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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".