  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.
