  Coupling dark matter to light new particles is an attractive way to combine
thermal production with strong velocity-dependent self-interactions. Here we
point out that in such models the dark matter annihilation rate is generically
enhanced by the Sommerfeld effect, and we derive the resulting constraints from
the Cosmic Microwave Background and other indirect detection probes. For the
frequently studied case of s-wave annihilation these constraints exclude the
entire parameter space where the self-interactions are large enough to address
the small-scale problems of structure formation.
