The resilience of the Etherington-Hubble relation

The Etherington reciprocity theorem, or distance duality relation (DDR), relates the mutual scaling of cosmic distances in any metric theory of gravity where photons are massless and propagate on null geodesics. In this paper, we make use of the DDR to build a consistency check based on its degeneracy with the Hubble constant, $H_0$. We parameterise the DDR using the form $\eta(z) = 1+ \epsilon z$, thus only allowing small deviations from its standard value. We use a combination of late time observational data to provide the first joint constraints on the Hubble parameter and $\epsilon$ with percentage accuracy: $H_0 = 68.6 \pm 2.5$ kms$^{-1}$Mpc$^{-1}$ and $\epsilon = 0.001^{+0.023}_{-0.026}$. We build our consistency check using these constraints and compare them with the results obtained in extended cosmological models using cosmic microwave background data. We find that extensions to $\Lambda$CDM involving massive neutrinos and/or additional dark radiation are in perfect agreement with the DDR, while models with non-zero spatial curvature show a preference for DDR violation, i.e., $\epsilon \ne 0 $ at the level of $\sim 1.5 \sigma$. Most importantly, we find a mild 2$\sigma$ discrepancy between the validity of the DDR and the latest publicly available Cepheid-calibrated SNIa constraint on $H_0$. We discuss the potential consequences of this for both the Etherington reciprocity theorem and the $H_0$ tension.