Effective volume of supernovae samples and sample variance

The source of the tension between local Supernova (SN) Ia based Hubble constant measurements and those from the cosmic microwave background or (baryon acoustic oscillation+big bang nucleosynthesis) measurements is one of the most interesting unknowns of modern cosmology. Sample variance forms a key component of the error on the local measurements, and will dominate the error budget in the future as more SNe Ia are observed. Many methods have been proposed to estimate sample variance in many contexts, and we compared results from a number of approximate methods to estimates from N-body simulations in a previous paper, confirming that sample variance for the Pantheon SNe Ia sample does not solve the Hubble tension. We now extend this analysis to include the more accurate analytic method based on calculating correlations between the radial peculiar velocities of SNe Ia, comparing this technique with results from numerical simulations. We consider the dependence of these errors on the linear power spectrum and how nonlinear velocities contribute to the error. Using this technique, and matching sample variance errors from more approximate methods, we can define an effective volume for SNe Ia samples, finding that the Pantheon sample is equivalent to a top-hat sphere of radius ~220 h^-1 Mpc. We use this link between sample-variance errors to compute ΔH₀ for idealized surveys with particular angular distributions of SNe Ia. For example, a half-sky survey at the Pantheon depth has the potential to suppress the sample variance of H₀ to ~0.1 km s^-1 Mpc^-1, a significant improvement compared with the current result. Finally, we consider the strength of large-scale velocity power spectrum required to explain the Hubble tension using sample variance, finding it requires an extreme model well beyond that allowed by other observations.