Getting More Out of Black Hole Superradiance: a Statistically Rigorous Approach to Ultralight Boson Constraints from Black Hole Spin Measurements

Black hole (BH) superradiance can provide strong constraints on the properties of ultralight bosons (ULBs). While most of the previous work has focused on the theoretical predictions, here we investigate the most suitable statistical framework to constrain ULB masses and self-interactions using BH spin measurements. We argue that a Bayesian approach based on a simple timescales analysis provides a clear statistical interpretation, deals with limitations regarding the reproducibility of existing BH analyses, incorporates the full information from BH data, and allows us to include additional nuisance parameters or to perform hierarchical modelling with BH populations in the future. We demonstrate the feasibility of our approach using mass and spin posterior samples for the X-ray binary BH M33 X-7 and, for the first time in this context, the supermassive BH IRAS 09149-6206. We explain the differences to existing ULB constraints in the literature and illustrate the effects of various assumptions about the superradiance process (equilibrium regime vs cloud collapse, higher occupation levels). As a result, our procedure yields the most statistically rigorous ULB constraints available in the literature, with important implications for the QCD axion and axion-like particles. We encourage all groups analysing BH data to publish likelihood functions or posterior samples as supplementary material to facilitate this type of analysis, and for theory developments to compress their findings to effective timescale modifications.

💡 Research Summary

The paper presents a statistically rigorous method for constraining ultralight bosons (ULBs), such as the QCD axion and axion‑like particles, using black‑hole (BH) spin measurements. While previous works have relied on theoretical superradiance (SR) rates and simple exclusion plots, they often ignored the full posterior information of BH mass and spin, treated uncertainties in an ad‑hoc way, and lacked reproducibility. Hoof et al. propose a Bayesian framework that directly incorporates the posterior samples of BH mass (M) and dimensionless spin (a*) as the likelihood, and compares the SR growth timescale τ_SR = Γ⁻¹ with an astrophysical spin‑down timescale τ_BH that encapsulates all other processes (accretion, mergers, etc.). If τ_SR < τ_BH, SR would have spun down the BH appreciably; therefore, any combination of boson mass μ and self‑interaction strength λ that predicts τ_SR < τ_BH is ruled out by the observed high spin.

The theoretical backbone follows the standard Kerr‑BH background. The massive scalar field obeys the Klein‑Gordon equation, leading to complex eigenfrequencies ω_nlm = ω_R + i ω_I. The SR condition α/l ≤ 1/2 (with α = GMμ) selects superradiant modes; the growth rate is Γ_nlm = 2 ω_I. The authors compute Γ using three approaches: the non‑relativistic approximation (NRA), a semi‑analytic continued‑fraction method (CFM) they implement themselves, and the publicly available SuperRad code. Cross‑validation shows agreement within an order of magnitude, with relativistic corrections at the 20 % level for the dominant |211⟩ mode. Next‑to‑leading‑order (NLO) corrections are included as a starting point for the root‑finding algorithm.

The cloud occupation number grows exponentially, N_cloud ∝ e^{Γt}, and extracts spin Δa* from the BH. The authors adopt the analytic estimate N_Δ ≈ 10^{76}(M/10 M_⊙)(Δa*/0.1) to relate a given occupation to a measurable spin reduction. The astrophysical spin‑down time τ_BH is taken conservatively as 10⁸–10⁹ yr, reflecting uncertainties in accretion histories and merger rates. By evaluating the ratio τ_SR/τ_BH for each posterior sample, they obtain a posterior probability distribution over (μ, λ) that fully accounts for measurement errors, correlations, and theoretical uncertainties.

The method is applied to two well‑studied BHs: the X‑ray binary M33 X‑7 (M ≈ 70 M_⊙, a* ≈ 0.84) and the supermassive BH IRAS 09149‑6206 (M ≈ 10⁸ M_⊙, a* ≈ 0.94). Both exhibit high spins, making them powerful probes of SR. For a free scalar (λ = 0), the Bayesian analysis excludes μ ≳ 1.2 × 10⁻¹¹ eV (M33 X‑7) and μ ≳ 8 × 10⁻¹⁹ eV (IRAS 09149‑6206) at the 95 % credible level. When self‑interactions are included, the exclusion region shrinks; only for λ ≳ 10⁻⁴⁰ GeV⁻² does SR remain fast enough to conflict with the data. The authors also explore alternative assumptions—equilibrium versus cloud‑collapse regimes, higher occupation levels, and multi‑mode contributions—and find that the constraints are robust against these variations.

Compared with earlier frequentist “trajectory” studies, the Bayesian timescale approach is computationally cheaper, fully utilizes the published posterior samples, and yields a transparent statistical interpretation. It also naturally extends to hierarchical modelling of BH populations, allowing future inclusion of many more sources and additional nuisance parameters (e.g., accretion efficiencies). The authors advocate that observational teams publish likelihoods or posterior samples alongside their papers, and that theorists compress their results into effective timescale modifications, to facilitate community‑wide adoption of this rigorous framework.

In summary, the paper delivers the most statistically sound ultralight‑boson limits to date, demonstrates how Bayesian timescale analysis can be applied to individual BH spin measurements, and sets a clear path toward population‑level constraints that will sharpen the search for the QCD axion and other light bosonic dark‑matter candidates.