lens_pairs¶

A screen for strongly lensed pairs among the catalog events. A galaxy close to the line of sight can split a gravitational wave into two or four images that reach the Earth days to months apart: the network records the same merger several times, as separate events. This mode tests every pair of the selected binary black holes for that, as the first (tier-1) stage of the LVK lensing searches does.

gwtc_analysis lens_pairs                                  # the O4a events (GWTC-4.0), as the LVK search
gwtc_analysis lens_pairs --catalogs GWTC-1 GWTC-2.1 GWTC-3 GWTC-4  # pairs across O1-O4a
gwtc_analysis lens_pairs --catalogs ALL --data-repo s3     # every run, PE files from the faster S3 mirror

It ranks pairs. It does not confirm lensing, which needs a joint parameter estimation of the two signals (the LVK tier-2 and tier-3 analyses, GOLUM and Hanabi).

Why lensed copies look alike¶

A lens changes neither the frequencies nor the frequency evolution of the signal (geometric optics), so the detector-frame masses and the spins of the images are the same, and the images are about an arcsecond apart, so their sky position is the same within the GW localization. What differs is the amplitude (each image has its own magnification μ, so its apparent distance is off by √μ), the arrival time, and for a saddle-point image a phase shift of π/2. See How the redshift is obtained for the (1 + z) degeneracy that hides the true masses.

Method¶

Posterior overlap (Haris et al. 2018 [108]). For two events with posteriors \(p_1, p_2\) and the PE prior \(\pi\),

\[ \mathcal{B}_\text{overlap} = \int \frac{p_1(\theta)\, p_2(\theta)}{\pi(\theta)}\, d\theta , \]

large when the two posteriors sit at the same place compared with the prior volume. θ is the set the LVK uses (GWTC-4.0 lensing [109], App. A.1): the detector-frame component masses, the spin magnitudes, the cosines of the spin tilts and the sky position.

The integral is factorized into an intrinsic part (masses and spins) and a sky part.
Intrinsic: a Gaussian kernel density of each event in the unbounded coordinates (ln 𝓜, logit q, logit a₁/0.99, logit a₂/0.99, artanh cos θ₁, artanh cos θ₂), and the Monte Carlo average of \(\hat p_2/\pi\) on the samples of event 1 and of \(\hat p_1/\pi\) on those of event 2, averaged.
Sky: HEALPix histograms (nside 64) of the samples, smoothed by a 2° beam; \(\mathcal{B}_\text{sky} = 4\pi \sum P_1 P_2 / A_\text{pix}\).
Prior: the LVK PE priors are uniform in the component masses (bilby UniformInComponents chirp mass and mass ratio), uniform in the spin magnitudes up to 0.99, isotropic in the tilts and on the sky. π is that prior over a fixed mass box (1–1000 M☉), so \(\log \mathcal{B}_\text{overlap}\) carries a constant offset, the same for every pair, which changes no ranking.
Pairs whose detector-frame chirp masses cannot overlap (99.8 % intervals disjoint, 5 % margin) get log₁₀ 𝓑 = −30, as in the LVK results.
PE samples: 2000 per event (--pe-samples) of the IMRPhenomXPHM analysis, as hubble_constant.

Time delay. The statistic is \(\mathcal{B}_\text{overlap}\,R\), with \(R = p_L(\Delta t)/p_U(\Delta t)\) (Haris et al. 2018; the LVK \(\mathcal{R}^L_U\)):

\(p_U\): the distribution of the delays of all the pairs of the selection. It follows the runs, the gaps between them and the duty cycle.
\(p_L\): galaxy lenses as singular isothermal spheres with the velocity-dispersion function of SDSS early-type galaxies (Choi et al. 2007 [111]), weighted by their cross-section (∝ σ⁴) and optical depth along the line of sight of each event (source redshift: the catalog median), source position uniform in the Einstein disk, and both images detectable (magnification bias \(\mu_-^{3/2}\) on the fainter one). The delay of the two images is \(\Delta t = (1+z_l)\,\frac{D_l D_s}{c\,D_{ls}}\,2y\,\theta_E^2\), \(\theta_E = 4\pi(\sigma/c)^2 D_{ls}/D_s\). Typical delays are days to a few months; pairs a year or more apart are disfavoured.
The lensed delays are weighted by the observing window, W(Δt) = ∫ I(t) I(t+Δt) dt over the runs of the selected events: a lens whose delay exceeds the runs gives no observed pair. The unlensed density, built from the pairs themselves, includes the window already.
Both densities keep 1 % uniform in log Δt (1.4 minutes to 27 years): where both vanish, a ratio of kernel tails would be arbitrary; this bounds |log₁₀ R| to about 2.

False-positive probability. The FPP of a pair is the fraction of the pairs of the selection with a statistic at least its own, the pair itself included (so FPP ≥ 1/N_pairs): the selection is its own background, valid while almost every pair is unrelated (the expected lensed fraction is ~10⁻³). The LVK computes it on simulated unlensed events instead. With a 1 % threshold, unrelated events alone flag about 1 % of the pairs, so a flagged pair is a candidate for joint analysis, not a detection; a lensed pair would show as an outlier at the top of the ranking.

What the mode does¶

The mode computes the screen itself from the released PE samples; it runs no parameter estimation and reads no strain. It uses scipy (gaussian_kde, at its default bandwidth), healpy and astropy's Planck18 cosmology.

Events. From the GWOSC event lists (eventapi/jsonfull) of the catalogs: FAR ≤ --far-threshold, both source-frame masses ≥ --min-mass, minus --exclude; an update (GWTC-4.1) replaces the entries of the catalog it re-analyses. The redshift of each event is that of the GWOSC list.
Samples. From each PE file, the first of C00:IMRPhenomXPHM-SpinTaylor, C01:IMRPhenomXPHM, C00:IMRPhenomXPHM (else another IMRPhenomXPHM label, not Mixed); the columns mass_1, mass_2, a_1, a_2, cos_tilt_1, cos_tilt_2 (or the cosine of the tilts), ra, dec, geocent_time, extracted once to <pe-cache>/samples_lens/ (the PE file is then deleted unless --keep-pe-files). Then --pe-samples random samples per event (--seed).
Delay of a pair: the difference of the median geocent_time of the two events, in days.
Chirp-mass gate. The detector-frame ln 𝓜 intervals from 0.1 to 99.9 % of the two events, widened by ln 1.05, must intersect; otherwise log₁₀ 𝓑 = −30 and no density is evaluated.
Intrinsic term. π in the unbounded coordinates is the box prior (uniform on \(m_2 \le m_1\) in 1–1000 M☉) times the Jacobian, with \(u_i = a_i/0.99\) and \(c_i = \cos\theta_i\): [ \ln\pi = \ln\frac{2}{999^2} + 2\ln\mathcal{M} + \tfrac25\ln(1+q) - \tfrac65\ln q + \ln q(1-q)
\sum_i \ln u_i(1-u_i) + \sum_i \ln\frac{1-c_i^2}{2} . ]
Sky term. Histogram of the samples at nside 64, smoothed (2° FWHM), floored at 10⁻¹² and normalized; \(\ln\mathcal{B}_\text{sky} = \ln(N_\text{pix}\sum P_1P_2)\).
Lensed delays. 200,000 draws: \(z_s\) among the event redshifts, \(z_l\) uniform in \((0, z_s)\), σ from φ(σ)σ⁴ on 20–600 km/s (φ: σ* = 161 km/s, α = 2.32, β = 2.67), \(y = \sqrt{U}\); weight \(z_s\,(D_lD_{ls}/D_s)^2(1+z_l)^2/E(z_l)\,\mu_-^{3/2}\), \(\mu_- = (1-y)/y\), times W(Δt) over the start–end spans of the runs of the selected events. 20,000 delays are resampled by weight for the kernel in log₁₀ Δt (delays floored at 10⁻³ day).
FPP of a pair: the fraction of the pairs whose statistic is at least its own (itself included).

Not recomputed: the PE posteriors and the event redshifts, taken as released.

Validation: the O4a search of the LVK¶

The GWTC-4.0 lensing paper [109] tested every pair of the 84 O4a binary black holes with FAR < 1 /yr, 3486 pairs, and its data release (Zenodo 18163632) gives the posterior-overlap results of each pair. gwtc_analysis lens_pairs (default catalog GWTC-4) selects the same 84 events.

Run of 14 October 2026 (PE files from the S3 mirror, 2000 samples per event), compared pair by pair with the LVK results (superevent names mapped to the catalog events through their GraceDB times):

	LVK (posterior overlap)	`lens_pairs`
Events, pairs	84, 3486	84, 3486 (the same)
Pairs without chirp-mass overlap (log₁₀ 𝓑 = −30)	2270	1803 (1777 in common)
Pairs flagged at FPP ≤ 1 %	38	34, 28 of them among the LVK's 38
Highest-ranked pair	S231113bw + S231223bg (GW231113_200417 + GW231223_202619); their joint analysis (Hanabi): log₁₀ B = −3.3, not lensed	the same
The LVK's 5 highest pairs	ranks 1–5	ranks 1, 2, 4, 5, 6

Overlap Bayes factor. Over the 429 pairs with LVK log₁₀ 𝓑_overlap > 0, ours equals the LVK's plus a constant 1.1 (the normalization of the reference prior, the same for every pair), with a scatter of 0.2 (robust standard deviation); Spearman rank correlation 0.89. Further down the tail the two differ more, where neither matters (pairs the LVK puts at −30 are at a median −3.5 here).
Time-delay factor. log₁₀ R against the LVK's: Spearman 1.000, Pearson 0.996, median offset −0.29: the isothermal-lens model, restricted to the observing window, reproduces their galaxy-lens delay distribution.
Differences. The FPP here is the rank among the pairs themselves, the LVK's comes from simulated unlensed events, which is why 34 pairs are flagged rather than 38. The largest single discrepancy, S231018by + S231118an (LVK rank 6, rank 78 here), comes from the sky term (ln 𝓑_sky = −0.8): our sky densities are smoothed histograms of 2000 samples of an earlier PE release version. The LVK passed 105 pairs (posterior overlap or Phazap) to tier 2; none survived tier 3.

Limits¶

Tier 1 only: no joint parameter estimation, and no phase consistency test (the LVK runs Phazap beside posterior overlap; a type-II image's π/2 phase is not used here).
Only events with PE: LVK marginal candidates have none. Faint second images below the detection threshold need a targeted search of the strain data (the LVK sub-threshold search), outside this mode.
Events of other catalogs (IAS, OGC), which the O1–O4a search of Barsode et al. [110] includes, are not read.
The lens model is a single isothermal sphere: no ellipticity (four images), no clusters (delays of years).

Outputs¶

lens_pairs.tsv: every pair, ranked: the two events, the delay (days), ln 𝓑 intrinsic and sky, log₁₀ 𝓑_overlap, log₁₀ R, log₁₀ of the statistic and the FPP; lens_pairs.events.tsv: the events.
The HTML report ends with the PE files of this run: for each event the file read, its type (nocosmo, cosmo, combined or the GW170817 bundle) and where it was taken from (Zenodo, the S3 mirror, Galaxy, a PE cache), then the options of the run; see Which PE file is read;
lens_pairs.html: the distribution of the statistic with the FPP threshold, the time-delay densities and R, and the highest-ranked pairs.

Options¶

Option	Default
`--catalogs`	`GWTC-4`	Catalogs whose events are paired; several, or `ALL`. With an update (GWTC-4.1) its events replace those of the catalog it re-analyses.
`--far-threshold`	1	FAR at most this, per year (the LVK lensing selection).
`--min-mass`	3	Both source-frame masses above it: binary black holes.
`--data-repo`	`zenodo`	`s3`, the MinIO mirror of the releases (much faster), or `galaxy`, the public usegalaxy.org history. The files are the same (the latest Zenodo versions, `nocosmo` up to GWTC-3): a file the mirror does not have comes from Zenodo (Which PE file is read).
`--pe-samples`	2000	PE samples per event for the kernel densities.
`--fpp-threshold`	0.01	Pairs flagged in the report.
`--exclude`	—	Events left out (full or common names).
`--pe-cache`	—	PE cache (default: that of hubble_constant).
`--workers`	`4`	Parallel PE downloads.
`--keep-pe-files`	off	Keep the downloaded PE files (only the extracted samples are kept by default).
`--seed`	`1`	Random seed (sample subsets, lens population).
`--out-report`	`lens_pairs.html`	Output HTML report path.
`--out-summary`	`lens_pairs.tsv`	Output TSV of all the pairs, ranked (the events go to <name>.events.tsv).
`--plots-dir`	`lens_pairs_plots`	Directory for the plots.

Every report ends with the options of its run, defaults included.