  We investigate the relationships between classical observables in cosmology
and the classical limit of quantum scattering amplitudes. We first look at the
relation between Bogoliubov transformations and the notion of classical limit.
Then, we compute the cosmological redshift for a particle in a cosmological
background and the emitted gravitational waveform from a quantum field theory
basis and its classical limit. We observe that there is no interpretation for
the geodesic redshift purely in terms of on-shell amplitudes in flat space,
given that the classical limit of a scalar 2-point vanishes when considering an
FRW background with two asymptotically flat in and out regions. We resolve this
apparent conundrum and recover the correct observable by showing that the
action of Hermitian operators differ between the in and out regions, unlike
standard approaches in flat spacetime. We then show that radiation reaction
corrections to the redshift enter already at order $G$. Furthermore, we
demonstrate that the emitted waveform can be represented solely in terms of an
on-shell 3-point amplitude in flat space without energy conservation, providing
a closed formula for the waveform in an impulsive FRW.
