Origin of discrepancies between EES and benchmark self-force results

Determine the origin of the differences between the frequency-domain gravitational extended-effective-source self-force results and the independent Lorenz-gauge benchmark calculation, and systematically quantify the contributions from Fourier, Chebyshev, radial-integration, and multipole-truncation errors.

Background

The calculation is benchmarked against an independent Lorenz-gauge frequency-domain result for an eccentric Schwarzschild orbit with (p,e)=(10,0.3)(p,e)=(10,0.3). The radial self-force component agrees to better than 0.1%0.1\%, while the temporal component differs by as much as 0.93%0.93\% at the sampled phases.

The reported values are truncated at ℓmax⁡=15\ell_{\max}=15 and contain no large-ℓ\ell tail correction. The authors state that the observed discrepancies are compatible with finite multipole truncation, but that the available data do not determine their origin uniquely. Resolving this issue requires a systematic error analysis covering Fourier truncation, Chebyshev interpolation, radial integration, and multipole truncation.

References

The observed differences are compatible with finite multipole truncation, although the present data do not uniquely establish their origin. A higher-precision calculation would require a systematic analysis of the Fourier, Chebyshev, radial-integration, and multipole-truncation errors.

— Frequency-domain extended-effective-source gravitational self-force for eccentric Schwarzschild orbits  (2609.02635 - Lu et al., 2 Sep 2026) in Results and validation discussion following Table \ref{tab:comparison}