Black-Hole–Host Scaling Relations

Black-hole mass correlates with host velocity dispersion, bulge luminosity, and stellar mass, but a fitted line is not automatically an intrinsic law or evidence of feedback. Dynamical targets are selected, host variables are correlated, morphology matters, and some measurement errors move both axes together. The scientific object is therefore a conditional distribution in a stated parent population, carried through targeting and measurement before it is compared with the published sample.

Required background. Probability and inference for gravity supplies errors-in-variables regression, censoring, covariance, and selection functions.

Helpful background. Black-Hole Geometry and Mechanics supplies the mass definition; Relativistic Matter and Compact Objects supplies stellar-population physics; Black-Hole Astrophysics supplies mass measurements and source selection; Relativistic Cosmology supplies distances, redshift, and galaxy assembly.

A relation is a conditional distribution

For one host observable $X$, write

$$y=\alpha+\beta(x-x_0)+\epsilon, \qquad y=\log_{10}\frac{M_\bullet}{M_\odot}, \qquad x=\log_{10}\frac{X}{X_0},$$

where $\epsilon$ has intrinsic dispersion $\sigma_{\rm int}$ only after measurement error and population structure are modeled. A multivariate relation may include velocity dispersion $\sigma_e$, bulge stellar mass $M_{\star,\rm bulge}$, effective radius, morphology, or environment. Because these predictors covary, the smallest one-dimensional scatter does not identify a fundamental cause.

Regression direction answers a question. Predicting $M_\bullet$ from $\sigma_e$ is not equivalent to predicting $\sigma_e$ from $M_\bullet$, and a symmetric bisector is not a substitute for either generative model. State which latent variable is conditional, how upper limits enter, and which population the scatter describes.

The sphere of influence makes the selection especially dangerous:

$$r_{\rm infl}\sim\frac{GM_\bullet}{\sigma_e^2}, \qquad \theta_{\rm infl}\sim\frac{GM_\bullet}{\sigma_e^2D}.$$

A requirement $\theta_{\rm infl}\gtrsim\theta_{\rm res}$ preferentially admits overmassive black holes at fixed host property and high-dispersion galaxies at fixed stellar mass. Cutting on the measured sphere of influence after fitting is also circular, because it depends on the response variable.

A forward selection model

Let $\mathbf x$ denote latent host variables, $y$ the latent black-hole mass, $d$ the measured photometry and kinematics, and $T$ the targeting and successful-measurement event. A coherent likelihood has the schematic form

$$p(d,T\mid\theta) = \int dy\,d\mathbf x\, p(d\mid y,\mathbf x,T)\, S(T\mid y,\mathbf x,D,\theta_{\rm res})\, p(y\mid\mathbf x,\theta)\, p_{\rm parent}(\mathbf x).$$

The first factor is the measurement distribution conditional on targeting and successful inclusion. For a likelihood conditioned on sample size, divide by $P(T\mid\theta)$, obtained by integrating this joint expression over the accepted data. In a count-based likelihood the same integral sets the expected count. When $T$ is determined by a cut on $d$ itself, the product $p(d\mid y,\mathbf x,T)S(T\mid y,\mathbf x,\ldots)$ reduces to the ordinary $p(d\mid y,\mathbf x)$ for accepted data; multiplying that ordinary likelihood by another selection probability would double-count the cut. The mass-function likelihood gives the corresponding derivation. Nondetections and upper limits contribute their appropriate likelihoods; dropping them conditions on success.

Covariance matters. A common distance enters a dynamical mass approximately as $M_\bullet\propto D$ and a luminosity as $L\propto D^2$. Bulge decomposition couples luminosity, effective radius, and stellar mass. A point-spread function can influence both the central kinematics and the photometric decomposition. Treating these as independent error bars can reduce apparent scatter and rotate a relation.

Posterior predictive checks should operate in observable space: reproduce the distributions of angular sphere of influence, distance, morphology, velocity dispersion, host mass, upper limits, and accepted mass quality. A relation that matches only the $M_\bullet$–$X$ panel may fail the selection variables that produced it.

The named local sample

The dataset version is the Kormendy–Ho 2013 dynamical compilation, machine-readable Tables 2 and 3, containing ellipticals, classical bulges, and pseudobulges with distances, masses, velocity dispersions, and bulge photometry. Kormendy and Ho explicitly separate morphology and exclude some systems from particular fits; their classical-bulge and elliptical relations should not be silently assigned to pseudobulges Kormendy and Ho 2013, § 6, pp. 577–604, Tables 2–3.

Calibration. Freeze the adopted distance and its uncertainty, astrometric and point-spread-function model, spectral resolution, velocity-dispersion aperture, stellar population and mass-to-light scale, bulge–disk–bar decomposition, and dynamical mass method. Preserve shared distance and decomposition covariance rather than rescaling only one column.

Selection. Record morphology and activity classes, the parent target list, mass detections and upper limits, angular resolution, sphere-of-influence sensitivity, distance and quality cuts, follow-up failures, and the rule for mergers or disturbed galaxies. The compilation is assembled from heterogeneous programs, so these probabilities are not recoverable from its accepted rows alone.

Model and statistics. Fit latent host and mass variables with correlated errors, censoring, intrinsic covariance, and a selection model. Compare $M_\bullet$–$\sigma_e$, $M_\bullet$–$M_{\star,\rm bulge}$, and multivariate forms under the same parent population. Report forward and inverse predictions and leave one morphology class out before claiming universality.

Shankar and collaborators compare local dynamical hosts with a large Sloan Digital Sky Survey galaxy population and show that black-hole hosts have unusually high dispersions at fixed stellar mass. Their Monte Carlo application of a sphere-of-influence selection produces scaling relations offset from the assumed intrinsic relation Shankar et al. 2016, §§ 2–4 and 6. That result demonstrates that the bias can be large under a plausible model; it does not uniquely determine the true relation because target selection, mass reliability, and the intrinsic multivariate form remain uncertain.

Results and causal interpretations

The evidence cutoff is 2026-08-18. Both the intrinsic relation and its physical interpretation remain unresolved at the precision needed for population extrapolation.

Relation or test	Evidence or result	Independent check	Scope, limitation, or open status
$M_\bullet$–$\sigma_e$ in classical bulges and ellipticals	Tight correlation in the selected dynamical compilation	Fit the parent dispersion distribution with angular-resolution selection	Selection and correlated host variables can shift slope, normalization, and scatter
$M_\bullet$–bulge mass or luminosity	Strong conditional trend after morphological decomposition	Propagate distance and decomposition covariance; leave morphology out	Pseudobulges and total versus bulge mass do not follow one interchangeable calibration
Multivariate mass–dispersion–size model	Can explain residual trends and multiple projected relations	Predict held-out radius, morphology, and resolution subsets	A better predictor need not be a causal variable
Active-galaxy comparison	Extends mass and host measurements into a differently selected population	Share the virial-factor and host-decomposition calibration in one hierarchy	Activity, broad-line selection, host contrast, and virial masses add new systematics

The competing positions are a direct self-regulating black-hole–bulge relation, common dependence on the central potential or assembly history, morphology-specific relations, merger averaging, and an important contribution from resolution and follow-up selection. A correlation by itself cannot distinguish them. A causal model must predict observables not used to tune the relation—for example residual gas fractions, time ordering, or held-out morphology—and must survive a no-feedback or altered-feedback counterfactual.

The reproduction pipeline should retrieve Tables 2–3 and their provenance, refit with common distance and correlated errors, construct a synthetic SDSS-like parent galaxy population, apply the declared target and angular-resolution selection, and verify coverage on injected relations. Repeat with upper limits and leave-morphology-out posterior predictions.

The interpretation could change after a volume-limited dynamical program or an independently documented sphere-of-influence selection function; it could also change when a stellar-mass calibration or high-redshift host release changes shared covariance or morphology coverage.

Interpretation and scope

Scaling relations are valuable compressors: they map galaxy censuses into mass functions and provide priors for individual sources. Their uncertainties must propagate into those uses. Extrapolation below the calibrated mass range, across morphology, or to high redshift is a model prediction rather than a measurement.

Generic host demographics without black-hole inference are outside this page’s scope. Physical feedback and causal coevolution are treated separately on Black-Hole Feedback and Coevolution; the treatment here concerns the statistical relation and how sample selection affects it.

Common pitfalls

Calling the smallest scatter fundamental. Correlated predictors, errors, and selection can make one projection appear tighter without identifying causation.

Resolving the sphere of influence and then ignoring that cut. The measurement criterion depends on mass, dispersion, distance, and angular resolution, exactly the variables in the regression.

Mixing bulge definitions. Classical bulges, pseudobulges, bars, and total stellar mass require explicit decompositions and morphology-dependent checks.

Follow a shared distance error

Suppose a galaxy’s adopted distance is revised upward by $20\%$. To leading order its dynamical black-hole mass scales as $D$ and its luminosity as $D^2$. Find the shifts in $\log_{10}M_\bullet$ and $\log_{10}L$. Along what slope does this shared error move the point in a plot of $\log M_\bullet$ versus $\log L$?

Solution

With $f_D=1.2$,

$$\Delta\log_{10}M_\bullet=\log_{10}(1.2)=0.0792,$$

while

$$\Delta\log_{10}L=2\log_{10}(1.2)=0.1584.$$

The displacement therefore has slope

$$\frac{\Delta\log M_\bullet}{\Delta\log L}=\frac12.$$

It is not two independent errors. Ignoring their covariance can make a relation aligned with this direction look artificially narrow and can distort both its slope and intrinsic-scatter estimate.

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, et al. “Selection Bias in Dynamically Measured Super-Massive Black Hole Samples: Its Consequences and the Quest for the Most Fundamental Relation.” Monthly Notices of the Royal Astronomical Society 460 (2016): 3119–3142. doi:10.1093/mnras/stw678.