Sean M. Carroll, University of Chicago
This is a short article I wrote for the Encyclopedia of Astronomy and Astrophysics (Institute of Physics) (PDF). See also The Preposterous Universe.
Cosmological Constant
The cosmological constant, conventionally denoted by the Greek letter 
, is a parameter describing the energy density of the vacuum (empty space), and a potentially important contributor to the dynamical history of the universe. Unlike ordinary matter, which can clump together or disperse as it evolves, the energy density in a cosmological constant is a property of spacetime itself, and under ordinary circumstances is the same everywhere. A sufficiently large cosmological constant will cause galaxies to appear to accelerate away from us, in contrast to the tendency of ordinary forms of energy to slow down the recession of distant objects. The value of in our present universe is not known, and may be zero, although there is some evidence for a nonzero value; a precise determination of this number will be one of the primary goals of observational cosmology in the near future.
The Cosmological Constant and Vacuum Energy
We live in an expanding universe: distant galaxies are moving away from us, such that the more distant ones are receding faster. Cosmologists describe this expansion by defining a SCALE FACTOR R(t), which specifies the relative distance of galaxies as a function of time. (If two galaxies are twice as far away at time 
as they were at time 
, we have 
.) The behavior of the scale factor is governed by the curvature of space (which can be positive, negative, or zero) and the average energy density of the universe (which is thought to be positive, although we should be open to exotic possibilities).
Imagine taking a region of space and removing from it all of the matter, radiation, and other substances we could conceivably remove. The resulting state is referred to as the “vacuum” — a somewhat stricter use of the word than that applied to the space in between planets and stars, which is actually filled with trace amounts of matter and radiation. The vacuum has the lowest energy of any state, but there is no reason in principle for that energy to be zero. In the absence of gravity there is no way of measuring energy on an absolute scale; the best we can do is to compare the relative energies of two different states. The vacuum energy is then arbitrary, unobservable. In GENERAL RELATIVITY, however, any form of energy affects the gravitational field, so the vacuum energy becomes a potentially crucial ingredient. To a good approximation (see below), we believe that the vacuum is the same everywhere in the universe, so the vacuum energy density is a universal number which we call the cosmological constant. (More precisely, the conventionally defined cosmological constant is proportional to the vacuum energy density 
; they are related by 
, where G is Newton’s constant of gravitation and c is the speed of light.)
The Cosmological Constant in Cosmology
The scale factor R(t), spatial curvature, and energy density of the universe are related by the FRIEDMANN EQUATION, which says that a positive energy density contributes positively to the curvature, while expansion contributes negatively. For simplicity, consider a flat universe — zero spatial curvature — so that the energy density and expansion are in perfect balance. As the universe expands, the matter within it becomes increasingly rarefied, so the energy density in matter diminishes. If matter is the dominant component of the energy, the expansion rate (as measured by the HUBBLE CONSTANT) will correspondingly decrease; if on the other hand the cosmological constant dominates, the energy density will be constant, and the expansion rate will attain a constant value. In a potentially confusing but nevertheless appropriate piece of nomenclature, a universe with a constant expansion rate is said to be “accelerating”. This is because, while the amount of expansion undergone in any one second by a typical cubic centimeter in such a universe is a constant, the number of centimeters between us and a distant galaxy will be increasing with time; such a galaxy will therefore be seen to have an apparent recession velocity that grows ever larger.
In a universe with both matter and vacuum energy, there is a competition between the tendency of to cause acceleration and the tendency of matter to cause deceleration, with the ultimate fate of the universe depending on the precise amounts of each component. This continues to be true in the presence of spatial curvature, and with a nonzero cosmological constant it is no longer true that negatively curved (“open”) universes expand indefinitely while positively curved (“closed”) universes will necessarily recollapse — each of the four combinations of negative/positive curvature and eternal expansion/eventual recollapse become possible for appropriate values of the parameters. There can even be a delicate balance, in which the competition between matter and vacuum energy is a draw and the universe is static (not expanding). The search for such a solution was Einstein’s original motivation for introducing the cosmological constant, as the data at the time did not indicate an expanding universe, but his solution was both unstable to small perturbations and unnecessary once HUBBLE‘S LAW was discovered.
The average energy density in the universe 
is often expressed in terms of the DENSITY PARAMETER 
, defined by 
, where H is the Hubble constant. The density parameter is directly related to the spatial curvature; space is negatively curved for 
, flat for 
, and positively curved for 
. We may decompose the density parameter into a sum of contributions from different sources of energy; we therefore speak of the density parameter for matter, 
, for the cosmological constant, 
, and so on. The figure indicates the spatial curvature and future history of expanding universes as a function of and , under the plausible (but by no means necessary) assumption that matter and vacuum energy are the only dynamically significant forms of energy in the universe today.
Figure: Geometry and evolution of universes with different amounts of matter and vacuum energy, as parameterized by the density parameters and . The diagonal line 
represents spatially flat universes. The circle centered on 
, 
represents very roughly the region favored by current observations of distant supernovae, the cosmic microwave background, and the dynamics of galaxies.
Note that a nonzero of the same order of magnitude as is in a sense quite unnatural, as the relative abundance of matter and vacuum energy changes rapidly as the universe expands. Indeed, since the energy density in matter decreases as 
while that in vacuum remains constant, we have 
. To have approximate equality between these two numbers at the present era would thus come as a great surprise, since the situation in the very early or very late universe would be much different.
Observational Prospects
The existence of a nonzero vacuum energy would, in principle, have an effect on gravitational physics on all scales; for example, it would alter the value of the precession of the orbit of Mercury. In practice, however, such effects accumulate over large distances, which makes cosmology by far the best venue for searching for a nonzero cosmological constant. Most of these effects depend not just on the vacuum energy but on the matter energy density as well, so a number of independent tests are necessary to pin down and separately.
There is insufficient space available to do justice to all of the ways in which we can constrain , and the reader is encouraged to consult the references. A paradigmatic example is provided by the statistics of GRAVITATIONAL LENSING. A positive cosmological constant increases the volume of space in between us and a source at any fixed redshift, and therefore the probability that such a source undergoes lensing by an intervening object. Limits on the frequency with which such lensing occurs can therefore put an upper limit on ; current data suggest that cannot be too close to 1, although upcoming surveys will provide much better data. A relatively new method for constraining various cosmological parameters, including , is the analysis of temperature anisotropies in the COSMIC MICROWAVE BACKGROUND. Such anisotropies have a distinctive power on any given angular scale which can be predicted, in any specified theory of structure formation, as a function of these parameters. Observations to date have provided some preliminary evidence in favor of an approximately flat universe, 
, if currently favored theories based on adiabatic scale-free primordial perturbations are correct. (Most versions of the INFLATIONARY UNIVERSE scenario robustly predict that 
is extremely close to 1.) Coupled with dynamical tests, which consistently indicate that 
, this can be construed as evidence in favor of a nonzero cosmological constant; once again, however, these conclusions are tentative, and will soon be superseded by a new generation of more precise data.
Perhaps the most direct way of measuring the cosmological constant is to determine the relationship between redshifts and distances of faraway galaxies, known as the HUBBLE DIAGRAM. Nearby galaxies have redshifts which are proportional to their distances (Hubble’s Law), but galaxies further away are expected to deviate slightly from this strict proportionality in a way which depends on both and . Measuring the distances to cosmological objects is notoriously difficult, but important progress has recently been made by using Type Ia SUPERNOVAE as standard candles. (In fact it is not necessary to get absolute distances, but only the relative distances to supernovae at different redshifts.) Supernovae are rare, but the number of distant galaxies is very large, and two independent groups have discovered dozens of high-redshift supernovae (as of late 1998) by carefully observing deep into small patches of the sky. The results of these studies thus far can be approximately expressed as 
; it must be stressed, however, that our understanding of the physics underlying supernova explosions and the environments in which they occur is very incomplete at this stage. Nevertheless, there is an impressive consistency between this result and those of the microwave background observations and dynamical measurements of the mass density, with agreement achieved for a universe with close to 0.3 and close to 0.7. Confirming or disproving this possibility is one of the foremost ambitions of contemporary cosmologists.
Physics of the Cosmological Constant
The value of the cosmological constant is an empirical issue which will ultimately be settled by observation; meanwhile, physicists would like to develop an understanding of why the energy density of the vacuum has this value, whether it is zero or not. There are many effects which contribute to the total vacuum energy, including the potential energy of scalar fields and the energy in “vacuum fluctuations” as predicted by quantum mechanics, as well as any fundamental cosmological constant. Furthermore, many of these contributions can change with time during a phase transition; for example, we believe that the vacuum energy decreased by approximately 
kg m
during the electroweak phase transition. (A change in the effective cosmological constant during a phase transition is a crucial ingredient in the inflationary universe scenario, which posits an exponential expansion in the very early universe driven by a large vacuum energy.)
From this point of view it is very surprising that the vacuum energy today, even if it is nonzero, is as small as the current limits imply (
kg m). Either the various contributions, large in magnitude but different in sign, delicately cancel to yield an extraordinarily small final result, or our understanding of how gravitation interacts with these sources of vacuum energy is dramatically incomplete. A great deal of effort has gone into finding ways in which all of the contributions may cancel, but it is unclear what would be special about the value 
; a vanishing vacuum energy could be demanded by a symmetry principle such as conformal invariance or supersymmetry, but unbroken symmetries of this type are incompatible with what we know of the other forces of nature. (One suggestion is to invoke the “anthropic principle”, which imagines that the constants of nature take on very different values in different regions of the universe, and intelligent observers only appear in those regions hospitable to the development of life. It is unclear, however, whether different regions of the universe really do have different fundamental constants, or what values of the cosmological constant are compatible with the existence of intelligent life.) The alternative, that our understanding of the principles underlying the calculation of the cosmological constant is insufficient (and must presumably await the construction of a complete theory of quantum gravity), is certainly plausible, although the vacuum energy manifests itself in a low-energy regime where it would have been reasonable to expect semiclassical reasoning to suffice. Understanding the smallness of the cosmological constant is a primary goal of string theory and other approaches to quantum gravity.
If the recent observational suggestions of a nonzero are confirmed, we will be faced with the additional task of inventing a theory which sets the vacuum energy to a very small value without setting it precisely to zero. In this case we may distinguish between a “true” vacuum, which would be the state of lowest possible energy which simply happens to be nonzero, and a “false” vacuum, which would be a metastable state different from the actual state of lowest energy (which might well have ). Such a state could eventually decay into the true vacuum, although its lifetime could be much larger than the current age of the universe. A final possibility is that the vacuum energy is changing with time — a dynamical cosmological “constant”. This alternative, which has been dubbed “quintessence”, would also be compatible with a true vacuum energy which was ultimately zero, although it appears to require a certain amount of fine-tuning to make it work. No matter which of these possibilities, if any, is true, the ramifications of an accelerating universe for fundamental physics would be truly profound.
Bibliography
Popular Expositions:
Goldsmith, D 1997 Einstein’s Greatest Blunder? (Cambridge: Harvard University Press)
Technical Reviews:
Carroll S M, Press W H, and Turner E L 1992 The cosmological constant Annu. Rev. Astron. Astrophys. 30 499
Carroll S M 2000 The cosmological constant Living Reviews in Relativity in press
Weinberg S 1989 The cosmological constant problem Rev. Mod. Phys. 61 1
Web pages for supernova groups:
Supernova Cosmology Project http://www-supernova.lbl.gov/
High-Z Supernova Search Team http://cfa-www.harvard.edu/cfa/oir/Research/supernova/HighZ.html
Sean Carroll
Mon Dec 14 16:40:19 PST 1998