The paper presents a method to turn existing “quasi-circular” gravitational-wave models into eccentric, multi-mode waveforms. Gravitational-wave signals from two merging black holes can be decomposed into a dominant quadrupole part and smaller higher-order parts, or modes. The authors exploit a simple, empirical pattern in how eccentricity changes the waveform — a set of “universal eccentric modulation functions” — to map a known eccentric quadrupole signal onto the higher-order modes. Using this idea they build a new non-spinning eccentric model called gwNRHME_NRSur_q4 that contains nine spherical-harmonic modes.

To make the model they combined a state-of-the-art quasi-circular surrogate model (NRHybSur3dq8) with an existing quadrupolar eccentric surrogate (NRSurE_q4NoSpin_22). They tested the result against 156 numerical-relativity waveforms from the SXS catalog. Using the Advanced LIGO design sensitivity as a benchmark, the new model gives very small differences from the numerical waveforms. The median frequency-domain mismatch — a standard, unitless measure of how different two waveforms are, where smaller is better — was about 9×10^−5 with a standard deviation of about 2×10^−4.

At a high level the method works because the modulation in amplitude and in instantaneous frequency caused by orbital eccentricity looks very similar across different modes. The authors quantify these modulations relative to a circular reference and find they are nearly identical for many modes and related by a roughly constant scaling factor (about 0.9). That means a robust estimate of the modulation from the quadrupole mode can be re-used to modify the other modes. The paper also demonstrates the framework’s modularity by grafting the same quadrupolar eccentric model onto two effective-one-body (EOB) quasi-circular models, producing alternate eccentric waveforms with somewhat larger mismatches.