On the response of non-linearly coupled, accelerated detectors in odd-dimensional flat spacetimes

In this note, we consider the response of a uniformly accelerated monopole detector that is coupled non-linearly to the nth power of a quantum scalar field in (D+1)-dimensional flat spacetime. We show that, when (D+1) is even, the response of the detector in the Minkowski vacuum is characterized by a Bose-Einstein factor for all n. Whereas, when (D+1) is odd, we find that a Fermi-Dirac factor appears in the detector response when n is odd, but a Bose-Einstein factor arises when n is even.