Abundance, timing, and black-hole-binary signatures of first-star formation at $z\gtrsim100$: identifying small-scale primordial fluctuation models that survive observational constraints

In the standard $\Lambda$CDM cosmology the first, metal-free (Population III) stars form at redshifts $z\sim20$–$30$, but the primordial density-fluctuation spectrum on sub-Mpc scales is observationally unconstrained and could be strongly enhanced by a blue tilt, a bump, or primordial-black-hole (PBH) induced isocurvature perturbations, permitting first-star formation at extremely early epochs $z\gtrsim100$. One-zone thermo-chemical calculations (Ito & Omukai 2024) show that the warm cosmic microwave background then suppresses $\mathrm{H}_2$ cooling, so that characteristic stellar masses rise sharply with formation epoch, from $\sim10^2\,\mathrm{M}_\odot$ at $z\lesssim100$ to supermassive stars of $\sim10^5\,\mathrm{M}_\odot$ forming by direct collapse at $z\gtrsim500$. The open problem, stated by that work as a crucial challenge for future research, is to integrate such thermal and mass results with specified small-scale fluctuation spectra, predict the abundance, formation timing, and remnant signatures (intermediate-mass-black-hole binaries and their gravitational-wave mergers) of these extremely early first stars, and identify which fluctuation models are consistent with the full observational constraint set (James Webb Space Telescope high-redshift galaxy counts, the CMB electron-scattering optical depth, dwarf-galaxy abundances, and pulsar-timing limits). Sub-uncertainties that must be resolved within the same objective are the mapping from core accretion rate to final stellar mass at the very high rates of $z\gtrsim100$, three-dimensional effects including baryon–dark-matter streaming motions, the possible role of Compton drag in ionized regions, and the multiplicity and binary-separation distributions that set merger delay times. While each ingredient is being advanced separately in the recent literature (2025–2026), no work to date carries out the coupled prediction-and-constraint program, so this remains an open problem.

Background

The first stars form out of pristine, metal-free gas (Population III, conventionally abbreviated Pop III) in dark-matter halos whose abundance at a given cosmic epoch is fixed by the spectrum of primordial density fluctuations. In the standard $\Lambda$CDM model — a universe dominated by a cosmological constant ($\Lambda$) and cold dark matter (CDM) — first-star formation is expected at redshifts $z\sim20$–$30$ (redshift $z$ measures cosmic time: the Universe's scale factor was smaller by a factor $1+z$, so $z\gtrsim100$ is within the first ~15 Myr after the Big Bang, shortly after recombination, the epoch at which electrons and protons combined and the cosmic microwave background (CMB) decoupled from matter). On scales above ~1 Mpc the fluctuation spectrum is precisely measured by the CMB and large-scale structure, but on sub-Mpc scales it is unconstrained: inflationary running could tilt it blue (more small-scale power, $P(k)\propto k^{m_{\rm s}}$ with $m_{\rm s}>1$, where $P(k)$ is the fluctuation power spectrum per wavenumber $k$), a feature (bump) could enhance it, or a primordial-black-hole (PBH) population — black holes formed from early density fluctuations — could add isocurvature fluctuations (perturbations in the relative matter/radiation distribution rather than the total density). Because the James Webb Space Telescope (JWST) has found high-redshift galaxies more abundant than standard models predict, several groups have invoked such enhanced small-scale spectra, which would also move first-star formation to much earlier epochs than $z\sim20$–$30$, a regime that remains scarcely explored.

Ito & Omukai (2024) studied this regime with a one-zone model — a spherically symmetric, single-cell thermo-chemical calculation following the density and temperature of a collapsing cloud — including CMB effects on radiative heating, Compton cooling, and photodissociation. They found that CMB photons photodetach $\mathrm{H}^-$ (the catalyst of the main $\mathrm{H}_2$-formation channel) at $130\lesssim z\lesssim500$, raising cloud temperatures, and that at $z\gtrsim500$ even the $\mathrm{H}_2^+$ channel is photodissociated so that gas cools only atomically and collapses almost isothermally at several thousand K. Estimating stellar masses from the Jeans mass (the mass above which gas cannot resist its own gravity; at the temperature minimum during collapse it sets the dense-core mass) and from the accretion rate $\dot{M}\simeq\phi c_{\rm s}^3/G$ during the loitering phase (the transient near-hydrostatic state before runaway collapse), they concluded that final masses rise steeply with formation epoch: $\sim10^2$–$10^3\,\mathrm{M}_\odot$ at $z<200$ rising beyond $1000\,\mathrm{M}_\odot$ at $z>200$, and monolithic direct collapse to $\sim10^5\,\mathrm{M}_\odot$ supermassive stars at $z>500$. Such massive stars would collapse quietly into intermediate-mass black holes (IMBHs, $\sim10^2$–$10^5\,\mathrm{M}_\odot$), whose binary mergers could be detected by the gravitational-wave detectors LISA (the Laser Interferometer Space Antenna, a space-based laser interferometer) or AEDGE (the Atomic Experiment for Dark Matter and Gravity Exploration, a proposed cold-atom space detector sensitive at even higher redshifts).

These mass estimates inherit several stated uncertainties. The mapping from accretion rate to final mass rests on simulation-based scalings calibrated at standard epochs: two-dimensional radiation-hydrodynamic simulations of minihalos (the low-mass dark-matter halos hosting the first star-forming clouds) (Hirano et al. 2015) give $M_\ast=250\,\mathrm{M}_\odot(\dot{M}/2.8\times10^{-3}\,\mathrm{M}_\odot\,{\rm yr}^{-1})^{0.7}$, while simulations of more massive atomically cooled halos (Toyouchi et al. 2023) give a steeper, non-power-law relation; at the high accretion rates of $z\gtrsim100$ cores ($\sim10^{-2}\,\mathrm{M}_\odot\,{\rm yr}^{-1}$) the two predictions diverge substantially and neither was calibrated there. The one-zone, spherically averaged treatment neglects non-spherical dynamics and fragmentation, and the supersonic baryon–dark-matter streaming motions (Tseliakhovich & Hirata 2010) — relic relative velocities of ~30 km/s between gas and dark matter after recombination that delay gas accumulation in halos — were only estimated semi-analytically at these epochs, with the paper concluding that their effect on cloud dynamics at the earliest redshifts remains a topic for future study. Compton drag (radiative viscosity: the drag force exerted by CMB photons on free electrons coupled to ions) would alter gas dynamics for significantly ionized gas at $z\gtrsim100$–$200$, though post-recombination gas is neutral ($x_{\rm e}\sim10^{-4}$). Finally, predicting the gravitational-wave signal requires the binary fraction and separation distribution of these stars, which set the delay time between stellar birth and black-hole merger and which the authors flag as very uncertain. The paper's own closing challenge, which defines this problem, is to identify which specific small-scale fluctuation spectrum models can explain the observed number density of high-redshift galaxies while also meeting the electron-scattering optical depth of the CMB (a measure of the total ionizing output integrated over cosmic history), the abundance of dwarf galaxies, and pulsar-timing limits on PBHs — using the mass–epoch relation now established for the thermal evolution at $z\gtrsim100$.

Problem Statement

For a specified model of each of the three named classes of the small-scale ($<$Mpc) primordial fluctuation spectrum — a blue-tilted power-law enhancement, a localized bump, or primordial-black-hole-induced isocurvature — determine, as one coupled program:

1. the abundance and formation timing of the first stars at $z\gtrsim100$, by combining the halo/structure formation predicted by that spectrum with the established thermal evolution of primordial gas at these epochs (including CMB photodissociation of $\mathrm{H}^-$ and $\mathrm{H}_2^+$ and the resulting mass–epoch relation, up to direct collapse to $\sim10^5\,\mathrm{M}_\odot$ supermassive stars at $z\gtrsim500$);

2. the characteristic final stellar masses at the high accretion rates ($\sim10^{-2}\,\mathrm{M}_\odot\,{\rm yr}^{-1}$) of $z\gtrsim100$ dense cores; the divergence between the simulation-based accretion-rate-to-mass scalings of Hirano et al. (2015) and Toyouchi et al. (2023) at these rates, characterized by the final mass (or allowed mass range) of a core accreting at $\sim10^{-2}\,\mathrm{M}_\odot\,{\rm yr}^{-1}$ under each of the two scalings and by the physical effects that dominate the difference between them; the three-dimensional effects neglected by one-zone models — non-spherical collapse, turbulent/rotational fragmentation, and baryon–dark-matter streaming motions — as well as the role of Compton drag where the gas is significantly ionized;

3. the observable signatures, in particular the formation rate and merger-delay distribution of the intermediate-mass black-hole binaries left by these stars, and their detectability by LISA/AEDGE, given the multiplicity and binary-separation distributions; and

4. which fluctuation-spectrum models survive the full set of current observational constraints: the JWST number counts of high-redshift galaxies, the electron-scattering optical depth of the CMB, dwarf-galaxy abundances, and pulsar-timing limits.

The abundance/timing, mass, and signature results of deliverables 1–3 and the constraint-survival result of deliverable 4 form one coupled objective, whose endpoint is the identification of the small-scale fluctuation models that are simultaneously consistent with all of these observations, given the predicted extremely-early first-star populations. The coupled program is therefore to be applied to an explicitly specified model in each of the three named classes — a blue-tilted power-law enhancement, a localized bump, and primordial-black-hole-induced isocurvature — carried through deliverables 1–3 and tested against the constraints of deliverable 4; a determination that a named class is excluded by those constraints resolves that class's contribution to deliverable 4, and the program yields a surviving-or-excluded verdict for each of the three named classes — an outcome in which no named class survives the constraint set is a possible result for deliverable 4. Models outside these three named classes lie outside this objective.

References

1. Ito, M. & Omukai, K. (2024), 'First star formation in extremely early epochs', Publications of the Astronomical Society of Japan 76, 850. DOI: 10.1093/pasj/psae054, https://doi.org/10.1093/pasj/psae054 (arXiv:2405.10073, https://arxiv.org/abs/2405.10073)
2. Hirano, S., Hosokawa, T., Yoshida, N., Omukai, K. & Yorke, H. W. (2015), 'Primordial star formation under the influence of far ultraviolet radiation: 1540 cosmological haloes and the stellar mass distribution', Monthly Notices of the Royal Astronomical Society 448, 568. DOI: 10.1093/mnras/stv044, https://doi.org/10.1093/mnras/stv044, arXiv:1501.01630, https://arxiv.org/abs/1501.01630
3. Toyouchi, D., Inayoshi, K., Li, W., Haiman, Z. & Kuiper, R. (2023), 'Radiative feedback on supermassive star formation: the massive end of the Population III initial mass function', Monthly Notices of the Royal Astronomical Society 518, 1601. DOI: 10.1093/mnras/stac3191, https://doi.org/10.1093/mnras/stac3191, arXiv:2206.14459, https://arxiv.org/abs/2206.14459
4. Tseliakhovich, D. & Hirata, C. (2010), 'Relative velocity of dark matter and baryonic fluids and the formation of the first structures', Physical Review D 82, 083520. DOI: 10.1103/PhysRevD.82.083520, https://doi.org/10.1103/PhysRevD.82.083520, arXiv:1005.2416, https://arxiv.org/abs/1005.2416
5. Hirano, S. & Yoshida, N. (2024), 'Early Structure Formation from Primordial Density Fluctuations with a Blue, Tilted Power Spectrum: High-Redshift Galaxies', The Astrophysical Journal 963, 2. DOI: 10.3847/1538-4357/ad22e0, https://doi.org/10.3847/1538-4357/ad22e0, arXiv:2306.11993, https://arxiv.org/abs/2306.11993
6. Kobayashi, M. U., Chiaki, G., Kimura, K., Akitsu, K., Kohri, K., Takahashi, T. & Omukai, K. (2026), 'Blue-tilted Runnings and the JWST Early Galaxy Tension'. arXiv:2605.22161, https://arxiv.org/abs/2605.22161
7. Zhang, S., Liu, B. & Bromm, V. (2025), 'A Novel Formation Channel for Supermassive Black Hole Binaries in the Early Universe via Primordial Black Holes', The Astrophysical Journal 992, 136. DOI: 10.3847/1538-4357/ae061c, https://doi.org/10.3847/1538-4357/ae061c, arXiv:2508.00774, https://arxiv.org/abs/2508.00774
8. Zhang, S., Liu, B., Bromm, V., Jeon, J., Boylan-Kolchin, M. & Kühnel, F. (2025), 'How do Massive Primordial Black Holes Impact the Formation of the First Stars and Galaxies?', The Astrophysical Journal 987, 185. DOI: 10.3847/1538-4357/adddb4, https://doi.org/10.3847/1538-4357/adddb4, arXiv:2503.17585, https://arxiv.org/abs/2503.17585
9. Venditti, A., Muñoz, J. B., Bromm, V., Fujimoto, S., Finkelstein, S. L. & Chisholm, J. (2025), 'Bursty or heavy? The surprise of bright Population III systems in the Reionization era', The Astrophysical Journal 994, 32. DOI: 10.3847/1538-4357/ae0610, https://doi.org/10.3847/1538-4357/ae0610, arXiv:2505.20263, https://arxiv.org/abs/2505.20263
10. Liu, B. & Bromm, V. (2025), 'Impact of primordial black holes on the formation of the first stars and galaxies', in: Byrnes, C. T., Franciolini, G., Harada, T., Pani, P. & Sasaki, M. (eds.), Primordial Black Holes, Springer Series in Astrophysics and Cosmology, pp. 269–301. DOI: 10.1007/978-981-97-8887-3_11, https://doi.org/10.1007/978-981-97-8887-3_11, arXiv:2312.04085, https://arxiv.org/abs/2312.04085
11. Hirano, S., Shen, Y., Nishijima, S., Sakai, Y. & Umeda, H. (2023), 'Formation of first star clusters under the supersonic gas flow -- I. Morphology of the massive metal-free gas cloud', Monthly Notices of the Royal Astronomical Society 525, 5737. DOI: 10.1093/mnras/stad2693, https://doi.org/10.1093/mnras/stad2693, arXiv:2306.11992, https://arxiv.org/abs/2306.11992
12. Hirano, S. (2025), 'Formation of first star clusters under the supersonic gas flow -- II. Critical halo mass and core mass function', Monthly Notices of the Royal Astronomical Society 540, 331. DOI: 10.1093/mnras/staf772, https://doi.org/10.1093/mnras/staf772, arXiv:2501.17491, https://arxiv.org/abs/2501.17491
13. Hirano, S., Sakai, Y. & Umeda, H. (2026), 'Formation of first star clusters under the supersonic gas flow -- III. Environmental trends and halo-to-halo scatter in the Pop III IMF', Monthly Notices of the Royal Astronomical Society 548, stag634. DOI: 10.1093/mnras/stag634, https://doi.org/10.1093/mnras/stag634, arXiv:2602.20690, https://arxiv.org/abs/2602.20690
14. Katz, O. Z., Redigolo, D. & Volansky, T. (2025), 'Closing in on Pop-III Stars: Constraints and Predictions Across the Spectrum', Journal of Cosmology and Astroparticle Physics 2025 (10), 047. DOI: 10.1088/1475-7516/2025/10/047, https://doi.org/10.1088/1475-7516/2025/10/047, arXiv:2502.03525, https://arxiv.org/abs/2502.03525
15. Cheng, H., Yin, Z., Di Valentino, E., Marsh, D. J. E. & Visinelli, L. (2025), 'Constraining exotic high-z reionization histories with Gaussian processes and the Cosmic Microwave Background'. arXiv:2506.19096, https://arxiv.org/abs/2506.19096
16. Das, S., Mondol, R., Singh, A. & Laha, R. (2026), 'Dark Secrets of Baryons: Illuminating Dark Matter-Baryon Interactions with JWST', Physical Review Letters 137, 011002. DOI: 10.1103/44nx-sgk4, https://doi.org/10.1103/44nx-sgk4, arXiv:2511.02906, https://arxiv.org/abs/2511.02906