  The construction of amplitudes on curved space-times is a major challenge,
particularly when the background has non-constant curvature. We give formulae
for all tree-level graviton scattering amplitudes in curved self-dual radiative
space-times; these are chiral, source-free, asymptotically flat spaces
determined by free characteristic data at null infinity. Such space-times admit
an elegant description in terms of twistor theory, which provides the powerful
tools required to exploit their underlying integrability. The tree-level
S-matrix is written in terms of an integral over the moduli space of
holomorphic maps from the Riemann sphere to twistor space, with the degree of
the map corresponding to the helicity configuration of the external gravitons.
For the MHV sector, we derive the amplitude directly from the Einstein-Hilbert
action of general relativity, while other helicity configurations arise from a
natural family of generating functionals and pass several consistency checks.
The amplitudes in self-dual radiative space-times exhibit many novel features
that are absent in Minkowski space, including tail effects. There remain
residual integrals due to the functional degrees of freedom in the background
space-time, but our formulae have many fewer such integrals than would be
expected from space-time perturbation theory. In highly symmetric special
cases, such as self-dual plane waves, the number of residual integrals can be
further reduced, resulting in much simpler expressions for the scattering
amplitudes.
