On the amount of peculiar velocity field information in supernovae from LSST and beyond

Peculiar velocities introduce correlations between supernova magnitudes, which implies that the supernova Hubble diagram residual carries information on both the matter power spectrum at the present time and its growth rate. By a combination of brute-force exact computations of likelihoods and Fisher matrix analysis, we investigate how this information, which comes from supernova data only, depends on different survey parameters such as covered area, depth, and duration. We show that, for a survey like The Rubin Observatory Legacy Survey of Space and Time (LSST) and a fixed redshift depth, the same observing time provides the same cosmological information whether one observes a larger area, or a smaller area during more years. We also show that although the peculiar velocity information is peaked in the range $z \in [0, 0.2]$, there is yet plenty of information in $z \in [0.2, 0.5]$, and for very high supernova number densities there is even more information in the latter range. We conclude that, after 5 years, LSST could measure $\sigma_8$ with an uncertainty of 0.17 with the current strategy, and that this could be improved to 0.09 if the supernova completeness is improved to 20%. Moreover, we forecast results considering the extra parameter $\gamma$, and show that this creates a non-linear degeneracy with $\sigma_8$ that makes the Fisher matrix analysis inadequate. Finally, we discuss the possibility of achieving competitive results with the current Zwicky Transient Facility.