Black-Hole Mass Functions

The black-hole mass function is the comoving number density per logarithmic mass interval, not a histogram of published masses. Turning targeted dynamical measurements or flux-limited active galaxies into that density requires a parent population, an occupation model, measurement-error convolution, and the probability that each object entered the sample. The central result is therefore a forward model: propose a latent population, pass it through mass measurement and selection, and compare only then with the observed catalog.

Required background. Probability and inference for gravity supplies hierarchical likelihoods, latent variables, censoring, and selection functions.

Helpful background. Black-Hole Geometry and Mechanics supplies the mass definition; Relativistic Matter and Compact Objects supplies compact-source alternatives; Black-Hole Astrophysics supplies mass estimators and accretion selection; Relativistic Cosmology supplies comoving volume and redshift evolution.

Mass as a population density

Define

$$\Phi_\bullet(m,z) \equiv \frac{dN}{dV_c\,dm}, \qquad m\equiv\log_{10}\left(\frac{M_\bullet}{M_\odot}\right),$$

with units $\mathrm{Mpc^{-3}\,dex^{-1}}$. An integral over $dm$ gives a number density; an integral of $10^mM_\odot\Phi_\bullet$ gives a mass density. Quoting one without the logarithm base, cosmology, redshift bin, and occupation convention makes comparisons ambiguous.

There are three main routes.

1. A dynamical census measures stellar, gas, maser, or orbital motions in inactive galaxies. It is close to the desired mass variable, but targets are strongly selected by distance, surface brightness, morphology, and the resolvability of the sphere of influence.

2. A galaxy census measures a host variable $x$, such as velocity dispersion or bulge mass, and convolves its distribution with a conditional black-hole relation:

   $$\Phi_\bullet(m) = \int dx\, \Phi_{\rm gal}(x)\, f_{\rm occ}(x)\, p(m\mid x).$$

   Intrinsic scatter belongs inside this integral. Applying only the mean relation suppresses the high-mass tail.

3. An active census uses broad-line virial masses or maps a luminosity function through an Eddington-ratio and duty-cycle model. It covers larger volumes and redshifts, but activity, obscuration, variability, and virial calibration enter explicitly.

Continuity-equation reconstructions add a fourth, evolutionary route: begin with a boundary mass function and move number density through mass space using accretion and mergers. Agreement among routes is a nontrivial check because their selections and calibrations differ.

The forward likelihood

Let $\eta$ collect host properties, activity state, distance, and measurement nuisances, and let $D$ denote inclusion in the catalog. The probability $S(m,\eta,z)=P(D\mid m,\eta,z)$ averages over possible noisy observations and targeting outcomes. For survey data $d_i$, the expected detected count is

$$\Lambda(\theta) = \int dz\,\frac{dV_c}{dz} \int dm\,d\eta\, \Phi_\bullet(m,z\mid\theta) p(\eta\mid m,z) S(m,\eta,z).$$

The Poisson point-process likelihood is

$$\mathcal L(\theta) \propto e^{-\Lambda(\theta)} \prod_i \int dm\,d\eta\, p(d_i\mid m,\eta,z_i,D) \Phi_\bullet(m,z_i\mid\theta) p(\eta\mid m,z_i) S(m,\eta,z_i).$$

Here redshifts are treated as known, the cosmology is fixed, and $dV_c/dz$ includes the survey solid angle; the corresponding parameter-independent volume factors in the event terms are suppressed. Uncertain redshifts or a fitted cosmology must instead remain inside the event integrals.

The likelihood in each event factor is conditioned on detection and is normalized over selected data. This distinction prevents counting selection twice. If inclusion is determined by a cut on the same recorded data, then, for an accepted $d_i$,

$$p(d_i\mid m,\eta,z_i,D)S(m,\eta,z_i) =p(d_i\mid m,\eta,z_i).$$

Thus the unconditioned measurement likelihood can be used with no extra $S$ in an accepted event’s integrand; $S$ still enters the expected count. If selection used other measurements that were discarded, include or marginalize those measurements in the joint observation model. These equivalent forms are derived by Mandel, Farr, and Gair 2019, §§ 2–4, equations (4)–(7) and (13). Dividing a mass histogram by a nominal volume accounts for neither noisy selection nor latent-mass uncertainty.

Occupation is not an infinitesimal mass. A useful model has a Bernoulli variable for whether a galaxy hosts a black hole, followed by a conditional mass distribution if occupied. Upper limits constrain both pieces but generally do not distinguish a low mass from nonoccupation without an observation model.

Eddington bias follows directly. Because the population falls steeply at high mass, more moderate-mass objects scatter upward than rare high-mass objects scatter downward. The correction depends on the entire error kernel, not a symmetric error bar attached after binning. Malmquist bias and sphere-of-influence selection add further asymmetry.

A named local reconstruction

The inactive-galaxy dataset version used here is the Kormendy–Ho 2013 machine-readable dynamical compilation, especially Table 3, joined for illustration to the Sloan Digital Sky Survey early-type velocity-dispersion function analyzed by Sheth et al. 2003. Kormendy and Ho tabulate black-hole masses, distances, host decompositions, and velocity dispersions for accepted dynamical measurements; their compilation calibrates conditional relations but is not a volume-limited black-hole census Kormendy and Ho 2013, §§ 2–3 and 6; machine-readable Table 3. Sheth and collaborators estimate $\Phi_{\rm gal}(\sigma)$ from a documented SDSS early-type sample and show that transforming a luminosity function through only the mean luminosity–dispersion relation gives the wrong shape Sheth et al. 2003, §§ 2–4, pp. 226–230.

Calibration. Use one distance scale, cosmology, comoving-volume convention, velocity-dispersion aperture, stellar mass-to-light scale, and black-hole mass calibration. Preserve the covariance introduced because a dynamical mass and host luminosity can share distance. For active objects, add the broad-line and continuum calibration and a declared virial factor.

Selection. The SDSS parent selection needs magnitude, redshift, early-type classification, spectral signal-to-noise, and velocity-dispersion quality cuts. The dynamical compilation needs the original target pool, morphology and activity cuts, angular-resolution and sphere-of-influence sensitivity, accepted upper limits, duplicates, and follow-up decisions. Because that target pool is heterogeneous, the table alone cannot supply a defensible $1/V_{\max}$ mass function.

Model and statistics. Fit the SDSS dispersion function and a latent $p(m\mid\sigma,{\rm morphology})$ with intrinsic scatter, then forward-apply the dynamical targeting and resolution probability. Retain censored observations and a host-dependent $f_{\rm occ}$. Compare a flexible binned or spline mass function with a parametric form using posterior predictive counts in observable space, not only residuals in inferred mass.

Statistical uncertainty includes finite counts, measurement covariance, and posterior width. Systematic uncertainty includes distance, stellar populations, aperture conversion, morphology, unresolved spheres of influence, occupation, virial factor, obscuration, variability, functional form, and extrapolated low- and high-mass tails.

Results and open questions

The evidence cutoff is 2026-08-18. The mass function remains unresolved where occupation and selection dominate; no single route is privileged without its forward model.

Route or dataset	Evidence or result	Independent check	Scope, limitation, or open status
Kormendy–Ho dynamical compilation	Provides source-level masses and host variables for conditional calibration	Refit with common distance, upper limits, morphology, and shared covariance	Heterogeneous targeting prevents a direct volume-density estimate
SDSS velocity-dispersion function plus a mass relation	Produces a local inactive-galaxy mass function after scatter and occupation are integrated	Recover a synthetic $\Phi_\bullet$ passed through the same galaxy and resolution selection	High-mass tail is scatter-sensitive; low mass is occupation- and morphology-sensitive
Broad-line active sample	Gives mass-dependent active counts across large volumes	Reconstruct the luminosity function with the same virial, obscuration, and variability model	Duty cycle and virial factor prevent direct identification with the total mass function
Accretion continuity model	Evolves a boundary function using luminosity-density constraints	Posterior-predict both active fractions and the local mass density	Efficiency, Eddington ratios, mergers, and seed boundary conditions are degenerate

Shankar, Weinberg, and Miralda-Escudé 2013, §§ 2–5 demonstrate this coupling: plausible continuity models can fit some luminosity, active-fraction, and Eddington-ratio constraints while no one model fits all of them. The competing positions are thus dynamical calibration plus host convolution, active virial estimation, occupation-corrected low-mass inference, and continuity reconstruction—not four interchangeable measurements.

A decisive pipeline first generates a mock parent galaxy population with known occupation and mass function, draws correlated measurement errors, applies the published targeting and resolution selection, and demonstrates calibrated recovery. It then downloads the frozen Table 3 and SDSS data products, reproduces the velocity-dispersion function, and repeats the real inference with alternative occupation, scatter, and error models.

The interpretation could change after a new homogeneous dynamical survey or major released selection function; a revised virial calibration or dwarf occupation constraint could also change the conclusion because either can move a different mass-function regime.

Interpretation and scope

The mass function is an input to integrated growth, merger rates, occupation studies, and detector forecasts. Its tails often dominate those derived quantities, exactly where extrapolation and scatter matter most. Publish the posterior covariance or samples, not just a smooth median curve.

Generic host demographics without black-hole inference are outside this page’s scope. The next page asks whether the conditional host relation used here is intrinsic or selected; the Soltan argument independently checks the integrated mass density against accretion light.

Common pitfalls

Treating a literature compilation as a survey. A list of secure masses does not encode the galaxies that were never targeted or never yielded a mass.

Deconvolving after selection. Error migration and selection act together in observable space; correcting a plotted histogram in two independent steps can be biased.

Setting occupation to one by definition. Nondetections and low-mass galaxies require a separate hosting probability.

Undo a two-bin Eddington bias

A toy population has true bin densities $(n_L,n_H)=(100,20)$ in common units. Measurement scatters $20\%$ of the low bin upward and $10\%$ of the high bin downward. Compute the observed bins and recover the truth using the response matrix. Why is this not yet a complete survey correction?

Solution

With columns denoting true bins,

$$\left(n_L^{\rm obs},n_H^{\rm obs}\right)^T = \begin{pmatrix}0.8&0.1\\0.2&0.9\end{pmatrix} \left(100,20\right)^T = \left(82,38\right)^T.$$

The high bin is inflated from $20$ to $38$ because the abundant low bin supplies more upward scatter. The inverse matrix has determinant $0.70$ and returns

$$\left(n_L,n_H\right)^T = \frac1{0.70} \begin{pmatrix}0.9&-0.1\\-0.2&0.8\end{pmatrix} \left(82,38\right)^T = \left(100,20\right)^T.$$

This only corrects a known measurement response. A real reconstruction must also include the mass- and host-dependent probability of entering the catalog, survey volume, occupation, and uncertainty in the response matrix itself.

References

• Kormendy, John, and Luis C. Ho. “Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies.” Annual Review of Astronomy and Astrophysics 51 (2013): 511–653. doi:10.1146/annurev-astro-082708-101811.
• Mandel, Ilya, Will M. Farr, and Jonathan R. Gair. “Extracting Distribution Parameters from Multiple Uncertain Observations with Selection Biases.” Monthly Notices of the Royal Astronomical Society 486 (2019): 1086–1093. doi:10.1093/mnras/stz896; Open PDF.
• Shankar, Francesco, David H. Weinberg, and Jordi Miralda-Escudé. “Accretion-Driven Evolution of Black Holes: Eddington Ratios, Duty Cycles, and Active Galaxy Fractions.” Monthly Notices of the Royal Astronomical Society 428 (2013): 421–446. doi:10.1093/mnras/sts026.
• Sheth, Ravi K., et al. “The Velocity Dispersion Function of Early-Type Galaxies.” The Astrophysical Journal 594 (2003): 225–231. doi:10.1086/376794.