We consider the dispersion on the supernova distance-redshift relation due to peculiar velocities and gravitational lensing , and the sensitivity of these effects to the amplitude of the matter power spectrum .
We use the MeMo lensing likelihood developed by Quartin et al. , which accounts for the characteristic non-Gaussian distribution caused by lensing magnification with measurements of the first four central moments of the distribution of magnitudes .
We build on the MeMo likelihood by including the effects of peculiar velocities directly into the model for the moments .
In order to measure the moments from sparse numbers of supernovae , we take a new approach using Kernel Density Estimation to estimate the underlying probability density function of the magnitude residuals .
We also describe a bootstrap re-sampling approach to estimate the data covariance matrix .
We then apply the method to the Joint Light-curve Analysis ( JLA ) supernova catalogue .
When we impose only that 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 , although we note that in tests on simulations , this model tends to overestimate the magnitude of the intrinsic dispersion , and underestimate \sigma _ { 8 } .
We note that the degeneracy between intrinsic dispersion and the effects of \sigma _ { 8 } is more pronounced when lensing and velocity effects are considered simultaneously , due to a cancellation of redshift dependence when both effects are included .
Keeping the model of the intrinsic dispersion fixed as a Gaussian distribution of width 0.14 mag , we find \sigma _ { 8 } = 1.07 ^ { +0.50 } _ { -0.76 } .