An analytical model of the large neutral regions during the late stage of reionization

In this paper, we investigate the nature and distribution of large neutral regions during the late epoch of reionization. In the 'bubble model' of reionization, the mass distribution of large ionized regions ('bubbles') during the early stage of reionization is obtained by using the excursion set model, where the ionization of a region corresponds to the first up-crossing of a barrier by random trajectories. We generalize this idea and develop a method to predict the distribution of large-scale neutral regions during the late stage of reionization, taking into account the ionizing background after the percolation of H II regions. The large-scale neutral regions, which we call 'neutral islands', are not individual galaxies or minihalos, but larger regions where fewer galaxies formed and hence ionized later and they are identified in the excursion set model with the first down-crossings of the island barrier. Assuming that the consumption rate of ionizing background photons is proportional to the surface area of the neutral islands, we obtained the size distribution of the neutral islands. We also take the 'bubbles-in-island' effect into account by considering the conditional probability of up-crossing a bubble barrier after down-crossing the island barrier. We find that this effect is very important. An additional barrier is set to avoid islands being percolated through. We find that there is a characteristic scale for the neutral islands, while the small islands are rapidly swallowed up by the ionizing background; this characteristic scale does not change much as the reionization proceeds.