The chemical evolution of galaxies within the IGIMF theory: the
  [alpha/Fe] ratios and downsizing
  The chemical evolution of galaxies is investigated within the framework of
the star formation rate (SFR) dependent integrated galactic initial mass
function (IGIMF). We study how the global chemical evolution of a galaxy and in
particular how [alpha/Fe] abundance ratios are affected by the predicted
steepening of the IGIMF with decreasing SFR. We use analytical and
semi-analytical calculations to evaluate the mass-weighted and
luminosity-weighted [alpha/Fe] ratios in early-type galaxies of different
masses. The models with the variable IGIMF produce a [alpha/Fe] vs. velocity
dispersion relation which has the same slope as the observations of massive
galaxies, irrespective of the model parameters, provided that the star
formation duration inversely correlates with the mass of the galaxy
(downsizing). These models also produce steeper [alpha/Fe] vs. sigma relations
in low-mass early-type galaxies and this trend is consistent with the
observations. Constant IMF models are able to reproduce the [alpha/Fe] ratios
in large elliptical galaxies as well, but they do not predict this change of
slope for small galaxies. In order to obtain the best fit between our results
and the observations, the downsizing effect (i.e. the shorter duration of the
star formation in larger galaxies) must be milder than previously thought.
