On the use of the local prior on the absolute magnitude of Type Ia supernovae in cosmological inference

A dark-energy which behaves as the cosmological constant until a sudden phantom transition at very-low redshift ($z 4$\sigma$ disagreement between the local and high-redshift determinations of the Hubble constant, while maintaining the phenomenological success of the $\Lambda$CDM model with respect to the other observables. Here, we show that such a hockey-stick dark energy cannot solve the $H_0$ crisis. The basic reason is that the supernova absolute magnitude $M_B$ that is used to derive the local $H_0$ constraint is not compatible with the $M_B$ that is necessary to fit supernova, BAO and CMB data, and this disagreement is not solved by a sudden phantom transition at very-low redshift. We make use of this example to show why it is preferable to adopt in the statistical analyses the prior on $M_B$ as an alternative to the prior on $H_0$. The three reasons are: i) one avoids potential double counting of low-redshift supernovae, ii) one avoids assuming the validity of cosmography, in particular fixing the deceleration parameter to the standard model value $q_0=-0.55$, iii) one includes in the analysis the fact that $M_B$ is constrained by local calibration, an information which would otherwise be neglected in the analysis, biasing both model selection and parameter constraints. We provide the priors on $M_B$ relative to the recent Pantheon and DES-SN3YR supernova catalogs. We also provide a Gaussian joint prior on $H_0$ and $q_0$ that generalizes the prior on $H_0$ by SH0ES.