Causal Reversal in the $M_\unicode{x25CF}\unicode{x2013}σ_0$ Relation: Implications for High-Redshift Supermassive Black Hole Mass Estimates

Benjamin L. Davis, Saakshi More, Zehao Jin, Mario Pasquato, Andrea Valerio Macciò, Feng Yuan

First listed 2026-02-18 | Last updated 2026-02-18

Abstract

The nascent methodology of applying the principles of causal discovery to astrophysical data has produced affirming results about deeply held theories concerning the causal nature behind the observed coevolution of supermassive black holes (SMBHs) with their host galaxies. The key results from observations have demonstrated an apparent causal reversal across different galaxy morphologies$\unicode{x2014}$SMBHs causally influence the evolution of the physical parameters of their spiral galaxy hosts, whereas SMBHs in elliptical galaxies are passive companions that grow in near lockstep with their hosts. To further explore and ascertain insights, it is necessary to utilize galaxy simulations to track the time evolution of the observed causal relations to learn more about the temporal nature of the changing SMBH/galaxy evolutionary directions. We conducted experiments with the NIHAO suite of cosmological zoom-in hydrodynamical simulations to follow the evolution of individual galaxies along with their central SMBH masses ($M_\unicode{x25CF}$) and properties, including central stellar velocity dispersion ($σ_0$). We reproduce the causal results from real galaxies, but add clarity by observing that the SMBH/galaxy causal directions are noticeably inverted between the epochs before and after the peak of star formation. The implications for causal reversal of the $M_\unicode{x25CF}\unicode{x2013}σ_0$ relation portend larger concerns about the reliability of SMBH masses estimated at high redshifts and presumptions of overmassive black holes at early epochs. Toward this problem, we apply updated causally-informed scaling relations that predict high-$z$ black hole masses that are approximately two orders of magnitude less massive, and thus not overmassive with respect to local $z=0$ SMBH$\unicode{x2013}$galaxy mass ratios.

Short digest

Using NIHAO cosmological zoom-ins, the authors apply causal-discovery tools to track the time evolution of the M•–σ0 connection and related host properties. Splitting 55 galaxies into 28 star-forming and 27 quenched via an sSFR cut and the SFR-turnover epoch, they find a temporal inversion of causality: during the star-forming phase M• drives host galaxy properties, while after quenching host properties drive M•—mirroring the observational spiral vs. elliptical trends. Leveraging this, they propose causally informed high‑z scaling relations that yield SMBH masses ≈2 dex lower than estimates based on the local M•–σ0, defusing claims of “overmassive” early black holes.

Key figures to inspect

• Figure 1: Inspect the sSFR–M* cut that separates 28 star-forming from 27 quenched NIHAO galaxies and the pairplot’s clear bimodality across variables; this sets the causal-regime bins used later.
• Figure 2: Follow the single-galaxy track in SFR vs M* with time coloring; the SFR peak and subsequent decline mark the transition from the SMBH-driving to host-driving phase.
• Figure 3: Read the edge/path marginal matrices and DAGs for star-forming vs quenched samples; verify the flip from M•→(host properties) to (host properties)→M• and note which variables connect directly to σ0 and M•.
• Figure 4: Compare the NIHAO-based pairplots to the observational sample from Paper I under the same variable set; check that the morphology-based trends align with the simulation-defined star-forming vs quenched splits.