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 the first analytic model that includes both factors. At first, recombinations can be ignored and ionized bubbles grow primarily through major mergers, because at any given moment the bubbles have a well-defined characteristic size. 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 grow until recombinations balance ionizations, when their expansion abruptly halts. If the IGM density structure is similar to that at moderate redshifts, this limits the bubble radii to similar to 20 comoving Mpc; however, if the IGM is significantly clumpier at higher redshifts (because of minihalo formation, for example), the limit could be much smaller. Once a bubble reaches saturation, that region of the Universe has for all intents and purposes entered the 'post-overlap' stage. Because different H II regions saturate over a finite time interval, the overlap epoch actually has a finite width. Our model also predicts a mean recombination rate several times larger than expected for a uniformly illuminated IGM. 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.