A Linear Perturbation Theory of Inhomogeneous Reionization

We develop an analytic approach to study inhomogeneous reionization on large scales by solving the equations of ionization balance and radiative transfer to first order in perturbations. Given the spatial distribution and spectrum of the ionizing sources, our formalism can be used to predict the large scale power spectra of fluctuations in the abundances of HII, HI and radiation. Our approach avoids common approximations/assumptions in existing analytic methods -- for instance, we do not assume a specific ionization topology from the outset; nor do we make a step-function bubble-like approximation to the HII distribution. Applying our formalism to sources biased according to the Press-Schechter prescription, we find: 1. reionization always proceeds ``inside--out'', with dense regions more highly ionized, at least on large scales; 2. on sufficiently large scales, HII, HI and radiation exhibits a scale independent bias relative to dark matter; 3. the bias is suppressed on scales comparable to or smaller than the mean free path of the ionizing photons; 4. if the ionizing source spectrum is sufficiently soft, the HII bias closely tracks the source bias for most of the reionization process but drops precipitously after percolation; 5. if the ionizing source spectrum is sufficiently hard, the HII bias drops in a more steady fashion throughout the reionization process. The tools developed here will be useful for interpreting future 21 cm, CMB and Lyman-alpha forest observations, both to learn about the reionization astrophysics (such as the hardness of the source spectrum and therefore the nature of the ionizing sources) and to possibly extract interesting cosmological information.