  We present the cosmological constraints on the cross section of baryon-dark
matter interactions for the dark matter mass below the MeV scale from the
Planck CMB (cosmic microwave background) and SDSS (Sloan Digital Sky Survey)
Lyman-$\alpha$ forest data. To explore the dark matter mass $m_{\chi}\lesssim
1$ MeV for which the dark matter's free-streaming effect can suppress the
observable small scale density fluctuations, in addition to the acoustic
oscillation damping in existence of the baryon-dark matter coupling, we apply
the approximated treatment of dark matter free-streaming analogous to that of
the conventional warm dark matter. We also demonstrate the mass dependence of
the baryon-dark matter cross section bounds (for the dark matter mass down to
$m_{\chi} \sim 5~{\rm keV}$), in contrast to the dark matter mass independence
of the cross section constraints for the light dark matter below the MeV scale
claimed in the previous literature.
