Modifications to the Etherington Distance Duality Relation and Observational Limits

The Etherington distance duality relation, which relates the luminosity distance, the angular diameter distance and the redshift of objects, depends only upon a conservation law for light that traces back directly to the Lorentzian spacetime geometry. We show that this duality relation indeed survives transition to the most general linear electrodynamics without birefringence, which rests on a spacetime geometry constituted by a Lorentzian metric and two scalar fields, a dilaton and an axion. By computing the Poynting vector and optical scalar transport in the geometrical optics limit of this framework, we derive the modification of the light flux in the presence of a dilaton field and present improved constraints on the gradient of the dilaton field from observations of the Cosmic Microwave Background spectrum. Although this flux modification may seem applicable also to fundamental modifications of the Etherington relation, we show that the distance duality relation still holds true. Thus any deviations within this classical theory would imply non-metricities, once astrophysical sources for attenuation, such as dust, are accounted for. Moreover, using the most up-to-date measurements of the luminosity distances to Supernovae of Type Ia, and the inferred angular diameter distances from the baryon acoustic feature measurements, we perform a scale-free and nearly model-independent test of the Etherington distance duality relation between redshifts of $0.38$ and $0.61$. We find consistency with the standard distance duality relation and constrain the optical depth of light between these two redshifts to $\Delta\tau=-0.006\pm0.046$.