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We present detections of [OIII]$\lambda$4363 and direct-method metallicitiesfor star-forming galaxies at $z=1.7-3.6$. We combine new measurements from theMOSFIRE Deep Evolution Field (MOSDEF) survey with literature sources toconstruct a sample of 18 galaxies with direct-method metallicities at $z>1$,spanning $7.5<1$2+log(O/H$)<8.2$ and log(M$_*$/M$_{\odot})=7-10$. We find thatstrong-line calibrations based on local analogs of high-redshift galaxiesreliably reproduce the metallicity of the $z>1$ sample on average. We constructthe first mass-metallicity relation at $z>1$ based purely on direct-method O/H,finding a slope that is consistent with strong-line results. Direct-method O/Hevolves by $\lesssim0.1$ dex at fixed M$_*$ and SFR from $z\sim0-2.2$. Weemploy photoionization models to constrain the ionization parameter andionizing spectrum in the high-redshift sample. Stellar models with super-solarO/Fe and binary evolution of massive stars are required to reproduce theobserved strong-line ratios. We find that the $z>1$ sample falls on the$z\sim0$ relation between ionization parameter and O/H, suggesting no evolutionof this relation from $z\sim0$ to $z\sim2$. These results suggest that theoffset of the strong-line ratios of this sample from local excitation sequencesis driven primarily by a harder ionizing spectrum at fixed nebular metallicitycompared to what is typical at $z\sim0$, naturally explained by super-solarO/Fe at high redshift caused by rapid formation timescales. Given the extremenature of our $z>1$ sample, the implications for representative $z\sim2$ galaxysamples at $\sim10^{10}$ M$_{\odot}$ are unclear, but similarities to $z>6$galaxies suggest that these conclusions can be extended to galaxies in theepoch of reionization.

The MOSDEF Survey: Direct-Method Metallicities and ISM Conditions at $z\sim1.5-3.5$