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

TL;DR

This paper explores how the maximum of the angular-diameter distance and its redshift can serve as independent constraints on cosmological parameters, especially the cosmological constant, in FLRW models, enhancing our understanding of the universe's composition.

Contribution

It derives explicit FLRW functional forms for distance measures with non-zero cosmological constant and shows how maximum distance data can determine key cosmological parameters independently.

Findings

The redshift of the maximum angular-diameter distance uniquely determines the cosmological constant in flat models.

For non-flat models, both the maximum distance and its redshift determine matter and vacuum energy densities.

The maximum distance provides a simple observational criterion for universe flatness.

Abstract

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. 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.