Revealing massive black hole astrophysics: The potential of hierarchical inference with extreme mass-ratio inspiral observations

TL;DR

This paper develops a hierarchical Bayesian framework to infer the population-level properties of extreme mass-ratio inspirals (EMRIs) observed by LISA, accounting for selection effects with an ML-emulated selection function. It compares two phenomenological population models, A and B, and explores their single-component and mixed-population implementations to assess how well LISA can constrain MBH mass spectra, CO mass spectra, MBH spins, and orbital parameters. The results show that sharp features in distributions yield tight constraints (e.g., MBH mass peaks and narrow spin widths), while broader distributions yield weaker but still informative constraints; mixtures can be disentangled with a modest number of detections, and mixed-model misspecification reveals systematic biases that depend on dominance and feature prominence. Overall, hierarchical population inference with EMRI catalogues proves resilient and capable of guiding realistic population modeling for LISA analyses, with practical implications for probing MBH growth and the dynamics of galactic nuclei.

Abstract

Gravitational waves from extreme mass-ratio inspirals (EMRIs) will enable sub-percent measurements of massive black hole parameters and provide access to the demographics of compact objects in galactic nuclei. During the LISA mission, multiple EMRIs are expected to be detected, allowing statistical studies of massive black hole populations and their formation pathways. We perform hierarchical Bayesian inference on simulated EMRI catalogues to assess how well LISA could constrain the astrophysical population using parametrised population models. We test our inference framework on a variety of populations, including heterogeneous and homogeneous mixtures of parametrised subpopulations, and scenarios in which the assumed model is deliberately misspecified. Our results show that population parameters governing distributions with sharp features can be tightly constrained. Mixed populations can be disentangled with as few as $\sim20$ detections, and even with model misspecification, the inference retains sensitivity to key population features. These results demonstrate the capabilities and limitations of EMRI population inference, providing guidance for constructing realistic astrophysical population models for LISA analysis.

Figures (6)

• Figure 1: Posterior predictive distributions (left) and one-dimensional hyperparameter posteriors (right) for Model A. In the predictive distribution panels, the coloured solid curves denote the $90$-th percentile and the dashed curves denote the mean. The colour scheme is consistent across panels: pink, green, and blue correspond to $10^2$, $10^3$, and $10^4$EMRI injections, respectively. For each set of injections $10^2$, $10^3$, and $10^4$ for these simulations we detect $18$, $188$ and $1886$ events, respectively. The solid black curve shows the true distribution reconstructed from the injected hyperparameters, whose values are marked by the vertical black dashed lines in the one-dimensional posteriors. The predictive distributions display the differential number density $\mathrm{d}N/\mathrm{d}\boldsymbol{\theta}_i$ for $\boldsymbol{\theta}_i \in \{\log_{10}(M/M_\odot), \log_{10}(\mu/M_\odot), a, e_0\}$, with symbols defined in Table \ref{['tab:models']}. The one-dimensional posteriors summarise the inferred hyperparameters from the hierarchical population analysis. Each subplot on the right reports the median and associated $90\%$ uncertainties, with colours indicating the number of injected events. The horizontal axis spans the relevant portion of the prior.
• Figure 2: Posterior predictive distributions (left) and one-dimensional hyperparameter posteriors (right) for Model B. The plotting follows the same structure as Figure \ref{['pop_A']}, where the only difference is the number of hyperparameters used to model population Model B, where the number of detections is $21$, $206$ and $2080$, for $10^2$, $10^3$, and $10^4$ injections, respectively.
• Figure 3: Posterior predictive distributions (upper left) and one-dimensional hyperparameter posteriors (right and bottom) for Model $\mathrm{A}+\mathrm{B}$. The plotting follows the same structure as Figure \ref{['pop_A']}, where the only difference is the number of hyperparameters used to model the mixed population, where the number of detections is $19$, $198$ and $1980$, for $10^2$, $10^3$, and $10^4$ injections, respectively. Results for other mixed populations are shown in Appendix \ref{['appendix:mixed-population']}.
• Figure 4: Posterior predictive distribution plots for the population Model $\mathrm{A}+\mathrm{B}$ inferred with Model A (left) and B (right). The coloured solid curves denote the $90$th percentile and the dashed curves denote the mean. The colour scheme is consistent across panels: pink, green, and blue correspond to $10^2$, $10^3$, and $10^4$EMRI injections, respectively. The solid black curve represents the true distribution reconstructed from the injected hyperparameters. We display the differential number density $\mathrm{d}N/\mathrm{d}\boldsymbol{\theta}_i$ for $\boldsymbol{\theta}_i \in \{\log_{10}(M/M_\odot), \log_{10}(\mu/M_\odot), a, e_0\}$, with symbols defined in Table \ref{['tab:models']}.
• Figure 5: Posterior predictive distributions (upper left) and one-dimensional hyperparameter posteriors (right and bottom) for Model $\mathrm{A^{(1)}}+\mathrm{A^{(2)}}$. The plotting follows the same structure as Figure \ref{['pop_A']}, where the only difference is the number of hyperparameters used to model the mixed population, where the number of detections is $19$, $196$ and $1988$, for $10^2$, $10^3$, and $10^4$ injections, respectively.