Light Seeds from Early Stellar Remnants

A light seed is a black hole left by an early star, typically with an initial mass of tens to a few hundred solar masses. Its appeal is economy: no new collapse channel is required. Its difficulty is quantitative. Reaching the massive-black-hole regime before the first billion years demands enough retained gas, for long enough, despite stellar, supernova, accretion, and recoil feedback. This page treats seed formation as an initial condition and asks whether a declared growth history actually closes the mass and time budgets; it does not assume that any one seed channel has been established.

Helpful background. Black-Hole Geometry and Mechanics supplies the mass and horizon concepts used for the remnant; Relativistic Matter and Compact Objects supplies stellar collapse and feedback physics; Black-Hole Astrophysics supplies accretion and host-galaxy language; and Relativistic Cosmology supplies the redshift-to-cosmic-time conversion.

The light-seed initial-value problem

The seed mass is not the progenitor’s zero-age mass. Metal-free stellar evolution separates stars that explode, stars affected by pair instability, and stars that collapse while retaining a large fraction of their mass. The mapping depends on rotation, mixing, binaries, and mass loss. In the classic Population III proposal, very massive stars above the pair-instability disruption range leave remnants of order $10^2 M_\odot$ in rare minihaloes; the authors also emphasized that ultraviolet and supernova feedback limits how many such sites can operate independently (Madau & Rees 2001, §§2–3). Modern stellar calculations broaden the allowed remnant distribution, but do not remove that environmental qualification.

A usable initial condition is therefore a tuple

$$\mathcal I=\{M_{\rm seed},z_{\rm seed},M_{\rm halo},v_{\rm kick},E_{\rm SN},Z,\boldsymbol j_{\rm gas}\},$$

not merely $M_{\rm seed}$. A natal kick comparable to the minihalo escape speed displaces the remnant from dense gas. A supernova can evacuate the central reservoir. Even direct collapse of the progenitor avoids only that explosion; it does not guarantee subsequent inflow. These distinctions explain why the relevant parameter regime is a selected tail of early environments, not the average Population III remnant.

Exponential growth, with its assumptions exposed

Let $\eta$ be radiative efficiency, $\lambda=L/L_{\rm Edd}$ the luminosity Eddington ratio, and $f_{\rm duty}$ the fraction of elapsed time spent in the stated accretion state. During an active interval, a rest-mass inflow $\dot M_0$ radiates $L=\eta\dot M_0c^2$ and increases the hole’s gravitational mass at $(1-\eta)\dot M_0$, assuming no additional outward energy loss. Eliminating $\dot M_0$ and averaging identical active intervals over time gives, for electron-scattering opacity in ionized hydrogen,

$$L_{\rm Edd}=\frac{4\pi G M m_p c}{\sigma_T},\qquad \frac{dM}{dt}=f_{\rm duty}\,\frac{1-\eta}{\eta}\,\lambda\, \frac{4\pi Gm_p}{\sigma_T c}M.$$

If $\eta$, $\lambda$, and $f_{\rm duty}$ are constant, then

$$M(t)=M_{\rm seed}\exp\!\left(\frac{\Delta t}{t_e}\right),\qquad t_e=\frac{\eta}{1-\eta}\frac{\sigma_T c}{4\pi Gm_p} \frac{1}{f_{\rm duty}\lambda}.$$

For $\eta=0.1$, $t_e\simeq 50\,\mathrm{Myr}/(f_{\rm duty}\lambda)$. Growing from $100M_\odot$ to $10^9M_\odot$ requires $\ln(10^7)=16.12$ e-folds, or about $0.81$ Gyr when $f_{\rm duty}\lambda=1$. For comparison, a spatially flat matter-plus-cosmological-constant model with $H_0=70\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and $\Omega_m=0.30$ gives approximately $0.62$ Gyr between illustrative redshifts $z=25$ and $z=7$, neglecting the small radiation correction. This is the standard light-seed timing problem, stated with its efficiency and duty-cycle assumptions (Inayoshi, Visbal & Haiman 2020, §1.2). Lower $\eta$, super-Eddington mass supply, an earlier start, or mergers can reduce the tension, but each changes a different term and must be modeled separately.

If the accretion state changes, integrate its instantaneous growth rate:

$$\ln\frac{M(t)}{M_{\rm seed}} =\int_{t_{\rm seed}}^t \frac{1-\eta(t')}{\eta(t')} \frac{\lambda(t')}{t_E}\,dt', \qquad t_E=\frac{\sigma_T c}{4\pi Gm_p}.$$

Inactive intervals have $\lambda=0$. A separate duty factor is unnecessary when those intervals are already in the integral. In particular, multiplying independently averaged efficiency, luminosity ratio, and duty cycle can be wrong when they are correlated.

The exponential is a capacity calculation, not a prediction. A $10^9M_\odot$ descendant has retained nearly $10^9M_\odot$ of gas regardless of its seed mass. The host must deliver that gas across halo, galactic, and disk angular-momentum barriers. Meanwhile luminosity and mechanical outflows heat or remove the reservoir. A consistent calculation therefore checks both the instantaneous accretion solution and the integrated host supply.

Feedback, mergers, and rare successful histories

Mergers add mass without radiative efficiency, but do not offer free exponential growth. Combining $N$ similar seeds gives at most a factor $N$ before gravitational-wave losses; dynamical friction must bring them together, and merger recoil can eject the remnant from a shallow potential. Thus a claimed merger factor belongs outside the accretion exponent unless a merger tree supplies its timing and retention probability.

High-resolution calculations now demonstrate possible short successes. Mehta, Regan, and Prole followed Population III remnants in cosmological zoom simulations with 1, 0.3, and 0.1 pc minimum cell sizes. In their highest-resolution run with thermal black-hole feedback, one remnant grew from about $66M_\odot$ to $1.39\times10^4M_\odot$ in bursts; only a small fraction of remnants grew substantially, and supernovae, black-hole heating, or starvation ended the episodes (Mehta, Regan & Prole 2026, Table 1 and Figs. 1–4). The result establishes existence within that numerical matter model and initial-data class. It does not establish a cosmic success fraction: the calculation ends near $z=21$, uses a Bondi–Hoyle–Lyttleton prescription, and does not include the progenitor’s ionizing-radiation feedback.

Results and limitations through 2026-08-18

The evidence cutoff is 2026-08-18. Results published later are outside this assessment. “Possible” below means that a mechanism works under stated conditions; it does not mean that it dominates the source population.

Evidence or result	What it supports	Scope, status, or limitation
Population III remnant argument	Metal-free massive stars can supply numerous light seeds at high redshift	Progenitor mass function, remnant mapping, radiative feedback, and natal kicks remain uncertain
Eddington growth integral	A declared $M_{\rm seed}$, $\eta$, $f_{\rm duty}$, and $\lambda$ give an exact timing test	The integral assumes the required gas and accretion state are available; it is not a host model
Resolved early-growth simulations	Rare remnants can undergo short, strongly super-Eddington growth bursts in dense gas	Outcomes depend on resolution, subgrid accretion, feedback coupling, initial data, and short simulated duration
Early massive-black-hole candidates	Their inferred masses and redshifts set demanding descendant boundary conditions	Virial masses, bolometric corrections, lensing, line broadening, and selection do not identify a unique seed channel

Credible competing positions therefore remain: a rare light seed may grow almost continuously near Eddington; intermittent super-Eddington bursts may supply the missing e-folds; mergers may assist after host assembly; or feedback and finite supply may prevent light seeds from reaching the earliest inferred masses. Current evidence does not establish a universal verdict, because observations constrain descendants while simulations sample particular environments and formulations.

The interpretation could change after a secure high-redshift mass with a controlled broadening and lensing model, a seed-host calculation that resolves the accretion radius while varying progenitor and accretion feedback, or a direct low-mass occupation constraint that separates central from wandering black holes. Any of those could change the allowed duty-cycle distribution rather than merely shift one illustrative track.

Formulation-choice diagnostic

A $100M_\odot$ seed has $0.60$ Gyr to grow. Model A sets $\eta=0.1$, $\lambda=1$, and $f_{\rm duty}=0.8$. Model B is continuously active and describes a photon-trapping state by a disk inflow rate $\dot M=10M/t_E$, where $t_E=\sigma_T c/(4\pi Gm_p)=0.45$ Gyr. It loses half of the supplied mass to winds; neglect further outward energy losses in this toy model. Which formulation should be inserted into the growth equation, and does either reach $10^9M_\odot$ without mergers?

Solution

For A, $t_e\simeq50/0.8=62.5$ Myr, so the seed gains $0.60/0.0625=9.6$ e-folds and reaches only $100e^{9.6}\simeq1.5\times10^6M_\odot$. For B, use the supplied mass and specified retention directly; a luminosity-based calculation would require a consistent luminosity and efficiency that have not been given. Half of $10M/t_E$ is retained, so $dM/dt=5M/t_E$ and the e-folding time is $0.45/5=0.090$ Gyr. The seed gains $6.67$ e-folds and reaches about $7.9\times10^4M_\odot$. Neither reaches $10^9M_\odot$. Calling B “ten times Eddington” without distinguishing supplied mass, emergent luminosity, and wind retention would overstate its growth.

References

• 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.
• Madau, Piero, and Martin J. Rees. “Massive Black Holes as Population III Remnants.” The Astrophysical Journal Letters 551 (2001): L27–L30. DOI; Open PDF.
• Mehta, Daxal H., John A. Regan, and Lewis Prole. “Growth of Light Seed Black Holes in the Early Universe.” Nature Astronomy 10 (2026): 583–592. DOI; Source data; analysis code.