Strong gravitational-wave lensing posterior odds

TL;DR

The paper provides a unified Bayesian treatment of strong gravitational-wave lensing detection, showing that posterior odds remain stable as the event catalog grows once lensing time-delay information is incorporated. It demonstrates that selection effects enter Bayes factors as a normalization, but cancel in the posterior odds when the prior is conditioned consistently on the population model, making the posterior odds dependent only on intrinsic rates and the data likelihood. The authors decompose the Bayes factor into a time-delay independent component and a rate-odds component, illustrating how arrival-time priors offset decreasing prior odds in large catalogs. They argue that the posterior odds, rather than the Bayes factor alone or p-values, should be the definitive statistic for lensing detections, and emphasize the necessity of explicit lens and population modeling to avoid biased inferences. The work reconciles previous debates on selection effects, population priors, and catalog size, and provides practical guidance for constructing robust lensing claims in gravitational-wave astronomy.

Abstract

Like light, gravitational waves are gravitationally lensed by intervening massive astrophysical objects, such as galaxies, clusters, black holes, and stars, resulting in a variety of potentially observable gravitational-wave lensing signatures. Searches for gravitational-wave lensing by the LIGO-Virgo-KAGRA (LVK) collaboration have begun. One common method focuses on strong gravitational-wave lensing, which produces multiple "images": repeated copies of the same gravitational wave that differ only in amplitude, arrival time, and overall "Morse phase." The literature identifies two separate approaches to identifying such repeated gravitational-wave events based on frequentist and Bayesian approaches. Several works have discussed selection effects and identified challenges similar to the well-known "birthday problem", namely, the rapidly increasing likelihood of false alarms in an ever-growing catalogue of event pairs. Here, we discuss these problems, unify the different approaches in Bayesian language, and derive the posterior odds for strong lensing. In particular, the Bayes factor and prior odds are sensitive to the number of gravitational-wave events in the data, but the posterior odds are insensitive to it once strong lensing time delays are accounted for. We confirm Lo et al.'s (2020) finding that selection effects enter the Bayes factor as an overall normalisation constant. However, this factor cancels out in the posterior odds and does not affect frequentist approaches to strong lensing detection.

Figures (6)

• Figure 1: Strong lensing produces multiple images (top panel), which can be detected with the ground-based gravitational-wave detectors as repeated events arriving at different times (bottom panel). The repeated gravitational waves ($i=1,2,3,\cdots$) are otherwise identical, but strong lensing can magnify the gravitational-wave amplitude by a factor $|\mu_i|^{1/2}$ (red), delay their arrival time by a time $t_i^d$ (blue), and induce an overall complex phase shift (orange), referred to as the Morse phase $\pi n_i$. The outcome is $n^{\rm i}$ gravitational waves with different amplitudes and (possibly) overall phases, arriving at the gravitational-wave detectors at different times. Strong lensing searches attempt to identify these repeated events.
• Figure 2: A binary black hole is strongly lensed by a galaxy, producing multiple images. The galaxy lens mass distribution is parametrised with $\vec{\theta}_L$, the binary black hole orbital parameters and position with $\vec{\theta}$ and $\vec{y}$, respectively. These multiple images each have a position $\vec{x}_i$, magnification $\mu_i$, arrival time $t_i$, and Morse phase $n_i$ associated with them. These parameters are collectively used to describe the lensed gravitational waves.
• Figure 3: The prior odds $P^L_U$ in favour of lensing as a function of the number of gravitational-wave signals $N$ for a variable number of strong lensing image configurations. Particularly noteworthy is that the prior odds in favour of lensing decrease as the number of events increases (or as time passes). Indeed, the prior probability that any given set of data contains a strongly lensed signal decreases over time. However, the arrival time priors (arrival time odds) increase so that they compensate for this decrease, leaving the posterior odds intact and independent of the number of events. The ratio of lensed-to-unlensed events is fixed at one to a thousand.
• Figure 4: A summary of our strongly lensed population model's mass, redshift, and velocity dispersion priors. The mass model follows the Powerlaw+Peak model with a primary mass slope index $\alpha=3.63$ and mass ratio power-law index $\beta_q=1.26$ (left panel), the redshift distribution traces the merger-rate density convolved with the strong lensing optical depth, the velocity dispersion of the galaxies follows the SDSS galaxy catalog (right panel) and we assume a simple singular isothermal sphere (SIS) lens model for illustrative purposes. The population model and the simulation procedure follow the ler gravitational-wave lensing package Phurailatpam2024LerSimulator. These priors are used to evaluate the ratio of arrival time priors in the lensed-to-unlensed hypothesis and the rate of strong lensing detections.
• Figure 5: Distribution of the expected lensing time-delay differences induced by a population of strong lenses and a binary black hole population tracing the star-formation rate density. Here, we presume a simple singular isothermal sphere (SIS) lens model, which creates two strong lensing images, as an easy illustrative example. We will use the time-delay distribution to estimate the posterior odds.