The angular-diameter distance maximum and its redshift as constraints on Λ≠ 0 Friedmann-Lemaître-Robertson-Walker models

Abstract

The plethora of recent cosmologically relevant data has indicated that our Universe is very well fitted by a standard Friedmann-Lemaitre-Robertson-Walker (FLRW) model, with Ω M ≈ 0.27 and Ω Λ ≈ 0.73 - or, more generally, by nearly flat FLRW models with parameters close to these values. Additional independent cosmological information, particularly the maximum of the angular-diameter (observer area) distance and the redshift at which it occurs, would improve and confirm these results, once sufficient precise Type Ia supernovae data in the range 1.5 < z < 1.8 become available. We obtain characteristic FLRW-closed functional forms for C = C(z) and M 0 = M 0 (z), the angular-diameter distance and the density per source counted, respectively, when A ≠ 0, analogous to those we have for A = 0. More importantly, we verify that for flat FLRW models z max - as is already known but rarely recognized - the redshift of C max , the maximum of the angular-diameter distance, uniquely gives Ω Λ , the amount of vacuum energy in the universe, independent of H 0 , the Hubble parameter. For non-flat models, determination of both z max and C max gives both Ω Λ and Ω M , the amount of matter in the universe, as long as we know H 0 independently. Finally, determination of C max automatically gives a very simple observational criterion for whether or not the universe is flat - presuming that it is FLRW.