Effective source for second-order self-force calculations: quasicircular orbits in Schwarzschild spacetime

TL;DR

This paper details the calculation of an effective source term for second-order gravitational self-force computations in Schwarzschild spacetime, crucial for improving gravitational waveform models for extreme mass ratio inspirals.

Contribution

It provides the first detailed derivation and validation of the effective source in Schwarzschild background for second-order self-force calculations, enabling more accurate gravitational wave modeling.

Findings

Validated the effective source through numerical tests.

Decomposed the source in tensor spherical harmonics.

Facilitated second-order self-force computations in Schwarzschild spacetime.

Abstract

Recent years have seen the first production of "post-adiabatic" 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 [Phys. Rev. Lett. 127, 151102 (2021); ibid. 128, 231101 (2022); ibid. 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.