Light propagation and the distance-redshift relation in a realistic inhomogeneous universe

We investigate the propagation of light rays in a clumpy universe constructed by a cosmological version of the post-Newtonian approximation. We show that the linear approximation to the propagation equations is valid in the region z1 even if the density contrast is much larger than unity. Based on a general order-of-magnitude statistical consideration, we argue that the linear approximation is still valid for z1. Then we give a general formula for the distance-redshift relation in a clumpy universe and derive an explicit expression for a simplified situation in which the effect of the gravitational potential of inhomogeneities dominates. In the light of the derived relation we discuss the validity of the Dyer-Roeder distance. Furthermore, we consider a simple model of an inhomogeneous universe and investigate statistical properties of light rays. We find that the result of this specific example also supports the validity of the linear approximation.