Growth, Demographics, and Host Galaxies

Black-hole growth is reconstructed from populations that are observed through different windows: dynamical masses in selected galaxies, flux-limited active nuclei, integrated radiation, merger rates, spin measurements, and disturbed gas. None is a direct movie of assembly. This chapter develops the forward models that connect those data to abundance, accretion, mergers, angular momentum, host correlations, and feedback, while keeping identifiability and causal claims visible.

Generic host demographics without black-hole inference are outside this chapter’s scope.

Helpful background. Black-Hole Geometry and Mechanics supplies the mass and spin parameters; Relativistic Matter and Compact Objects supplies compact-remnant physics; Black-Hole Astrophysics supplies the source and measurement setting; Probability and inference for gravity supplies hierarchical and selection-aware inference; Relativistic Cosmology supplies comoving volume, redshift, and cosmic time.

Enter this chapter

The population state can be summarized by a black-hole mass function per unit physical mass, $n(M,t)=dN/(dV_c\,dM)$, and a continuity equation,

$$\frac{\partial n}{\partial t} +\frac{\partial}{\partial M} \left[n(M,t)\langle\dot M\rangle\right] =S_{\rm merge}(M,t)+S_{\rm seed}(M,t)-S_{\rm eject}(M,t).$$

The mean growth rate is not observed directly. It is assembled from an occupied population, a duty cycle $U(M,z)$, an Eddington-ratio distribution $P(\lambda\mid M,z)$, radiative efficiency, and obscuration. The observed luminosity function is a convolution of those ingredients with measurement and survey selection. Its integral over cosmic time constrains retained mass through the Soltan argument, while mergers move number and spin through parameter space without contributing the same radiative term.

The mean accretion rate is averaged over active and inactive black holes at that mass. The signed merger term removes both progenitors and adds a remnant; it is not a positive source in every mass bin. Given an initial mass function and fluxes at the mass boundaries, the equation expresses conservation of number under accretion. For a per-dex mass function $\Phi(m,t)$, where $m=\log_{10}(M/M_\odot)$, use $\Phi=M\ln(10)n$ and mass-space speed $\dot m=\dot M/[M\ln(10)]$. Inserting a per-dex function directly into the displayed per-unit-mass equation would miss this Jacobian.

Three distinctions organize the chapter:

• Intrinsic versus selected populations. A catalog of secure masses or luminous active nuclei is not a comoving mass function.
• Instantaneous versus integrated growth. Activity at one epoch constrains a censored distribution; luminosity integrated over time constrains a mass-energy total.
• Correlation versus causal regulation. A black-hole–host relation can reflect feedback, common fuel, halo or bulge potential, mergers, measurement selection, or several mechanisms together.

Throughout, $M_\bullet$ is physical mass and $a_*=J/(GM_\bullet^2)$. Luminosities, masses, and densities are tied to a declared cosmology and logarithmic measure. The observational and frontier snapshot closes on 2026-08-18; later catalog or calibration releases require reassessment.

The six-page route

The pages form an inference chain. Read them in order when reconstructing a population; enter later where a specific input is already available.

1. Black-Hole Mass Functions defines abundance per comoving volume and logarithmic mass, then forward-models occupation, measurement error, censoring, and heterogeneous targeting. It explains why neither a dynamical compilation nor an active-mass histogram is a mass function by itself.
2. Black-Hole–Host Scaling Relations treats the mass–dispersion, mass–bulge, and related correlations as selected conditional distributions. Forward and inverse fits, morphology, distance covariance, sphere-of-influence selection, and residual tests separate description from causal interpretation.
3. The Soltan Argument and Integrated Growth derives the luminosity-density-to-retained-mass relation with the correct time, luminosity-measure, efficiency, and obscuration factors. It provides an integral consistency check, not an object-by-object accretion history.
4. Duty Cycles and Eddington-Ratio Distributions decomposes the luminosity function into abundance, active probability, and $P(\lambda\mid M,z)$. A frozen SDSS DR7 example shows how flux, broad-line, virial-mass, and obscuration selection enter the likelihood.
5. Mergers, Accretion, and Spin Evolution evolves mass and angular-momentum vectors through coherent disks, finite episodes, mergers, and recoil. It then forward-selects the intrinsic spin population through electromagnetic and gravitational-wave measurements.
6. Black-Hole Feedback and Coevolution asks what would establish causal regulation: a closed multiphase budget, spatial and temporal coupling, a credible counterfactual, and predictions beyond a tuned host relation.

The order prevents two shortcuts. One cannot infer a duty cycle without a parent mass function and an activity threshold. Nor can one interpret a scaling relation as feedback before testing selection, integrated growth, timing, energy deposition, and alternatives.

The common forward model

For each data stream, begin with latent black-hole and host variables $\theta$. Let $\Phi(\theta\mid\psi)$ be their expected number density over the modeled survey domain, and let $D$ denote detection. Distinguish the density of particular detected data from its integral, the expected count:

$$\begin{aligned} \lambda_{\rm det}(d\mid\psi) &=\int d\theta\,\Phi(\theta\mid\psi)S(\theta)\, p(d\mid\theta,D),\\ \Lambda_{\rm det} &=\int_{D} dd\,\lambda_{\rm det}(d\mid\psi) =\int d\theta\,\Phi(\theta\mid\psi)S(\theta). \end{aligned}$$

Here $S(\theta)=P(D\mid\theta)$ and the observation model is conditional on detection, so it integrates to one over selected data. If detection is a cut on the same recorded data, the product $S(\theta)p(d\mid\theta,D)$ reduces to the ordinary $p(d\mid\theta)$ for accepted data. The expected count $\Lambda_{\rm det}$ enters the Poisson factor $e^{-\Lambda_{\rm det}}$ and each observation supplies $\lambda_{\rm det}(d_i\mid\psi)$, as developed in the mass-function likelihood. Correcting a plotted histogram after fitting generally cannot reproduce this normalization or the migration caused by correlated measurement error.

The population parameters $\psi$ should be tested in native observables. A mass-function model must predict the targeted kinematics and nondetections; an active-population model must predict fluxes and line widths; a spin-history model must predict method-specific spectra or gravitational-wave posteriors; a feedback model must predict gas and star-formation responses in matched apertures. Synthetic recovery asks whether known latent functions can be recovered after the same selection.

Independent checks constrain different combinations. The local mass density and integrated luminosity density test radiative efficiency Soltan 1982, pp. 115–120. Active fractions and Eddington ratios test whether a continuity solution also reproduces snapshot demographics Shankar, Weinberg, and Miralda-Escudé 2013, §§ 2–5. Dynamical host samples test conditional mass relations but require morphology and resolution selection Kormendy and Ho 2013, §§ 5–9. None supplies all missing functions alone.

What can and cannot be combined

Mass density, luminosity density, duty cycle, and radiative efficiency share variables and should be inferred jointly or with their covariance propagated. Host mass and black-hole mass often share distance; continuum luminosity enters both a virial mass and its Eddington ratio; mass, distance, and inclination can be common nuisances across spin methods. Multiplying posteriors that fixed those quantities separately counts information twice.

Other comparisons need a translation rather than a common posterior. An X-ray spin sample of luminous accretors and a gravitational-wave sample of merging stellar remnants occupy different mass scales, states, epochs, and selection functions. A cavity enthalpy averaged over a buoyancy time and an instantaneous nuclear luminosity probe different clocks. A high correlation coefficient and an energy surplus answer different questions from a causal intervention.

The chapter therefore reports conditional conclusions:

• a mass function is conditional on occupation, mass calibration, and selection;
• a scaling relation is conditional on sample, morphology, regression direction, and parent population;
• an integrated efficiency is conditional on bolometric, obscuration, kinetic-output, merger, and boundary assumptions;
• a duty cycle is conditional on an activity threshold and timescale;
• a spin history is conditional on alignment, episode, merger, recoil, and measurement models; and
• feedback is causal only for the spatial, temporal, and counterfactual contrast actually demonstrated.

Reconstructing the source population

Freeze the data and calibration releases first. State the cosmology, logarithmic measure, source frame, and parent denominator. Build a generative model that maps latent masses, accretion states, spins, hosts, and gas into the native observables, including nondetections. Normalize through the selection function, perform mock recovery, and compare flexible and physically motivated alternatives.

Then make the cross-checks close: number and mass density; radiated and retained energy; mass and angular momentum; injected and coupled feedback energy; direct and selected population rates. Report extrapolated regions separately. A model that fits the luminosity function but misses active fractions, or fits a host relation but misses gas and star-formation responses, has failed a useful test rather than merely acquired another nuisance parameter.

Chapter diagnostic: interrogate a coevolution claim

A study combines a flux-limited broad-line quasar sample at $z\simeq1$, a heterogeneous local dynamical-mass compilation, several high-spin X-ray measurements, and cool-core cavities with $P_{\rm cav}\simeq L_{\rm cool}$. It concludes that long coherent accretion built most black holes, created the black-hole–bulge relation, and quenched massive galaxies. Which conclusions are directly supported, which are conditional, and what sequence of tests is required?

Solution

The direct inputs are calibrated quasar fluxes, spectra and redshifts under their targeting rules; source-level local dynamical measurements; method-specific X-ray data; and cavity and hot-gas observables. The broad-line catalog does not directly give the total mass function or duty cycle. The local compilation is not volume limited. The X-ray sources are a luminous, method-selected spin sample. $P_{\rm cav}\simeq L_{\rm cool}$ is a time-averaged energetic comparison, not proof of where heat was deposited or of galaxy-wide quenching.

First reconstruct the total and active mass functions with occupation, measurement errors, parent samples, and selection. Fit $U(M,z)$ and $P(\lambda\mid M,z)$ by forward-predicting the luminosity function and active fraction together. Apply the Soltan integral to test the implied radiative efficiency and mass density. Next evolve coherent, episodic, and merger histories through alignment, recoil, and the X-ray selection; ask whether one model predicts all spin and demographic observables. Refit the host relation with morphology and resolution selection before assigning causality. Finally close the cavity energy in a matched aperture, locate dissipation, compare gas and star-formation histories with matched controls or paired simulations, and test held-out host properties. Until those steps agree, the coherent-growth and universal-quenching conclusions remain conditional.

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.
• 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.
• Soltan, Andrzej. “Masses of Quasars.” Monthly Notices of the Royal Astronomical Society 200 (1982): 115–122. doi:10.1093/mnras/200.1.115.