The Mass Function of Supermassive Black Holes in the Direct-collapse Scenario
Bibliographic record
Abstract
One of the ideas that explains the existence of supermassive black holes (SMBHs) that are in place by z∼ 7 is that there was an earlier phase of very rapid accretion onto direct-collapse black holes (DCBHs) that started their lives with masses ∼104-5 . Working in this scenario, we show that the mass function of SMBHs after such a limited time period, with growing formation rate paired with super-Eddington accretion, can be described as a broken power law with two characteristic features. There is a power law at intermediate masses whose index is the dimensionless ratio α ≡ λ/γ, where λ is the growth rate of the number of DCBHs during their formation era, and γ is the growth rate of DCBH masses by super-Eddington accretion during the DCBH growth era. A second feature is a break in the power-law profile at high masses, above which the mass function declines rapidly. The location of the break is related to the dimensionless number β = γ T, where T is the duration of the period of DCBH growth. If the SMBHs continue to grow at later times at an Eddington-limited accretion rate, then the observed quasar luminosity function can be directly related to the tapered power-law function derived in this Letter.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".