Gravitational wave forms for extreme mass ratio collisions from supersymmetric gauge theories

We study the wave form emitted by a particle moving along an arbitrary (in general open) geodesic of the Schwarzschild geometry. The mathematical problem can be phrased in terms of quantities in $\mathcal{N}=2$ supersymmetric gauge theories that can be calculated by using localization and the AGT correspondence. In particular through this mapping, the post-Newtonian expansion of the wave form is expressed as a double instanton sum with rational coefficients that resums all tail contributions into Gamma functions and exponentials. The formulae we obtain are valid for generic values of the orbital quantum numbers $\mathrm{\ell }$ and $m$. For $\mathrm{\ell }=2,3$ we check explicitly that our results agree with the small mass ratio limit of the wave forms derived in the Multipole Post-Minkowskian and the amplitudes approaches. We show how the so-called tail and tail of tail contributions to the wave form arise in our approach. Finally, we derive a universal formula for the soft limit of the wave form that resums all logarithmic divergent terms of the form ${\omega }^{n-1}(\log \omega {)}^n$.