Testing the distance duality relation with present and future data

The assumptions that "light propagates along null geodesics of the spacetime metric" and "the number of photons is conserved along the light path" lead to the distance duality relation (DDR), $\eta = D_L(z) (1 + z)^{-2}/D_A(z) = 1$, with $D_L(z)$ and $D_A(z)$ the luminosity and angular diameter distances to a source at redshift $z$. In order to test the DDR, we follow the usual strategy comparing the angular diameter distances of a set of clusters, inferred from X - ray and radio data, with the luminosity distance at the same cluster redshift using the local regression technique to estimate $D_L(z)$ from Type Ia Supernovae (SNeIa) Hubble diagram. In order to both strengthen the constraints on the DDR and get rid of the systematics related to the unknown cluster geometry, we also investigate the possibility to use Baryon Acoustic Oscillations (BAO) to infer $D_A(z)$ from future BAO surveys. As a test case, we consider the proposed Euclid mission investigating the precision can be afforded on $\eta(z)$ from the expected SNeIa and BAO data. We find that the combination of BAO and the local regression coupled allows to reduce the errors on $\eta_a = d\eta/dz|_{z = 0}$ by a factor two if one $\eta_0 = \eta(z = 0) = 1$ is forced and future data are used. On the other hand, although the statistical error on $\eta_0$ is not significantly reduced, the constraints on this quantity will be nevertheless ameliorated thanks to the reduce impact of systematics.