  We propose a geometric interpretation for the Stokes phenomenon in de Sitter
spacetime that particles are produced in even dimensions but not in odd
dimensions. The scattering amplitude for a quantum field between the in-vacuum
and the transported one along a closed path in the complex-time plane gives the
particle-production rate that explains not only the Boltzmann factor from the
simple pole at infinity, corresponding to the cosmological horizon, but also
the sinusoidal behavior from simple poles at the north and south poles of the
Euclidean geometry. The Stokes phenomenon is a consequence of interference
among four independent closed paths in the complex plane.
