Scalar field fluctuations in Schwarzschild-de Sitter space-time

We calculate quantum fluctuations of a free scalar field in the Schwarzschild-de Sitter space-time, adopting the planar coordinates that is pertinent to the presence of a black hole in an inflationary universe. In a perturbation approach, doing expansion in powers of a small black hole event horizon compared to the de Sitter cosmological horizon, we obtain time evolution of the quantum fluctuations and then derive the scalar power spectrum.