Stellar Mass–BH Mass Relations

• Stellar Mass–BH Mass Relations is a topic that defines how black hole mass scales with host stellar mass, emphasizing morphology, redshift evolution, and density dependencies.
• The relations vary between early- and late-type galaxies, with nonlinear growth patterns and distinct bulge-disk dynamics informing the co-evolution of galaxies and their central black holes.
• Studies link central stellar density and feedback processes to black hole growth, offering practical insights into galaxy assembly from the local universe to high redshift.

The relation between stellar mass and black hole (BH) mass is fundamental to understanding the co-evolution of galaxies and their central BHs across the full mass spectrum and cosmic epochs. Empirical scaling relations—connecting either the bulge, spheroid, or total stellar mass of the host to the BH mass—emerge in both the local and high-redshift universe, exhibiting morphology dependence, nonlinear structure, and cosmic evolution. These relations are also modulated by halo potential, feedback processes, and the growth histories of both galaxies and BHs.

1. Foundations: Empirical Scaling Relations and Morphology Dependence

The canonical relation between supermassive black hole (SMBH) mass and host spheroid stellar mass is described by distinct regimes according to galaxy morphology. In the local universe, for massive early-type (core-Sérsic) galaxies, the $M_{BH}$–$M_{sph,*}$ relation is approximately linear: $$\log(M_{BH}/M_\odot) = (0.97 \pm 0.14) \log \left( \frac{M_{sph,*}}{3 \times 10^{11} M_\odot} \right) + (9.27 \pm 0.09)$$ implying BHs constitute a roughly constant fraction ($\sim0.55\%$) of the spheroid stellar mass (Scott et al., 2013).

In contrast, for Sérsic (lower-mass, late-type) galaxies: $$\log(M_{BH}/M_\odot) = (2.22 \pm 0.58) \log \left( \frac{M_{sph,*}}{2 \times 10^{10} M_\odot} \right) + (7.89 \pm 0.18)$$ This implies a dramatic increase in $M_{BH}/M_{sph,*}$ at higher spheroid masses and that BHs in these systems grow super-linearly relative to their host (Scott et al., 2013); similar quadratic scaling is observed specifically in detailed spiral galaxy studies (Davis et al., 2018).

For spiral galaxies, sophisticated multicomponent decompositions yield: $$\log(M_{BH}/M_\odot) = [2.44_{-0.31}^{+0.35}] \log \frac{M_{*,sph}}{\upsilon(1.15 \times 10^{10} M_\odot)} + (7.24 \pm 0.12)$$ with $\upsilon$ the $M/L$ normalization (Davis et al., 2018). Notably, AGN-hosting bulges follow the same relation, with no support for systematic offsets in active versus inactive bulges. The implications are that secular, bar, or merger-driven bulge and BH growth are closely coupled in both quiescent and AGN-active systems.

When scaling with total stellar mass in late-type hosts,

$$\log(M_{BH}/M_\odot) = [3.05_{-0.49}^{+0.57}] \log \frac{M_{*,tot}}{\upsilon(6.37 \times 10^{10} M_\odot)} + [7.25_{-0.14}^{+0.13}]$$

demonstrating a slope more than twice that obtained for core-Sérsic (early-type) galaxies (for which the slope is $\sim1.3$) (Davis et al., 2018). This steepening primarily arises from the varying bulge-to-total ratios in late-type systems, reflecting distinct evolutionary pathways.

Table 1: Summary of Morphology-Dependent $M_{BH}$–Host Scaling Relations

Host Type	$M_{BH}$ Relation	Slope	Scatter (dex)
Core-Sérsic (early-type)	$M_{BH} \propto M_{sph,*}^{0.97}$	$\sim1$	$\sim0.3$
Sérsic (late-type)	$M_{BH} \propto M_{sph,*}^{2.22}$	$>2$	$\sim0.7$
Spirals ($M_{*,tot}$)	$M_{BH} \propto M_{*,tot}^{3.05}$	$\sim3$	$0.66$
Dwarf (bulge mass)	$M_{BH} \propto M_{bulge}^{1.24}$	$\sim1$	$0.47$

This structure underscores the necessity of considering galaxy morphological type (and whether scaling uses bulge, spheroid, or total stellar mass) in any quantitative or evolutionary paper, as mixing types produces arbitrary slopes with no physical meaning (Davis et al., 2018).

2. Stellar Mass–BH Mass Evolution with Redshift

Scaling relations are manifestly not static with cosmic epoch. The ratio

$$\Gamma(z) = \left( \frac{M_{BH}}{M_*} \right)_z \Bigg / \left( \frac{M_{BH}}{M_*} \right)_{z=0}$$

quantifies the redshift evolution of the normalized BH-to-stellar mass ratio (Lamastra et al., 2010). Theoretical models trace this ratio's dependence on both BH mass and redshift.

Key results:

• At $z \gtrsim 4$, the most massive BHs ($M_{BH} \gtrsim 10^9 M_\odot$) reach $\Gamma \sim 5$—i.e., their BH-to-stellar mass ratio is five times the local value, implying more efficient (relative) BH growth in dense early environments.
• At $1 < z < 2$, broad-line AGNs show predicted $\Gamma \sim 2$, in line with observations of luminous type 1 AGN samples.
• Gas-rich submillimeter galaxies at $2 < z < 3$ display much lower values ($0.3 < \Gamma < 1.5$), indicating that SMBH growth in these systems lags stellar assembly (Lamastra et al., 2010).

Empirical studies confirm that the normalization and scatter in the $M_{BH}$–$M_*$ relation (both bulge and total) increase with redshift, especially for massive systems (Li et al., 2023, Zhang et al., 2023). Recent reverberation-mapping AGN samples at $z \sim 0.5$ and HETDEX AGN at $z \sim 2$ reveal a relation: $$\log(M_{BH}/M_\odot) = 7.01^{+0.23}_{-0.33} + 1.74^{+0.64}_{-0.64} \log \left( \frac{M_{*,host}}{10^{10} M_\odot} \right)$$ at $z_{\text{med}} = 0.5$, essentially indistinguishable from the local relation, but at $z \sim 2$ there is a positive 0.52 dex offset in normalization, indicating that SMBHs were comparatively overmassive for a given $M_*$ at earlier epochs (Zhang et al., 2023).

This evolution is interpreted in merger-driven models as a manifestation of the downsizing scenario: the most massive SMBHs undergo most of their accretion early and subsequently relax towards the local scaling relation as further stellar mass assembles and major accretion slows (Lamastra et al., 2010).

3. Dependence on Galaxy Structure and Stellar Density

Recent large-sample decompositions of the host galaxy light using HST or Spitzer IR imaging and advanced codes (GALFIT, isofit, cmodel) have revealed further nuance: $M_{BH}$ correlates not just with integrated stellar mass, but with central stellar densities, surface brightness profiles, and Sérsic index (Sahu et al., 2021).

For both projected ($\Sigma$) and deprojected ($\rho$) stellar densities, relations of the form

$$\log(M_{BH}/M_\odot) = a \log(\Sigma_R / \Sigma_0) + b$$

or

$\log(M_{BH}/M_\odot) = a \log(\rho_r / \rho_0) + b$

have been established, with slopes often $>2$ for measures within small radii (compactness), decreasing at larger aperture, and the scatter correspondingly decreasing. The tightest relations are obtained when using larger physical apertures (e.g., 5 kpc), supporting the conclusion that global spheroid structure is tightly linked to central BH mass (Sahu et al., 2021).

Moreover, for high-Sérsic index (core-Sérsic) galaxies, the $M_{BH}$–density relation at the BH sphere of influence radius is notably tight: core-Sérsic systems show a shallow slope ($-0.68 \pm 0.06$) and minimal scatter ($0.21$ dex), while low-$n$ systems show a steeper relation and higher scatter [$2110.05037$].

These density-based relations indicate that the BH "knows" not just about the host mass, but its spatial distribution—likely an imprint of assembly history and feedback mechanisms.

4. Theoretical Context: Growth Mechanisms and Feedback

Semi-analytic and empirical models track the build-up of $M_{BH}$ and $M_*$ via:

• Galaxy-Galaxy Interactions: SMBH accretion is predominantly triggered by encounters—mergers and fly-bys—in high-density environments, leading to early, rapid BH growth in the most massive objects (Lamastra et al., 2010).
• Secular Evolution: Late-type galaxies with lower $B/T$ and higher disk fractions can fuel central BH growth via bars, pseudo-bulge formation, and minor mergers.
• Feedback Processes: BH accretion and star formation are regulated by stellar- and AGN-driven feedback, with two regimes emerging as a function of halo mass and BH mass (Marasco et al., 2021, Delvecchio et al., 2019). Below $M_{halo} \sim 2 \times 10^{12} M_\odot$, SN feedback suppresses accretion; above, efficient cold gas inflows fuel both SFR and BHAR with scaling $BHAR/SFR \sim M_{DM}^{0.3}$.

A succinct empirical model: $BHAR/SFR \propto M_*^{0.73}$ at $z = 0$, leading at late times to a superlinear $M_{BH} \propto M_*^{1.7}$ relation (Delvecchio et al., 2019), in qualitative agreement with both the observed quadratic relations at low mass and linear scaling at high mass.

5. Universality, Scatter, and Outliers

The $M_{BH}$–$M_*$ scaling relations are not universal across all morphological types or evolutionary states:

• Late-type, low $B/T$ disk galaxies show systematically lower $M_{BH}$ values for a given total $M_*$, both locally and at high $z$; this is observed in AGN and megamaser spiral samples, with offsets of $-0.6$ to $-0.8$ dex relative to early-type, bulge-dominated systems (Läsker et al., 2016, Greene et al., 2016).
• The scatter in the $M_{BH}$–host relations is minimized when accounting for host morphology, bulge-disk decomposition, and when using global (rather than compactness) measures of density (Sahu et al., 2021).
• Dwarf galaxies with $M_{BH} \lesssim 10^{5} M_\odot$ and $M_* \sim 10^9 M_\odot$ follow near-linear $M_{BH}$–$M_{bulge}$ relations—indicative that BH seeding and early growth mechanisms establish a scalable relation across five orders of magnitude (Schutte et al., 2019).

6. Cosmological and Simulation Implications

These scaling relations impact:

• BH Mass Function and Low-Mass End: The distribution of stellar-remnant BHs is nearly flat to $m_{BH} \sim 50 M_\odot$, then falls log-normally; local density in stellar BHs is $\rho_{BH} \sim 5 \times 10^7 M_\odot$ Mpc$^{-3}$, over $100\times$ that in SMBHs (Sicilia et al., 2021).
• Interpretation of High-$z$ AGN: At $z > 6$, using low-normalization scaling fit to AGN samples ($\mu = 1.05 s - 4.10$) with $M_{BH}/M_* \sim 10^{-4}$–$10^{-3}$, resolves the dearth of faint AGN in deep X-ray surveys. Adopting the high-normalization "bulge scaling" would otherwise predict substantially more luminous AGN in LBGs than are observed (Volonteri et al., 2016).
• Feedback Regulation: The equilibrium between cosmic gas accretion, cooling, star formation, and AGN feedback predicts a transition at $M_{halo} \sim 10^{12} M_\odot$ (and $M_{BH} \sim 10^{7−8} M_\odot$), separating a rapid, unregulated regime from a slow, self-regulated one (Marasco et al., 2021).

7. Current Controversies and Future Prospects

• Morphological mixing: Studies emphasizing total stellar mass scaling (particularly at high $z$ or in unresolved data) must account for variable bulge-to-total ratios. Uncritical mixing of early- and late-type systems yields scaling slopes lacking physical interpretability (Davis et al., 2018).
• High-Redshift Efficiency: At $z \sim 2$, empirical relation offsets imply that BHs at fixed host mass are significantly more massive than in today's universe (Zhang et al., 2023)—disagreeing with some simulation predictions that invoke strong SN feedback to suppress central BH growth in low-mass galaxies.
• Probing the Scatter: Selection effects in local dynamical samples (favoring massive, quiescent galaxies with well-resolved spheres of influence) must be compared with AGN and megamaser samples to avoid misinterpreting the width or location of the intrinsic scaling (Greene et al., 2016, Reines et al., 2015).

Deep, high-resolution surveys at high redshift and further expansions of precision dynamical samples in late-types and dwarfs (via masers, IFU kinematics, and advanced photometric decomposition) are essential to refining the universality and drivers of the stellar mass–BH mass relation.

• "The Building Up of the Black Hole Mass - Stellar Mass Relation" (Lamastra et al., 2010)
• "The black hole - bulge mass relation of Active Galactic Nuclei in the Extended Chandra Deep Field - South Survey" (Schramm et al., 2012)
• "The Supermassive Black Hole Mass - Spheroid Stellar Mass Relation for Sérsic and Core-Sérsic Galaxies" (Scott et al., 2013)
• "Relations Between Central Black Hole Mass and Total Galaxy Stellar Mass in the Local Universe" (Reines et al., 2015)
• "Black Hole Mass Scaling Relations for Spiral Galaxies. I. $M_{BH}$–$M_{*,sph}$" (Davis et al., 2018)
• "Black Hole Mass Scaling Relations for Spiral Galaxies. II. $M_{BH}$–$M_{*,tot}$ and $M_{BH}$–$M_{*,disk}$" (Davis et al., 2018)
• "A universal relation between the properties of supermassive black holes, galaxies, and dark matter halos" (Marasco et al., 2021)
• "The Stellar Mass - Black Hole Mass Relation at $z\sim2$ Down to $\mathcal{M}_\mathrm{BH}\sim10^7 M_\odot$ Determined by HETDEX" (Zhang et al., 2023)
• "The Sloan Digital Sky Survey Reverberation Mapping Project: The Black Hole Mass$-$Stellar Mass Relations at $0.2\lesssim z\lesssim 0.8$" (Li et al., 2023)
• "The Black Hole -- Bulge Mass Relation including Dwarf Galaxies Hosting Active Galactic Nuclei" (Schutte et al., 2019)
• "Dark matter halos and the M-σ relation for supermassive black holes" (Larkin et al., 2016)
• "Megamaser Disks Reveal a Broad Distribution of Black Hole Mass in Spiral Galaxies" (Greene et al., 2016)
• "The Black Hole Mass Function Across Cosmic Times I. Stellar Black Holes and Light Seed Distribution" (Sicilia et al., 2021)
• "A local baseline of the black hole mass scaling relations for active galaxies. IV. Correlations between $M_{BH}$ and host galaxy $σ$, stellar mass, and luminosity" (Bennert et al., 2021)
• "The (Black Hole Mass)-(Spheroid Stellar Density) Relations: $M_{BH}$–$\mu$ (and $M_{BH}$–$\Sigma$) and $M_{BH}$–$\rho$" (Sahu et al., 2021)

This compendium defines the landscape and dynamical evolution of the stellar mass–black hole mass relations across galaxy types, mass scales, and epochs, anchoring theoretical, simulation, and observational efforts to understand the assembly of galaxies and their central black holes.