The stellar mass versus stellar metallicity relation of star-forming galaxies at $1.6\le z\le3.0$ and implications for the evolution of the $\alpha$-enhancement

We measure the relationship between stellar mass and stellar metallicity, the stellar mass--metallicity relation (MZR), for 1336 star-forming galaxies at $1.6\le z\le3.0$ ( =2.2) using rest-frame far-ultraviolet spectra from the zCOSMOS-deep survey. High signal-to-noise composite spectra containing stellar absorption features are fit with population synthesis model spectra of a range of metallicity. We find stellar metallicities, which mostly reflect iron abundances, scaling as $(Z_{Fe,\ast}/Z_{Fe,\odot})=-(0.81\pm0.01)+(0.32+0.03)\log(M_\ast/10^{10}M_\odot)$ across the mass range of $10^9\lesssim M_\ast/M_\odot\lesssim10^{11}$, being $\approx6\times$ lower than seen locally at the same masses. The instantaneous oxygen-to-iron ratio ($\alpha$-enhancement) inferred using the gas-phase oxygen MZRs, is on average found to be [O/Fe]$\approx0.47$, being higher than the local [O/Fe]$\approx0$. The observed changes in [O/Fe] and [Fe/H] are reproduced in simple flow-through gas-regulator models with steady star-formation histories (SFHs) that follow the evolving main sequence. Our models show that the [O/Fe] is determined almost entirely by the instantaneous specific star formation rate alone while being independent of the SFHs, mass, and the gas-regulation characteristics of the systems. We find that the locations of $\sim10^{10}M_\odot$ galaxies at z~2 in the [O/Fe]--metallicity planes are in remarkable agreement with the sequence of low-metallicity thick-disk stars in our Galaxy. This manifests a beautiful concordance between the results of Galactic archaeology and observations of high-redshift Milky Way progenitors. However, there remains a question of how and when the old metal-rich, low-$\alpha$/Fe stars seen in the bulge had formed by z~2 because such a stellar population is not seen in our data and difficult to explain in the context of our models.