Annu. Rev. Astron. Astrophys. 1991. 29: 325-362 Copyright © 1991 by Annual Reviews. All rights reserved

6. THE COSMOLOGICAL CONSTANT

The source term for Einstein's equations is any conserved stress tensor. If this stress tensor is isotropic with strength (corresponding to an equation of state p = - c² = - , implying either pressure or density to be negative), then the quantity is called the cosmological constant. Such a term was originally postulated by Einstein and has an interesting history (see e.g. 124).

The value of such a constant is severely constrained by cosmological considerations. Since contributes = (8 G / 3H₀²) to the critical density, one can safely conclude (in spite of any astronomical uncertainties!) that

50.

As discussed below, this small value is a deep mystery.

To begin with, nothing prevents the existence of a -term (say, ₀) in Einstein's equations (as postulated by him). This will make the gravitational part of the Lagrangian dependent on two fundamental constants ₀ and G, which differ widely in scale; the dimensionless combination made out of these fundamental constants, (G / c³)₀, has a value less than 10^-126!

The surprise in the smallness of || is mainly due to the following: We do not know of any symmetry mechanism that requires it to be zero. In fact, we know of several independent, unrelated phenomena that contribute to . To produce such a small , these terms have to be fine-tuned to a bizarre accuracy.

Quantum field theory provides a wide variety of contributions to . For example, consider a scalar field with a potential V (). The particle physics predictions do not change (except possibly in theories with exact supersymmetry, a condition not realized in nature) if we add a constant term V₀ to this potential. A potential like

and

will give the same effects, though they differ by the constant term (µ⁴ / 4), but such a shift in the energy-density will contribute to . According to currently accepted scenarios, the value of this constant changes in every phase transition by E⁴ where E is the energy scale at which the phase transition occurs: at the GUTs transition, it is 10⁵⁶ (GeV)⁴; at the Salam-Weinberg transition, it changes by 10¹⁰ (GeV)⁴. These are enormous numbers compared to the present value of 10^-47 (GeV)⁴. How a physical quantity can change by such a large magnitude and finally adjust itself to be zero at such fantastic accuracy is not clear.

Finally, one should not forget that the ``zero-point energy'' of quantum fields will also contribute to gravity (91, 131). Each degree of freedom contributes an amount

51.

where k_max is an ultravioqlet cut-off. If we take general relativity to be valid up to Planck energies, then we may take k_max 10¹⁹ GeV and will be 10⁷⁶ (GeV)⁴.

If we assume that all the contributions are indeed there, then they have to be fine-tuned to cancel each other, for no good reason. Before the entry of GUTs into cosmology, we needed to worry only about the first and last contribution, both of which could be tackled in an ad hoc manner. One arbitrarily sets ₀ = 0 in the Lagrangian defining gravity and tries to remove the zero-point contribution by complicated regularization schemes. (Neither argument is completely water-tight but both seem plausible.) With the introduction of GUTs and inflationary scenarios however, the cosmological constant becomes a dynamical entity and the situation becomes more serious. Notice that it is precisely the large change in the V () that leads to a successful inflation; it has to be large to inflate the universe and change to a small value in the end for a graceful exit from the inflationary phase.

Several mechanisms have been suggested in the literature to make the cosmological constant zero: supersymmetry (26, 79, 80, 135), complicated dynamical mechanisms (37, 97, 110, F. Wilczek and A. Zee, unpublished), probabilistic arguments from quantum gravity (12, 24, 53, 89), and anthropic principle (123) are only a few of them. None of these seems to provide an entirely satisfactory solution. A somewhat different approach in which scale invariance was used to set = 0 was provided by the conformal theory of gravity (61).

The smallness of the cosmological constant is probably the most important single problem that needs to be settled in cosmology. We have no idea as to what this mechanism is, but if it is based on some general symmetry consideration, it may demand vanishing of at all epochs. This can wipe out the entire inflationary picture.