The Mystery of the Cosmological Constant

According to theory, the constant, which measures the energy of the , should be much greater than it is. An understanding of the disagreement could revolutionize fundamentalphysics

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. ? 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 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 , 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 : 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 . In , 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 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.


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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 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 afterEinstein's paper appeared by an argument advanced by the Dutch physicist , who tional field produced not by 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. 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 dissatisfiedwith model. The 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


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-


© 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. First there is the bare cle accelerators). able to provide a value for its mass, cosmological constant: the value the The standard model is a quantum because this is another free param­ cosmological constant would have if field theory. This means that for ev­ eter of the theory. A key challenge in none of the known particles existed ery distinct type of fundamental par­ today is to develop and if the only force in the universe ticle in nature there exists a corre­ a more powerful theory based on a were gravity. The bare cosmological sponding field in the model used to smaller number of free parameters constant is a free parameter that can describe the properties and interac­ that nonetheless incorporates all be determined only by experimental­ tions of that particle. Thus in the the successes of the standard model. ly measuring the true value of the standard model there is an electron Such a theory would be able to deter­ cosmological constant. field, a field for the photon (the elec­ mine the values of some of the pa­ The second type of contribution to tromagnetic field) and a field for each rameters that cannot be predicted by the total energy density of the vacu­ of the known particles. the standard model. In their search um arises in part from quantum fluc­ The standard model depends on a for such a theory, physicists are con­ tuations. The fieldsin the standard fairly large number of free parame­ stantly looking for relations among model, such as the electron field, ters: numbers that must be deter­ the parameters of the standard mod­ experience fluctuations even in the mined by experiment and fed into el that might reveal a deeper struc­ vacuum. Such fluctuations manifest the theory before definite predic­ ture. As we shall see, the cosmolog­ themselves as pairs of so-called vir­ tions can be made. Examples of free ical constant will provide us with tual particles, which appear spon­ parameters include the values of the such a relation, but in this case we taneously, briefly interact and then masses of the particles and numbers shall get more than we bargained for. disappear. (Each pair of virtual par­ characterizing the strengths of their ticles consists of a particle and its interactions. Once the numbers have n the standard model, as in any corresponding antiparticle, such as been determined the model can be Iquantum fieldtheory, the vacuum the electron and the positron, which used to predict the results of fur­ is defined as the state of lowest ener­ have identical masses but opposite ther experiments, and it can be tested gy, or more properly as the state of electric charges.) Although virtual on the basis of its predictions. In the least energy density. This does not particles cannot be detected by a past such tests have been spectacu­ imply that the energy density of the casual glance at empty space, they larly successful. vacuum is zero, however. The ener­ have measurable impacts on phys­ The free parameters of the stan­ gy density can in fact be positive, ics, and in particular they contribute dard model will play a central role in negative or zero depending on the to the vacuum energy density. The our discussion. Although the stan­ values of various parameters in the contribution made by vacuum fluc­ dard model is highly successful,the theory. Regardless of its value, there tuations in the standard model de­ fact that it depends on such a large are many complex processes that pends in a complicated way on the


al particles can appear spontaneously in the vacuum (b), interact in an abstract and highly symbolic manner. Each pair of virtual briefly (e) and then disappear (d). Here fluctuations are depicted particles consists of a particle and corresponding antiparticle.


© 1988 SCIENTIFIC AMERICAN, INC masses and interaction strengths of constant-a free parameter that can (an assumption we are continual­ all the known particles. be determined only by carrying out ly checking in our search for .deep­ The second type of term also de­ the very measurement we are at­ er structure), it seems unlikely that pends on at least one additional field tempting to predict-and by the sen­ these apparently unrelated terl1\s known as the Higgs field, which rep­ sitivity of the vacuum energy to un­ would cancel one another. As a con­ resents a massive particle, the Higgs known physics. All is not lost, how­ sequence it seems reasonable to as­ boson, that has not yet been detect­ ever, at least not yet. Although all the sume that the total cosmological con­ ed. The Higgs field should have a par­ terms that go into making up the cos­ stant will be at least as large as or ticularly dramatic effect on the ener­ mological constant depend in a com­ larger than the individual terms we gy density of the vacuum state [see plicated way on all the parameters can compute. Such an argument is "The ," by Martinus]. G. of the standard model, the values too crude to predict whether the Veltman; SCIENTIFIC AMERICAN, No­ of many of the terms can be fairly cosmological constant should be vember, 1986]. accurately estimated. The constitu­ positive or negative, but we would The last type of term that must be ents of protons and neutrons, the conservatively estimate that its mag­ included is essentially a fudge factor "up" and "down" quarks, contribute nitude should be at least I/O kilome­ representing the contributions to the an amount of about 1/0 kilometer)2 ter)2, that it could well be something vacuum energy density from addi­ to the cosmological constant, for in­ on the order of 1/( 10 centimeters)2 tional particles and interactions that stance, and the Higgs field contrib­ and perhaps that it is even larger. may exist but we do not yet know utes an even larger amount, roughly In other words, we expect the grav­ about. The value of this term is of 1/00 centimeters)2. itational effects of a nonzero vacu­ course unknown. Each of the terms that contributes um energy density to appear as dis­ The cosmological constant is de­ to the cosmological constant de­ tortions in spacetime geometry over termined by adding together the pends on the parameters of the stan­ distances of one kilometer or less. three terms we have discussed. Our dard model in a distinct and inde­ ability to predict its value using the pendent way. If we assume that the t does not require any sophisticat­ standard model is frustrated by the parameters of the standard mod­ Ied experimentation to show that existence of the bare cosmological el are really free and independent the theoretical estimate we have just


HIGGS FIELD, if it exists, would make a particularly large contri­ the value of the field, 4>.Although the Higgs potential is com­ bution to the energy density of the vacuum. The Higgs field is pletely symmetricabout the vertical axis, the vacuum must the conjectured field corresponding to the particle called the break the symmetry by choosing a certain position in the trough Higgs boson, which is thought to give rise to particle masses. (ball). Such a selection is known as spontaneous symmetry Here the Higgs potential-the part of the vacuum energy density breaking, and it plays a key role in the standard model: the the­ that depends on the value of the Higgs field-isplotted against ory that describes elementary particles and their interactions.


© 1988 SCIENTIFIC AMERICAN, INC given is wildly wrong. We all know that ordinary Euclidean geometry ' provides a perfectly adequate de­ scription of space over distances much greater than one kilometer. While walking around the block none of us has ever noticed large dis­ tortions in the spacetime structure of our neighborhood. If the magnitude of the cosmological constant were as large as our standard model estimate, ordinary Euclidean geometry would not be valid over distance scales of one kilometer or even less. If the cosmological constant were negative with a magnitude of 1/( 1 kilometer)2, then the sum of the angles of a tri­ angle with sides on the order of one kilometer would be significantly less than 180 degrees, and the volume of a sphere of radius one kilometer would be significantly greater than 47T/3 cu­ bic kilometers. A positive cosmological constant of order 1/( 1 kilometer)2 would have even more bizarre consequences. If the cosmological constant were that large, we would not be able to see objects more than a few kilometers away from us owing to the tremen­ dous distortions in spacetime struc­ ture. In addition, if we walked farther GEOMETRIC DISTORTIONS produced by a nonzero cosmological constant can affect than a few kilometers away from both space and time. Here the effect on spatial geometry is shown, assuming that the home to see what the rest of the distortions are independent of time. A negative cosmological constant would produce world looked like, the gravitation­ a space with negative constant curvature (left); a positive constant would produce posi­ al distortion of spacetime would be tive constant curvature (right). (The positive case corresponds to the illustration on so great that we could never return page 107.) In a space with negative curvature the sum of the angles of a triangle would home no matter how hard we tried. be less than 180 degrees; with positive curvature the sum would be greater than 180. What if the cosmological constant is nonzero but quite small? In this case we would have to look over universe in the "" cosmolo­ dence for either a positive or a nega­ large distances to see its effects on gy currently accepted. The ordinary tive cosmological constant. spacetime structure. Of course, we gravitational attraction among galax­ A good example of how astrono­ cannot draw triangles the size of the ies tends to slow this expansion. As mers can measure the geometry of universe and measure their angles, the galaxies get farther away from the universe and look for a nonzero but we can observe the positions and one another their gravitational at­ cosmological constant is provided motions of distant gaJaxies. By care­ traction weakens, and so the rate by the recently published work of fully charting the distribution and at which the expansion slows de­ Edwin D. Loh and Earl]. Spillar of velocities of distant galaxies, astron­ creases with time. Thus the effect of Princeton University. Their survey omers can deduce the geometric ordinary matter on the expansion of counts the numbers of galaxies in re­ structure of the spacetime in which the universe is to decelerate the ex­ gions of a speCific size at various lo­ they exist and move. pansion at an ever decreasing rate. cations in space. Ifwe assume that on It has long been recognized that the average the number of galaxies the dominant source of gravitational hat effects would a nonzero per unit volume is the same every­ distortion in the spacetime geometry Wcosmological constant have on where, then by counting galaxies in a of the universe at large scales ap­ the expansion rate of the universe? region we are estimating the volume pears to be the energy density of mat­ A negative cosmological constant of that region. By measuring volumes ter and not that of the vacuum. Al­ would tend to slow the expansion of regions far from us we are deter­ though the energy density of matter of the galaxies, but at a rate that is mining the relation between distance and that of the vacuum both affect constant, not decreasing with time. and volume over very large scales the geometric structure of the uni­ A positive cosmological constant, on and at earlier times, since the light verse, they do so in different and dis­ the other hand, would tend to make from distant galaxies takes a long tinguishable ways. Numerous obser­ the galaxies accelerate away from time to reach us-billions of years in vations have shown that the galax­ one another and increase the expan­ the case of this survey. ies in the universe are moving away sion rate of the universe. Compre­ Although such surveys contain from one another, a fact that is one of hensive studies of the expansion many subtle sources of potential er­ the cornerstones of the expanding rates of distant galaxies show no evi- ror, the results differ so startlingly


© 1988 SCIENTIFIC AMERICAN, INC stant add up to a quantity more than 46 orders of magnitude smaller than the individual terms in the sum. In other words, the small value of t\;1e 3 cosmological constant is telling us that a remarkably precise and total­ ly unexpected relationship exists among all the parameters of the stan­ N dard model, the bare cosmological Via:: w constant and unknown physics. f- w 2 ::;;: 0...J A relationship among the free pa­ >2 l-\.rameters of the standard model M '0 is just what we seek in our quest to discover deeper and more predictive ::::::: :=. theories. How could such a complex f- z relationship among what we thought

V'l� were free and unconstrained param­ Z eters arise, and what does it mean? 0 U ...J In answering this question it is well


© 1988 SCIENTIFIC AMERICAN, INC netic theory it was generally as­ The second example, in which the standard model or the misconcep­ sumed that light waves traveled velocity of the ether was related to tion about the vacuum, which once ·through a medium known as the the velocity of the earth, is what to­ understood will either eliminate the ether. Using an interferometer, Al­ day would be called an "unnatural" problem or at least turn it into a natu­ bert A. Michelson and Edward W. relation: one that involves many ral one. As long as the problem of the Morley attempted to measure the parameters, some of which are cosmological constant remains un­ velocity of the earth as it traveled unknown or even unknowable. It natural, the only hope we have for through the ether. They found that seems unlikely, for instance, that finding a solution is to stumble on an the relative velocity was zero: the ve­ we will ever know and understand all-encompassing theory capable of locity of the earth and the velocity all the many factors that determine accounting for all particle-physics of the ether were identical. This is what the velocity of the earth is in parameters with nearly perfect accu­ another relationship involving what relation to the distant galaxies. Any racy. If we can change the relation was then thought to be a fundamen­ unified theory developed to account required to produce an acceptably ' tal parameter of nature, namely the for an unnatural relation would have small vacuum energy density into a velocity of the ether. Did the discov­ to explain the values of many known natural one, then, even though we ery point the way to a unified theory and unknown parameters all at once. have not yet accounted for its val­ relating a fundamental property of It seems quite unlikely that such a ue, we at least reduce the issue of electromagnetism to the motion of theory could be discovered even if it the cosmological constant to a more the earth? did exist. manageable problem involving a rea­ Although the idea that the ether Our example indicates that an un­ sonable number of known parame­ drifted with the earth was suggested, natural relation suggests a deep mis­ ters that only have to be predicted the zero result of the Michelson-Mor­ understanding about the essence of with a moderate degree of accuracy. ley experiment is actually explained what is being measured and related, There is little to report to date about by Einstein's special theory of rela­ rather than the existence of an un­ this effort. In spite of a lot of hard tivity, which showed that the con­ derl¥ing unified theory. As a conse­ work and creative ideas we still do ception of the ether being used in quence an unnatural relation may not know why the cosmological con­ that era was inconsistent with the point to an even more dramatic revo­ stant is so small. symmetries of space and time. No lution in our thinking than a natural theory providing a fundamental rela­ one would. ven though nature does not, in tion between the velocity of the ether If we discount the possibility that Ethe words of Aristotle, "abhor and something as idiosyncratic as the vanishingly small value of the a vacuum," perhaps it does abhor a the velocity of the earth has sur­ cosmological constant is accidental, vacuum that is not empty. By intro­ vived. That is hardly surprising. The we must accept that it has profound ducing the ether in the early days of velocity of the earth is affected by implications for physics. Before we electromagnetic theory, Maxwell and many things-the shape and size of launch into constructing new unified others cluttered the vacuum with a its orbit around the sun, the mass of models, however, we must face the hypothetical fluid that had complex the sun and the motion of the sun in dilemma that the relation implied by properties. Michelson and Morley the galaxy, for instance-that seem the vanishing of the cosmological showed that this view of the vacuum completely unrelated to issues in the constant is unnatural. The miracu­ was inconsistent with experimental theory of electromagnetism. There is lous cancellations required to pro­ reality, and Einstein showed that it no fundarriental relation between the duce an acceptably small cosmologi­ was inconsistent with the symme­ velocity of the ether and the velocity cal constant depend on all the param­ tries of the universe. of the earth because the ether itself eters relevant to particle physics, Quantum field theories also fill the as the 19th-century theorists imag­ known and unknown. To predict a emptiness of the vacuum, this time ined it does not even exist. zero (or small) value for the cos­ with quantum fluctuations and fields mological constant, a unified theory rather than ether. These modern n both examples a surprjsing rela­ would face the imposing task of ac­ forms of clutter are consistent with Ition between parameters of nature counting for every parameter affect­ the special theory of relativity, but foreshadowed dramatic and revolu­ ing particle physics. Even worse, they seem to cause problems when tionary new discoveries. We have ev­ achieving a suffiCiently small cos­ they are viewed in the framework of ery reason to believe the mysterious mological constant requires that ex­ the general theory. With the mystery relation implied by the vanishingly tremely precise (one part in 1046 or of the cosmological constant, per­ small value of the cosmological con­ more) cancellations take place; the haps we are again paying the price stant indicates that discoveries as im­ parameters would have to be predict­ for dumping too much into the vacu­ portant as these remain to be made. ed by the theory with extraordinary um. The standard model, which has The two examples we have consid­ accuracy before any improvement in a large number of fluctuating quan­ ered are quite different. The first rela­ the situation regarding the cosmo­ tum fields including a Higgs field, is a tion, which involves two parameters logical constant would even be no­ particularly egregious polluter of the of electromagnetism and one from ticeable. Constructing such a theory, vacuum. There is no doubt that the light propagation, is what physicists even if it does exist, seems to be an resulting theory is a beautiful and today would call a "natural" relation: awesome if not impossible task. highly successful structure, but it one that involves a small number Although certain theories of the may be based on a conception of of well-known parameters. The exis­ "ether drift" variety have been pro­ the vacuum or of spacetime that is tence of a natural relation may indi­ posed, most efforts concerning the flawed. It is our challenge to repair cate that a unifying theory exists, cosmological constant now focus on that faulty foundation without de­ and, more important, it suggests that finding the underlying misunder­ stroying the towering edifice we such a theory can be discovered. standing, the missing piece of the have built on it.