Calculation of distances in cosmological models with small-scale inhomogeneities and their use in observational cosmology: a review

Abstract: The Universe is not completely homogeneous. Even if it is sufficiently so on large scales, it is very inhomogeneous at small scales, and this has an effect on light propagation, so that the distance as a function of redshift, which in many cases is defined via light propagation, can differ from the homogeneous case. Simple models can take this into account. I review the history of this idea, its generalization to a wide variety of cosmological models, analytic solutions of simple models, comparison of such solutions with exact solutions and numerical simulations, applications, simpler analytic approximations to the distance equations, and (for all of these aspects) the related concept of a "Swiss-cheese" universe.