How do nonlinear voids affect light propagation?

Propagation of light in a clumpy universe is examined. As an inhomogeneous matter distribution, we take a spherical void surrounded by a dust. shell, where the lost mass in the void is compensated by the shell. We study how the angular-diameter distance behaves when such a structure exists. The angular-diameter distance is calculated by integrating the Raychaudhuri equation including the shear. An explicit expression for the junction condition for the massive thin shell is calculated. We apply these results to a dust shell embedded in a Friedmann universe and determine how the distance-redshift relation is modified compared with that in the purely Friedmann universe. We also study the distribution of distances in a universe filled with voids. We show that the void-filled universe gives a larger distance than the FRW universe by similar to 5% at z similar to 1 if the size of the void is similar to 5% of the Horizon radius.