Constraining smoothness parameter and the DD relation of Dyer-Roeder equation with supernovae

Our real universe is locally inhomogeneous. Dyer and Roeder introduced the smoothness parameter 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, DL (z) (1+z) ^-2/DA (z) =1, should be valid for all cosmological models that are described by Riemannian geometry, where DL and DA 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 and test whether the Dyer-Roeder equation satisfies the DD relation. By using ² minimization, we get =0. 92-₀. ₃₂^+0. 08 at 1 and 0. 92-₀. ₆₅^+0. 08 at 2, and our results show that the Dyer-Roeder equation is in good consistency with the DD relation at 1.