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

Technical Summary: Identification and Characterization of Distorted Gravitational Waves by Lensing using Deep Learning

Problem Statement
Gravitational waves (GWs) propagating through the universe can be distorted by intervening mass distributions, leading to gravitational lensing. This phenomenon produces frequency-dependent modulations, time delays, and phase shifts in observed waveforms, offering a unique probe for small compact lens populations, halo mass functions, and tests of general relativity. However, confirming the lensed nature of a candidate event is computationally prohibitive using conventional Bayesian inference methods (e.g., Markov chain Monte Carlo or nested sampling). The computational cost of parameter estimation (PE) for lensed GWs is high, and assessing statistical significance requires generating millions of simulations to estimate background noise fluctuations and waveform systematics. With the rapidly growing GW catalog from the LIGO-Virgo-KAGRA (LVK) network, standard samplers cannot keep pace with the volume of data or the need for extensive background estimation.

Methodology
The authors present DINGO-lensing, a novel inference pipeline that extends the neural posterior estimation (NPE) framework DINGO to perform simulation-based inference on lensed GWs.

• Framework: The method utilizes a dual-network structure comprising an embedding network and a normalizing flow. The embedding network reduces raw strain data to a 128-dimensional representation using singular value decomposition (SVD), which is then mapped to a full Bayesian posterior distribution by the flow network.
• Waveform Model: The training dataset consists of $5 \times 10^6$ simulated waveforms using the IMRPhenomXPHM phenomenological model (including higher modes and precession) injected into Gaussian noise corresponding to LIGO's design sensitivity (O5). The lensing model assumes two overlapping images with a constant $\pi/2$ phase shift, characterized by a time delay ($\Delta t$) and a relative magnification ($\mu_{rel}$). This model approximates the strong-lensing regime for point, fold, and cusp singularities.
• Training Strategy: The network is trained in two stages. Stage 0 freezes the SVD layer to stabilize early optimization, while Stage 1 unfreezes all layers for fine-tuning. The loss function minimizes the forward Kullback–Leibler divergence between the true posterior and the neural approximation.
• Importance Sampling: To correct for mismatches between the training distribution and specific data likelihoods without retraining, the authors employ importance sampling. Samples drawn from the neural posterior are reweighted using the true likelihood and prior, allowing for accurate evidence calculation and posterior refinement.
• Validation: The authors validate the model using Probability-Probability (P-P) plots, comparisons with the standard Bayesian sampler bilby, and tests on signals diffracted by point-mass lenses (which include wave-optics effects not explicitly parametrized in the training model).

Key Results

• Computational Efficiency: DINGO-lensing reduces the parameter estimation time for lensed GWs from weeks (using conventional samplers) to approximately 40 seconds per event, while preserving the accuracy of posterior distributions and evidence ratios.
• Parameter Recovery: The network successfully recovers intrinsic binary parameters (masses, spins, sky location) and lensing parameters ($\Delta t$, $\mu_{rel}$) with high fidelity. Time delays are recovered with millisecond precision.
• Generalization to Diffraction: Despite being trained on a geometric optics approximation (two interfering chirps), the network accurately recovers signals generated by point-mass lenses that exhibit wave-optics diffraction effects. The inferred lensing parameters for these signals align with geometric optics predictions, demonstrating the model's flexibility.
• Detectability and Background: By simulating thousands of lensed and non-lensed events, the authors characterize the detectability of lensing signatures. They find that lensed events generally produce large lensing Bayes factors ($B_{lens}$) and significant distances from the non-lensing hypothesis. However, the statistical significance is sensitive to the background; non-lensed events with high component masses or large spins can mimic lensing signatures more frequently, increasing the false alarm rate.
• Calibration: P-P plots confirm that the neural posteriors are well-calibrated, with Kolmogorov-Smirnov test p-values indicating no significant systematic bias.

Significance and Claims
The paper claims that DINGO-lensing provides a scalable and efficient pathway for identifying and characterizing gravitationally lensed GW events in upcoming observing runs. Its primary contribution is overcoming the computational bottlenecks that have historically limited lensing searches, enabling the generation of the massive background simulations required to establish statistical significance (e.g., $3\sigma$ or $5\sigma$ detections) for lensed candidates.

The authors position the method as a two-stage tool for future LVK analyses:

1. Candidate Identification: Using networks trained on average power spectral densities to rapidly screen events and identify candidates via posterior distances or Bayes factors.
2. Specialized Analysis: Training specialized networks for significant candidates to account for non-stationary noise and specific waveform approximants, thereby computing dedicated backgrounds for rigorous statistical validation.

The work concludes that while current results assume Gaussian and stationary noise, the framework establishes a robust foundation for future extensions to more complex lens models, waveform systematics, and real-time identification pipelines. The authors explicitly note that non-Gaussianity and non-stationarity in realistic noise may affect detectability estimates, a topic reserved for future study.