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

Technical Summary: Revealing Massive Black Hole Astrophysics via Hierarchical Inference of EMRI Observations

Problem Statement
The Laser Interferometer Space Antenna (LISA) mission is expected to detect gravitational waves (GWs) from extreme mass-ratio inspirals (EMRIs), where compact objects (COs) such as stellar-mass black holes spiral into massive black holes (MBHs). While individual EMRI detections will allow for sub-percent precision measurements of source parameters (MBH mass, spin, CO mass, orbital eccentricity), the primary scientific goal is to constrain the underlying astrophysical population and formation channels. However, the detected EMRI population is subject to strong selection biases dependent on the signal-to-noise ratio (SNR), meaning it does not perfectly represent the true astrophysical distribution. Furthermore, the computational cost of evaluating the selection function (the probability of detecting an event given a population model) is prohibitive due to the need for repeated waveform generation. Additionally, the true astrophysical population may be a complex mixture of formation channels (e.g., loss-cone dynamics, AGN disc migration, tidal capture) that are not perfectly captured by a single parametric model. This paper addresses the challenge of performing robust hierarchical Bayesian inference on EMRI populations to recover hyperparameters governing these distributions, accounting for selection effects and potential model misspecification.

Methodology
The authors employ a hierarchical Bayesian inference framework to estimate population-level hyperparameters ($\Lambda$) from a catalogue of detected EMRIs.

1. Inference Framework: The hyperposterior $p(\Lambda|d)$ is computed by marginalizing over individual event parameters ($\theta_i$) and accounting for the selection function $\alpha(\Lambda)$. The selection function is defined as the integral of the detection probability over the population model.
2. Selection Function Emulation: To overcome the computational bottleneck of calculating the selection function (which requires $\sim 10^5$–$10^6$ waveform evaluations), the authors utilize poplar, a machine-learning-based emulator. This emulator uses multi-layer perceptron (MLP) neural networks trained to map astrophysical hyperparameters directly to the selection function $\alpha$, bypassing the need for repeated waveform generation during the inference loop.
3. Population Models: The study tests the framework on phenomenological population models representing different formation scenarios:
   • Model A: A complex model motivated by hierarchical galaxy assembly and accretion, featuring a Schechter distribution for MBH mass, a skew-normal distribution for CO mass, a Beta distribution for MBH spin, and a Beta distribution for initial eccentricity.
   • Model B: A simplified model using power-law distributions for MBH and CO masses, a truncated normal distribution for spin, and a uniform distribution for eccentricity.
   • Mixed Models: Heterogeneous mixtures (A + B) and homogeneous mixtures (A(1) + A(2), B(1) + B(2)) with varying hyperparameters to test the ability to disentangle subpopulations.
4. Simulation and Analysis: The authors generate simulated EMRI catalogues with varying numbers of injections ($10^2$ to $10^4$). They assume an SNR threshold of 20 for detection. Parameter estimation for individual events is approximated using a Fisher matrix analysis with a multivariate Gaussian likelihood, offset by noise realizations. Nested sampling (nessai) is used to explore the hyperparameter space.

Key Contributions

• Validation of ML-Emulated Inference: The paper demonstrates that integrating ML-based emulators for the selection function allows for unbiased and computationally efficient hierarchical inference of EMRI populations.
• Quantification of Constraining Power: The study quantifies how the precision of recovered hyperparameters scales with the number of detections ($N_{det}$) and the structural complexity of the underlying distributions.
• Analysis of Mixed Populations: The work investigates the framework's ability to disentangle heterogeneous (different functional forms) and homogeneous (same functional form, different parameters) subpopulations within a single catalogue.
• Robustness to Misspecification: The authors systematically test the inference framework under model misspecification (e.g., inferring a mixed population using a single-component model) to identify systematic biases and assess the resilience of population-level conclusions.

Results

• Single-Component Recovery: For single-component models (A and B), the framework accurately recovers injected hyperparameters. Parameters governing distributions with sharp features (e.g., the peak of the Schechter mass distribution in Model A, or the narrow width of the Gaussian spin distribution in Model B) are constrained with high precision (sub-percent to few percent) even with $\sim 20$ detections. Parameters governing broad distributions (e.g., power-law slopes) show larger uncertainties.
• Mixed Population Disentanglement:
  • Heterogeneous Mixtures (A + B): The framework successfully separates distinct subpopulations. With $\sim 20$ detections, the branching fraction ($w$) and the hyperparameters of the dominant component are recovered with percent-level accuracy.
  • Homogeneous Mixtures (A(1) + A(2)): The framework can identify internal diversity within a single model type. While correlations between subpopulation parameters exist, the inference retains sensitivity to multimodality, avoiding a simple averaging of the subpopulations when sufficient data ($>100$ detections) is available.
• Model Misspecification: When the assumed model does not match the true population (e.g., inferring a mixed A+B population using only Model A), systematic biases arise. The inference tends to recover the dominant component's features while smoothing over or misrepresenting subdominant features. For instance, a single-component model inferring a mixed population will shift hyperparameters to approximate the dominant distribution, leading to broadened credible intervals and shifted means. However, the framework retains sensitivity to key physical trends (e.g., spin distributions) even under incorrect assumptions, suggesting that qualitative insights remain accessible.

Significance and Claims
The paper claims that hierarchical population inference is a robust tool for extracting astrophysical insights from future LISA EMRI catalogues, even in the presence of model misspecification. The results suggest that:

1. Precision is Feature-Dependent: The ability to constrain population parameters is intrinsically linked to the presence of sharp features in the underlying distributions; populations with distinct peaks or narrow widths are more easily constrained than those with broad power-law tails.
2. Disentanglement is Feasible: It is possible to distinguish between different formation channels and quantify their relative contributions with as few as $\sim 20$ detections, provided the channels produce distinct signatures in the parameter space.
3. Resilience to Imperfect Models: While incorrect functional assumptions lead to systematic biases, the inference framework does not fail catastrophically. It preserves sensitivity to key population trends, implying that simplified phenomenological models can still yield meaningful constraints on EMRI formation channels, provided that model selection is performed alongside the inference.
4. Methodological Guidance: The study provides guidance for constructing realistic astrophysical population models for LISA, emphasizing the need for flexible models that can capture sharp features and the importance of accounting for selection effects via ML emulation to avoid biased population estimates.