  A new type of integral representation is proposed for the propagators of the
massive Klein-Gordon field minimally coupled to the gravity of the de Sitter
expanding universe. This representation encapsulates the effects of the
Heaviside step functions of the Feynman propagators making possible for the
first time the calculation of Feynman diagrams involving scalar particles on
this background. In order to convince that, a simple example is given outlining
the amplitudes in the second order of perturbations of the Compton effect in
the de Sitter scalar quantum electrodynamics.
