  We investigate the properties of the Lyman alpha forest as predicted by
numerical simulations for a range of currently viable cosmological models. This
is done in order to understand the dependencies of the forest on cosmological
parameters. Focusing on the redshift range from two to four, we show that: (1)
most of the evolution in the distributions of optical depth, flux and column
density can be understood by simple scaling relations, (2) the shape of optical
depth distribution is a sensitive probe of the amplitude of density
fluctuations on scales of a few hundred kpc, (3) the mean of the b distribution
(a measure of the width of the absorption lines) is also very sensitive to
fluctuations on these scales, and decreases as they increase. We perform a
preliminary comparison to observations, where available. A number of other
properties are also examined, including the evolution in the number of lines,
the two-point flux distribution and the HeII opacity.
