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

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.

💡 Research Summary

The paper investigates the causal relationship between supermassive black hole (SMBH) mass (M·) and the central stellar velocity dispersion (σ₀) of their host galaxies, focusing on how this relationship evolves over cosmic time. Building on a previous observational study (Paper I) that found a morphology‑dependent causal direction—SMBHs appear to drive σ₀ in elliptical galaxies while σ₀ drives SMBH growth in spirals—the authors turn to the NIHAO suite of cosmological zoom‑in hydrodynamical simulations to add a temporal dimension.

NIHAO simulations employ the Gasoline2 code with a flat ΛCDM cosmology, particle masses ranging from 6 × 10² to 3 × 10⁵ M⊙ for gas and 3 × 10³ to 2 × 10⁶ M⊙ for dark matter, and gravitational softenings of 50–1000 pc. Black holes are seeded in halos above 5 × 10¹⁰ M⊙ with an initial mass of 10⁵ M⊙, grow via Bondi–Hoyle–Lyttleton accretion capped at the Eddington limit, and inject thermal feedback calibrated to reproduce observed scaling relations. For each central galaxy the authors extract five quantities at every simulation snapshot: SMBH mass (M·), stellar mass (M*), effective radius (Rₑ), central velocity dispersion (σ₀), and specific star formation rate (sSFR).

The causal analysis mirrors Paper I: a full Bayesian score‑based evaluation of all 29,281 possible directed acyclic graphs (DAGs) formed from the five variables. By computing the exact posterior distribution over DAGs, the method identifies the most probable causal structure without imposing prior directional assumptions. Crucially, the simulation provides a natural time ordering, allowing the authors to compare causal structures before and after the epoch of peak star formation (T_peak). They define “star‑forming” phases as the five snapshots leading up to T_peak and “quenched” phases as the final five snapshots approaching z = 0.

Results reveal a clear reversal of causal direction across the star‑formation peak. During the star‑forming phase—characteristic of gas‑rich, spiral‑like systems—the dominant causal link is M· → σ₀ (≈68 % of posterior weight). This suggests that active SMBH feedback can reshape the central stellar kinematics, increasing σ₀. After the peak, when galaxies have exhausted much of their cold gas and are transitioning to quiescence (often elliptical‑like), the causal arrow flips to σ₀ → M· (≈74 % of posterior weight). Here the host galaxy’s structural evolution, reflected in σ₀, becomes the primary driver of further SMBH growth, likely through stellar dynamical processes and minor mergers.

The authors argue that this temporal reversal has profound implications for high‑redshift (high‑z) SMBH mass estimates. Conventional practice extrapolates the local M·–σ₀ relation to early epochs, implicitly assuming the same causal direction as at z = 0. The simulation‑based causal analysis shows that applying the local scaling to galaxies observed near their star‑forming peak would overestimate SMBH masses by roughly 1–2 dex. By fitting a “causally‑informed” scaling relation that accounts for the pre‑peak direction (M· → σ₀), they predict high‑z SMBH masses that are about two orders of magnitude lower, bringing them into line with the local mass‑to‑host ratios.

The discussion acknowledges several limitations. NIHAO’s sub‑grid treatment of the interstellar medium does not resolve a multiphase cold component, potentially under‑representing bursty star formation that could introduce short‑timescale fluctuations in σ₀ and SMBH accretion. The sample is not volume‑complete; halos were selected to span a wide mass range rather than to reproduce the observed stellar mass function, which may bias statistical conclusions. Moreover, the feedback parameters are calibrated to match observed scaling relations, raising the possibility that the causal reversal is partially model‑dependent. The authors propose future work with higher‑resolution ISM physics, larger and more representative galaxy samples, and cross‑validation with other simulation suites (e.g., Illustris‑TNG, EAGLE) to test the robustness of the inferred causal structures.

In conclusion, the paper provides the first direct, time‑resolved demonstration that the causal link between SMBH mass and host‑galaxy velocity dispersion flips as galaxies evolve from active star formation to quiescence. This finding refines our understanding of AGN feedback, challenges the straightforward use of local scaling relations at high redshift, and showcases causal discovery as a powerful tool for interpreting complex astrophysical datasets.