Chemical evolution of ultra-faint dwarf galaxies: testing the IGIMF

We test the integrated galactic initial mass function (IGIMF) on the chemical evolution of 16 ultra-faint dwarf (UFD) galaxies discussing in detail the results obtained for three of them: Bo\"otes I, Bo\"otes II and Canes Venatici I, taken as prototypes of the smallest and the largest UFDs. These objects have very small stellar masses ($\sim 10^3-10^4 \mathrm{M_{\odot}}$) and quite low metallicities ([Fe/H]$<-1.0$ dex). We consider four observational constraints: the present-day stellar mass, the [$\alpha$/Fe] vs. [Fe/H] relation, the stellar metallicity distribution function and the cumulative star formation history. Our model follows in detail the evolution of several chemical species (H, He, $\alpha$-elements and Fe). We take into account detailed nucleosynthesis and gas flows (in and out). Our results show that the IGIMF, coupled with the very low star formation rate predicted by the model for these galaxies ($\sim 10^{-4}-10^{-6}\ \mathrm{M_{\odot}yr^{-1}}$), cannot reproduce the main chemical properties, because it implies a negligible number of core-collapse SNe and even Type Ia SNe, the most important polluters of galaxies. On the other hand, a constant classical Salpeter IMF gives the best agreement with data, but we cannot exclude that other formulations of the IGIMF could reproduce the properties of these galaxies. Comparing with Galaxy data we suggest that UFDs could not be the building blocks of the entire Galactic halo, although more data are necessary to draw firmer conclusions.