Recoil velocity at second post-Newtonian order for spinning black hole binaries

Abstract

We compute the flux of linear momentum carried by gravitational waves emitted from spinning binary black holes at second post-Newtonian (2PN) order for generic orbits. In particular we provide explicit expressions of three new types of terms, namely, next-to-leading order spin-orbit terms at 1.5 post-Newtonian (1.5PN) order, spin-orbit tail terms at 2PN order, and spin-spin terms at 2PN order. Restricting ourselves to quasicircular orbits, we integrate the linear-momentum flux over time to obtain the recoil velocity as function of orbital frequency. We find that in the so-called superkick configuration the higher-order spin corrections can increase the recoil velocity up to a factor $\ensuremath{\sim}3$ with respect to the leading-order PN prediction. Whereas the recoil velocity computed in PN theory within the adiabatic approximation can accurately describe the early inspiral phase, we find that its fast increase during the late inspiral and plunge, and the arbitrariness in determining until when it should be trusted, makes the PN predictions for the total recoil not very accurate and robust. Nevertheless, the linear-momentum flux at higher PN orders can be employed to build more reliable resummed expressions aimed at capturing the nonperturbative effects until merger. Furthermore, we provide expressions valid for generic orbits, and accurate at 2PN order, for the energy and angular momentum carried by gravitational waves emitted from spinning binary black holes. Specializing to quasicircular orbits we compute the spin-spin terms at 2PN order in the expression for the evolution of the orbital frequency and found agreement with Mik\'oczi, Vas\'uth, and Gergely. We also verified that in the limit of extreme mass ratio our expressions for the energy and angular momentum fluxes match the ones of Tagoshi, Shibata, Tanaka, and Sasaki obtained in the context of black hole perturbation theory.