Abstract
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. Appropriate boundary conditions, required by the elliptic nature of part of the equations of motion, lead to the elimination of the apparent instability associated with this extra mode.
Highlights
Scalar-tensor theories have always played a prominent role in providing alternative theories of gravity
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 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
Summary
Scalar-tensor theories have always played a prominent role in providing alternative theories of gravity. For quadratic theories (in second derivatives of the scalar field), we find that their Lagrangian L can be written as the sum of a totally U-degenerate Lagrangian, by which we mean a Lagrangian whose kinetic terms (for the scalar and tensor modes) vanish in the unitary gauge, and another term that does not involve the metric curvature and can be written in a simple way that makes the degeneracy in the unitary gauge manifest. Both terms of the Lagrangian correspond to DHOST Lagrangians separately, but their sum is not a DHOST Lagrangian.
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