An introduction to the Vainshtein mechanism

We introduce the Vainshtein mechanism which plays a crucial role in massive gravities, as well as in related theories such as Galileons and their extensions. This mechanism, also known as k-mouflage, allows to hide via nonlinear effects—typically for source distances smaller than a so-called Vainshtein radius which depends on the source and on the theory considered—some degrees of freedom whose effects are then only left important at large distances, e.g. for cosmology. It is introduced here in nonlinear Fierz–Pauli theories (massive gravities), including the dRGT theories, in their decoupling limits, as well as in other models such as the DGP model or generalized Galileons. This presentation is self-contained, and before discussing the Vainshtein mechanism we introduce some useful results and concepts concerning massive gravity, such as the vDVZ discontinuity, the decoupling limits or the Boulware–Deser ghost. (paper)