Effective source for second-order self-force calculations: Quasicircular orbits in Schwarzschild spacetime
Phys. Rev. D 113, 064013 – Published 4 March, 2026
DOI: https://doi.org/10.1103/f9j2-k64r
Abstract
Recent years have seen the first production of “postadiabatic” gravitational waveform models based on second-order gravitational self-force theory. These models rely on calculations of an effective source in the perturbative second-order Einstein equation. Here, for the first time, we detail the calculation of the effective source in a Schwarzschild background, which underlies the second-order self-force results in [N. Warburton et al., Gravitational-wave energy flux for compact binaries through second order in the mass ratio, Phys. Rev. Lett. 127, 151102 (2021); S. Isoyama et al., Adiabatic waveforms from extreme-mass-ratio inspirals: An analytical approach, Phys. Rev. Lett. 128, 231101 (2022); B. Wardell et al., Gravitational waveforms for compact binaries from second-order self-force theory, Phys. Rev. Lett. 130, 241402 (2023)]. The source is designed for use in the multiscale form of the Lorenz-gauge Einstein equation, decomposed in tensor spherical harmonics, or in the analogous second-order Teukolsky equation. It involves, among other things, contributions from (i) quadratic coupling of first-order field modes, (ii) the slow evolution of first-order fields, (iii) quadratic products of a first-order puncture field, and (iv) the second-order puncture field. We validate each of these pieces through numerical and analytical tests.
