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In this global study of solutions to the linear wave equation onSchwarzschild de Sitter spacetimes we attend to the cosmological region ofspacetime which is bounded in the past by cosmological horizons and to thefuture by a spacelike hypersurface at infinity. We prove an energy estimatecapturing the expansion of that region which combined with earlier results forthe static region yields a global boundedness result for linear waves. Itasserts that a general finite energy solution to the global initial valueproblem has a limit on the future boundary at infinity that can be viewed as afunction on the standard cylinder with finite energy, and that moreover anydecay along the cosmological horizon is inherited along the future boundary. Inparticular, we exhibit an explicit nonvanishing quantity on the future boundaryof the spacetime consistent with our expectations for the nonlinear stabilityproblem. Our results apply to a large class of expanding cosmologies near theSchwarzschild de Sitter geometry, in particular subextremal Kerr de Sitterspacetimes.

Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies