Charged scalar field instability between the event and cosmological horizons

Recently, a new interesting instability of a charged scalar field in the Reissner-Nordstr\"om-de Sitter background has been found (arXiv:1405.4931v2) through the time-domain integration of the perturbation equation. We investigate further properties of this instability, confirm its existence by concordant frequency-domain and time-domain calculations and show that it occurs at however small value of the coupling eQ, where e and Q are charges of a scalar field and black hole respectively. We also investigate the parametric region of instability and show that the critical values of eQ at which the stabilization happens strongly depends on the value of cosmological constant $\Lambda$ and softly on Q. We show that all the unstable modes are superradiant, but not all the superradiant modes are unstable. We analytically prove that superradiance is necessary (but not sufficient) condition for the instability in this case and, thereby, demonstrate the superradiant origin of the instability.