Observing the tail of reionization: neutral islands in the $z=5.5$ Lyman-$\alpha$ Forest

Previous studies have noted difficulties in modeling the highest opacities of the $z>5.5$ Ly$\alpha$ forest, epitomized by the extreme Ly$\alpha$ trough observed towards quasar ULAS J0148+0600. One possibility is that the most opaque regions at these redshifts contain significant amounts of neutral hydrogen. This explanation, which abandons the common assumption that reionization ended before $z=6$, also reconciles evidence from independent observations of a significantly neutral Universe at $z=7.5$. Here we explore a model in which the neutral fraction is still $\approx 10\%$ at $z=5.5$. We confirm that this model can account for the observed scatter in Ly$\alpha$ forest opacities, as well as the observed Ly$\beta$ transmission in the J0148 trough. We contrast the model with a competing "earlier" reionization scenario characterized by a short mean free path and large fluctuations in the post-reionization ionizing background. We consider Ly$\alpha$ and Ly$\beta$ effective optical depths, their correlations, trough size distributions, dark pixel fractions, the IGM thermal history, and spatial distributions of Lyman-$\alpha$ emitters around forest sight lines. We find that the models are broadly similar in almost all of these statistics, suggesting that it may be difficult to distinguish between them definitively. We argue that improved constraints on the mean free path and the thermal history at $z>5$ could go a long way towards diagnosing the origin of the $z>5.5$ opacity fluctuations.