Optimal cross-correlation technique to search for strongly lensed gravitational waves

TL;DR

The paper tackles the challenge of efficiently identifying strongly lensed gravitational-wave events without resorting to computationally expensive joint parameter estimation. It introduces OCCAM, an optimal cross-correlation statistic with mild model dependence that leverages the detectors’ noise PSD via a time-domain inner product and normalizes outputs to a 0–1 scale. On simulated astrophysical populations, OCCAM achieves high lensed-detection performance and favorable ROC curves, outperforming cheaper chi-squared baselines, and demonstrates significant gains when combining a network of detectors. The approach substantially reduces false positives and enables rapid scanning of thousands of CBC events, including sub-threshold candidates, while remaining computationally light and compatible with sky-localization information for further refinement.

Abstract

As the number of detected gravitational wave (GW) events increases with the improved sensitivity of the observatories, detecting strongly lensed pairs of events is becoming a real possibility. Identifying such lensed pairs, however, remains challenging due to the computational cost and/or the reliance on prior knowledge of source parameters in existing methods. This study investigates a novel approach, Optimal Cross-Correlation Analysis for Multiplets (OCCAM), applied to strain data from one or more detectors for Compact Binary Coalescence (CBC) events identified by GW searches, using an optimal, mildly model-dependent, low computation cost approach to identify strongly lensed candidates. This technique efficiently narrows the search space, allowing for more sensitive, but (much) higher latency, algorithms to refine the results further. We demonstrate that our method performs significantly better than other computationally inexpensive methods. In particular, we achieve 97 percent (80 percent) lensed event detection at a pairwise false positive probability of approximately 13 percent (7 percent) for a single detector with LIGO design sensitivity, assuming an SNR greater than or equal to 10 astrophysically motivated lensed and unlensed populations. Thus, this method, using a network of detectors and in conjunction with sky-localisation information, can enormously reduce the false positive probability, making it highly viable to efficiently and quickly search for lensing pairs among thousands of events, including the sub-threshold candidates.

Figures (10)

• Figure 1: Flowchart illustrating the application of the cross-correlation method for identifying strongly lensed GW candidates. Before cross-correlating $s_1$ and $s_2$ according to the formalism developed in this paper, we perform time slicing on $s_1$ and $s_2$, as shown in Fig. \ref{['fig:cross-correlation']}. After performing cross-correlation, we normalize the CC output, $\rho_\text{CC}$ using Eq. \ref{['eq:cc-SNR-max-norm']} to obtain the final output, $\hat{\rho}_\text{CC}$. We take this output to be the result of our method.
• Figure 2: Figure displaying chopping width in time-slicing operation. The two subfigures on the left(right) designate lensed(unlensed) scenario, while the top(bottom) row shows $s_1(s_2)$. The data (light blue) contains noise and the signal (blue). We chop the data such that $t-t_\text{coa} \in [-\tau_\text{chirp}, 10\tau_\text{QNM}]$, which is depicted by the shaded area (light blue). We take these chopped $s_1$ and $s_2$ and perform the cross-correlation (Eq. \ref{['eq:cc-SNR-maximized']}), after making sure that the time of coalescences, $t_\text{coa}$ (shown as blue dashed line), of the events are aligned. For the above example, the CC statistic, $\hat{\rho}_\text{CC}$, was $0.97$ for lensed and $0.17$ for unlensed.
• Figure 3: Histogram (top) and ROC with AUC (bottom) shown for cross-correlation analysis performed on simulated lensed and unlensed events based on a realistic astrophysical population with $\rho_\text{MF}^\text{opt} \geqslant 10$. $\hat{\rho}_\text{CC}$ (blue) corresponds to the method described in the text of background analysis, while we compare our method with that of the $\chi^2$ based analysis gholapStatisticIdentification2025 (orange). Histogram is shown only for $\hat{\rho}_\text{CC}$. The light blue ROC curves correspond to several different runs with the same injections but in different noise realizations. We also indicate a threshold (black dotted vertical line) in the histogram, and the corresponding (FPP, TPP) coordinate (black dot) in the ROC plot.
• Figure 4: ROC shown after performing multi-detector cross-correlation for a network of HLV detectors for network $\rho_\text{MF}^\text{opt} \geqslant 8$ and individual $\rho_\text{MF}^\text{opt} \geqslant 4$. As desired, the ROC for the network (blue) is much better compared to individual detector responses (black). We have also added the ML-based SLICK pipeline magareSLICKStrongLensing2024 ROC curve (orange) for reference.
• Figure 5: The distribution of $\mathcal{M}$ and $\hat{\rho}_\text{CC}$ values for all unlensed pairs. This clearly shows that the optimal statistic outperforms the simple match by significantly reducing false positives. This partly explains why our method can outperform a direct cross-correlation without an optimized estimator.