Rainbow scattering in the gravitational field of a compact object

We study the elastic scattering of a planar wave in the curved spacetime of a compact object such as a neutron star, via a heuristic model: a scalar field impinging upon a spherically-symmetric uniform density star of radius $R$ and mass $M$. For $R < r_c$, there is a divergence in the deflection function at the light-ring radius $r_c = 3GM/c^2$, which leads to spiral scattering (orbiting) and a backward glory; whereas for $R > r_c$ there instead arises a stationary point in the deflection function which creates a caustic and rainbow scattering. As in nuclear rainbow scattering, there is an Airy-type oscillation on a Rutherford-like cross section, followed by a shadow zone. We show that, for $R \sim 3.5 GM/c^2$, the rainbow angle lies close to $180^\circ$, and thus there arises enhanced back-scattering and glory. We explore possible implications for gravitational wave astronomy, and dark matter models.