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

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 Omega(M) approximate to 0.27 and Omega(Lambda) approximate to 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) over cap (0) = (M) over cap (0)(z), the angular-diameter distance and the density per source counted, respectively, when Lambda not equal 0, analogous to those we have for Lambda = 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 Omega(Lambda), 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 Omega(Lambda) and Omega(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.