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We examine the correlations of star formation rate (SFR) and gas-phasemetallicity $Z$. We first predict how the SFR, cold gas mass and $Z$ willchange with variations in inflow rate or in star-formation efficiency (SFE) ina simple gas-regulator framework. The changes $\Delta {\rm log}$SFR and $\Delta{\rm log} Z$, are found to be negatively (positively) correlated when drivingthe gas-regulator with time-varying inflow rate (SFE). We then study thecorrelation of $\Delta {\rm log}$sSFR (specific SFR) and $\Delta {\rmlog}$(O/H) from observations, at both $\sim$100 pc and galactic scales, basedon two 2-dimensional spectroscopic surveys with different spatial resolutions,MAD and MaNGA. After taking out the overall mass and radial dependences, whichmay reflect changes in inflow gas metallicity and/or outflow mass-loading, wefind that $\Delta {\rm log}$sSFR and $\Delta {\rm log}$(O/H) on galactic arefound to be negatively correlated, but $\Delta {\rm log}$sSFR and $\Delta {\rmlog}$(O/H) are positively correlated on $\sim$100 pc scales within galaxies. Ifwe assume that the variations across the population reflect temporal variationsin individual objects, we conclude that variations in the star formation rateare primarily driven by time-varying inflow at galactic scales, and driven bytime-varying SFE at $\sim$100 pc scales. We build a theoretical framework tounderstand the correlation between SFR, gas mass and metallicity, as well astheir variability, which potentially uncovers the relevant physical processesof star formation at different scales.

Gas-phase Metallicity as a Diagnostic of the Drivers of Star-formation on Different Spatial Scales