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We have developed a technique of simulating Ly$\alpha$ absorption in QSO spectra in a linear approximation regime, in which the collapse of baryonic matter is treated by a Press-Schecter-like criterion. 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. Within a reasonable range of $J_\nu$, the UV background radiation at high redshift, and $\delta_{th}$, the linear threshold of the onset of gravitational collapse of the baryonic matter, the LCDM model is generally in agreement with observational data in all above-mentioned aspects. The model of SCDM seems to show some lack of Ly$\alpha$ lines at redshifts less than 2.5, but it can survive if we take a smaller $J_\nu$ and a higher $\delta_{th}$. This suggests that whether a significant part of the Ly$\alpha$ forest lines is given by halos of galaxies or galaxy-like objects would be crucial to the success of SCDM. The $``$standard" CHDM model, i.e. 60\% cold and 30\% hot dark matters and 10\% baryons, is found to be unable to survive under the Ly$\alpha$ forest tests, because it produces structures too late and favors to form structures on large scales instead of small scale objects like Ly$\alpha$ clouds.
