An introduction to the Vainshtein mechanism

TL;DR

The paper surveys the Vainshtein mechanism in massive gravity and closely related theories (e.g., DGP, dRGT, Galileons), starting from the vDVZ discontinuity of linear Fierz-Pauli theory and the BD ghost that afflicts nonlinear extensions. It explains how nonlinearities, captured in decoupling limits and St"uckelberg formalisms, generate the Vainshtein screening that restores General Relativity near sources, with explicit scaling relations for the Vainshtein radius $r_V$ in various models. The discussion extends from the DL to complete nonlinear theories, showing that GR-like behavior can emerge in realistic settings, though some potentials still face challenges such as global solution existence or black-hole behavior. The work also covers time-dependent and astrophysical/cosmological contexts, stressing the need for UV completions and highlighting open questions about the full dynamical robustness of the mechanism. Overall, the Vainshtein mechanism remains a cornerstone of phenomenologically viable modifications of gravity, enabling nonlinear hiding of extra degrees of freedom while preserving agreement with solar-system tests and cosmological observations.

Abstract

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 non linear 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 non linear Fierz-Pauli theories (massive gravities), including the dRGT theories, in their decoupling limits, as well as in other models such as 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.