Ellis proposed a proof of this equation in the context of Riemannian geometry.
The Etherington's distance-duality equation has been validated from astronomical observations based on the X-ray surface brightness and the Sunyaev–Zel'dovich effect of galaxy clusters.
[6] Any violation of the distance duality would be attributed to exotic physics provided that astrophysical effects altering the cosmic distance measurements are well below the statistical errors.
For instance, an incorrect modelling of the three-dimensional gas density profile in galaxy clusters may introduce systematic uncertainties in the determination of the cluster angular diameter distance from X-ray and/or SZ observations, thus altering the outcome of the distance-duality test.
[7] Similarly, unaccounted extinction from a diffuse dust component in the inter-galactic medium can affect the determination of luminosity distances and cause a violation of the distance-duality relation.