GoKawiilOur cosmological initial conditions are based on the simulation Phi-4096 of ref. 51. In this work, a dark-matter-only N-body simulation was performed in a comoving box of side length 16 h−1 Mpc. The initial condition is generated by MUSIC52 at a redshift of 127. The cosmological parameters used follow the latest measurement by Planck53: Ω m = 0.31, Ω Λ = 0.69, Ω b = 0.048, h = 0.68, n s = 0.96 and σ 8 = 0.83. The simulation used 4,0963 dark-matter particles, corresponding to a particle mass of 5.13 × 103 h−1 M ⊙ . This high resolution allows the identification of mini-halos with masses down to 105 h−1 M ⊙ , sufficient to resolve the sites of Population III (hereafter Pop III) star formation and to track chemical enrichment across cosmic time. Halo merger trees were constructed from simulation snapshots between z= 35 and z = 7.5 using the ROCKSTAR phase space halo/subhalo finder54 and the CONSISTENT-TREES merger tree code55. On top of these, a semi-analytic galaxy-formation model was implemented to follow gas cooling, Pop II and Pop III star formation, supernova feedback, metal enrichment and local Lyman–Werner (LW) radiation fields17. We use the result of their model with no baryon streaming motion (0σ vbc ).
The details of the semi-analytic model largely follow those described in ref. 56 with recent updates of the treatment of early star formation57. Star formation in Pop II halos follows the standard prescription used in galaxy-formation models, in which the cold-gas component is converted into stars on a timescale t SF = t dyn /α * with efficiency α * = 0.03 (ref. 58). Along with star formation, the model assumes that the rate at which cold gas is reheated into the hot-gas phase by associated supernova feedback is proportional to the star-formation rate, \(\gamma {\dot{M}}_{* }\), in which γ = (V halo /110 km s−1)−1.74 (refs. 56,59). When the progenitor halos never formed any stars, the stellar population follows a Pop III initial stellar mass function (IMF). The characteristic Pop III stellar mass is determined by the halo mass growth rate, which correlates with the final stellar mass60. Stellar populations emit LW radiation that photodissociates molecular hydrogen and delays star formation in nearby halos. The LW radiation intensity is calculated separately for Pop II and Pop III stars61:
$${J}_{21,{\rm{III}}}=\sum _{i}15{\left(\frac{{r}_{i}}{1{\rm{kpc}}}\right)}^{-2}\left(\frac{{M}_{{\rm{PopIII}},i}}{1000\,{M}_{\odot }}\right),$$ (1)
$${J}_{21,{\rm{II}}}=\sum _{i}3{\left(\frac{{r}_{i}}{1{\rm{kpc}}}\right)}^{-2}\left(\frac{{M}_{{\rm{PopII}},i}}{1000\,{M}_{\odot }}\right),$$ (2)
in which r i is the distance to halo i and M PopIII,i and M PopII,i are the masses of Pop III and Pop II stars formed within the past 5 Myr in halo i, respectively. The sums run over all halos that host Pop III or Pop II star formation. The coefficient in the Pop II expression is derived assuming a Scalo IMF (ref. 62) and a stellar metallicity of Z = 0.001 (refs. 63,64).
The LW background is primarily contributed by Pop II stars and its amplitude depends on assumptions about the IMF and the stellar models used. We note that uncertainties in the LW intensity modelling have only a small impact on our results, as we later test explicitly. A halo begins to form Pop III stars once its mass exceeds the critical threshold determined by the local LW intensity65. Under strong LW irradiation, star formation is delayed until the halo virial temperature reaches about 8,000 K, at which point Lyα cooling triggers rapid collapse. Such halos typically experience high mass accretion rates and host more massive Pop III stars, extending to supermassive stars with M * ≈ 105 M ⊙ , which subsequently collapse into heavy BH seeds. Indeed, Ishiyama and Hirano17 identified more than about 104 supermassive stars with masses larger than 105 M ⊙ as heavy-seed candidates in their simulation volume.
From this dataset, we select one representative halo predicted to form a massive seed, which serves as the target of our follow-up radiation-hydrodynamic simulations described in this work. The halo is chosen according to two criteria: (1) the local LW intensity at the time of seed formation exceeds the critical value of J 21,crit = 1,000 and (2) the halo is not tidally disrupted by nearby massive galaxies. Here J 21 denotes the FUV intensity normalized to 10−21 erg s−1 Hz−1 cm−2 sr−1. The second condition is necessary because candidate halos are often located near luminous galaxies that provide strong LW irradiation, but their gravitational collapse may be suppressed by external tidal fields66. To exclude such cases, we select halos whose distances from the nearest massive galaxy satisfy r dist > 10d tidal , in which d tidal is defined as the separation at which the tidal radius imposed by the neighbouring galaxy equals the virial radius of the candidate halo. Within the simulated volume, we identify 60 halos that meet both criteria. Among them, we focus on the one that exhibits the most rapid mass growth, reaching a virial temperature of T vir ≃ 8 × 103 K, which marks the onset of atomic cooling and subsequent collapse.
We trace back the initial position of the selected candidate halo to the cosmological initial conditions at redshift z = 127. Extended Data Fig. 1 shows the time evolution of the halo mass, in which MBH1 forms. The white star marks the redshift at which the virial temperature of the halo reaches 8,000 K, when it is identified as a DCBH halo. A zoom-in region is defined to be 40 times larger than the Lagrangian radius of the target halo, corresponding to a comoving size of about 400 kpc. We confirm that this region is sufficiently large to encompass all material relevant to the formation of the heavy-seed BH in the selected halo. We measured the overdensity of the specified zoom-in region by computing the mean density within its bounding box of about 400 comoving kpc and found it to correspond to a 3.8σ fluctuation.
Radiation-hydrodynamic calculation
The radiation-hydrodynamic calculation is performed within this zoom-in region using the moving-mesh code AREPO (ref. 16), extended to include prescriptions for star formation and BH accretion physics. The resolution of the zoom-in region at the initial snapshot is identical to that used in ref. 51, in which the dark-matter particle and baryonic cell masses are 4.33 × 103 and 7.94 × 102 h−1 M ⊙ , respectively. At this resolution, star formation within mini-halos can be reliably resolved. Adaptive mesh refinement is applied whenever the local cell size falls below 16 times the local Jeans length, ensuring that gravitational collapse is properly captured and that artificial fragmentation is avoided67. Unlike the semi-analytic model in ref. 17, the formation and properties of primordial stars are directly followed by the high-resolution radiation-hydrodynamic simulations described here. Uncertainties in the stellar mass and the resulting FUV background intensity in the semi-analytic model may affect whether a heavy-seed BH forms or not, but we later show that these uncertainties do not substantially alter our main conclusion about the rapid emergence of overmassive BHs.
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