  Damped Ly-alpha absorbers (DLAs) can be used to measure gas-phase
metallicities at large cosmological lookback times with high precision.
Relative abundances can still be measured accurately deep into the reionization
epoch (z > 6) using transitions redward of Ly-alpha. Here we study chemical
evolution of DLAs using a constrained model for evolution of galaxies and IGM
to determine the degree to which DLA abundance measurements can probe
Population III enrichment. We find that if the critical metallicity of
Population III to II transition is < 10^-4 Zsun, the cosmic Population III SFR
is zero for z<8. Nevertheless, at high redshift (z ~ 6) Population III chemical
signatures are retained in low-mass galaxies (halo mass < 10^9 Msun). This is
because photoionization feedback suppresses star formation in these galaxies
until relatively low redshift (z ~ 10), and the chemical record of early
Population III star formation is retained. We model DLAs as these galaxies by
assigning to them a mass-dependent H I absorption cross section and predict
distribution of DLA abundance ratios. We find that these distributions are
anchored towards abundance ratios set by Population II yields, but exhibit a
tail that depends on the Population III IMF for z > 5. Thus, a sample of DLA
abundance measurements at high redshift holds the promise to constrain
Population III IMF. A sample of just 10 DLAs with relative abundances measured
to an accuracy of 0.1 dex is sufficient to constrain the Population III IMF at
4-sigma. These constraints may prove stronger than other probes such as
metal-poor stars and individual DLAs. Our results provide a global picture of
the cosmic thermal, ionization, and chemical evolution, and can rule out
certain Population III scenarios.
