The Effect of Large-Scale Inhomogeneities on the Luminosity Distance

We study the form of the luminosity distance as a function of redshift in the presence of large scale inhomogeneities, with sizes of order 10 Mpc or larger. We approximate the Universe through the Swiss-cheese model, with each spherical region described by the Tolman-Bondi metric. We study the propagation of light beams in this background, assuming that the locations of the source and the observer are random. We derive the optical equations for the evolution of the beam area and shear. Through their integration we determine the configurations that can lead to an increase of the luminosity distance relative to the homogeneous cosmology. We find that this can be achieved if the Universe is composed of spherical void-like regions, with matter concentrated near their surface. For inhomogeneities consistent with the observed large scale structure, the relative increase of the luminosity distance is of the order of a few percent at redshifts near 1, and falls short of explaining the substantial increase required by the supernova data. On the other hand, the effect we describe is important for the correct determination of the energy content of the Universe from observations.