Spin-Inversion Degeneracies in Restricted Inspiral Waveforms for LISA

Abstract: Spin inversions appear in several settings. Analytical work predicts a single spin flip during unequal-mass supermassive binary inspiral, while numerical relativity and post-Newtonian calculations show repeated flip-flop motion in comparable-mass binaries. Secular spin evolution also predicts additional cases driven by spin-induced mass quadrupoles. Whether these effects can be distinguished in gravitational-wave data is still unclear. We combine the secular spin angle equations with a quasi-circular second post-Newtonian frequency evolution and build a restricted waveform weighted by the sky-averaged LISA sensitivity. We study five near-equal-mass injections with detector-frame total mass $M=2\times10<sup>5M_\odot$, including Kerr flip-flops and one quadrupole-induced case. Each injection is compared with physically evolving waveforms constrained to have no orbital-plane crossings. We search all four no-inversion sectors, vary the masses, spin magnitudes, and initial spin angles, and maximize over time, phase, and overall amplitude. Large spin motion does not by itself lead to a clearly different waveform in this restricted model. A weak spin that sweeps through $133.26<sup>\circ$ leaves a residual SNR of $0.806$ at reference signal SNR $ρ<em>\star=100$. A case where both spins cross the orbital plane many times gives the largest residual, $1.641$. The quadrupole case has a secondary-spin range of $128.11<sup>\circ$ with five crossings, yet the best no-inversion candidate found leaves a residual SNR of only $0.454$. All largest matches found exceed $0.999865$. Within this restricted model, the spin inversion cases are therefore strongly degenerate with no-inversion binaries at $ρ</em>\star=100$. More complete waveforms, including observer-frame precession modulations, higher harmonics, separate polarizations, and the full LISA response, are needed to test whether this degeneracy can be broken.

• The paper demonstrates that one-off flips, repeated Kerr flip-flops, and quadrupole-induced inversions are highly degenerate with no-inversion binaries, producing matches above 0.999865 and residual SNRs below 1.7 at a reference SNR of 100.
• The authors evolve near-equal-mass binaries with orbit-averaged 2PN spin dynamics and TaylorT4 inspiral phasing, then optimize masses, spins, angles, and waveform shifts across all four no-inversion sectors using LISA noise weighting.
• The results show that even large spin excursions and numerous orbital-plane crossings may be unobservable in the restricted model, while full precessing waveforms, higher harmonics, sidebands, and the time-dependent LISA response remain potential ways to break the degeneracy.

Motivation and scope

Spin inversions in compact binaries arise through three theoretically established mechanisms: a single large reorientation of the primary spin during unequal-mass supermassive black hole (SMBH) inspiral, predicted analytically at intermediate mass ratios $1/30 \lesssim q \lesssim 1/3$; the recurrent "flip-flop" motion of comparable-mass binaries found in numerical relativity (NR) and continued with 3.5PN evolution; and a quadrupole-induced flip-flop class emerging from the closed secular spin system when a dominant component carries a non-Kerr spin-induced mass quadrupole, $w_1 \simeq 1 + 2/q$ (2608.20174). Whether any of these geometric signatures can be identified in gravitational-wave data is an open question. The paper addresses it by asking a specific, well-posed question: can a restricted waveform generated by a spin-inverting binary be reproduced, to high match, by a physically evolving binary whose spins never cross the orbital plane?

The astrophysical motivation for the intermediate-mass-ratio regime is substantial — statistical models place roughly 48.9% of coalescing SMBH encounters within $1/30 \lesssim q \lesssim 1/3$ — but the study deliberately restricts itself to near-equal masses ($q \approx 0.94$–$0.97$), where the NR flip-flop phenomenology and its PN extensions are concentrated. The authors are explicit that they do not claim verification of the unequal-mass SMBH tracking.

Dynamical framework

The internal spin configuration is described by three angles: $\kappa_1$, $\kappa_2$ (spin polar angles relative to the Newtonian orbital angular momentum $\bm L_N$) and the relative azimuth $\Delta\zeta$. A spin inversion is defined operationally as a sign change of $\cos\kappa_i$. The evolution uses the circular limit of the orbit-averaged 2PN secular equations including spin-orbit, spin-spin, and quadrupole-monopole couplings, coupled to an adiabatic 2PN frequency evolution of TaylorT4 type. In the weak-secondary-spin limit, the flip-flop dynamics reduces to a harmonic oscillator in $w_1 \simeq 1 + 2/q$0 with frequency $w_1 \simeq 1 + 2/q$1; large-amplitude oscillations require $w_1 \simeq 1 + 2/q$2, which for Kerr quadrupoles is satisfied when the dominant spin lies near the orbital plane ($w_1 \simeq 1 + 2/q$3), and for the quadrupole-induced case when $w_1 \simeq 1 + 2/q$4.

A key structural observation governs the entire analysis: each spin's nutation rate is proportional to the other spin's magnitude ($w_1 \simeq 1 + 2/q$5, $w_1 \simeq 1 + 2/q$6), so a dynamically dominant spin can drive large excursions of a weak companion while barely moving itself. Moreover, in the chirp equation the leading spin-orbit contribution enters through the signed projection $w_1 \simeq 1 + 2/q$7; a weak spin crossing the plane changes sign but contributes negligibly to $w_1 \simeq 1 + 2/q$8, while the quadratic-in-spin terms at 2PN depend on $w_1 \simeq 1 + 2/q$9 or on $1/30 \lesssim q \lesssim 1/3$0 and are not directly sensitive to the sign of a projection. This provides an analytic reason to expect weak degeneracy before any numerical search is performed.

Waveform model and search protocol

The waveform is a restricted stationary-phase-approximation model: Newtonian quadrupole amplitude with a 2PN chirp whose $1/30 \lesssim q \lesssim 1/3$1 and $1/30 \lesssim q \lesssim 1/3$2 coefficients are recomputed from the evolving spin geometry at every frequency sample, plus a leading inclination factor $1/30 \lesssim q \lesssim 1/3$3 evaluated about the fixed direction of $1/30 \lesssim q \lesssim 1/3$4. Noise weighting uses the sky-averaged LISA sensitivity curve including one year of Galactic confusion foreground. No precession sidebands, higher harmonics, separate polarizations, merger-ringdown, or time-dependent detector response are included.

Five injections at detector-frame total mass $1/30 \lesssim q \lesssim 1/3$5, $1/30 \lesssim q \lesssim 1/3$6 Mpc, over $1/30 \lesssim q \lesssim 1/3$7 were studied:

Case	Mechanism	$1/30 \lesssim q \lesssim 1/3$8 / $1/30 \lesssim q \lesssim 1/3$9 (deg)	Crossings $q \approx 0.94$0/$q \approx 0.94$1
K0	Kerr, two active spins	79.56 / 154.75	3 / 6
KW	Kerr, weak secondary	5.27 / 133.26	5 / 9
KH1	Kerr, many crossings	83.56 / 90.09	14 / 14
KH2	Kerr, large sweeps	138.78 / 138.81	6 / 6
Q	Quadrupole-induced ($q \approx 0.94$2, $q \approx 0.94$3)	10.10 / 128.11	0 / 5

For each injection, all four no-inversion sectors $q \approx 0.94$4, $q \approx 0.94$5, $q \approx 0.94$6, $q \approx 0.94$7 were searched, varying total mass, mass ratio, spin magnitudes, and initial angles, maximizing over time shift, phase shift, and an overall amplitude factor. Candidates were rejected unless both signed projections remained bounded away from zero ($q \approx 0.94$8). Optimization proceeded in two differential-evolution stages followed by Powell refinement, with final verification at 8192 frequency samples and relative integration tolerance $q \approx 0.94$9; results are stable under doubling the sample count (largest change in $0.97$0 of $0.97$1).

Results

All best no-inversion matches exceed 0.999865, and the residual SNRs at reference SNR $0.97$2 are small:

Case	Sector	$0.97$3	$0.97$4
K0	$0.97$5	0.9999720034	0.748
KW	$0.97$6	0.9999674995	0.806
KH1	$0.97$7	0.9998652957	1.641
KH2	$0.97$8	0.9999731354	0.733
Q	$0.97$9	0.9999897071	0.454

Three findings stand out. First, angular excursion is not a proxy for observability: KW's weak secondary sweeps $\kappa_1$0 with nine plane crossings yet leaves a residual of only 0.806, and KH2's roughly $\kappa_1$1 two-spin motion leaves only 0.733. Second, even KH1 — fourteen crossings per spin, the largest residual in the sample at 1.641 — is closely matched once source parameters are varied. Third, the quadrupole-induced case is essentially fully absorbed: a $\kappa_1$2-sector source with $\kappa_1$3 reaches $\kappa_1$4, reproducing the carrier phase and inclination-amplitude evolution without any orbital-plane crossing. A control confirms that the Q inversion is genuinely quadrupole-driven: replacing both $\kappa_1$5 by their Kerr values eliminates the secondary's five crossings for otherwise identical parameters, and holding $\kappa_1$6 fixed while varying $\kappa_1$7 shows the effect is controlled primarily by $\kappa_1$8.

The search-depth comparison in the appendix is instructive: deeper searches reduced the residuals for KH1, KH2, and Q by 72%, 80%, and 89% respectively relative to moderate searches, demonstrating that shallow optimization can substantially overestimate distinguishability. The authors correctly note that the reported values are largest matches found numerically, not proofs of global maxima — meaning the true degeneracy could be still stronger.

Limitations

The central caveat, stated plainly by the authors, is that the conclusion holds within this restricted model. The waveform contains only the dominant harmonic, the accumulated 2PN carrier phase, and the leading inclination factor; complete precessing waveforms add inertial-frame mode rotation, precession sidebands, higher harmonics, separate polarization responses, and the time-dependent LISA response, any of which could break the degeneracy. The noise weighting is sky-averaged rather than a full moving-detector response. Crossing counts for trajectories starting at or passing very close to $\kappa_1$9 (notably K0 and KW primaries) depend on the sampling rule used to count sign changes and should not be read as complete flip-flop cycle counts. The value $\kappa_2$0 is used as a dynamical test case, not as an astrophysical model of a known object. Eccentricity, the unequal-mass SMBH regime, and connection to NR/effective-one-body descriptions are left unexamined, and the reported residuals scale linearly with the assumed $\kappa_2$1 without addressing parameter estimation.

Conclusion

Within a restricted quasi-circular 2PN inspiral waveform weighted by the sky-averaged LISA sensitivity, one-off flips, repeated Kerr flip-flops, and quadrupole-induced inversions are strongly degenerate with physically evolving no-inversion binaries at $\kappa_2$2: all best matches exceed 0.999865 and all residual SNRs fall below 1.7. The paper establishes both a dynamical result — confirmation of the distinct quadrupole-induced inversion branch near $\kappa_2$3, separable from the coplanar-spin branch — and a negative observational result within its model class. Whether full precessing waveforms with higher modes, sidebands, and the complete LISA response can break these degeneracies remains the specific open question, for which the injections and verified no-inversion fits provided here serve as numerical benchmarks.