Cosmological simulations with gas provide a detailed description of th
e intergalactic medium, which makes possible predictions of neutral hy
drogen absorption in the spectra of background QSOs. We present result
s from a high-resolution calculation of an Omega = 1 cold dark matter
model. Our simulation reproduces many of the observed properties of th
e Ly alpha forest surprisingly well. The distribution of H I column de
nsities agrees with existing data to within a factor of similar to 2 o
ver most of the range from 10(14) cm(-2) to 10(22) cm 2; i.e., from un
saturated Ly alpha forest lines to damped Ly alpha systems. The equiva
lent width distribution matches the observed exponential form with a c
haracteristic width W-double dagger approximate to 0.3 Angstrom. The d
istribution of b-parameters appears consistent with that derived from
QSO spectra. Most of the low column density absorption arises in large
, flattened structures of moderate or even relatively low overdensity,
so there is no sharp distinction between the Ly alpha forest and the
''Gunn-Peterson'' absorption produced by the smooth intergalactic medi
um. Our results demonstrate that a Ly alpha forest like that observed
develops naturally in a hierarchical clustering scenario with a photoi
onizing background. Comparison between simulations and high-resolution
QSO spectra should open a new regime for testing theories of cosmic s
tructure formation.