  We analyse the instantaneous transition rate of an accelerated Unruh-DeWitt
particle detector whose coupling to a quantum field on Minkowski space is
regularised by a finite spatial profile. We show, under mild technical
assumptions, that the zero size limit of the detector response is well defined,
independent of the choice of the profile function, and given by a manifestly
finite integral formula that no longer involves epsilon-regulators or limits.
Applications to specific trajectories are discussed, recovering in particular
the thermal result for uniform acceleration. Extensions of the model to de
Sitter space are also considered.
