Connection between galaxy morphology and dark-matter halo structure II: predicting disk structure from dark-matter halo properties

1. Problem Statement

While the $\Lambda$CDM framework posits that galaxies form within dark-matter (DM) halos, the quantitative link between specific halo properties and the detailed structural parameters of galactic disks (size, thickness, mass fraction) remains uncertain. Previous studies have established that halo mass is the primary driver of galaxy size, but the roles of secondary properties (spin, concentration, assembly history, shape) are less clear. Furthermore, it is unknown whether disk vertical height and mass fractions can be robustly inferred from halo properties alone, and how baryonic processes (gas cooling, feedback) alter the relationship between the inner halo and the disk compared to dark-matter-only (DMO) simulations.

2. Methodology

Data Source:

• Simulation: The study utilizes the TNG50 simulation from the IllustrisTNG suite, which includes high-resolution hydrodynamics and sub-grid physics (cooling, star formation, feedback).
• Comparison: Results are compared against matched Dark-Matter-Only (DMO) runs to isolate the impact of baryonic restructuring.
• Sample: Central halos at redshifts $z = 0, 1, 2, 4$. Systems are selected based on resolution criteria ($>1000$ stellar particles, half-mass radius $>2\times$ softening length).

Measurements:

• Halo Properties (37 features): A comprehensive set of descriptors including:
  • Mass & Density: Virial mass ($M_{vir}$), concentration ($c$), Einasto shape index ($\alpha$).
  • Angular Momentum: Global spin ($\lambda$) and inner-halo spin ($\lambda_{inner}$).
  • Shape: 3D axis ratios ($q, p, s$) and triaxiality ($T$), measured globally and in the inner halo ($0.1 R_{vir}$).
  • Assembly History: Formation redshift ($z_{1/2}$), mass accretion rate ($\dot{M}_{norm}$), and Principal Components (PC1, PC2) of the mass assembly history.
  • Environment: Cosmic web classification, local density, and merger statistics.
• Galaxy Properties:
  • Global: Stellar mass ($M_*$), Star Formation Rate (SFR), half-stellar-mass radius ($r_{1/2}$).
  • Disk-specific: Using the kinematic decomposition method from Paper I, the authors measure the size ($R_{1/2}$), scale height ($Z_{1/2}$), and mass fraction ($f_{disk}$) for total, thin, and thick disks.

Machine Learning Framework:
The authors employ a three-step ML pipeline:

1. Random Forest (RF) Regression: Used to predict galaxy properties from halo features. RF serves as the "gold standard" for accuracy and is used to rank feature importance via SHAP (SHapley Additive exPlanations) values.
2. Symbolic Regression (SR): Implemented via the PySR package. SR uses the top 10 features identified by RF to derive compact, interpretable empirical equations.
   • Loss Function: A combination of Normalized Mean Squared Error (NMSE) and Normalized Quantile Loss (NQL) to handle outliers and asymmetric distributions.
   • Selection: A score function balancing loss and complexity ($w=0.7$) to select the simplest equation with sufficient accuracy.
3. Evaluation: Models are tested on an 80/20 train/test split. Performance is measured using $R^2$, Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE).

3. Key Contributions

1. Comprehensive Feature Set: The study moves beyond simple mass-spin relations by incorporating a high-dimensional set of 37 halo properties, including inner-halo structure and detailed assembly history.
2. Baryonic vs. DMO Comparison: By comparing hydro and DMO runs, the paper quantifies how baryonic physics modifies the halo structure (specifically the inner halo) and how this impacts the predictability of disk properties.
3. Discovery of Disk Height Predictability: The authors demonstrate that disk vertical thickness ($Z_{1/2}$) is easier to predict from halo properties than disk size ($R_{1/2}$), a finding not previously established.
4. Empirical Prescriptions: The paper provides explicit, analytic formulas (via Symbolic Regression) that map halo properties to disk structures, offering practical tools for semi-analytic models (SAMs) and empirical galaxy formation models.

4. Key Results

A. Global Galaxy Properties:

• Accuracy: RF models predict stellar mass ($R^2 \approx 0.87$), SFR ($R^2 \approx 0.69$), and galaxy size ($R^2 \approx 0.76$) with high accuracy using hydro halos.
• SR Performance: Symbolic regression achieves slightly lower but still strong accuracy ($R^2 \approx 0.65-0.78$) with simple, interpretable equations involving only 3-5 parameters (e.g., $M_{vir}$, $z_{1/2}$, $c$, $s$).
• Hydro vs. DMO: Predictions based on hydro halos generally outperform DMO-based predictions, particularly for size, indicating the imprint of baryons on the halo.

B. Disk Structure Predictions:

• Disk Size ($R_{1/2, Disk}$):
  • Hydro: The most predictive features are the Einasto shape index ($\alpha$), concentration ($c$), and inner-halo spin ($\lambda_{inner}$). The strong link to $\alpha$ suggests it is a consequence of disk-halo interaction rather than a primordial driver.
  • DMO: $\alpha$ loses predictive power. The best predictors are global spin ($\lambda$), mass ($M_{vir}$), redshift ($z$), and concentration ($c$).
  • SR Equations: The derived SR relations significantly outperform previous analytic prescriptions (e.g., Mo et al. 1998, Kravtsov 2013) in terms of correlation and scatter.
• Disk Thickness ($Z_{1/2, Disk}$):
  • Surprising Finding: Disk height is predicted with higher accuracy ($R^2 \approx 0.83$ for hydro) than disk size.
  • Drivers: The primary drivers are halo mass, redshift, and recent mass assembly (accretion rate $\dot{M}_{norm}$ and formation redshift $z_{1/2}$).
  • Physical Interpretation: Thicker disks are associated with rapid, recent halo growth (dynamical heating from mergers/accretion). Disk height correlates tightly with disk size, suggesting that factors setting the disk extent also regulate its vertical scale.
• Disk Mass Fraction: Largely determined by halo mass ($M_{vir}$) and concentration, with secondary dependence on inner shape parameters.

C. Redshift and Mass Trends:

• Low-Mass Halos ($M_{vir} < 10^{11.6} M_\odot$): Disks become significantly more extended relative to their host halos at high redshift ($R_{1/2}/R_{vir}$ increases from $\sim0.02$ at $z=0$ to $\sim0.08$ at $z=4$). This is driven by the shrinking virial radius while the star-forming radius remains roughly constant.
• High-Mass Halos ($M_{vir} > 10^{11.6} M_\odot$): The trend reverses; high-redshift massive halos host more compact disks due to rapid assembly and gas-rich mergers triggering central starbursts.

D. Inner Halo Importance:

• Inner-halo parameters ($\lambda_{inner}$, inner shape) are critical predictors in hydro simulations because the disk gravitationally reshapes the inner DM halo.
• In DMO simulations, inner-halo parameters offer no advantage over global parameters, confirming that the "disk-halo coupling" is a baryonic effect.

5. Significance and Implications

• Modeling Tools: The provided Symbolic Regression equations offer a computationally efficient, physically motivated method for populating DM halos with galaxies in semi-analytic models and empirical frameworks, surpassing traditional linear scaling relations.
• Baryonic Feedback: The study provides concrete evidence that baryonic processes fundamentally alter the inner halo structure (changing $\alpha$ and spin profiles), creating a tighter coupling between the disk and the inner halo that is absent in DMO simulations.
• Predictive Power: The finding that disk thickness is more predictable than size suggests that vertical heating mechanisms (linked to assembly history) are more directly tied to halo properties than the angular momentum acquisition mechanisms that set disk size.
• Future Directions: The authors note that while the specific relations are TNG50-dependent, the methodology is modular and can be applied to other simulations (e.g., FIRE, EAGLE) to test the universality of these connections.

In summary, this paper establishes a robust, data-driven framework for predicting galaxy disk morphology from halo properties, highlighting the critical role of baryonic physics in shaping the inner halo and the surprising ease of predicting disk thickness via assembly history.