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.
