The initial mass function (IMF) of the first, metal-free (Population III) stars can in principle be reconstructed by fitting metal-free supernova yield models to the abundance patterns of extremely metal-poor (EMP) stars. Jiang et al. (2024) applied an explodability filter — retaining only supernova models predicted to explode rather than collapse to black holes — to a 406-star EMP sample, inferring an extremely top-heavy Population III IMF, a steep pair-instability-supernova (PISN) mass slope, and a large explosion-energy exponent, while explicitly acknowledging that the inference is unvalidated against several systematics: explodability prescriptions anchored on theory calibrated largely at solar metallicity, with their metallicity dependence unvalidated for metal-free stars, an upward PISN selection bias because PISNe pollute much larger star-forming volumes, unresolved per-star enrichment multiplicity, an uncertain maximum neutron-star mass that sets the remnant-mass explodability threshold, 1D LTE abundance systematics over a restricted element set, and the phenomenological character of the modified IMF. The later literature leaves this open: the same group's 2025 Bayesian follow-up narrowed to mono-enrichment and dropped the PISN channel rather than correcting it; independent work supplies probabilistic multiplicity classifiers, zero-metallicity explosion grids still calibrated at solar metallicity, and PISN-descendant simulations that do not quantify the required selection correction. No published work delivers the bias-corrected IMF determination or a demonstrated non-identifiability; the problem remains open.

Background

The first generation of stars, formed from primordial metal-free gas (Population III, or Pop III, stars), are thought to have been predominantly massive and short-lived, so none survive for direct observation. Their accessible fossil record is chemical: a subset of present-day long-lived low-mass stars with extremely low iron content ([Fe/H]<−2.5\mathrm{[Fe/H]}<-2.5, conventionally called extremely metal-poor, or EMP, stars) formed from gas enriched by one or a few of the first supernovae, and their measured elemental abundances can be compared with nucleosynthesis yields computed for metal-free progenitors — an approach called stellar archaeology. Matching observed abundance patterns to yield grids of core-collapse supernovae (CCSNe) and pair-instability supernovae (PISNe — complete disruptions of roughly 140140–260 M⊙260\ M_\odot stars driven by electron–positron pair production, which synthesize and eject far more metal than CCSNe) recovers each EMP star's progenitor zero-age main-sequence mass and explosion energy, from which the Pop III initial mass function (IMF, the distribution of stellar birth masses) and explosion-energy distribution can in principle be inferred. Earlier analyses (Fraser et al. 2017; Ishigaki et al. 2018) found that the resulting progenitor-mass distribution deviates from a Salpeter-like power law, including a spike near 25 M⊙25\ M_\odot. 'Explodability' theory predicts, from progenitor structure — e.g. the two-parameter criterion of Ertl et al. (2016) based on neutrino-driven explosion models, and remnant-mass classifications from the Sukhbold et al. (2016) grid — which massive stars successfully explode rather than collapse directly to black holes; the predicted 'islands' of failed explosions motivated Jiang et al. (2024) to filter allowed supernova models by a binary explodability indicator ζ(M,E)\zeta(M,E) before fitting a modified power-law IMF P~M∝M−αm∫ζ(M,E)E−αedE\widetilde{P}_M\propto M^{-\alpha_m}\int\zeta(M,E)E^{-\alpha_e}dE to 406 EMP stars. Under a two-supernova ('duo') enrichment scenario they inferred an extremely top-heavy CCSN IMF (αm,CC=−0.32\alpha_{m,\mathrm{CC}}=-0.32 to −0.69-0.69, depending on whether the maximum neutron-star gravitational mass is taken as 1.7 or 2.0 M⊙M_\odot; this mass sets the baryonic remnant-mass threshold separating successful explosions from black-hole formation), a steep PISN slope αm,PI≈5.3\alpha_{m,\mathrm{PI}}\approx5.3, and an energy exponent αe≈3.2\alpha_e\approx3.2–3.6. The study itself, however, acknowledges that this inference is not yet trustworthy: because the compactness-based explodability criteria are calibrated almost entirely on solar-metallicity progenitors, the study replaced them with a baryonic-remnant-mass criterion grounded in metal-free supernova evolution calculations, yet the metallicity dependence of explodability predictions in general remains unvalidated for metal-free stars and the remnant-mass threshold hinges on the assumed maximum neutron-star mass; the abundances used were one-dimensional local-thermodynamic-equilibrium (1D LTE — the standard but systematically biased atmospheric approximation, improved by non-LTE, or NLTE, modeling where available) values over a restricted element set; enrichment multiplicity (whether a given star's birth gas was polluted by one or by several supernovae) was treated as an assumed scenario rather than measured per star; and because a PISN ejects vastly more metal into a much larger volume than a CCSN, a disproportionally large fraction of second-generation stars carries a PISN imprint, biasing the inferred PISN contribution upward relative to true PISN birth rates. EMP stars are also rare and require dedicated surveys (Bonifacio et al. 2025), limiting sample sizes.

Problem Statement

Determine the initial mass function of the first, metal-free (Population III) stars — together with the coupled explosion-energy distribution — from the abundance patterns of extremely metal-poor stars under explodability filtering, and establish whether that determination is robust against, or must be corrected for, the systematic channels acknowledged by the source analysis (Jiang et al. 2024): (i) the explodability prescriptions ζ(M,E)\zeta(M,E) and the locations of the failed-explosion islands rest on explosion theory calibrated largely on solar-metallicity progenitors — the source analysis substituted a baryonic-remnant-mass criterion based on metal-free evolution calculations, but the metallicity dependence of explodability predictions remains unvalidated for metal-free stars — while the remnant-mass explodability threshold additionally depends on the uncertain maximum neutron-star gravitational mass; (ii) the inferred pair-instability-supernova contribution (the slope αm,PI\alpha_{m,\mathrm{PI}}) is expected to be biased upward because PISNe eject far more metal into far larger star-forming volumes than core-collapse supernovae, so the fraction of second-generation stars carrying a PISN imprint exceeds the true PISN birth fraction; (iii) the enrichment multiplicity (mono- versus multi-supernova) of individual observed stars is unmeasured, so the progenitor-mass distribution itself depends on an assumed enrichment scenario; (iv) the abundance inputs are inhomogeneous one-dimensional LTE measurements over a restricted element set, with known NLTE/3D systematics and incomplete species coverage; and (v) the explodability-modified IMF remains phenomenological, its inferred mass slopes and energy exponent αe\alpha_e not yet connected to first-principles star-formation and explosion physics. These channels are not separate questions: each is a route through which the single inference — the explodability-filtered Pop III IMF and energy distribution recovered from stellar archaeology — could be wrong. The determination requires the IMF and energy-distribution parameters to be recovered with the systematic channels (i)–(iv) each quantified and propagated and channel (v) assessed non-quantitatively — whether the favoured IMF and energy parameter values are consistent with first-principles star-formation and explosion expectations, and how the inferred parameters depend on the phenomenological form of the modified IMF — or, failing that, a demonstration from the full yield-and-IMF model family, with the systematic channels (i)–(iv) left unrestricted, that present stellar-archaeology data cannot identify these parameters and which channels dominate the degeneracy.

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