Gravitational Bremsstrahlung from Reverse Unitarity

We compute the total radiated momentum carried by gravitational waves during scattering of two spinless black holes at lowest order in Newton's constant, $\mathcal{O}({G}^{3})$, and all orders velocity. By analytic continuation into bound state regime, we obtain $\mathcal{O}({G}^{3})$ energy loss elliptic orbits. This provides an essential step toward complete understanding third-post-Minkowskian binary dynamics. employ formalism Kosower, Maybee, O'Connell (KMOC), which relates classical observables to quantum amplitudes, derive relevant integrands using generalized unitarity. The subsequent phase-space integrations are performed via reverse unitarity method familiar from collider physics, differential equations exact velocity dependence near-static boundary conditions.