3.3. Probes of Spacetime Geometry
If spacetime is close to homogeneous and isotropic and described by a single line element then the geometry is represented by two functions of redshift: r (z) fixes the angle subtended by an object of given linear size at redshift z, and dt / dz fixes proper world time as a function of redshift. The latter determines H0 t0, the former the z-m relation. In the measurements of the z-m relation by Perlmutter et al. (1998a, b) and Reiss et al. (1998) the cosmologically flat Friedmann-Lemaître model with m = 0.25 and = 0.75 is within one standard deviation, the open model with = 0 and m = 0.25 is about 3 off, and the Einstein-de Sitter model is some 7 off. I explained in Section 2.2 why I suspect the case against the open low density model is serious but maybe premature: we should await further consideration of these new measurements by the authors and the community. The Einstein-de Sitter model would require a more substantial reconsideration, so it gets a more serious demerit in line 1c in Table 2.
Both functions, r (z) and dt / dz, enter galaxy counts and the rate of lensing of quasars by the gravitational deflection of the masses in foreground galaxies. The importance of the latter was demonstrated by Fukugita et al. (1990) and Turner (1990). The analysis by Falco et al. (1998) indicates that, in a cosmologically flat model, m 0.38 at 2. An open low density model does better: m = 0.25 is at the 2 contour.
This constraint from lensing depends on the galaxy mass function. The predicted peak of the lensing rate at angular separation ~ 1 arc sec is dominated by the high surface density branch of early-type galaxies at luminosities L ~ L* (where the galaxy mass function is approximated as dn / dL L e-L / L*, with ~ -1). The number density of these objects is not well known, because it is difficult to separate counts of early-type galaxies in the high surface density branch from a low density branch that is likely to be irrelevant for lensing (Kormendy 1987). Masataka Fukugita and I have been unable to find a reliable way around this ambiguity using available surveys.
If further tests of the lensing and redshift-magnitude constraints confirmed the apparent inconsistency in entries 1c and 1d the lesson could be that the cosmological constant is dynamical, rolling to zero, as Ratra & Quillen (1992) point out.
I keep a line in Table 2 for counts because galaxies are observed at redshifts greater than unity, where the predicted counts are quite sensitive to the cosmological parameters. The counts are quite sensitive to galaxy evolution, too, but people may learn how to deal with that as the understanding of galaxy evolution improves.