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, $m_a \simeq 10^{-24}\hbox{--}10^{-22} \, \rm eV$ 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-$\alpha$ cloud data in the redshift range $2 < z < 4.2$. The one-dimensional posterior probabilities of the relevant masses peak at $m_a \simeq 5 \times 10^{-23} \, \rm eV$ and $m_{\rm wdm} \simeq 1.1 \, \rm keV$. The posterior probabilities remain flat for larger masses, which shows that the data is compatible with the $\Lambda$CDM model.