Identification and characterization of distorted gravitational waves by lensing using deep learning

TL;DR

This work tackles the computational bottleneck of identifying and characterizing gravitational-wave lensing by introducing DINGO-lensing, a simulation-based neural posterior estimation framework. By training neural networks to learn the posterior $q_{\phi}(\boldsymbol{\theta}|d)$ for lensed GW signals and employing importance sampling to correct to the true likelihood, the authors achieve Bayesian-like inference in seconds rather than weeks. The method accurately recovers lensing observables such as the time delay $\Delta t$ and relative magnification $\mu_{\rm rel}$, validates posterior calibration against traditional pipelines, and demonstrates capability to identify lensing signatures including diffraction by point masses. The approach enables scalable background estimation and population studies for upcoming observing runs, with potential extensions to more complex lens models and real-time analyses.

Abstract

Gravitational waves (GWs) can be distorted by intervening mass distributions while propagating, leading to frequency-dependent modulations that imprint a distinct signature on the observed waveforms. Bayesian inference for GW lensing with conventional sampling methods is costly, and the problem is exacerbated by the rapidly growing GW catalog. Moreover, assessing the statistical significance of lensed candidates requires thousands, if not millions, of simulations to estimate the background from noise fluctuations and waveform systematics, which is infeasible with standard samplers. We present a novel method, \texttt{DINGO-lensing}, for performing inference on lensed GWs, extending the neural posterior estimation framework \texttt{DINGO}. By comparing our results with those using conventional samplers, we show that the compute time of parameter estimation of lensed GWs can be reduced from weeks to seconds, while preserving accuracy both in the posterior distributions and the evidence ratios. We train our neural networks with LIGO detector noise at design sensitivity and a lens model that accommodates two overlapping chirps with opposite parity. We show that the lensing parameters are recovered with millisecond precision for the time delays. We also demonstrate that our network can identify signals diffracted by point masses, highlighting its flexibility for searches. By simulating thousands of lensed and nonlensed events, we determine how the detectability changes with different source properties. \texttt{DINGO-lensing} provides a scalable and efficient avenue for identifying and characterizing gravitationally lensed GW events in the upcoming observing runs.

Figures (12)

• Figure 1: The loss as a function of the number of training epochs of the lensed network, shown for both the training set (dashed) and testing set (solid).
• Figure 2: P-P plot for the lensed neural posterior estimation model using $10^3$ lensed injections and no importance sampling. For each injection, we generate a posterior with our model and compute the percentile value of the injected parameter. Each colored line corresponds to the cumulative distribution function (CDF) of the corresponding parameters (see label). The Kolmogorov-Smirnov test $p$-values are given in the legend.
• Figure 3: Posterior distribution of an injection recovered using our DINGO-lensing (solid blue curves) and bilby (dashed gray curves), respectively. The two methods agree with each other and both recovered the injected values (solid gray lines; also tabulated in Tab. \ref{['tab:priors']}) well.
• Figure 4: P-P plot showing the calibration of the posterior distributions for the lensing parameters $\Delta t$ and $\mu_{\rm rel}$, obtained from $10^3$ simulated injections of point-mass–lensed GW signals. Each curve compares the cumulative fraction of true parameter values enclosed within given credible levels against the ideal uniform expectation (black dashed line). The light-gray band indicates the expected $3\sigma$ statistical uncertainty for a perfectly calibrated inference. The analysis uses the geometrical optics approximation to recover signals generated with a point-lens model, demonstrating accurate recovery of both the time delay and relative magnification distributions.
• Figure 5: The injection corresponds to a point-mass lens with $M_{\mathrm{Lz}} = 900M_{\odot}$ and $y = 0.3$. The inferred $\Delta t = 10.47 \pm 0.57~\mathrm{ms}$ and $\mu_{\rm rel} = 0.66\pm0.17$ are consistent with the geometrical optics predictions of $\Delta t = 11~\mathrm{ms}$ and $\mu_{\rm rel} = 0.5$, demonstrating the capability of the geometrical optics approximation to accurately reproduce lensing-induced distortions.