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$\Lambda$-Warm Dark Matter (WDM) has been proposed as alternative scenario to$\Lambda$ cold dark matter (CDM), motivated by discrepancies at the scale ofdwarf galaxies, with less small-scale power and realized by collisionlessparticles with energies in the range $1-3$ keV. We present a new approach toconstrain the viability of such WDM models using star formation histories ofthe dwarf spheroidal galaxies (dSphs) in the Local Group. We compare their hightime-resolution star formation histories (SFHs) obtained with HST-based colormagnitude diagrams with the range of possible collapse redshifts of their darkmatter halos expected in CDM and in different WDM scenarios. The collapseredshift is inferred after determining a plausible infall mass of the subhalo.This is based on the current mass of individual dwarf inferred from stellarkinematics combined with results of cosmological simulations providinginformation on the subhalo evolution. Since WDM subhalos close to the filteringmass scale form significantly later than CDM, we show that they are in thefirst place difficult to reconcile with a truncation of star formationoccurring as early as $z\geq 3$. The Ultra-Faint Dwarfs (UFDs) provide the moststringent constraints. Using 6 UFDs with the best determination of the SFHs, weshow that we can exclude a 1 keV warm particle to a 2-$\sigma$ confidenceinterval consistently with other methods reported in the literature. For someobjects the $2$ keV model is also excluded. We discuss the various caveats ofthe method, most notably the low number of dwarfs with accurately determinedstar formation histories and the uncertainties in the determination of theinfall mass of the subhalos. Our preliminary analysis serves as a pathfinderfor future investigations that will combine upcoming accurate SFHs for morelocal dSphs with direct analysis of WDM cosmological simulations with baryons.

Constraining the nature of dark matter with the star formation history of the faintest Local Group dwarf galaxy satellites