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