Monopole stability of time-dependent, spherically symmetric backgrounds in Horndeski theory

Abstract

We study linear perturbations about an arbitrary time-dependent, spherically symmetric background in the most general Horndeski theory. The odd-parity sector of such perturbations, analyzed previously, contains no scalar field perturbation and, hence, leaves the related degree of freedom unconstrained. In this paper, we address this issue by analyzing even-parity monopole ($\ell = 0$) perturbations, which isolate this degree of freedom. We derive the quadratic master action for the only propagating physical mode associated with the scalar field and formulate stability criteria which ensure the absence of ghost and radial gradient instabilities for the high-energy modes. These conditions are necessary for the stability of the background and complement the odd-parity ones. We show that they reproduce the known results in the static and cosmological limits. We also demonstrate that, in the cubic subclass of Horndeski theory on a dynamical background, the monopole criteria impose no constraints beyond those previously obtained for modes with large multipoles ($\ell \gg 1$). As an illustration, we apply the derived stability conditions for the monopole mode to a non-minimally coupled scalar field theory and to a black hole model with linearly time-dependent scalar hair; in the latter case, the monopole conditions further restrict the parameter space allowed by odd-parity stability.