Cosmic SMBH Density Evolution

• Cosmic number density of SMBHs is a measure of the abundance of supermassive black holes per unit comoving volume, crucial for testing formation and growth models.
• The measurement utilizes HST WFC3/IR variability in the HUDF, applying rigorous aperture photometry and difference imaging to correct for incompleteness and false positives.
• High, nearly constant density from z ~0 to z >8 challenges traditional seeding models, favoring efficient Pop III.1 scenarios and widespread SMBH formation.

The cosmic number density of supermassive black holes (SMBHs) quantifies the abundance of these objects per unit comoving volume as a function of redshift, serving as a fundamental constraint on models of SMBH seeding, growth, and co-evolution with galaxies. Recent observational campaigns, particularly those using deep photometric variability in the Hubble Ultra Deep Field (HUDF), have refined direct constraints on $n_{\rm SMBH}(z)$ across $z\sim0$–9, yielding robust measurements that challenge and inform theoretical scenarios for the origin and assembly of SMBHs.

1. Measurement Techniques and Observational Strategies

The most comprehensive direct measurement of the SMBH number density over cosmic time is based on the identification of active galactic nuclei (AGN) via photometric variability within the HUDF as reported in (Cammelli et al., 29 Jan 2025). This methodology leverages time-resolved, high-angular resolution imaging from the HST’s WFC3/IR in F105W, F140W, and F160W filters, spanning epochs in 2009–2012 and a key new dataset from 2023. The technique consists of:

• Aperture photometry centered on the galaxy nucleus using dual-mode Source Extractor runs to ensure precise centroiding across epochs.
• Difference imaging, where images from two epochs are convolution-matched and subtracted to highlight transient nuclear flux.
• Significance assessment: Variables are selected as AGN candidates if photometric differences between epochs exceed $2$, $2.5$, or $3\,\sigma$ in at least one filter, after empirical calibration of uncertainty using the relation:

$$\delta_{\Delta m} = \frac{2.5}{\ln 10} \sqrt{ \left(\frac{\delta_{F_1}}{F_1}\right)^2 + \left(\frac{\delta_{F_2}}{F_2}\right)^2 }$$

with corrections for correlated noise and filter-dependent scale factors.

Redshifts for variable candidates are established by matching to comprehensive spectroscopic and photometric catalogs (from VLT/MUSE, JWST/NIRSpec, and public HUDF compilations), ensuring robust $z$ estimates especially at $z>6$.

2. Results: Cosmic SMBH Number Density as a Function of Redshift

The variability-selected sample yields the following constraints on the co-moving SMBH number density $n_{\rm SMBH}(z)$:

Redshift Range	$n_{\rm SMBH}$ $^{-3}$" title="" rel="nofollow" data-turbo="false" class="assistant-link">raw	$n_{\rm SMBH,lum}$ $^{-3}$" title="" rel="nofollow" data-turbo="false" class="assistant-link">completed
$z=6-9$	$\gtrsim 10^{-2}$	$\gtrsim 10^{-2}$
$z = 0$	$\gtrsim 6\times 10^{-2}$	$\gtrsim 6\times 10^{-2}$

• "Raw" values subtract expected false-positives (noise triggers), while "completed" values further correct for both variability incompleteness (recovery rate of variable AGN candidates, empirically $\sim2$–$3\times$) and luminosity incompleteness (using a correction down to $M_{\rm UV} = -17$ at high-$z$, appropriate for the HUDF sensitivity).
• At $z = 6-9$, the completeness- and luminosity-corrected number density is $n_{\rm SMBH,lum} \gtrsim 10^{-2}$ cMpc$^{-3}$.
• At $z \sim 0$, the completeness-corrected number density increases to $n_{\rm SMBH,lum} \gtrsim 6\times 10^{-2}$ cMpc$^{-3}$, due to sensitivity to much fainter AGN.

These empirically determined number densities are an order of magnitude (or more) higher than the values adopted in prior theoretical treatments of the cosmic SMBH census, which often anchored at $n_{\rm SMBH}\sim 4.6-8.8\times10^{-3}$ cMpc$^{-3}$ based on galaxy luminosity functions and spheroid occupation arguments (Banik et al., 2016Singh et al., 2023). Variability selection, sensitive to faint and low-luminosity AGN, reveals a much larger active SMBH population, particularly at high redshift.

3. Theoretical Interpretation and Constraints on Seeding Models

The near-constant, high value of $n_{\rm SMBH}(z)$ across $z=0$ to $z\gtrsim8$, after all corrections, places stringent constraints on SMBH formation models:

• Direct collapse models produce extremely low seeding rates, $n_{\rm SMBH}\sim 10^{-4}-10^{-7}$ cMpc$^{-3}$ (Habouzit et al., 2016Hartwig, 2018), falling short by orders of magnitude compared to the observed densities, even when adopting optimistic Lyman–Werner flux thresholds or weak supernova feedback scenarios.
• Halo threshold ("HMT") seeding models with mass limits of $M_{\rm th}\sim 7\times10^{10}\,M_\odot$ also result in densities substantially below the measured $n_{\rm SMBH}$ at high redshift (Singh et al., 2023).
• Pop III.1 seeding models—in which supermassive ($\sim10^5\,M_\odot$) primordial stars collapse to SMBHs in early, isolated minihalos—naturally produce near-constant, high number densities for plausible isolation distances $d_{\rm iso}\sim 50$–$75$ kpc proper (Singh et al., 2023Banik et al., 2016). The empirical $n_{\rm SMBH}(z)$ reported in the HUDF surveys is consistent with these model predictions.

Such observations imply that SMBH seeding must be highly efficient and relatively ubiquitous at early epochs, consistent only with Pop III.1-like or very low halo-threshold models but inconsistent with the low efficiencies of direct collapse scenarios.

4. Systematic Uncertainties, Corrections, and Comparison with Historical Values

The dominant sources of uncertainty in $n_{\rm SMBH}$ arise from variability completeness, luminosity function corrections, and redshift misidentification:

• Variability completeness ($F_\text{var}$): Empirically measured by cross-matching to previously known AGN samples, typically $F_\text{var}\sim2$–$3$ (Cammelli et al., 29 Jan 2025).
• Luminosity completeness ($F_\text{lum}$): Correction to account for objects fainter than the survey limits (extrapolating the observed AGN luminosity function to $M_{\rm UV}=-17$).
• False positive rate: Statistically estimated using the error distributions and significance thresholds, then subtracted from the candidate variable count.

Comparison with prior literature:

• Local "canonical" estimates for $z=0$ (using galaxy-bulge or velocity-dispersion-based SMBH occupation) yield $n_{\rm SMBH}\sim4.6-8.8\times10^{-3}$ cMpc$^{-3}$ (Banik et al., 2016Singh et al., 2023), an order of magnitude below the HUDF variability-based estimate, which is $n_{\rm SMBH,lum} \gtrsim 6\times 10^{-2}$ cMpc$^{-3}$ (Cammelli et al., 29 Jan 2025).

5. Consequences for SMBH–Galaxy Co-evolution and Host Properties

The observational result of a high, nearly flat $n_{\rm SMBH}$ from $z\gtrsim8$ to $z=0$ impacts several areas of SMBH–galaxy co-evolution:

• SMBH occupation fraction: The high $n_{\rm SMBH}$ implies that SMBHs are present in a majority, possibly all, luminous galaxies over cosmic time.
• Downsizing and mass function evolution: While the number density remains roughly constant, evolutionary features such as an invariant high-mass end and rapidly evolving low-mass end (cosmic downsizing) are confirmed by bulge-spheroid demographic studies (1109.00891212.2187).
• Population across environments: The local estimate likely undercounts SMBHs in systems such as ultra-compact dwarfs, stripped nuclei, or faint galaxies (Voggel et al., 2018), which may each contribute a nontrivial fraction to the total $n_{\rm SMBH}$.

A plausible implication is that the integrated SMBH mass density—and hence the baryonic fraction locked in black holes—may be higher than previously assumed, affecting models of feedback, star formation, and AGN duty cycles.

6. Future Directions and Open Issues

Key areas for advancement as outlined in (Cammelli et al., 29 Jan 2025) include:

• Spectroscopically complete redshifts: Essential for precisely binning $n_{\rm SMBH}(z)$ and performing spatial pair statistics relevant for seeding model discrimination.
• Longer time-baseline and wider area variability surveys: To refine variability completeness factors and reduce sample variance.
• Host galaxy characterization: To map the relation between SMBH occupation and host mass, star formation, and environment.
• Mitigation of contamination: Further exclusion of stellar transients such as supernovae through deeper, multi-wavelength temporal and spectroscopic data.

Improvements in these areas will enable a more refined census and direct empirical test of seeding theories, occupation statistics, and evolutionary mechanisms, further anchoring the cosmic number density of SMBHs as a cornerstone of galaxy formation physics.

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Summary Table: Empirical SMBH Number Densities from HUDF Variability (Cammelli et al., 29 Jan 2025)

Redshift Range	$n_{\rm SMBH}$ (cMpc$^{-3}$, corrected)	Method	Consistency with Pop III.1 Seeding?
$z = 6-9$	$\gtrsim 10^{-2}$	HST variability	Yes
$z \sim 0$	$\gtrsim 6\times 10^{-2}$	HST variability	Yes
$z = 0$ (prior)	$4.6-8.8\times10^{-3}$	Bulge/host stats	No (too low)

These data robustly indicate a near-constant, high cosmic number density of SMBHs, supportive of highly efficient and widespread seed formation at early times, as described by Pop III.1 scenarios and inconsistent with most direct collapse or high halo-threshold models.