An analytic model for [O III] fine structure emission from high redshift galaxies

Recent ALMA measurements have revealed bright [O III] 88 micron line emission from galaxies during the Epoch of Reionization (EoR) at redshifts as large as z ~ 9. We introduce an analytic model to help interpret these and other upcoming [O III] 88 micron measurements. Our approach sums over the emission from discrete Str$\ddot{\mathrm{o}}$mgren spheres and considers the total volume of ionized hydrogen in a galaxy of a given star-formation rate. We estimate the relative volume of doubly ionized oxygen and ionized hydrogen and its dependence on the spectrum of ionizing photons. We then calculate the level populations of O III ions in different fine-structure states for H II regions of specified parameters. In this simple model, a galaxy's [O III] 88 μm luminosity is determined by the typical number density of free electrons in H II regions (n_e), the average metallicity of these regions (Z), the rate of hydrogen ionizing photons emitted (Q_{H I}), and the shape of the ionizing spectrum. We cross-check our model by comparing it with detailed CLOUDY calculations, and find that it works to better than 15${{\ \rm per\ cent}}$ accuracy across a broad range of parameter space. Applying our model to existing ALMA data at z ~ 6-9, we derive lower bounds on the gas metallicity and upper bounds on the gas density in the H II regions of these galaxies. These limits vary considerably from galaxy to galaxy, with the tightest bounds indicating Z ≳ 0.5Z_⊙ and n_H ≲ 50 cm^-3 at 2 - σ confidence.