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The latest improvements in the scale and calibration of Type Ia supernovaecatalogues allow us to constrain the specific nature and evolution of darkenergy through its effect on the expansion history of the universe. We presentthe results of Bayesian cosmological model comparison on the SNe~Ia cataloguePantheon+, where Flat $\Lambda$CDM is preferred by the data over all othermodels and we find moderate evidence ($\Delta \log \mathcal{Z} \sim 2.5$) toreject a number of the alternate dark energy models. The effect of peculiarvelocity corrections on model comparison is analysed, where we show thatremoving the peculiar velocity corrections results in a varying fit onnon-$\Lambda$CDM parameters. As well as comparing cosmological models, theBayesian methodology is extended to comparing the scatter model of the data,testing for non-gaussianity in the Pantheon+ Hubble residuals. We find thatadding a scale parameter to the Pantheon+ covariances, or alternately using amultivariate Student's t-distribution fits the data better than the fiducialanalysis, producing a cosmology independent evidence increase of $\Delta \log\mathcal{Z} = 2.29 $ and $2.46$ respectively. This improved treatment of thescatter decreases the uncertainty in the constraint on the Hubble constant,finding $H_0 = 73.67 \pm 0.99 $ km s$^{-1}$ Mpc$^{-1}$, in $ 5.7 \sigma$tension with Planck. We also explore $M_B$ transition models as a potentialsolution for the Hubble tension, finding no evidence to support these modelsamong the SNe data.

Non-Gaussian Likelihoods for Type Ia Supernovae Cosmology: Implications for Dark Energy and $H_0$