The Angular-Diameter-Distance-Maximum and Its Redshift as Constraints on $Λ\neq 0$ FLRW Models
The plethora of recent cosmologically relevant data has indicated that our universe is very well fit by a standard Friedmann-Lemaître-Robertson-Walker (FLRW) model, with $Ω_{M} \approx 0.27$ and $Ω_Λ \approx 0.73$ -- 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 Supernovae Ia data in the range $1.5 < z < 1.8$ become available. We obtain characteristic FLRW closed functional forms for $C = C(z)$ and $\hat{M}_0 = \hat{M}_0(z)$, the angular-diameter distance and the density per source counted, respectively, when $Λ\neq 0$, analogous to those we have for $Λ= 0$. 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, independently 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.