  We develop and test with cosmological simulations a physically motivated
theory for how the interplay between gravity, pressure, cooling, and
self-shielding set the redshift--dependent mass scale at which halos can
accrete intergalactic gas. This theory provides a physical explanation for the
halo mass scale that can accrete unshocked intergalactic gas, which has been
explained with ad hoc criteria tuned to reproduce the results of a few
simulations. Furthermore, it provides an intuitive explanation for how this
mass scale depends on the reionization redshift, the amplitude of the ionizing
background, and the redshift. We show that accretion is inhibited onto more
massive halos than had been thought because previous studies had focused on the
gas fraction of halos rather than the instantaneous mass that can accrete gas.
A halo as massive as 10^11 Msun cannot accrete intergalactic gas at z=0, even
though typically its progenitors were able to accrete gas at higher redshifts.
We describe a simple algorithm that can be implemented in semi-analytic models,
and we compare the predictions of this algorithm to numerical simulations.
