Effective Field Theory of Black Hole Perturbations with Timelike Scalar Profile: Formulation

Understand

We formulate the Effective Field Theory (EFT) of perturbations within scalar-tensor theories on an inhomogeneous background.

• The EFT is constructed while keeping a background of a scalar field to be $\textit{timelike}$, which spontaneously breaks the time diffeomorphism.
• We find a set of consistency relations that are imposed by the invariance of the EFT under the 3d spatial diffeomorphism.
• This EFT can be generically applied to any inhomogeneous background metric as long as the scalar profile is everywhere timelike.