Duty Cycles and Eddington-Ratio Distributions

An active-galaxy luminosity function does not by itself reveal how often black holes accrete or how rapidly an active hole grows. The same counts can arise from many black holes active rarely, fewer black holes active often, or a different distribution of Eddington ratios. The useful decomposition therefore keeps the black-hole abundance, occupation, active fraction, accretion-rate distribution, obscuration, variability, and survey selection as distinct parts of one forward model.

Required background. Probability and inference for gravity supplies latent-variable likelihoods, selection normalization, and identifiability tests.

Helpful background. Black-Hole Geometry and Mechanics supplies the mass and spin parameters; Relativistic Matter and Compact Objects supplies compact-source alternatives; Black-Hole Astrophysics supplies the accretion observables; Relativistic Cosmology supplies comoving volumes and cosmic-time conversion.

From active fraction to a latent duty cycle

Let

$$m=\log_{10}(M_\bullet/M_\odot), \qquad \lambda=\frac{L_{\rm bol}}{L_{\rm Edd}(M_\bullet)},$$

and let $n_\bullet(m,z)$ be the comoving black-hole mass function per dex. For occupied black holes, define $U(m,z)$ as the probability of being in an explicitly declared active state and $P(\log_{10}\lambda\mid m,z)$ as the normalized Eddington-ratio distribution within that state. These definitions require a lower boundary $\lambda_{\min}$ or an equivalent luminosity criterion:

$$\int_{\log_{10}\lambda_{\min}}^\infty P(\log_{10}\lambda\mid m,z)\,d\log_{10}\lambda=1.$$

Changing $\lambda_{\min}$ moves probability between $U$ and $P$ without changing the predicted luminosity function. Thus a quoted duty cycle has no meaning until the activity threshold, occupation convention, and averaging timescale are stated. The instantaneous probability sampled by a survey is also not automatically the fraction of cosmic time spent accreting: source variability, obscuration changes, and population turnover break that identification.

Ignoring scatter in bolometric correction for one line, write $\ell=\log_{10}L_{\rm bol}$ and $\ell_{\rm Edd}(m)=\log_{10}L_{\rm Edd}(10^mM_\odot)$. The intrinsic active luminosity function is

$$\Phi_L(\ell,z) = \int dm\,n_\bullet(m,z)U(m,z) P\!\left(\ell-\ell_{\rm Edd}(m)\mid m,z\right).$$

The observed catalog requires another layer. If $x$ denotes spectral class, obscuration, host contamination, variability state, and other nuisances, then

$$\Phi_{\rm det}(\ell,z) = \int dm\,d\log_{10}\lambda\,dx\; n_\bullet U P(\log_{10}\lambda\mid m,z) p(x\mid m,\lambda,z) S(\ell,x,z) \delta\!\left[\ell-\ell_{\rm Edd}(m)-\log_{10}\lambda\right].$$

Here $S$ is the probability of detection and classification, not a post-hoc weight. Radiative efficiency enters when luminosity is translated into mass growth, $\dot M_\bullet=(1-\eta)L_{\rm bol}/(\eta c^2)$; it does not change the luminosity mapping at fixed $L_{\rm bol}$, but it changes the continuity history and hence the mass function used in the integral.

What an active fraction actually measures

Suppose a survey is complete only above a mass-dependent threshold $\lambda_{\rm lim}(m,z)$. Even before obscuration and classification losses, its active fraction is

$$f_{\rm active,obs}(m,z) = U(m,z) \int_{\log_{10}\lambda_{\rm lim}(m,z)}^\infty P(\log_{10}\lambda\mid m,z)\,d\log_{10}\lambda.$$

Consequently $f_{\rm active,obs}$ is a censored tail probability, not $U$. A narrow distribution centered above the limit and a broad distribution with a long faint tail give different corrections. Measurement scatter in virial mass couples the axes because both $M_\bullet$ and $\lambda=L_{\rm bol}/L_{\rm Edd}$ use the same continuum luminosity. Treating their plotted errors as independent can manufacture an anticorrelation.

The luminosity function constrains principally the product and convolution $n_\bullet U P$. It cannot, alone, identify all three functions. A total mass function constrains $n_\bullet$; a selection-corrected active mass function adds information on $n_\bullet U$; the joint flux and line-width distribution constrains the shape of $P$; and variability or demographic continuity can supply time information. Even this combined program depends on occupation, bolometric correction, obscured and Compton-thick populations, and the virial-mass scale. Shankar, Weinberg, and Miralda-Escudé 2013, §§ 2–5 show explicitly that models fitting one demographic projection need not fit the observed active fractions and Eddington-ratio distributions simultaneously.

A frozen broad-line data example

The named source data product is the SDSS DR7 quasar-property catalog, VizieR J/ApJS/194/45, containing the Shen et al. measurements for 105,783 cataloged quasars. It includes redshift, targeting flags, continuum and emission-line measurements, broad-absorption flags, and single-epoch virial mass estimates Shen et al. 2011, §§ 2–4 and Tables 1–2. The complete table is a value-added catalog, not a statistical sample. A demographic analysis must restrict it to the uniformly targeted subset and reproduce the angular and flux selection; Kelly and Shen used about 58,000 uniformly selected Type 1 quasars to infer broad-line mass and Eddington-ratio functions with an explicit selection model Kelly and Shen 2013, §§ 2–3.

Calibration. Freeze the DR7 spectroscopic reduction and target flags, the Hβ, Mg II, and C IV line-fitting prescriptions, continuum luminosities, redshifts, Galactic extinction, and photometric zero points. Declare the cosmology, $K$-correction, virial relations and factor, bolometric correction, obscuration model, and comoving-volume convention. The line used changes with redshift, so cross-line offsets cannot be absorbed silently into evolution.

Selection. Record sky mask, targeting class, magnitude and color boundaries, redshift interval, spectral signal-to-noise and fit-quality cuts, broad-line definition, BAL rejection, duplicate-epoch rule, and the probability of successful spectroscopy. State whether the denominator is all occupied black holes, all galaxies in a host-mass bin, or only unobscured Type 1 quasars. Include nondetections and variability epoch rather than conditioning on successful mass estimates.

Likelihood. Generate latent $(m,\lambda)$ from $n_\bullet U P$, map them to continuum fluxes and line widths with correlated virial scatter, apply obscuration and the actual target selection, and compare predicted counts to the native observables. The Poisson normalization over all detectable objects is part of the likelihood. Test spline smoothness, parametric tails, and obscuration priors by recovering known mock populations. A modern bolometric luminosity-function fit provides a wider redshift cross-check, but its heterogeneous survey compilation and spectral-energy-distribution corrections must not be mistaken for independent object-level masses Shen et al. 2020, §§ 2–4.

The leading systematics are correlated virial-mass errors, the virial factor, bolometric corrections, luminosity-dependent obscuration, Compton-thick incompleteness, source variability, radiative efficiency, occupation, faint-end extrapolation, and the abundance–activity degeneracy. Statistical intervals from catalog counts do not include those model choices.

Results and open questions

The evidence cutoff is 2026-08-18. The inference remains unresolved because each observable is a projection of the same latent population under a different selection.

Position or evidence stream	Evidence or result	Counterexample or independent check	Scope, limitation, or open status
Narrow Eddington-ratio sequence	Bright Type 1 samples can display a comparatively concentrated observed $\lambda$ range	Inject a broad faint tail and pass it through the same flux and line-width cuts	Truncation can make a broad intrinsic distribution appear narrow
Broad mass- and redshift-dependent $P(\lambda)$	Flexible forward models can reproduce luminosity and active-mass distributions across bins	Predict the active fraction and luminosity function jointly from one fitted population	Smoothness priors and low-$\lambda$ extrapolation remain influential
Luminosity-dependent obscuration	X-ray and multi-band censuses add objects missing from optical Type 1 selection	Require one intrinsic model to predict optical, X-ray, and nondetection counts	Compton-thick corrections and cross-band bolometric mappings remain model dependent
Snapshot variability as a distribution	Short-timescale fluctuations can broaden the observed luminosity at fixed long-term supply	Convolve a latent long-term rate with a measured variability kernel	Variability does not uniquely determine episode duration or recurrence
Continuity plus radiative efficiency	Integrated growth connects $P(\lambda)$ and $U$ to evolution of $n_\bullet$	Reproduce both the luminosity function and local mass density without retuning each redshift bin	Mergers, seeds, efficiency, and boundary conditions are degenerate

A reproducible pipeline begins from the machine-readable VizieR table and the published uniform-target flags, rebuilds the binned luminosity function from the source weights, and validates its accessible volume against the survey mask. It then injects known $n_\bullet$, $U$, and $P(\lambda)$ into simulated fluxes and line widths, applies the same selection, and checks calibrated recovery. The real-data inference is repeated under alternative smoothness, virial-scatter, bolometric, and obscuration priors; posterior predictive counts are reported in observable space.

The interpretation could change after a new luminosity- or mass-function data release, a variability-informed lifetime constraint, or a revised obscured census; a direct accretion-rate distribution could also change the conclusion because it can break a different part of the abundance–activity degeneracy.

Interpretation and scope

The duty cycle is a population probability under a chosen activity definition. It becomes a lifetime only under an additional stationarity or cohort-evolution model. Likewise, an Eddington-ratio distribution describes luminosity per mass, not the supplied mass rate unless radiative efficiency and losses are specified.

Generic host demographics without black-hole inference are outside this page’s scope. The black-hole mass function supplies the abundance term, while the Soltan argument tests the integrated luminosity-to-mass conversion.

Common pitfalls

Equating detected active fraction with duty cycle. The detected fraction is the duty cycle multiplied by the fraction of $P(\lambda)$ visible to the survey and by classification completeness.

Normalizing away the faint tail. Defining the active distribution only over the observed interval makes $U$ depend on the flux limit and prevents comparisons among surveys.

Reading variability as a lifetime. A power spectrum over days or years does not determine the recurrence of accretion episodes over millions of years without a generative connection across timescales.

Recover a censored duty cycle

In one mass and redshift bin, the total black-hole density is $n_\bullet=10^{-4}\,\mathrm{Mpc^{-3}\,dex^{-1}}$. The intrinsic duty cycle is $U=0.10$, but only $25\%$ of active objects lie above the survey’s Eddington-ratio threshold. What active density is observed? What duty cycle would a naive ratio infer, and how is it corrected?

Solution

The detected active density before any additional incompleteness is

$$n_{\rm active,obs} =n_\bullet U\times0.25 =(10^{-4})(0.10)(0.25) =2.5\times10^{-6}\,\mathrm{Mpc^{-3}\,dex^{-1}}.$$

Dividing this density by the total density gives $0.025$, which is the observed active fraction, not the intrinsic duty cycle. Correcting for the known tail probability gives

$$U=\frac{0.025}{0.25}=0.10.$$

If the shape of $P(\lambda)$ is not independently constrained, the correction is not identified: $U=0.05$ with one half of active objects visible predicts the same observed fraction. Obscuration and classification completeness would multiply the visible fraction and add further degeneracy.

References

• Kelly, Brandon C., and Yue Shen. “The Demographics of Broad-Line Quasars in the Mass–Luminosity Plane. II. Black Hole Mass and Eddington Ratio Functions.” The Astrophysical Journal 764 (2013): 45. doi:10.1088/0004-637X/764/1/45.
• 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.
• Shen, Xuejian, et al. “The Bolometric Quasar Luminosity Function at $z=0$–7.” Monthly Notices of the Royal Astronomical Society 495 (2020): 3252–3275. doi:10.1093/mnras/staa1381.
• Shen, Yue, et al. “A Catalog of Quasar Properties from Sloan Digital Sky Survey Data Release 7.” The Astrophysical Journal Supplement Series 194 (2011): 45. doi:10.1088/0067-0049/194/2/45; VizieR J/ApJS/194/45.