Accelerated inference of microlensed gravitational waves with machine learning

TL;DR

The paper addresses the challenge of efficiently inferring microlensed gravitational-wave signals in the wave-optics regime by integrating an accurate diffraction-lensing solver (GLoW) with a simulation-based neural posterior estimator (DINGO). By training normalizing-flow-based posteriors on simulated lensed and unlensed signals and employing GNPE for arrival-time alignment plus IS for corrective sampling, the authors achieve rapid parameter estimation that closely matches traditional Bilby results while reducing inference time from days to hours (with IS) or minutes (standalone). The framework enables fast identification of lensed events and scalable population studies, while also providing a diagnostic via sampling efficiency to flag out-of-distribution data. The study also discusses limitations in highly lensing-extreme cases and outlines clear avenues for extending the method to more complex lens models and multi-signal scenarios, facilitating real-time GW lensing analyses in upcoming observing runs.

Abstract

Gravitational waves (GWs) propagating through the universe can be microlensed by stellar and intermediate-mass objects. Lensing induces frequency-dependent amplification of GWs, which can be computed using \texttt{GLoW}, an accurate code suitable for evaluating this factor for generic lens models and arbitrary impact parameters depending on the lens configuration. For parameter inference, we employ the DINGO algorithm, a machine learning framework based on neural posterior estimation, a simulation-based inference method that uses normalizing flows to efficiently approximate posterior distributions of the physical parameters. As a proof-of-principle, we demonstrate that it enables efficient parameter estimation of diffracted GW signals using an isolated point mass lens model. This method can be useful for rapidly identifying microlensed events within large GW catalogs and for conducting population studies of compact binaries. Compared to traditional parameter estimation techniques, we find that combining DINGO with importance sampling can provide efficient estimation of the background Bayes-factor distribution, which is required in evaluating the significance of candidate lensed events. However, for foreground (lensed) events, care must be taken, as sampling efficiency can decrease when the lensed data lie outside the distribution learned by the unlensed DINGO network. Our framework can be naturally extended to more complex and realistic lens models, allowing detailed analyses of the microlensed GWs.

Figures (10)

• Figure 1: Lensing Amplification factor $F(w)$ for a point mass lens at different impact parameters $y$, in units of the Einstein radius $R_E$. The WO regime is near $w\sim1$, whereas for $w\gg1$ the GO approximation is valid. The smaller impact parameters modulate the waveform much more than the larger impact parameters, such that for $y\gg1$ the lensing effect is negligible $F(w) \sim 1$.
• Figure 2: Frequency domain GW strain amplitude of a GW150914-like binary black-hole merger lensed by a point mass with $M_{Lz}=1000 M_\odot$ and different values of $y$. The lensed waveforms have a richer modulation structure than unlensed (black) one, as they contain the WO modulations $F(f)$. We show the noise spectral density in grey. The $y=5$ (green) lensed waveform is similar to the unlensed one, as the effect of lensing is minimal and non-identifiable at the current network sensitivities. The parameters of the binary are mentioned in the appendix.
• Figure 3: Evolution of the training and validation loss for both DINGO (unlensed) and DINGO (lensed) as a function of epochs. It demonstrates similar training and validation losses, indicating no overfitting. Note that one cannot compare directly the losses of the two networks as they have different priors and number of parameters.
• Figure 4: $p-p$ plots comparing the posterior distributions predicted by DINGO without IS. The alignment with the diagonal line indicates a good calibration of the model, that is the inferred parameters fall within the predicted posterior probability regions at the correct rates. Except the coalescence time $t_c$, all the parameters are recovered well. We divided in extrinsic and intrinsic parameters for better visualization.
• Figure 5: A lensed injection with $y=1.2$ analysed using DINGO (lensed). Posterior distributions comparing DINGO (orange), DINGO-IS (blue), and Bilby (black). Contours represent 50% and 90% credible regions. Vertical and horizontal lines mark the true injected values (see Tab. \ref{['tab:injected_params_comparison']}), corresponding to an optimal SNR of 18. For this injection, $\epsilon=0.74\%,$$n_{\rm eff}=7442$, Bilby $\log Z_L=134.7$ compared to DINGO (lensed) $\log Z=133.9$.