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The Mystery of the Cosmological Constant The Mystery of the Cosmological Constant According to theory, the constant, which measures the energy of the vacuum, should be much greater than it is. An understanding of the disagreement could revolutionize fundamental physics by Larry Abbott hat determines the struc­ swer to this question at present. In­ By convention, energies are often ture of space and time in the deed, a comparison of our theoretical measured in relation to the vacuum. universe? According to Ein­ and experimental understanding of When it is defined in this way, the stein'sW general theory of relativity, the cosmological constant leads to vacuum automatically has zero ener­ the geometric properties of space are one of the most intriguing and frus­ gy in relation to itself. The traditional related to the density of energy (and trating mysteries in particle physics approach will not work if we want to momentum) in the universe. To un­ and relativity today. discuss the energy of the vacuum in derstand the structure of spacetime, an absolute and significant way. We therefore, we must identify potential­ ost people are unaccustomed to must use a different technique to ly relevant sources of energy and Mthe idea that the vacuum might measure its value. evaluate their contributions to the to­ have a nonzero energy density: How The only way to establish an abso­ tal energy (and momentum) density. can a unit volume of empty space lute measure of energy is by using The most obvious energy sources contain energy? The answer in part gravity. In general relativity, energy that come to mind are ordinary mat­ lies in the fact that, according to is the source of gravitational fields in ter and radiation. A much less obvi­ quantum mechanics, physical quan­ the same way that electric charge is ous source of energy that can have tities tend to fluctuate unavoidably. the source of electric fields in the an enormous impact on the structure Even in the apparent quiet of the vac­ Maxwell theory of electromagnet­ of the universe is empty space itself: uum state pairs of particles are con­ ism. An energy density of any kind, the vacuum. stantly appearing and disappearing. including that produced by fluctu­ The notion that the vacuum can be Such fluctuations contribute energy ations in the vacuum, generates a a source of energy may seem coun­ to the vacuum. gravitational field that reveals itself terintuitive. But present theories of The notion of a vacuum energy is as a change in the geometry of space­ elementary particles and forces not also unfamiliar because that energy time. The gravitational field of the only allow for a nonzero vacuum en­ cannot be detected by normal tech­ earth, for instance, is produced by its ergy density but also strongly sug­ niques. Energies are usually deter­ rest energy, which equals the mass of gest that it should have a large value. mined by measuring the change in the earth multiplied by the square Is the vacuum energy density really the energy of a system when it is of the speed of light (as given by the as large as these theories appear to modified in some way, or by measur­ famous formula E = me2). The gravi­ suggest it is? ing a difference in energy between tational field produces a small dis­ The answer is most emphatically two systems. For example, we might tortion in the spacetime geometry no. The geometric structure of the measure the energy released when near the earth, resulting in the attrac­ universe is extremely sensitive to the two chemicals react. Because of this, tive force that pulls us all toward the value of the vacuum energy density. energy as we normally define it is a ground. In general relativity the en­ So important is this value that a con­ relative quantity. The energy of any ergy density of the vacuum has an stant proportional to the vacuum en­ state of a system only has meaning in absolute meaning, and it can be de­ ergy density has been defined. It relation to some other state. termined by measuring the gravita- is called the cosmological constant. If the vacuum energy density, or UNIVERSE with a large cosmological constant would be vastly different from the exist­ equivalently the cosmological con­ ing one. Here an artist has painted a scene as it might appear if the constant were as stant, were as large as theories of ele­ large as theoretical estimates suggest it could be. The illustration is based on a positive mentary particles suggest, the uni­ value for the constant on the order of 1/(1 kilometer)2. With such a value the structure verse in which we live would be dra­ of space would be so distorted that the radiation from distant objects would be red­ matically different, with properties shifted, or shifted toward longer wavelengths. The farther an object is from an observ­ we would find both bizarre and un­ er, the greater the red shift would be. A spectral blue object about a kilometer away settling. What has gone wrong with would look red; objects more than a kilometer or so away would have such large red our theories? We do not know the an- shifts that they would be invisible. Distant objects would appear spatially distorted. 106 © 1988 SCIENTIFIC AMERICAN, INC 1/4 KM '''----= o KILOMETER (KM) 107 © 1988 SCIENTIFIC AMERICAN, INC ter directly to the distribution of en­ COSMOLOGICAL CONSTANT = 8TTG/C4 X VACUUM ENERGY DENSITY ergy and momentum in the un�verse, in a manner first suggested by the Here G is Newton's gravitational constant and c is the speed of light. Defined in Austrian physicist and philosoph�r such a way, the cosmological constant has units of lover distance squared. Ernst Mach. The hope was dashed soon after Einstein's paper appeared by an argument advanced by the Dutch physicist Willem de Sitter, who tional field produced not by matter ry of relativity. First, he thought that discovered the spacetime we shall but by the vacuum itself. without a cosmological constant the discuss. Of course, determining the energy general theory could not account for After such an ignominious start it is density of the vacuum is tantamount a homogeneous and isotropic uni­ not surprising that in 1923 Einstein to determining the cosmological con­ verse: one that looks much the same wrote, perhaps somewhat bitterly, stant, since one is proportional to the everywhere. (It is remarkable that "away with the cosmological term." other. It turns out that the cosmolog­ Einstein even cared about such mat­ As we shall see, it has not been so ical constant can be assigned units ters in 1917, since at the time there easy to eliminate the cosmological of lover distance squared. In other was no evidence that the universe constant-it has survived to frustrate words, the square root of the recipro­ was homogeneous and isotropic, many theoretical physicists since cal of the cosmological constant is a which indeed it is.) Unfortunately Einstein. George Gamow has written distance. This distance has a direct Einstein's reasoning was incorrect. that Einstein felt "the introduction of physical meaning. It is the length In 1922 Alexander A. Friedmann the cosmological term was the big­ scale over which the gravitational ef­ showed that the general theory does gest blunder he ever made in his fects of a nonzero vacuum energy allow for a homogeneous and iso­ life," but once introduced by Einstein density would have an obvious and tropic universe, although not a static "the cosmological constant. ..rears highly visible effect on the geometry one: the universe must be expand­ its ugly head again and again." of space and time. By studying the ing (or contracting). Subsequent as­ At the present time we would ap­ geometric properties of the universe tronomical observations have con­ pear to be in an excellent position to over length scales on the order of vincingly demonstrated that models address the issue of the cosmological that distance, the value of the cosmo­ based on Friedmann's work accu­ constant, because we possess one of logical constant can be measured. rately describe the large-scale struc­ the most successful physical theories ture of the universe. ever developed, namely the standard hysicists have been struggling Einstein was also dissatisfied with model. The standard model is the Pwith the issue of the cosmological his original formulation because the rather unimaginative name given to constant for more than 70 years. The theory did not provide an explana­ a collection of theories that success­ constant was first introduced by Ein­ tion of inertia. He believed that by fully describes all the known elemen­ stein in 19 17 in an attempt to elim­ adding a cosmological constant he tary particles and their interactions. inate two "problems" in his origi­ might produce a theory capable of re­ The remarkable ability of the stan­ nal formulation of the general theo- lating the inertial properties of mat- dard model to interpret and predict a QUANTUM FLUCTUATIONS are among the phenomena that quantum mechanics, the values of physical quantities tend to contribute to the energy density of the vacuum (a). According to fluctuate unavoidably. As a consequence pairs of so-called virtu- 108 © 1988 SCIENTIFIC AMERICAN, INC the results of an enormous range of number of free parameters seriously contribute to the total vacuum ener­ particle-physics experiments leaves limits its predictive power. The mod­ gy density. it unchallenged as a model for parti­ el, for example, predicts that an ad­ In essence the total energy density cle physics (at least up to the highest ditional particle called the top quark of the vacuum is the sum of three energies accessible to current parti­ remains to be discovered, but is un­ types of terms.
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