Pre-localization of Massive Black Hole Binaries in the Millihertz Band

Author(s)

Zhang, Xue-Ting, Gair, Jonathan, Messenger, Chris, Korsakova, Natalia, Hu, Yi-Ming, Chen, Hong-Yu

Abstract

The space-borne gravitational-wave (GW) detectors will open a new mass and redshift regime, allowing us to observe massive black hole binaries (MBHBs) throughout the Universe. A subset of these systems is expected to produce electromagnetic (EM) counterparts, offering a unique opportunity to follow the continuous evolution of massive black holes through joint GW and EM observations. Realizing this potential, however, requires low-latency, high-throughput data-analysis pipelines that can extract reliable source parameters and sky localizations from space-borne data streams fast enough to trigger EM follow-up. In this work we develop a fast, normalising flow-based inference pipeline designed for early-warning analysis of MBHB signals in a TianQin-like configuration. Our method combines a learned embedding of the detector time series with a neural spline flow (NSF) to perform amortized Bayesian inference, producing posterior samples for the main source parameters in roughly one minute per event. For a representative MBHB whose merger occurs $\sim 15$ minutes after the end of the analyzed GW observation, the pipeline achieves pre-merger sky localizations of order $\sim 20~\mathrm{deg}^2$, recovers the same number of sky modes as a reference parallel-tempered Markov chain Monte Carlo (PTMCMC) analysis, and yields parameter uncertainties of comparable scale, while still operating within a practically useful pre-merger warning window. These results demonstrate that NSF-based inference can deliver accurate, near-real-time parameter estimation for space-borne MBHB GW signals, and that the resulting early-warning localizations are sufficiently precise to make rapid EM follow-up.

Figures

P--P plot assessing the calibration of the NSF posteriors over 1000 events. The dashed black line indicates perfect calibration, and the shaded bands denote the $1\sigma$, $2\sigma$, and $3\sigma$ binomial credible regions. The legend displays the Kolmogorov-Smirnov (K-S) test results for each parameter, listing the \textit{p}-value first to indicate statistical consistency with a uniform distribution, followed by the K-S statistic ($D$) which quantifies the maximum absolute deviation from the diagonal.
Caption P--P plot assessing the calibration of the NSF posteriors over 1000 events. The dashed black line indicates perfect calibration, and the shaded bands denote the $1\sigma$, $2\sigma$, and $3\sigma$ binomial credible regions. The legend displays the Kolmogorov-Smirnov (K-S) test results for each parameter, listing the \textit{p}-value first to indicate statistical consistency with a uniform distribution, followed by the K-S statistic ($D$) which quantifies the maximum absolute deviation from the diagonal.
Comparison of posterior distributions for a representative MBHB injection: \ac{PTMCMC} (slate blue) computed on noise-free data versus trained NSF (terracotta) evaluated on noisy data. The NSF samples represent unfiltered raw outputs without rejection of discrete outliers. Contours indicate joint $3\sigma$ credible regions, with dark cross-hairs marking the injected parameter values.
Caption Comparison of posterior distributions for a representative MBHB injection: \ac{PTMCMC} (slate blue) computed on noise-free data versus trained NSF (terracotta) evaluated on noisy data. The NSF samples represent unfiltered raw outputs without rejection of discrete outliers. Contours indicate joint $3\sigma$ credible regions, with dark cross-hairs marking the injected parameter values.
NSF sky map for the representative MBHB event in ecliptic coordinates. The yellow triangles mark theoretical symmetry-related sky locations induced by the TianQin detector configuration, and the black points show the TianQin's orbit.
Caption NSF sky map for the representative MBHB event in ecliptic coordinates. The yellow triangles mark theoretical symmetry-related sky locations induced by the TianQin detector configuration, and the black points show the TianQin's orbit.
Histogram of the recovered 90\% sky-localization areas for 1000 simulated signals. The source parameters are sampled from the prior distribution, and each signal is injected into an independent random noise realization.
Caption Histogram of the recovered 90\% sky-localization areas for 1000 simulated signals. The source parameters are sampled from the prior distribution, and each signal is injected into an independent random noise realization.
Posterior comparison for case~II, in which the chirp mass is modified relative to the reference injection. The posteriors obtained with \ac{PTMCMC} (slate blue) and the trained NSF (terracotta) are shown for comparison. The dark cross-hairs mark the injected parameter values. As in the main analysis, \ac{PTMCMC} is evaluated on the noise-free dataset, whereas the NSF is trained and evaluated on data that include detector noise.
Caption Posterior comparison for case~II, in which the chirp mass is modified relative to the reference injection. The posteriors obtained with \ac{PTMCMC} (slate blue) and the trained NSF (terracotta) are shown for comparison. The dark cross-hairs mark the injected parameter values. As in the main analysis, \ac{PTMCMC} is evaluated on the noise-free dataset, whereas the NSF is trained and evaluated on data that include detector noise.
Posterior comparison for case~III, in which the symmetric mass ratio is modified relative to the reference injection. The posteriors obtained with \ac{PTMCMC} (slate blue) and the trained NSF (terracotta) are shown for comparison. The dark cross-hairs mark the injected parameter values. As in the main analysis, \ac{PTMCMC} is evaluated on the noise-free dataset, whereas the NSF is trained and evaluated on data that include detector noise.
Caption Posterior comparison for case~III, in which the symmetric mass ratio is modified relative to the reference injection. The posteriors obtained with \ac{PTMCMC} (slate blue) and the trained NSF (terracotta) are shown for comparison. The dark cross-hairs mark the injected parameter values. As in the main analysis, \ac{PTMCMC} is evaluated on the noise-free dataset, whereas the NSF is trained and evaluated on data that include detector noise.
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