M_BH–M_* Relation and Galaxy Coevolution

• M_BH–M_* relation is an empirical scaling law connecting a galaxy’s central black hole mass with its stellar mass, exhibiting distinct regimes for massive (core-Sérsic) and low-mass (Sérsic) systems.
• The relation’s bend arises from differing M_*–σ and M_BH–σ dependencies, where massive systems follow near-linear scaling while low-mass galaxies show super-quadratic growth.
• This scaling law underpins models of galaxy evolution, AGN feedback, and black hole demographics, driving revised theoretical frameworks and simulation strategies.

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 (Scott et al., 2013, Graham et al., 2014, Graham et al., 2012, Davis et al., 2018):

• 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 (Davis et al., 2018).

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 (Graham et al., 2014). 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 (Graham et al., 2012).

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 (Sani et al., 2010).

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 (Scott et al., 2013).

4. Evolution with Redshift

Redshift evolution of the $M_{\rm BH}$–$M_*$ relation is constrained by deep imaging and spectroscopic studies (Lamastra et al., 2010, Li et al., 2023, Sarria et al., 2010, Trakhtenbrot et al., 2010, Sun et al., 5 Mar 2025, Delvecchio et al., 2019):

• For massive systems ($\log M_*/M_\odot > 10$): The canonical local scaling persists up to $z \sim 4$ (slope $\alpha \sim 1.6$ (Sun et al., 5 Mar 2025)) with negligible evolution in normalization or slope. SDSS-RM reverberation mapping at $z \sim 0.5$ yields $$\log(M_{\rm BH}/M_\odot) = 7.01^{+0.23}_{-0.33} + 1.74^{+0.64}_{-0.64}\log(M_*/10^{10}M_\odot)$$ (Li et al., 2023).
• 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 (Sun et al., 5 Mar 2025). 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 $M_{\rm BH}=10^8$–$10^9~M_\odot$ (Trakhtenbrot et al., 2010).
• 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 (Sun et al., 5 Mar 2025).
• 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 (Lamastra et al., 2010).

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 (Soker et al., 2010). 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 (Feoli et al., 2010).

• 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 $M_{\rm BH} \propto M_*^{1.7}$ (Delvecchio et al., 2019).

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 (Davis et al., 2018, Graham et al., 2012, Sahu et al., 2019).

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 (Lützgendorf et al., 2013, Graham et al., 2012). NSCs exhibit distinct scaling exponents, reinforcing the physical dichotomy between black hole and star cluster assembly (Scott et al., 2013).

Locally, total galaxy stellar mass can effectively proxy bulge mass for black hole scaling, with similar scatter, especially where bulge/disk decompositions are impractical (Sahu et al., 2019). Mid-infrared mass-to-light ratios ($\Upsilon_*^{3.6\mu m} \approx 0.6$–$1.1$) are robust for cross-paper comparisons (Sani et al., 2010, Sahu et al., 2019).

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 (Trakhtenbrot et al., 2010).
• 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 (Li et al., 2023, Sun et al., 5 Mar 2025).

The $M_{\rm BH}$–$M_*$ relation, in its bent, mass-dependent, and evolving incarnations, remains central to the paper 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.