Comparison of 4.5PN and 2SF gravitational energy fluxes from quasicircular compact binaries

Recent years have seen significant advances in models of gravitational waveforms emitted by quasicircular compact binaries in two regimes: the weak-field, post-Newtonian regime, in which the gravitational wave energy flux has now been calculated to fourth-and-a-half post-Newtonian order (4.5PN) [Phys. Rev. Lett. 131}, 121402 (2023)]; and the small-mass-ratio, gravitational self-force regime, in which the flux has now been calculated to second perturbative order in the mass ratio (2SF) [Phys. Rev. Lett. 127, 151102 (2021)]. We compare these results and find agreement, showing consistency between the two (very distinct though both first-principle) perturbative calculations.

💡 Research Summary

This paper presents a detailed comparison between two state‑of‑the‑art perturbative calculations of the gravitational‑wave (GW) energy flux emitted by non‑spinning black‑hole binaries on quasicircular orbits: the 4.5‑post‑Newtonian (PN) result, which expands in the small orbital velocity parameter x = (m ω)^{2/3}, and the second‑order self‑force (2SF) result, which expands in the small mass‑ratio ν = m₂/m₁. Both approaches are rooted in first‑principles general relativity but rely on very different approximations—weak‑field, slow motion for PN, and extreme‑mass‑ratio for self‑force.

The authors first review the PN derivation. Using the multipolar‑post‑Minkowskian (MPM) formalism, they construct the external metric from canonical mass and current multipole moments, match it to the near‑zone PN field, and obtain the radiative moments U_L and V_L. The calculation incorporates all known nonlinear effects up to 4.5PN: the leading quadrupole tail at 1.5PN, memory at 2.5PN and 3.5PN, tail‑of‑tail at 3PN, tails‑of‑memory at 4PN, spin‑quadrupole tail at 4PN, and finally the quartic tail‑of‑tail‑of‑tail interaction at 4.5PN. Black‑hole horizon absorption, which contributes only at effective 5PN order, is shown to be irrelevant for the flux at the order considered. The final PN flux takes the schematic form

 F = (32/5) ν² x⁵