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. 
Ryotaro Kase, L\'aszl\'o \'Arp\'ad Gergely and Shinji Tsujikawa
hep-th astro-ph.CO gr-qc hep-ph