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

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.

Figures (7)

• Figure 1: Left: A plot of specific star-formation rate vs. stellar mass. Here, we use a cut ([0.35ex]8mm1pt1pt) at $\log(\textrm{sSFR}/\textrm{yr}^{-1})=-11$ to demarcate the 28 star-forming galaxies ($\bullet$) from the 27 quenched galaxies ($\bullet$). Right: A pairplot of all the investigated parameters for the galaxies in this study. This pairplot illustrates the effectiveness of the $\log(\textrm{sSFR}/\textrm{yr}^{-1})=-11$ cut, which creates a clear bimodal distribution across all of the variables between star-forming galaxies and quenched galaxies.
• Figure 2: Evolution of a typical NIHAO galaxy that transitions from star-forming to quenching (one of 27). The plot shows the star-formation rate as a function of stellar mass. The stellar mass increases approximately monotonically as a function of time (indicated by the color of the marker). The star-formation rate initially increases until it reaches a maximum and then it steadily decreases. We use this turnover in the star-formation rate as a discriminator between a galaxy's star-forming epoch and its quenching epoch.
• Figure 3: Edge (top row) and path (middle row) marginal matrices and the most probable directed acyclic graphs (DAGs; bottom row) for the 28 simulated star-forming galaxies (left column) and 27 simulated quenched galaxies (right column) at $z=0$. These matrices clearly indicate a change in color between the first rows of each matrix and also between the first columns of each matrix. These changes represent higher probabilities for $M_\bullet$causing host galaxy properties for star-forming galaxies (left matrices) and $M_\bullet$ being caused by properties of their host galaxies in quenched galaxies (right matrices). Indeed, the corresponding DAGs reflect $M_\bullet$ directly or indirectly causing all the simulated properties of its host galaxy in star-forming galaxies (left DAG) and $M_\bullet$ being caused by all simulated properties of its host galaxy in quenched galaxies (right DAG).
• Figure 4: Comparison pairplot with the observational galaxy sample used in Jin:2025, restricted to the same five variables considered in the present causal analysis ($M_\bullet$, $\sigma_0$, $R_e$, $M^\ast$, and sSFR). The sample is separated into spiral galaxies ($\bullet$) and elliptical galaxies ($\bullet$), with lenticulars excluded for clarity. This figure data is analogous to Fig. 2 from Jin:2025, but restricted to the parameter set used here. Also, we plot again the star-forming galaxies ($\bullet$) and quenched galaxies ($\bullet$) from the pairplot in Fig. \ref{['fig:categorize']} to show a visual comparison and facilitate interpretation of the posterior marginal comparisons presented in Table \ref{['tab:compare']}.
• Figure 5: Time evolution of the causal $M_\bullet$--$\sigma_0$ relation. Here, we reduce the full $5\times5$ edge marginal matrices to only the $2\times2$ matrices concerning SMBH mass and central stellar velocity dispersion for our simulated galaxies, with 27 galaxies in our sample completing their evolution from star-forming to quenched by $z=0$. These marginal matrices are listed across the top of the plot, with $x$-axes depicting causal children and the $y$-axes depicting causal parents. We select eleven matrices from successive snapshots, centered at the peak of star formation at $T_\mathrm{peak}$ (the vertical [0.35ex]8mm1pt1mm). The plot below the matrices connects the snapshot times of the matrices via the vertical [0.35ex]8mm1pt1mm. For each snapshot, we illustrate the probabilities of the $P(M_\bullet\rightarrow\sigma_0)$ causal direction ($\bullet$ connected by [0.35ex]8mm1pt), its opposing $P(M_\bullet\leftarrow\sigma_0)$ represented by $\blacksquare$ connected by [0.35ex]8mm1pt, and the probability, $P(M_\bullet\perp \!\!\! \perp\sigma_0)$, that $M_\bullet$ and $\sigma_0$ are independent ($\blacktriangle$ connected by [0.35ex]8mm1pt1pt); $P(M_\bullet\rightarrow\sigma_0)+P(M_\bullet\leftarrow\sigma_0)+P(M_\bullet\perp \!\!\! \perp\sigma_0)=1$. The horizontal lines at $P=0.4$ ([0.35ex]8mm1pt1pt) and $P=0.3$ ([0.35ex]8mm1pt) represent the null probabilities for the independency case and causal directional cases, respectively. Thus, significance occurs when the plotted solid lines are further away from the solid horizontal line and the plotted dotted line is significantly different from the horizontal dotted line. For example, the snapshot at $T_\mathrm{peak}-216$ Myr demonstrates a period of transition that lacks any meaningful causal information because all values are near their null values.