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A stepping stone toward detecting gravitational wave memory: a cumulative analysis with the full $(\ell=2, m=0)$ spherical harmonic using events from GWTC-4.0 and GWTC-5.0

T0 review · 2 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Stacking current binary-black-hole events gives only moderate evidence for the full (2,0) gravitational-wave mode; decisive detection is projected after roughly 166 events.

desk verdict Solid cumulative search for the full (2,0) mode on O4 data; the stacked log10 B = 1.38 ± 0.79 and memory SNR ~0.9 are clean, while the N_events ~166 forecast is explicitly optimistic and systematics-limited. read the letter →

arxiv 2607.04909 v1 pith:RWWPH3G3 submitted 2026-07-06 gr-qc

Maria Rosselló-Sastre , Sascha Husa , Yumeng Xu , Jorge Valencia , Joan Llobera-Querol , Antoni Ramos-Buades This is my paper · ORCID
classification gr-qc
keywords gravitational-wavememory(20)modeBayesianmodelselectionIMRPhenomTHM_20GWTC-4.0SNRstackingbinaryblackholes
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 tests whether the subdominant (ℓ=2,m=0) spherical-harmonic mode—containing both its oscillatory ringdown piece and the permanent displacement memory—is already present in the binary-black-hole signals of GWTC-4.0 and a few loud GWTC-5.0 events. By comparing two otherwise identical waveform models, one with and one without that mode, the authors obtain a cumulative log-10 Bayes factor of 1.38 ± 0.79 for the full O4a catalog—moderate support that falls short of decisive. Stacking the mode’s signal-to-noise ratio yields a memory-only value near 0.9 after 7.5 months of O4a data, consistent with earlier forecasts. The authors argue that including the oscillatory contribution first strengthens the evidence relative to memory-only searches and therefore serves as a practical intermediate step toward a memory detection. Under the observed distribution of per-event Bayes factors they estimate that decisive evidence (log-10 B − 1σ ≥ 2) would require roughly 166 events, while cautioning that waveform systematics remain uncontrolled.

What carries the argument

Paired Bayesian model comparison of IMRPhenomTHM_20 (full (2,0) mode = oscillatory ringdown + displacement memory) versus IMRPhenomTHM (identical model without the mode), with per-event log-10 Bayes factors stacked under independence and validated by SNR stacking and high-SNR scaling relations.

What would settle it

Accumulate a catalog of ~170 binary-black-hole events at O4/O5 sensitivity, recompute the stacked log-10 Bayes factor with the same two models (and with models that include the (3,±2) multipole), and check whether log-10 B − 1σ reaches or exceeds 2.

Watch

Extended reading notes

Core claim

For the 84 binary-black-hole events of GWTC-4.0 the stacked log-10 Bayes factor between the full-(2,0) model IMRPhenomTHM_20 and the model without that mode is 1.38 ± 0.79; under the fitted empirical distributions of the individual Bayes factors, decisive evidence is expected after N_events = 166^{+82}_{-55} detections at current sensitivity.

Load-bearing premise

That the distribution of Bayes factors measured on the present GWTC-4.0 sample will continue to describe future detections, and that residual waveform modelling errors (especially the missing (3,±2) multipole) do not systematically bias those Bayes factors.

Editorial extensions

If this is right

  • Decisive statistical evidence for the full (2,0) mode is expected before a clean detection of the memory component alone.
  • Controlling the presence of the full (2,0) mode supplies a practical intermediate step that also helps isolate systematics before memory-only searches.
  • Current individual events remain statistically consistent with either model; the (2,0) contribution is still too weak to shift recovered source parameters appreciably.
  • An optimistic total of ~166 events at present sensitivity would suffice for decisive evidence under the measured Bayes-factor distribution.

Reading between the lines

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

  • Once the (2,0) mode is established, the same stacking pipeline can be re-run with the oscillatory piece subtracted to isolate the pure memory contribution and quantify how much extra catalog size it requires.
  • The mild preference already seen for the loudest events (especially GW230814_230901) suggests that a handful of future high-SNR, edge-on systems could accelerate the climb to decisive evidence faster than the average-event forecast.
  • Any future claim of memory detection that does not first demonstrate control of the full (2,0) mode will be vulnerable to the same systematics the authors flag for the missing higher multipoles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript reanalyzes the 84 BBH events of GWTC-4.0 (plus six selected high-SNR GWTC-5.0 events) with the quasi-circular, non-precessing IMRPhenomTHM_20 model that includes both the oscillatory ringdown and displacement-memory pieces of the (2,0) mode. Bayesian model comparison against IMRPhenomTHM yields a stacked log10 B = 1.38 ± 0.79 for the GWTC-4.0 sample; the corresponding stacked memory SNR after ~7.5 months of O4a is 0.89^{+0.29}_{-0.11}. A single precessing candidate is also examined with IMRPhenomTPHM_20 and shows no additional support. From the empirical distribution of the per-event Bayes factors the authors forecast that decisive evidence (log10 B - 1σ ≥ 2) would require N_events = 166^{+82}_{-55} under the assumptions of the present analysis.

Significance. The work supplies the first systematic, catalog-level Bayesian assessment of the full (2,0) multipole (oscillatory + memory) rather than memory alone, using public O4 data, measured PSDs, calibration envelopes, and a carefully documented nested-sampling pipeline. The zero-noise NR and model injections that recover the expected high-SNR scaling of the Bayes factor, the explicit quadrature propagation of evidence uncertainties, and the transparent stacking of both Bayes factors and SNRs are concrete strengths. The cumulative numbers and the N_events forecast give the community a concrete, if optimistic, benchmark for when the (2,0) mode (and ultimately memory) may become detectable with second-generation detectors.

major comments (2)
  1. Sec. V D and the abstract: the optimistic forecast N_events = 166^{+82}_{-55} is obtained by fitting a Student-t (2.85 dof) and a skew-normal to the 84 GWTC-4.0 log10 B values and drawing 500 Monte-Carlo trajectories. The same section notes that the loudest event, GW230814_230901, drops from log10 B = 0.63 ± 0.09 (vs THM) to 0.18 ± 0.10 when compared with the multipole-complete IMRPhenomXHM. Because the stacking formulae (Eqs. 4.3–4.4) treat every event as i.i.d., a coherent bias of even ~0.1–0.2 in the high-SNR tail would shift the median N_events by tens of events. The independence and representativeness assumptions are therefore load-bearing for the central forecast, yet they are not stress-tested against a multipole-complete model for more than one event. A short additional comparison (or an explicit statement that the forecast is conditional on the present mode content) is needed be
  2. Sec. V A / Fig. 1 and Tab. III: the cumulative log10 B = 1.38 ± 0.79 is dominated by a handful of high-SNR events, most notably GW230814_230901. The paper already shows that this event’s Bayes factor is sensitive to the missing (3,±2) multipole. Without a quantitative assessment of how residual waveform systematics propagate into the stacked evidence, the claim that the catalog provides “moderate support” for the (2,0) mode remains provisional. At minimum the authors should recompute the stack after replacing or down-weighting the systematics-sensitive events, or clearly label the result as model-dependent.
minor comments (4)
  1. Fig. 1: the dual color-coding (fill = full SNR, edge = (2,0) SNR) is hard to read once printed in grayscale; a second panel or a different visual encoding would help.
  2. Eq. (4.7)–(4.8): the high-SNR scaling is validated only with zero-noise injections; a brief remark that real noise can flip the sign of O(1) Bayes factors (already noted in Sec. IV E) would strengthen the interpretation of the small measured values.
  3. Appendix A: the relative-runtime plot is useful but the absolute wall-clock times (color bar) are given only for ΔlnZ = 0.10; quoting the corresponding numbers for ΔlnZ = 0.05 would make the cost of the tighter threshold clearer.
  4. Throughout: a few typographical inconsistencies remain (“W A VEFORM”, occasional missing spaces around math mode). A final copy-edit pass would remove them.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: Bayes factors and SNR stacks are direct measurements on real data; the N_events forecast is an explicit empirical bootstrap from the observed distribution, not a tautological derivation.

  1. self citation load bearing [Sec. III (waveform model) and abstract / Sec. V A (model comparison)]
    "As our signal model we use the quasi-circular, non-precessing IMRPhenomTHM_20 waveform model, which includes the oscillatory and displacement memory contributions. … To incorporate the memory effect in this analysis, we employ the phenomenological waveform model of the (ℓ=2,m=0) spherical harmonic mode … [24]. … Both the aligned-spin and the precessing models are implemented within IMRPhenomTHM and IMRPhenomTPHM, respectively, in the Python package phenomxpy [63]."

    The signal hypothesis H_20 is defined by the authors’ own prior papers that construct IMRPhenomTHM_20 / TPHM_20. While the subsequent Bayes-factor computation on real data is independent, the very existence of the mode content being tested rests on that self-citation chain; without it there is no H_20 to compare against H_∅. This is minor (normal model-building practice) and does not force the numerical value of the stacked log10 B.

full rationale

The central results (per-event and stacked log10 B = 1.38 ± 0.79, SNR stacking of the full (2,0)/memory/oscillatory pieces) are obtained by nested sampling of real GWOSC strain against two nested, previously calibrated waveform models (IMRPhenomTHM_20 vs IMRPhenomTHM) with fixed priors and measured PSDs; no free parameter is adjusted to force a positive stacked evidence. The N_events = 166^{+82}_{-55} figure is obtained by maximum-likelihood fitting of a Student-t (2.85 dof) and skew-normal to the same 84 observed (log10 B, σ) pairs, then drawing 500 Monte-Carlo trajectories until the cumulative reaches the Jeffreys threshold of 2 (Sec. V D). This is an ordinary empirical forecast under the stated i.i.d. and representativeness assumptions; the paper itself labels it “optimistic” and “based on the specific assumptions adopted in this work.” Self-citations supply the waveform models and the prior PE pipeline, but those models are NR-calibrated and the data analysis is independent of the citations. No equation reduces by construction to its own input, no uniqueness theorem is imported to forbid alternatives, and no ansatz is smuggled in. Residual waveform systematics (missing (3,±2)) are a correctness risk, not circularity. Score 1 only for the minor self-citation of the authors’ own (2,0) models that define the signal hypothesis being tested.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The analysis rests on standard Bayesian model comparison, public LVK data products, and previously published phenomenological waveform models. The only free parameters introduced for the forecast are the shape parameters of the empirical distributions fitted to the observed log B and σ(log B). No new physical entities are postulated.

free parameters (2)
  • Student-t degrees of freedom for log10 B distribution = 2.85
    Fitted by maximum likelihood to the 84 GWTC-4.0 Bayes factors; used to generate the Monte-Carlo realizations that produce N_events = 166^{+82}_{-55}.
  • skew-normal shape parameter for σ(log10 B) = 8.33
    Fitted to the observed evidence uncertainties; controls the width of the cumulative-error band in the forecast.
assumptions (3)
  • domain assumption Events are statistically independent so that log Bayes factors add.
    Stated explicitly before Eq. (4.3); standard for LVK catalog stacking but ignores possible common calibration or noise systematics.
  • domain assumption IMRPhenomTHM_20 accurately captures the (2,0) mode (oscillatory + memory) for quasi-circular, non-precessing binaries up to mass ratio 10.
    Taken from the authors’ prior calibration paper; residual systematics (especially missing (3,±2)) are acknowledged but not quantified for the full catalog.
  • standard math Jeffreys scale thresholds (log10 B ≥ 2 decisive) remain the appropriate decision criterion.
    Adopted without modification in Sec. IV A and used to set the forecast threshold.

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Cite this review

Pith. "Pith review of A stepping stone toward detecting gravitational wave memory: a cumulative analysis with the full $(\ell=2, m=0)$ spherical harmonic using events from GWTC-4.0 and GWTC-5.0." pith.science (2026). https://pith.science/paper/RWWPH3G3

@misc{pith2026260704909,
  author       = {Pith},
  title        = {Pith review of: A stepping stone toward detecting gravitational wave memory: a cumulative analysis with the full $(\ell=2, m=0)$ spherical harmonic using events from GWTC-4.0 and GWTC-5.0},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWWPH3G3}},
  note         = {Machine review of arXiv:2607.04909}
}
abstract

We perform Bayesian model selection to test for the presence of the $(\ell=2,m=0)$ spherical harmonic mode in gravitational wave events that have previously been identified as binary black hole mergers. As our signal model we use the quasi-circular, non-precessing IMRPhenomTHM_20 waveform model, which includes the oscillatory and displacement memory contributions. Including the oscillatory component of the (2,0) mode increases the signal-to-noise ratio and evidence for this mode, compared to testing only for the presence of gravitational wave memory. Our analysis thus constitutes a natural stepping stone toward detecting gravitational wave memory. We perform our analysis for the binary black hole signals identified in the GWTC-4.0 catalog, and for selected GWTC-5.0 events. In our Bayesian model comparison we find a cumulative $\log_{10}\mathcal{B}=1.38\pm0.79$ in favor of the presence of the (2,0) mode for the GWTC-4.0 catalog. We also stack the signal-to-noise ratio of the full (2,0) mode and of its individual contributions, obtaining results consistent with previous studies and reaching $\mathrm{SNR}_{\mathrm{memory}} = 0.89^{+0.29}_{-0.11}$ after approximately 7.5 months of O4a observations. In addition, we study the precessing candidate GW241127_061008, and find no additional evidence for the (2,0) mode when precession is included in IMRPhenomTPHM_20. Overall, our results provide an assessment of the observational support for the (2,0) mode in current gravitational wave data and allow us to discuss prospects for its future detection. We find that decisive statistical evidence will likely require a larger catalog, with an optimistic estimated number of events of $N_{\mathrm{events}} = 166^{+82}_{-55}$, based on the specific assumptions adopted in this work. We also expect that decisive evidence will require a more extensive waveform systematics study.

Figures

Figures reproduced from arXiv: 2607.04909 by the authors.

Figure 1
Figure 1. FIG. 1. Log-10 Bayes factors for each individual event in increasing SNR of the [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cumulative sum of log-10 Bayes factors as a function of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Posterior distributions of the SNR of the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Cumulative sum of the SNR of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Posterior distributions for the event GW241127_061008. The [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Distribution of the number of events required for the cumu [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Relative runtime difference between the runs performed with [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Corner plot for the NR injection with SXS:BBH:2497, recovering with [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Posterior distributions for the events listed in Tab. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Whitened waveforms for GW230814_230901 (only detected in L1) and GW250114_082203 (detected in both L1 and H1). The gray [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.