Multidimensionality of the Hubble tension: the roles of ${\mathrm{\Omega }}_m$ and ${\omega }_c$

The Hubble tension is inherently multidimensional, and bears important implications for parameters beyond $H_0$. We discuss the key role of the matter density parameter ${\mathrm{\Omega }}_m$ and the physical cold dark matter density ${\omega }_c$. We argue that once ${\mathrm{\Omega }}_m$ and the physical baryon density ${\omega }_b$ are calibrated, through Baryon Acoustic Oscillations (BAO) and/or Type Ia Supernovae (SNeIa) for ${\mathrm{\Omega }}_m$, and via Big Bang Nucleosynthesis for ${\omega }_b$, any model raising $H_0$ requires raising ${\omega }_c$ and, under minimal assumptions, also the clustering parameter $S_8$. We explicitly verify that this behaviour holds when analyzing recent BAO and SNeIa data. We argue that a calibration of ${\mathrm{\Omega }}_m$ 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.