"Shadowy" modes in Higher-Order Scalar-Tensor theories

Understand

We consider Higher-Order Scalar-Tensor theories which appear degenerate when restricted to the unitary gauge but are not degenerate in an arbitrary gauge.

• We dub them U-degenerate theories.
• We provide a full classification of theories that are either DHOST or U-degenerate and that are quadratic in second derivatives of the scalar field, and discuss its extension to cubic and higher order theories.
• Working with a simple example of U-degenerate theory, we find that, for configurations in which the scalar field gradient is time-like, the apparent extra mode in such a theory can be understood as a generalized instantaneous, or "shadowy" mode, which does not propagate.