Rapid inference of gravitational-wave signals in the time domain using a heterodyned likelihood

TL;DR

The paper tackles the prohibitive cost of time-domain gravitational-wave parameter estimation caused by non-diagonal noise covariance. It introduces a heterodyned likelihood that uses a complex, mode-by-mode ratio $r^{\ell m}(t)$ approximated as a linear function within adaptive time bins, enabling waveform downsampling to bin edges and a reformulated, data-driven summary-data–based likelihood. This approach decouples cost from waveform duration, achieving large speedups (e.g., $\sim$19x for 2 s, $\sim$124–$\sim$340x for 16–128 s signals) while preserving posterior accuracy, demonstrated via BBH and BNS injections and p-p percentile tests. The framework relies on precomputed summary data and efficient handling of Toeplitz covariance matrices, with extensions to subdominant modes, precession, and time-dependent detector responses discussed for future work. Practically, this method makes time-domain analyses of long GW signals tractable, paving the way for real-time or near-real-time Bayesian inference with future detectors.

Abstract

Parameter estimation of gravitational wave signals is computationally intensive and typically requires millions of likelihood evaluations to construct posterior probability distributions. This computational cost increases significantly in the time domain, which requires non-diagonal covariance matrices to compute the likelihood. Consequently, parameter estimation of long-duration gravitational wave signals, such as binary neutron star mergers, becomes computationally infeasible in time domain. In this work, we detail a framework for the heterodyned likelihood that enables rapid inference in the time domain. Our method is applicable to signals with arbitrary mode content, and leverages the smoothness of the ratio of complex-valued waveform modes, approximating the ratio as a linear function within appropriately chosen time bins. This allows downsampling of the waveform modes and a reformulation of the likelihood, such that it depends only on the bin edges. We demonstrate that this likelihood recovers posteriors that are indistinguishable from those obtained using the standard likelihood in the time domain. We also observe dramatic improvement in speed - for a 128 seconds-long gravitational wave signal, our method is at least $\sim 400$ times faster than the standard time-domain analysis, reducing the wall clock time to just a few hours. We also demonstrate the reliability and unbiasedness of the likelihood using percentile-percentile tests for binary black hole and binary neutron star injections. We use the Gohberg-Semencul representation of the inverse of Toeplitz covariance matrix to accelerate matrix-vector products, which has potential applications even in non heterodyned time-domain inference.

Figures (8)

• Figure 1: The ratio of sampled to fiducial (reference) waveform modes, $r^{\ell m} = h^{\ell m}/h^{\ell m}_o$, behaves as a slowly varying function of time. (a) The ratio for (2,2) mode, $r^{22}(t)$, plotted for different sampled chirp masses. (b) The ratio for various $(\ell, m)$ modes, $r^{\ell m}(t)$, at a fixed sampled chirp mass of $28\;M_\odot$. The insets in both figures show three small time bins, demonstrating that the ratio $r^{\ell m}(t)$ can be approximated as a straight line within each time bin. In both panels, the fiducial chirp mass is fixed at $30 \; M_\odot$.
• Figure 2: An illustration of time bins constructed for the inspiral and merger-ringdown part of a GW signal. The upper panel shows that the time bins for early inspiral part are wider. The bin width progressively reduces towards merger, with a few bins very close to merger containing only one data point. The lower panel shows the time bins for merger and ringdown part of signal. Near merger, the bins are narrow, some containing only one data point. The bins widen as the signal decays rapidly during the late ringdown phase.
• Figure 3: Comparison between posterior probability distribution of intrinsic (left panel) and extrinsic (right panel) parameters of a 2-second long GW signal, estimated using time-domain heterodyning, frequency domain heterodyning Krishna:2023bug and full time-domain likelihood method. The time-domain analyses use IMRPhenomT and frequency domain analysis uses IMRPhenomXASPratten:2020fqn waveform approximant. All three methods produce identical posteriors. The maximum JS divergence between both time-domain methods is 0.00191 across all sampled parameters, while the divergence between frequency-domain and time-domain heterodyning is 0.00397. The time-domain heterodyning achieves this accuracy while being 19$\times$ faster than the full likelihood method. The time required for a single likelihood evaluation, and the injection parameters are listed in \ref{['Table:Speedup']} and \ref{['Table:Injection_parameters']}, respectively.
• Figure 4: Comparison between posterior probability distribution of intrinsic (left panel) and extrinsic (right panel) parameters of a 16-second long GW signal, estimated using time-domain heterodyning and frequency domain heterodyning Krishna:2023bug. The time-domain analysis uses IMRPhenomT and frequency domain analysis uses IMRPhenomXASPratten:2020fqn waveform approximant. Both methods produce identical posteriors, with maximum JS divergence of 0.00196 across all sampled parameters. The time required for a single likelihood evaluation, and the injection parameters are listed in \ref{['Table:Speedup']} and \ref{['Table:Injection_parameters']}, respectively.
• Figure 5: Similar to \ref{['fig:16_second_example']}, we show comparison between the posterior probability distribution of intrinsic (left panel) and extrinsic (right panel) parameters of a 128-second aligned-spin BBH signal, recovered using time-domain and frequency-domain heterodyning. Both methods produce identical posteriors, with maximum JS divergence of 0.00133 across all sampled parameters.