Understanding the Dyer-Roeder approximation as a consequence of local cancellations of projected shear and expansion rate fluctuations

TL;DR

This paper demonstrates that the Dyer-Roeder approximation holds when local cancellations occur between shear and expansion rate fluctuations along light rays, but can fail in models lacking such cancellations, affecting redshift-distance predictions.

Contribution

It clarifies the conditions under which the Dyer-Roeder approximation is valid, linking it to local shear-expansion cancellations in various cosmological models.

Findings

Dyer-Roeder approximation valid with local shear-expansion cancellations

Invalid in models without local cancellations, requiring alternative averaging methods

Applicable to perturbed FLRW, N-body, and Swiss-cheese cosmologies

Abstract

It is shown with several concrete examples that the Dyer-Roeder approximation is valid in spacetimes which fulfill the condition that fluctuations in the expansion rate along a light ray locally cancels with the shear contribution to the redshift. This is the case for standard cosmological scenarios including perturbed FLRW spacetimes, N-body simulations and Swiss-cheese models. With another concrete example it is then illustrated that it is possible to construct statistically homogeneous and effectively statistically isotropic cosmological models which do not fulfill the condition. In this case, the Dyer-Roeder approximation is invalid. Instead, the mean redshift-distance relation can be described using a relation based on the spatial averages of the transparent part of the spacetime.