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

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 incorporate 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.