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The supernova (SN) Hubble diagram residual contains valuable information onboth the present matter power spectrum and its growth history. In this paper weshow that this information can be retrieved with precision by combining bothpeculiar velocity and weak-lensing analysis on the data. To wit, peculiarvelocity induces correlations on the nearby SN while lensing induces anon-Gaussian dispersion in faraway objects. We show that both effects havealmost orthogonal degeneracies and discuss how they can be extractedsimultaneously from the data. We analyze the JLA supernova catalog in a14-dimensional parameter space, assuming a flexible growth-rate index $\gamma$.We arrive at the following marginalized constraints: $\sigma_8 =0.65^{+0.23}_{-0.37}$ and $\gamma = 1.38^{+1.7}_{-0.65}$. Assuming instead GRas the correct gravitation theory (and thus $\gamma \equiv 0.55$), theconstraints in $\sigma_8$ tighten further: $\sigma_8 = 0.40^{+0.21}_{-0.23}$.We show that these constraints complement well the ones obtained from otherdatasets and that they could improve substantially with more SNe.

Turning noise into signal: learning from the scatter in the Hubble diagram