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.
