ERGO-ML: A continuous organization of the X-ray galaxy cluster population in TNG-Cluster with contrastive learning

The physical properties of the intracluster medium (ICM) trace the underlying gravitational potential, cluster mergers, interactions with halos and satellites, and galactic feedback from supernovae and supermassive black holes (SMBHs). Clusters are often described by summary statistics, such as halo mass, X-ray luminosity, cool-core state, active galactic nuclei (AGN) activity, or number of mergers. In this paper of the Extracting Reality from Galaxy Observables with Machine Learning (ERGO-ML) series, we instead explore the full information content of X-ray maps of the ICM. We apply Nearest Neighbour Contrastive Learning (NNCLR) to build a low-dimensional representation space of such images. Using X-ray maps from the 352 clusters in the TNG-Cluster cosmological magnetohydrodynamical simulation, we take three orthogonal projections at eight snapshots in the redshift range 0 ≤ z < 1, producing ∼8,000 images. The learned representation forms a continuous distribution from relaxed to merging systems, and from centrally peaked to flat profiles. It also shows clear correlations with redshift, halo and gas mass, stellar and SMBH mass, time since last major merger, and indicators of dynamical state. We further demonstrate that an 8-dimensional representation suffices to predict cluster properties, identify analogues, and capture relationships between physical quantities. Our results establish contrastive learning as a powerful framework for characterizing clusters from images alone, providing constraints on their properties and formation histories using cosmological hydrodynamical simulations.