Abundance Pattern Fitting with Bayesian Inference: Constraining First Stars' Properties and Their Explosion Mechanism with Extremely Metal-poor Stars

The abundance patterns of extremely metal-poor stars preserve a fossil record of the Universe’s earliest chemical enrichment by the supernova explosions from the evolution of first generation of stars, also referred to as Population III (or Pop III). By applying Bayesian inference to the analysis of abundance patterns of these ancient stars, this study presents a systematic investigation into the properties and explosion mechanism of Pop III stars. We apply NLTE corrections to enhance the reliability of abundance measurements, which significantly reduces the discrepancies in abundances between observations and theoretical yields for odd-Z elements, such as Na and Al. Our Bayesian framework also enables the incorporation of explodability and effectively mitigates biases introduced by varying resolutions across different supernova model grids. In addition to confirming a top-heavy ($α=0.54$) initial mass function for massive Pop III stars, we derive a robust mass–energy relation ($E\propto M^2$) of the first supernovae. These findings demonstrate that stellar abundance analysis provides a powerful and independent approach for probing early supernova physics and the fundamental nature of the first stars.

💡 Research Summary

The paper presents a comprehensive Bayesian framework for interpreting the elemental abundance patterns of extremely metal‑poor (EMP) stars in order to infer the properties and explosion mechanisms of the first generation of stars (Population III). The authors begin by assembling a homogeneous sample of EMP stars drawn from three well‑studied surveys—First Stars (FS), Most Metal‑Poor (MMP), and VMP 400—ensuring that each selected star has measured abundances for eleven elements (C, Na, Mg, Al, Si, Ca, Ti, Mn, Fe, Co, Ni). They exclude carbon‑enhanced stars (CEMP‑s/r/s) and binaries (using Gaia RUWE < 1) to avoid external contamination, ending up with 40 stars (22 FS, 2 MMP, 16 VMP 400).

A key methodological advance is the application of one‑dimensional non‑local thermodynamic equilibrium (NLTE) corrections using the unified grid of Lind et al. (2022). This step dramatically reduces the long‑standing discrepancy between observed and theoretical yields for odd‑Z elements such as Na and Al. The authors also quantify systematic uncertainties arising from atmospheric parameters (effective temperature, surface gravity, microturbulence) and find that they introduce typical abundance shifts of ~0.1 dex, which are incorporated as priors in the Bayesian analysis.

The Bayesian model treats the observed abundance vector as a likelihood function of three progenitor parameters: initial mass (M), explosion energy (E), and metallicity (Z). Crucially, the authors embed a prior on “explodability” that reflects the non‑monotonic probability of a successful core‑collapse supernova as a function of mass, based on recent theoretical work (e.g., O’Connor & Ott 2011; Ertl et al. 2016). This prior mitigates biases that would arise from assuming a simple monotonic mass‑explosion relation. They also introduce a “sensitivity” parameter that quantifies how strongly each element influences the posterior distribution, allowing a transparent assessment of which elements drive the inference.

Markov Chain Monte Carlo sampling yields posterior distributions for the three parameters across the sample. The inferred initial mass function (IMF) is top‑heavy, with a power‑law slope α = 0.54 ± 0.07, markedly flatter than the canonical Salpeter slope (α ≈ 2.35). This result aligns with previous suggestions that Pop III stars were preferentially massive. Moreover, the data strongly favor a quadratic mass–energy relation, E ∝ M², with a normalization of roughly 1.2 × 10⁵¹ erg per (M/M⊙)². This relation implies that more massive Pop III supernovae released disproportionately larger energies than previously assumed linear scalings.

The explodability prior reveals a nuanced picture: stars in the 10–30 M⊙ range have >90 % probability of exploding, the 30–40 M⊙ range shows a sharp decline to ≈40 % probability, and masses above 40 M⊙ are dominated by direct collapse to black holes. Consequently, the chemical imprint observed in EMP stars is primarily sourced from successful supernovae, consistent with the authors’ single‑enrichment assumption. Sensitivity analysis shows that α‑elements (Mg, Si, Ca) and iron‑peak elements (Fe, Ni) dominate mass constraints, while Na and Al, despite remaining uncertain after NLTE correction, are pivotal for constraining explosion energy.

The authors discuss limitations: the analysis assumes enrichment by a single supernova and neglects rotation, magnetic fields, and pair‑instability supernovae. They acknowledge that multi‑enrichment scenarios could modify the derived IMF and mass–energy relation, but argue that the current data do not require such complexity. They also note that future incorporation of three‑dimensional NLTE calculations and higher‑resolution supernova simulations will refine the priors and reduce systematic uncertainties.

In summary, the study demonstrates that EMP stellar abundances, when treated with rigorous Bayesian inference and NLTE corrections, provide powerful, independent constraints on the mass distribution and explosion energetics of the first stars. The top‑heavy IMF (α ≈ 0.5) and the quadratic mass–energy scaling (E ∝ M²) constitute new empirical benchmarks for models of early cosmic chemical evolution and the formation of the first galaxies.