  Perturbatively around flat space, the scattering amplitudes of gravity are
related to those of Yang-Mills by colour-kinematic duality, under which
gravitational amplitudes are obtained as the 'double copy' of the corresponding
gauge theory amplitudes. We consider the question of how to extend this
relationship to curved scattering backgrounds, focusing on certain 'sandwich'
plane waves. We calculate the 3-point amplitudes on these backgrounds and find
that a notion of double copy remains in the presence of background curvature:
graviton amplitudes on a gravitational plane wave are the double copy of gluon
amplitudes on a gauge field plane wave. This is non-trivial in that it requires
a non-local replacement rule for the background fields and the momenta and
polarization vectors of the fields scattering on the backgrounds. It must also
account for new 'tail' terms arising from scattering off the background. These
encode a memory effect in the scattering amplitudes, which naturally double
copies as well.
