First measurement of $\sigma_8$ using supernova magnitudes only

A method was recently proposed which allows the conversion of the weak-lensing effects in the supernova Hubble diagram from noise into signal. Such signal is sensitive to the growth of structure in the universe, and in particular can be used as a measurement of $\sigma_8$ which is independent from more traditional methods such as those based on the CMB, cosmic shear or cluster abundance. We extend here that analysis to allow for intrinsic non-Gaussianities in the supernova PDF, and discuss how this can be best modelled using the Bayes Factor. Although it was shown that a precise measurement of $\sigma_8$ requires ~$10^5$ supernovae, current data already allows an important proof of principle. In particular we make use of the 732 supernovae with z < 1 of the recent JLA catalog and show that a simple treatment of intrinsic non-Gaussianities with a couple of nuisance parameters is enough for our method to yield the values $\sigma_8 = 0.84^{+0.28}_{-0.65}$ or $\sigma_8 < 1.45$ at a $2\sigma$ confidence level. This result is consistent with mock simulations and it is also in agreement with independent measurements and presents the first ever measurement of $\sigma_8$ using supernova magnitudes alone.