H0 tension or T0 tension?

We study if the discrepancy between the local and cosmological measurements of the Hubble constant $H_0$ can be reformulated as a tension in the cosmic microwave background (CMB) monopole temperature $T_0$. The latter is customarily fixed to the FIRAS best-fit value in CMB data analyses. Although this value was confirmed by several independent experiments, it is interesting to see how much parameter constraints depend on this prior. We first provide a detailed pedagogical description of the $T_0$ effects on cosmological observables. We show that the recombination history and transfer functions do not depend on $T_0$, provided they are parametrized by the energy scale rather than redshift, and at fixed dark matter and baryon densities per CMB photon. Thus, $T_0$ is only a property of the observer, quantifying the amount of expansion between key cosmological events and today. As a consequence, the sole effect of $T_0$ on small-scale primary CMB anisotropies is through the angular diameter distance to the epoch of last scattering, resulting in a near-perfect degeneracy between $T_0$ and $H_0$. This geometric degeneracy is partially lifted by the late-time integrated Sachs-Wolfe effect and CMB lensing. Still, Planck data alone is consistent with a broad region in the $H_0-T_0$ plane, implying that removing the FIRAS prior on $T_0$ can make Planck and SH0ES less discrepant, without introducing new physics beyond $\Lambda$CDM. One may break the degeneracy by combining Planck with SH0ES, yielding an independent measurement of $T_0$, which happens to be in $3\sigma$ tension with FIRAS. Therefore, the Hubble tension indeed can be recast into the $T_0$ tension. The agreement with FIRAS is restored when combining Planck with the baryon acoustic oscillation data instead of SH0ES. Thus, the tension between SH0ES and cosmological measurements of $H_0$ persists even if we discard the FIRAS $T_0$ prior.