  We discuss the production and evolution of cosmological gravitons showing how
the cosmological background affects their dynamics. Besides, the detection of
cosmological gravitons could constitute an extremely important signature to
discriminate among different cosmological models. Here we consider the cases of
scalar-tensor gravity and $f(R)$ gravity where it is demonstrated the
amplification of graviton amplitude changes if compared with General
Relativity. Possible observational constraints are discussed.
