Galaxies interstellar media (ISM) are observed to be supersonically-turbulent, but the ultimate power source that drives turbulent motion remains uncertain. The two dominant models are that the turbulence is driven by star formation feedback and/or that it is produced by gravitational instability in the gas. Here we show that, while both models predict that the galaxies ISM velocity dispersions will be positively correlated with their star formation rates, the forms of the correlation predicted by these two models are subtly but measurably different. A feedback-driven origin for the turbulence predicts a velocity dispersion that rises more sharply with star formation rate, and that does not depend on the gas fraction (i.e. $dot{M}_* propto sigma^2$), while a gravity-driven model yields a shallower rise and a strong dependence on gas fraction (i.e. $dot{M}_* propto f_g^2 sigma$). We compare the models to a collection of data on local and high-redshift galaxies culled from the literature, and show that the correlation expected for gravity-driven turbulence is a better match to the observations than a feedback-driven model. This suggests that gravity is the ultimate source of ISM turbulence, at least in the rapidly-star-forming, high velocity dispersion galaxies for which our test is most effective. We conclude by discussing the limitations of the present data set, and the prospects for future measurements to enable a more definitive test of the two models.
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