[Paper Review] Ain't No Mountain High Enough: Semi-Parametric Modeling of LIGO-Virgos Binary Black Hole Mass Distribution

B. Edelman, Z. Doctor+2 more|arXiv (Cornell University)|Sep 13, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy71 references61 citations

TL;DR

This paper introduces a semi-parametric cubic spline perturbation model to refine the primary mass distribution of LIGO-Virgo binary black holes (BBHs) using GWTC-2 data. The method recovers a 35M⊙ peak with >97% credibility, corroborating prior findings linked to pulsational pair-instability supernovae, while revealing potential low-mass features consistent with multi-channel formation or stellar evolution effects.

ABSTRACT

We introduce a semi-parametric model for the primary mass distribution of binary black holes (BBHs) observed with gravitational waves (GWs) that applies a cubic-spline perturbation to a power law. We apply this model to the 46 BBHs included in the second gravitational wave transient catalog (GWTC-2). The spline perturbation model recovers a consistent primary mass distribution with previous results, corroborating the existence of a peak at $35\,M_\odot$ ($>97\%$ credibility) found with the extsc{Powerlaw+Peak} model. The peak could be the result pulsational pair-instability supernovae (PPISNe). The spline perturbation model finds potential signs of additional features in the primary mass distribution at lower masses similar to those previously reported by Tiwari and Fairhurst (2021). However, with fluctuations due to small number statistics, the simpler extsc{Powerlaw+Peak} and extsc{BrokenPowerlaw} models are both still perfectly consistent with observations. Our semi-parametric approach serves as a way to bridge the gap between parametric and non-parametric models to more accurately measure the BBH mass distribution. With larger catalogs we will be able to use this model to resolve possible additional features that could be used to perform cosmological measurements, and will build on our understanding of BBH formation, stellar evolution and nuclear astrophysics.

Motivation & Objective

• To develop a flexible, data-driven method for modeling the BBH primary mass distribution that avoids strong parametric assumptions.
• To test whether observed features in the BBH mass distribution—particularly a peak near 35M⊙—are robust beyond simple parametric models.
• To assess the presence of additional substructures in the mass distribution, especially at low masses, using a model that balances interpretability and flexibility.
• To enable future cosmological measurements by identifying calibrated mass scales such as PISN/PPISN features.

Proposed method

• Uses a truncated power law as the base parametric model for the primary mass distribution.
• Applies a non-parametric cubic spline perturbation to the power law to capture deviations from smooth power-law behavior.
• Employs Bayesian inference with hierarchical modeling to estimate the spline knot locations and amplitudes.
• Performs posterior predictive checks to validate model fit against simpler parametric models like Powerlaw+Peak and BrokenPowerlaw.
• Uses the bilby and GWPopulation software frameworks for likelihood evaluation and sampling.
• Enables adaptive resolution by allowing knot locations to vary in future extensions.

Experimental results

Research questions

• RQ1 Does the 35M⊙ peak in the BBH primary mass distribution persist when using a flexible, non-parametric perturbation to a power law?
• RQ2 Are there additional features in the BBH mass distribution—particularly at low masses—beyond the standard power-law or peak models?
• RQ3 Can the semi-parametric model detect deviations from parametric models that may signal multiple formation channels or stellar physics effects?
• RQ4 How well does the spline model fit the data compared to established parametric models like Powerlaw+Peak and BrokenPowerlaw?
• RQ5 Can this method be extended to multi-dimensional parameter spaces to uncover correlations (e.g., mass-spin) indicative of hierarchical mergers?

Key findings

• The spline perturbation model recovers a 35M⊙ peak in the primary mass distribution with >97% credibility, confirming prior results from the Powerlaw+Peak model.
• The model reveals potential low-mass features in the distribution, consistent with hints reported by Tiwari & Fairhurst (2021) using a non-parametric Gaussian Mixture model.
• Posterior predictive checks show the spline model fits the high-mass structure at least as well as the Powerlaw+Peak model, while offering greater flexibility to capture low-mass excesses.
• The observed features are consistent with astrophysical origins such as pulsational pair-instability supernovae (PPISNe), which could produce a pile-up near 35M⊙.
• The method provides a robust, bias-minimizing approach to measuring mass distribution structure without assuming a specific functional form.
• With larger catalogs, this model can resolve finer features and enable cosmological measurements using calibrated mass scales from PISN/PPISN physics.