A binary black hole merger rate comparison within the same metallicity-star formation rate framework

The gist: This work performs an extended analysis by considering BBH merger efficiencies coming from multiple population synthesis codes across a wide range of physical assumptions and parameter’s choices, finding that the tension between theoretical predictions and observational constraints persists.

Tension in Merger Rates

Recent studies suggest that binary population synthesis models, when coupled with observationally based, metallicity-dependent star formation rate density, overpredict the observed local binary black hole (BBH) merger rate density The significance of this tension might vary depending on the specific code and parameters adopted In almost all our metallicity variations, 14 out of 18, none of the models considered predicts a local BBH merger rate within or below the observed 90% credible interval Even in the four most favorable metallicity variations, only ≲ 10% of the models are consistent with the observational constraints

Origin of Discrepancy

The paper shows that such a discrepancy originates from several factors related to metallicity and binary evolution Specifically, it is attributed to the low-metallicity tail contributed by low-mass galaxies and starbursts, as well as from the use of iron abundance rather than oxygen abundance in deriving the metallicity distribution Even literature models that predict moderate BBH merger rates shift toward higher merger rates when combined with observationally motivated metallicity distributions

Methodology and Analysis

The researchers extended previous analyses by standardizing the contribution of the SFRD(z, Z) to multiple binary population synthesis models They combined BBH merger efficiencies, η(Z), compiled by van Son et al. (2025), comprising 101 models based on different population synthesis codes and physical assumptions, with the parametric SFRD(z, Z) introduced by B26 The aim was to obtain R0 predictions from different binary population synthesis codes with the same SFRD(z, Z), derived from galaxy observations They also explored a suite of metallicity variations corresponding to different parameters of the SFRD(z, Z), bracketing observational uncertainties in metallicity measurements

Key Findings and Model Sensitivity

The analysis confirms and extends the findings that the long low-metallicity tail pushes the value of R0 of almost all binary evolution models considered outside of the observed range The paper demonstrates that using an observationally-derived SFRD implies a tension between models’ predictions and data Furthermore, different treatments of delay time distributions can affect the results; flattening the delay time distribution worsens the tension, while steepening it helps in alleviating it The study suggests that models featuring stronger natal kicks and/or non-standard treatments of mass-transfer and common-envelope physics provide the most promising avenue for alleviating the tension with the observed local BBH merger rate

Caveats and Future Directions

The analysis includes caveats regarding binary population-synthesis models, noting that this sample is not comprehensive of all literature models Some literature models predict particularly low R0 values ≲ 1 Gpc−3 yr−1 (e.g., Mennekens & Vanbeveren 2014; Dominik et al. 2015; Kruckow et al. 2018; Belczynski et al. 2020; van Son et al. 2022; Boesky et al. 2024a) A further extension of the study to more population synthesis codes and different model variations is required Moreover, comparisons between theory and observations should be extended to more observables such as redshift evolution, masses, and spins The tension can be slightly alleviated only under extreme assumptions on the minimum galaxy mass (> 108 M⊙), starburst fraction (fsb = 0), and scatter around the fundamental metallicity relation, σFMR = 0.05 Finally, an alternative possibility is to consider binary evolution models predicting delay time distributions steeper than t−1d

Conclusion Summary

The study concludes that metallicity distributions derived from galaxy observations feature a significant low-metallicity tail which naturally increases the local BBH merger rates predicted by binary evolution models Adopting an iron abundance-based metallicity distribution leads to even more populated low-metallicity tails Using metallicity distributions consistent with the latest galaxy observations leads only to ≲ 10% of the binary evolution models to reproduce local BBH merger rates and only in the most favorable metallicity variations Simplified symmetric metallicity distributions tend to underestimate the low-Z tail if the log Z scatter is less than 0.8 dex, leading to lower merger rate densities The tension between models and gravitational-wave data can be slightly alleviated only under extreme assumptions on the minimum galaxy mass (> 108 M⊙), starburst fraction (fsb = 0), and scatter around the fundamental metallicity relation, σFMR = 0.05 An alternative, more intriguing possibility is to consider binary evolution models predicting delay time distributions steeper than t−1d The rates reported here only account for the isolated binary evolution channel

How it works

The BBH merger rate density at cosmic time t is computed using the formula R(t) = Z / d log Z η(Z) / dtd dp / dtd SFRD(t − td, Z) (1) This calculation involves convolving the BBH merger efficiency of different binary evolution models with the metallicity-dependent star formation rate density The SFRD(z, Z) is computed in two ways: (i) the fiducial configuration using galaxy observed statistics and empirical scaling relations, and (ii) the symmetric configuration using a mean redshift-dependent metallicity with a symmetric scatter in log Z

Key Parameters Explored

The researchers explored several variations to assess robustness against uncertainties These variations included:

1. Metallicity distributions derived from galaxy observations, including 18 variations by varying parameters like ZO/H,0 and aMZR

2. The use of iron abundance instead of oxygen abundance in deriving the metallicity distribution

3. Different delay time distribution exponents (αD = −0.5, −1, −1.5) to see the effect on R0

4. Variations in minimum galaxy mass (M⋆,min = 107 M⊙ vs M⋆,min = 108 M⊙), starburst fraction (fSB = 0), and scatter around the fundamental metallicity relation (σFMR = 0.1 vs σFMR = 0.05)

5. Comparison between the fiducial configuration and a symmetric configuration with different scatter values (σZ = 0.1, 0.2, 0.4, 0.8)

Model Comparison

The work compares R0 predicted by the 101 different binary evolution models against the LVK values at z = 0.2 Figure A.1 shows the predicted R0 for the different works considered, and all codes overpredict R0 with respect to observations The comparison between our derived R0 and the original R0 taken at face value can be found in Appendix B This comparison reveals that when a SFRD with extended low-metallicity tail is used, as in the fiducial configuration, all the models predict R0 above the LVK 90% credible interval The effect of the iron correction has a substantial effect, increasing the average R0 by a factor ∼ 3, as predicted by Chrusli´nska et al. (2025), and making the distribution narrower

Conclusion Summary

The study concludes that metallicity distributions derived from galaxy observations feature a significant low-metallicity tail which naturally increases the local BBH merger rates predicted by binary evolution models The tension between models and gravitational-wave data can be slightly alleviated only under extreme assumptions on the minimum galaxy mass (> 108 M⊙), starburst fraction (fsb = 0), and scatter around the fundamental metallicity relation, σFMR = 0.05 An alternative, more intriguing possibility is to consider binary evolution models predicting delay time distributions steeper than t−1d The rates reported here only account for the isolated binary evolution channel The tension between models and gravitational-wave data can be slightly alleviated only under extreme assumptions on the minimum galaxy mass (> 108 M⊙), starburst fraction (fsb = 0), and scatter around the fundamental metallicity relation, σFMR = 0.05 A further extension of the study to more population synthesis codes and different model variations is required The tension between models and gravitational-wave data can be slightly alleviated only under extreme assumptions on the minimum galaxy mass (> 108 M⊙), starburst fraction (fsb = 0), and scatter around the fundamental metallicity relation, σFMR = 0.

1. Improved model selection for binary evolution: The AI system can now prioritize binary population synthesis codes that feature stronger natal kicks and/or non-standard treatments of mass-transfer and common-envelope physics, as these provide the most promising avenue for alleviating the tension with the observed local BBH merger rate.

2. Enhanced metallicity distribution modeling: The system can generate a more robust metallicity distribution by incorporating variations in parameters like ZO/H,0, aMZR, and ∇FMR,0 to encompass observational uncertainties and explicitly account for the low-metallicity tail contributed by low-mass galaxies and starbursts.

3. Tension quantification across parameter space: The AI can perform an extended analysis by considering BBH merger efficiencies coming from multiple population synthesis codes across a wide range of physical assumptions and parameter’s choices, allowing it to quantify how the tension persists or is alleviated based on specific input parameters.

4. Delay time distribution sensitivity analysis: The system can test the impact of different delay time distribution exponents (e.g., αD = −0.5, −1, −1.5) on the predicted merger rate density, revealing that steepening the distribution, instead, helps in alleviating the tension.

5. Iron abundance prioritization: The AI can be tuned to use iron abundance rather than oxygen abundance in deriving the metallicity distribution because it recognizes this is more sensitive to stellar evolution effects, as iron group elements set the strengths of linedriven winds in massive stars.