Systematic dependence of direct-wave frequency evolution on the rational-filter choice

Determine systematically how the instantaneous complex-frequency evolution of filtered direct waves depends on the set of quasinormal modes removed by the rational filter, including the effects of filtering different overtone ranges.

Background

The authors note that the extracted direct-wave complex frequency changes with the set of quasinormal modes removed by the rational filter. Their examples use filters removing prograde and retrograde modes through the second overtone, while the paper also discusses spin-dependent pole–zero pairing and the possible effects of filtering additional overtones.

Because additional filtering can suppress the waveform near corresponding Matsubara frequencies and increase the relative importance of numerical noise or other artifacts, a systematic study is needed to establish which filter prescriptions most reliably recover the direct-wave evolution. The authors explicitly defer this study to future work.

References

The extracted direct-wave complex frequency depends on the set of modes removed by the rational filter (see more details in Appendix~\ref{app:zeros_poles}); a systematic study of this dependence will be presented in future work.

— Complex frequency evolution of direct waves from binary black hole mergers  (2608.23209 - Sun et al., 24 Aug 2026) in Section “Benchmark with NR and CCE waveforms,” subsection “Particle plunges from ISCO”

A quantitative analysis based on the full BH S-matrix is left for future work.

— Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis  (2609.11908 - Nakamoto et al., 10 Sep 2026) in Section 3, Numerical test of direct waves via the pole-splitting method, immediately before Section 4, Conclusion