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

TL;DR

This study constrains the smoothness parameter in the Dyer-Roeder equation using supernova data and tests its consistency with the Distance-Duality relation, finding good agreement within 1 sigma.

Contribution

It provides the first constraints on the smoothness parameter and verifies the Dyer-Roeder equation's compatibility with the DD relation using observational data.

Findings

The smoothness parameter α is approximately 0.92 with uncertainties.

The Dyer-Roeder equation is consistent with the DD relation at 1 sigma.

The results support the validity of the Dyer-Roeder approximation in inhomogeneous universes.

Abstract

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 $1\sigma$.