Relations between physical observables: what is better?

TL;DR

This paper examines the differences between using redshift and angular-distance as relations between physical observables, highlighting how the choice impacts theoretical uncertainties and the measurement of the Hubble constant.

Contribution

It introduces a detailed analysis of the redshift-angular-distance relation, revealing significant differences at linear perturbation levels and implications for H_0 tension.

Findings

Redshift/angular-distance relation differs significantly at linear order.

The effect is suppressed by 2 orders of magnitude compared to the inverse relation.

Results may reduce uncertainties in local H_0 measurements.

Abstract

We investigate some possible relations between physical observables and estimate the "cosmic variance" which affects these measurements. We focus on redshift and angular-distance and we discuss the difference in considering the redshift as function of the angular-distance rather than the usually considered inverse relation. Already at linear level in metric perturbations, we find a significant difference. Indeed, even if both relations are led by source radial velocity for close enough sources, this effect is suppressed by 2 orders of magnitude in the redshift/angular-distance relation. This fact can significantly reduce the theoretical uncertainty for close sources already investigated in the literature for the angular-distance/redshift relation and open a new scenario for clarifying the tension in the measurement of $H_0$ from local sources rather than from the CMB.