Effective field theory of modified gravity on the spherically symmetric background: leading order dynamics and the odd-type perturbations

TL;DR

This paper develops a comprehensive effective field theory framework for analyzing perturbations in spherically symmetric modified gravity models, focusing on odd-type perturbations and stability conditions, applicable to Horndeski and beyond theories.

Contribution

It introduces a 2+1+1 canonical formalism for perturbations in spherically symmetric backgrounds, extending analysis to generic scalar-tensor theories beyond Horndeski.

Findings

Derived background equations of motion for static spherically symmetric spacetimes.

Established stability conditions to avoid ghosts and Laplacian instabilities.

Unified treatment encompassing Horndeski and beyond-Horndeski theories.

Abstract

We consider perturbations of a static and spherically symmetric background endowed with a metric tensor and a scalar field in the framework of the effective field theory of modified gravity. We employ the previously developed 2+1+1 canonical formalism of a double Arnowitt-Deser-Misner (ADM) decomposition of space-time, which singles out both time and radial directions. Our building block is a general gravitational action that depends on scalar quantities constructed from the 2+1+1 canonical variables and the lapse. Variation of the action up to first-order in perturbations gives rise to three independent background equations of motion, as expected from spherical symmetry. The dynamical equations of linear perturbations follow from the second-order Lagrangian after a suitable gauge fixing. We derive conditions for the avoidance of ghosts and Laplacian instabilities for the odd-type perturbations. We show that our results not only incorporates those derived in the most general scalar-tensor theories with second-order equations of motion (the Horndeski theories) but they can be applied to more generic theories beyond Horndeski.