The MillenniumTNG Project: The hydrodynamical full physics simulation and a first look at its galaxy clusters

Cosmological simulations are an important theoretical pillar for understanding nonlinear structure formation in our Universe and for relating it to observations on large scales. In several papers, we introduce our MillenniumTNG (MTNG) project that provides a comprehensive set of high-resolution, large volume simulations of cosmic structure formation aiming to better understand physical processes on large scales and to help interpreting upcoming large-scale galaxy surveys. We here focus on the full physics box MTNG740 that computes a volume of $(740\,\mathrm{Mpc})^3$ with a baryonic mass resolution of $3.1\times~10^7\,\mathrm{M_\odot}$ using \textsc{arepo} with $80.6$~billion cells and the IllustrisTNG galaxy formation model. We verify that the galaxy properties produced by MTNG740 are consistent with the TNG simulations, including more recent observations. We focus on galaxy clusters and analyse cluster scaling relations and radial profiles. We show that both are broadly consistent with various observational constraints. We demonstrate that the SZ-signal on a deep lightcone is consistent with Planck limits. Finally, we compare MTNG740 clusters with galaxy clusters found in Planck and the SDSS-8 RedMaPPer richness catalogue in observational space, finding very good agreement as well. However, {\it simultaneously} matching cluster masses, richness, and Compton-$y$ requires us to assume that the SZ mass estimates for Planck clusters are underestimated by $0.2$~dex on average. Thanks to its unprecedented volume for a high-resolution hydrodynamical calculation, the MTNG740 simulation offers rich possibilities to study baryons in galaxies, galaxy clusters, and in large scale structure, and in particular their impact on upcoming large cosmological surveys.