The Lame${}^{\mathrm{\prime }}$ Equation for Distance-Redshift in Partially Filled Beam Friedmann-Lema\^\i tre-Robertson-Walker Cosmology

The differential equation governing distance-redshift for partially filled-beam optics in pressure-freeFriedmann-Lema^\i tre-Robertson-Walker Cosmology (FLRW) is shown to be the Lame${}^{\mathrm{\prime }}$ equation. The distance-redshift, $D(z)$, discussed is appropriate for observations in inhomogeneous cosmologies for which lensing by masses external to the observing beam are 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 Lame${}^{\mathrm{\prime }}$ equation are given in terms of Weierstrass elliptic integrals. A new simplified and useful expression for filled-beam $D(z)$ in standard flat FLRW is also given.