small-scale power and realized by collisionless
particles with energies in the range $1-3$ keV. We present a new approach to
constrain the viability of such WDM models using star formation histories of
the dwarf spheroidal galaxies (dSphs) in the Local Group. We compare their high
time-resolution star formation histories (SFHs) obtained with HST-based color
magnitude diagrams with the range of possible collapse redshifts of their dark
matter halos expected in CDM and in different WDM scenarios. The collapse
redshift is inferred after determining a plausible infall mass of the subhalo.
This is based on the current mass of individual dwarf inferred from stellar
kinematics combined with results of cosmological simulations providing
information on the subhalo evolution. Since WDM subhalos close to the filtering
mass scale form significantly later than CDM, we show that they are in the
first place difficult to reconcile with a truncation of star formation
occurring as early as $z\geq 3$. The Ultra-Faint Dwarfs (UFDs) provide the most
stringent constraints. Using 6 UFDs with the best determination of the SFHs, we
show that we can exclude a 1 keV warm particle to a 2-$\sigma$ confidence
interval consistently with other methods reported in the literature. For some
objects the $2$ keV model is also excluded. We discuss the various caveats of
the method, most notably the low number of dwarfs with accurately determined
star formation histories and the uncertainties in the determination of the
infall mass of the subhalos. Our preliminary analysis serves as a pathfinder
for future investigations that will combine upcoming accurate SFHs for more
local dSphs with direct analysis of WDM cosmological simulations with baryons.
