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 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.
