Relativistic SZ temperature scaling relations of groups and clusters derived from the BAHAMAS and MACSIS simulations

The Sunyaev-Zeldovich (SZ) effect has long been recognized as a powerful cosmological probe. Using the BAHAMAS and MACSIS simulations to obtain $>10,000$ simulated galaxy groups and clusters, we compute three temperature measures and quantify the differences between them. The first measure is related to the X-ray emission of the cluster, while the second describes the non-relativistic thermal SZ (tSZ) effect. The third measure determines the lowest order relativistic correction to the tSZ signal, which is seeing increased observational relevance. Our procedure allows us to accurately model the relativistic SZ (rSZ) contribution and we show that a $\gtrsim 10\%-40\%$ underestimation of this rSZ cluster temperature is expected when applying standard X-ray relations. The correction also exhibits significant mass and redshift evolution, as we demonstrate here. We present the mass dependence of each temperature measure alongside their profiles and a short analysis of the temperature dispersion as derived from the aforementioned simulations. We also discuss a new relation connecting the temperature and Compton-$y$ parameter, which can be directly used for rSZ modelling. Simple fits to the obtained scaling relations and profiles are provided. These should be useful for future studies of the rSZ effect and its relevance to cluster cosmology.