Quantum scalar field in the FRW Universe with a constant electromagnetic background

We discuss a massive scalar field with conformal coupling in the Friedmann-Robertson-Walker (FRW) Universe of a special type with a constant electromagnetic field. Treating an external gravitational-electromagnetic background exactly, at the first time the proper-time representations for out-in, in-in and out-out scalar Green functions are explicitly constructed as proper-time integrals over the corresponding (complex) contours. The vacuum-to-vacuum transition amplitudes and the number of created particles are found and vacuum instability is discussed. The mean values of the current and the energy-momentum tensor are evaluated, and different approximations for them are investigated. The back reaction of the particles created to the electromagnetic field is estimated in different regimes. The connection between the proper-time method and the effective action is outlined. The effective action in scalar QED in the weakly curved FRW Universe (de Sitter space) with a weak constant electromagnetic held is found as a derivative expansion over curvature and electromagnetic field strength. Possible further applications of the results are mentioned.