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 $\ensuremath{\sim}5\ensuremath{\sigma}$ (statistical error) from predictions cosmic microwave background (CMB), or baryon acoustic oscillations (BAO) and big-bang nucleosynthesis (BBN) constraints, which are themselves consistent. The effective volume covered is small compared to other probes, it therefore interesting consider whether sample variance (often also called variance) significant contributor offset. We four ways calculating variance: (i) perturbation theory applied luminosity distance, most common method considered in literature; (ii) cosmological parameters, as commonly used alleviate supersample covariance sets N-body simulations; (iii) new based on between perturbed spherical top-hat regions; (iv) numerical simulations. All give consistent results showing that, for Pantheon supernova sample, can only lead fluctuations ${H}_{0}$ order $\ifmmode\pm\else\textpm\fi{}1\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ less. While this not itself result, agreement methods adds its robustness. Furthermore, instructive see how different fit together. investigate internal measurement SH0ES data. By searching an offset opposite hemispheres, we find direction coincident with CMB dipole has higher than hemisphere roughly $4\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. compare large number simulations size asymmetry statistically likely, but preference may indicate further calibration needed.