Sample variance in the local measurements of the Hubble constant

TL;DR

This study tests whether local sample variance can explain the tension between local and CMB-derived measurements of the Hubble constant by explicitly modeling the inhomogeneous 3D distribution of supernovae within a large-volume $N$-body simulation. Using the Dark Sky simulations and the Supercal SN sample, the authors place observers on Milky Way–mass haloes, rotate SN frames across 3240 orientations, and compute the local Hubble constant displacement $\Delta H_0^{\rm loc}$ with an inverse-variance weighted estimator. They find a total sample-variance dispersion of $\sigma(\Delta H_0^{\rm loc}) \approx 0.31\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$, far smaller than the $\sim 6\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$ gap between Riess et al. and Planck values, and demonstrate that reproducing the gap would require an unrealistically large local underdensity $\delta \simeq -0.8$ on $\sim 120\,h^{-1}\mathrm{Mpc}$ scales, inconsistent with observations. The work concludes that sample variance from local measurements cannot resolve the H0 tension in the $\Lambda$CDM framework, underscoring the need to scrutinize systematics or new physics. The analysis robustly incorporates the actual SN selection geometry and supports the interpretation that the tension is not a local-variance artifact.

Abstract

The current $>3σ$ 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 $σ(H_0^{\rm loc})=0.31\,{\rm km\ s^{-1}Mpc^{-1}}$, 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 $(δ\simeq -0.8)$ makes it very unlikely in a $Λ$CDM universe, and it also violates existing observational constraints. Therefore, sample variance in a $Λ$CDM universe cannot appreciably alleviate the tension in $H_0$ measurements even after taking into account the inhomogeneous selection of type Ia supernovae.

Figures (6)

• Figure 1: Flowchart depicting our simulation procedure. The inner loop runs over all mutual rotations between the SN and the simulation coordinate systems. The outer loop runs over all sub-volumes from the Dark Sky simulations, each sub-volume representing a realization of the local Universe.
• Figure 2: Top: redshift distributions of the Supercal SN sample (blue histogram) and the dark matter haloes in Dark Sky simulations (green histogram), for $0.023 <z <0.15$, split into 20 bins. Bottom: angular distribution of the Supercal SN sample in the Galactic coordinates, shown with the Hammer (equal-area) projection. The two symbols correspond to two ranges of redshift as shown. Note that the angular distribution is highly sparse.
• Figure 3: Sample variance in $\Delta H_0^{\rm loc}$ from our simulations, compared to P16 and R16 error bars (assuming P16 is the true global value). The blue histogram shows 3240 rotations of the SN coordinate system from 512 sub-volumes in the Dark Sky simulations, corresponding to $\sim$1.5 million SN-to-halo coordinate system configurations. The green, slightly more jagged, histogram shows the results of a particularly underdense sub-volume with a high $\Delta H_0^{\rm loc}$ at the 2-$\sigma$ level relative to all sub-volumes. The two histograms are separately normalized. Note that the sample variance in $H_0^{\rm loc}$ is much smaller than the difference between R16 and P16 measurements.
• Figure 4: Correlation between $\Delta H_0^{\rm loc}$ for $z_{\rm max}=0.15$ (corresponding to the SN sample) and dark matter density contrast $\delta$ for $z_{\rm max}=0.04$ (corresponding to the distance scale for local density measurements); both are measured from 512 sub-volumes of the Dark Sky simulations. Left: $\Delta H_0^{\rm loc}$ measurements from matching the 3D coordinates of SNe and haloes in sub-volumes (green points with error bars), compared to inference from all haloes in sub-volumes (blue points). The error bars on the green points reflect the variances from rotations of the SN coordinate system within each sub-volume. The solid line shows the linear fit to the green points. Right: zoomed-out version of the left-hand panel. We additionally mark the location of several $\delta$ values from observations, as well as the 1-$\sigma$ range favoured by the R16 analysis. We note that none of the observations of $\delta$ can account for the 6 $\,{\rm km\ s^{-1}Mpc^{-1}}$ difference between $H_0^{\rm loc}$ and $H_0^{\rm CMB}$.
• Figure 5: Deviations of the locally-measured Hubble constant as a function of maximum redshift (bottom $x$-axis) and distance (top $x$-axis). Observers and SNe are centred on haloes within $10^{12.3} < M_{\rm vir}/\, \rm M_\odot < 10^{12.4}$. The blue curves and shades correspond to no weighting, and the black dash curves correspond to weighting with the redshift distribution of SNe in R16. The curves correspond to the median and the 68% and 95% intervals. We see that, at this halo mass, $H_0^{\rm loc}$ is essentially unbiased on all scales. Using the realistic redshift distribution leads to a larger statistical error.