Multi-messenger Analysis of Supermassive Black Hole Binaries: The Joint-likelihood Approach

TL;DR

The paper develops a true joint-likelihood MMA to jointly analyze EM time-domain lightcurves and nanohertz GW data from PTAs for supermassive black hole binaries. Using 208 LSST-like EM signals and IPTA-like PTA realizations, the study shows that the MMA yields tighter, more information-rich parameter posteriors than EM-only, GW-only, or EM-prior approaches, with the largest gains in the total mass $M_{ m tot}$ and inclination cosines. The analysis demonstrates that the GW frequency $f_{ m gw}$ is well constrained across methods, while the MMA notably improves constraints on $M_{ m tot}$ and $ heta$, as quantified by higher KL divergences from priors. While the study relies on simplifying assumptions (e.g., Doppler-boost EM signature, circular orbits, Earth-term PTA analysis), it provides a robust proof-of-concept that true joint multi-messenger inference can enhance SMBHB parameter estimation and motivate more realistic MMA pipelines in the future.

Abstract

Supermassive black hole binaries (SMBHBs) formed in galaxy mergers are promising multi-messenger sources. They can be identified as quasars with periodic variability in electromagnetic (EM) time-domain surveys. The most massive of those systems can be detected by Pulsar Timing Arrays (PTAs) in the nanohertz frequency gravitational-wave (GW) band. We present a method to simultaneously analyze EM lightcurves and PTA observations as a multi-messenger data stream. For this, we employ a joint likelihood analysis, in which the likelihood of the EM data and the PTA likelihood are multiplied. We test this approach by simulating 208 binary signals that can be detected both by the Rubin Observatory in the nominal ten-year survey and by a PTA dataset with a ~30-year baseline, which resembles our expectations for a dataset of the International Pulsar Timing Array (IPTA) collaboration in ~2035. We compare our multi-messenger analysis with analyses that take into account the EM and PTA data separately. We find that the joint likelihood approach results in improved parameter estimation with smaller percent errors compared to the distinct analyses that consider only EM or PTA data separately. Among the SMBHB parameters, the binary total mass and the orbital inclination show the greatest improvement. We also compare our multi-messenger pipeline with an analysis, in which the EM constraints are used as priors to the PTA analysis. We demonstrate that the joint likelihood approach delivers tighter constraints on all binary parameters, with systematically higher values of Kullback-Leibler divergence, which measures the deviation of the posterior distribution from the prior.

Figures (4)

• Figure 1: Posterior distributions of the binary parameters, obtained with four different analyses. With dark orange we show the GW Only analysis, dark green the EM Only analysis, purple the MMA and gray the GW + EM Prior analyses, respectively. The dashed lines show the simulated parameters. We also show a zoom-in version of the 2D posterior distribution of the total mass and the orbital inclination, for which the MMA analysis provided the tightest constraints. The simulated source for which this analysis was performed has $SNR\sim6$ in the 30-year IPTA dataset that we consider.
• Figure 2: Distributions of the percent error $\delta X[\%]$ for the six binary parameters we examine among the 208 binary simulations. With dotted dark green lines we show the EM Only analysis, with dashed dark orange lines the GW Only analysis, and with solid purple lines the MMA analysis. Vertical dashed gray lines indicate 0% error.
• Figure 3: Distributions of KL divergence for the six binary parameters we examine among the 208 binary simulations. The KL divergence quantifies the deviations observed in the posteriors compared to the priors, with high KL divergence values indicating that the data are informative, while low values indicate that the data are not very informative. We observe the highest KL divergence values for the GW frequency $f_{\rm gw}$, which is the best constrained parameter, and the lowest values for the polarization angle, $\psi$, which is hard to constrain. Color coding and line styles as in Figure \ref{['fig:percent_error']}.
• Figure 4: Comparison of KL Divergence of the posteriors of the MMA analysis versus KL divergence from the GW + EM Prior analysis for each binary parameter we consider. Each point corresponds to a distinct simulation and is color-coded according to its $SNR$ in the 30-year IPTA-like array. Simulations that fall in the blue/red shaded regions of the plots, above/below the equality line obtain better constraints in the MMA/GW + EM Prior analysis.