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 $\alpha$ 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 $1\sigma$.
