The Pantheon+ Analysis: SuperCal-Fragilistic Cross Calibration, Retrained SALT2 Light Curve Model, and Calibration Systematic Uncertainty

We present here a re-calibration of the photometric systems used in the Pantheon+ sample of Type Ia supernovae (SNe Ia) including those used for the SH0ES distance-ladder measurement of H$_0$. We utilize the large and uniform sky coverage of the public Pan-STARRS stellar photometry catalog to cross-calibrate against tertiary standards released by individual SN Ia surveys. The most significant updates over the `SuperCal' cross-calibration used for the previous Pantheon and SH0ES analyses are: 1) expansion of the number of photometric systems (now 25) and filters (now 105), 2) solving for all filter offsets in all systems simultaneously in order to produce a calibration uncertainty covariance matrix that can be used in cosmological-model constraints, and 3) accounting for the change in the fundamental flux calibration of the HST CALSPEC standards from previous versions on the order of $1.5\%$ over a $\Delta \lambda$ of 4000~\AA. The re-calibration of samples used for light-curve fitting has historically been decoupled from the retraining of the light-curve model. Here, we are able to retrain the SALT2 model using this new calibration and find the change in the model coupled with the change to the calibration of the light-curves themselves causes a net distance modulus change ($d\mu/dz$) of 0.04 mag over the redshift range $0<z<1$. We introduce a new formalism to determine the systematic impact on cosmological inference by propagating the covariance in fitted calibration offsets through retraining simultaneously with light-curve fitting and find a total calibration uncertainty impact of $\sigma_w=0.013$, which is roughly half the size of the sample statistical uncertainty. Similarly, we find a systematic SN calibration contribution to the SH0ES H$_0$ uncertainty is less than 0.2~km/s/Mpc, suggesting that SN Ia calibration cannot resolve the current level of the `Hubble Tension'.