In this talk I will present recent results for solutions to the linear wave equation on Schwarzschild de Sitter black hole spacetimes. We focus here on the expanding region of the spacetime, and exhibit a stability mechanism which manifests itself in a global redshift effect. I shall describe how this can be combined with earlier results to obtain a global linear stability result. If time permits, applications to Kerr de Sitter spacetimes will be discussed.The talk is based on the preprint: http://arxiv.org/abs/1207.6055