Telling tails in the presence of a cosmological constant

We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a nonzero cosmological constant. Considering test fields on both Schwarzschild–de Sitter and Reissner–Nordström–de Sitter backgrounds, we demonstrate the existence of exponentially decaying tails at late times. Interestingly, the $\mathcal{l}=0\mathrm{}$ mode asymptotes to a nonzero value, contrasting the asymptotically flat situation. We also compare these results, for $\mathcal{l}=0\mathrm{}$, with a numerical integration of the Einstein-scalar field equations, finding good agreement between the two. Finally, the significance of these results to the study of the Cauchy horizon stability in black-hole–de Sitter spacetimes is discussed.