Gravitational wave double copy of radiation from gluon shockwave collisions

In a recent paper, we showed that the gravitational wave spectrum from trans-Planckian shockwave scattering in Einstein gravity is determined by the gravitational Lipatov vertex expressed as the bilinear double copy $\Gamma^{\mu\nu} = \frac12 C^\mu C^\nu - \frac12 N^\mu N^\nu$ where $C^\mu$ is the QCD Lipatov vertex and $N^\mu$ is the QED soft photon factor. We show here that this result can be directly obtained by careful application of the classical color-kinematic duality to the spectrum obtained in gluon shockwave collisions.