Sample Variance for Supernovae Distance Measurements and the Hubble tension

Recent local measurements of the Hubble constant made using supernovae have delivered a value that differs by $\sim$5$\sigma$ (statistical error) from predictions using the Cosmic Microwave Background (CMB), or using Baryon Acoustic Oscillations (BAO) and Big-Bang Nucleosynthesis (BBN) constraints, which are themselves consistent. The effective volume covered by the supernovae is small compared to the other probes, and it is therefore interesting to consider whether sample variance (often also called cosmic variance) is a significant contributor to the offset. We consider four ways of calculating the sample variance: (i) perturbation theory applied to the luminosity distance, which is the most common method considered in the literature; (ii) perturbation of cosmological parameters, as is commonly used to alleviate super-sample covariance in sets of N-body simulations; (iii) a new method based on the variance between perturbed spherical top-hat regions; (iv) using numerical N-body simulations. All give consistent results showing that, for the Pantheon supernova sample, sample variance can only lead to fluctuations in $H_0$ of order $\pm1$ km s$^{-1}$Mpc$^{-1}$ or less. While this is not in itself a new result, the agreement between the methods used adds to its robustness. Furthermore, it is instructive to see how the different methods fit together. We also investigate the internal variance of the $H_{0}$ measurement using SH0ES and Pantheon data. By searching for an offset between measurements in opposite hemispheres, we find that the direction coincident with the CMB dipole has a higher $H_{0}$ measurement than the opposite hemisphere by roughly 4 km s$^{-1}$Mpc$^{-1}$. We compare this with a large number of simulations and find that the size of this asymmetry is statistically likely, but the preference of direction may indicate that further calibration is needed.