High-Redshift Seeding Scenarios

• High-redshift seeding scenarios are pathways that explain the formation of the first black hole seeds and nuclear star clusters in early protogalaxies.
• They rely on critical conditions such as atomic cooling, suppressed H2 formation by UV feedback, and minimal metal enrichment to confine star formation and induce runaway stellar collisions.
• Predicted statistics include a seed fraction of ~0.05 and typical seed masses around 10^3–2×10^3 M⊙, supporting their role in the early assembly of supermassive black holes.

High-redshift seeding scenarios refer to the proposed physical pathways for producing the first generation of black hole seeds and nuclear star clusters in the early universe, typically at redshifts $z \sim 10$–20. These scenarios are motivated by the need to explain the observed rapid appearance of supermassive black holes (SMBHs) and luminous quasars when the universe was less than 1 Gyr old, as well as to understand the overarching process of hierarchical galaxy and structure formation. The theoretical framework is rooted in models of cooling and fragmentation in protogalactic halos, the interplay of metallicity and radiative feedback, and stellar-dynamical evolution in the earliest stellar systems.

1. Physical Context: High-Redshift Halos and Cooling Conditions

High-redshift seeding in this scenario occurs in protogalaxies embedded in halos with virial temperatures $T_{\rm vir} > 10^4$ K at $z \gtrsim 10$–20. These are halos that have already hosted the first (Population III) stars whose death enriches the ambient gas with trace amounts of metals ($Z \sim 10^{-5} Z_\odot$) and produces a background field of ultraviolet (UV) radiation.

Key physical conditions:

• Atomic hydrogen cooling is enabled at $T_{\rm vir} > 10^4$ K, permitting gas to cool to $\sim 4000$ K via atomic lines, but not lower unless further cooling agents are present.
• The newly established UV background rapidly destroys $H_2$ at low densities; thus, molecular hydrogen cooling is suppressed in most of the halo.
• Metal-line cooling is possible only at high densities because the metallicity is low; the gas fragments only when a critical, metallicity-dependent density $n_{\rm crit,Z}$ is reached.

As a result, efficient fragmentation into stars is restricted to the innermost, densest core of the protogalaxy, preventing widespread star formation and producing extremely compact nuclear star clusters with typical masses $\sim 10^5$ $M_\odot$ and half-mass radii $\sim 1$ pc.

2. Stellar-Dynamical Pathway and the Formation of Massive Black Hole Seeds

The densest central clusters formed in these environments become unstable to rapid core collapse on a timescale $t_{\rm cc}$:

$$t_{\rm cc} \simeq 3\,\mathrm{Myr}\,\left(\frac{R_h}{1\,\mathrm{pc}}\right)^{3/2} \left(\frac{M_{\rm cl}}{5 \times 10^5\,M_\odot}\right)^{1/2} \left(\frac{10\,M_\odot}{\langle m \rangle}\right) \left(\frac{8.5}{\ln \lambda_C}\right)$$

where $R_h$ is the cluster half-mass radius, $M_{\rm cl}$ is the cluster mass, $\langle m \rangle$ is the mean stellar mass, and $\ln \lambda_C$ is the Coulomb logarithm [Equation (7) of the paper].

If $t_{\rm cc} < t_{\rm MS} \sim 3$ Myr (the main sequence lifetime for massive stars), then massive stars do not yet have time to evolve and explode, so stellar collisions can proceed unimpeded. This triggers runaway collisions at the cluster center, culminating in the formation of a very massive star (VMS), which then collapses quickly (with reduced mass loss due to low metallicity) into a massive black hole remnant. The mass of the resultant black hole seed is given by:

$$M_{\rm BH} = m_* + 4 \times 10^{-3} M_{\rm cl}\, f_c\, \ln(\lambda_C) \ln\left(\frac{t_{\rm MS}}{t_{\rm cc}}\right)$$

with $m_*$ the initial seed star mass, $f_c$ a numerical calibration factor, and other quantities as above [Equation (15)].

Clusters that meet this rapid-collapse criterion are always those in which metallicity remains very low and dynamical inflows are strong.

3. Role of Early UV Radiation and Metal Enrichment

The interplay of metal enrichment and the early UV background is critical to regulating fragmentation and cluster formation:

• The UV field, produced by the first (Population III) stars, destrys $H_2$ outside the densest regions, so atomic cooling dominates.
• The first supernovae inject metals at $Z \sim 10^{-5} Z_\odot$, but at such low metallicity, efficient low-mass star formation (via metal-line and dust cooling) is only possible above a high critical density $n_{\rm crit,Z}$. This spatially confines fragmentation.
• As metallicity increases over time, this threshold density drops, fragmentation spreads, and nuclear inflows weaken due to enhanced star formation over a larger area, extending $t_{\rm cc}$ and suppressing rapid VMS formation.

Thus, the seeding scenario is strongly dependent on the timing of early metal injection and the intensity of the UV background—both inherited from the first stellar generation.

4. Demographics and Statistical Outcomes

The scenario predicts that only a subset of halos—those in which all conditions coincide—produce massive black hole seeds:

• The fraction of protogalaxies forming such seeds is $\sim 0.05$ at $z \sim 10$–20.
• Typical seed masses are in the range $\sim 10^3$–$2 \times 10^3\,M_\odot$.
• The resulting comoving mass density in seeds is $\simeq 10^2\,M_\odot\,{\rm Mpc}^{-3}$ (order-of-magnitude estimate).
• The clusters that produce seeds have a peaked mass function around $10^5\,M_\odot$ and half-mass radii near 1 pc; the mass function of all clusters spans $\sim10^4$–$10^6\,M_\odot$.

This relatively “abundant” population of seed black holes provides a large enough initial mass density to permit further SMBH assembly by gas accretion and mergers during the canonical quasar epoch.

Quantity	Typical Value	Context
Seed fraction	$\sim 0.05$	Protogalaxies at $z \sim 10$–20
Seed mass	$\sim 10^3$–$2 \times 10^3\,M_\odot$	Individual seed BHs
Seed mass density	$\simeq 10^2\,M_\odot\,{\rm Mpc}^{-3}$	Integrated over halo population
Cluster mass (seeding)	$\sim10^5\,M_\odot$	Peak of seeding cluster mass function
Core collapse timescale	$t_{\rm cc} < 3\,\mathrm{Myr}$	Condition for runaway collisions

5. Implications for Later Supermassive Black Hole Growth

The initial population of $\sim 10^3 \,M_\odot$ seeds with a total mass density of a few hundred $M_\odot {\rm \,Mpc}^{-3}$ is sufficient—when integrated with gas accretion and merger rates over cosmic time—to explain the observed SMBHs in bright high-redshift quasars ($z \sim 6$) and in local galaxy nuclei. This dynamical seeding channel yields comparable efficiency to classic Population III remnant scenarios and naturally complements them.

A crucial aspect is that, by requiring rapid core collapse before enhanced fragmentation and star formation dilute cluster densities, this pathway leads to massive seeds in an environment where mass loss (via stellar winds or supernovae) is minimized by both youth and extremely low metallicity.

The available time for accretion is thus maximized, and the conditions favor robust, early SMBH formation.

6. Model Limitations and Connections to Other Pathways

The outlined mechanism is sensitive to several physical conditions:

• The timing and intensity of the early metal and UV enrichment episodes;
• The inflow efficiency and disk instabilities shaping central cluster assembly;
• The critical density for fragmentation as a function of metallicity.

As a result, only the earliest and most compact clusters, in halos that are massive enough to permit atomic cooling, will undergo the required rapid dynamical evolution. More “typical” halos or later-forming halos with higher metallicity will see fragmentation spread, core collapse times increase, and the probability of VMS and seed black hole formation sharply drop.

This seeding mechanism thus forms a distinct and complementary path within the broader landscape of high-redshift black hole formation, alongside direct collapse channels, Population III remnants, and later nuclear cluster mergers.

7. Summary: Synthesis and Theoretical Significance

High-redshift seeding by rapid core collapse in metal-poor nuclear star clusters provides a physically motivated and quantitatively viable pathway for generating intermediate-mass “heavy” black hole seeds ($\gtrsim 10^3\,M_\odot$) at $z \sim 10$–20. The mechanism critically depends on atomic hydrogen cooling, UV-dissociated $H_2$, and modest pre-enrichment by Population III stars, with a strong dynamical instability driving runaway collisions and VMS assembly. The resulting seeds occur in a minority of protogalaxies, with masses and densities sufficient for subsequent SMBH growth, and statistical properties that naturally accommodate quasar and local black hole demographics. This scenario establishes a direct link between the physics of the first stars and metals, the formation of the densest star clusters, and the seeding and evolution of the first massive black holes.