# The variance of the locally measured Hubble parameter explained with different estimators

## TL;DR

This paper compares different estimators for the variance of local Hubble constant measurements, highlighting how their differences impact the interpretation of observational discrepancies.

## Contribution

It clarifies how estimator choice affects variance calculations of Hubble constant measurements in perturbation theory and N-body simulations.

## Key findings

- N-body simulation estimators give larger weights to distant sources.
- Different estimators lead to different variance estimates.
- Both approaches produce variances too small to explain the Hubble tension.

## Abstract

We study the expected variance of measurements of the Hubble constant, $H_0$, as calculated in either linear perturbation theory or using non-linear velocity power spectra derived from $N$-body simulations. We compare the variance with that obtained by carrying out mock observations in the N-body simulations, and show that the estimator typically used for the local Hubble constant in studies based on perturbation theory is different from the one used in studies based on N-body simulations. The latter gives larger weight to distant sources, which explains why studies based on N-body simulations tend to obtain a smaller variance than that found from studies based on the power spectrum. Although both approaches result in a variance too small to explain the discrepancy between the value of $H_0$ from CMB measurements and the value measured in the local universe, these considerations are important in light of the percent determination of the Hubble constant in the local universe.