A signal can look the same for different reasons
An arXiv preprint reports that five simulated compact-binary inspirals containing spin inversions could be matched extremely closely by physically evolving waveforms in which neither spin projection crossed zero. The best numerical matches for all five cases exceeded 0.999865. At a reference signal-to-noise ratio (SNR) of 100, the remaining difference after fitting ranged from a residual SNR of 0.454 to 1.641. Within this restricted calculation, the finding means that a signal’s apparent spin history may not uniquely identify the trajectory that produced it.
The question is narrower than whether a binary’s spins literally reverse direction. Here, the relevant test is the sign of each spin’s projection on the orbital-angular-momentum direction: an inversion is registered when that projection reaches and crosses zero. The comparison waveforms were allowed to evolve, but were constrained to remain in one of four no-inversion sign sectors. That gave the researchers a direct test of whether different spin histories could produce nearly the same modeled signal.
What was actually compared
The calculation used a restricted numerical waveform: it kept the dominant quadrupole harmonic of a quasi-circular inspiral, used a Newtonian source amplitude and a TaylorT4-type second-order post-Newtonian, or 2PN, chirp, and evolved the spin angles with secular equations. In plain language, it modeled the main carrier of the signal and its leading strength while using a controlled approximation for how the frequency changes.
Five injections formed the test set: K0, KW, KH1 and KH2 used Kerr quadrupole coefficients, while Q was a non-Kerr quadrupole-induced configuration with w1 = 3.0619 and w2 = 3. The no-inversion search covered all four sign sectors for every injection. It varied the masses, spin magnitudes and initial spin angles, while fixing the quadrupole coefficients to those of the injection; time and phase shifts were optimized, and an overall amplitude factor was fitted.
To compare the signals, the study used a LISA-weighted match—an overlap that gives more weight to frequencies where the assumed detector is more sensitive. The calculation used analytic, sky-averaged LISA sensitivity and included a one-year Galactic confusion foreground for frequency weighting, but it did not simulate a moving-detector response. Final candidates were reintegrated with 8,192 frequency samples and a relative tolerance of 10−10. The reported matches were the largest obtained numerically, not proof of exact global maxima.
The number of flips was not the whole story
At the reference SNR of 100, the residual SNR was 0.748 for K0, 0.806 for KW, 1.641 for KH1, 0.733 for KH2 and 0.454 for Q. KH1 therefore produced the largest residual in the set, while Q produced the smallest. Yet all five matches cleared the 0.999865 mark. The pattern shows why a dramatic-looking spin trajectory need not create an equally dramatic difference in this restricted waveform.
KW provides a useful warning against reading the waveform from spin size alone. Its weak secondary spin had magnitude χ2 = 0.05, swept through 133.26 degrees and made nine crossings, but its best no-inversion comparison still left a residual SNR of 0.806. KH1 had fourteen crossings for each spin and left the largest residual, 1.641. Those cases show that repeated crossings can coincide with a larger residual, but the crossing count alone does not set it.
KH2 makes the same point from another angle. Both spins underwent approximately 139 degrees of motion, yet the residual SNR was 0.733—well below KH1’s 1.641 despite KH1’s fourteen crossings per spin. In this five-case sample, angular excursion and crossing count were not reliable stand-alone guides to whether the restricted waveform would be distinguishable.
A non-Kerr case remained easy to mimic
The Q injection tested dynamics induced by a non-Kerr quadrupole configuration. In the selected case, the mass ratio was q = 0.97, with w1 = 3.0619 and w2 = 3. The secondary spin moved through 128.11 degrees and crossed five times, while the otherwise identical Kerr-coefficient evolution had no such crossings. The scan placed the case in a secondary-inversion region near w1 ≃ 1 + 2/q.
Even there, the waveform fit remained highly degenerate. The best no-inversion Q fit stayed in the (−, −) sector, retained w1 = 3.0619 and w2 = 3, and reached a match of 0.9999897071. At reference SNR 100, it left a residual SNR of 0.454 while both fitted spins stayed anti-aligned with the orbit. In practical terms, the restricted calculation reproduced the carrier and leading amplitude evolution closely without reproducing the inversion itself.
A narrow result, with a wider test still needed
That shortcut is also the study’s key limitation. The model did not include the full set of observer-frame precession modulations, higher harmonics, separate polarizations, merger and ringdown, or the moving full LISA response. Its LISA weighting was analytic and sky-averaged. A result that survives a complete precessing waveform and the full detector response could therefore look different.
The exercise also covered only five selected near-equal-mass injections, not an astrophysical population. Eccentric systems and the unequal-mass supermassive-black-hole regime were outside the test. The crossing totals came from sampled sign changes rather than continuous root-finding, so they should be read as numerical diagnostics rather than complete counts of every possible flip-flop cycle.
The numerical search was followed by high-resolution verification, but the fits remain the best values obtained in that search rather than demonstrated global optima. The open question is whether complete precessing waveforms and full source-parameter recovery would break the degeneracy. Broader tests using global optimization or Bayesian inference, along with eccentric and unequal-mass systems, could change how strongly the two classes can be separated.
On its own, the study provides no direct evidence of spin inversions in observed gravitational-wave data. Its contribution is a set of numerical benchmarks showing that, under the restricted assumptions, even large or repeated spin motion can be closely mimicked by a physically evolving no-inversion fit. Source code, the notebook, numerical data and reproduction scripts are openly available in Zenodo version 0.1.0 at https://doi.org/10.5281/zenodo.22025538.
Paper data and sources
Original title: Spin-Inversion Degeneracies in Restricted Inspiral Waveforms for LISA
Authors: Kata Karácsonyi, László Árpád Gergely
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text