Constraints on the Evolution of the Ionizing Background and Ionizing Photon Mean Free Path at the End of Reionization

The variations in Ly$\alpha$ forest opacity observed at $z>5.3$ between lines of sight to different background quasars are too strong to be caused by fluctuations in the density field alone. The leading hypothesis for the cause of this excess variance is a late, ongoing reionization process at redshifts below six. Another model proposes strong ionizing background fluctuations coupled to a short, spatially varying mean free path of ionizing photons, without explicitly invoking incomplete reionization. With recent observations suggesting a short mean free path at $z\sim6$, and a dramatic improvement in $z>5$ Ly$\alpha$ forest data quality, we revisit this latter possibility. Here we apply the likelihood-free inference technique of approximate Bayesian computation to jointly constrain the hydrogen photoionization rate $\Gamma_{\rm HI}$ and the mean free path of ionizing photons $\lambda_{\rm mfp}$ from the effective optical depth distributions at $z=5.0$-$6.1$ from XQR-30. We find that the observations are well-described by fluctuating mean free path models with average mean free paths that are consistent with the steep trend implied by independent measurements at $z\sim5$-$6$, with a concomitant rapid evolution of the photoionization rate.