The Small Scatter in BH-Host Correlations & The Case for Self-Regulated BH Growth

Supermassive black holes (BHs) obey tight scaling relations between their mass and their host galaxy properties such as total stellar mass, velocity dispersion, and potential well depth. This has led to the development of self-regulated models for BH growth, in which feedback from the central BH halts its own growth upon reaching a critical threshold. However, models have also been proposed in which feedback plays no role: so long as a fixed fraction of the host gas supply is accreted, relations like those observed can be reproduced. Here, we argue that the scatter in the observed BH-host correlations, and its run with scale, presents a demanding constraint on any model for these correlations, and that it favors self-regulated models of BH growth. We show that the scatter in the stellar mass fraction within a radius R in observed ellipticals and spheroids increases strongly at small R. At fixed total stellar mass (or host velocity dispersion), on very small scales near the BH radius of influence, there is an order-of-magnitude scatter in the amount of gas that must have entered and formed stars. In short, the BH appears to ‘know more’ about the global host galaxy potential on large scales than the stars and gas supply on small scales. This is predicted in self-regulated models; however, models where there is no feedback would generically predict order-of-magnitude scatter in the BH-host correlations. Likewise, models in which the BH feedback in the ‘bright’ mode does not regulate the growth of the BH itself, but sets the stellar mass of the galaxy by inducing star formation or blowing out a mass in gas much larger than the galaxy stellar mass, are difficult to reconcile with the scatter on small scales.

💡 Research Summary

The paper tackles one of the most striking empirical facts in extragalactic astronomy: super‑massive black holes (SMBHs) obey remarkably tight scaling relations with their host galaxies, such as the M‑BH–σ, M‑BH–M* and M‑BH–potential‑depth correlations. Two broad classes of theoretical explanations have been advanced. In “self‑regulated” models, feedback from the accreting black hole (radiative pressure, winds, jets) injects enough energy or momentum into the surrounding interstellar medium to halt further accretion once the black hole reaches a critical mass that is set by the depth of the galaxy’s potential well. In contrast, “fixed‑fraction” models assume that a constant proportion of the galaxy’s available gas is always funneled onto the black hole, with no explicit role for feedback; the observed correlations then arise simply because the total gas reservoir scales with the host’s mass or velocity dispersion.

The authors argue that the scatter of these correlations, especially its dependence on spatial scale, provides a decisive test between the two scenarios. Using a large sample of elliptical and spheroidal galaxies with high‑resolution photometry, they measure the enclosed stellar‑mass fraction f*(<R)=M*(<R)/M_tot as a function of radius R. At a fixed total stellar mass (or σ), the dispersion in f* is modest (≈0.1 dex) on kiloparsec scales but grows dramatically as R approaches the black‑hole sphere of influence (∼10–100 pc). Near the influence radius the scatter reaches ≳0.5–1 dex, meaning that galaxies with essentially identical global properties can have central stellar (and by inference gas) masses that differ by an order of magnitude.

If black‑hole growth were governed solely by the amount of gas present on these small scales, the black‑hole mass itself would inherit this large scatter, contradicting the observed tightness of the M‑BH–σ relation (scatter ≈0.3 dex). Instead, the data imply that the black hole “knows” about the global potential of its host rather than the stochastic local gas supply. This is precisely what self‑regulated models predict: once the black hole reaches the mass at which its feedback can unbind or heat the surrounding gas, further accretion is shut off, and the final black‑hole mass is set by the depth of the galaxy’s potential well, not by the instantaneous central gas reservoir.

The paper backs up this qualitative argument with simple analytic estimates and with numerical experiments. In the self‑regulated case, feedback energy scales with M‑BH and couples to the gas with an efficiency that yields a critical mass proportional to σ^4 (or to the binding energy of the bulge). This naturally reproduces both the slope and the low scatter of the observed relations, and it also predicts that the scatter in f* should increase toward smaller radii – exactly what is seen. By contrast, in the fixed‑fraction scenario the black‑hole mass is directly proportional to the central gas mass; the large observed variation of f* on sub‑kiloparsec scales would then translate into a comparable variation in M‑BH, which is not observed. Simulated galaxy populations built under this assumption invariably show a scatter in the M‑BH–σ plane that is at least a factor of two larger than the data.

The authors also examine a hybrid idea in which bright‑mode feedback does not regulate the black‑hole growth itself but instead determines the final stellar mass of the galaxy by triggering star formation or ejecting a gas mass many times larger than the stellar mass. They argue that such a mechanism would require unrealistically high coupling efficiencies and finely tuned timing to reconcile the small‑scale scatter with the tight global correlations, making it an implausible explanation.

In summary, the paper presents three key pieces of evidence favoring self‑regulated black‑hole growth: (1) the observed increase of stellar‑mass‑fraction scatter toward the black‑hole sphere of influence, (2) the consistency of this trend with feedback‑limited growth models, and (3) the inability of non‑feedback, fixed‑fraction models to reproduce the low scatter of the M‑BH–σ relation. The work thus strengthens the case that active‑galactic‑nucleus feedback is not a peripheral detail but a central, governing process that couples the evolution of super‑massive black holes to that of their host galaxies.