Taxing the rich: recombinations and bubble growth during reionization

Reionization is inhomogeneous for two reasons: the clumpiness of the intergalactic medium (IGM) and clustering of the discrete ionizing sources. While numerical simulations can in principle take both into account, they are at present limited by small box sizes. On the other hand, analytic models have only examined the limiting cases of a clumpy IGM (with uniform ionizing emissivity) and clustered sources (embedded in a uniform IGM). Here, we present an analytic model for the evolving topology of reionization that includes both factors. At first, recombinations can be ignored and ionized bubbles grow primarily through major mergers. As a result, reionization resembles "punctuated equilibrium," with a series of well-separated sharp jumps in the ionizing background. These features are local effects and do not reflect similar jumps in the global ionized fraction. We then combine our bubble model with a simple description of recombinations in the IGM. We show that the bubbles stop growing when recombinations balance ionizations. If the IGM density structure is similar to that at moderate redshifts, this limits the bubble radii to ~20 comoving Mpc; however, it may be much smaller if the IGM is significantly clumpier at higher redshifts. Once a bubble reaches saturation, that region of the universe has for all intents and purposes entered the "post-overlap" stage, so the overlap epoch actually has a finite width. This picture naturally explains the substantial large-scale variation in Lyman-series opacity along the lines of sight to the known z>6 quasars. More quasar spectra will shed light on the transition between the "bubble-dominated" topology characteristic of reionization and the "web-dominated" topology characteristic of the later universe. Abridged