3.1 Spacetime Geometry

In the relativistic Friedmann-Lemaître cosmological model the mean spacetime geometry (ignoring curvature fluctuations produced by local mass concentrations in galaxies and systems of galaxies) may be represented by the line element

(12)

where the expansion rate satisfies the equation

(13)

which might be approximated as

(14)

The last equation defines the fractional contributions to the square of the present Hubble parameter H₀ by matter, space curvature, and the cosmological constant (or a term in the stress-energy tensor that acts like one). The time-dependence assumes pressureless matter and constant . Other notations are in the literature; a common practice in the particle physics community to add the matter and terms in a new density parameter, ' = + . I prefer keeping them separate, because the observational signatures of and can be quite different.

By 1930 people understood how one would test the space-time geometry in these equations, and as I mentioned there is at last direct evidence for the detection of one of the effects, the curvature of the relation between redshift and apparent magnitude ([1], [2]). As indicated in line 1b, the measured curvature is inconsistent with the Einstein-de Sitter model in which = 1 and = 0 = . The measurements also disagree with a low density model with = 0, though the size of the discrepancy approaches the size of the error flags, so I assign a weaker failing grade for this case. The measurements are magnificent. The issue yet to be thoroughly debated is whether the type Ia supernovae observed at redshifts 0.5 z 1 are drawn from essentially the same population as the nearer ones.

In a previous volume in this series Krauss [28] discusses the time-scale issue. Stellar evolution ages and radioactive decay ages do not rule out the Einstein-de Sitter model, within the still considerable uncertainties in the measurements, but the longer expansion time scales of the low models certainly relieve the problem of interpretation of the measurements. Thus I enter a tentative negative grade for the Einstein-de Sitter model in line 1a.

In the analysis by Falco et al. [29] of the rate of lensing of quasars by foreground galaxies (line 1d) for a combined sample of lensing events detected in the optical and radio, the 2 bound on the density parameter in a cosmologically flat ( = 0) universe is > 0.38. The SNeIa redshift-magnitude relation seems best fit by = 0.25, = 0.75, a possibly significant discrepancy. A serious uncertainty in the analysis of the lensing rate is the number density of early-type galaxies in the high surface density branch of the fundamental plane at luminosities L ~ L_*, the luminosity of the Milky Way. If further tests confirm an inconsistency of the lensing rate and the redshift-magnitude relation the lesson may be that is dynamical, rolling to zero, as Ratra & Quillen [30] point out.