A new study has investigated whether spin inversion events, where the rotation axis of a binary black hole crosses the orbital plane, can be distinguished in gravitational waves detected by the future LISA observatory. The results indicate a strong degeneracy: signals from binaries with spin inversion are almost indistinguishable from those without inversion within the restricted model used, even for significant spin motion. This poses a challenge for the interpretation of future LISA observations.

Spin inversions are a predicted phenomenon in the evolution of supermassive binary black hole systems. While analytical work suggests a single inversion in unequal-mass binaries, numerical relativity simulations and post-Newtonian calculations show repeated inversions in comparable-mass binaries. Secular spin evolution also predicts additional cases driven by spin-induced mass quadrupoles. The ability to discern these effects in gravitational-wave data is crucial for understanding the dynamics of these systems.

For this study, secular spin angle equations were combined with a quasi-circular second post-Newtonian frequency evolution, building a restricted waveform weighted by the sky-averaged LISA sensitivity. Five near-equal-mass injections were analyzed, with a detector-frame total mass of 2x10^5 solar masses, including Kerr flip-flops and one quadrupole-induced case. Each injection was compared with physically evolving waveforms constrained to have no orbital-plane crossings. The results showed that the largest matches between inversion and no-inversion signals exceeded 0.999865, even for a case with multiple orbital plane crossings, implying very high degeneracy.

The main implication is that, with the restricted waveforms employed, detecting spin inversions in supermassive binary black holes with LISA will be extremely difficult. To break this degeneracy, more complete waveforms are needed, including observer-frame precession modulations, higher harmonics, separate polarizations, and the full LISA response. This is fundamental for extracting detailed information about the spin dynamics of these cosmic systems from future observations.