Quantum Effects in 4D Schwarzschild-de Sitter Spacetime: Properties of the Hadamard Two-Point Function

Schwarzschild-de Sitter spacetime offers a classical background structure in which quantum fields and their correlations can be studied in a case where there is both a black hole and cosmological horizon. To investigate these correlations, the Hadamard two-point function which will be computed for a massless minimally coupled scalar field for the Unruh state in four dimensions. It was previously found in two dimensions that the Hadamard function grows linearly in time if the points are split in a spatial direction for coordinates that are natural near the black hole horizon. The growth is proportional to the sum of the black hole and cosmological horizon surface gravities. A main goal in extending the analysis to four dimensions is to determine if this growth persists, or is modified by scattering effects due to the existence of an effective potential in the mode equation. If this growth in the two-point function is indeed present in four dimensions, this could be indicative of an instability which might have interesting physical implications, for example modification of the power spectrum if primordial black holes existed during inflation.