Understand
The plethora of recent cosmologically relevant data has indicated that our universe is very well fit by a standard Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) model, with $\Omega_{M} \approx 0.27$ and $\Omega_{\Lambda} \approx 0.73$ -- 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 $\Lambda \neq 0$, 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, independently of $H_0$, the Hubble parameter.