Using the redshift evolution of the Lyman-$\alpha$ effective opacity as
  a probe of dark matter models
  Lyman-$\alpha$ forest data are known to be a good probe of the small scale
matter power. In this paper, we explore the redshift evolution of the
observable effective optical depth $\tau_{\rm eff} (z)$ from the Lyman-$\alpha$
data as a discriminator between dark matter models that differ from the
$\Lambda$CDM model on small scales. We consider the thermal warm dark matter
(WDM) and the ultra-light axion (ULA) models for the following set of
parameters: the mass of ULA, $m_a \simeq 10^{-24}\hbox{--}5 \times 10^{-22} \,
\rm eV$ and WDM mass, $m_{\rm wdm} = 0.1 \hbox{--} 4.6 \, \rm keV$. We simulate
the line-of-sight HI density and velocity fields using semi-analytic methods.
The simulated effective optical depth for the alternative dark matter models
diverges from the $\Lambda$CDM model for $z \gtrsim 3$, which provides a
meaningful probe of the matter power at small scales. Using likelihood
analysis, we compare the simulated data with the high-resolution Lyman-$\alpha$
forest data in the redshift range $2 < z < 4.2$. The analysis yields the
following 1$\sigma$ bounds on dark matter masses: $m_{\rm wdm} > 0.7\, {\rm
keV}$ and $m_{\rm a} > 2 \times 10^{-23} \, {\rm eV}$. To further test the
efficacy of our proposed method, we simulate synthetic data sets compatible
with the $\Lambda$CDM model in the redshift range $2 \leq z \leq 6.5$ and
compare with theory. The 1$\sigma$ bounds obtained are significantly tighter:
$m_{\rm wdm} > 1.5 \, {\rm keV}$ and $m_{\rm a} > 7 \times 10^{-23} \, {\rm
eV}$. Although our method provides an alternative way of constraining dark
matter models, we note that these bounds are weaker than those obtained by
high-resolution hydrodynamical simulations.
