  Recent studies have highlighted that galaxies at z = 6-8 fall short of
producing enough ionizing photons to reionize the IGM, and suggest that
Population III stars could resolve this tension, because their harder spectra
can produce ~10x more ionizing photons than Population II. But this argument
depends critically on the duration of the Population III era, and because
Population III stars form from pristine gas, in turn depends on the rate of
galactic enrichment. We use a semi-analytic model of galaxy formation which
tracks galactic chemical evolution, to gauge the impact of Population III stars
on reionization. Population III SNe produce distinct metal abundances, and we
argue that the duration of the Population III era can be constrained by precise
relative abundance measurements in high-z damped Ly{\alpha} absorbers (DLAs),
which provide a chemical record of past star-formation. We find that a single
generation of Population III stars can self-enrich galaxies above the critical
metallicity Zcrit=10^-4 Zsun for the Population III-to-II transition, on a very
short timescale of ~10^6 yr, owing to the large metal yields and short
lifetimes of Population III stars. This subsequently terminates the Population
III era, hence they contribute >~ 50% of the ionizing photons only for z >~ 30,
and at z=10 contribute <1%. The Population III contribution can be increased by
delaying metal mixing into the ISM. However comparing the resulting metal
abundance pattern to existing measurements in z <~ 6 DLAs, we show that the
fractional contribution of high-mass Population III stars to the ionization
rate must be <~ 10% at z = 10. Future abundance measurements of z~7-8 QSOs and
GRBs should probe the era when the chemical vestiges of Population III star
formation become detectable.
