Stability analysis of new solutions of the EYM system with a cosmological constant

We analyze the stability properties of the purely magnetic, static solutions to the Einstein-Yang-Mills equations with a cosmological constant. It is shown that all three classes of solutions found in a recent study are unstable under spherical perturbations. Specifically, we argue that the configurations have $n$ unstable modes in each parity sector, where $n$ is the number of nodes of the magnetic Yang-Mills amplitude of the background solution. The "sphaleronlike" instabilities (odd parity modes) decouple from the gravitational perturbations. They are obtained from a regular Schr\"odinger equation after a supersymmetric transformation. The body of the work is devoted to the fluctuations with even parity. The main difficulty arises because the Schwarzschild gauge, which is usually imposed to eliminate the gravitational perturbations from the Yang-Mills equation, is not regular for solutions with compact spatial topology. In order to overcome this problem, we derive a gauge-invariant formalism by virtue of which the unphysical (gauge) modes can be isolated.