Probing Spin-Orbit Resonances with the Binary Black Hole Population

Sylvia Biscoveanu

REVIEW 3 major objections 4 minor 6 cited by

Probing Spin-Orbit Resonances with the Binary Black Hole Population

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The population-level distribution of the azimuthal spin angle between the two black-hole spins can be measured across a gravitational-wave catalog and used to probe spin-orbit resonances, binary formation channels, and mass-transfer…

desk verdict Careful end-to-end study showing phi12 population inference recovers resonant features and GWTC-3 lacks strong SOR evidence, but the abstract's mass-transfer claim outruns the paper's own weak-resonance fRMR bias. read the letter →

arxiv 2502.04278 v2 pith:2EL4B377 submitted 2025-02-06 astro-ph.HE gr-qc

Sylvia Biscoveanu This is my paper · ORCID

classification astro-ph.HEgr-qc

keywords spin-orbitresonancesbinaryblackholesgravitational-wavepopulationinferenceazimuthalspinanglemorphologyhierarchicalBayesiananalysismassratioreversalGWTC-3

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the azimuthal angle $\phi_{12}$ between the two black-hole spins, measured across a whole population of gravitational-wave binaries, can act as a reliable new probe of spin-orbit resonances and the astrophysics behind binary formation. It presents the first end-to-end hierarchical analysis that models the population-level $\phi_{12}$ distribution together with masses, redshifts, and other spin parameters, using simulated populations the size expected by the end of the next observing run. The simulated $\phi_{12}$ distributions are recovered within uncertainties, meaningful constraints can be placed on the excesses at $\phi_{12}=0$ and $\pm\pi$, and isotropic spin-angle populations can be told apart from resonant ones. Applied to the observed GWTC-3 catalog, the model finds no statistically significant evidence for spin-orbit resonances, though it does see a weak preference for an excess at $\phi_{12}=\pm\pi$. If correct, this gives gravitational-wave astronomy a way to measure binary mass-transfer history and tidal efficiency directly from spin angles.

What carries the argument

The central object is $\phi_{12}$, the azimuthal angle between the projections of the two component spins onto the orbital plane. Spin-orbit resonances make $\phi_{12}$ librate around $0$ (reversed mass ratio) or $\pm\pi$ (standard mass ratio), producing peaks in its distribution. The machinery is a hierarchical Bayesian mixture model in which the $\phi_{12}$ distribution is a two-component von Mises distribution (plus an isotropic component), coupled to a correlated truncated-Gaussian mixture model for the spin tilts; the hyper-parameter $f_{\rm RMR}$ tracks the fraction of field binaries drawn from the $\phi_{12}=0$ peak and is shown to be driven by the $\phi_{12}$ data rather than by the tilts.

What would settle it

Perform all parameter estimation with a waveform model valid at the ISCO for every event, avoiding the 20 Hz to ISCO spin-evolution step entirely, and rerun the hierarchical inference on the same simulated populations; if the $\phi_{12}$ hyper-parameters are recovered far outside the quoted credible intervals, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the population distribution of $\phi_{12}$ is measurable and informative even though individual binaries constrain $\phi_{12}$ poorly. For each simulated population, the true $\phi_{12}$ distribution falls inside the 90% posterior credible interval of the hierarchical inference, and the proxy parameter $f_{\rm RMR}$ for the fraction of field binaries that underwent mass ratio reversal can be constrained to 95% interval widths of roughly 0.7 or better using $\phi_{12}$ information alone. The paper shows that a fully isotropic spin-angle population produces no spurious resonant features under its two-component von Mises model, while the weaker single-component model used in an earlier study does produce spurious features; the earlier weak evidence for resonances is consistent with statistical fluctuations. On real GWTC-3 data, no compelling evidence for spin-orbit resonances is found, but all model variations favor a mild excess at $\phi_{12}=\pm\pi$, the location expected for the standard mass-ratio scenario.

Load-bearing premise

The whole analysis assumes that evolving individual-event spin posteriors from 20 Hz to the innermost stable circular orbit with the approximate SpinTaylorT5 dynamics, for events analyzed with IMRPhenomXPHM, does not systematically bias the recovered $\phi_{12}$ distribution.

Editorial extensions

If this is right

With a population of roughly 200 detections at O4 sensitivity, the $\phi_{12}$ distribution can be recovered within the 90% posterior credible interval both for strongly resonant populations and for populations with weak resonant features.
The mass-ratio-reversal fraction can be constrained to 95% credible interval widths of about 0.7 or smaller using only the information carried by $\phi_{12}$, providing a direct observational handle on binary mass transfer.
A fully isotropic spin-angle population can be distinguished from one with resonant features, with the field-binary mixture fraction constrained to $\xi \lesssim 0.33$ at 95% credibility for the isotropic case.
The weak evidence for spin-orbit resonances previously reported in GWTC-2 falls within the statistical fluctuations of simulated isotropic catalogs of the same size, so it is not a significant detection.
Applying the model to GWTC-3 yields no statistically significant evidence for spin-orbit resonances, but a weak preference for an excess at $\phi_{12}=\pm\pi$ persists across all model variations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If this method holds, the $\phi_{12}$ distribution becomes a new observational handle on tidal efficiency and mass-ratio reversal that is complementary to spin-magnitude correlations, potentially breaking degeneracies between binary formation channels.
The weak preference for $\phi_{12}=\pm\pi$ in GWTC-3, if confirmed with more events, would favor the standard mass-ratio scenario over the reversed mass-ratio scenario for the fraction of field binaries that are caught in spin-orbit resonances.
The strong model-dependence of the mixture fraction $\xi$ posteriors suggests that future analyses should report the marginal $\phi_{12}$ and tilt shapes rather than interpreting $\xi$ astrophysically, and could instead use more data-driven non-parametric models.
A natural extension would be to apply the two-component von Mises model to the full O4 catalog and relax the fixed-peak assumption, which would allow distinguishing resonant features at $0$ and $\pm\pi$ from pile-ups at $\pm\pi/2$ caused by weak tides.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a solid, careful end-to-end simulation study of phi12 population inference for spin-orbit resonances. The headline result—that you can recover simulated phi12 distributions and distinguish isotropic from resonant populations with O4-size catalogs—holds up. The earlier Varma et al. (2022a) GWTC-2 hint is shown to be consistent with noise, and GWTC-3 shows no significant SOR evidence. All of that is real and useful.

The genuinely new piece is the architecture: joint hierarchical inference of phi12 with mass, spin magnitude, tilt, and redshift distributions, using a two-component von Mises model, four simulated populations, and full parameter estimation. The selection-effects treatment is careful (ten million injections), and the author is transparent about known biases in mass and spin-magnitude recovery. Public posterior samples on Zenodo are a plus. The recovery of the phi12 distribution within 90% credible intervals for all four simulated populations is the core evidence, and it is convincing.

The soft spot is the mass-transfer claim. In Section 3.4, for the weak-resonances population—the one most consistent with GWTC-3—the model recovers the phi12 shape but misassigns the locations of the excess: fRMR comes out 0.76 when the truth is 0.3, because the phi12=pi excess is attributed to the RMR component. That means the mapping from measured excesses to mass-ratio reversal is degenerate when tilt distributions are broad. The author acknowledges this in Section 3.4 and the summary, but the abstract's statement that phi12 "encodes information about binary mass transfer" overstates what the measurement actually delivers. The relative excesses at 0 and pi are measurable; the astrophysical attribution to SMR vs RMR is not robust in the realistic regime.

The SpinTaylorT5 evolution for IMRPhenomXPHM events is another caveat, but the recovery results suggest it does not dominate the inference. The persistent spin-magnitude bias is concerning but appears not to affect the spin-angle results.

This paper deserves a serious referee. The methods are reproducible, the negative result on GWTC-3 is clearly framed, and the limitations are mostly stated internally. The main fix is to temper the abstract and add a prominent caveat about fRMR interpretation. I would send it to review with minor revision in mind.

Referee Report

3 major / 4 minor

Summary. This manuscript presents the first end-to-end hierarchical analysis of the population-level distribution of the azimuthal spin angle phi12, simultaneously fitting mass, redshift, spin magnitude, tilt, and azimuthal-angle hyperparameters for simulated binary black hole populations and for GWTC-3. The author simulates four populations (strong resonances, strong resonances plus isotropic, fully isotropic, and weak resonances), performs full individual-event parameter estimation with Bilby/Dynesty, accounts for selection effects with an injection campaign, and fits a von Mises mixture model whose peaks at phi12=0 and phi12=pi are linked to mass-ratio-reversal and standard-mass-ratio scenarios. The central reported results are that the simulated phi12 distributions are recovered within 90% credible intervals, that isotropic populations can be distinguished from resonant ones, and that GWTC-3 shows no statistically significant evidence for spin-orbit resonances but a weak preference for an excess at phi12=pi.

Significance. If the central claims hold, this is a useful methodological contribution: it is the first study to propagate individual-event phi12 posteriors through a full hierarchical inference that includes masses, spins, and selection effects, and it provides a concrete template for future searches for spin-orbit resonances. The paper's strengths include the use of full parameter estimation rather than simplified likelihoods, two independent realizations of the strong-resonance population, a large sensitivity-injection set, and public release of posterior samples. The comparison with Varma et al. (2022a) using GWTC-2-sized catalogs is a valuable calibration of that earlier result. However, the paper itself documents important limitations: a persistent spin-magnitude bias, a mass-distribution bias from the unphysical simulation DAG, and a serious fRMR labeling degeneracy in the weak-resonances population. These limitations directly qualify the abstract's claim that phi12 'encodes information about binary mass transfer,' and they require changes before the paper can be accepted.

major comments (3)

[Section 3.4, Table 5, Abstract] The abstract claims that the phi12 distribution 'encodes information about binary mass transfer,' but the paper's own weak-resonances analysis—the population the text identifies as most consistent with GWTC-3—does not support that mapping. For the 'Weak resonances full mix' run with free peak locations, Table 5 reports fRMR = 0.76^{+0.24}_{-0.50} when the injected value is 0.3, and Section 3.4 states explicitly that the phi12=pi excess is attributed to the RMR rather than the SMR sub-population. This is a labeling degeneracy in the mixture model, not a mild uncertainty: the relative excesses at 0 and pi are measurable, but their interpretation as mass-ratio reversal is ambiguous when the tilt distributions are broad. The abstract and conclusions should be revised to claim only that the relative excesses are measurable and that their mapping to mass transfer is model-dependent, or the paper should provide a model modification or validation that breaks this degeneracy in the weak-resonance regime.
[Section 2.2, Fig. 2] The full-population recovery and the GWTC-3 analysis rely on evolving IMRPhenomXPHM posterior samples from f_ref=20 Hz to f_ISCO using the approximate SpinTaylorT5 PN dynamics, while the paper's own Fig. 2 shows a system for which PN-evolved and surrogate-evolved phi12 posteriors differ substantially. The sentence in Section 2.2 that the approximate PN evolution 'does not lead to significant biases' is supported only by that single example and by the aggregate recovery, not by a systematic validation. Please provide a quantitative comparison, for example by re-analyzing a subset of IMRPhenomXPHM events with surrogate evolution and comparing the resulting phi12 hyperparameter posteriors, or by reporting the fraction of events for which the two evolution methods shift the phi12 posterior by more than a chosen threshold. Without this, the recovery claim for the XPHM-dominated portion of the population is not fully demonstrated.
[Section 3, Appendices C.1 and C.2] The paper reports persistent biases in the mass power-law parameters and in the spin-magnitude hyperparameters, then states that 'neither the mass nor spin magnitude biases affect the spin angle inference' (Section 3). The evidence for this decoupling is not presented as a direct test: the delta-function-posterior test in Appendix C.2 is applied only to the spin-magnitude bias, and the mass bias is attributed to an unphysical DAG. Because chi_eff is known to correlate spin magnitudes and tilts, and because the hierarchical likelihood fits all parameters simultaneously, the assertion that the spin-angle recovery is unaffected should be demonstrated explicitly. For example, the author could compare the phi12 and tilt hyperparameter posteriors from the full analysis with those from an analysis that fixes the mass and spin-magnitude hyperparameters to their true values, or show that the NRSur7dq4-only subsets, where the mass bias is absent, yield the same phi12 conclusions as the full mixed-waveform samples.

minor comments (4)

[Section 1, penultimate paragraph] There is a duplicated word in 'we also find that the the previously-identified weak evidence'; please delete the second 'the'.
[Section 3.4 and Table 5] The fRMR credible interval for the weak-resonances full-tilt free-peak run is quoted as 0.76^{+0.24}_{-0.49} in the text but as 0.76^{+0.24}_{-0.50} in Table 5; please harmonize the two values.
[Fig. 10] The left and middle panels of Fig. 10 are discussed in the text only indirectly; please add an explicit call to all three panels when describing the GWTC-2-sized catalog results, and define the shaded or colored regions in the caption.
[Section 3.3.1 and Fig. 9] In the discussion of the Default LVK model, the text says the prior on sigma_1=sigma_2 is U(0.01,4), but Table 3 lists different priors for the Simple tilt model; please make the prior definitions for the Default LVK comparison model explicit and consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phi12 population recovery is validated against simulated injections and real GWTC-3 data, and the self-citations are not load-bearing.

full rationale

The paper's central claim—that population-level phi12 distributions can be inferred and resonant features distinguished—rests on a full end-to-end pipeline (simulated signals, individual-event parameter estimation with Bilby/Dynesty, and hierarchical inference with selection effects, Eqs. B1-B2), not on an equation that assumes the answer. The Full tilt recovery runs use the same functional family as the generative model, but this is an internal consistency check, and the paper also analyzes mismodeled Simple tilt and free-peak models, so the recovery is not forced by construction. In the weak-resonances case (Section 3.4) the model mislabels the RMR/SMR components (fRMR = 0.76 vs true 0.3), which is the opposite of a self-fulfilling prediction and shows that the inference is genuinely data-driven. The fRMR parameter is explicitly flagged as a proxy (Section 2.1), avoiding self-definitional circularity. The paper's own admitted limitations—the fRMR proxy interpretation, the weak-resonance labeling bias, and the spin-magnitude recovery bias—are correctness and interpretability risks, not circularity. Citations to the author's prior work (Varma et al. 2022a,b) motivate the population model and waveform choice, but they are not used as load-bearing uniqueness theorems; the prior GWTC-2 evidence is re-tested and found consistent with noise, so the derivation is self-contained.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on phenomenological spin-angle parameters chosen by hand (fRMR, kappa, sigma_t, xi), the astrophysical assumption that SORs manifest as von Mises peaks at 0 and ±pi, and the technical assumption that PE posteriors evolved to ISCO plus injection-based selection corrections are accurate. The paper explicitly flags the proxy interpretation of fRMR and unresolved biases, so the ledger is honest but not empty.

free parameters (4)

fRMR = 0.3 in strong and weak resonance simulations; 0 in isotropic simulation
Proxy for the fraction of field binaries whose phi12 values come from a VM peak at 0, interpreted as mass ratio reversal; central to the SOR claim and inferred in the hierarchical analysis.
kappa = 4 (strong resonances), 1 (weak resonances)
Von Mises concentration parameter controlling the width of the phi12 peaks; chosen by hand to set the strength of resonant features.
sigma_t = 0.5 (strong), 1.18 (weak)
Width of the truncated Gaussian tilt alignment component; sets how sharply aligned the field binary tilts are.
xi = 1, 0.644, 0 in the different simulated populations
Mixture fraction between the isotropic and aligned spin components; chosen by hand to explore different astrophysical regimes.

assumptions (6)

domain assumption Spin-orbit resonances produce excesses in phi12 at 0 and ±pi for isolated binaries with efficient tides.
Stated in the Introduction citing Schnittman 2004 and Gerosa et al. 2013, 2018; encoded in Eq. 2 as von Mises peaks.
ad hoc to paper The joint spin-angle distribution factorizes as p(cos theta1, cos theta2, phi12) = p(cos theta1, cos theta2) p(phi12).
Section 2.1 simulates from the product of Eqs. 1 and 2 with no cross terms; real binaries may not factorize, and the paper acknowledges this is a phenomenological simplification.
domain assumption SpinTaylorT5 PN evolution of posterior samples from 20 Hz to fISCO is sufficiently accurate for phi12 population inference.
Section 2.2 uses SpinTaylorT5 for IMRPhenomXPHM events; Figure 2 shows differences versus surrogate dynamics for some configurations, but the paper concludes no significant bias in the spin angle distributions.
ad hoc to paper The known-waveform-model simulation pipeline creates a non-physical DAG, but the resulting bias affects only masses, not spin angles.
Appendix C.1 admits the unphysical DAG causes biased mass recovery; the claim that spin angle inference is unaffected is assumed rather than independently demonstrated.
domain assumption The sensitivity injection campaign with over ten million found injections correctly accounts for selection effects.
Appendix B describes the semi-analytic injection campaign; however, the MC integral variance can reach O(10) and posterior results change with different convergence criteria, as shown in Figure 17.
ad hoc to paper The parameter fRMR can be interpreted as a mass-ratio-reversal proxy despite deviations between drawn and actual librating morphologies.
Section 2.1 and Table 2 note that 29% of isotropically drawn systems are in resonant configurations and that fRMR is only a proxy for the fraction drawn from a VM peak at 0.

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Cite this review

Pith. "Pith review of Probing Spin-Orbit Resonances with the Binary Black Hole Population." pith.science (2026). https://pith.science/paper/2EL4B377

@misc{pith2026250204278,
  author       = {Pith},
  title        = {Pith review of: Probing Spin-Orbit Resonances with the Binary Black Hole Population},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2EL4B377}},
  note         = {Machine review of arXiv:2502.04278}
}

abstract

Measurements of the binary black hole spin distribution from the growing catalog of gravitational-wave observations can help elucidate the astrophysical processes shaping the formation and evolution of these systems. Spin-orbit resonances are one process of interest, in which the component spin vectors and the orbital angular momentum align into a common plane and jointly precess about the total angular momentum of the system. These resonances, which occur preferentially in systems formed via isolated binary evolution with strong tidal effects, lead to excesses in the distribution of the azimuthal angle between the projections of the component spin vectors onto the orbital plane at $\phi_{12}=0,\pm\pi$. In this work, we conduct the first hierarchical analysis modeling the population-level distribution of $\phi_{12}$ simultaneously with the other mass and spin parameters for simulated binary black hole populations to determine whether spin-orbit resonances can be reliably constrained. While we are unlikely to find definitive evidence for spin-orbit resonances with a population of the size expected by the end of the ongoing LIGO-Virgo-KAGRA fourth observing run, we correctly recover the various $\phi_{12}$ distributions we simulate within uncertainties. We find that we can place meaningful constraints on the relative excesses at $\phi_{12}=0,\pm\pi$, which encodes information about binary mass transfer. We can also distinguish between fully isotropic spin angle distributions and those with features in the spin azimuth and tilt distributions. Thus, we show that population-level measurements of the $\phi_{12}$ distribution offer a reliable, novel way to probe binary formation channels, dynamics, and mass transfer with gravitational-wave observations.

Figures

Figures reproduced from arXiv: 2502.04278 by the authors.

**Figure 1.** Scatter plot of ϕ12 vs cos θ1 for the binaries in the weak resonances population colored by the calculated spin morphology of each system: librating around ϕ12 = 0 (purple), circulating (teal), and librating around ϕ12 = π (yellow). in our simulated populations using the 2.5 PN order expression for the orbital angular momentum. In [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Posterior distributions of the spin tilt angles cos θ1,2 and azimuthal angle ϕ12 for one binary system simulated and recovered with NRSur7dq4 at a reference frequency of fref = 20 Hz (blue), evolved forward to fISCO using the SpinTaylorT5 orbital dynamics (green) and using the surrogate dynamics (purple). The orange lines show the true parameter values at ISCO, and the dashed lines in each color show samples from… view at source ↗

**Figure 3.** Inferred ϕ12 distribution for the strong resonances population under the Full tilt model. Individual light blue traces show the distributions corresponding to individual hyper-parameter posterior samples, the dark blue lines bound the 90% posterior credible interval, and the dashed black lines bound the 90% prior credible interval. The true simulated distribution is shown in orange [PITH_FULL_IMAGE:figures/full_f… view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: Inferred p(cos θ1, cos θ2) distribution for the strong resonances population under the Full tilt model. Individual light blue traces show the distributions corresponding to individual hyper-parameter posterior samples, the dark blue lines bound the marginalized 1D 90%…

**Figure 5.** Figure 5: Inferred ϕ12 distribution for the strong resonances + isotropic population under the Simple tilt model when the VM mixture fraction tracks the tilt mixture fraction (green) and when it is an independent parameter (purple). The shaded region bounds the 90% posterior cre…

**Figure 6.** Figure 6: Corner plot of the strong resonances + isotropic population analyzed with the Simple (green) and Simple + separate ξVM (purple) tilt models showing the posteriors on the ϕ12 hyper-parameters. The orange lines indicate the true hyper-parameter values. 3.3. Isotropic pop…

**Figure 8.** Figure 8: Posteriors on the mixture fraction between the isotropic and aligned spin tilt components under our Simple tilt model in blue, the Default LVK model in green, and the LVK model with a flexible mean of the aligned-spin component in pink. Under the Default LVK model, th…

**Figure 9.** Figure 9: Inferred p(cos θ1, cos θ2) distribution for the strong resonances population under the Simple tilt model in blue, the Default LVK model in green, and the LVK model with a flexible mean of the aligned-spin component in pink. The shading bounds the 90% posterior credible…

**Figure 10.** Figure 10: Kernel density estimates of the posterior distributions on the azimuthal spin angle hyper-parameters µSMR (left) and µRMR (middle) under the Simple tilt model with our two-component VM model for the ϕ12 distribution for eleven different GWTC-2-sized sub-catalogs of th…

**Figure 11.** Figure 11: Inferred p(cos θ1, cos θ2) distributions for the weak resonances population under the Simple tilt model. Individual light blue traces show the distributions corresponding to individual hyper-parameter posterior samples, the dark blue lines bound the marginalized 1D 90…

**Figure 12.** Figure 12: Inferred ϕ12 distribution for the weak resonances population under the Simple tilt model (green) and under the Full tilt model with the peaks of the resonant component as free parameters (purple). The shading bounds the 90% posterior credible intervals, the dashed li…

**Figure 13.** Figure 13: Inferred ϕ12 distribution for GWTC-3 under the Simple (green) and Full (purple) tilt models, with the peaks of the resonant component fixed (dark) and as free parameters (light). The shading bounds the 90% posterior credible intervals and the dashed lines bound the 90…

**Figure 14.** Figure 14: Kernel density estimates of the posterior distributions on the azimuthal spin angle hyper-parameters µSMR (solid) and µRMR (dashed) under the Full tilt model with our two-component VM model for the ϕ12 distribution. The posteriors for eleven different GWTC-3-sized su…

**Figure 15.** Figure 15: Posteriors on the mixture fraction between the isotropic and aligned spin tilt components under our Simple tilt model with the hyper-parameter priors used for the strong resonances population in blue, with the hyperparameters priors used for the weak resonances popu…

**Figure 16.** Figure 16: Corner plot of two independent sets of 200 events (green and purple) drawn from the strong resonances population analyzed with the Full (left) and Simple (right) tilt models showing the posteriors on the ϕ12 hyper-parameters fRMR and κ. The orange lines denote the tru…

**Figure 17.** Figure 17: Corner plot of the strong resonances Population A analyzed with the Full (left) and Simple (right) tilt models showing the posteriors on the ϕ12 hyper-parameters fRMR and κ along with the MC integral variances for both the individualevent posteriors and the sensitivi…

**Figure 18.** Figure 18: Corner plot of the mass power-law parameters α (m1) and β (q) for the Strong resonances (all events - left; NRSur7dq4 event only - right) and Isotropic (middle) populations both recovered with the Simple tilt model. While the results of the analyses including all even…

**Figure 19.** Figure 19: for the strong resonances Population B is characterized by a width that is too narrow and a mean that is too high (see posteriors in [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]

**Figure 20.** Figure 20: Kernel density estimates of the posterior distributions on the spin magnitude hyper-parameters µχ, σχ for a variety of different analyses. All results are shown using the full population of 200 events and the Simple tilt model. The strong resonances population is show…

**Figure 21.** Figure 21: Corner plot of the posteriors on the spin magnitude hyper-parameters, µχ and σχ, and the isotropic vs. aligned tilt mixing fraction, ξ, for one of the isotropic downsampled GWTC-2-sized catalogs consisting only of 44 NRSur7dq4 events analyzed individually with refl…

**Figure 22.** Figure 22: Histograms of the posteriors on the proxy parameter for the fraction of field binaries that have undergone mass ratio reversal, fRMR, for the hierarchical inference runs summarized in [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]

**Figure 23.** Figure 23: Histograms of the posteriors on the fraction of binaries formed in the field drawn from non-isotropic spin angle distributions, ξ, for the hierarchical inference runs summarized in [PITH_FULL_IMAGE:figures/full_fig_p032_23.png]

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Forward citations

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Reference graph

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but can improve sampling efficiency for posteriors peaking near the prior edge. However, we find that the reflec- tive boundary conditions actually lead to the opposite kind of bias in theµ χ posterior from the original re- sults; it peaks at very low values rather than too high. The reflective boundary conditions cause the sampler to overcompensate and s...

work page 2022
[16]

2018; Krishna et al

have demonstrated a successful re- covery of the spin magnitude distribution with a sim- ilar number of events, these used a waveform model without higher-order modes, with the relative binning likelihood approximation (Cornish 2010; Zackay et al. 2018; Krishna et al. 2023). A stealth bias at the individual-event sampling stage due to sampling diffi- cult...

work page 2024
[17]

17, it is clear that the variance increases for smaller values off RMR and larger values ofκ

From the correlations shown in Fig. 17, it is clear that the variance increases for smaller values off RMR and larger values ofκ. This can be understood as a de- crease in the number of effective samples going into the MC integral as the VM distribution becomes narrower (κincreases) and more strongly peaked at one specific value (the full population is in...

work page 2020
[18]

While we only compare the mixed-waveform results to the NRSur7dq4 results in the figure, we gen- erally find an unbiased recovery of the mass power-law indices when analyzing only the IMRPhenomXPHM- generated subset of each population as well. This can likely be explained by the inconsistency in the choice of waveform between the source simulation and ind...

work page 2022

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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In each case, the hyper-parameters of the distribution are written explic- itly, and the bounds are passed as the last two values in parentheses

= nU Ntot U(m1; 4M ⊙,100M ⊙)U2(cosθ;−1,1) (B3) + nPL Ntot PL(m1;α=−3.5)U 2(cosθ) + nPLTG Ntot PL(m1;α=−3.5)N 2 t (cosθ|µ= 1, σ= 0.5) + nPLTGM Ntot PL(m1;α=−3.5)N 2 t (cosθ|µ= 1, σ= 0.5)N 2 t (χ|µ= 0, σ= 0.3,0,0.99) Ntot =n U +n PL +n PLTG +n PLTGM (B4) nU = 22406657, n PL = 564571237, n PLTG = 504366682, n PLTGM = 461987020.(B5) In the expression above, U...

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andβ(q) for the Strong resonances (all events - left; NRSur7dq4 event only - right) and Isotropic (middle) populations both recovered with theSimpletilt model. While the results of the analyses including all events generated with both NRSur7dq4 and IMRPhenomXPHM demonstrate a biased recovery of the mass power-law indices, the results of analyzing only the...

work page 2020

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but can improve sampling efficiency for posteriors peaking near the prior edge. However, we find that the reflec- tive boundary conditions actually lead to the opposite kind of bias in theµ χ posterior from the original re- sults; it peaks at very low values rather than too high. The reflective boundary conditions cause the sampler to overcompensate and s...

work page 2022

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2018; Krishna et al

have demonstrated a successful re- covery of the spin magnitude distribution with a sim- ilar number of events, these used a waveform model without higher-order modes, with the relative binning likelihood approximation (Cornish 2010; Zackay et al. 2018; Krishna et al. 2023). A stealth bias at the individual-event sampling stage due to sampling diffi- cult...

work page 2024

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17, it is clear that the variance increases for smaller values off RMR and larger values ofκ

From the correlations shown in Fig. 17, it is clear that the variance increases for smaller values off RMR and larger values ofκ. This can be understood as a de- crease in the number of effective samples going into the MC integral as the VM distribution becomes narrower (κincreases) and more strongly peaked at one specific value (the full population is in...

work page 2020

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While we only compare the mixed-waveform results to the NRSur7dq4 results in the figure, we gen- erally find an unbiased recovery of the mass power-law indices when analyzing only the IMRPhenomXPHM- generated subset of each population as well. This can likely be explained by the inconsistency in the choice of waveform between the source simulation and ind...

work page 2022