Bound or blown: the fate of hot gas in galaxy groups
The impact of AGN feedback on the hot gas content of galaxy groups remains a key uncertainty in galaxy formation and its connection to the large scale structure of the Universe. We aim to compare the XMM-Newton Group AGN Project (X-GAP) sample to the hydrodynamical FLAMINGO simulations, which span a wide range of AGN feedback prescriptions. We construct X-GAP analogues by forward-modelling the full selection function, including detection and observational systematics, and generate end-to-end XMM-Newton mock observations analysed consistently with the data. We study multiple observables, including the L--T and Mgas--T relations, number of groups, mean temperature, and velocity dispersion, accounting for their covariance. The forward model accurately recovers input luminosities, gas masses, and core-excised temperatures for regular systems, enabling direct comparison in observable space. The normalisation of the scaling relations is the best discriminator between feedback models, while cosmic variance introduces > 20% fluctuations in the number of detected systems, making counts alone a weak discriminator. Models with intermediate feedback strength provide the best agreement with X-GAP, with the fgas-2sigma model yielding the lowest tension of only 0.8sigma, while the most extreme feedback scenario (fgas-8sigma) is ruled out at > 4sigma. Our results indicate that the thermodynamic properties of galaxy groups favour feedback stronger than the fiducial FLAMINGO calibration, but disfavour the most ejective models. This highlights the importance of combining forward modelling and multi-observable constraints to probe the fate of hot baryons in low-mass haloes.