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Bridging Optical and Far-Infrared Emission-Line Diagrams of Galaxies from Local to the Epoch of Reionization: Characteristic High [O III] 88 $\mathrm{μm}$/SFR at $z > 6$

We present photoionization modeling of galaxy populations at $z\sim0$, $2$, and $> 6$ to bridge optical and far-infrared (FIR) emission-line diagrams. We collect galaxies with measurements of optical and/or FIR ([O III] 88 $\mathrm{μm}$ and [C II] 158 $\mathrm{μm}$) emission line fluxes and plot them on the [O III]$\lambda5007/\mathrm{Hβ}$--[N II]$\lambda6585/\mathrm{Hα}$ (BPT) and L([O III]88)/SFR--L([C II]158)/SFR diagrams, where SFR is the star-formation rate and L([O III]88) and L([C II]158) are the FIR line luminosities. We aim to explain the galaxy distributions on the two diagrams with photoionization models that employ three nebular parameters: the ionization parameter $U$, hydrogen density $n_\text{H}$, and gaseous metallicity $Z_\text{gas}$. Our models successfully reproduce the nebular parameters of local galaxies, and then predict the distributions of the $z\sim0$, $2$, and $> 6$ galaxies on the diagrams. The predicted distributions illustrate the redshift evolution on all the diagrams; e.g., [O III]$/\mathrm{Hβ}$ and [O III]88/[C II]158 ratios continuously decrease from $z > 6$ to $0$. Specifically, the $z > 6$ galaxies exhibit $\sim\!0.5$ dex higher $U$ than low-redshift galaxies at a given $Z_\text{gas}$ and show predicted flat distributions on the BPT diagram at $\log{\mathrm{[O III]/Hβ}} = 0.5$-$0.8$. We find that some of the $z > 6$ galaxies exhibit high L([O III]88)/SFR ratios. To explain these high ratios, our photoionization models require a low stellar-to-gaseous metallicity ratio or bursty/increasing star-formation history at $z > 6$. The James Webb Space Telescope will test the predictions and scenarios for the $z > 6$ galaxies proposed by our photoionization modeling.