  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.
