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.
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