Keeping the Universe ionized: photoheating and the clumping factor of the high-redshift intergalactic medium

The critical star formation rate (SFR) density required to keep the intergalactic hydrogen ionized depends crucially on the average rate of recombinations in the intergalactic medium (IGM). This rate is proportional to the clumping factor C ≡ <ρ²_b>_IGM/<ρ_b>², where ρ_b and <ρ_b> are the local and cosmic mean baryon density, respectively, and the brackets <>_IGM indicate spatial averaging over the recombining gas in the IGM. We perform a suite of cosmological smoothed particle hydrodynamic simulations that include radiative cooling to calculate the volume-weighted clumping factor of the IGM at redshifts z >= 6. We focus on the effect of photoionization heating by a uniform ultraviolet background and find that photoheating strongly reduces the clumping factor because the increased pressure support smoothes out small-scale density fluctuations. Photoionization heating is often said to provide a negative feedback on the re-ionization of the IGM because it suppresses the cosmic SFR by boiling the gas out of low-mass haloes. However, because of the reduction of the clumping factor it also makes it easier to keep the IGM ionized. Photoheating therefore also provides a positive feedback which, while known to exist, has received much less attention. We demonstrate that this positive feedback is in fact very strong. Using conservative assumptions, we find that if the IGM was reheated at z >~ 9, the observed population of star-forming galaxies at z ~ 6 may be sufficient to keep the IGM ionized, provided that the fraction of ionizing photons that escape the star-forming regions to ionize the IGM is larger than ~0.2.