4 Summary

We have investigated the best-fit values of the homogeneity-parameter $\alpha$in the Dyer-Roeder distance-redshift relation. For a variety of inhomogeneity models, redshifts and cosmological parameter values, the relation between $\alpha$ and the fraction of compact objects, $f_{\rm p}$, is approximately linear and we can parametrize the relation by

$$1-\alpha =a\cdot f_{\rm p},$$

where $a\approx 0.6$. We expect the dispersion in the angular-diameter distance to be 5.9/h Mpc for z=0.5, 22/h Mpc for z=1 and 43/h Mpc for z=1.5 in doing this approximation. In the limit of no point-like compact objects ( $f_{\rm p}=0$), our results agree with those of Tomita (1998), namely that the best-fit value of $\alpha$ is close to one.

Acknowledgements

The author would like to thank Ariel Goobar for useful comments.