The effective volume of supernovae samples and sample variance
First author: Zhongxu Zhai
The source of the tension between local SN Ia based Hubble constant measurements and those from the CMB or BAO+BBN 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 supernovae are observed. Many methods have been proposed to estimate sample variance in many contexts, and we compared results from a number of them in Zhai & Percival (2022), confirming that sample variance for the Pantheon supernovae sample does not solve the Hubble tension. We now extend this analysis to include a method based on analytically calculating correlations between the radial peculiar velocities of supernovae, comparing this technique with results from numerical simulations, which can be considered a non-linear Monte-Carlo solution that works similarly. We consider the dependence of these errors on the linear power spectrum and how non-linear velocities contribute to the error. Using this technique, and matching sample variance errors, we can define an effective volume for supernovae samples, finding that the Pantheon sample is equivalent to a top-hat sphere of radius $\sim220~h^{-1}$Mpc. We use this link between sample-variance errors to compute $\Delta H_{0}$ for idealised surveys with particular angular distributions of supernovae. For example, a half-sky survey at the Pantheon depth has the potential to suppress the sample variance of $H_{0}$ to $\sim0.1$ 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.