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

5.2 Other Explanations

Although quantum cosmology has attracted significant attention recently, there are many other proposed alternative solutions to the cosmological constant problem. We briefly review several here, noting in advance that while many are provocative, none could be described as compelling. More details and references to many of these proposals may be found in Zee (1985) and Weinberg (1989).

A popular explanation for various unusual coincidences in physics is the anthropic principle, which holds that life (and scientists) will exist only if the laws of physics so allow; therefore, constants of nature must have friendly values. This argument has been applied to the cosmological constant by Banks (1985), Abbott (1985), Brown & Teitelboim (1987), and Linde (1989). An interesting consequence of this argument is that should not be zero, but only small enough for life to exist. Weinberg (1987, 1989) argues that this bound is very close to the observational limits. The possibility that is small for anthropic reasons is therefore of interest to astronomers. since they should then be able to detect a nonvanishing value.

Another suggestion which allows for a non-zero cosmological constant today is to let vary smoothly with time (Freese et al 1987, Özer & Taha 1987, Peebles & Ratra 1988, Ratra & Peebles 1988, Chen & Wu 1990, Abdel-Rahman 1990, Berman 1991, Fujii & Nishioka 1991). (Even in conventional theories, varies rapidly with time during cosmological phase transitions.) The extra degree of freedom introduced allows models to be constructed in which is appreciable, either today or in the early universe. Unfortunately, attempts at constructing a realistic field theory incorporating such features run into difficulty with cosmological nucleosynthesis and observations of cosmic background radiation (Freese et al 1987, Weinberg 1989).

Many authors have proposed the existence of a scalar field which serves to cancel out the cosmological constant (Dolgov 1982, Zee 1985, Ford 1987, Peccei et al 1987, Barr & Hochberg 1988, Sola 1989, Tomboulis 1990). A similar procedure has found theoretical, if not observational, success with the CP-violating parameter of QCD (Peccei & Quinn 1977, Weinberg 1978, Wilczek 1978). Unfortunately, none of these models has proven to be workable. Weinberg (1989) argues that there is a good reason for this failure: The condition that a scalar field relax the cosmological constant to zero will generally overdetermine the field equations, such that no solution can be found without fine-tuning. On the other hand, he notes that this argument relies on technical assumptions which may be in error.

It is natural to wonder, given that the cosmological constant problem involves the overlap of quantum theory with general relativity, whether a solution will eventually be provided by a true quantum theory of gravity. Although such a theory is not available at present, progress has been made in understanding the cosmological constant problem in the context of supersymmetry and superstring theory. In supersymmetry, every boson is associated with a fermion of equal mass. Both bosons and fermions contribute identically to the energy of the vacuum, as given in Equation 6; however, they contribute with opposite signs! Therefore, in the presence of supersymmetry the net contribution of quantum fluctuations to the vacuum energy is zero (Zumino 1975). Unfortunately, we do not observe supersymmetry in the real world - if it exists, it must be spontaneously broken, in which case the vacuum energies of the bosons and fermions will no longer cancel. Nevertheless, the startling cancellation has led many workers to search for supergravity or superstring theories in which the cosmological constant remains zero even after supersymmetry breaking (Christensen et al 1980, Cremmer et al 1983, Witten 1985, Dine et al 1985, Moore 1987, Siopsis 1989). It is probably safe to say that no firm conclusions can be drawn until the theories themselves are better understood.

Other proposals have been made. Linde (1988), in a precursor to the wormhole proposal, suggested a model in which two interacting universes contain particles with energies of opposite sign, in which the effective cosmological constant in each universe vanished. Pagels (1984) proposed a theory of gravitation in which the metric did not enter into the action. Many groups (Zee 1985, Buchmuller & Dragon 1989, Henneaux & Teitelboim 1989, Unruh 1989a) have noted that multiplies (-g)^1/2 in the action for general relativity; they therefore suggest changing gravity in a way which demands -g be fixed, so that becomes a Langrangian multiplier. Weinberg (1989) notes that this ``does not solve the cosmological constant problem, but it does change it in a suggestive way,'' while Ng & Dam (1990) maintain that, in the context of quantum cosmology, it does provide a solution. La (1991) has proposed an ``elastic vacuum theory,'' in which the vacuum energy oscillates rapidly but averages to zero. Taylor & Veneziano (1989) propose that quantum gravity corrections (not involving wormholes) can serve to rearrange the vacuum energy to produce a vanishing cosmological constant today. Finally, it has been argued that quantum fluctuations could destabilize a universe dominated by a cosmological constant, although there are many issues still to be resolved (Mottola 1986, Traschen & Hill 1986, Ford 1985, Isaacson & Rogers 1991).

The multiplicity of proposed solutions to the cosmological constant problem is telling - one correct solution would be enough. However, the search for a solution has led in some instances to increased understanding of the relationship between gravitation, field theory, and cosmology. While it is difficult to judge the relative likelihood that any of the above proposals will ultimately succeed, one can predict with confidence that the cosmological constant problem will continue to produce creative, and sometimes interesting, speculations.