In a four-dimensional Schwarzschild–de Sitter background, the spherically symmetric ($\ell =0$) contribution to the Hadamard two-point correlation function is computed for a massless minimally coupled scalar field in the Unruh state. Consideration is given to spacetime points located between the black hole and cosmological horizons. Previously it was found in two dimensions at late times for spatially separated points that the Hadamard function exhibits unbounded linear growth in time, with a rate of growth proportional to the sum of the black hole and cosmological surface gravities. Here it is shown numerically that this instability persists in four dimensions, but with a modification of the two-dimensional result due to scattering effects associated with the scalar field modes. An analytic approximation is derived for the growth rate in four dimensions and, in the limit that the black hole vanishes, is found to be equivalent to the rate of growth for the Hadamard function found previously for de Sitter space in cosmological coordinates.

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• Cosmology
• Quantum aspects of black holes
• Quantum fields in curved spacetime
• Differential equations