Reduced uncertainties up to 43\% on the Hubble constant and the matter density with the SNe Ia with a new statistical analysis

Type Ia Supernovae (SNe Ia) are considered the most reliable \textit{standard candles} and they have played an invaluable role in cosmology since the discovery of the Universe's accelerated expansion. During the last decades, the SNe Ia samples have been improved in number, redshift coverage, calibration methodology, and systematics treatment. These efforts led to the most recent \textit{``Pantheon"} (2018) and \textit{``Pantheon +"} (2022) releases, which enable to constrain cosmological parameters more precisely than previous samples. In this era of precision cosmology, the community strives to find new ways to reduce uncertainties on cosmological parameters. To this end, we start our investigation even from the likelihood assumption of Gaussianity, implicitly used in this domain. Indeed, the usual practise involves constraining parameters through a Gaussian distance moduli likelihood. This method relies on the implicit assumption that the difference between the distance moduli measured and the ones expected from the cosmological model is Gaussianly distributed. In this work, we test this hypothesis for both the \textit{Pantheon} and \textit{Pantheon +} releases. We find that in both cases this requirement is not fulfilled and the actual underlying distributions are a logistic and a Student's t distribution for the \textit{Pantheon} and \textit{Pantheon +} data, respectively. When we apply these new likelihoods fitting a flat $\Lambda$CDM model, we significantly reduce the uncertainties on $\Omega_M$ and $H_0$ of $\sim 40 \%$. This boosts the SNe Ia power in constraining cosmological parameters, thus representing a huge step forward to shed light on the current debated tensions in cosmology.