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

TL;DR

This paper develops an effective field theory framework for analyzing perturbations in scalar-tensor theories with a timelike scalar field background, applicable to inhomogeneous spacetimes including black holes.

Contribution

It introduces a novel EFT formulation for scalar-tensor theories with a timelike scalar profile, establishing consistency relations and connecting to Horndeski theories.

Findings

Derived a set of invariance-imposed consistency relations.

Provided a dictionary linking EFT parameters to Horndeski theories.

Computed background equations for spherically symmetric black hole solutions.

Abstract

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. For completeness, we report a dictionary between our EFT parameters to those of Horndeski theories. Finally, we compute background equations for a class of spherically symmetric, static black hole backgrounds, including a stealth Schwarzschild-de Sitter solution.