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We examine the cosmic evolution of a stellar initial mass function (IMF) ingalaxies that varies with the Jeans mass in the interstellar medium, payingparticular attention to the K-band stellar mass to light ratio (M/L_K) ofpresent-epoch massive galaxies. We calculate the typical Jeans mass usinghigh-resolution hydrodynamic simulations coupled with a fully radiative modelfor the ISM, which yields a parameterisation of the IMF characteristic mass asa function of galaxy star formation rate (SFR). We then calculate the starformation histories of galaxies utilising an equilibrium galaxy growth modelcoupled with constraints on the star formation histories set by abundancematching models. We find that at early times, energetic coupling between dustand gas drive warm conditions in the ISM, yielding bottom-light/top- heavy IMFsassociated with large ISM Jeans masses for massive star-forming galaxies. Owingto the remnants of massive stars that formed during the top-heavy phases atearly times, the resultant M/L_K(sigma) in massive galaxies at the presentepoch is increased relative to the non- varying IMF case. At late times, lowercosmic ray fluxes allow for cooler ISM temperatures in massive galaxies, andhence newly formed clusters will exhibit bottom-heavy IMFs, further increasingM/L_K(sigma). Our central result is hence that a given massive galaxy may gothrough both top-heavy and bottom-heavy IMF phases during its lifetime, thoughthe bulk of the stars form during a top-heavy phase. Qualitatively, thevariations in M/L_K(sigma) with galaxy mass are in agreement with observations,however, our model may not be able to account for bottom-heavy mass functionsas indicated by stellar absorption features.

The Cosmic Evolution of the IMF Under the Jeans Conjecture with Implications for Bottom-Heavy Ellipticals