Classifying Supermassive Black Hole Growth Regimes to Observables Across Cosmological Simulations with Forecasts for LSST

1. Problem Statement

The early growth of Supermassive Black Holes (SMBHs) in the first billion years of the universe remains a major challenge in astrophysics. Recent JWST observations of "little red dots" suggest the existence of over-massive black holes that challenge standard "light seed" formation models (remnants of Population III stars). However, distinguishing between different seeding and growth regimes (e.g., light vs. heavy seeds, or over-massive vs. under-massive growth relative to the host) is difficult because:

• Direct black hole mass measurements are impossible for the vast majority of high-redshift galaxies.
• The Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) will detect billions of galaxies via broadband photometry alone, lacking spectroscopic mass constraints.
• Current cosmological simulations (Simba, IllustrisTNG, Eagle) employ different sub-grid physics for SMBH seeding and feedback, making it unclear if observational signatures are universal or simulation artifacts.

The core problem is whether purely photometric observables can robustly distinguish between "over-massive" and "under-massive" SMBH growth regimes across different theoretical models, providing a pathway to interpret future LSST data.

2. Methodology

The authors developed a forward-modeling machine learning framework to bridge theoretical simulations and observational predictions.

A. Data Sources

• Simulations: Data from Simba, IllustrisTNG (TNG100, TNG300, TNG50 variants), and Eagle. These simulations utilize "heavy seed" prescriptions ($\ge 10^4 M_\odot$), allowing the study of post-seeding growth regimes.
• Observational Baseline: LSST Data Preview 1 (DP1) from the Extended Chandra Deep Field South (ECDFS) was used to validate the synthetic photometry.

B. Forward Modeling Pipeline

To generate LSST-compatible synthetic data:

1. Luminosity Calculation: Bolometric AGN luminosity ($L_{bol}$) was derived from the accretion rate ($\dot{M}$) assuming a radiative efficiency $\eta=0.1$.
2. Spectral Energy Distribution (SED): A multi-break power-law SED (based on Vanden Berk et al. 2001) was applied, with UV slope corrections and Eddington-ratio-dependent bolometric corrections.
3. Extinction & Obscuration: 50% of AGN were modeled as Type 2 (obscured) with dust reddening (SMC extinction curve). Intergalactic medium (IGM) absorption was applied for $z > 3$.
4. Host Galaxy: Host luminosities were computed using stellar mass-to-light ratios ($M/L$) dependent on star formation rates.
5. Noise Injection: Gaussian scatter was added to magnitudes to match LSST DP1 observational uncertainties.

C. Machine Learning Framework

• Task: Binary classification of SMBHs as "Over-massive" or "Under-massive" based on their deviation from a simulation-fitted $M_{BH} - M_*$ relation (using a 50th percentile split).
• Features: 16 purely observable features including broadband magnitudes ($g, r, i, z$), colors, redshift, clipped Eddington ratio ($\lambda_{Edd}$), and engineered features (e.g., SED curvature, spectral range).
• Model Architecture: An ensemble of four models (Neural Network, FT-Transformer, Random Forest, XGBoost) stacked via a logistic regression meta-learner.
• Training Strategy:
  • Cross-Simulation Transfer: To test physical universality, models were trained on one simulation (e.g., Simba) and tested on another (e.g., TNG50-1).
  • Rank Normalization: To handle differing absolute luminosity scales between simulations, photometric features were converted to percentile ranks within redshift bins, creating domain-invariant features.
  • Validation: 5-fold cross-validation with strict splitting by unique Black Hole ID to prevent temporal data leakage.

3. Key Contributions

1. Photometric Classification of Growth Regimes: Demonstrated that broadband photometry alone can distinguish between over-massive and under-massive SMBH growth with high accuracy, without direct mass measurements.
2. Cross-Simulation Universality: Proved that the relative photometric ordering of growth regimes is preserved across fundamentally different sub-grid physics (Simba's torque-limited accretion vs. IllustrisTNG's Bondi-Hoyle accretion) when using rank-normalized features.
3. Signal Decomposition: Rigorously ruled out "circularity" (the model simply inverting the forward model's analytical luminosity prescription) through ablation studies.
4. LSST Forecasts: Provided specific accuracy forecasts for the upcoming LSST survey, establishing a baseline for interpreting billions of future detections.

4. Key Results

Classification Performance

• Within-Simulation Accuracy: The ensemble achieved 91%–94% accuracy (AUC $\approx$ 0.98) in distinguishing growth regimes under realistic LSST magnitude limits for both Simba and IllustrisTNG.
• Cross-Simulation Transfer: When training on one simulation and testing on another using rank-normalized features, the model maintained 83%–89% accuracy (intrinsic mode). Under LSST magnitude limits, this dropped to 74%–82%, attributed to flux truncation effects.
• Resolution Independence: High-resolution runs (TNG50) and lower-resolution runs (TNG300) yielded consistent results, whereas extremely low-resolution runs showed artificially high accuracy (>99%) due to coarse mass discretization (an artifact).

Signal Decomposition & Physical Drivers

• Host Galaxy Dominance: Ablation tests showed that classifying using host galaxy photometry alone (setting AGN luminosity to zero) still yielded 82%–87% accuracy. This confirms that the signal is driven by the host galaxy's color evolution (mass-to-light ratio) imprinted by the black hole's co-evolution.
• SED Shape: Attention heatmaps from the FT-Transformer revealed that the model prioritizes the spectral energy distribution (SED) shape (specifically the absolute magnitude proxy and spectral range $m_g - m_z$). Over-massive systems tend to have flatter, bluer SEDs due to higher accretion rates.
• Non-Circularity: Shuffling the accretion rate ($\dot{M}$) only reduced accuracy by 2–4%, proving the model relies on complex physical correlations rather than a trivial inversion of the $L \propto \dot{M}$ relation.

Systematic Errors

• The total systematic uncertainty due to forward model parameter variations (e.g., radiative efficiency, dust reddening, mass-to-light ratios) was quantified at $\pm 1.91\%$ in magnitude-limited mode.
• The classification was robust to changes in the label boundary (splitting at 40th vs. 60th percentile), with accuracy variations $< 2\%$.

5. Significance and Implications

• Interpreting JWST and LSST: This framework provides a validated method to interpret the "little red dots" and future LSST detections. It suggests that even without direct mass measurements, the growth history of SMBHs can be inferred from their host galaxy colors and composite SEDs.
• Heavy Seed Paradigm: Since the simulations used heavy seeds ($\ge 10^4 M_\odot$), this work establishes a robust baseline for identifying over-massive growth within heavy-seed scenarios. It sets the stage for future work to distinguish between light and heavy seeds by comparing these signatures against light-seed simulations.
• Methodological Advance: The use of rank-normalized features allows machine learning models trained on specific cosmological simulations to generalize to real observational data and other simulations, overcoming the "domain shift" problem caused by different sub-grid physics implementations.
• Future Directions: The authors note that while the current framework works for heavy seeds, future iterations must incorporate light-seed simulations (e.g., BRAHMA) and more sophisticated spectral population synthesis codes to fully disentangle seed origins from growth regimes.

In conclusion, the paper successfully demonstrates that broadband photometry encodes the physical history of SMBH-host co-evolution, enabling the classification of growth regimes across diverse cosmological models and providing a critical tool for the upcoming LSST era.