Distance‐Redshift in Inhomogeneous Ω0 = 1 Friedmann‐Lemaitre‐Robertson‐Walker Cosmology

Distance-redshift relations are given in terms of associated Legendre functions for partially filled beam observations in spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies. These models are dynamically pressure-free and flat FLRW on large scales but, because of mass inhomogeneities, differ in their optical properties. The partially filled beam area-redshift equation is a Lamé equation for arbitrary FLRW models and is shown to simplify to the associated Legendre equation for the spatially flat, i.e., Ω 0 = 1, case. We fit these new analytic Hubble curves to recent supernovae (SNe) data in an attempt to determine both the mass parameter Ω m and the beam-filling parameter ν. We find that current data are inadequate for limiting ν. However, we are able to estimate what limits are possible when the number of observed SNe is increased by factor of 10 or 100, sample sizes achievable in the near future with the proposed SuperNova Acceleration Probe satellite.