Static de-Sitter Black Holes Abhor Charged Scalar Hair

We prove a no charged scalar hair theorem for static black holes in de-Sitter spacetime in the region between the event horizon and the cosmological horizon. The proof does not depend on the assumption of spherical symmetry. It allows for general non-minimal coupling functions of the scalar field to gravity and electromagnetic fields, and for higher curvature term corrections to Einstein gravity. The extension to other asympitotic spacetimes is applicable by requiring appropriate boundary conditions. Our result excludes the possibility for spontaneous scalarization of charged scalar around static charged de-Sitter black holes.Comment: 8 pages, 0 figure