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We consider the dispersion on the supernova distance-redshift relation due topeculiar velocities and gravitational lensing, and the sensitivity of theseeffects to the amplitude of the matter power spectrum. We use the MeMo lensinglikelihood developed by Quartin, Marra & Amendola (2014), which accounts forthe characteristic non-Gaussian distribution caused by lensing magnificationwith measurements of the first four central moments of the distribution ofmagnitudes. We build on the MeMo likelihood by including the effects ofpeculiar velocities directly into the model for the moments. In order tomeasure the moments from sparse numbers of supernovae, we take a new approachusing Kernel Density Estimation to estimate the underlying probability densityfunction of the magnitude residuals. We also describe a bootstrap re-samplingapproach to estimate the data covariance matrix. We then apply the method tothe Joint Light-curve Analysis (JLA) supernova catalogue. When we impose onlythat the intrinsic dispersion in magnitudes is independent of redshift, we find$\sigma_8=0.44^{+0.63}_{-0.44}$ at the one standard deviation level, althoughwe note that in tests on simulations, this model tends to overestimate themagnitude of the intrinsic dispersion, and underestimate $\sigma_8$. We notethat the degeneracy between intrinsic dispersion and the effects of $\sigma_8$is more pronounced when lensing and velocity effects are consideredsimultaneously, due to a cancellation of redshift dependence when both effectsare included. Keeping the model of the intrinsic dispersion fixed as a Gaussiandistribution of width 0.14 mag, we find $\sigma_8 = 1.07^{+0.50}_{-0.76}$.

The effects of velocities and lensing on moments of the Hubble diagram