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One of the most robust observations of the stellar initial mass function(IMF) is its near-universality in the Milky Way and neighboring galaxies. Butrecent observations of early-type galaxies can be interpreted to imply abottom-heavy IMF, while others of ultra-faint dwarfs could imply a top-heavyIMF. This would impose powerful constraints on star formation models. Weexplore what sort of cloud-scale IMF models could possibly satisfy theseconstraints. We utilize simulated galaxies which reproduce (broadly) theobserved galaxy properties, while they also provide the detailed star formationhistory and properties of each progenitor star-forming cloud. We then considergeneric models where the characteristic mass of the IMF is some arbitrarypower-law function of progenitor cloud properties, along with well-knownliterature IMF models, which scale with Jeans mass, turbulent Bonnor-Ebertmass, temperature, the opacity limit, metallicity, or the protostellar heatingmass. We show that no IMF models currently in the literature - nor any modelwhere the turnover mass is an arbitrary power-law function of a combination ofcloud temperature/density/size/metallicity/velocity dispersion/magnetic field -can reproduce the claimed IMF variation in ellipticals or dwarfs withoutseverely violating observational constraints in the Milky Way. Specifically,they predict too much variation in the extreme environments of the Galaxy,compared to that observed. Either the IMF varies in a more complicated manner,or alternative interpretations of the extragalactic observations must beexplored.

Is it possible to reconcile extragalactic IMF variations with a universal Milky Way IMF?