The chemical evolution of galaxies with a variable IGIMF
  Standard analytical chemical evolution modelling of galaxies has been
assuming the stellar initial mass function (IMF) to be invariant and fully
sampled allowing fractions of massive stars to contribute even in dwarf
galaxies with very low star formation rates (SFRs). Recent observations show
the integrated galactic initial mass function (IGIMF) of stars, i.e. the
galaxy-wide IMF, to become systematically top-heavy with increasing SFR. This
has been predicted by the IGIMF theory, which is here used to develop the
analytical theory of the chemical evolution of galaxies. This theory is
non-linear and requires the iterative solution of implicit integral equations
due to the dependence of the IGIMF on the metallicity and on the SFR. It is
shown that the mass-metallicity relation of galaxies emerges naturally,
although at low masses the theoretical predictions overestimate the
observations by 0.3--0.4 dex. A good agreement with the observation can be
obtained only if gas flows are taken into account. In particular, we are able
to reproduce the mass--metallicity relation observed by Lee et al. (2006) with
modest amounts of infall and with an outflow rate which decreases as a function
of the galactic mass. The outflow rates required to fit the data are
considerably smaller than required in models with invariant IMFs.
