Multidimensionality of the Hubble tension: the roles of Ωm and ωc

The Hubble tension is inherently multidimensional and bears important implications for parameters beyond H0. We discuss the key role of the matter density parameter ωm and the physical cold dark matter density ωc. We argue that once ωm and the physical baryon density ωb are calibrated, through baryon acoustic oscillations (BAO) and/or type Ia supernovae (SNeIa) for ωm, and via big bang nucleosynthesis for ωb, any model raising H0 requires raising ωc and, under minimal assumptions, also the clustering parameter S8. We explicitly verify that this behavior holds when analyzing recent BAO and SNeIa data. We argue that a calibration of ωm as reliable and model-independent as possible should be a priority in the Hubble tension discussion, and an interesting possibility in this sense could be represented by galaxy cluster gas mass fraction measurements.