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 , flat FLRW on large scales but , due to mass inhomogeneities , differ in their optical properties .
The partially filled beam area-redshift equation is a Lame ^ { \prime } equation for arbitrary FLRW and is shown to simplify to the associated Legendre equation for the spatially flat , i.e. , \Omega _ { 0 } = 1 case .
We fit these new analytic Hubble curves to recent supernovae ( SNe ) data in an attempt to determine both the mass parameter \Omega _ { m } and the beam filling parameter \nu .
We find that current data are inadequate to limit \nu .
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 .