SAMI and TNG-Cluster: tracing galaxy spin and environmental transformation across cluster phase-space and cosmic time

The dense environment of galaxy clusters suppresses star formation and alters the kinematic properties of infalling satellites through gas stripping, tidal interactions, gravitational harassment and starvation. Projected phase-space diagrams connect the present-day distribution of cluster galaxies to their accretion histories. We combine SAMI Galaxy Survey integral field spectroscopy with the TNG-Cluster simulation to investigate how the stellar spin parameter (λ R), (g-i) colour, and sSFR vary across projected phase-space infall regions. At z = 0, TNG-Cluster reproduces the direction and broad strength of the phase-space trends observed in SAMI, including the weak yet significant λ R-clustercentric distance correlation. Leveraging this agreement, we extend the analysis across the last 8 Gyr, tracing the statistical evolution of galaxy properties within each infall region, and complement this with individual orbital histories of representative satellites. While colour and sSFR show clear monotonic gradients with both phase-space position and cosmic time, λ R behaves differently: it remains largely uniform across the outer infall regions, with only the virialised core exhibiting systematically lower values, and displays a slow monotonic decline toward the present day across all regions. We find that angular momentum suppression driven by the cluster environment is a slow, cumulative process requiring several Gyr of exposure to the cluster core, modulated by orbital history and stellar mass. This gradual nature explains why the λ R-environment correlation appears weak across phase-space regions in statistical samples; the effect emerges when individual satellite histories are tracked, revealing a sustained dynamical response to prolonged cluster residence.