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We compare cluster scaling relations published for three different samples selected via X-ray and Sunyaev–Zel'dovich (SZ) signatures. We find tensions driven mainly by two factors: (i) systematic differences in the X-ray cluster observables used to derive the scaling relations and (ii) uncertainty in the modelling of how the gas mass of galaxy clusters scales with total mass. All scaling relations are in agreement after accounting for these two effects. We describe a multivariate scaling model that enables a fully self-consistent treatment of multiple observational catalogues in the presence of property covariance and apply this formalism when interpreting published results. The corrections due to scatter and observable covariance can be significant. For instance, our predicted YSZ-LX scaling relation differs from that derived using the naive ‘plug in’ method by ≈25 per cent. Finally, we test the mass normalization for each of the X-ray data sets we consider by applying a space density consistency test: we compare the observed ROSAT-ESO Flux-Limited X-ray (REFLEX) luminosity function to expectations from published LX–M relations convolved with the mass function for a Wilkinson Microwave Anisotropy Probe 7 flat Λ cold dark matter model.
Rozo et al. (Sat,) studied this question.