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 .