How to read the O3 search for lensed gravitational waves

We read the arXiv version 1 of the paper, dated 17 April 2023: Search for gravitational-lensing signatures in the full third observing run of the LIGO–Virgo network. We did not compare it with a journal version. Section, equation and figure numbers below refer to that version. Before reading, work through the exact amplification factor and the matched-filtering chapter.

How to label what you read

Keep four labels apart while reading.

• Published result. A measurement or limit the collaboration reports from data, such as a Bayes factor for a named event.
• Preprint. A claim in a paper we read only as an arXiv version, or read only in part.
• Model prediction. A number that depends on assumed lens and source populations, such as an expected rate.
• Open problem. Something the authors, or the papers they cite, say has not yet been worked out.

Labels describe the evidence behind a statement, not whether it is true. A model prediction can be excellent and a published result can later be revised.

What data were searched

Section 2 describes the catalog. The run's second half added 35 events to the 55 already reported from earlier data, all from the third gravitational-wave transient catalog, GWTC-3. Every test below starts from events that were already detected as ordinary binary mergers. A lensed signal would enter this sample only if the standard searches found it first.

Section 3: two events, one source

Strong lensing by a galaxy can produce several images of one merger, arriving minutes to months apart. Eq. 1 gives each image the factor $\sqrt{\lvert\mu_j\rvert}\,e^{-i\pi n_j}$, with $n_j=0$, $\tfrac12$ and 1 for type I, II and III images. This is the same Morse phase as the saddle-point $-i$ in our chapter on the amplification factor.

Two degeneracies shape everything in this section, and the paper states both.

• A magnification $\mu$ makes the signal look like an unlensed source at distance $D/\sqrt\mu$. A delay looks like a different coalescence time. Neither can be detected from one image alone.
• A type III image, with phase $-\pi$, is the same as rotating the binary's polarization by $\pi/2$, so it too is invisible in one image.

The tests therefore compare pairs. Read them in this order:

1. Fig. 1, quick screening of pairs. Two fast statistics, one based on how much the parameter posteriors of two events overlap and one from machine learning, give a false-positive probability for every pair. Pairs below $10^{-2}$ go to follow-up.
2. Fig. 2, joint analysis of 17 pairs. A full joint fit with the hanabi pipeline gives a lensed-versus-unlensed Bayes factor for each pair. All are below 1. The highest, for GW190620_030421 and GW200216_220804, is still about 100 to 1 against lensing under the paper's fiducial merger-rate model.
3. Table 1, faint companions. A targeted search looks for weaker counterparts of known events with a reduced template bank and a sky-overlap veto. Table 1 lists the top five candidates. The top two passed the false-alarm threshold of one in 69 years and went to a joint analysis with the golum pipeline; both gave mildly negative coherence ratios, so neither supports lensing. The paper stresses that these false-alarm rates say how often noise ranks as high, not how likely a trigger is to be a lensed counterpart.
4. Fig. 3, single images of type II. Of the three types, only type II distorts the waveform of a single image, so it can be tested one event at a time. For GW190412 the paper reports $\log_{10}B=0.60\pm0.16$ for type II against type I and $0.22\pm0.16$ against type III; for GW200129_065458, $0.38\pm0.14$ and $0.24\pm0.14$. Simulations show that GW190412 carries enough higher-harmonic power to test the idea, but its preference could be a waveform-modelling effect, even though it remains with two other waveform families. GW200129_065458 lacks that power and may have a significant glitch under subtraction. The authors conclude the analysis cannot settle either case.

Status: published result, read in the arXiv version. Appendix A, Figs. 7 to 9, describes the simulation campaigns behind the thresholds; we read it only to identify what it contains.

Section 4: microlensing, the subject of this course

Eq. 2 is the model every chapter here has built toward:

where $h_{\rm U}$ is the unlensed waveform and $F$ is the isolated point-mass amplification factor. The paper calls the isolated point mass a first approximation. What to check:

• Priors. $M_{Lz}$ is log-uniform from 1 to $10^5\,M_\odot$, and $p(y)\propto y$ on $[0.1,3]$. The $y$ prior means most of the prior volume has a faint second image, the case our tutorial found hardest to tell from noise.
• Waveform. The unlensed model is IMRPhenomXPHM, which includes precession and higher harmonics, sampled with bilby and dynesty. Unlike our Lab, the unlensed fit can bend masses and spins to absorb part of a lens pattern.
• Mass range. The paper expects detectable effects for lenses up to about $10^5\,M_\odot$.
• Fig. 4. The Bayes factor $\log_{10}B^{\rm Micro}_{\rm U}$ for each event, plotted against the same statistic for simulated unlensed signals. The real events sit within the simulated spread. The authors find no compelling evidence for microlensing.

Status: published result. Fig. 4 is the paper's version of the noise-only row in our tutorial: a lens statistic means nothing until you know what unlensed signals give.

Appendix B: a beating pattern made by noise

Fig. 10 lists the posterior on $M_{Lz}$ and the Bayes factor for each event of the run's second half. Several events have a narrow lens-mass posterior. The paper's example is GW200208_130117, with $\log_{10}B=0.8$. Fig. 11 shows its best-fit waveform, where the lens appears as a lower amplitude for two cycles in the middle of the signal. The authors hypothesize that short-duration noise fluctuations produced an apparent dip that mimics a beating pattern, and call the data inconclusive about microlensing.

Status: the Bayes factor is a published result; the noise explanation is the authors' hypothesis. Compare it with our tutorial, where 14 of 120 noise-only trials beat the lens threshold, the largest preferring a $1150\,M_\odot$ lens.

Section 5: how often lensing should happen

Section 5.1 and Table 2 give the rate of lensed doubles with both images detected, relative to unlensed events, as 90 percent intervals: $1.9$ to $11.0\times10^{-4}$ for galaxy lenses and $0.8$ to $4.4\times10^{-4}$ for cluster lenses. The authors conclude that observing a lensed double is unlikely at current sensitivity. Multiplying by about 90 catalog events gives at most about 0.1 expected pairs, our own arithmetic, so a null result is what the model predicts.

Status: model prediction. The ranges come from assumed lens and binary populations.

Microlenses in a galaxy: an open problem

Most stars that could microlens a gravitational wave sit inside a galaxy, so an isolated point mass is rarely the whole story. Cheung, Gais, Hannuksela and Li studied this case: Stellar-mass microlensing of gravitational waves. We read their abstract and parts of Sections 1 to 3 of arXiv version 1, not the full paper. From what we read:

• They consider stars in the nuclear star cluster of a lensing galaxy.
• In the LIGO band diffraction matters for microlenses below about $100\,M_\odot$.
• The host galaxy changes the configuration qualitatively, so microlenses cannot be treated as isolated point masses when strong lensing is involved.
• For stars of about one solar mass, diffraction significantly suppresses the microlensing magnification. The authors suggest that waves lensed by typical galaxies may therefore be fairly free of contamination by stars.
• More complicated configurations need further study.

Status: preprint as read, with an open problem stated by the authors. An isolated point lens, as in the O3 microlensing search and in our Lab, is the simplest case of this larger problem.

What we reproduced, and what we did not

We did not reproduce any figure or number from the O3 paper. Our recompute guide recomputes the point-lens $F$ of Eq. 2 by two independent routes and runs a toy search with a leading-order chirp in synthetic Gaussian noise. It shares the physics of Section 4 and none of its data, waveform model or sampler.

Checklist for reading a lensing claim

• Which paper version did you read, and which sections in full?
• Is the lens model an isolated point mass, or a lens inside a galaxy?
• Can the unlensed model absorb the effect through distance, coalescence time, polarization, masses or spins?
• Is the lens statistic compared with what unlensed signals and noise produce, as in Fig. 4?
• Have glitches and their subtraction been checked for the event in question?
• Is each number a published result, a preprint claim, a model prediction or an open problem?