  This is the second in a series of papers investigating the formulation of the
simplified Dark Matter models with graviton mediators in cosmological
backgrounds. We address here the crucial problem of the fundamental observable
of interest, namely the graviton spectrum in a FRW cosmological background with
an arbitrary Dark Matter background component. We calculate the correction to
the free graviton two-point function up to the second order in the coupling
constant between the Dark Matter and the graviton in the simplified Dark Matter
model with graviton mediators approach in the de Sitter space. Our result is
model independent in the sense that it does not depend on the particular form
of the Dark Matter fields. Also, due to the universality of the interaction
between the Dark Matter and the graviton, the result obtained here apply to the
interaction between the baryonic matter and the gravitons. As an application,
we discuss in detail the massive scalar Dark Matter model and calculate the
first order correction to the two-point function due to two Dark Matter modes
in the adiabatic regime.
