Constraining Dark Matter -- Dark Radiation interactions with CMB, BAO,
  and Lyman-$\alpha$
  Several interesting Dark Matter (DM) models invoke a dark sector leading to
two types of relic particles, possibly interacting with each other:
non-relativistic DM, and relativistic Dark Radiation (DR). These models have
interesting consequences for cosmological observables, and could in principle
solve problems like the small-scale cold DM crisis, Hubble tension, and/or low
$\sigma_8$ value. Their cosmological behaviour is captured by the ETHOS
parametrisation, which includes a DR-DM scattering rate scaling like a
power-law of the temperature, $T^n$. Scenarios with $n=0$, $2$, or $4$ can
easily be realised in concrete dark sector set-ups. Here we update constraints
on these three scenarios using recent CMB, BAO, and high-resolution
Lyman-$\alpha$ data. We introduce a new Lyman-$\alpha$ likelihood that is
applicable to a wide range of cosmological models with a suppression of the
matter power spectrum on small scales. For $n=2$ and $4$, we find that
Lyman-$\alpha$ data strengthen the CMB+BAO bounds on the DM-DR interaction rate
by many orders of magnitude. However, models offering a possible solution to
the missing satellite problem are still compatible with our new bounds. For
$n=0$, high-resolution Lyman-$\alpha$ data bring no stronger constraints on the
interaction rate than CMB+BAO data, except for extremely small values of the DR
density. Using CMB+BAO data and a theory-motivated prior on the minimal density
of DR, we find that the $n=0$ model can reduce the Hubble tension from
$4.1\sigma$ to $2.7\sigma$, while simultaneously accommodating smaller values
of the $\sigma_8$ and $S_8$ parameters hinted by cosmic shear data.
