Constraints on the mean free path of ionising photons at $z\sim6$ using limits on individual free paths

The recent measurement of an ionising mean free path $\lambda_{\text{mfp}}<1$ pMpc at $z=6$ challenges our understanding of the small-scale structure of the intergalactic medium (IGM) at the end of reionisation. We introduce a new method to constrain \mfp at $z=6$ by using lower limits on the individual free paths of ionisation around quasars. Lyman-limit absorbers with a density sufficient to halt ionising photons produce strong absorption in the 6 lowest-energy Lyman transitions, in the absence of which a robust lower limit can be placed on the individual free path. Applying this method to a set of $26$ quasars at $5.5<z<6.5$, we find that $80\%$ of bright quasars ($M_{1450}<-26.5$) require individual free paths larger than $2$ pMpc. We model the relation between opacity $\kappa$ and photo-ionisation rate $\Gamma$ via the parameter $\xi$ such that $\kappa\propto\Gamma^{-\xi}$, and pose joint limits on \mfp and $\xi$. For the nominal value of $\xi=2/3$, we constrain $\lambda_{\text{mfp}}>0.31 \ (0.18)$ pMpc at $2\sigma \ (3\sigma)$: a much tighter lower bound than obtained through traditional stacking methods. Our constraints get significantly stronger for lower values of $\xi$. New constraints on \mfp and $\xi$ are crucial to our understanding of the reionisation-era IGM.