The Predicament of Absorption-dominated Reionization II: Observational Estimate of the Clumping Factor at the End of Reionization

The history of reionization reflects the cumulative injection of ionizing photons by sources and the absorption of ionizing photons by sinks. The latter process is traditionally described in terms of a "clumping factor" which encodes the average quadratic increase in the recombination rate of dense gas within the cosmic web. The recent measurement of a short mean free path of ionizing photons from stacked quasar spectra at $z\simeq6$ has placed the importance of sinks under increased scrutiny, but its connection to the recombination rate is not immediately obvious. Here we present analytic arguments to connect the clumping factor to the mean free path by invoking ionization equilibrium within the ionized phase of the intergalactic medium at the end of (and after) reionization. We find that the latest mean free path and hydrogen photoionization rate measurements at $z=5-6$ imply a global clumping factor $C\approx12$, much higher than previous determinations from radiation-hydrodynamic simulations of the reionization process. Similar values of $C$ are also derived when applying the same procedure to observations at $2<z<5$. Compared to the traditional assumption of $C=3$, high-redshift galaxies must produce roughly twice as many ionizing photons ($\approx3$ photons per baryon) to reionize the universe by $z\sim6$. This additional requirement on the ionizing photon budget may help to reconcile the reionization history with JWST observations that suggest a far greater output of ionizing photons by the most distant galaxy populations.