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.
