Any viable cosmological model must produce enough structure at early e
pochs to explain the amount of gas associated with high-redshift dampe
d Ly alpha systems. We study the evolution of damped Ly alpha systems
at z greater than or equal to 2 in cold dark matter (CDM) and in cold
plus hot dark matter (CDM+HDM) models using both N-body and hydrodynam
ic simulations, Our approach incorporates the effects of gas dynamics,
and we find that all earlier estimates that assumed that all the bary
ons in dark matter halos would contribute to damped Ly alpha absorptio
n have overestimated the column density distribution, f(N), and the fr
action of neutral dense gas, Omega(g), in damped Ly alpha systems. The
differences are driven by ionization of hydrogen in the outskirts of
galactic halos and by gaseous dissipation near the halo centers, and t
hey tend to exacerbate the problem of late galaxy formation in CDM+HDM
models. We include systems only up to the highest observed column den
sity N similar to 10(21.8) cm(-2) in the estimation of Omega(g) for a
fair comparison with data. If the observed f(N) and Omega(g) inferred
from a small number of confirmed and candidate absorbers are robust, t
he amount of gas in damped Ly alpha systems at high redshifts in the O
mega(v) = 0.2 CDM+HDM model falls well below the observations.