Sample variance in the local measurements of the Hubble constant

The current $>3\sigma$ tension between the Hubble constant $H_0$ measured from local distance indicators and from cosmic microwave background is one of the most highly debated issues in cosmology, as it possibly indicates new physics or unknown systematics. In this work, we explore whether this tension can be alleviated by the sample variance in the local measurements, which use a small fraction of the Hubble volume. We use a large-volume cosmological $N$-body simulation to model the local measurements and to quantify the variance due to local density fluctuations and sample selection. We explicitly take into account the inhomogeneous spatial distribution of type Ia supernovae. Despite the faithful modelling of the observations, our results confirm previous findings that sample variance in the local Hubble constant $(H_0^{\rm loc})$ measurements is small; we find $\sigma(H_0^{\rm loc})=0.31\,{\rm km\ s^{-1}Mpc^{-1}}$, a nearly negligible fraction of the $\sim6\,{\rm km\ s^{-1}Mpc^{-1}}$ necessary to explain the difference between the local and the global $H_0$ measurements. While the $H_0$ tension could in principle be explained by our local neighbourhood being a underdense region of radius $\sim 150 \,\rm Mpc$ , the extreme required underdensity of such a void $(\delta\simeq -0.8)$ makes it very unlikely in a $\Lambda$CDM universe, and it also violates existing observational constraints. Therefore, sample variance in a $\Lambda$CDM universe cannot appreciably alleviate the tension in $H_0$ measurements even after taking into account the inhomogeneous selection of type Ia supernovae.