The differential equation governing the distance redshift for partially filled-beam optics in pressure-free Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology is shown to be the Lamé equation. The distance redshift $D\left(z\right)\mathrm{}$ discussed is appropriate for observations in inhomogeneous cosmologies for which lensing by masses external to the observing beam is negligible and for which lensing by transparent matter within the beam can be approximated by a homogeneous mass density expanding with the FLRW background. Some solutions of the derived Lamé equation are given in terms of Weierstrass elliptic integrals. A new simplified and useful expression for filled-beam $D\left(z\right)\mathrm{}$ in standard flat FLRW is also given.

• Received 20 August 2003

DOI:https://doi.org/10.1103/PhysRevD.68.123516