Finding the boundary: Using galaxy membership to inform galaxy cluster extent through machine learning

1. Problem Statement

Galaxy clusters are the largest gravitationally bound structures in the universe, significantly influencing the evolution of member galaxies through processes like ram-pressure stripping, strangulation, and tidal interactions. However, defining the precise spatial boundary of a cluster's influence is challenging.

• The Issue: Traditional definitions of cluster boundaries (e.g., $R_{200,mean}$, $R_{200,crit}$, and the dynamically motivated Splashback Radius $R_{sp}$) often assume a sharp transition between "cluster" and "field" galaxies.
• The Reality: Observations and simulations suggest a complex, mixed population in cluster outskirts where galaxies undergo "pre-processing" before reaching the core. The transition is likely probabilistic and extended rather than a hard physical wall.
• Goal: The authors aim to move beyond rigid geometric boundaries to define a probabilistic transition region where intrinsic galaxy properties shift from field-like to cluster-like, using machine learning to quantify this extent.

2. Methodology

Data Source

• Simulation: The study utilizes the IllustrisTNG cosmological simulation suite, specifically the TNG300-1 volume (the largest high-resolution box).
• Sample Selection:
  • Clusters: Selected at redshift $z=0$ with total mass $M_{200,mean} > 10^{13} M_\odot$.
  • Galaxies: Subhalos with stellar mass $M_* > 10^9 M_\odot$ and dark matter mass $M_{dm} > 10^{9.5} M_\odot$.
  • Mass Bins: Clusters are grouped into four mass bins ($10^{13}$ to $10^{15} M_\odot$) to analyze mass dependence.
  • Distance Metric: For each galaxy, the distance to the closest cluster is calculated in a periodic box and normalized by the host cluster's $R_{200,mean}$.

Machine Learning Framework

• Model Architecture: A supervised Deep Neural Network (DNN) (specifically a 4-layer Multilayer Perceptron) is employed for binary classification.
• Task: Classify galaxies as either "Cluster Members" (Label 0) or "Field Galaxies" (Label 1).
• Input Features: The model uses intrinsic galaxy properties sensitive to environmental effects. The study tests various combinations:
  • Fiducial Set (6 properties): Stellar-to-subhalo mass ratio ($M_*/M_{tot}$), gas fraction ($M_{gas}/M_{baryon}$), specific star formation rate (sSFR), color ($g-r$), gas metallicity ($Z_{gas}$), and stellar metallicity ($Z_*$).
  • Extended Sets: Includes all 15 available properties, categorized into Stellar, Gas, and Dynamical subsets to isolate physical processes.
• Labeling Schemes: To train the model, "ground truth" labels are generated using two methods:
  1. Distance-based: Galaxies within a threshold radius (e.g., $0.5 R_{sp}$) are labeled "Cluster," and those beyond a larger threshold (e.g., $5.0 R_{sp}$) are labeled "Field." Intermediate galaxies remain unlabelled.
  2. Merger Tree-based (SUBLINK): Galaxies associated with a host group ($M > 10^{13} M_\odot$) in the merger tree are labeled "Cluster."
• Training Strategy:
  • Preprocessing: Features undergo arcsinh or log transformations to symmetrize distributions.
  • Validation: Clusters are split strictly (70% train, 15% val, 15% test) to prevent environmental contamination (i.e., galaxies from the same cluster cannot appear in both training and validation sets).
  • Regularization: Dropout, weight decay (AdamW), and batch normalization are used to prevent overfitting.

Analysis Approach

• Probabilistic Mapping: The trained model outputs a logit (probability) for every galaxy in the full dataset.
• Transition Radius ($r_0$): Defined as the radial distance where the predicted probability (logit) crosses zero (i.e., the model is equally likely to classify a galaxy as cluster or field).
• Convergence Tests: The authors vary the labeling thresholds to ensure the predicted transition region is robust and not an artifact of the specific label definition.

3. Key Contributions

1. First Probabilistic Map: The paper presents the first quantitative, data-driven map of the transition region between cluster and field galaxies, characterizing it as a broad, intrinsically scattered region rather than a sharp boundary.
2. Category-Specific Boundaries: By training separate models on Stellar, Gas, and Dynamical properties, the authors isolate how different physical processes respond to the cluster environment at different radii.
3. Mass Dependence Quantification: The study establishes a scaling relation between the transition radius and cluster mass, linking it to the physics of ram-pressure stripping and halo concentration.
4. Validation of Boundary Definitions: The work critically evaluates standard definitions ($R_{200}$, $R_{sp}$), demonstrating that they do not perfectly align with the transition in intrinsic galaxy properties.

4. Key Results

• Nature of the Transition:

  • The transition is probabilistic and extended. The "ambiguous" population (where the model is uncertain, logit $\approx 0$) spans a wide radial range, confirming that galaxies do not switch states instantaneously upon crossing a specific radius.
  • Intrinsic Scatter: There is significant scatter in the transition radius across individual clusters, reflecting the diversity of cluster environments and galaxy histories.

• Mass Dependence:

  • The transition radius ($r_0$) scales with cluster mass as $r_0 \propto M_{200,mean}^{0.1}$.
  • This trend is opposite to the mass dependence of the Splashback Radius ($R_{sp}$), which decreases with mass.
  • Physical Interpretation: This scaling correlates with the halo mass-concentration relation. Lower-mass halos are more concentrated, leading to steeper gas density profiles where ram-pressure stripping occurs closer to the center. Higher-mass halos have shallower profiles, allowing environmental effects to extend further out (larger $r_0$).

• Property-Specific Transitions:

  • Dynamical Properties: Change deepest within the cluster (smallest radii), responding quickly to the gravitational potential.
  • Gas vs. Stellar:
    • In low-mass clusters, gas and stellar transitions occur at similar radii.
    • In high-mass clusters, the gas transition occurs in the outermost regions (further out than stellar), likely due to the extended reach of ram-pressure stripping and the cosmic accretion shock.
  • Ordering: Generally, dynamical changes happen first, followed by gas depletion, and finally stellar/quenching changes (due to the delay in star formation cessation).

• Comparison with Standard Definitions:

  • Standard boundaries ($R_{200,mean}$, $R_{sp}$) fail to cleanly separate cluster and field populations based on intrinsic properties.
  • The ML-predicted transition region often lies beyond $R_{200,mean}$ for high-mass clusters, suggesting that environmental preprocessing begins well before the virial radius.

5. Significance and Implications

• Refining Cluster Definitions: The study argues against using a single "hard" radius for cluster membership in both simulations and observations. Instead, a probabilistic approach better captures the physical reality of galaxy evolution.
• Environmental Preprocessing: The results highlight that significant environmental processing (quenching, gas stripping) occurs in the "pre-processing" zone of cluster outskirts, potentially extending beyond the splashback radius.
• Methodological Advancement: The paper demonstrates the efficacy of supervised machine learning in astrophysics to uncover complex, non-linear relationships between galaxy properties and their environment, offering a tool to refine cluster boundary definitions in future surveys (e.g., DESI, Roman).
• Future Applications: The framework can be adapted to observational data using photometric redshifts and multi-band colors, though projection effects will need to be accounted for. It also opens the door to studying the specific impact of AGN feedback and dwarf galaxy evolution in cluster environments.

In summary, Hao et al. successfully utilize machine learning to demonstrate that the boundary of a galaxy cluster is not a sharp line but a broad, mass-dependent, and physically distinct transition zone where galaxy properties evolve gradually under the influence of the intracluster medium.