Weak lensing and the Dyer-Roeder approximation

The distance-redshift relation plays an important role in cosmology. In the standard approach to cosmology, it is assumed that this relation is the same as in a homogeneous universe. As the real Universe is not homogeneous, there are several methods used to calculate the correction. The weak-lensing approximation and the Dyer-Roeder relation are among them. This paper establishes a link between these two approximations. It is shown that if the Universe is homogeneous with only small density fluctuations along the line of sight that vanish after averaging, then the distance correction is negligible. It is also shown that a vanishing three-dimensional average of density fluctuations does not imply that the mean of density fluctuations along the line of sight is zero. In this case, even within the linear approximation, the distance correction is not negligible. A modified version of the Dyer-Roeder relation is presented and it is shown that this modified relation is consistent with the correction obtained within the weak-lensing approximation. The correction to the distance for a source at z similar to 2 is of the order of a few per cent. Thus, with the increasing precision of cosmological observations, an accurate estimation of the distance is essential. Otherwise errors due to miscalculation of the distance can become a major source of systematics.