Our real universe is locally inhomogeneous .
Dyer and Roeder introduced the smoothness parameter \alpha to describe the influence of local inhomogeneity on angular diameter distance , and they obtained the angular diameter distance-redshift approximate relation ( Dyer-Roeder equation ) for locally inhomogeneous universe .
Furthermore , the Distance-Duality ( DD ) relation , D _ { L } ( z ) ( 1 + z ) ^ { -2 } / D _ { A } ( z ) = 1 , should be valid for all cosmological models that are described by Riemannian geometry , where D _ { L } and D _ { A } are , respectively , the luminosity and angular distance distances .
Therefore , it is necessary to test whether if the Dyer-Roeder approximate equation can satisfy the Distance-Duality relation .
In this paper , we use Union2.1 SNe Ia data to constrain the smoothness parameter \alpha and test whether the Dyer-Roeder equation satisfies the DD relation .
By using \chi ^ { 2 } minimization , we get \alpha = 0.92 _ { -0.32 } ^ { +0.08 } at 1 \sigma and 0.92 _ { -0.65 } ^ { +0.08 } at 2 \sigma , and our results show that the Dyer-Roeder equation is in good consistency with the DD relation at 1 \sigma .