On the relationship between mean observations, spatial averages and the Dyer-Roeder approximation in Einstein-Straus models

TL;DR

This paper compares the Dyer-Roeder approximation and spatial averaging in Einstein-Straus models, showing their agreement in certain limits and highlighting conditions where the Dyer-Roeder approximation outperforms other methods in predicting redshift-distance relations.

Contribution

The study demonstrates the conditions under which the Dyer-Roeder approximation accurately predicts redshift-distance relations in Einstein-Straus models and introduces an expression for redshift drift in these models.

Findings

Dyer-Roeder approximation agrees with spatial averages in specific limits.

In some regimes, Dyer-Roeder provides a better approximation of the true mean.

Boundary terms and Sachs-Wolfe effects influence the accuracy of approximations.

Abstract

The redshift and redshift-distance relation in different Einstein-Straus models are considered. Specifically, the mean of these observables along 1000 light rays in different specific models are compared with predictions based on the Dyer-Roeder approximation and relations based on spatial averaging. It is shown that in certain limits, including those studied earlier in the literature, the Dyer-Roeder approximation and relations based on spatial averages agree with each other to a good precision regarding the redshift and redshift-distance relation and make good predictions of the mean of the exact relations. In limits where the two methods disagree, the Dyer-Roeder approximation clearly yields the better approximation of the true mean. This is explained by demonstrating the effect of boundary terms and integrated Sachs-Wolfe contributions but it is pointed out that the result seems to be valid for other Swiss-cheese models as well. Lastly, an expression for the redshift drift in Einstein-Straus models is presented and used for studying the behavior of this quantity in particular Einstein-Straus models.