  One of the most dramatic signatures of the reionization era may be the
enormous ionized bubbles around luminous quasars (with radii reaching ~40
comoving Mpc), which may survive as "fossil'' ionized regions long after their
source shuts off. Here we study how the inhomogeneous intergalactic medium
(IGM) evolves inside such fossils. The average recombination rate declines
rapidly with time, and the brief quasar episode significantly increases the
mean free path inside the fossil bubbles. As a result, even a weak ionizing
background generated by galaxies inside the fossil can maintain it in a
relatively highly and uniformly ionized state. For example, galaxies that would
ionize 20-30% of hydrogen in a random patch of the IGM can maintain 80-90%
ionization inside the fossil, for a duration much longer than the average
recombination time in the IGM. Quasar fossils at z<10 thus retain their
identity for nearly a Hubble time, and will appear "gray,'' distinct from both
the average IGM (which has a "swiss-cheese" ionization topology and a lower
mean ionized fraction), and from fully-ionized bubbles around active quasars.
More distant fossils, at z>10 have a weaker galaxy-generated ionizing
background and a higher gas density, so they can attain a swiss-cheese topology
similar to the rest of the IGM, but with a smaller contrast between the ionized
bubbles and the partially neutral regions separating them. Analogous
HeIII-fossils should exist around the epoch of HeII/HeIII reionization at z~3,
although rapid recombinations inside the HeIII-fossils will be more common. Our
model of inhomogeneous recombination also applies to "double reionization''
models and shows that a non-monotonic reionization history is even more
unlikely than previously thought.
