Charged scalar field instability between the event and cosmological horizons

Abstract

Recently, a new interesting instability of a charged scalar field in the Reissner-Nordstr\"om-de Sitter background has been found [Z. Zhu et al. Phys. Rev. D 90, 044042 (2014)] 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 $\mathrm{\ensuremath{\Lambda}}$ 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 a necessary (but not sufficient) condition for the instability in this case and, thereby, demonstrate the superradiant origin of the instability.