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

In cosmology, one often assumes that the universe is homogeneous and isotropic. While originally a simplifying assumption, today there is observational evidence that this is a good approximation in our Universe on scales above a few hundred megaparsecs. This approximation is often used when calculating various distances as a function of redshift, even though the scales probed by a beam of light are much smaller than the scale of homogeneity. Since our Universe is obviously not homogeneous and isotropic on small scales, it is at least conceivable that this could affect distance calculation.

Two models have been proposed in order to take such small-scale inhomogeneities into account in a relatively simple way. One, due to Zel'dovich, involves a two-component universe where one component is smoothly distributed and the other in clumps, with the assumption that, when calculating distance from redshift, light propagates far from all clumps. Under those assumptions, one can derive a second-order differential equation for the distance. This is a simple ansatz but it is not obvious how valid it is. Another approach, originally due to Einstein and Straus but developed with regard to cosmological-distance calculation by Kantowski, involves removing material from a spherical region of an otherwise smooth universe and redistributing it inside this sphere (e.g. as a point mass at the centre, as a shell at the boundary, or in a more complicated manner). This ansatz is more difficult for calculations, but is an exact solution of the Einstein equations, so there is no question about its validity (how realistic such a mass distribution is as a model of our Universe is a separate question). Long after both had been investigated in detail, Fleury showed that they are equivalent at a well controlled level of approximation.

After a review of the history of those two approaches, I present my own work in this area:~an efficient numerical implementation for the solution of the most general form of the differential equation (i.e. arbitrary values of λ₀, Ω₀, and the homogeneity parameter η, the last indicating the fraction of matter distributed smoothly), a discussion of the uncertainty in distance calculation due to uncertainty in the value of η, the effect of η on the calculation of H₀ from gravitational-lens time delays, the effect of η on the separation between images in a gravitational-lens system, and the effect of η on the determination of λ₀ and Ω₀ from the magnitude—redshift relation for type Ia supernovae—including evidence that observations indicate that, in our Universe, the standard distance is a good approximation, even though small-scale inhomogeneities can be appreciable, probably because the Zel'dovich model does not accurately describe our Universe.