Next Contents Previous


In this, the final section of our review, we must ask the following question: do we expect the present value of Lambda to be fundamental (= defined by the parameters of a physical theory) or accidental (= determined by initial conditions in the early Universe)? At present, we have no answer to this question. Models for Lambda considered in previous sections admit both possibilities. For instance, in the class of minimally-coupled lambda-field models with an inverse power-law potential, the present value of Lambda is fundamental (= defined by the parameters of V(phi) only) if "initially" (at the end of inflation or at a later moment when the lambda-field becomes a separate degree of freedom of matter) phi was sufficiently small, so that the corresponding solution for phi(t) had time to reach a future attractor (the "tracker" solution of [215, 188]) by the present epoch. On the other hand, if the initial value phiin is large, then the present value phi0 approx phiin, and the current value of the Lambda-term is accidental. Note that in the latter case Lambda is practically time independent now. Such a large value of phiin may, for instance, be generated during an early inflationary stage, in which case stochastic methods [182, 183, 196] may be used to derive probability distributions for phiin and Lambda. As a byproduct of such a mechanism, small quasi-static inhomogeneous perturbations of Lambda will also be generated. (15)

If Lambda is accidental, then a wide range of "explanations" for its currently (small) value can be given, based on the most reliable form of the anthropic principle - the weak anthropic principle. However, even if Lambda is fundamental and can be expressed through other microphysical constants, one may still try to use a more controversial form of this principle - the strong anthropic principle. (16)

An anthropic argument for Lambda > 0 has been suggested by Banks (1985) and Weinberg (1987), who felt that the extraordinary difference between likely values of the vacuum energy rhoLambda ~ rhom ~ 10-29 g/cm3 and the expected value (from a consideration of Planck scale physics) rhoP ~ 1093 g/cm3 could only be understood through anthropic arguments, since, in the absence of a fundamental symmetry which set the value of Lambda to precisely zero, it would be extremely fortuitous if particle physics determined a value for rhoLambda which was comparable to the matter density at this precise moment in the history of the universe. The case for the anthropic principle as a viable means for understanding properties of the universe has received a strong measure of support from recent developments in inflationary cosmology. A self-consistent treatment of quantum effects in inflationary models has shown that the entire universe may consist of an ensemble of sub-universes (separated from each other by particle horizons) having `all possible types of vacuum states and all possible types of compactification' of extra space-time dimensions [131]. According to this picture our observable universe is but one of an infinite number of universes each having its own set of conserved quantities and dimensions. Since in each sub-universe physical fields determining the value of Lambda have distinct values it is reasonable to expect that the value of Lambda varies from one sub-universe to another.

Weinberg (1987) showed that large values of Lambda were unlikely to be `observed' since the presence of observers demanded the existence of galaxies and galaxy formation was strongly suppressed if the energy in the cosmological constant greatly exceeded the matter density (also see [55, 198]). Martel, Shapiro & Weinberg (1998) have suggested that the probability that observers living in a given sub-universe will measure a value rhoLambda for the `vacuum energy' be given by the expression

Equation 125 (125)

where F(rhoLambda) is the fraction of matter in galaxies in a sub-universe with vacuum energy rhoLambda = Lambda / 8piG [137]. The value of F(rhoLambda) is calculated assuming Gaussian initial fluctuations at recombination, with a COBE-normalized cold dark matter spectrum with a cosmological constant (LambdaCDM). The requirement that the observed value OmegaLambda, * in our sub-universe equal the statistical mean or median evaluated over all sub-universes (i.e. OmegaLambda, * = <OmegaLambda>, where OmegaLambda = Lambda / 3H02) gives a value which peaks in the region OmegaLambda, * ~ 0.6 - 0.9 for a broad region of parameter space and assuming fairly reasonable conditions for galaxy formation [137]. Thus small observed values of OmegaLambda appear to be strongly disfavoured by the anthropic argument !

15 Previous discussions involving quantum cosmology also held the possibility that the value of Lambda is not determined uniquely. For instance Hawking (1984) showed that the wave function for the universe could contain a superposition of terms with different values for the cosmological constant. Investigating the effect of wormholes on quantum gravity, Coleman (1988a,b) subsequently showed that coupling constants whose values were not fixed by symmetries in the Lagrangian could take on all possible values in the superposition of terms describing the state vector in quantum cosmology. Back.

16 The weak anthropic principle in the narrow sense states that our location in space and time should be such that it admits the existence of intelligent life. An extension of this principle is that initial conditions allow the existence of such a region in space-time. On the other hand the strong anthropic principle states that laws of nature should permit the existence of intelligent life. It may be noted that the border between these two versions of the anthropic principle is not absolutely rigid. Namely, by generalizing a physical theory (say, the electroweak model) with fixed constants into a more general theory where these constants may have arbitrary values depending upon initial conditions, we make a step from the strong to the weak anthropic principle (also see [11]). Back.

Next Contents Previous