  We study the imprints of secluded dark sectors with a mass gap and
self-interactions on the matter power spectrum. When Dark Matter (DM) is
sufficiently light, in the ballpark of a few KeV, and self-interacting we find
qualitative difference with respect to $\Lambda$CDM and also to free streaming
DM. In order to emphasize the role of interactions for the evolution of the
primordial perturbations we discuss various regimes: ranging from the ideal
case of a tightly coupled perfect fluid to the free case of Warm Dark Matter,
including the realistic case of small but non-vanishing self-interactions. We
compute the matter power spectrum in all these regimes with the aid of
Boltzmann solvers. Light dark sectors with self-interactions are efficiently
constrained by Lyman-$\alpha$ data and we find that the presence of
self-interactions relaxes the bound on the DM mass. As a concrete realization
we study models with dark QCD-like sectors, where DM is made of light
dark-pions.
