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