  In previous work we argued that the correct procedure to predict the
gravitational wave signal from preheating after inflation rests on the in-in
formalism. We extend here our previous analysis to include the backreaction of
the produced matter on the motion of the inflaton and the expansion of the
universe, and study how the latter affect the spectrum of the resulting
gravitational waves. The addition of backreaction demands the appropriate
renormalization of divergent expectation values, which we regularize by
preserving diffeomorphism invariance in a manner that is amenable to numerical
integration. The very same calculation also allows us to determine for which
strength of the inflaton to matter couplings reheating is successful. We
illustrate our results with the scalar version of the Starobinsky model of
inflation, and observe that it is hard to reheat the universe while keeping
radiative corrections under control.
