  Recent results by the WMAP satellite suggest that the intergalactic medium
was significantly reionized at redshifts as high as z~17. At this early epoch,
the first ionizing sources likely appeared in the shallow potential wells of
mini-halos with virial temperatures T < 10^4 K. Once such an ionizing source
turns off, its surrounding HII region Compton cools and recombines.
Nonetheless, we show that the ``fossil'' HII regions left behind remain at high
adiabats, prohibiting gas accretion and cooling in subsequent generations of
mini-halos. Thus, early star formation is self-limiting. We quantify this
effect to show that star formation in mini-halos cannot account for the bulk of
the electron scattering opacity measured by WMAP, which must be due to more
massive objects. We argue that gas entropy, rather than IGM metallicity,
regulates the evolution of the global ionizing emissivity, and impedes full
reionization until lower redshifts. We discuss several important consequences
of this early entropy floor for reionization. It reduces gas clumping,
curtailing the required photon budget for reionization. An entropy floor also
prevents H2 formation and cooling, due to reduced gas densities: it greatly
enhances feedback from UV photodissociation of H2. An early X-ray background
would also furnish an entropy floor to the entire IGM; thus, X-rays impede
rather than enhance H2 formation. Future 21cm observations may probe the
topology of fossil HII regions.
