  Observational and theoretical arguments increasingly suggest that the initial
mass function (IMF) of stars may depend systematically on environment, yet most
galaxy formation models to date assume a universal IMF. Here we investigate
simulations of the formation of Milky Way analogues run with an empirically
derived metallicity-dependent IMF and the moving-mesh code AREPO in order to
characterize the associated uncertainties. In particular, we compare a constant
Chabrier and a varying metallicity-dependent IMF in cosmological,
magneto-hydrodynamical zoom-in simulations of Milky Way-sized halos. We find
that the non-linear effects due to IMF variations typically have a limited
impact on the morphology and the star formation histories of the formed
galaxies. Our results support the view that constraints on stellar-to-halo mass
ratios, feedback strength, metallicity evolution and metallicity distributions
are in part degenerate with the effects of a non-universal,
metallicity-dependent IMF. Interestingly, the empirical relation we use between
metallicity and the high mass slope of the IMF does not aid in the quenching
process. It actually produces up to a factor of 2-3 more stellar mass if
feedback is kept constant. Additionally, the enrichment history and the z = 0
metallicity distribution are significantly affected. In particular, the alpha
enhancement pattern shows a steeper dependence on iron abundance in the
metallicity-dependent model, in better agreement with observational
constraints.
