  We study the evolution of cosmological perturbations in dark-matter models
with elastic and velocity-independent self interactions. Such interactions are
imprinted in the matter-power spectrum as dark acoustic oscillations, which can
be experimentally explored to determine the strength of the self scatterings.
Models with self interactions have similarities to warm dark matter, as they
lead to suppression of power on small scales when the dark-matter velocity
dispersion is sizable. Nonetheless, both the physical origin and the extent of
the suppression differ for self-interacting dark matter from conventional warm
dark matter, with a dark sound horizon controlling the reduction of power in
the former case, and a free-streaming length in the latter. We thoroughly
analyze these differences by performing computations of the linear power
spectrum using a newly developed Boltzmann code. We find that while current
Lyman-$\alpha$ data disfavor conventional warm dark matter with a mass less
than 5.3 keV, when self interactions are included at their maximal value
consistent with bounds from the Bullet Cluster, the limits are relaxed to 4.4
keV. Finally, we make use of our analysis to set novel bounds on light scalar
singlet dark matter.
