  Our goal in this paper is to test some popular dark matter models by Ly-alpha
forest in QSO spectra. Recent observations of the size and velocity of Ly-alpha
forest clouds have indicated that the Ly-alpha absorption is probably not given
by collapsed objects, but pre-collapsed regions in the baryonic density field.
Therefore, a linear approximation description would be able to provide valuable
information. We developed a technique to simulate Ly-alpha forest as the
absorption of such pre-collapsed regions under linear approximation regime. The
simulated Ly-alpha forests in models of the standard cold dark matter (SCDM),
the cold plus hot dark matter (CHDM), and the low-density flat cold dark matter
(LCDM) have been confronted with observational features, including 1) the
number density of Ly-alpha lines and its dependencies on redshift and
equivalent width; 2) the distribution of equivalent widths and its redshift
dependence; 3) clustering; and 4) the Gunn-Peterson effect. The "standard" CHDM
model, i.e. 60% cold and 30% hot dark matters and 10\% baryons, is found to be
difficult to pass the Ly-alpha forest test, probably because it produces
structures too late and favors to form structures on large scales instead of
small scale objects like Ly-alpha clouds. Within a reasonable range of J_nu,
the UV background radiation at high redshift, and delta_th, the threshold of
the onset of gravitational collapse of the baryonic matter, the LCDM model is
consistent with observational data in all above-mentioned four aspects. The
model of SCDM can also fit with observation, but it requires a smaller J_nu and
a higher delta_th. This suggests that whether a significant part of the
Ly-alpha forest lines is located in the halos of collapsed objects would be
crucial to the success of SCDM.
