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Cosmological Constant

The cosmological constant, conventionally de-

noted by the Greek letter , is a parame-ter describing the energy density of the vac-uum (empty space), and a potentially impor-tant contributor to the dynamical history of theuniverse. Unlike ordinary matter, which canclump together or disperse as it evolves, theenergy density in a cosmological constant is aproperty of spacetime itself, and under ordinarycircumstances is the same everywhere. A suf-ficiently large cosmological constant will cause

galaxies to appear to accelerate away from us,in contrast to the tendency of ordinary formsof energy to slow down the recession of distantobjects. The value of in our present universeis not known, and may be zero, although thereis some evidence for a nonzero value; a precisedetermination of this number will be one of theprimary goals of observational cosmology in thenear future.

The Cosmological Constant andVacuum Energy

We live in an expanding universe: distantgalaxies are moving away from us, such thatthe more distant ones are receding faster. Cos-mologists describe this expansion by defininga scale factor R(t), which specifies the relativedistance of galaxies as a function of time: whenthe value of the scale factor doubles, the dis-tance between any two galaxies doubles. The

behavior of the scale factor is governed by thecurvature of space (which can be positive, neg-ative, or zero) and the average energy densityof the universe (which is thought to be positive,although we should be open to exotic possibil-ities).

Imagine taking a region of space and remov-ing from it all of the matter, radiation, and

other substances we could conceivably remove.The resulting state is referred to as the vac-uum a somewhat stricter use of the word

than that applied to the space in between plan-ets and stars, which is actually occupied bytrace amounts of matter and radiation. Thevacuum has the lowest energy of any state, butthere is no reason in principle for that energyto be zero. In the absence of gravity thereis no way of measuring energy on an absolutescale; the best we can do is to compare the rel-ative energies of two different states. The vac-uum energy is then arbitrary, unobservable. In

the general theory of relativity, how-ever, any form of energy affects the gravita-tional field, so the vacuum energy becomes apotentially crucial ingredient. To a good ap-proximation (see below), we believe that thevacuum is the same everywhere in the universe,so the vacuum energy density is a universalnumber which we call the cosmological con-stant. (More precisely, the conventionally de-fined cosmological constant is proportional to

the vacuum energy density

; they are relatedby = (8G/3c2), where G is Newtons con-stant of gravitation and c is the speed of light.It was not until years after Einstein introduced as a parameter in cosmology that it was re-alized that the same parameter measured theenergy density of the vacuum.)

The Cosmological Constant in

Cosmology

The scale factor R(t), spatial curvature, and en-ergy density of the universe are related by theFriedmann equation, which says that a positiveenergy density contributes positively to the cur-vature, while expansion contributes negatively.For simplicity, consider a flat universe zerospatial curvature so that the energy densityand expansion are in perfect balance. (If the

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universe starts out with zero spatial curvature,it will remain that way throughout its evolu-tion.) As the universe expands, the matter

within it becomes increasingly rarefied, so theenergy density in matter diminishes. If mat-ter is the dominant component of the energy,the expansion rate (as measured by the Hub-ble constant) will correspondingly decrease;if on the other hand the cosmological constantdominates, the energy density will be constant,and the expansion rate will attain a constantvalue. In a potentially confusing but neverthe-less appropriate piece of nomenclature, a uni-

verse with a constant expansion rate is said tobe accelerating. This is because, while theamount of expansion undergone in any one sec-ond by a typical cubic centimeter in such a uni-verse is a constant, the number of centimetersbetween us and a distant galaxy will be increas-ing with time; such a galaxy will therefore beseen to have an apparent recession velocity thatgrows ever larger.

In a universe with both matter and vac-

uum energy, there is a competition betweenthe tendency of to cause acceleration andthe tendency of matter to cause deceleration,with the ultimate fate of the universe depend-ing on the precise amounts of each component.This continues to be true in the presence of spa-tial curvature, and with a nonzero cosmologi-cal constant it is no longer true that negativelycurved (open) universes expand indefinitelywhile positively curved (closed) universes willnecessarily recollapse each of the four combi-

nations of negative/positive curvature and eter-nal expansion/eventual recollapse become pos-sible for appropriate values of the parameters.There can even be a delicate balance, in whichthe competition between matter and vacuumenergy is a draw and the universe is static (notexpanding). The search for such a solution wasEinsteins original motivation for introducing

the cosmological constant, as the data at thetime did not indicate an expanding universe,but his solution depended on careful fine-tuning

and became unnecessary once Hubbles Lawwas discovered. Since that time, astrophysicistshave occasionally invoked a nonzero cosmolog-ical constant in order to explain puzzling ob-servations; in the 1960s there was an apparentexcess in the number of quasars at a redshift ofz 1.95, and more recently there has been dis-agreement between the ages of the oldest starsand that of the universe as inferred from itsexpansion rate. Subsequently, however, these

observations have either not held up to closerscrutiny or have been explained by more con-ventional means.

The average energy density in the universe is often expressed in terms of the densityparameter , defined by = (8G/3H2c2),where H is the Hubble constant. The densityparameter is directly related to the spatial cur-vature; space is negatively curved for < 1,flat for = 1, and positively curved for > 1.

We may decompose the density parameter intoa sum of contributions from different sources ofenergy; we therefore speak of the density pa-rameter for matter, M, for the cosmologicalconstant, , and so on. The figure indicatesthe spatial curvature and future history of ex-panding universes as a function of M and ,under the plausible (but by no means neces-sary) assumption that matter and vacuum en-ergy are the only dynamically significant formsof energy in the universe today.

Note that a nonzero of the same orderof magnitude as M is in a sense quite unnat-ural, as the relative abundance of matter andvacuum energy changes rapidly as the universeexpands. Indeed, since the energy density inmatter decreases as R3 while that in vacuumremains constant, we have /M R

3. Tohave approximate equality between these two

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0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

Figure 1: Geometry and evolution of universeswith different amounts of matter and vacuumenergy, as parameterized by the density param-eters M and . The diagonal line M+ =1 represents spatially flat universes. The cir-cle centered on M = 0.3, = 0.7 repre-

sents very roughly the region favored by currentobservations of distant supernovae, the cosmicmicrowave background, and the dynamics ofgalaxies.

numbers at the present era would thus comeas a great surprise, since the situation in thevery early or very late universe would be much

different.

Observational Prospects

The existence of a nonzero vacuum energywould, in principle, have an effect on gravi-tational physics on all scales; for example, itwould alter the value of the precession of theorbit of Mercury. In practice, however, sucheffects accumulate over large distances, which

makes cosmology by far the best venue forsearching for a nonzero cosmological constant.Most of these effects depend not just on the vac-uum energy but on the matter energy densityas well, so a number of independent tests arenecessary to pin down and M separately.

There is insufficient space available to dojustice to all of the ways in which we can con-strain , and the reader is encouraged to con-sult the references. A paradigmatic example isprovided by the statistics of gravitationallensing. A positive cosmological constant in-creases the volume of space in between us anda source at any fixed redshift, and thereforethe probability that such a source undergoeslensing by an intervening object. Limits onthe frequency with which such lensing occurscan therefore put an upper limit on ; cur-rent data suggest that cannot be too closeto 1, although upcoming surveys will providemuch better data. A relatively new method for

constraining various cosmological parameters,including , is the analysis of temperatureanisotropies in the cosmic microwave back-ground. Such anisotropies have a distinctivepower on any given angular scale which can bepredicted, in any specified theory of structureformation, as a function of these parameters.Observations to date have provided some pre-

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liminary evidence in favor of an approximatelyflat universe, + M 1, if currently favoredtheories based on adiabatic scale-free primor-

dial perturbations are correct. (Most versionsof the inflationary universe scenario ro-bustly predict that + M is extremely closeto 1.) Coupled with dynamical tests, whichconsistently indicate that 0.1 M 0.4,this can be construed as evidence in favor ofa nonzero cosmological constant; once again,however, these conclusions are tentative, andwill soon be superseded as a new generationof telescopes and satellites provides more ac-

curate observations of microwave backgroundanisotropies and large-scale structure in theuniverse.

Perhaps the most direct way of measuringthe cosmological constant is to determine therelationship between redshifts and distances offaraway galaxies, known as the Hubble dia-gram. Nearby galaxies have redshifts whichare proportional to their distances (HubblesLaw), but galaxies further away are expected

to deviate slightly from this strict proportion-ality in a way which depends on both andM. Measuring the distances to cosmologicalobjects is notoriously difficult, but importantprogress has recently been made by using TypeIa supernovae as distance indicators. (Suchsupernovae are not precisely standard candles,but variations in their luminosity are correlatedwith the rate of decay of their light curves, andcan be accounted for.) Supernovae are rare, butthe number of distant galaxies is very large, and

two independent groups have discovered dozensof high-redshift supernovae (as of 1999) by care-fully observing deep into small patches of thesky. The results of these studies thus far wouldbe consistent with zero cosmological constantonly if the matter density were lower than thatdetermined by dynamical measurements, andare consistent with a spatially flat universe only

if a substantial fraction of the total energy den-sity is due to a positive cosmological constant.It must be stressed, however, that our under-

standing of the physics underlying supernovaexplosions and the environments in which theyoccur is very incomplete at this stage. Never-theless, there is an impressive consistency be-tween this result and those of the microwavebackground observations and dynamical mea-surements of the mass density, with agreementachieved for a universe with M close to 0.3 and close to 0.7. Confirming or disproving thispossibility is one of the foremost ambitions of

contemporary cosmologists.

Physics of the Cosmological Con-

stant

The value of the cosmological constant is anempirical issue which will ultimately be settledby observation; meanwhile, physicists wouldlike to develop an understanding of why theenergy density of the vacuum has this value,

whether it is zero or not. There are many effectswhich contribute to the total vacuum energy,including the potential energy of scalar fieldsand the energy in vacuum fluctuations as pre-dicted by quantum mechanics, as well as anyfundamental cosmological constant. Further-more, many of these contributions can changewith time during a phase transition; for ex-ample, we believe that the effective cosmo-logical constant decreased by approximately1047 erg cm3 during the electroweak phasetransition. (A change in the effective cosmo-logical constant during a phase transition is acrucial ingredient in the inflationary universescenario, which posits an exponential expansionin the very early universe driven by a large vac-uum energy.)

From this point of view it is very surpris-ing that the vacuum energy today, even if it is

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nonzero, is as small as the current limits imply(|| 10

9 erg cm3). Either the various con-tributions, large in magnitude but different in

sign, delicately cancel to yield an extraordinar-ily small final result, or our understanding ofhow gravitation interacts with these sources ofvacuum energy is dramatically incomplete. Agreat deal of effort has gone into finding waysin which all of the contributions may cancel,but it is unclear what would be special aboutthe value = 0; a vanishing vacuum energycould be demanded by certain symmetry princi-ples, but unbroken symmetries of the appropri-

ate type are incompatible with what we knowof the other forces of nature. (One suggestionis to invoke the anthropic principle, whichimagines that the constants of nature take onvery different values in different regions of theuniverse, and intelligent observers only appearin those regions hospitable to the developmentof life. It is unclear, however, whether differentregions of the universe really do have differentfundamental constants, or what values of the

cosmological constant are compatible with theexistence of intelligent life.) The alternative,that our understanding of the principles under-lying the calculation of the cosmological con-stant is insufficient (and must presumably awaitthe construction of a complete theory of quan-tum gravity), is certainly plausible, althoughthe vacuum energy manifests itself in a low-energy regime where it would have been reason-able to expect semiclassical reasoning to suffice.Understanding the smallness of the cosmologi-

cal constant is a primary goal of string theoryand other approaches to quantum gravity.

If the recent observational suggestions of anonzero are confirmed, we will be faced withthe additional task of inventing a theory whichsets the vacuum energy to a very small valuewithout setting it precisely to zero. In this casewe may distinguish between a true vacuum,

which would be the state of lowest possible en-ergy which simply happens to be nonzero, anda false vacuum, which would be a metastable

state different from the actual state of lowestenergy (which might well have = 0). Sucha state could eventually decay into the truevacuum, although its lifetime could be muchlarger than the current age of the universe. Afinal possibility is that the vacuum energy ischanging with time a dynamical cosmologi-cal constant. This alternative, which is some-times called quintessence, would also be com-patible with a true vacuum energy which was

ultimately zero, although it appears to requirea certain amount of fine-tuning to make it work.No matter which of these possibilities, if any, istrue, the ramifications of an accelerating uni-verse for fundamental physics would be trulyprofound.

Bibliography

Popular Expositions:

Goldsmith, D 1997 Einsteins Greatest Blun-der? (Cambridge: Harvard University Press)

Technical Reviews:

Carroll S M, Press W H, and Turner E L 1992The cosmological constant Annu. Rev. Astron.Astrophys. 30 499Kirshner R P 1999 Supernovae, an acceleratinguniverse and the cosmological constant Proc.

Natl. Acad. Sci. 96 4224Weinberg S 1989 The cosmological constantproblem Rev. Mod. Phys. 61 1

Web pages for supernova groups:

Supernova Cosmology Project http://www-supernova.lbl.gov/

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High-Z Supernova Search Team http://cfa-www.harvard.edu/cfa/oir/Research/super

nova/HighZ.html

SEAN M. CARROLL

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