Quasar Sightline and Galaxy Evolution (QSAGE) -- III. The mass-metallicity and fundamental metallicity relation of $z \approx$ 2.2 galaxies

We present analysis of the mass-metallicity relation (MZR) for a sample of 67 [OIII]-selected star-forming galaxies at a redshift range of $z=1.99 - 2.32$ ($z_{\text{med}} = 2.16$) using \emph{Hubble Space Telescope} Wide Field Camera 3 grism spectroscopy from the Quasar Sightline and Galaxy Evolution (QSAGE) survey. Metallicities were determined using empirical gas-phase metallicity calibrations based on the strong emission lines [OII]3727,3729, [OIII]4959,5007 and H$\beta$. Star-forming galaxies were identified, and distinguished from active-galactic nuclei, via Mass-Excitation diagrams. Using $z\sim0$ metallicity calibrations, we observe a negative offset in the $z=2.2$ MZR of $\approx -0.51$ dex in metallicity when compared to locally derived relationships, in agreement with previous literature analysis. A similar offset of $\approx -0.46$ dex in metallicity is found when using empirical metallicity calibrations that are suitable out to $z\sim5$, though our $z=2.2$ MZR, in this case, has a shallower slope. We find agreement between our MZR and those predicted from various galaxy evolution models and simulations. Additionally, we explore the extended fundamental metallicity relation (FMR) which includes an additional dependence on star formation rate (SFR). Our results consistently support the existence of the FMR, as well as revealing an offset of $0.28\pm0.04$ dex in metallicity compared to locally-derived relationships, consistent with previous studies at similar redshifts. We interpret the negative correlation with SFR at fixed mass, inferred from an FMR existing for our sample, as being caused by the efficient accretion of metal-poor gas fuelling SFR at cosmic noon.