Revisiting the fundamental metallicity relation with observation and simulation

The gas-phase metallicity of galaxies is regulated by multiple astrophysical processes, which makes it a crucial diagnostic of galaxy formation and evolution. Beyond the fundamental mass-metallicity relation, a debate about the secondary galaxy property to predict the metallicity of galaxies arises. Motivated by this, we systematically examine the relationship between gas-phase metallicity and other galaxy properties, i.e. star formation rate (SFR) and galaxy size, in addition to stellar mass in both observation and simulation. We utilize the data from the MaNGA (Mapping Nearby Galaxies at Apache Point Observatory) survey and the TNG50 simulations. We find that the combination of $M_*/R_{\rm e}^\beta$ with $\beta\sim 0.6-1$ is in much stronger correlation to the metallicity than stellar mass alone, regardless of whether the SFR is included or not, in both observation and simulation. This indicates that galaxy size plays a more important role in determining gas-phase metallicity of galaxies than SFR. In addition, the TNG simulation predicts that the SFR, although being a subdominant role, becomes increasingly important in high-$z$ universe. Finally, we speculate that SFR modulates metallicity on the temporal dimension, synchronized with time-varying gas inflows, and galaxy size regulates metallicity on the spatial dimension by affecting the gravitational potential and the mass loading factor.