Causal structure of supermassive black hole-galaxy coevolution: unobserved confounders, cyclic feedback, and generality of the morphology-dependent causal reversal

Exact Bayesian causal discovery applied to 101 nearby galaxies with dynamically measured supermassive black hole (SMBH) masses revealed a morphology-dependent causal reversal: in spiral galaxies the SMBH (mass $M_\bullet$ ) causally drives host properties, while in elliptical galaxies the host bulge drives SMBH growth (Jin et al. 2025). The inference rests on untested assumptions - causal sufficiency (no unobserved confounders such as central gas fraction or merger history), acyclicity (no simultaneous SMBH-galaxy feedback loops, hypothesized for lenticular galaxies), and a sample of only ~100 galaxies - and its ultimate payoff, causally-informed black-hole mass scaling relations, has only begun to be realized (Davis et al. 2026). The open problem is to establish the true causal structure of SMBH-galaxy coevolution: test whether the inferred causal edges survive direct inclusion of gas fraction and merger history as measured variables, determine whether any population or epoch is genuinely cyclic rather than acyclic with a temporal reversal across the star-formation peak, and test the generality of the causal mode on larger homogeneous samples - the prerequisite for physically motivated scaling relations and quantitative AGN feedback models.

Background

Supermassive black holes (SMBHs, masses $10^6$ - $10^{10}$ solar masses) sit at the centers of most galaxies, and their masses correlate tightly with host-galaxy properties - most famously the central stellar velocity dispersion $\sigma_0$ (the $M_\bullet$ - $\sigma_0$ relation). Correlation does not say which quantity drives which: host bulges could build SMBHs through mergers and gas accretion, or growing SMBHs could regulate their hosts through active galactic nucleus (AGN) feedback, the injection of accretion energy into surrounding gas that can heat or expel it and thereby quench star formation. Because astrophysics is purely observational, this direction-of-causation question must be attacked with causal discovery: statistical methods that infer causal structure from observational data. A directed acyclic graph (DAG) encodes hypothesized causal links; graphs implying the same conditional independencies form a Markov equivalence class (MEC). Constraint-based algorithms such as Peter-Clark (PC) and Fast Causal Inference (FCI) test conditional independencies; FCI relaxes causal sufficiency (the assumption of no unobserved confounders, i.e. no variables outside the dataset that cause two or more studied variables) and outputs a partial ancestral graph (PAG) whose special edge marks can flag possible confounding. Score-based Bayesian approaches instead compute a posterior probability for every possible DAG from a marginal-likelihood score such as the Bayesian Gaussian equivalent (BGe); summing over DAGs yields edge and path marginals, the posterior probabilities that a direct, or direct-plus-indirect, causal link exists between two variables. The number of DAGs grows super-exponentially with the variable count (about $1.14\times10^9$ for seven variables), so exact enumeration becomes impractical beyond eight or nine variables, where approximate posterior samplers such as DAG-GFN (Deleu et al. 2022), which builds graphs edge-by-edge with a Generative Flow Network, a learned sequential sampler over compositional objects, become necessary.

Jin et al. (2025) applied exact Bayesian causal discovery to 101 nearby galaxies with dynamically measured SMBH masses, using seven variables ( $M_\bullet$ , $\sigma_0$ , bulge effective radius $R_e$ , mean projected stellar density within it $\langle\Sigma_e\rangle$ , total stellar mass $M_*$ , mid-infrared color $W2-W3$ , and specific star formation rate sSFR, the star formation rate per unit stellar mass), split by morphology into 35 ellipticals, 38 lenticulars (S0), and 28 spirals. This extended the first constraint-based attempt on 83 galaxies by Pasquato et al. (2023). They found a morphology-dependent causal reversal: in ellipticals, bulge properties built by major mergers drive SMBH growth ( $\sigma_0 \to M_\bullet$ ), while in spirals the SMBH causally influences host properties ( $M_\bullet \to \sigma_0$ and other links), plausibly via AGN feedback in gas-rich systems; quenching of star formation is in progress in spirals and complete in ellipticals. Cross-checks with PC and FCI, with semi-analytic model galaxies (including a control run with black-hole feedback switched off, which recovered the designed absence of the SMBH-to-galaxy direction), with observational-error resampling, leave-one-out tests, and alternative priors all supported the main result. Three assumptions nevertheless remained untested: causal sufficiency (FCI left open that all relations between $M_\bullet$ and its inferred parents, notably the $M_\bullet$ -sSFR link, could be produced by an unobserved confounder such as the central gas fraction); acyclicity (lenticulars, hosting both mergers and feedback, might genuinely be cyclic, with both causal directions acting simultaneously); and generality beyond the small local sample.

A first follow-up by overlapping authors, Davis et al. (2026), took the simulation route Jin et al. (2025) proposed: applying the same exact-posterior method per snapshot to 55 galaxies from the NIHAO suite of cosmological zoom-in hydrodynamical simulations, they reproduced the spiral/elliptical dichotomy as a star-forming/quenched dichotomy and showed that the causal direction of the $M_\bullet$ - $\sigma_0$ relation reverses around the epoch of peak star formation - from $M_\bullet \to \sigma_0$ in actively star-forming phases to $\sigma_0 \to M_\bullet$ after quenching - and they turned this into causally-informed scaling relations that substantially lower estimated high-redshift black-hole masses. Their analysis, however, still fits acyclic DAGs at each snapshot, does not include gas fraction among its five variables, and leaves the cycle-versus-epoch-mixture question, confounder control, and sample generality open. In parallel, Desmond & Ramsey (2026) demonstrated scalable, confounder-aware causal discovery (the FCIT algorithm) on roughly 450,000 galaxies from the NASA Sloan Atlas (a homogeneous photometric catalogue of nearby galaxies) - confirming that the large-sample regime is methodologically reachable, though that dataset contains no SMBH masses.

Problem Statement

Determine the causal structure connecting supermassive black holes and their host galaxies beyond the static, causally sufficient, small-sample regime in which the morphology-dependent causal reversal was discovered. Concretely, with respect to the variable set of Jin et al. (2025) - dynamically measured $M_\bullet$ , $\sigma_0$ , $R_e$ , $\langle\Sigma_e\rangle$ , $M_*$ , $W2-W3$ , and sSFR - extended by the candidate confounders:

(1) Confounder control. Test whether the inferred causal edges - above all the $M_\bullet$ -sSFR relation whose FCI analysis admitted an unobserved confounder - survive when central gas fraction and merger history are directly measured and included as variables. The operationalization of the two added confounders must be stated explicitly, since it fixes the enlarged variable set: for central gas fraction, the proxy used by Jin et al. (2025) is the gas mass within 5 kpc relative to the total stellar mass (Section 10.4, Figure 17); for merger history, candidate summary statistics include merger counts, mass ratios, or time since the last major merger.

(2) Cyclicity. Determine whether the SMBH-galaxy causal structure contains genuine feedback loops - both $M_\bullet \to$ host and host $\to M_\bullet$ acting simultaneously, as hypothesized for lenticular galaxies - or whether it is acyclic at each epoch with the causal direction reversing across the star-formation peak, as the NIHAO snapshot analysis of Davis et al. (2026) suggests but cannot establish while fitting acyclic DAGs; in particular, separate a true cycle from a mixture of galaxies observed before and after the reversal.

(3) Generality. Establish whether the morphology- (or quenching-state-) dependent causal mode persists in larger, homogeneously measured samples of galaxies with dynamically measured SMBH masses, including subpopulation structure such as mass-dependent transitions, scaling the causal inference to the larger variable sets that (1) and (2) require.

These deliverables are complementary open directions toward one shared objective - a validated causal account of SMBH-galaxy coevolution - which is the prerequisite for physically motivated black-hole mass scaling relations and quantitative AGN feedback models. They need not all be completed together: any one direction can be advanced on its own.

References

Jin, Z., Pasquato, M., Davis, B. L., Deleu, T., Luo, Y., Cho, C., Lemos, P., Perreault-Levasseur, L., Bengio, Y., Kang, X., Macciò, A. V., & Hezaveh, Y. 2025, 'Causal Discovery in Astrophysics: Unraveling Supermassive Black Hole and Galaxy Coevolution', The Astrophysical Journal 979, 212, DOI: 10.3847/1538-4357/ad9ded, https://doi.org/10.3847/1538-4357/ad9ded (arXiv:2410.00965)
Davis, B. L., More, S., Jin, Z., Pasquato, M., Macciò, A. V., & Yuan, F. 2026, 'Causal Reversal in the M•–σ0 Relation: Implications for High-redshift Supermassive Black Hole Mass Estimates', The Astrophysical Journal 1000, 100, DOI: 10.3847/1538-4357/ae47ef, https://doi.org/10.3847/1538-4357/ae47ef (arXiv:2602.16775)
Desmond, H., & Ramsey, J. 2026, 'The causal structure of galactic astrophysics', Open Journal of Astrophysics 9, DOI: 10.33232/001c.159080, https://doi.org/10.33232/001c.159080 (arXiv:2510.01112)
Pasquato, M., Jin, Z., Lemos, P., Davis, B. L., & Macciò, A. V. 2023, 'Causa prima: cosmology meets causal discovery for the first time', arXiv:2311.15160, https://arxiv.org/abs/2311.15160
Deleu, T., Gois, A., Emezue, C., Rankawat, M., Lacoste-Julien, S., Bauer, S., & Bengio, Y. 2022, 'Bayesian Structure Learning with Generative Flow Networks', Proceedings of the 38th Conference on Uncertainty in Artificial Intelligence (UAI 2022), arXiv:2202.13903, https://arxiv.org/abs/2202.13903
Jin, Z., Lu, Y., Ting, Y.-S., Zheng, Y., & Buck, T. 2025, 'Causal Discovery of Latent Variables in Galactic Archaeology', arXiv:2507.00134, https://arxiv.org/abs/2507.00134
Davis, B. L., Ali-Dib, M., Zheng, Y., Jin, Z., Zhang, K., & Macciò, A. V. 2025, 'Causal evidence for the primordiality of colours in trans-Neptunian objects', Monthly Notices of the Royal Astronomical Society: Letters 543, L34-L42, DOI: 10.1093/mnrasl/slaf089, https://doi.org/10.1093/mnrasl/slaf089 (arXiv:2507.03760)
Waters, T. K., Gültekin, K., Gebhardt, K., Foskic, S., & Kadri, A. 2026, 'A Supermassive Black Hole Mass Measurement in NGC 5102 with Schwarzschild Orbit-superposition Modeling', The Astrophysical Journal 1007, 63, DOI: 10.3847/1538-4357/ae8793, https://doi.org/10.3847/1538-4357/ae8793 (arXiv:2607.26186)