Comparing Kerr Coordinates for Parameterizing Fundamental Ringdown Damping in Numerical Relativity

This paper compares physically motivated Kerr coordinate parameterisations for the dominant quasinormal-mode quality factor Q220 of binary black hole ringdown, using a two-layer benchmark. Exact Kerr control layer: Coordinate efficiency is evaluated directly on the (2,2,0) spectrum from perturbation theory (qnm package). The transform (1−af)−0.50 best linearises the exact quality factor over af∈[0.30,0.95] (LOOCV RMSE 0.004), while 1/κM is best for the pure damping rate τ220/Mf, consistent with the near-extremal expectation |Im(Mfω220)| ∝ κM. NR pipeline layer: Q220 is extracted from 382 SXS binary black hole simulations and compared under a common two-parameter linear model and LOOCV RMSE. Surface gravity κM yields lower LOOCV RMSE than both ΩHMf and af (bootstrap 95% CIs exclude zero), with ranking stable under blocked out-of-sample validation (6/6 protocols) and leverage removal. An extraction validation shows a systematic +8.5% positive bias vs exact Kerr, correlated with progenitor spin χ1z, indicating the NR advantage of κM reflects pipeline-level properties rather than the Kerr spectrum. Key contributions:• Two-layer benchmark: exact Kerr primary control + NR pipeline secondary• Exact Kerr: (1−af)−0.50 best for Q; 1/κM best for τ/Mf• 382-simulation NR benchmark, pre-registered, bootstrap 95% CIs• Extraction validation: +8.5% bias correlated with progenitor spin• Blocked out-of-sample validation: κM wins 6/6 protocols• Weighted RMSE: ranking stable under empirical, uniform-spin, and astrophysical-prior weights• Leverage diagnostics confirming ranking robustness