Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories.

We study U-DHOST theories, i.e., higher-order scalar-tensor theories which are degenerate only in the unitary gauge and yield an apparently unstable extra mode in a generic coordinate system. We show that the extra mode satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface, and hence it does not propagate. We clarify how to treat this "shadowy" mode at both the linear and the nonlinear levels.