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 $\Delta\tau=-0.006\pm0.046$.
