Super-Eddington and Obscured Early Growth

“Super-Eddington” can describe a mass supply, a mass crossing the horizon, or an emergent luminosity, and those quantities cease to be proportional in an optically thick flow. Early growth succeeds only if gas reaches the disk, survives winds, is retained by the black hole, and repeats often enough before feedback empties the host. This page compares those budgets. It does not infer rapid mass growth from luminosity alone, and it treats obscuration and geometrical beaming as observational transformations rather than extra fuel. Undated claims that one seed channel is established are outside scope.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon and mass scales; Relativistic Matter and Compact Objects supplies radiation and fluid physics; Black-Hole Astrophysics supplies accretion, winds, and spectra; and Relativistic Cosmology supplies the available early cosmic time.

Three rates that must not be conflated

Write the host supply as $\dot M_{\rm sup}$, the disk inflow as $\dot M_{\rm in}$, and the retained horizon growth as $\dot M_{\rm BH}$. A minimal mass budget is

$$\dot M_{\rm in}=f_{\rm feed}\dot M_{\rm sup},\qquad \dot M_{\rm BH}=(1-f_{\rm w})(1-\eta_{\rm out})\dot M_{\rm in},$$

where $f_{\rm w}$ is the wind-carried fraction and $\eta_{\rm out}$ is the net outward energy extracted from the post-wind inflow per unit rest mass in the site’s $c=1$ bookkeeping. Heat or trapped radiation advected through the horizon is retained in the black hole’s mass–energy rather than subtracted as an outward loss. Authors also normalize the Eddington accretion rate differently: either $\dot M_{\rm E}=L_{\rm Edd}$ in the site’s $c=1$ units, or $\dot M_{\rm E}=L_{\rm Edd}/\eta$. A quoted dimensionless $\dot m$ is meaningless until that choice is stated.

For a radiatively efficient thin disk, $L=\eta\dot M_{\rm in}$ makes luminosity a growth proxy. At high optical depth, the diffusion time can exceed the inflow time. Photons are advected inward inside a trapping radius with scale

$$r_{\rm trap}\sim\frac{\kappa\dot M_{\rm in}}{4\pi},$$

where $\kappa$ is opacity per unit mass; restoring units gives $r_{\rm trap}\sim\kappa\dot M_{\rm in}/(4\pi c)$. This spherical or geometrically thick estimate follows from $t_{\rm diff}\sim\tau r/c$, $t_{\rm in}\sim r/|v_r|$, $\tau\sim\kappa\rho r$, and $\dot M\sim4\pi r^2\rho|v_r|$. A disk’s vertical escape path introduces geometry factors, so this is a scale rather than a universal radius. Trapping permits luminosity to grow more slowly than mass supply. A slim-disk estimate often has $L/L_{\rm Edd}$ of order $1+\ln\dot m$ rather than $\dot m$. Conversely, a polar funnel can make an observer infer a large isotropic-equivalent luminosity even when the angle-integrated power is smaller. Neither effect fixes $\dot M_{\rm BH}$ without the wind and viewing-angle model.

From the Bondi scale to the horizon

At large radius, radiative heating can build an ionized bubble and choke accretion. One-dimensional radiation-hydrodynamic calculations identify a contrasting dense-gas regime: when the ionized region remains inside the Bondi radius, ram pressure maintains a nearly isothermal inflow and photon trapping limits the escaping luminosity. Inayoshi, Haiman, and Ostriker found steady solutions above roughly $5{,}000L_{\rm Edd}$ of supplied mass per unit time under their normalization and spherical, low-angular-momentum assumptions (Inayoshi, Haiman & Ostriker 2016, §§3.2–3.4, Fig. 7, Eqs. 28–29 and 43).

That result is an existence proof in a sharply defined parameter regime. Rotation creates a disk; multidimensional radiation escapes preferentially; magnetic stresses launch winds and jets; dust changes opacity; and the host supply can end before a steady state develops. The original analysis identifies angular momentum, outflows, and multidimensional instability as its main extrapolation boundaries (Inayoshi, Haiman & Ostriker 2016, §5.1). Radiation-MHD calculations generally find substantial outflows. The relevant question is therefore not “can $\dot m$ be large?” but “what fraction reaches the horizon after the large-to-small-scale solution is matched?”

The integrated test is simple:

$$\Delta M_{\rm BH}=\int f_{\rm duty}\,f_{\rm feed}\,(1-f_{\rm w})(1-\eta_{\rm out}) \dot M_{\rm sup}\,dt.$$

Feedback closes the other half of the problem. Thermal energy, radiation momentum, and mechanical power can unbind or stir the central gas. A model must compare deposited energy and momentum with the host binding budget, while allowing cooling and anisotropic leakage. Prescribing a long duty cycle independently of that calculation is circular.

What obscured early sources do and do not show

Weak X-rays can arise from a suppressed corona, shielding by a thick inner disk, or absorption outside it. Red continua can reflect dust, dense ionized gas, or a composite stellar and accretion spectrum. These alternatives predict different high-ionization lines, Balmer decrements, variability, polarization, and far-infrared reprocessing.

Lambrides and collaborators assembled 14 broad-Hα candidates at $4<z<7$ with deep X-ray coverage. Their X-ray upper limits lay at least $5\sigma$ below a standard sub-Eddington ultraviolet-to-X-ray relation; seven JADES sources with the needed NIRSpec coverage also lacked detected high-ionization ultraviolet lines. Slim-disk plus photoionization models reproduced the relative weakness and large Balmer decrement in a super-Eddington interpretation (Lambrides et al. 2026, “A surprising lack of X-ray continuum and UV line emission,” Figs. 1–4). The sample supplies valuable multiwavelength constraints, but no X-ray detection measures the spectral slope directly, and the conversion to Eddington ratio depends on virial masses and an accretion model. It supports a viable interpretation, not a direct measurement of retained growth.

Results and limitations through 2026-08-18

The evidence cutoff is 2026-08-18. Results published later are outside this assessment. The central unresolved issue is the joint distribution of supply, retention, duration, and viewing angle.

Evidence or result	Supported conclusion	Scope, status, or limitation
Photon-trapping solutions	Mass inflow can exceed the luminosity Eddington scale without proportional emergent power	Geometry, opacity, angular momentum, and inner boundary determine the applicable branch
Dense spherical feeding calculations	Hyper-Eddington inflow can overcome an ionized feedback bubble	One-dimensional low-angular-momentum matter model; host supply and multidimensional winds are external
Radiation-MHD disk calculations	Thick disks permit rapid inflow but often launch powerful winds and funnels	Retention and beaming depend on domain, magnetic flux, resolution, and radiative closure
Cosmological seed simulations	Short super-Eddington bursts occur in rare dense environments	Subgrid accretion, feedback coupling, resolution, and short duration control the inferred success fraction
JWST plus deep X-ray samples	Broad-line sources can be X-ray and high-ionization-line weak	Non-detections admit super-Eddington, shielding, obscured, and alternative broadening interpretations

Credible competing positions are sustained photon-trapping growth, wind-regulated retention, geometrically beamed emission, obscured near-Eddington accretion, bursty feedback-limited episodes, and merger-assisted assembly. Current evidence does not establish one universal state. The answer depends on the parameter regime and on whether $\dot m$ denotes supplied, disk, or horizon-crossing mass.

The interpretation could change after a radiation-MHD calculation connects resolved host feeding to the horizon while varying net magnetic flux, a survey directly constrains obscured duty cycles and bolometric corrections, or a secure early mass-and-host measurement is paired with variability and multiwavelength detections. Any of those could change retained growth rather than only the apparent luminosity.

Formulation-choice diagnostic

A $10^3M_\odot$ seed receives disk inflow at $0.20M_\odot\,\mathrm{yr^{-1}}$ during active intervals spread over 20 Myr. The state is active 25% of the time, 75% of disk inflow is lost to winds, and the remaining gas reaches the hole. Its observed luminosity at one active epoch is $2L_{\rm Edd}$. Should growth be computed from a luminosity-based exponential with a fixed $\eta=0.1$, or from the supplied-mass budget? First neglect outward energy losses after the wind, then assess that approximation.

Solution

The problem specifies supply and wind retention directly, so inserting the observed $L/L_{\rm Edd}$ into a thin-disk exponential would add an unsupported efficiency assumption. The active supplied mass is $0.20\times20\times10^6\times0.25=1.0\times10^6M_\odot$. Retaining 25% gives $\Delta M_{\rm BH}\simeq2.5\times10^5M_\odot$ and $M_f\simeq2.51\times10^5M_\odot$ at leading order in outward energy losses.

Nonzero radiation carries energy, so the exact gravitational mass is smaller if that energy comes from the inflow. A single luminosity measurement does not give its time integral. As an additional check, suppose the angle-integrated luminosity stays at $2L_{\rm Edd}$ throughout all active intervals, is negligible otherwise, and is the only further energy loss. With total active time $T=5\ {\rm Myr}$ and $t_E=450\ {\rm Myr}$,

$$\frac{dM}{dt_{\rm active}} =0.05M_\odot\,{\rm yr}^{-1}-\frac{2M}{t_E}.$$

Its solution gives $M_f\simeq2.48\times10^5M_\odot$, about $1.1\%$ below the leading estimate. This explicitly checks the approximation under the extra luminosity-history assumption. A complete model must also test whether feedback permits the active intervals and whether wind retention changes during growth.

References

• Inayoshi, Kohei, Zoltán Haiman, and Jeremiah P. Ostriker. “Hyper-Eddington Accretion Flows on to Massive Black Holes.” Monthly Notices of the Royal Astronomical Society 459 (2016): 3738–3755. DOI; Open PDF.
• Lambrides, Erini, Rebecca L. Larson, Kristen Garofali, et al. “The Case for Super-Eddington Accretion in JWST Broad-Line Active Galactic Nuclei during the First Billion Years.” Nature Astronomy 10 (2026): 868–879. DOI.