Authors: Alister W. Graham
The correlation between black hole mass, Mbh, and stellar velocity dispersion, σ0, is revisited using 137 galaxies with quantitative bar strengths and enhanced morphological awareness. Interpreted within the `Triangal' evolutionary framework, gas-rich and gas-poor assembly pathways emerge in the Mbh–σ0 diagram. To robustly quantify these scaling relations, a versatile Bayesian hierarchical regression code, dubbed the Symmetric COvariance Population Estimator (SCOPE), is introduced. Unlike conditional estimators (e.g., LINMIX), SCOPE derives the intrinsic population covariance, natively accommodating asymmetric measurement errors while guaranteeing directional invariance between axes. Applying SCOPE reveals that the current sample of primeval, dust-poor S0 galaxies (including dwarf early-type galaxies with Re,gal≈1 kpc) follows a shallow relation (Mbh∝σ02.5–3.1). Explained via the virial theorem, this flattening reframes expectations for intermediate-mass black holes. In contrast, tracing the `Disc Down-sizing' sequence—where dry mergers erase discs—yields a steep relation for massive elliptical and ellicular galaxies (Mbh∝σ07.8±1.4). The historical practice of applying a single, monolithic scaling relation across all morphological types averages over different formation histories, potentially skewing AGN virial f-factor calibrations and systematically under-predicting the ultra-massive black holes needed to generate the nanohertz gravitational wave background. Furthermore, strongly barred, dust-poor S0 galaxies appear offset to higher velocity dispersions, supporting theoretical expectations that bars kinematically heat galaxies. This dynamical signature is lost in the complexities of spiral galaxies. Ultimately, these morphology-dependent relations provide physically-motivated benchmarks for cosmological simulations and a framework for disentangling regimes driven by AGN feedback from those driven by collisionless mergers.