On the [$\alpha$/Fe]-[Fe/H] relations in early-type galaxies
  We study how the predicted [$\alpha$/Fe]-[Fe/H] relations in early-type
galaxies vary as functions of their stellar masses, ages and stellar velocity
dispersions, by making use of cosmological chemodynamical simulations with
feedback from active galactic nuclei. Our model includes a detailed treatment
for the chemical enrichment from dying stars, core-collapse supernovae (both
Type II and hypernovae) and Type Ia supernovae. At redshift $z=0$, we create a
catalogue of $526$ galaxies, among which we determine $80$ early-type galaxies.
From the analysis of our simulations, we find [$\alpha$/Fe]-[Fe/H] relations
similar to the Galactic bulge. We also find that, in the oldest galaxies, Type
Ia supernovae start to contribute at higher [Fe/H] than in the youngest ones.
On the average, early-type galaxies with larger stellar masses (and,
equivalently, higher stellar velocity dispersions) have higher [$\alpha$/Fe]
ratios, at fixed [Fe/H]. This is qualitatively consistent with the recent
observations of Sybilska et al., but quantitatively there are mismatches, which
might require stronger feedback, sub-classes of Type Ia Supernovae, or a
variable initial mass function to address.
