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The Sunyaev-Zeldovich (SZ) effect has long been recognized as a powerfulcosmological probe. Using the BAHAMAS and MACSIS simulations to obtain$>10,000$ simulated galaxy groups and clusters, we compute three temperaturemeasures and quantify the differences between them. The first measure isrelated to the X-ray emission of the cluster, while the second describes thenon-relativistic thermal SZ (tSZ) effect. The third measure determines thelowest order relativistic correction to the tSZ signal, which is seeingincreased observational relevance. Our procedure allows us to accurately modelthe relativistic SZ (rSZ) contribution and we show that a $\gtrsim 10\%-40\%$underestimation of this rSZ cluster temperature is expected when applyingstandard X-ray relations. The correction also exhibits significant mass andredshift evolution, as we demonstrate here. We present the mass dependence ofeach temperature measure alongside their profiles and a short analysis of thetemperature dispersion as derived from the aforementioned simulations. We alsodiscuss a new relation connecting the temperature and Compton-$y$ parameter,which can be directly used for rSZ modelling. Simple fits to the obtainedscaling relations and profiles are provided. These should be useful for futurestudies of the rSZ effect and its relevance to cluster cosmology.

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