1 Introduction

Assuming the validity of the cosmological principle, i.e., that the Universe is isotropic and homogeneous, we can apply the Robertson-Walker (RW) metric to the field equations of general relativity to derive a relation of cosmological distance measures to redshift for various cosmological parameter values. However, we know that our Universe is very far from homogeneous on scales smaller than galaxy clusters. It is generally assumed that this does not affect the large scale expansion rate of the Universe. Still, inhomogeneities will affect measured distances through the effect of gravitational lensing. There will be different amounts of matter along different lines-of-sight, causing different amounts of focusing of the light-rays. It is not possible to obtain exact solutions of the field equations for general inhomogeneous models, thus one is referred to numerical simulations to compute gravitational lensing effects on distance measurements.

Sometimes, we would like to be able to use simpler methods to compute at least approximate distances. The Dyer-Roeder (DR) distance-redshift relation (Dyer & Roeder 1973) assumes that the expansion rate of the Universe is governed by the total matter density whereas the focusing of light is only affected by a fraction $\alpha$ of the total matter density. The DR distance thus contains an additional parameter, namely the homogeneity-parameter, $\alpha$. The approximation should be fair if a fraction $1-\alpha$ of the matter density is in very compact objects and the light-ray travels far from all matter accumulations, i.e., if one can neglect the effect of gravitational lensing.

In this paper we investigate properties of the DR distance-redshift relation by comparing with numerical simulations of the distance-redshift relation in inhomogeneous universes, including the effect from gravitational lensing. More specifically, we compute the best fit value of the homegeneity-parameter $\alpha$ for different cosmologies and inhomogeneity models. These values can be used, e.g., with some of the publicly available routines for computing cosmological distances (Kayser et al. 1995).

In an earlier study using inhomogeneity models derived from N-body simulations, Tomita (1998) has found the best-fit value of $\alpha$ to be close to one in most cases, with a dispersion in $\alpha$ dependent on the cosmological model, the physical radius of the inhomogenities, the redshift and separation of light-rays. This paper is complementary in the respect that we study the effect of including point-like compact objects on the propagation of infinitesimal light-rays.