Distance-Redshift in Inhomogeneous FLRW

We give distance–redshift relations in terms of elliptic integrals for three different mass distributions of the Friedmann-Lema^\i tre-Robertson-Walker (FLRW) cosmology. These models are dynamically pressure free FLRW on large scales but, due to mass inhomogeneities, differ in their optical properties. They are the filled-beam model (standard FLRW), the empty-beam model (no mass density exists in the observing beams) and the 2/3 filled-beam model. For special $\text{OM}$– $\text{OL}$ values the elliptic integrals reduce to more familiar functions. These new expressions for distance-redshift significantly reduce computer evaluation times.