Galaxy morphology-dependent (black hole mass)-(velocity dispersion) relations: implications for gravitational-wave forecasts and cosmological simulations

Abstract

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 proposed ‘Triangal’ evolutionary framework, gas-rich and gas-poor assembly pathways emerge in the Mbh–σ0 diagram. To robustly quantify these scaling relations, a symmetric 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) follow a shallow relation ($M_{\rm bh}\propto \sigma _0^{2.5\text{--}3.1}$). 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 ($M_{\rm bh}\propto \sigma _0^{7.8\pm 1.3}$). Applying a single, monolithic scaling relation across all morphologies inadvertently averages over these distinct galaxy formation histories, potentially influencing AGN virial f-factor calibrations and systematically under-predicting estimates of the ultra-massive black hole population thought to contribute to the nanohertz gravitational wave background. Furthermore, although lost in the complexity of spiral galaxies, strongly barred, dust-poor S0 galaxies appear offset to higher velocity dispersions, qualitatively consistent with theoretical expectations of kinematic heating by stellar bars. 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 mergers.