On the existence of conformally coupled scalar field hair for black holes in (anti-)de Sitter space

The Einstein-conformally coupled scalar field system is studied in the presence of a cosmological constant. We consider a massless or massive scalar field with no additional self-interaction, and spherically symmetric black hole geometries. When the cosmological constant is positive, no scalar hair can exist and the only solution is the Schwarzschild- de Sitter black hole. When the cosmological constant is negative, staple scalar field hair exists provided the mass of the scalar field is not too large.