Seed Signatures and Occupation Fractions

Seed models predict distributions, not a unique mass at one epoch. Light remnants are expected to form in many small haloes; heavier seeds require rarer conditions and may produce a low-mass occupation threshold. Accretion, galaxy mergers, gravitational recoil, tidal stripping, and wandering then blur those initial differences. The useful question is which combinations of central occupation, black-hole mass function, host properties, and merger rates remain identifiable after this evolution and after survey selection. 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 and population inference.

Define occupation before measuring it

An occupation fraction needs three qualifiers:

$$f_{\rm occ}(M_\star,z;M_{\rm min},R_{\rm ap}) =P\!\left(M_{\rm BH}>M_{\rm min},\ r_{\rm BH}<R_{\rm ap}\mid M_\star,z\right).$$

Changing the black-hole threshold $M_{\rm min}$ or the aperture $R_{\rm ap}$ changes the answer. A central X-ray survey, a spatially resolved dynamical survey, and a gravitational-wave catalog measure different projections. A kicked or tidally stripped black hole may count toward total halo occupation but not nuclear occupation; a quiet central hole is occupied but inactive.

For an object $i$, a simple detection model is

$$P(d_i=1)=f_{{\rm occ},i}\,q_i+ (1-f_{{\rm occ},i})\,b_i,$$

Here $q_i=P(d_i=1\mid\mathrm{occupied})$ is the total probability of a positive classification in an occupied galaxy, while $b_i=P(d_i=1\mid\mathrm{unoccupied})$ is the false-positive probability in an unoccupied galaxy. If contaminants can trigger a positive result even in occupied galaxies, include them in $q_i$ as well; it is not automatically just the completeness for the black hole’s own emission. After contaminants are controlled, the expected detected black-hole fraction cannot exceed occupation. Converting a measured fraction to occupation still requires sampling uncertainty, an accretion-rate distribution, and instrument response; assuming every occupied hole has the same Eddington ratio can manufacture a mass trend.

How seed memory is changed by assembly

An abundant light-seed model tends to begin with high low-mass occupation and a broad tail below $10^4M_\odot$. A rare heavy-seed model tends to leave more empty low-mass haloes and a characteristic initial floor. Neither qualitative pattern is immutable. Light seeds can remain too faint to detect; heavy seeds can be delivered by mergers; recoil can remove either; and gas accretion can erase the initial mass scale. The low-redshift mass relation may depend more strongly on the accretion prescription than on seeding itself (Inayoshi, Visbal & Haiman 2020, §7.3).

The correct comparison is a forward model,

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

Here $s$ is the seed state, $K$ propagates accretion, mergers, wandering, and ejection into descendant state $x$, and $S$ applies the electromagnetic or gravitational-wave selection. Identifiability means that competing $\theta_{\rm seed}$ values remain distinguishable after marginalizing over assembly and selection. Comparing a raw seed curve directly with detected galaxies skips both kernels.

Several observables help because their degeneracies differ. Central occupation in dwarf galaxies tests the empty-halo fraction. A low-mass black-hole mass function tests a floor or tail but is highly censored. Host morphology and nuclear-star-cluster coexistence constrain delivery and dynamics. Redshift-resolved massive-black-hole mergers probe pairs that electromagnetic surveys may miss, while extreme mass-ratio inspirals and wandering accretors test off-nuclear retention. No single axis is decisive.

A local X-ray example

The combined AMUSE-Virgo and AMUSE-Field programs targeted 203 optically selected early-type galaxies within 30 Mpc. Miller and collaborators modeled nuclear X-ray luminosity jointly with host stellar mass, nondetections, intrinsic scatter, and a mass-dependent occupation curve. Their clean sample contained 194 galaxies and 78 X-ray detections; for $M_\star<10^{10}M_\odot$, occupation below 20% was excluded at 95% confidence, but full occupation remained allowed (Miller et al. 2015, §§2–3, Eq. 1, Fig. 2, and Table 1).

That wide interval is scientifically informative: detection fraction alone cannot separate empty galaxies from occupied but faint ones. Nuclear low-mass X-ray binaries are a contamination channel, and the sample covers early-type galaxies rather than every dwarf population. The authors’ survey forecasts further show that sensitivity and sample size trade differently against the occupation–luminosity degeneracy (Miller et al. 2015, §4 and Figs. 3–4). This is why a multiwavelength, morphology-controlled selection is more useful than simply adding heterogeneous detections.

Results and limitations through 2026-08-18

The evidence cutoff is 2026-08-18. Results published later are outside this assessment. Population statements below distinguish direct measurements from assembly-dependent interpretations.

Evidence or signature	Supported conclusion	Scope, status, or limitation
Nuclear X-ray active fractions	Some low-mass galaxies contain accreting central black holes	Duty cycle, luminosity distribution, binaries, and aperture prevent direct conversion to full occupation
Optical, radio, and dynamical candidates	Multiple independent tracers can reduce contamination	Each has distinct mass, distance, host, and activity selection; dynamical resolution is restrictive
Descendant mass functions and scaling relations	Low-mass floors and tails can retain seed information	Accretion histories and measurement censoring can mimic either light- or heavy-seed patterns
Wandering and ejected populations	Recoil and galaxy assembly move holes outside nuclei	Poorly known detectability means nuclear absence need not imply halo absence
Massive-black-hole merger distributions	Mass, mass ratio, and redshift provide an independent assembly channel	Event rates depend on delays, binary hardening, detector selection, and seed occupation jointly

Credible competing positions are broadly occupied light-seed descendants, a heavy-seed occupation threshold, near-total erasure of seed memory by later growth, and apparent thresholds created by selection. Evidence does not establish a universal low-mass occupation curve; results depend on the source population, aperture, mass threshold, accretion prior, and calibration domain.

The interpretation could change after a statistically controlled dwarf survey combines X-ray, radio, optical, and dynamical information, a secure set of low-mass detections constrains the censored mass function, or a next-generation gravitational-wave catalog measures merger masses and redshifts. A major assembly-model revision affecting recoil, wandering, or accretion also changes the inference even without new detections.

Formulation-choice diagnostic

A nuclear survey contains 100 dwarf galaxies and 15 detections. Three detections are expected contaminants. A model combining an assumed accretion distribution with injection and recovery tests gives an occupied black hole probability $q=0.20$ of passing the selection; residual false positives after subtraction are negligible. What contamination-corrected detected fraction and plug-in occupation estimate follow, and may this alone select a light-seed model?

Solution

The estimated number of detected black holes is $15-3=12$, giving a corrected detected fraction $0.12$. A true detected black-hole fraction is a lower limit on occupation in expectation, but this finite-sample estimate with uncertain contamination is not itself a confidence bound. Using the modeled $q$ gives $\widehat f_{\rm occ}=12/(100\times0.20)=0.60$. A binomial detection likelihood, contamination uncertainty, and the uncertainty and mass dependence of $q$ are needed for an interval. Injection tests measure recovery of a supplied signal; they do not independently establish the accretion distribution used to average that recovery. Sixty percent does not by itself select light seeds: a heavy-seed model propagated through assembly and the same selection could overlap it, and wandering holes change nuclear versus halo occupation.

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.
• Miller, Brendan P., Elena Gallo, Jenny E. Greene, et al. “X-Ray Constraints on the Local Supermassive Black Hole Occupation Fraction.” The Astrophysical Journal 799 (2015): 98. DOI; Open PDF.