JADES: Detecting [OIII]$\lambda 4363$ Emitters and Testing Strong Line Calibrations in the High-$z$ Universe with Ultra-deep JWST/NIRSpec Spectroscopy up to $z \sim 9.5$

We present 10 novel [OIII]$\lambda 4363$ auroral line detections up to $z\sim 9.5$ measured from ultra-deep JWST/NIRSpec MSA spectroscopy from the JWST Advanced Deep Extragalactic Survey (JADES). We leverage the deepest spectroscopic observations yet taken with NIRSpec to determine electron temperatures and oxygen abundances using the direct T$_e$ method. We directly compare against a suite of locally calibrated strong-line diagnostics and recent high-$z$ calibrations. We find the calibrations fail to simultaneously match our JADES sample, thus warranting a self-consistent revision of these calibrations for the high-$z$ Universe. We find weak dependence between R2 and O3O2 with metallicity, thus suggesting these line-ratios are ineffective in the high-$z$ Universe as metallicity diagnostics and degeneracy breakers. We find R3 and R23 still correlate with metallicity, but we find tentative flattening of these diagnostics, thus suggesting future difficulties when applying these strong-line ratios as metallicity indicators in the high-$z$ Universe. We also propose and test an alternative diagnostic based on a different combination of R3 and R2 with a higher dynamic range. We find a reasonably good agreement (median offset of 0.002 dex, median absolute offset of 0.13 dex) with the JWST sample at low metallicity. Our sample demonstrates higher ionization/excitation ratios than local galaxies with rest-frame EWs(H$\beta$) $\approx 200 -300$ Angstroms. However, we find the median rest-frame EWs(H$\beta$) of our sample to be $\sim 2\text{x}$ less than the galaxies used for the local calibrations. This EW discrepancy combined with the high ionization of our galaxies does not present a clear description of [OIII]$\lambda 4363$ production in the high-$z$ Universe, thus warranting a much deeper examination into the factors affecting production.