This is the draft of an article I have written for the Encyclopedia of Astronomy and Astrophysics (Institute of Physics, forthcoming). You can also several related articles, as well as some talks I have given.
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.
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 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.
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.
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.
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
