No-Go theorem in the cubic subclass of Horndeski theory for spherically symmetric dynamical background

We consider a general dynamical, spherically symmetric background in the cubic subclass of Horndeski theory and obtain the quadratic action for the perturbations. We analyze the stability conditions for high-energy modes and study the issue of the no-go theorem. We formulate the no-go theorem for weak dependence on one variable (time or radial) and derive its generalization to the cases which could be reduced by coordinate transformation to scenarios where the scalar field has weak dependence on one of the coordinates. Moreover, we show that a wide class of singular solutions is also prohibited within the cubic subclass of Horndeski theory.