Constraints on Dark Matter interactions from structure formation: Damping lengths

(Shortened) Weakly Interacting Massive Particles are often said to be the best Dark Matter candidates. Studies have shown however that rather large Dark Matter-photon or Dark Matter-baryon interactions could be allowed by cosmology. Here we address the question of the role of the Dark Matter interactions in more detail to determine at which extent Dark Matter has to be necessarily weakly interacting. To this purpose, we compute the collisional damping (and free-streaming) lengths of generic interacting Dark Matter candidates and compare them to the scale of the smallest primordial structures known to exist in the Universe. We obtain necessary conditions that any candidate must satisfy. We point out the existence of new Dark Matter scenarios and exhibit new damping regimes. For example, an interacting candidate may bear a similar damping than that of collisionless Warm Dark Matter particles. The main difference is due to the Dark Matter coupling to interacting (or even freely-propagating) species. Our approach yields a general classification of Dark Matter candidates which extends the definitions of the usual Cold, Warm and Hot Dark Matter scenarios when interactions, weak or strong, are considered.