Variations of the initial mass function in semi-analytical models:
  implications for the mass assembly and the chemical enrichment of galaxies in
  the GAEA model
  In this work, we investigate the implications of the Integrated Galaxy-wide
stellar Initial Mass Function (IGIMF) approach in the framework of the
semi-analytic model GAEA (GAlaxy Evolution and Assembly), which features a
detailed treatment of chemical enrichment and stellar feedback. The IGIMF
provides an analytic description of the dependence of the stellar IMF shape on
the rate of star formation in galaxies. We find that our model with a universal
IMF predicts a rather flat [$\alpha$/Fe]-stellar mass relation. The model
assuming the IGIMF, instead, is able to reproduce the observed increase of
$\alpha$-enhancement with stellar mass, in agreement with previous studies.
This is mainly due to the fact that massive galaxies are characterized by
larger star formation rates at high-redshift, leading to stronger
$\alpha$-enhancement with respect to low-mass galaxies. At the same time, the
IGIMF hypothesis does not affect significantly the trend for shorter star
formation timescales for more massive galaxies. We argue that in the IGIMF
scenario the [$\alpha$/Fe] ratios are good tracers of the highest star
formation events. The final stellar masses and mass-to-light-ratio of our model
massive galaxies are larger than those estimated from the synthetic photometry
assuming a universal IMF, providing a self-consistent interpretation of similar
recent results, based on dynamical analysis of local early type galaxies.
