Feature: November 2000
Cosmologists have proposed that a mysterious substance called quintessence can explain why our universe is accelerating. But what is it made of?
A revolution is taking place in cosmology. New ideas are usurping traditional notions about the composition of the universe, the relationship between geometry and destiny, and Einstein's greatest blunder. As numerous observations and experiments reshape the field, many cosmologists are exploring the possibility that the vast majority of the energy in the universe is in the form of a hitherto undiscovered substance called "quintessence".
Quintessence has the striking physical characteristic that it causes the expansion of the universe to speed up. Most forms of energy, such as matter or radiation, cause the expansion to slow down due to the attractive force of gravity. For quintessence, however, the gravitational force is repulsive, and this causes the expansion of the universe to accelerate.
The name has historical precedents. In philosophy, quintessence refers to the fifth element - after air, earth, fire and water - proposed by the ancient Greeks to describe a sublime, perfect substance. In literature, Quintessence is the queen of a land of speculative science in Rabelais' Gargantua.
In cosmology, quintessence is a real form of energy distinct from any normal matter or radiation, or even "dark matter". Its bulk properties - energy density, pressure and so forth - lead to novel behaviour and unusual astrophysical phenomena. So far its existence has only been inferred indirectly from a range of observations, but a number of current and planned experiments will make direct searches for this elusive form of energy.
Although cosmological quintessence bears some superficial resemblance to the historical version, there is plenty of substance in the modern invocation of this classical name.
Some 15 billion years ago, the universe was filled with a hot, dense, uniformly distributed gas of matter and radiation. Over the intervening years, space has been stretching, and as the gas has expanded to fill the growing volume, the matter has condensed to form atoms, molecules, planets, stars, galaxies and everything else we see in the universe today. But where is all this going?
According to Einstein's equations, the expansion of the universe is governed by the amount and type of energy in the universe, and by the geometry of space. Until recently, big-bang cosmologists assumed that almost all of the energy in the universe today consists of the mass energy (E = mc2) of the matter contained within.
As for geometry, space may be flat and obey the laws of Euclidean geometry, or it may be curved. The curvature may be negative, in which case parallel light beams diverge and the universe is open; or it may be positive, in which case the beams ultimately converge like lines of longitude on a globe and the universe is closed.
As the universe expands, the matter spreads out, with its density decreasing in inverse proportion to the volume. The strength of the curvature effect decreases less rapidly, as the inverse of the surface area. So, in the standard picture of cosmology, geometry ultimately gains control of the expansion of the universe.
In a flat or open universe, the expansion continues forever, albeit at an ever-decreasing rate because of the gravitational self-attraction of the matter. In a closed universe, on the other hand, the expansion eventually comes to a halt and the universe starts to contract. In this standard picture, the universe is decelerating in all cases, and its ultimate fate is decided by the choice of geometry.
But perhaps the geometry is not a free choice. In the 1980s Alan Guth of the Massachusetts Institute of Technology introduced the inflationary theory of the universe to address a number of flaws in the standard big-bang picture (see further reading). According to the inflationary concept, the universe underwent a fantastic burst of hyperexpansion during the first instants after the big bang, stretching unimaginably faster than the conventional picture would predict. This hyperexpansion can explain why energy is spread so uniformly throughout the universe and how tiny deviations from perfect uniformity can arise. These deviations eventually led to the formation of galaxies and large-scale structure.
A by-product of hyperexpansion is that the geometry of the universe is ironed out (figure 1) and space is made extraordinarily flat. Since the geometric effect on the expansion today is negligibly small, matter alone accounts for the current expansion rate. Therefore, by measuring the expansion rate, the matter density can be predicted from general relativity to be approximately 10-29 grams per cubic centimetre. This value is known as the critical density, rhocritical = 3H02/8¼G, where H0 is the Hubble constant (which is closely related to the expansion rate) and G is the gravitational constant. Cosmologists could go to sleep at night knowing that inflation had tamed the geometry, determined the matter density, and decided the fate of the universe: space expands forever at an ever-decelerating rate.
In the last decade, three new discoveries have awoken cosmologists to the possibility that one of their key assumptions about the composition and behaviour of the universe might be wrong (see Bahcall et al. in further reading). While evidence for a flat universe and the inflationary theory has grown, a new element that breaks the chain of logic between inflation, geometry and destiny has been added. That element is "dark energy".
First, a census of the total matter density of the universe has revealed that it adds up to considerably less than expected. Cosmologists have known for decades that the sum of all the ordinary or "baryonic" matter - that is all the matter made of protons and neutrons - is only about 5% of the critical value predicted for a flat universe. Numerous measurements, dating as far back as the 1930s, have indicated that there must be other invisible or "dark" matter in the universe, to explain, for example, how stars remain in rapid orbit around galaxies and how galaxies orbit around galaxy clusters.
This dark matter might consist of exotic new elementary particles suggested by various unified theories of particle physics, and it might add up to the missing 95% needed to reach the critical density. However, a series of diverse measurements have converged on the common conclusion that while some "exotic" dark matter exists, it adds up to less than half of the critical density. (Some baryonic matter, such as that in asteroids or brown dwarfs in distant galaxies, does not shine and is therefore "dark", but its density in insignificant compared with exotic dark matter.)
One of the simplest observational methods takes advantage of the fact that galaxy clusters, the largest objects in the universe, contain a fair sample of the relative proportions of dark matter and baryons. The ratio of cosmological dark matter to baryonic matter can be inferred from the gravitational mass of the cluster (that is the sum of the baryonic and dark matter) and its luminosity (which is determined by just the baryonic or ordinary matter). Knowing that baryonic matter accounts for at most 5% of the critical density, astronomers expected the clusters to contain at least 20 times as much dark matter as ordinary matter. However, the observed ratio is only about ten to one. Therefore, the total amount of matter of all kinds in the universe is less than half the critical density.
Confidence in inflation and its prediction of a flat universe might have been shaken if not for the emergence in the early 1990s of precise measurements of the cosmic microwave background that strongly supported the inflationary predictions (see figure 2). The microwave background is a bath of radiation emitted when the universe was just 300 000 years old and the hot plasma of electrons and protons condensed to form the first atoms - leaving behind hydrogen, helium and traces of other elements, as well as photons - in an event known as recombination.
These photons, now observed at microwave and radio frequencies, have an average energy that corresponds to a black-body spectrum of temperature T = 2.726 K. Most remarkably, as one scans across the sky there are slight variations in the temperature at the level of 1 part in 105. These fluctuations are due to the varying conditions in the distribution of matter that existed at recombination. In effect, these microwave-background photons provide a snapshot of the inhomogeneities in the dust and radiation that later collapsed to form clusters and galaxies.
Inflation provides a very specific, detailed prediction about the pattern of hot and cold patches on the sky: it predicts how many patches there should be of each angular size, how much hotter or colder than the average temperature they should be, and so forth. The most important prediction is the angular scale of the hottest and coldest patches.
These hot and cold spots are due to photons climbing out of the most overdense and underdense regions at recombination. The characteristic size of these regions can be calculated, based on the Jeans length (which is determined by the balance between gravity and pressure) at recombination.
The relationship between the physical size and the apparent angular size as observed on the sky depends crucially on the geometry of the space-time. If the universe is flat, the spots subtend an angle of about 1º on the sky. However, negative spatial curvature makes the apparent size on the sky smaller, while positive curvature makes it larger. In 1999 the ground-based MAT/TOCO experiment observed the stunning result that the hottest and coldest spots are at just the angular size consistent with a flat, Euclidean geometry (figure 3). This was later confirmed with better precision by the balloon-based BOOMERANG and MAXIMA experiments (see further reading and Physics World July 2000 pp23-24).
How can it be that the matter density is only one-third of the critical value, yet the universe is flat? Does this mean that Einstein's general theory of relativity is wrong? Most likely not. By the mid-1990s, based on the results of various observations, several groups, including Jeremiah Ostriker of Princeton University and one of the authors (PJS), foresaw the problem and pointed to its resolution (see further reading). The missing two-thirds of the critical density might consist of an exotic form of "dark energy", quite distinct from dark matter in that it does not cluster under the influence of an attractive gravitational force to form galaxies and large-scale structure. Hence, the total energy density could add up to the critical value, consistent with the evidence for a flat universe, but any census of the matter density would only find one-third of the critical value.
This proposal seemed to fit all the existing data beautifully, resolving many of the discrepancies associated with previous models. However, the proposal also made a shocking prediction. Although dark energy accounts for two-thirds of the energy density in the universe today, it must have been an insignificant fraction just a short time ago, otherwise its gravitational influence would have made it almost impossible for ordinary matter to form the stars, galaxies and large-scale structure that we see in the universe today.
It follows that any form of energy that dominates today, but was insignificant in the recent past, must have a density that decreases much more slowly with time than the matter density. That is, as the expanding universe doubles in volume and the matter density decreases by a factor of two, the density of this dark energy must decrease by a smaller factor. According to Einstein's equations, dark energy with this property is quite possible, but it must have an unusual property - it must be gravitationally self-repulsive.
Unlike normal matter, this self-repulsive dark energy will cause the expansion of the universe to accelerate. If the dark-energy proposal was correct, therefore, the universe should be accelerating today - a prediction that ran contrary to the accepted wisdom, and data, at that time.
Then, in 1998, two independent groups - the Supernovae Cosmology Project and the High-Z Supernova Search - announced a spectacular result based on significantly more precise measurements of cosmic expansion. Their observations of the brightening and dimming of distant type 1a supernovae revealed that the expansion of the universe is in fact accelerating (see further reading).
Their findings relied on the discovery that there is a relationship between the intrinsic brightness of a supernova and the rate at which it brightens and dims. Having determined the brightness of a supernova by measuring its "light curve" (i.e. how the observed brightness varies with time), the observed flux of photons is used to determine the physical distance to the supernova. Then, by comparing the distance with the redshift (which tells us how fast the supernova is moving away from us), the expansion history of the universe is reconstructed, one supernova at a time. An analogy can be made with mileposts viewed from a moving vehicle: the rate at which the mileposts pass by and recede tells us how fast the vehicle is moving.
The measurements indicate that the distant supernovae are dimmer than they ought to be if the universe was expanding at a steady pace. Exhaustive efforts have been made to demonstrate that no systematic effects are confounding the measurements, such as obscuring dust or variations in the supernovae themselves, but no effects have been found so far.
For cosmologists who had been studying the issue closely, the supernova result was the last key observation to fall into place. Now it can be said that a cosmological model based on the big bang, inflationary cosmology, and a universe that is composed of one-third matter and two-thirds dark energy is consistent with all current astrophysical and cosmological measurements. For the broader scientific community and the public-at-large, the discovery that the universe is accelerating came as a stunning surprise.
Although cosmologists can be justly proud of having a model that fits a dazzling array of observations, they cannot rest for long. A new mystery immediately arises. What is the dark energy that composes two-thirds of the present energy in the universe?
One fact we know about the dark energy is that it has negative pressure: cosmic acceleration can only occur if the pressure is sufficiently negative. The reason for this is found in general relativity, which tells us that energy and momentum, and therefore pressure, all gravitate. (Indeed, the deflection of light by gravity is exploited by astronomers in the well-established technique known as gravitational lensing.) The strength of this gravitational force is determined by rho + 3P, where rho is the energy density (including all forms of energy) and P is the pressure.
Normally, the pressure is negligibly small compared with the energy density, rho, so the greater the energy density, the more attractive the force. A large energy density will therefore cause space to bend and contract around it - this is how a black hole forms. However, if rho + 3P is negative - which can happen for negative pressures - then the gravitational force is repulsive.
When we apply this to the universe, we find that a ubiquitous energy substance with negative pressure causes space to repel itself: every point in space flees from its neighbours and the cosmic expansion accelerates. The bottom line for quintessence is that its pressure must be negative enough to overcome the attractive gravitational force of all the energy density in the universe.
Negative pressure may seem extraordinarily exotic, but it can actually be caused by rather straightforward physical processes. An ordinary gas composed of atoms and radiation acts like a compressed spring, pushing outwards in all directions. However, a bubble of interacting gas atoms in a metastable state (i.e. with higher energy than the surrounding gas), can act like a stretched spring under tension and exert an inward force or negative pressure.
The counterintuitive aspect is the gravitational response, which is an unappreciated feature of Einstein's general theory of relativity (see figure 4). Filling the universe with a fluid (e.g. matter and radiation) that has positive pressure and a positive rho + 3P slows the expansion. On the other hand, a fluid with sufficiently negative pressure will have a negative rho + 3P, and this will cause the expansion to accelerate.
The other piece of information we know about dark energy is that it somehow resists the gravitational pull of galaxies. The negative pressure is sufficient to explain why dark energy is spatially uniform on average, but why don't small inhomogeneities grow in, for instance, the dense regions at the centres of galaxies? There is not a unique answer to this question, but it seems likely that the particles composing this dark energy are so light and relativistic that nothing short of a black hole can disturb them.
Perhaps the dark energy is not made of particles at all. One candidate is vacuum energy, the energy of empty space. Einstein introduced this possibility in 1917 in his first attempt to apply his new theory of gravity to cosmology (see further reading). Einstein was convinced that the universe was static, but he could not construct a static universe if there was only matter and curvature because rho + 3P was positive. He therefore introduced an additional term to his theory, the so-called cosmological constant or Lambda. This was a form of energy with constant negative pressure P = -rho and, therefore, negative rho + 3P. By carefully choosing the amount of matter and the value of the cosmological constant, he could balance the forces to obtain a static universe. Several years later, after Hubble showed that the universe was truly expanding, Einstein described the cosmological constant as his "greatest blunder".
Today's cosmologists find Lambda to be just as objectionable, but for a different reason. All quantum fields possess a finite amount of "zero-point" vacuum energy as a result of the uncertainty principle. A naive estimate of the zero-point energy predicts a vacuum energy density that is 120 orders of magnitude greater than the energy density of all the other matter in the universe. If the vacuum energy density really is so enormous, it would cause an exponentially rapid expansion of the universe that would rip apart all the electrostatic and nuclear bonds that hold atoms and molecules together. There would be no galaxies, stars or life. Since we cannot ignore quantum mechanics, some other mechanism must nullify this vacuum energy. One of the major goals of unified theories of gravity has been to explain why the vacuum energy is zero.
Einstein's blunder has been resurrected as a possible solution to the dark-energy problem. Maybe there is a miraculous cancellation mechanism, but perhaps it is slightly imperfect. Instead of making Lambda exactly zero, the mechanism only cancels to 120 decimal places. Then, vacuum energy would comprise the missing two-thirds of the critical density. The requirements seem bizarre, though. Some constant that is naturally enormous must be cut down by 120 orders of magnitude, but with such precision that today it has just the right value to account for the missing energy.
Extrapolating back in time to the early universe, the story seems even more bizarre. When the volume of the universe was 100 orders of magnitude smaller, say, the mass density was 100 orders of magnitude greater, but the vacuum energy density had to have the same value as today. In other words, the vacuum energy density remained constant as the universe expanded, but the total vacuum energy increased as the volume of space increased. This extra energy came from the gravitational potential energy of the universe. Whatever physical processes created the initial energy in the universe had to arrange for an exponentially large difference between the two forms of energy, but somehow this difference had to have exactly the right value for the vacuum energy to become important 15 billion years later.
It would seem more natural for the dark energy to start with an energy density similar to the density of matter and radiation in the early universe. The dark energy and matter density could both then decrease at similar rates as the universe expanded, with the dark energy density overtaking the matter density only after structure has formed in the universe. However, if the dark energy density has been changing, it cannot consist of vacuum energy. Therefore, the concept of quintessence was introduced to overcome this problem by ourselves and Rahul Dave, then at the University of Pennsylvania, in 1998 (see Caldwell et al. in further reading). Quintessence is a dynamic, time-evolving and spatially dependent form of energy with negative pressure sufficient to drive the accelerating expansion. Whereas the cosmological constant is a very specific form of energy - vacuum energy - quintessence encompasses a wide class of possibilities.
The simplest model proposes that the quintessence is a quantum field with a very long wavelength, approximately the size of the observable universe. Some examples had been explored almost a decade earlier by Bharat Ratra and James Peebles at Princeton University, and by Chris Wetterich at the University of Heidelberg in Germany. A particle is usually thought of as a bundle of oscillations in a quantum field, but since this bundle is much larger than any conventional length scale, the particle description is impractical.
The energy is composed of kinetic energy, which depends on the rate of oscillations in the field strength, and potential energy, which depends on the interaction of the field with itself and matter. The pressure is determined by the difference between the kinetic and potential energy, with kinetic energy contributing positively to the pressure. However, since the oscillation has an extremely long wavelength and period - essentially the size and age of the universe - its kinetic energy is negligible. The behaviour of the quintessence field is therefore dominated by how it interacts with itself. Much like a stretched spring, this self-interaction potential leads to negative pressure.
Within certain models that seek to unify the four fundamental forces of nature there exist fields, called "tracker fields", that can make quintessence behave in this way (see Zlatev et al. in further reading). First, the dark energy density in the early universe can be comparable with the matter density. The model is insensitive to the precise initial value because the dynamical equations that determine the time evolution of the tracker field have solutions that cause the energy to follow the same evolution, independent of initial conditions (similar to the classical attractor solutions found in conventional nonlinear dynamics). In particular, the energy density tracks the radiation and matter density (see figure 5). For most of the history of the universe, the quintessence occupies a very small fraction of the critical density, but the fraction grows slowly until it catches up with and ultimately overtakes the matter density.
It seems natural to ask if there are any direct gravitational interactions between ordinary matter and dark energy. If the dark energy is vacuum energy, then the two do not interact because vacuum energy is inert and unchanging. But if the dark energy is quintessence, they can interact under certain conditions. Ordinary particles are physically very small compared with the Compton wavelength, lambdac = h/mc, of the quintessence particles, so individual particles have a completely negligible effect on quintessence (and vice versa). However, very large clumps of ordinary matter - spread out over a region comparable with the Compton wavelength - can interact gravitationally with quintessence and create inhomogeneities in its distribution that may produce detectable signals in the cosmic microwave background.
What cosmologists find most difficult to explain is why the acceleration should begin at this particular moment in cosmic history. Is it a coincidence that, just when thinking beings have evolved, the universe suddenly shifts into overdrive? The situation is peculiar because the energy associated with the cosmological constant or quintessence is very tiny, less than a millielectron-volt. If new ultra-low-energy physics is responsible, it should have already been observed in other experiments.
Some physicists and astronomers have proposed an anthropic argument (see Weinberg in further reading). Perhaps there is a multitude of universes, all with different values for the vacuum energy density, with larger values being more probable than smaller values. Then universes with a vacuum energy much greater than a millielectron-volt would be more probable, but they would expand too rapidly to form stars, planets or life. At the same time, universes with much smaller values are less probable. The anthropic argument would say that our universe has the optimal value. Physicists disagree about whether this kind of explanation, which makes bold assumptions about the existence of universes that can never be tested, and about the probability distribution of the vacuum energy, is an acceptable explanation.
Perhaps a more satisfying possibility is that the acceleration is triggered by natural events in the recent history of the universe. According to the big-bang model, the energy density in the universe was predominantly in the form of hot, relativistic particles until the universe was a few tens of thousands of years old. At that time, the universe had cooled enough that the mass energy of non-relativistic particles became more important than both their kinetic energy and the energy of radiation, resulting in a change in the cosmic expansion rate. This marked the beginning of the "matter-dominated epoch". Only then could gravity begin to clump matter together to form stars, galaxies and large-scale structure. Is it possible that this transition triggered the onset of quintessence?
Tracker quintessence has a pressure that adjusts to the form of energy that dominates the universe - up to a point. When the universe is radiation dominated, the tracker field mimics radiation: the energy density falls at the same rate as the radiation energy density, and the pressure for quintessence is given by P = rho/3, the same as for radiation. When the universe becomes matter dominated, the tracker field mimics the matter, for which the pressure is nearly zero. The tracker field is able to follow the radiation and matter energy densities because the time variation of the tracker-field energy and pressure are controlled by a frictional effect, Hubble damping, that is determined, in turn, by the radiation or matter. As long as the mimicking continues, the tracker-field energy is a small fixed fraction of the total energy and the expansion decelerates.
To stop this mimicking and begin a period of acceleration, the tracker potential must possess some feature that causes the field to become locked into a nearly constant value at some later time. If the tracker field is constant, the kinetic energy is negligible compared with the potential energy, which is precisely the condition required for a negative pressure component and cosmic acceleration. The problem is that the feature of the potential that locks the tracker field must be delicately tuned so that the acceleration begins at the right time.
The tuning problem can be circumvented for a novel form of tracker field called "k-essence", short for kinetic-energy-driven quintessence (see Armendariz et al. in further reading). In these models, the kinetic energy depends nonlinearly on the time variation of the tracker field, which causes novel dynamical features. In the early universe, when the universe is radiation dominated, these features are not evident, and the energy mimics the radiation. However, when the universe undergoes the transition to matter domination, unlike the examples discussed above, the k-essence field refuses to track the matter. At first, the field slows down and the kinetic energy density drops sharply, but it soon converges to a fixed value and begins to act as a source of negative pressure. It is then just a short time (about the present epoch) before it overtakes the matter density (which continues to fall) and drives the universe into cosmic acceleration. The crucial point is that there does not have to be some special feature in the potential energy for this to happen. Rather, it is an automatic dynamical response to the onset of matter domination.
In this picture, the fact that thinking beings and cosmic acceleration occur at nearly the same time in cosmic history is not a coincidence. Both the formation of the stars and planets necessary to support life and the transformation of quintessence into a negative pressure component are triggered by the onset of matter domination. This explanation is decidedly non-anthropic.
Quintessence leaves its mark on the universe in several ways, so experimenters have a number of methods they can use to test for this exotic form of energy. The acceleration effect of a dark-energy component depends on the ratio of its pressure to its energy density. More negative values of this ratio, w, lead to greater acceleration. Quintessence and vacuum energy have different values of w, so more precise measurements of supernovae over a longer span of distances may be able to separate these two possibilities.
This challenge is the motivation for two proposals - the Earth-based Large-Aperture Synoptic Survey Telescope (LSST) and the space-based Supernova Acceleration Project (SNAP) - that will monitor the sky for supernovae and other time-varying astrophysical phenomena. The proposers of these projects are currently seeking funding.
Cosmic acceleration also affects the number of galaxies to be found as one explores deeper and deeper into space. With appropriate corrections for evolution and other effects, the average density of galaxies is uniform throughout space. Consequently, for a fixed range of distances, one should find the same number of galaxies nearby and far away. But cosmologists measure the redshift of distant galaxies, not their distance. The conversion from redshift to distance follows a simple linear relation (the Hubble law) if the distances are small, but a nonlinear relation depending on the acceleration of the universe if the distances are large. The nonlinear relation will cause the number of galaxies found for a fixed range of redshifts to change systematically as one probes deeper into space. The Deep Extragalactic Evolutionary Probe (DEEP), an advanced spectrograph on the Keck II telescope in Hawaii, is poised to test this prediction with an accuracy that may be sufficient to distinguish between quintessence and a cosmological constant.
Quintessence should also have an effect on the cosmic microwave background because differences in the acceleration rate will produce small differences in the angular size of hot and cold spots. Moreover, unlike a cosmological constant, quintessence is not spatially homogeneous. Small variations in the amount of quintessence across the sky should be seen as ripples in the microwave background temperature. Measurements by the MAP and Planck satellites (launch dates 2001 and 2007, respectively) may be able to detect these effects, which will be at the level of a few per cent, although it will be difficult to separate them from other effects.
In many cases, quintessence interacts with matter in a way that affects the forces between particles. Then, if the quintessence field is varying temporally or spatially, it will cause the strengths of the forces between particles to change as well. Hence, ongoing tests for changes in the values of the fundamental physical constants with time could be another source of evidence for quintessence. It might be possible to search for such effects with astrophysical observations (e.g. a variation of hyperfine splitting with redshift) or in ultrahigh-precision laser-spectroscopy experiments.
The revolution in cosmology, driven by observations and experiments, has changed more than our understanding of the composition of the universe - it has changed our expectations for the future. Quintessence, a sublime substance, may permeate the universe, marking an end to the epoch that saw the formation of stars and galaxies, and the beginning of an epoch of cosmic acceleration. In the short term, space will stretch ever faster, and galaxies will fly apart from one another, leaving a colder, emptier universe. As for the ultimate fate of the universe, the nature of quintessence, not geometry, will be the determining factor.
The universe may accelerate forever, or the quintessence could decay into new forms of hot matter and radiation that could repopulate the universe with new structure. The various experiments described above will provide key tests of the quintessence hypothesis and supply new information about the future of the universe. To obtain a more definitive answer, however, physicists will have to understand how quintessence fits within the fabric of the still-elusive unified theory of all the fundamental forces.
About the author
Robert R Caldwell is in the Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755-3528, USA ( www.dartmouth.edu/artsci/physics/faculty/Caldwell.html). Paul J Steinhardt is in the Department of Physics, Princeton University, Princeton, New Jersey 08544, USA (feynman.princeton.edu/~steinh/)
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