Measuring the sound horizon and absolute magnitude of SNIa by maximizing the consistency between low-redshift data sets

The comoving sound horizon at the baryon drag epoch, $r_{d}$, encapsulates very important physical information about the pre-recombination era and serves as a cosmic standard ruler. On the other hand, the absolute magnitude of supernovae of Type Ia (SNIa), $M$, is pivotal to infer the distances to these standard candles. Having access to (at least) one of these two quantities is crucial to measure the Hubble parameter $H_0$ from BAO/SNIa data. In this work we present a new method to measure how long is the cosmic ruler and how bright are the standard candles independently from the main drivers of the $H_0$ tension, namely by avoiding (i) the use of CMB data; (ii) the calibration of SNIa in the first steps of the cosmic distance ladder; and (iii) the assumption of any concrete cosmological model. We only assume that SNIa can be safely employed as standard candles and $r_d$ as a standard ruler, together with the validity of the Cosmological Principle and the metric description of gravity, with photons propagating in null geodesics and the conservation of the photon number. Our method is based on the minimization of a loss function that represents the level of inconsistency between the low-redshift data sets employed in this study, to wit: SNIa, BAO and cosmic chronometers. In our main analysis we obtain: $r_d=(146.0^{+4.2}_{-5.1})$ Mpc, $M=-19.362^{+0.078}_{-0.067}$. The former is fully compatible with Planck's $\Lambda$CDM best-fit cosmology, but still leaves plenty of room for new physics before the decoupling, whereas our constraint on $M$ lies closer to the value preferred by the concordance model, although it is only $\sim 1.4\sigma$ below the SH0ES measurement.