  Recent observations have constrained the galaxy UV luminosity function up to
z~10. However, these observations alone allow for a wide range of reionization
scenarios due to uncertainties in the abundance of faint galaxies and the
escape fraction of ionizing photons. We show that requiring continuity with
post-reionization (z<6) measurements, where the Lya forest provides a complete
probe of the cosmological emissivity of ionizing photons, significantly reduces
the permitted parameter space. Models that are simultaneously consistent with
the measured UV luminosity function, the Thomson optical depth to the CMB, and
the Lya forest data require either: 1) extrapolation of the galaxy luminosity
function down to very faint UV magnitudes M_lim ~ -10, corresponding roughly to
the UV background suppression scale; 2) an increase of f_esc by a factor > ~10
from z=4 (where the best fit is 4%) to z=9; or 3) more likely, a hybrid
solution in which undetected galaxies contribute significantly and f_esc
increases more modestly. Models in which star formation is strongly suppressed
in low-mass, reionization-epoch haloes of mass up to ~10^10 M_sun (e.g., owing
to a metallicity dependence) are only allowed for extreme assumptions for the
evolution of f_esc. However, variants of such models in which the suppression
mass is reduced (e.g., assuming an earlier or higher metallicity floor) are in
better agreement with the data. Concordance scenarios satisfying the available
data predict a consistent redshift of 50% ionized fraction z_reion(50%) ~ 10.
On the other hand, the duration of reionization is sensitive to the relative
contribution of bright versus faint galaxies, with scenarios dominated by faint
galaxies predicting a more extended reionization event. Scenarios relying too
heavily on high-z dwarfs are disfavored by kinetic Sunyaev-Zeldovich
measurements, which prefer a short reionization history.
