Geometric origin of pair production by electric field in de sitter space

Particle production in de Sitter space provides an interesting model for understanding the curvature effect on Schwinger pair production by a constant electric field or the Schwinger mechanism on de Sitter radiation. For that purpose, we employ the recently introduced complex analysis method in which the quantum evolution in complex time explains pair production via the geometric transition amplitude and gives the pair-production rate as a contour integral. We compare the result from the contour integral with that of the phase-integral method.