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

Abstract

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 N=2 supersymmetric gauge theories that can be calculated by using localization and the Alday-Gaiotto-Tachikawa 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 formulas we obtain are valid for generic values of the orbital quantum numbers ℓ and m. For ℓ=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 tails and tails-of-tails 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 terms of the form ωn−1(logω)n. Published by the American Physical Society 2025