  Observations of star-forming galaxies at high-z have suggested discrepancies
in the inferred star formation rates (SFRs) either between data and models, or
between complementary measures of the SFR. These putative discrepancies could
all be alleviated if the stellar IMF is systematically weighted toward more
high-mass star formation in rapidly star-forming galaxies. Here, we explore how
the IMF might vary under the central assumption that the turnover mass in the
IMF, Mc, scales with the Jeans mass in giant molecular clouds (GMCs), M_J. We
employ hydrodynamic and radiative transfer simulations of galaxies to predict
how the typical GMC Jeans mass, and hence the IMF, varies with galaxy property.
We then study the impact of such an IMF on the star formation law, the SFR-M*
relation, submillimetre galaxies (SMGs), and the cosmic SFR density. Our main
results are: The H2 mass-weighted Jeans mass in a galaxy scales with the SFR
when the SFR is greater a few M_sun/yr. SPS modeling shows that this results in
a nonlinear relation between SFR and Lbol, such that SFR Lbol^0.88. Using this
model relation, the inferred SFR of local ULIRGs decreases by ~2, and that of
high-z SMGs decreases by ~3-5. At z 2, this results in a lowered normalisation
of the SFR-M* relation in better agreement with models, a reduced discrepancy
between the observed cosmic SFR density and stellar mass density evolution, and
SMG SFRs that are easier to accommodate in current hierarchical structure
formation models. It further results in a Schmidt relation with slope of ~1.6
when utilising a physically motivated form for the CO-H2 conversion factor.
While each of the discrepancies considered here could be alleviated without
appealing to a varying IMF, the modest variation implied by assuming Mc M_J is
a plausible solution that simultaneously addresses numerous thorny issues
regarding the SFRs of high-z galaxies.
