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For radiative efficiency $ϵ$, luminosity and black-hole growth rate are related byIf the absolute accretion rate is constant, the lifetime needed to reach $M_f=5\times 10^8{M}_{\odot }$ isExpressing the constant luminosity as $L=f{L}_{\mathrm{Edd}}(M_f)$ gives, when $M_{\mathrm{seed}}\ll M_f$,Sub-Eddington feeding can therefore last a substantial fraction of the roughly $1\mathrm{Gyr}$ available by $z=6$, while super-Eddington feeding can assemble the hole much faster than $50\mathrm{Myr}$. Because an arbitrary constant rate can be traded against its starting time and black-hole seed mass, the final mass alone provides essentially no separate constraint on either seed mass or seeding redshift.

For continuously Eddington-limited growth the luminosity rises with the mass, so growth is exponential rather than linear:ConsequentlyFor seed masses from roughly $1M_{\odot }$ to $10^6{M}_{\odot }$, the supplied logarithms give about $20$ to $6.2$ e-foldings, or approximately $0.9$ to $0.3\mathrm{Gyr}$ if the common $t_{\mathrm{Sal}}\simeq 0.045\mathrm{Gyr}$ convention neglects the small $(1-ϵ)$ correction. Unlike constant absolute-rate growth, Eddington-limited growth therefore spends many e-folding times at low mass and strongly links viable seed mass to seeding epoch.

A sufficiently deep high-redshift quasar survey measures the numerous faint progenitors that precede the rare luminous objects. Its luminosity function and duty cycle can distinguish long exponential growth from short rapid episodes and constrain the distribution of seed masses, formation redshifts, Eddington ratios, and obscured growth.

Two useful host-mass arguments are the empirical black-hole--bulge mass relation and an independent dynamical or halo estimate. Applying the low-redshift ratio $M_{\mathrm{BH}}/M_{\mathrm{bulge}}\sim 10^{-3}$--$10^{-2}$ suggests $M_{\ast }\sim 5\times 10^{10}$--$5\times 10^{11}{M}_{\odot }$. Spatially resolved line widths and sizes give $M_{\mathrm{dyn}}\sim R{v}^2/G$, while the abundance of such rare quasars and the cosmic baryon fraction constrain their halo's available stellar mass. The existence of these systems before the Universe is one gigayear old implies exceptionally rapid assembly of both stars and black holes; departures from the local mass relation are therefore plausible and scientifically informative.

For adiabatic collapse, $T{\rho }^{1-\gamma }$ is constant. With $\gamma =5/3$,The rising Jeans mass produces adiabatic suppression of fragmentation: smaller subregions become more pressure-supported as density increases.

For isothermal fragmentation, $T$ stays approximately constant, andThe instability scale then falls during collapse, allowing hierarchical fragmentation until cooling fails, opacity rises, or another source of support intervenes.

Primordial metal-free gas cools inefficiently, principally through molecular hydrogen, and remains relatively hot. It therefore has a larger Jeans mass and tends toward a top-heavy initial mass function of massive Population III stars. Metal lines and dust let enriched gas remain cool to higher density, so Population II stars extend to much lower birth masses.

These alternatives map directly onto black-hole seed channels. Massive Population III remnants produce light Population III remnant black-hole seeds. If cooling and fragmentation are strongly suppressed while a primordial halo supplies rapid inflow, near-monolithic collapse can produce a heavy direct-collapse black-hole seed. Intermediate cooling and fragmentation in a dense cluster can instead permit a runaway stellar-collision black-hole seed. The Jeans argument selects plausible mass scales; angular momentum, feedback, chemistry, and accretion determine which channel actually operates.

The three principal black-hole seed channels occupy different mass ranges.

For radiative efficiency $\eta =0.1$, growth at Eddington ratio $f_{\mathrm{Edd}}$ obeyswhere $t_E=0.45\mathrm{Gyr}$. Between $z=25$ and GN-z11 at $z=10.6$,so the exponent is $6.04f_{\mathrm{Edd}}$. Equivalently, the Salpeter time is about $0.05\mathrm{Gyr}/f_{\mathrm{Edd}}$.

Reaching $10^6{M}_{\odot }$ requiresRepresentative values areContinuous Eddington-limited growth supplies only $e^{6.04}\simeq 420$, so a $100M_{\odot }$ seed reaches about $4\times 10^4{M}_{\odot }$, whereas a seed above roughly $2.4\times 10^3{M}_{\odot }$ can reach the target at a unit duty cycle. Light seeds require sustained mildly super-Eddington accretion, an earlier start, mergers, or lower effective efficiency. Heavy direct-collapse seeds need only a moderate time-averaged Eddington ratio and are therefore easier to reconcile with the short available time. No channel is ruled out by the mass alone because seed masses, obscuration, duty cycles, super-Eddington episodes, mergers, and the observational mass estimate are uncertain.