Machine learning classification of black holes in the mass--spin diagram

TL;DR

The paper proposes a mass–spin diagram as a unifying, HR-like framework to classify black holes across the stellar, intermediate, and supermassive regimes, tracing their evolution through formation, accretion, and mergers over cosmic time. It combines theoretical tracks with a Black Hole Continuum and three synthetic stellar-mass populations to explore how spin evolution models shape the diagram, and applies advanced machine-learning workflows—variational autoencoders, unsupervised clustering, and supervised random forests—to classify overlapping BH populations in latent space. The key findings show that simple unsupervised clustering struggles with overlapping subpopulations, whereas latent-space representations learned by VAEs paired with supervised classifiers can achieve high accuracy on distinct cases and reveal the potential of semi-supervised approaches for more complex datasets. This framework promises a practical bridge between electromagnetic and gravitational-wave observations and theoretical models, enabling improved BH population inference as catalogs expand with future multi-messenger facilities.

Abstract

We present the mass--spin diagram for classifying black holes and studying their formation pathways providing an analogue to the Hertzsprung-Russell diagram. This allows for black hole evolutionary tracks as a function of redshift, combining formation, accretion, and merger histories for the variety of black hole populations. A realistic black hole continuum constructed from initial mass and spin functions and approximate redshift evolution reveals possible black hole main sequences, such as sustained coherent accretion through cosmic time or hierarchical merger trees. In the stellar-mass regime, we use a binary population synthesis software to compare three spin prescriptions for tidal evolution of Wolf-Rayet progenitors, showing how the mass--spin diagram exposes interesting modeling differences. We then classify black hole populations by applying supervised and unsupervised machine learning clustering methods to mass--spin datasets. While bare unsupervised clustering can nearly recover canonical population boundaries (stellar-mass, intermediate-mass, and supermassive), a more sophisticated approach utilizing deep learning via variational autoencoders for latent space representation learning aids in clustering of realistic datasets with subclasses that highly overlap in mass--spin space. We find that a supervised random forest can accurately recover the correct clusters from the learned latent space representation depending on the complexity of the underlying dataset, semi-supervised methods show potential for further development, and the performance of unsupervised classifiers is a great challenge. Our findings motivate future machine learning applications and demonstrate that the mass--spin diagram can be used to connect gravitational-wave and electromagnetic observations with theoretical models.

Figures (7)

• Figure 1: A Hertzsprung-Russell diagram (luminosity vs effective surface temperature) showing the evolutionary tracks of a handful of stars modeled with the COMPAS binary population synthesis software COMPAS2022 through the stellar main sequence and Hertzsprung gap (red dashed line), giant branches (purple dash-dot line), and core helium burning (green dotted line) phases. Some of these stars form into white dwarfs (gray dots) or neutron stars (blue stars).
• Figure 2: A sketch of the mass--spin diagram for astrophysical black holes with dimensionless spin angular momentum (i.e. $\chi = c|\mathbf{S}|/Gm^2$) on the vertical axis. Regions are colored, as indicated in the legend, according to a selection of formation channels of black holes in single, binary, or higher systems across the mass spectrum: binary stars in isolation, dense stellar clusters, canonical population III stars, and hierarchical galaxy mergers. In reality, these formation channels will overlap and their prevalence will depend on redshift. The two green vertical patches with red hatch symbolize the commonly used boundaries between the stellar-mass, intermediate-mass, and supermassive regimes. Typical astrophysical processes that drive the mass and spin growth across the diagram are written in each regime, with small arrows indicating uncertainty in the region borders for mass and spin predictions of formation channels. The solid blue line extending from the stellar-mass regime to the upper right corner of the diagram represents a possible main-sequence analogue for black holes from sustained accretion episodes over cosmic time.
• Figure 3: Theoretical evolutionary tracks of accreting black holes across the mass--spin diagram. Black holes are drawn from initial mass functions and simple natal spin prescriptions, and are evolved from redshift $z = 10$ to $z = 0$ along the trajectories. Stellar (blue dots and lines) and supermassive (red triangles and lines) black holes undergo an approximate thin disk evolution, while intermediate-mass black holes (green squares and lines) experience chaotic accretion modeled with a Gaussian drift. The clusters that result from the flow of points toward the upper-right corner of the panel is a quantitative example of a Cosmic Accretion main sequence (i.e., the blue solid line of Fig. \ref{['F:cartoon']}).
• Figure 4: The predictions of three models for the spin magnitude of the Wolf-Rayet star tidally synchronized with a black hole companion in the isolated formation channel. The masses $m_2$ and spin magnitudes $\chi_2$ correspond to those of the secondary black holes (i.e., formed from the initially less massive star) in binaries from the COMPAS population COMPASdata with color corresponding to the semi-major axis $a_{\rm preSN2}$ prior to its formation in a supernova explosion. The three spin models are indicated by triangles (Qin et al. (2018)), circles (Bavera et al. (2020)) and x's (Steinle & Kesden (2021)). The models of Qin et al. (2018) and Bavera et al. (2020) and of Steinle & Kesden (2021) are numerical and analytical adaptations of the model of 1981AA....99..126H, respectively.
• Figure 5: A simple example of a black hole mass gap, i.e. no intermediate-mass black holes, and distinct and separated mass--spin correlations for stellar-mass and supermassive black holes. Given to bare unsupervised clustering algorithms 2011JMLR...12.2825P, k-means (right panel) and a Gaussian mixture model (left panel) cluster with 100% accuracy.