On the correlation between Ly$\alpha$ forest opacity and galaxy density in late reionization models

The relationship between Ly$\alpha$ forest opacity and local galaxy density (the opacity-density relation) is a key observational test of late reionization models. Using narrow-band surveys of z=5.7 Ly$\alpha$ emitters centered on quasar sight lines, Christenson et al. (2023) showed that two of the most transmissive forest segments at this redshift intersect galaxy underdensities. This is in tension with models of a strongly fluctuating ionizing background, including some late reionization models, which predict that the vast majority of these segments should intersect overdensities where the ionizing intensity is strongest. We use radiative transfer simulations to explore in detail the opacity-density relation in late reionization models. Fields like the one toward quasar PSO J359-06 -- the more underdense of the two transmissive sight lines in Christenson et al. (2023) -- typically contain recently reionized gas in cosmic voids where the hot temperatures and low densities enhance Ly$\alpha$ transmission. The opacity-density relation's transmissive end is sensitive to the amount of neutral gas in voids, and its morphology, set by the reionization source clustering. These effects are, however, degenerate. We demonstrate that models with very different source clustering can yield similar opacity-density relations when their reionization histories are calibrated to match Ly$\alpha$ forest mean flux measurements at z<6. In models with fixed source clustering, a lower neutral fraction increases the likelihood of intersecting hot, recently reionized gas in voids, increasing the likelihood of observing PSO J359-06. For instance, the probability of observing this field is 15% in a model with neutral fraction $x_{\rm HI}=5\%$ at z=5.7, three times more likely than in a model with $x_{\rm HI}=15\%$. The opacity-density relation may thus provide a complementary probe of reionization's end.