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

Frederick B. Davies, Sarah E. I. Bosman+1 more|arXiv (Cornell University)|Jun 26, 2024

Solid-state spectroscopy and crystallographyMaterials Science3 citations

TL;DR

This paper derives an observational estimate of the clumping factor $ C \approx 12 $ at $ z \approx 5-6 $ using the measured mean free path of ionizing photons and ionization equilibrium in the intergalactic medium, challenging the conventional assumption of $ C = 3 $ in reionization models. The higher clumping factor implies that high-redshift galaxies must produce roughly twice as many ionizing photons per baryon to complete reionization, reconciling the ionizing photon budget with recent JWST observations of early galaxies.

ABSTRACT

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

Motivation & Objective

• To reconcile the observed short mean free path of ionizing photons at $ z \approx 6 $ with the reionization history.
• To re-evaluate the clumping factor $ C $, traditionally assumed to be $ \sim 3 $, using observational constraints rather than simulations.
• To assess the implications of a higher $ C $ for the ionizing photon budget required to complete reionization by $ z \lesssim 6 $.
• To investigate whether the discrepancy between observed ionizing output from high-redshift galaxies and reionization constraints can be resolved by a revised $ C $.

Proposed method

• Derives the clumping factor $ C $ by linking the observed mean free path of ionizing photons to the recombination rate in the ionized IGM via ionization equilibrium.
• Uses the relation $ t_{\text{rec}} = 1 / (C \langle n_{\text{H}} \rangle \alpha_{\text{HII}}) $, where $ \alpha_{\text{HII}} $ is the hydrogen recombination rate, to connect $ C $ to the mean free path and photoionization rate.
• Applies the formalism to observational data from stacked quasar spectra at $ z \approx 6 $, incorporating measurements of the mean free path and hydrogen photoionization rate.
• Considers the ionization equilibrium condition $ dQ/dt = 0 $ at $ Q = 1 $ to derive the critical ionizing emissivity and infer $ C $.
• Compares the derived $ C \approx 12 $ with standard simulation-based estimates of $ C \approx 3 $, highlighting the role of dense gas in halos that is often excluded in simulations.
• Validates the result using data from $ 2 < z < 5 $, confirming consistency across redshifts.

Experimental results

Research questions

• RQ1 What is the observational value of the clumping factor $ C $ at the end of reionization, given the measured mean free path of ionizing photons?
• RQ2 How does a higher $ C $ affect the required ionizing photon budget to complete reionization by $ z \lesssim 6 $?
• RQ3 Why do standard simulations underestimate $ C $, and what physical processes are missed when excluding dense gas in halos?
• RQ4 Can the high ionizing photon output inferred from JWST observations be reconciled with reionization constraints if $ C \approx 12 $?
• RQ5 How does the clumping factor influence the critical ionizing emissivity needed to maintain reionization at different ionized fractions?

Key findings

• The observational estimate of the clumping factor at $ z \approx 5-6 $ is $ C \approx 12 $, significantly higher than the commonly assumed $ C = 3 $.
• This higher $ C $ implies that high-redshift galaxies must produce roughly twice as many ionizing photons per baryon—about 3 photons per baryon—to complete reionization by $ z \sim 6 $.
• The result is consistent across redshifts, as similar values of $ C \approx 12 $ are derived when applying the method to observations at $ 2 < z < 5 $.
• The elevated clumping factor resolves tension with JWST observations, which suggest a much greater ionizing photon output from early galaxies than previously thought.
• The study attributes the discrepancy with simulations to the exclusion of dense gas in galaxy halos, which still absorbs external ionizing photons and contributes to the total recombination rate.
• The critical ionizing emissivity required to maintain reionization at $ Q = 1 $ is consistent with the observed emissivity from UV luminosity functions when $ C = 12 $, especially with $ f_{\text{esc}} \xi_{\text{ion}} \approx 10^{24.8} $.