Direct-Collapse Black-Hole Seeds

A direct-collapse seed begins with a large, self-gravitating gas inflow rather than an ordinary stellar remnant. The proposed advantage is a head start, often $10^4$–$10^6M_\odot$; the hard part is keeping enough gas warm, weakly enriched, and dynamically connected while it loses angular momentum without being converted into normal stars. “Direct” therefore does not mean structureless or instantaneous: many viable formulations pass through a rapidly accreting supermassive star or a radiation-supported envelope before a black hole forms.

Helpful background. Black-Hole Geometry and Mechanics supplies the collapse endpoint; Relativistic Matter and Compact Objects supplies gas, stellar, and instability physics; Black-Hole Astrophysics supplies accretion and feedback concepts; and Relativistic Cosmology supplies early-halo assembly and abundance.

A thermochemical corridor, not one threshold

The usual starting point is a metal-poor halo with virial temperature near the atomic-hydrogen cooling threshold, $T_{\rm vir}\sim10^4\,\mathrm{K}$. Atomic lines can then hold primordial gas near $5{,}000$–$8{,}000\,\mathrm{K}$. If molecular hydrogen instead cools it to a few hundred kelvin, the Jeans mass falls and multiple ordinary protostars become more likely. Lyman–Werner photons can dissociate $\mathrm H_2$, while infrared photons can detach the intermediate $\mathrm H^-$; collisions can also destroy $\mathrm H_2$ in sufficiently hot, dense shocks. No universal scalar $J_{\rm crit}$ describes all of this, because the result depends on spectrum, time dependence, self-shielding, chemistry, and density history.

Metals add a second boundary. Fine-structure and dust cooling may restore fragmentation even when molecular hydrogen is suppressed. Conversely, some rapidly assembled or shock-heated systems need not be perfectly pristine. The physical requirement is a complete trajectory $T(n,Z,J_\nu)$ that remains on a high-inflow branch, not the label “metal free.” The review synthesis maps the competing radiative, shock, streaming-velocity, and rapid-assembly routes and explicitly notes where simulations disagree (Inayoshi, Visbal & Haiman 2020, §§5.2–5.3 and Figs. 7–10).

Inflow and angular-momentum transport

Warm gas supplies a useful scale. With sound speed $c_s\sim8\,\mathrm{km\,s^{-1}}$,

$$\dot M_{\rm dyn}\sim\frac{c_s^3}{G} \simeq0.1\left(\frac{c_s}{8\,\mathrm{km\,s^{-1}}}\right)^3 M_\odot\,\mathrm{yr^{-1}}.$$

Maintained for $10^6$ yr, this can assemble $10^5M_\odot$. But the estimate presumes that torques carry angular momentum outward. Global bars, spiral modes, turbulent stresses, and mergers have all been proposed. Bromm and Loeb’s pioneering simulations found a central condensation containing millions of solar masses when molecular cooling was disabled; nonzero halo spin produced two clumps rather than one, making angular momentum an outcome-defining axis (Bromm & Loeb 2003, §§3–4 and Figs. 4–8). That calculation demonstrated a mechanism under chosen chemistry and resolution, not a channel frequency.

At disk scales, the relevant diagnostic changes from a spherical Jeans mass to the Toomre parameter

$$Q=\frac{c_s\kappa}{\pi G\Sigma}.$$

Here $\Sigma$ is disk surface density and $\kappa$ is the epicyclic frequency, which measures small radial oscillations about a circular orbit; $\kappa=\Omega$ in a Keplerian potential. This $\kappa$ is a frequency, not a radiative opacity. The ideal razor-thin, locally isothermal fluid disk is unstable to local axisymmetric perturbations when $Q\lt1$: self-gravity overcomes the combined support of pressure and rotation. Thickness, turbulence, magnetic fields, and nonaxisymmetric modes modify that simple boundary.

Fragmentation does not follow from $Q\lt1$ alone, and it need not terminate central growth: clumps can migrate inward faster than they contract into independent stars. A convincing calculation must compare cooling, orbital, migration, Kelvin–Helmholtz, and inflow times, then follow radiation and chemistry across the transition. The Kelvin–Helmholtz time measures how long a clump or star takes to radiate its gravitational binding energy and contract. These times distinguish transient spiral structure from fragments that survive as separate stars.

The supermassive-star or quasistar stage

Once the central inflow becomes optically thick, radiation pressure changes the problem. One formulation builds a bloated supermassive star that accretes at $\gtrsim0.1M_\odot\,\mathrm{yr^{-1}}$ and later encounters relativistic instability. Another builds a lower-entropy core inside a massive “quasistar” envelope; the core can collapse first and the embedded black hole then grow using the envelope’s Eddington scale. Begelman, Volonteri, and Rees derived an inflow scale $\dot M\sim v^3/G$, a radiation-trapping radius, and a compact quasistar sequence, while warning that fragmentation, feedback, or interrupted supply could stop the process (Begelman, Volonteri & Rees 2006, §§2–7, especially Eqs. 7–10).

These are different matter models and boundary conditions, not interchangeable names. Rotation, magnetic transport, multidimensional radiation escape, nuclear burning, and winds determine the retained seed mass. The robust statement is that a warm inflow can assemble a massive central object quickly. Whether that object remains single and how much of it enters the horizon remain unresolved.

Results and limitations through 2026-08-18

The evidence cutoff is 2026-08-18; later results are outside this assessment. The matrix separates demonstrated processes from extrapolations to a cosmic population.

Evidence or mechanism	Supported conclusion	Scope, status, or limitation
Atomic-cooling thermochemistry	Warm primordial gas permits inflow rates near $0.1$–$1M_\odot\,\mathrm{yr^{-1}}$	Chemistry, spectrum, shielding, metals, dust, and shocks make the boundary multidimensional
Three-dimensional collapse calculations	Selected haloes can avoid early fragmentation and build dense central structures	Initial data, radiation transport, resolution, and imposed backgrounds differ across simulations
Disk fragmentation and migration studies	Fragmentation can coexist with continued central feeding	Long integrations with stellar evolution, magnetic fields, and feedback remain limited
Supermassive-star and quasistar models	A massive inflow can pass through an intermediate object and yield a heavy seed	Rotation, mass loss, relativistic instability, and envelope coupling change the retained mass
High-redshift candidates	Early massive descendants motivate a heavy-seed hypothesis	Spectra and inferred masses do not uniquely distinguish direct collapse from rapid light-seed growth

Competing positions include intense-irradiation atomic collapse, shock- or merger-driven rapid inflow, a supermassive-star intermediate, a quasistar-like envelope, and fragmentation-dominated failure after enrichment. Evidence does not establish which position supplies most massive-black-hole ancestors, and abundance estimates depend sensitively on how the required radiation field and assembly history are selected.

The interpretation could change after a radiation-hydrodynamic calculation follows non-equilibrium chemistry and fragmentation from halo to protostar, a revised distribution replaces an imposed critical radiation field, or a candidate spectrum and host constraint jointly test the predicted thermal state. A robust low-mass occupation measurement could also change how rare the heavy-seed initial-data class may be.

Formulation-choice diagnostic

A central structure is a rotating disk with $c_s=8\,\mathrm{km\,s^{-1}}$, epicyclic frequency $\kappa=3.0\times10^{-14}\,\mathrm{s^{-1}}$, and surface density $\Sigma=10^3M_\odot\,\mathrm{pc^{-2}}$. Its halo supplies $0.10M_\odot\,\mathrm{yr^{-1}}$ for $0.80$ Myr, but only 30% reaches the center after star formation and outflows. Should one use a spherical Jeans criterion or disk $Q$, and can the retained inflow make a $10^5M_\odot$ seed?

Solution

Because rotation organizes the object into a disk, Toomre $Q$ is the relevant first diagnostic. Converting $\Sigma$ gives about $0.21\,\mathrm{g\,cm^{-2}}$, hence $Q=c_s\kappa/(\pi G\Sigma)\simeq0.55$: the ideal thin-fluid-disk criterion predicts local gravitational instability, although $Q\lt1$ alone does not decide whether clumps migrate or survive. The delivered mass is $0.10\times0.80\times10^6=8.0\times10^4M_\odot$; retaining 30% leaves $2.4\times10^4M_\odot$. The stated history therefore cannot assemble a $10^5M_\odot$ seed from that inflow alone, even before collapse losses. A spherical Jeans calculation would miss both the angular-momentum barrier and the explicit retention shortfall.

References

• Begelman, Mitchell C., Marta Volonteri, and Martin J. Rees. “Formation of Supermassive Black Holes by Direct Collapse in Pregalactic Halos.” Monthly Notices of the Royal Astronomical Society 370 (2006): 289–298. DOI; Open PDF.
• Bromm, Volker, and Abraham Loeb. “Formation of the First Supermassive Black Holes.” The Astrophysical Journal 596 (2003): 34–46. DOI; Open PDF.
• Inayoshi, Kohei, Eli Visbal, and Zoltán Haiman. “The Assembly of the First Massive Black Holes.” Annual Review of Astronomy and Astrophysics 58 (2020): 27–97. DOI; Open PDF.