The Soltan Argument and Integrated Growth

The Soltan argument converts the light radiated by accreting black holes over cosmic history into retained black-hole mass. It is an energy-accounting identity plus observational corrections, not a detailed growth model. Its power is that quasar lifetimes and individual light curves disappear from the integrated result; its limitation is that bolometric luminosity functions, obscured accretion, radiative efficiency, boundary mass, and the local mass density must all be supplied.

Required background. Probability and inference for gravity supplies likelihoods, covariance propagation, and selection-aware population inference.

Helpful background. Black-Hole Geometry and Mechanics supplies binding energy and spin-dependent efficiency; Relativistic Matter and Compact Objects supplies radiation and outflow physics; Black-Hole Astrophysics supplies luminosity and mass functions; Relativistic Cosmology supplies $H(z)$, cosmic time, and comoving volume.

Energy accounting from luminosity to mass

Let the bolometric luminosity function be

$$\Phi(L,z) \equiv \frac{dN}{dV_c\,d\log_{10}L},$$

and define its comoving luminosity density

$$j_{\rm bol}(z) = \int_{\log L_{\min}}^{\log L_{\max}} L\,\Phi(L,z)\,d\log_{10}L.$$

Because $\Phi$ is per dex, no extra factor of $\ln10$ belongs in this integral. A luminosity function per unit $L$ would instead be integrated with $dL$.

Suppose rest mass is supplied at rate $\dot M_{\rm in}$, radiation is the only energy escaping the accretion system, and no supplied rest mass is lost in a wind. With radiative efficiency $\eta$,

$$L=\eta\dot M_{\rm in}c^2, \qquad \dot M_\bullet=(1-\eta)\dot M_{\rm in}.$$

Restoring $c$ to expose the dimensions, the retained accreted mass density is therefore

$$\Delta\rho_{\bullet,\rm acc} = \int_{z_{\min}}^{z_{\max}} \left\lvert\frac{dt}{dz}\right\rvert dz\, \frac{1-\eta}{\eta c^2}\, j_{\rm bol}(z),$$

with

$$\left\lvert\frac{dt}{dz}\right\rvert = \frac{1}{(1+z)H(z)}.$$

The factor $(1-\eta)/\eta$ is essential: luminosity measures the radiated fraction, while the black hole retains the rest. Soltan’s original calculation expresses the same logic in terms of quasar counts and emitted energy per comoving volume Soltan 1982, § 2, equations (1)–(7), pp. 116–118.

If efficiency varies with mass, spin, accretion state, or redshift, the integrand needs a luminosity-weighted distribution:

$$\dot\rho_{\bullet,\rm acc}(z) = \frac{1}{c^2} \int d\log L\,d\eta\, \frac{1-\eta}{\eta}\, L\,\Phi(L,z)\,p(\eta\mid L,z).$$

Replacing this by a simple arithmetic mean efficiency is generally wrong because the nonlinear quantity being averaged is $(1-\eta)/\eta$.

Boundary terms and what the identity omits

The local mass density closes a broader budget,

$$\rho_\bullet(0) = \rho_{\rm seed} +\Delta\rho_{\bullet,\rm acc} -\rho_{\rm GW} -\rho_{\rm eject} +\rho_{\rm boundary}.$$

Black-hole mergers redistribute objects across mass bins but do not create mass density; gravitational radiation reduces the retained total. Ejection reduces a census restricted to galactic nuclei, but an ejected hole still contributes to the cosmic density of all black holes. Set $\rho_{\rm eject}=0$ for that all-hole census unless an actual spatial boundary is crossed. Here $\rho_{\rm seed}$ includes the initial seed population and subsequent seed formation; $\rho_{\rm boundary}$ collects other declared contributions omitted from the luminosity integral, such as growth below its luminosity limit, without counting the seeds again.

Radiatively inefficient growth adds mass with little bolometric light. Mechanical output changes the conversion even when $\eta$ denotes radiation alone. For example, if there is no rest-mass outflow and $\eta_{\rm kin}$ is the fraction of the same supplied rest-mass energy carried off mechanically, then

$$\dot M_\bullet =\frac{1-\eta-\eta_{\rm kin}}{\eta c^2}L.$$

This expression assumes a consistent energy boundary and no additional energy reservoir. Winds carrying rest mass and jets extracting stored black-hole spin require a fuller mass and energy balance. One cannot simply add observed jet power to bolometric luminosity while retaining the radiative-only prefactor.

The approximation regime is population-averaged growth dominated by radiatively efficient accretion over the integrated luminosity and redshift domain. It does not justify assigning one efficiency to every black hole, inferring individual spins, or ignoring dark growth. The limit $\eta\to0$ makes a fixed observed radiated energy require unbounded supplied mass, signaling that the radiatively efficient premise has failed. As $\eta\to1$, almost no mass is retained.

The argument also constrains an integrated history, not when or at what Eddington ratio the mass was acquired. Those require the continuity equation and the duty-cycle distribution developed on the following page.

A named luminosity and mass-density comparison

The luminosity-function dataset version used here is the Shen et al. 2020 bolometric quasar luminosity-function compilation at $z=0$–$7$, with the surveys listed in Table A1, the public analysis products described in Appendix C, and global fits A and B in Table 4. The two fits deliberately bracket different high-redshift faint-end behavior. The local comparison is the Shankar, Weinberg, and Miralda-Escudé 2009 electronic black-hole mass-function tabulation and its documented local mass-density range.

Shen and collaborators combine rest-frame infrared, optical, ultraviolet, soft-X-ray, and hard-X-ray luminosity functions with luminosity-dependent spectral and extinction corrections. They caution that overlapping samples are not fully independent and that the high-redshift faint end remains poorly constrained Shen et al. 2020, §§ 2–4, Tables 4 and A1, Appendix C. Shankar and collaborators show how the resulting integrated emissivity, an assumed efficiency, and a local mass function enter a self-consistent population calculation Shankar, Weinberg, and Miralda-Escudé 2009, §§ 2–4.

Calibration. Put every contributing survey on one cosmology, rest-frame luminosity definition, redshift scale, $K$-correction, host-subtraction rule, and comoving-volume convention. Preserve band-dependent bolometric-correction scatter, absorption and obscuration—including the Compton-thick component—and the covariance of the fitted luminosity-function parameters. Calibrate the local density with the same distance, host-mass, and black-hole scaling choices.

Selection. State each survey’s band, flux and color limits, AGN classification, redshift and luminosity domain, nondetection treatment, sky area, completeness weight, overlapping objects, and extrapolation below the flux limit. On the local side, state occupation, morphology, sphere-of-influence selection, upper limits, and integration bounds.

Model and statistics. Sample the luminosity-function coefficient covariance, bolometric and obscuration nuisance parameters, and the local mass-density likelihood. Integrate each draw over the same $(L,z)$ domain, add declared seed, kinetic, merger-loss, and ejection terms, and infer $\eta$ or a parameterized efficiency distribution. Posterior predictive checks should recover the contributing band luminosity functions and the local density, not only their final ratio.

Statistical uncertainty comes from survey counts and fit covariance. Systematic uncertainty is dominated by bolometric correction, obscured and Compton-thick fractions, faint- and high-redshift extrapolation, radiatively inefficient phases, kinetic output, efficiency variation, local mass calibration, occupation, and boundary terms.

The evidence cutoff is 2026-08-18. Results or releases after that date are outside this treatment and require a renewed integral rather than a verbal update.

Reproduction and independent checks

A reproducible pipeline downloads the frozen Shen et al. observational tables and public code, evaluates both global fits on a declared $(\log L,z)$ grid, and integrates with the exact per-dex measure and $dt/dz$. It saves luminosity-density contributions by band, luminosity, and redshift so faint-end extrapolation is visible. It then samples the published covariance and repeats with alternative obscuration corrections and the Shankar et al. electronic local mass functions.

Independent checks are:

• recover the original Soltan relation from the count-based form;
• verify that $j_{\rm bol}dt/c^2$ has mass per comoving volume;
• reproduce the fixed-$\eta$ limit and the behavior as $\eta\to0$ and $\eta\to1$;
• confirm that a merger-only remapping conserves integrated mass except for declared gravitational-wave loss;
• compare global fits A and B before extrapolating beyond their data support.

Generic host demographics without black-hole inference are outside this page’s scope. The argument determines the integrated energy-to-mass conversion; disk models supply the microphysical efficiency, and the black-hole mass-function page develops the selection-corrected local census.

Common pitfalls

Using luminosity per dex with $dL$. Match the integration measure to the published definition of $\Phi$.

Writing $\rho=E/(\eta c^2)$. The black hole retains only $(1-\eta)$ of the supplied rest mass in this model.

Calling the inferred efficiency a spin measurement. The result is a population-weighted accounting parameter entangled with obscuration, dark growth, and local mass calibration.

Infer a mean efficiency in a toy universe

Suppose the integrated, selection-corrected AGN radiation density has mass equivalent $u_{\rm rad}/c^2=5.0\times10^4\,M_\odot\,{\rm Mpc^{-3}}$. Seeds, ejections, mergers, and dark growth are negligible, and the measured local density is $4.5\times10^5\,M_\odot\,{\rm Mpc^{-3}}$. What constant $\eta$ closes the budget? What retained density would $\eta=0.20$ predict?

Solution

The budget is

$$\rho_\bullet = \frac{1-\eta}{\eta}\frac{u_{\rm rad}}{c^2},$$

so

$$\eta = \frac{u_{\rm rad}/c^2} {\rho_\bullet+u_{\rm rad}/c^2} = \frac{5.0\times10^4}{4.5\times10^5+5.0\times10^4} =0.10.$$

For $\eta=0.20$,

$$\rho_{\bullet,\rm acc} = \frac{0.8}{0.2}(5.0\times10^4) =2.0\times10^5\,M_\odot\,{\rm Mpc^{-3}}.$$

It undershoots the stated local density. That mismatch could mean the assumed efficiency is too high, but it could also signal missing obscured luminosity, dark accretion, a seed boundary term, or an overestimated local density.

References

• Shankar, Francesco, David H. Weinberg, and Jordi Miralda-Escudé. “Self-Consistent Models of the AGN and Black Hole Populations: Duty Cycles, Accretion Rates, and the Mean Radiative Efficiency.” The Astrophysical Journal 690 (2009): 20–41. doi:10.1088/0004-637X/690/1/20.
• 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.
• Soltan, Andrzej. “Masses of Quasars.” Monthly Notices of the Royal Astronomical Society 200 (1982): 115–122. doi:10.1093/mnras/200.1.115.