The Angular-Diameter-Distance-Maximum and Its Redshift as Constraints on $\Lambda \ne 0$ FLRW Models

Abstract

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 ${\Omega }_M\approx 0.27$ 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 $\Lambda \ne 0$, 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. 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.

17 Pages, 1 Figure

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