---
title: M_BH–M_* Relation and Galaxy Coevolution
url: https://www.emergentmind.com/topics/m_bh-m_-relation
type: topic
---

# M_BH–M_* Relation and Galaxy Coevolution

The $M_{\rm BH}$–$M_*$ relation (also known as the Magorrian relation) quantifies the empirical and theoretical connection between the mass of a galaxy’s central black hole ($M_{\rm BH}$) and its stellar content ($M_*$). This scaling law underpins models of galaxy formation, AGN feedback, and black hole–galaxy coevolution, with different functional forms and evolutionary trends emerging in disparate morphological regimes and cosmic epochs.

## 1. Functional Form and Morphological Regimes

Large samples of nearby galaxies reveal that the $M_{\rm BH}$–$M_*$ relation is not a single power-law but is strongly bent, consisting of two distinct regimes governed by structural type, accretion history, and mass scale [1303.5490, 1411.1438, 1211.3199, 1810.04887]:

- **Core-Sérsic (massive, “core” galaxies):** 
  \[
  M_{\rm BH} \propto M_{*,\rm sph}^{0.97 \pm 0.14}
  \]
  or, in logarithmic form, 
  \[
  \log(M_{\rm BH}/M_\odot) = 9.27 \pm 0.09 + (0.97 \pm 0.14)\,\log(M_{*,\rm sph}/3.0 \times 10^{11} M_\odot)
  \]
  The mass fraction is nearly constant: $M_{\rm BH}/M_{*,\rm sph} \approx 0.5\%$.

- **Sérsic (lower-mass, “cusp” galaxies):** 
  \[
  M_{\rm BH} \propto M_{*,\rm sph}^{2.22 \pm 0.58}
  \]
  or
  \[
  \log(M_{\rm BH}/M_\odot) = 7.89 \pm 0.18 + (2.22 \pm 0.58)\,\log(M_{*,\rm sph}/2.0 \times 10^{10} M_\odot)
  \]
  Here, $M_{\rm BH}/M_{*,\rm sph} \propto M_{*,\rm sph}^{1.22}$; the mass fraction increases steeply with $M_{*,\rm sph}$, ranging from $\sim 10^{-4}$ at $10^{9} M_\odot$ to $\sim 5 \times 10^{-3}$ at $10^{11} M_\odot$.

- **Spiral galaxies (late-type):** 
  \[
  \log(M_{\rm BH}/M_\odot) = 7.24 \pm 0.12 + (2.44_{-0.31}^{+0.35})\,\log[M_{*,\rm sph}/(\upsilon(1.15 \times 10^{10} M_\odot))]
  \]
  with $\upsilon$ a mass-to-light ratio term. This slope robustly excludes pure merger-driven growth (which would yield $\alpha \approx 1$). AGNs in spirals occupy the same locus as inactive bulges, requiring revision of feedback prescriptions based on linear scaling [1810.04887].

## 2. Origin of the Bent Relation and Physical Interpretation

The bend in $M_{\rm BH}$–$M_*$ arises from a combination of the galaxy’s $M_*$–$\sigma$ relation and the approximately log-linear $M_{\rm BH}$–$\sigma$ relation [1411.1438]. For low/intermediate-mass spheroids (Sérsic), $M_{*,\rm sph} \propto \sigma^{2-3}$; for high-mass spheroids (core-Sérsic), $M_{*,\rm sph} \propto \sigma^{5-6}$. A log-linear $M_{\rm BH}$–$\sigma$ relation ($M_{\rm BH} \propto \sigma^{5.5 \pm 0.3}$) then yields a quadratic scaling at low masses and a linear scaling at high masses.

The regime distinction has direct implications for feeding and growth histories:
- **High masses / dry mergers:** Both stellar and black hole mass add linearly, preserving a fixed ratio.
- **Low masses / gas-rich growth:** BHs grow disproportionately fast, likely via secular accretion, cold flows, or gas-rich mergers, producing a super-quadratic relation.

A revised cold-gas "quasar"-mode feeding law for semi-analytic models is proposed for Sérsic systems to mimic quadratic growth:
\[
\delta M_{\rm BH} \propto \left(\frac{M_{\rm min}}{M_{\rm maj}}\right) \frac{M_{\rm cold}^2}{1 + \left(280~{\rm km\,s}^{-1}\right)/V_{\rm virial}}
\]
where $M_{\rm cold}$ is the total cold-gas mass available for accretion [1211.3199].

## 3. Intrinsic Scatter, Sample Composition, and Secondary Correlations

The observed log-normal scatter (rms in $\log M_{\rm BH}$) is regime-dependent:
- $0.44$–$0.47$ dex for core-Sérsic;
- $0.90$–$0.95$ dex for Sérsic galaxies;
- $0.60$–$0.70$ dex for spiral galaxies;
- $0.35$–$0.38$ dex in combined bulge-disk decompositions at mid-IR wavelengths [1012.3073].

Complementary scaling laws, such as $M_{\rm BH}$–Sérsic index and $M_{*,\rm sph}$–spiral-arm-pitch-angle, yield comparably tight fits and provide additional predictors for $M_{\rm BH}$, particularly in late-type systems. Nuclear star clusters (NSCs) follow a much shallower mass–scaling law, confirming distinct evolutionary pathways for NSCs and SMBHs [1303.5490].

## 4. Evolution with Redshift

Redshift evolution of the $M_{\rm BH}$–$M_*$ relation is constrained by deep imaging and spectroscopic studies [1001.5407, 2301.04177, 1010.0768, 1005.2408, 2503.03675, 1910.08114]:

- **For massive systems ($\log M_*/M_\odot > 10$):** The canonical local scaling persists up to $z \sim 4$ (slope $\alpha \sim 1.6$ [2503.03675]) with negligible evolution in normalization or slope. SDSS-RM reverberation mapping at $z \sim 0.5$ yields $\log(M_{\rm BH}/M_\odot) = 7.01^{+0.23}_{-0.33} + 1.74^{+0.64}_{-0.64}\log(M_*/10^{10}M_\odot)$ [2301.04177].

- **For low-mass systems:** Significant positive offsets in $M_{\rm BH}/M_*$ are observed at $z\gtrsim4$, with galaxies hosting “overmassive” SMBHs during accretion episodes, then reverting toward the local relation as stellar mass builds during later quiescent periods [2503.03675]. Strong mass dependence persists, especially for star-forming AGN hosts: $M_*/M_{\rm BH}$ grows by factors $\sim$4–8 between $z \sim 2$ and $z \sim 0$ for $M_{\rm BH}=10^8$–$10^9~M_\odot$ [1005.2408].

- **High-z quasars:** At $z \gtrsim 6$, the most luminous quasars can reach $M_{\rm BH}/M_*$ ratios well above local averages but remain consistent with rare, early massive-seed growth tracks [2503.03675].

- **Interaction-driven models:** Hierarchical scenarios predict a ratio $\Gamma(z) = (M_{\rm BH}/M_*)(z) / (M_{\rm BH}/M_*)(z=0)$ rising to $\Gamma \sim 5$ for $M_{\rm BH} > 10^9 M_\odot$ at $z > 4$, supporting observed high-redshift QSO offsets [1001.5407].

## 5. Theoretical Models and Physical Mechanisms

Several physical models underpin the form and scaling of $M_{\rm BH}$–$M_*$:

- **Penetrating-jet feedback models:** These produce a proportionality of $M_{\rm BH} \propto M_G \sigma / c$ (“momentum parameter”), tightly correlated with classical relations and physically motivated by balancing jet momentum against the bulge [1005.5701]. The fitted normalization yields
  \[
  M_{\rm BH} \sim 3.3 \frac{M_G \sigma}{c}
  \]
  with intrinsic scatter $\sim0.32$ dex and efficiency parameter $\eta_p \sim 0.04$.

- **Angular momentum conservation models:** Feoli & Mancini derive $M_{\rm BH} \propto R_e \sigma^3$, which translates to
  \[
  M_{\rm BH} = 10^{8.74 \pm 0.03}\left(\frac{M_*}{10^{11}\;M_\odot}\right)^{1.21 \pm 0.03}
  \]
  based on structural scaling relations, with scatter $\sim0.45$ dex [1012.3160].

- **Empirical models with gas-regulated growth:** Delvecchio et al. show that halo mass sets a critical threshold. Below $M_{\rm DM} \sim 2\times10^{12}~M_\odot$, feedback suppresses BH feeding (${\rm BHAR/SFR} \propto M_{\rm DM}^{1.6}$); above it, cold flows fuel coeval BH and stellar mass growth (${\rm BHAR/SFR} \propto M_{\rm DM}^{0.3}$), yielding an asymptotic scaling $M_{\rm BH} \propto M_*^{1.7}$ [1910.08114].

## 6. Implications for AGN Feedback, Simulations, and Black Hole Demographics

The recognition of a strongly bent $M_{\rm BH}$–$M_*$ relation demands revision of AGN feedback prescriptions in hydrodynamic and semi-analytic models. Particularly in late-type and gas-rich systems, the quadratic regime cannot arise from additive merging but requires efficient fueling and/or gas-rich compaction episodes [1810.04887, 1211.3199, 1903.04738]. 

Demographically, the scaling law influences gravitational wave predictions (via black hole mass functions), black hole occupation fractions in dwarf galaxies, and intermediate-mass black hole (IMBH) candidate analyses in globular clusters and low-luminosity AGN hosts [1304.7156, 1211.3199]. NSCs exhibit distinct scaling exponents, reinforcing the physical dichotomy between black hole and star cluster assembly [1303.5490].

Locally, total galaxy stellar mass can effectively proxy bulge mass for black hole scaling, with similar scatter, especially where bulge/disk decompositions are impractical [1903.04738]. Mid-infrared mass-to-light ratios ($\Upsilon_*^{3.6\mu m} \approx 0.6$–$1.1$) are robust for cross-study comparisons [1012.3073, 1903.04738].

## 7. Key Parametric Relations Table

| Morphological Type   | Relation                                 | Slope      | Scatter (dex)   |
|---------------------|-------------------------------------------|------------|-----------------|
| Core-Sérsic         | $M_{\rm BH} \propto M_{*,\rm sph}^{0.97}$| 0.97±0.14  | 0.44–0.47       |
| Sérsic              | $M_{\rm BH} \propto M_{*,\rm sph}^{2.22}$| 2.22±0.58  | 0.90–0.95       |
| Spiral/Late-Type    | $M_{\rm BH} \propto M_{*,\rm sph}^{2.44}$| 2.44±0.35  | 0.60–0.70       |
| Early-Type (ETG)    | $M_{\rm BH} \propto M_{*,\rm sph}^{1.27}$| 1.27±0.07  | 0.52            |
| Magorrian (Combined)| $M_{\rm BH} = 0.5\%\times M_{*,\rm sph}$ | ~1         | ~0.5            |
| Angular Mom. Model  | $M_{\rm BH} \propto M_*^{1.21}$          | 1.21±0.03  | 0.45            |

## 8. Controversies and Outstanding Challenges

- The existence and universality of a single $M_{\rm BH}/M_*$ ratio at all cosmic epochs is refuted by consistent demonstration of strong mass and evolutionary dependence, especially in low-mass, high-redshift systems.
- Previous studies often failed to account for $M_{\rm BH}$-dependence or host star-forming status, yielding erroneously shallow evolution [1005.2408].
- Selection biases (“Lauer bias”) in AGN and QSO samples can spuriously inflate the normalization and slope; careful modeling of completeness and Eddington ratio distributions is required to recover intrinsic scaling [2301.04177, 2503.03675].

The $M_{\rm BH}$–$M_*$ relation, in its bent, mass-dependent, and evolving incarnations, remains central to the study of galaxy and supermassive black hole cosmic history, with attendant implications for gravitational wave astronomy, AGN feedback theory, and the interpretation of high-redshift scaling relations from JWST and future missions.

Source: https://www.emergentmind.com/topics/m_bh-m_-relation