Using the near-IR spectroscopy of the MOSFIRE Deep Evolution Field (MOSDEF) survey, we investigate the role of local environment in the gas-phase metallicity of galaxies. The local environment measurements are derived from accurate and uniformly calculated photometric redshifts with well-calibrated probability distributions. Based on rest-frame optical emission lines, [NII]$lambda6584$ and H$alpha$, we measure gas-phase oxygen abundance of 167 galaxies at $1.37leq zleq1.7$ and 303 galaxies at $2.09leq zleq2.61$, located in diverse environments. We find that at $zsim1.5$, the average metallicity of galaxies in overdensities with $M_*sim10^{9.8}M_odot, 10^{10.2}M_odot$ and $10^{10.8}M_odot$ is higher relative to their field counterparts by $0.094pm0.051$, $0.068pm0.028$ and $0.052pm0.043$ dex, respectively. However, this metallicity enhancement does not exist at higher redshift, $zsim2.3$, where, compared to the field galaxies, we find $0.056pm0.043$, $0.056pm0.028$ and $0.096pm 0.034$ dex lower metallicity for galaxies in overdense environments with $M_*sim10^{9.8}M_odot, 10^{10.2}M_odot$ and $10^{10.7}M_odot$, respectively. Our results suggest that, at $1.37leq zleq2.61$, the variation of mass-metallicity relation with local environment is small ($<0.1$dex), and reverses at $zsim2$. Our results support the hypothesis that, at the early stages of cluster formation, owing to efficient gas cooling, galaxies residing in overdensities host a higher fraction of pristine gas with prominent primordial gas accretion, which lowers their gas-phase metallicity compared to their coeval field galaxies. However, as the Universe evolves to lower redshifts ($zlesssim2$), shock-heated gas in overdensities cannot cool down efficiently, and galaxies become metal-rich rapidly due to the suppression of pristine gas inflow and re-accretion of metal-enriched outflows in overdensities.
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