Data compression of measurements of peculiar velocities of Supernovae Ia

We study the compression of information present in the correlated perturbations to the luminosity distance in the low-redshift ($z<0.1$) supernovae Ia due to peculiar velocities of these supernovae. We demonstrate that the na\"{i}ve compression into angular velocity power spectrum does not work efficiently, due to thickness of the spherical shell over which the supernovae are measured. Instead, we show that measurements can be compressed into measurements of $f^2P(k)$, where $f$ is the logarithmic rate of growth of linear perturbations and $P(k)$ is their power spectrum. We develop an optimal quadratic estimator and show that it recovers all information for $\Lambda CDM$ models for surveys of $N\sim10,000$ or more supernovae. We explicitly demonstrate robustness with respect to the assumed fiducial model and the number of power spectrum bins. Using mock catalogues of SNe Ia we estimate that future low redshift surveys will be able to probe $\sigma_8$ to 6% accuracy with $10,000$ SNe Ia.