Distance-redshift relation in an isotropic inhomogeneous universe:   Spherically symmertic dust-shell universe
  The relation between angular diameter distance and redshift in a spherically symmetric dust-shell universe is studied. This model has large inhomogeneities of matter distribution on small scales. We have discovered that the relation agrees with that of an appropriate Friedmann-Lemaitre(FL) model if we set a ``homogeneous'' expansion law and a ``homogeneous'' averaged density field. This will support the averaging hypothesis that a universe looks like a FL model in spite of small-scale fluctuations of density field, if its averaged density field is homogeneous on large scales. 
Norimasa Sugiura, Ken-ichi Nakao and Tomohiro Harada