Massive Black-Hole Seeds

The first massive black holes pose an inverse problem. Observations provide masses, luminosities, spectra, host properties, and merger events after an unknown interval of growth; theory proposes several ways to create the initial objects. This chapter compares stellar-remnant, direct-collapse, and dense-cluster channels, then follows how accretion and assembly alter their signatures. Its central lesson is that a seed is specified by a distribution of masses, birth environments, formation times, and abundances—not by one memorable mass scale. Undated claims that one seed channel is established are outside scope.

Helpful background. Black-Hole Geometry and Mechanics supplies the black-hole mass concept; Relativistic Matter and Compact Objects supplies stellar-remnant formation; Black-Hole Astrophysics supplies accretion, hosts, and selection; and Relativistic Cosmology supplies halo assembly, redshift, and cosmic time.

Enter the seed problem

Three broad channel families organize the subject. Light seeds from early stellar remnants inherit the compact-remnant mass scale and may be common, but natal kicks, binary interactions, and radiative feedback determine whether they remain in shallow haloes and receive fuel. Direct-collapse seeds seek much larger initial masses by keeping an atomic-cooling inflow hot and suppressing fragmentation; their advantage in mass is offset by restrictive chemical, radiative, angular-momentum, and dynamical conditions. Runaway collisions in dense clusters occupy an intermediate route: mass segregation and repeated mergers must outrun stellar evolution, binary heating, mass loss, and cluster expansion.

These are physical mechanisms, not disjoint boxes. A dense cluster may form from gas assembled in an atomic-cooling halo; a rapidly accreting supermassive star may resemble the transient central object in a direct-collapse calculation; stellar remnants can later merge in the same nucleus. The robust comparison therefore records the state at black-hole formation:

$$\mathcal S_{\rm seed} =\{M_{\rm seed},z_{\rm form},n_{\rm seed}, M_{\rm halo},Z,J_{\rm rad},\mathbf v_{\rm kick},\mathcal E\},$$

Here $M_{\rm seed}$ is the birth mass, $z_{\rm form}$ the formation redshift, $n_{\rm seed}$ the comoving number density, $M_{\rm halo}$ the host-halo mass, $Z$ the metallicity, $J_{\rm rad}$ the radiation-field intensity, and $\mathbf v_{\rm kick}$ the recoil velocity; $\mathcal E$ summarizes the remaining gas, stellar, and dynamical environment. Quoted mass ranges are conditional on choices of stellar evolution, chemical network, radiation field, transport, and collapse endpoint. They are useful scales, not universal definitions.

The formation calculation also needs a survival test. A seed that forms but is kicked from its halo, starved by feedback, or stranded away from dense gas contributes differently from a central seed that grows immediately. That distinction links formation physics to the occupation fraction and to the observed active fraction.

Growth can erase the initial mass hierarchy

After birth, let $\epsilon=L/(\dot M_0c^2)$ be the radiative efficiency relative to the rest-mass inflow $\dot M_0$, $\lambda=L/L_{\rm Edd}$ the Eddington luminosity ratio during active intervals, and $f_{\rm duty}$ the fraction of time spent active. With constant active-state $\epsilon$ and $\lambda$, no mergers or additional energy loss, and fully ionized hydrogen, the idealized growth law in ordinary units is

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

The Thomson cross-section $\sigma_T$ and proton mass $m_p$ set $t_{\rm Edd}\simeq450$ Myr. The factor $1-\epsilon$ appears because only the rest-mass energy remaining after radiation increases the black-hole mass. For varying efficiency or luminosity, the exponent must instead integrate the instantaneous growth rate, as derived on the light-seed page.

For $\epsilon=0.1$ and continuous Eddington-limited luminosity, the e-folding time is about $50$ Myr. Growing from $100M_\odot$ to $10^9M_\odot$ requires $\ln(10^7)\simeq16.1$ e-folds, roughly $0.8$ Gyr in that ideal limit; starting from $10^5M_\odot$ saves $\ln(10^3)\simeq6.9$ e-folds, about $0.35$ Gyr. These estimates reveal the leverage of the initial mass, but not the answer: spin-dependent efficiency, intermittent feeding, mergers, winds, obscuration, and supercritical inflow all change the clock.

Super-Eddington and obscured early growth separates mass supply from emergent luminosity. Photon trapping can make $\dot M$ exceed the conventional Eddington mass rate without making the observed luminosity rise proportionally. Outflows can remove much of the supplied gas, and anisotropy changes the flux inferred along one line of sight. The page therefore compares models at fixed definitions of $\dot M$, $\epsilon$, and luminosity rather than treating “super-Eddington” as a single state. The growth bottleneck and photon trapping are reviewed by Inayoshi, Visbal & Haiman 2020, §§1.2 and 3.1.

Read signatures as a forward model

The descendant population is obtained by propagating seeds through accretion, mergers, recoil, wandering, and galaxy assembly, then applying an instrument and selection function:

$$p(\mathcal D\mid\theta_{\rm seed}) =\int p(\mathcal D\mid x,\theta_{\rm inst})\, K(x\mid s,\theta_{\rm asm})\, p(s\mid\theta_{\rm seed})\,dx\,ds .$$

The kernel $K$ is why a high observed mass does not uniquely imply a heavy seed. A light seed with sustained supercritical episodes and a heavy seed with delayed feeding can reach the same point. Conversely, abundance and host information can retain differences even when the masses overlap.

Seed signatures and occupation fractions develops the low-redshift and gravitational-wave side of this inference. Central occupation in dwarf galaxies, the censored low-mass mass function, wandering populations, host structure, and merger masses probe different combinations of formation and assembly. An active fraction is not automatically an occupation fraction: it must be corrected for duty cycle, sensitivity, contaminants, and aperture. The AMUSE analysis illustrates how nondetections and an accretion prior enter a quantitative bound rather than a simple detection count (Miller et al. 2015, §§2–4, Eq. 1, and Figs. 2–4).

High-redshift evidence for early black holes gives the object-level validation chain. A candidate must survive release-specific detector calibration, line identification, PSF and LSF modeling, lens reconstruction, contamination tests, and competing mass estimators. Spectroscopic redshifts and resolved velocity fields are comparatively durable. Virial masses, Eddington ratios, host masses, and channel preferences are more mutable because lensing, line transfer, geometry, and growth assumptions intervene. That page maintains a dated evidence assessment through 2026-08-18; later results require a new review rather than silent incorporation.

Two routes through the chapter

For a formation-first route, begin with stellar remnants, then compare direct collapse and dense-cluster runaway. Record for each channel the required initial conditions, characteristic timescale, failure boundary, birth abundance, and retention probability. Continue to rapid growth only after defining what exits the formation calculation. Finish with descendant signatures and high-redshift evidence, where the channel predictions meet selection and calibration.

For an observation-first route, begin with the high-redshift validation workflow. Ask which quantities are measured and which are inferred. Then use rapid-growth timescales to translate each allowed present mass into a family of seed masses. The occupation page adds abundance and descendant constraints. Finally inspect the three formation pages to see which physical environments can supply those priors. This reverse order is often more effective for a new candidate because it prevents a favored formation story from dictating the data reduction.

Whichever route is used, compare models on the same axes: seed mass distribution, comoving formation rate, halo and metallicity domain, retention, accretion prescription, merger delays, observable duty cycle, and survey selection. A claim that changes several axes at once has not isolated the effect of seeding.

Chapter synthesis diagnostic

A black hole is inferred to have $M_{\rm obs}=10^8M_\odot$ after $\Delta t=300$ Myr of possible growth. Assume $\epsilon=0.1$, $\lambda=1$, and no mergers. Compare the minimum seed mass for continuous activity with that for $f_{\rm duty}=0.5$. Does either result identify a unique formation channel?

Solution

With $t_{\rm grow}\simeq50$ Myr, continuous activity supplies $N=300/50=6$ e-folds. Thus

$$M_{\rm seed}=10^8e^{-6}M_\odot \simeq2.5\times10^5M_\odot.$$

At $f_{\rm duty}=0.5$, only three effective e-folds occur, giving

$$M_{\rm seed}=10^8e^{-3}M_\odot \simeq5.0\times10^6M_\odot.$$

The second requirement is larger than the characteristic range of many standard seed calculations, which signals that at least one assumption—mass, available time, efficiency, Eddington ratio, duty cycle, or merger growth—must change. The first value is compatible with some heavy or extreme cluster-origin priors, but it does not uniquely select them: a lighter seed with earlier formation or supercritical episodes can reach the same mass. The calculation constrains a combination of formation and growth, not a channel label.

References

• Inayoshi, K., Visbal, E., & Haiman, Z. “The Assembly of the First Massive Black Holes.” Annual Review of Astronomy and Astrophysics 58, 27–97 (2020). DOI.
• Miller, B. P., Gallo, E., Greene, J. E., et al. “X-Ray Constraints on the Local Supermassive Black Hole Occupation Fraction.” The Astrophysical Journal 799, 98 (2015). DOI.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Origins of Massive Black-Hole Seeds