Effective Description of Higher-Order Scalar-Tensor Theories
  Most existing theories of dark energy and/or modified gravity, involving a
scalar degree of freedom, can be conveniently described within the framework of
the Effective Theory of Dark Energy, based on the unitary gauge where the
scalar field is uniform. We extend this effective approach by allowing the
Lagrangian in unitary gauge to depend on the time derivative of the lapse
function. Although this dependence generically signals the presence of an extra
scalar degree of freedom, theories that contain only one propagating scalar
degree of freedom, in addition to the usual tensor modes, can be constructed by
requiring the initial Lagrangian to be degenerate. Starting from a general
quadratic action, we derive the dispersion relations for the linear
perturbations around Minkowski and a cosmological background. Our analysis
directly applies to the recently introduced Degenerate Higher-Order
Scalar-Tensor (DHOST) theories. For these theories, we find that one cannot
recover a Poisson-like equation in the static linear regime except for the
subclass that includes the Horndeski and so-called "beyond Horndeski" theories.
We also discuss Lorentz-breaking models inspired by Horava gravity.
