Annu. Rev. Astron. Astrophys. 1992. 30: 499-542 Copyright © 1992 by Annual Reviews. All rights reserved

4.4 Dynamical Tests of

Since a non-zero might be thought of as producing significant non-gravitational long range forces in the evolution of the universe, it is natural to hope that the large-scale dynamics of the material in the universe (i.e. large-scale galaxy clustering) might be sensitive to its value and thus provide some useful tests. Unfortunately, as Martel & Wasserman (1990), Martel (1991), and Lahav et al (1991) have shown in detail, the properties of present day structures and galaxy clusters are remarkably insensitive to ; it is doubtful that anything significant can be learned about the cosmological constant from their study.

On the other hand, if one considers not merely the present day clustering but also some information on its derivatives (time evolution), there is hope of some purchase on the issue. For example, Carlberg (1991) has shown that the expected rate of galaxy mergers increases much more rapidly with redshift (at z 1) for zero models than for -dominated ones, at least for conventional models of structure formation, and has interpreted some evidence for a high rate of galaxy mergers at moderate redshifts as evidence against a significant value of . However, since it may be reasonably doubted that galaxy mergers were ever a common process (Ostriker 1980) and since it is anything but clear how cosmic structure formed (Peebles & Silk 1990), it is probably more sensible to regard this test as an interesting idea for further investigation than as yet giving any clear result.

Similarly, Richstone et al (1992) have pointed out that the mean density (in absolute units or relative to the critical density) of just collapsing structures are expected to be somewhat lower in -dominated cosmologies (assuming only gravity-driven structure formation) than in conventional ones, because the increased age of the universe allows time for their slower dynamical evolution. Thus, if one could use the presence of unrelaxed substructure, galaxy populations, or some other indicator to identify just post-collapse clusters and could measure their mean cluster densities accurately enough, a test might be feasible. Again, available data and our current understanding of cluster evolution are still far from up to the task.

Recently, a non-zero term has been advocated (Efstathiou et al 1990, M.S. Turner 1991) as a means of saving the cold dark matter (CDM) model of structure formation (e.g. Davis et al 1985, Bardeen et al 1986) from the contrary discoveries of excess matter perturbations on large scales (e.g. Maddox et al 1990a, Geller & Huchra 1989). In CDM theory, there is a change of logarithmic slope in the perturbation spectrum, caused by suppression of the growth of perturbations that are smaller than the size of the horizon during the radiation-dominated era. The length scale of this break becomes larger if the epoch of matter dominance is made more recent. i.e. if _M is decreased. Therefore, for fixed (observed) normalization of the perturbations at the small-scale end, the amplitudes of large-scale matter perturbations increases as _M decreases. A value _M 0.2 is found to give best agreement with observation.

If = 0, such a value is incompatible not only with theoretical prejudices in favor of inflationary models with _tot = 1, but also directly with anisotropy measurements of the cosmic microwave background (Bond et al 1991, Vittorio et al 1991). One can see the problem in Figure 5, whose ordinate is proportional (by Equation 19) to the proper size of a scale that subtends a fixed angle : Models A, B, and D, with progressively decreasing _M and = 0, subtend respectively larger proper scales, which are therefore less correlated, implying increasing anisotropies. The sequence A, C, F, where _tot = 1 and decreasing _M is compensated by increasing , yields much smaller increases in scale, therefore smaller increases in anisotropy (Vittorio & Silk 1985, Kofman & Starobinskii 1985, Sugiyama et al 1990, Gorski et al 1992). A model with _M = 0.2, = 0.8 is claimed to be compatible with both observed large-scale structure and present microwave anisotropy limits. Whether this model can be confirmed or ruled out by other tests - e.g. gravitational lensing (see below) or the X-ray temperature distribution of clusters of galaxies (Lilje 1992) - is an important current question.

The patching up of CDM, by itself, can hardly be taken as firm evidence of a non-zero . CDM theory has been perhaps unjustifiably wed to the assumption of a single, constant bias factor b relating mass to light. Large-scale structure is no more direct evidence of a non-zero than it is evidence of a scale-dependent value of b. In fact, scale - and velocity-dependent biasing is seen in recent, as yet unpublished, numerical simulations by Carlberg (1991), and by Cen & Ostriker (1992); these simulations include hydrodynamical and radiative effects and attempt to calculate, rather than assume, biasing effects.