Annu. Rev. Astron. Astrophys. 1992. 30: 311-358 Copyright © 1992 by Annual Reviews. All rights reserved

4.2 Deceleration Parameter

If a lensing potential is ever specified to better than a few per cent accuracy, it will be possible to infer the distance ratio D_ds / D_s observationally and consequently to solve for q₀ (Refsdal 1966a, Lacroix & Schneider 1982). It is also possible, in principle, to measure q₀ by comparing the locations and magnifications of arc images of galaxies at different redshifts. In the simple case of a circularly symmetric mass distribution, the arcs will trace the variation of the Einstein radius with source redshift. Unfortunately, for an optimal lens redshift of z_d ~ 0.6, the Einstein radius only varies by about 12% as q₀ increases from 0 to 0.5. This is much less than the expected uncertainty in the radial variation of the mass distribution in the cluster even if a large number of galaxy velocities are acquired (Tyson 1990). Under extreme conditions, gravitational lensing may complicate alternative determinations of q₀ (e.g. Omote & Yoshida 1990).

Undoubtedly, the best test of determinations of H₀ and q₀ through gravitational lensing will be to obtain similar estimates of these parameters from observations of different objects.

4.3 Alternative Cosmologies

Gravitational lenses have certainly affirmed our belief in the redshift distance relation. High redshift sources do appear to lie beyond lower redshift lenses, even when the a posteriori probability of small angular separations occurring is small (Burbidge 1985). Also, the approximate consistency of the Hubble constant derived from Q0957+561 with values from more traditional approaches affirms the FRW model and is incompatible with radical alternatives like the chronometric cosmology (Segal 1982) which predicts t 1 month (Blandford & Falco 1989, unpublished).

Even in the context of FRW models, some extreme effects are, in principle, possible and certainly worth seeking. If the mean density of the universe were so large (or the cosmological constant, ₀, so negative) that a typical backward propagating congruence had conjugate points or foci (beyond which all images are inverted, cf Section 3.3) at a modest antipodal redshift, the character of the imaging of higher redshift sources would be radically altered (Petrosian & Salpeter 1968, Lebedev & Lebedeva 1985, Gott et al. 1989). For instance, intervening galaxy lenses would typically form dim, single images. The existence of multiple images in Q2016+112 may then be used to set a lower bound of z = 3.27 on the antipodal redshift (Gott et al. 1989). This argument is, however, somewhat problematical in the case of this object. There are at least two lensing galaxies (probably at different redshifts) in the line-of-sight to Q2016+112 and there is no robust imaging model of the object within the framework of a conventional cosmology (but see Narasimha et al. 1987). An antipodal model that invokes two additional unseen galaxies near images A and B may be able to reproduce the observations. The argument of Gott et al. is certainly valid in the case of Q0142-100 which, at a redshift of z = 2.72, sets a reliable lower bound on the antipodal redshift.

Gravitational lenses can also be used to limit the value of a cosmological constant (Fukugita et al. 1990; Turner 1990; Fukugita & Turner 1991; Kochanek 1992). The choice of ₀ = ₀ / 3H₀² = 1 - ₀, which allows the universe to remain flat and consistent with inflationary cosmological models, is particularly interesting (Peebles 1984). In this case, the optical depth to lensing increases more dramatically with redshift than in an Einstein-De Sitter universe. We defer to the accompanying article by Carroll et al. (1992) for a critical discussion of this point.