The Cosmological Constant
Sean M. Carroll, University of
Chicago
This is a short article I wrote for
the Encyclopedia of Astronomy and
Astrophysics (Institute of Physics). See also The Preposterous
Universe, or related reviews, lectures,
and talks.
Here is the postscript version,
and the pdf version.
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
close
to 0.3 and close
to 0.7. Confirming or disproving this possibility is one of the foremost
ambitions of contemporary cosmologists )......
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