THE SEARCH FOR EXTRA DIMENSIONS - The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis

The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)


Given that a decade has passed without a major revelation on the theoretical front, string theory partisans now find themselves under increasing pressure to link their ethereal ideas to something concrete. Hovering above all their fantastical notions is one question that won’t go away: Do these ideas actually describe our universe?

That’s a legitimate issue to raise in view of the provocative ideas broached here, any one of which might give the average citizen pause. One such claim is that everywhere in our world, no matter where we go, there is a higher-dimensional space always within reach, yet so small we’ll never see it or feel it. Or that our world could implode in a violent Big Crunch or explode in a fleeting spurt of cosmic decompactification, during which the realm we inhabit instantly changes from four large dimensions to ten. Or simply that everything in the universe—all the matter, all the forces, and even space itself—is the result of the vibrations of tiny strings moving in ten dimensions. There’s a second question we also ought to consider: Do we have a prayer of verifying any of this—of gleaning any hints of extra dimensions, strings, branes, and the like?

The challenge facing string theorists remains what it has been since they first tried to re-create the Standard Model: Can we bring this marvelous theory into the real world—the goal being not just to make contact with our world, but also to show us something new, something we haven’t seen before?

There’s a huge chasm, at present, between theory and observation: The smallest things we can resolve with current technology are roughly sixteen orders of magnitude bigger than the Planck scale on which strings and the extra dimensions are thought to reside, and there appears to be no conceivable way of bridging that gap. With the “brute force” approach of direct observation apparently ruled out, it will take extraordinary cleverness, and some measure of luck, to test these ideas by indirect means. But that challenge must be met if string theorists are ever going to win over the doubters, as well as convince themselves that their ideas add up to more than just grandiose speculation on a very small scale.

So where do we start? Peer through our telescopes? Smash particles together at relativistic speeds and sift through the rubble for clues? The short answer is that we don’t know which avenue, if any, will pay off. We still haven’t found that one make-or-break experiment that’s going to settle our questions once and for all. Until we do, we need to try all of the above and more, pursuing any lead that might furnish some tangible evidence. That is exactly what researchers are gearing up for right now, with string phenomenology becoming a growth area in theoretical physics.

A logical starting point is to look upward, to the skies, as Newton did in devising his theory of gravity and as astrophysicists have done to test Einstein’s theory of gravity. A meticulous scan of the heavens may, for instance, shed light on one of string theory’s more recent, and strangest, ideas—the notion that our universe is literally housed within a bubble, one of countless bubbles floating around within the cosmic landscape. While this may not seem like the most promising line of inquiry—being one of the more speculative ideas out there—it is pretty much where we left off in this narrative. And the example does illustrate some of the difficulties involved in translating these far-flung ideas into experiment.

When we discussed bubbles in the last chapter, it was in the context of decompactification, a process both extremely unlikely to be witnessed, taking possibly as long as e(10120) years to unfold, and probably not worth waiting for, anyway, as we wouldn’t see a decompactification bubble coming until it literally hit us. And once it hit us, we wouldn’t be “us” anymore, nor would we be capable of figuring out what the heck did us in. But there may be other bubbles out there. In fact, many cosmologists believe that right now, we’re sitting in one that formed at the end of inflation, a fraction of a second after the Big Bang, when a tiny pocket of lower-energy material formed amid the higher-energy inflationary vacuum, and that has expanded since then to become the universe we presently know. It is widely believed, moreover, that inflation never fully ends—that once started, it will continue to spin off an endless number of bubble universes that differ in their vacuum energies and other physical attributes.

The hope held by proponents in the obscure (and sparsely populated) realm of bubble phenomenology is not to see our own bubble but rather signs of another bubble, filled with an entirely different vacuum state, that careened into ours at some time in the past. We might find evidence of such an encounter lurking, for instance, in the cosmic microwave background (CMB), the background radiation that bathes our universe. An aftereffect of the Big Bang, the CMB is remarkably homogeneous, uniform to 1 part in 100,000. From our point of view, the CMB is said to be isotropic, which means that no matter in what direction we look, the view is the same. A violent collision with another bubble, which deposits a huge amount of energy in one portion of the universe but not in another, would produce a localized departure from such uniformity called an anisotropy. This would impose a direction on our universe—an arrow that points straight toward the center of the other bubble, just before it slammed into us. Despite the hazards associated with our own universe’s decompactifying, a collision with another universe inside another bubble would not necessarily be fatal. (Our bubble wall, believe it or not, would afford some protection.) Such a collision may, however, leave a discernible mark in the CMB that is not merely the result of random fluctuations.

That is just the sort of calling card that cosmologists look for, and a potential anisotropy, referred to as “the axis of evil” by its discoverers Joao Magueijo and Kate Land of Imperial College London, may have been uncovered within the CMB data. Magueijo and Land claim that hot spots and cold spots in the CMB appear to be aligned along a particular axis; if this observation is correct, it would suggest that the universe has a specific orientation, which would clash with hallowed cosmological principles avowing that all directions are equivalent. But for the moment, no one knows whether the putative axis is anything more than a statistical fluke.

If we could obtain firm evidence that another bubble crashed into us, what exactly would that prove? And would it have anything to do with string theory? “If we didn’t live in a bubble, there wouldn’t be a collision, so we’d know, for starters, that we really are in a bubble,” explains New York University physicist Matthew Kleban. Not only that, but we’d also know, by virtue of the collision, that there is at least one other bubble out there. “While that wouldn’t prove that string theory is right, the theory makes a bunch of weird predictions, one being that we live in a bubble”—one of a vast number of such bubbles strewn across the string theory landscape. So at a minimum, says Kleban, “we’d be seeing something strange and unexpected that string theory also happens to predict.”1

There’s an important caveat, however, as Cornell’s Henry Tye points out: Bubble collisions can also arise in quantum field theories that have nothing to do with strings. If traces of a collision were observed, Tye says, “I know of no good way of telling whether it came from string theory or quantum field theory.”2

Then there’s the question of whether we could ever see something like this, regardless of the source. The likelihood of such a detection, of course, depends on whether any other stray bubbles are within our path or “light cone.” “It could go either way,” says Ben Freivogel, a physicist at the University of California, Berkeley. “It’s a question of probabilities, and our understanding is not good enough yet to determine those probabilities.”3 While no one can rate the exact odds of such a detection, most experts would probably rate it as very low.

If calculations eventually suggest that bubbles are not apt to be a fruitful avenue of investigation, many physicists still believe that cosmology offers the best chance of testing string theory, given that the almost-Planck-scale energies from which strings are thought to arise are so enormous, they could never be reproduced in the lab. Perhaps the best hope of ever seeing strings—presumed to be as small as 10-33 centimeters long—is if they were formed at the time of the Big Bang and have grown in step with the universe’s expansion ever since. We’re talking now about hypothetical entities called cosmic strings—an idea that originated before string theory took hold, but has gained renewed vigor through its association with that theory.

According to the traditional view (which is compatible with the string theory view), cosmic strings are slender, ultradense filaments formed during a “phase transition” within the first microsecond of cosmic history. Just as cracks inevitably appear in ice when water freezes, the universe in its earliest moments also went through phase transitions that were likely to have produced defects of various kinds. A phase transition would occur in different regions at the same time, and linear defects would form at junctures where these regions run into each other, leaving behind wispy filaments of unconverted material forever trapped in a primordial state.

Cosmic strings would emerge from this phase transition in a spaghetti-like tangle, with individual threads moving at speeds near the speed of light. They’re either long and curvy, with a complex assortment of wiggles, or fragmented into smaller loops that resemble taut rubber bands. Far narrower than subatomic particles, cosmic strings are expected to be almost immeasurably thin, yet almost boundless in length, stretched by cosmic expansion to span the universe. These elongated filaments are physically characterized by their mass per unit length or tension, which provides a measure of their gravitational heft. Their linear density can reach incredibly high values—about 1022 grams per centimeter for strings formed at the so-called grand unified energy scale. “Even if we squeezed one billion neutron stars into the size of an electron, we would still hardly reach the matter-energy density characteristic of grand unified cosmic strings,” says University of Buenos Aires astronomer Alejandro Gangui.4

These bizarre objects gained currency in the early 1980s among cosmologists who saw them as potential “seeds” for galaxy formation. However, a 1985 paper by Edward Witten argued that the presence of cosmic strings would create density inhomogeneities in the CMB far larger than observed, thus apparently ruling out their existence.5

Since then, cosmic strings have hit the comeback trail, owing much of their recent popularity to string theory, which has prompted many people to view these objects in a new light. Cosmic strings now seem to be common by-products of string-theory-based inflation models. The most recent versions of the theory show that so-called fundamental strings—the basic units of energy and matter in string theory—can reach astronomical sizes without suffering from the problems Witten identified in 1985. Tye and his colleagues explained how cosmic strings could be produced at the end of inflation and thus would not be diluted to oblivion during the brief period of runaway expansion, during which the universe doubled in size perhaps fifty to a hundred or more times. These strings, Tye demonstrated, would be less massive than the strings Witten and others contemplated in the 1980s, which means their influence on the universe would not be so pronounced as to have been already ruled out by observations. Meanwhile, Joe Polchinski of the University of California, Santa Barbara, showed how the newly conceived strings could be stable on cosmological timescales. The efforts of Tye, Polchinski, and others deftly addressed the objections Witten had raised two decades ago, leading to a resurgence of interest in cosmic strings.


12.1—This image comes from a simulation that shows a network of cosmic strings when the universe was about ten thousand years old. (Courtesy of Bruce Allen, Carlos Martins, and Paul Shellard)

Thanks to their postulated density, cosmic strings ought to exert a noticeable gravitational influence on their surroundings, and this ought to make them detectable. If a string ran between us and another galaxy, for example, light from that galaxy would go around the string symmetrically, producing two identical images close to each other in the sky. “Normally you’d expect three images, if lensing is due to a galaxy,” explains Alexander Vilenkin, a cosmic string theorist at Tufts University.6 Some light would pass straight through the lensing galaxy, and other rays would travel around on either side. But light can’t go through a string, because the string’s diameter is much smaller than the light’s wavelength; thus strings, unlike galaxies, would produce just two images rather than three.

Hopes were stirred in 2003, when a Russian-Italian group led by Mikhail Sazhin of Moscow State University announced that they had taken double images of a galaxy in the Corvus constellation. The images were at the same distance or redshift and were spectrally identical at a 99.96 percent confidence level. Either two extremely similar galaxies were, by chance, closely aligned or it was the first case of lensing by a cosmic string. In 2008, a more detailed analysis drawing on Hubble Space Telescope data—which yielded sharper pictures than the ground-based telescope Sazhin and colleagues used—showed that what first appeared to be a lensed galaxy was, in fact, two different galaxies, thereby ruling out the cosmic string explanation.

A related approach called microlensing is based on the premise that a loop formed from the breakup of a cosmic string could lens individual stars in potentially detectable ways. Although it might be impossible to actually see two identical stars, astronomers might instead detect a star that periodically doubles in brightness, while remaining constant in color and temperature, which could signal the presence of a cosmic string loop oscillating in the foreground. Depending on where it is, how fast it’s moving, its tension, and its precise oscillation mode, the loop would produce a double image at some times and not others—with stellar brightness changes occurring over the course of seconds, hours, or months. Such a signature might be secured by the Gaia Satellite, scheduled for launch in 2012, which is due to survey a billion stars in the Milky Way and its immediate vicinity. The Large Synoptic Survey Telescope (LSST) now under construction in Chile might also spot a similar signature. “Direct astronomical detection of superstring relics would constitute an experimental verification of some of the basic ingredients of string theory,” claims Cornell astronomer David Chernoff, a member of the LSST Science Collaboration.7

Meanwhile, researchers continue to explore other means of detecting cosmic strings. Theorists believe, for instance, that cosmic strings could form cusps and kinks, in addition to loops, emitting gravitational waves as these irregularities straighten out or decay. The waves so produced might be at just the right frequency to be detected by the Laser Interferometer Space Antenna (LISA), the proposed orbital observatory now being developed for NASA. According to current plans, LISA will consist of three identical spacecraft separated from each other by 5 million kilometers in the configuration of an equilateral triangle. By closely monitoring changes in the distance between these spacecraft, LISA could sense the passage of gravitational waves. Vilenkin and Thibault Damour (of IHES in France) proposed that precise measurements of these waves could reveal the presence of cosmic strings. “Gravitational waves produced from a cosmic string source would have a specific waveform that would look very different from that produced by black hole collisions or other sources,” Tye explains. “The signal would start at zero and then rapidly increase and decrease. The way it increases and decreases, which is what we mean by the ‘waveform,’ would be peculiar to cosmic strings alone.”8

Another approach involves looking for distortions in the CMB produced by strings. A 2008 study along these lines, headed by Mark Hindmarsh of Sussex University, found that cosmic strings might account for the clumpy distribution of matter observed by the Wilkinson Microwave Anisotropy Probe (WMAP). This phenomenon of clumpiness is known as non-Gaussianity. While the data, according to Hindmarsh’s team, suggests the presence of cosmic strings, many are skeptical, regarding the apparent correlation as a mere coincidence. This matter should be clarified as more sensitive CMB measurements become available. Investigating the potential non-Gaussianity of the matter distribution in the universe is, in fact, one of the goals of the Planck mission, launched by the European Space Agency in 2009.

“Cosmic strings may or may not exist,” says Vilenkin. But the search for these entities is under way, and assuming they do exist, “their detection is very feasible within the next few decades.”9

In some models of string inflation, the exponential growth of space occurs in a region of the Calabi-Yau manifold known as a warped throat. In the abstract realm of string cosmology, warped throats are considered both fundamental and generic features “that arise naturally from six-dimensional Calabi-Yau space,” according to Princeton’s Igor Klebanov.10 While that doesn’t guarantee that inflation takes place in such regions, warped throats offer a geometrical framework that can nevertheless help us understand inflation and other mysteries. For theorists, it’s a setting rich in possibilities.

The throat, the most common defect seen in a Calabi-Yau, is a cone-shaped bump, or conifold, that juts out from the surface. The rest of space—often described as the bulk—can be thought of as a large scoop of ice cream sitting atop a slender and infinitely pointy cone, suggests Cornell physicist Liam McAllister. This throat becomes even more distended when the fields posited by string theory (technically called fluxes) are turned on. Cornell astronomer Rachel Bean argues that because a given Calabi-Yau space is likely to have more than one warped throat, a rubber glove makes a better analogy. “Our three-dimensional universe is like a dot moving down the finger of a glove,” she explains. Inflation ends when the brane, or “dot,” reaches the tip of the glove, where an antibrane, or a stack of antibranes, sits. Because the motion of the brane is constrained by the shape of the finger or throat, she says, “the specific features of inflation stem from the geometry of that throat.”11

Regardless of the chosen analogy, different warped throat models lead to different predictions about the cosmic string spectrum—the full range of strings, of different tensions, expected to arise under inflation—which, in turn, could give us an indication of what Calabi-Yau geometry underlies our universe. “If we’re lucky enough to see that [entire spectrum of cosmic strings],” Polchinski says, “we may be able to say that some specific picture of a warped throat looks right, whereas another does not.”12

Even if we’re not so lucky as to see a cosmic string—or, better yet, to see a whole network of them—we might still constrain the shape of Calabi-Yau space through cosmological observations that rule out some models of cosmic inflation while allowing others. At least that’s the strategy being pursued by University of Wisconsin physicist Gary Shiu and his colleagues. “How are the extra dimensions of string theory curled up?” asks Shiu. “We argue that precise measurements of the cosmic microwave background (CMB) can give us clues.”13

As Shiu suggests, the latest string-theory-based models of cosmic inflation are nearing the point where they can make detailed predictions about our universe. These predictions, which vary depending on the specific Calabi-Yau geometry wherein inflation is assumed to originate, can now be tested against CMB data.

The basic premise is that inflation is driven by the motions of branes. And the thing we call our universe actually sits on a brane (of three dimensions) as well. In this scenario, a brane and its counterpart, an antibrane, slowly move toward each other in the extra dimensions. (In a more refined version of the story, the brane motions take place in a warped throat region within those extra dimensions.) Because of the mutual attraction, the separation of these branes represents a source of potential energy that drives inflation. The fleeting process, during which our four-dimensional spacetime expands exponentially, continues until the branes smack into each other and annihilate, unleashing the heat of the Big Bang and making an indelible imprint on the CMB. “The fact that the branes have been moving allows us to learn much more about that space than if they were just sitting in the corner,” Tye says. “Just like at a cocktail party: You don’t pick up much if you stay in one corner. But if you move around, you’re bound to learn more.”14

Researchers like Tye are encouraged by the fact that data is getting precise enough that we can say that one Calabi-Yau space is consistent with experiment, whereas another space is not. In this way, cosmological measurements are starting to impose constraints on the kind of Calabi-Yau space we might live in. “You can take inflation models and divide them in half—those that are favored by observations and those that are not,” says Perimeter Institute physicist Cliff Burgess. “The fact that we can now distinguish between inflation models means we can also distinguish between the geometric constructions that give rise to those models.”15

Shiu and his former graduate student Bret Underwood (now at McGill University) have taken some additional steps in this direction. In a 2007 paper in Physical Review Letters, Shiu and Underwood showed that two different geometries for the hidden six dimensions—variations of familiar Calabi-Yau conifolds with warped throats—would lead to different patterns in the distribution of cosmic radiation. For their comparison, Shiu and Underwood picked two throat models—Klebanov-Strassler and Randall-Sundrum, whose geometries are reasonably well understood—and then looked at how inflation under these distinct conditions would affect the CMB. In particular, they focused on a standard CMB measurement, temperature fluctuations in the early universe. These fluctuations should appear roughly the same on small and large scales. The rate at which the fluctuations change as you go from small to large scales is called the spectral index. Shiu and Underwood found a 1 percent difference in the spectral index between the two scenarios, showing that the choice of geometry has a measurable effect.

Although that might seem inconsequential, a 1 percent difference can be significant in cosmology. And the recently launched Planck observatory should be able to make spectral index measurements to at least that level of sensitivity. In other words, the Klebanov-Strassler throat geometry might be allowed by Planck data, while the Randall-Sundrum throat might not, or vice versa. “Away from the tip of the throat, the two geometries look almost identical, and people used to think they could be used interchangeably,” notes Underwood. “Shiu and I showed that the details do matter.”16

However, going from the spectral index, which is a single number, to the geometry of the extra dimensions is a huge leap. This is the so-called inverse problem: If we see enough data in the CMB, can we determine what Calabi-Yau it is? Burgess doesn’t think this will be possible in “our lifetime,” or at least in the dozen or so years he has left before retirement. McAllister is also skeptical. “We’ll be lucky in the next decade just to be able to say inflation did or did not occur,” he says. “I don’t think we’ll get enough experimental data to flesh out the full shape of the Calabi-Yau space, though we might be able to learn what kind of throat it has or what sort of branes it contains.”17

Shiu is more optimistic. While the inverse problem is much harder, he acknowledges, we still ought to give it our best shot. “If you can only measure the spectral index, it’s hard to say something definitive about the geometry. But you’d get much more information if you could measure something like non-Gaussian features of the CMB.” A clear indication of non-Gaussianity, he says, would impose “much more constraints on the underlying geometry. Instead of being one number like the spectral index, it’s a whole function—a whole bunch of numbers that are all related to each other.” A large degree of non-Gaussianity, Shiu adds, could point to a specific version of brane inflation—such as the Dirac-Born-Infeld (or DBI) model—occurring within a well-prescribed throat geometry. “Depending on the precision of the experiment, such a finding could, in fact, be definitive.”18

Columbia physicist Sarah Shandera notes that a string-theory-motivated inflation model like DBI may, ironically, prove important even if we find out that string theory is not the ultimate description of nature. “That’s because it predicts a kind of non-Gaussianity that cosmologists hadn’t thought of before,” Shandera says.19 And in any experimental endeavor, knowing what questions to ask, how to frame them, and what to look for is a big part of the game.

Other clues for string inflation could come from gravitational waves emitted during the violent phase transition that spawned inflation. The largest of these primordial spacetime ripples cannot be observed directly, because their wavelengths would now span the entire visible universe. But they would leave a mark in the microwave background. While this signal would be hard to extract from CMB temperature maps, say theorists, gravitational waves would create a distinctive pattern in maps of the polarization of the CMB’s photons.

In some string inflation models, the gravitational wave imprint would be detectable; in others, it would not. Roughly speaking, if the brane moves a small distance on the Calabi-Yau during inflation, there’ll be no appreciable gravitational wave signal. But if the brane travels a long way through the extra dimensions, says Tye, “tracing out small circles like the grooves on a record, the gravitational signal could be big.” Getting the brane to move in this tightly circumscribed manner, he adds, “takes a special type of compactification and a special type of Calabi-Yau. If you see it, you know it must be that kind of manifold.” The compactifications we’re talking about here are ones in which the moduli are stabilized, implying the presence of warped geometry and warped throats in particular.20

Getting a handle on the shape of Calabi-Yau space, including its throaty appendages, will require precise measurements of the spectral index and, hopefully, sightings of non-Gaussianity, gravitational waves, and cosmic strings as well. Patience is also in order, suggests Shiu. “Although we now have confidence in the Standard Model of physics, that model did not materialize overnight. It came from a whole sequence of experiments over the course of many years. In this case, we’ll need to bring a lot of measurements together to get an idea of whether extra dimensions exist or whether string theory seems to be behind it all.”21

The overall goal here is not just to probe the geometry of the hidden dimensions. It’s also to test string theory as a whole. McAllister, among others, believes that this approach may offer our best shot at an experimental test of the theory. “It’s possible that string theory will predict a finite class of models, none of which are consistent with the observed properties of the early universe, in which case we could say the theory is excluded by observation. Some models have already been excluded, which is exciting because it means the cutting-edge data really does make a difference.” While that kind of statement is not at all novel for physics, it is novel for string theory, which has yet to be experimentally verified. At the moment, he adds, warped throat inflation is one of the best models we’ve produced so far, “but in reality, inflation may not occur in warped throats even though the picture looks quite compelling.”22

In the end, Bean agrees, “models of inflation in warped throats may not be the answer. But these models are based on geometries coming out of string theory for which we can make detailed predictions that we can then go out and test. In other words, it’s a way of making a start.”23

The good news is that there is more than one way of making a start. While some investigators are scouring the night (or day) sky for signs of extra dimensions, other eyes are trained on the Large Hadron Collider (LHC). Finding hints of extra dimensions may not be the top priority of the LHC, but it still ranks high on the list.

The most logical starting point for string theorists is to look for the supersymmetric partners of the particles we know. Supersymmetry is of interest to many physicists, string theory aside: The lowest-mass supersymmetric partner—which could be the neutralino, gravitino, or sneutrino, among others—is extremely important in cosmology because it’s considered a leading candidate for dark matter. The presumed reason we haven’t seen these particles yet—which is why they have remained invisible and hence “dark”—is that they are more massive than ordinary particles. Particle colliders up until now have not been powerful enough to produce these heavier “superpartners,” whereas the LHC is on the cusp of being able to do so.

In the string-theory-inspired models of Harvard’s Cumrun Vafa and Jonathan Heckman of the Institute for Advanced Study, the gravitino—which is the hypothetical superpartner of the graviton (the particle responsible for gravity)—is the lightest partner. Unlike heavier superpartners, the gravitino would be completely stable, because there is nothing lighter that it could decay into. The gravitino, in their model, accounts for the bulk of the universe’s dark matter. Although it would be too weakly interacting to be observable at the LHC, Vafa and Heckman believe that another theorized supersymmetric particle—the stau, the superpartner of the so-called tau lepton—would be stable from anywhere from a second to an hour, more than long enough to leave a recognizable track in LHC’s detectors.

Finding such particles would corroborate an important aspect of string theory. Calabi-Yau manifolds, as we’ve seen, were handpicked by string theorists as a suitable geometry for the extra dimensions partly because supersymmetry is automatically built into their internal structure. Discovering signs of supersymmetry at LHC would thus be encouraging news, to say the least, for string theory and the whole Calabi-Yau picture. For one thing, the attributes of the supersymmetric particles could tell us about the hidden dimensions themselves, explains Burt Ovrut, “because how you compactify the Calabi-Yau manifold affects the kind of supersymmetry you get and the degree of supersymmetry you get. You can find compactifications that preserve supersymmetry or break it completely.”24


12.2—Experiments at the Large Hadron Collider (LHC) at the CERN lab in Geneva could find hints of extra dimensions or the existence of supersymmetric particles. Apparatus from the LHC’s ATLAS experiment is shown here. (Courtesy of CERN)

Confirmation of supersymmetry would not confirm string theory per se, but it would at least point in the same direction, showing that part of the story that string theory tells is correct. Not observing supersymmetric particles, on the other hand, would not bury string theory, either. It could mean that we’ve miscalculated and the particles are just beyond the reach of the LHC. (Vafa and Heckman, for instance, allow for the possibility that the LHC might generate a semistable and electrically neutral particle instead of the stau, which could not be seen directly.) If it turns out that the superpartners are slightly more massive than can be produced at this collider, it would take still higher energies to reveal them—and a long wait for the new machine that will eventually replace it.

Although it’s a long shot, the LHC might turn up more direct, and less ambiguous, evidence of the extra dimensions predicted by string theory. In experiments already planned at the facility, researchers will look for particles bearing signs of the extra dimensions from where they came—so-called Kaluza-Klein particles. The idea here is that vibrations in higher dimensions would manifest themselves as particles in the four-dimensional realm we inhabit. We might either see remnants of the decay of these Kaluza-Klein particles or perhaps even hints of such particles (along with their energy) disappearing from our world and then crossing over to the higher-dimensional realm.

Unseen motion in the extra dimensions would confer momentum and kinetic energy to a particle, so Kaluza-Klein particles are expected to be heavier than their slower, four-dimensional counterparts. Kaluza-Klein gravitons are an example. They’d look like ordinary gravitons, which are the particles that transmit the gravitational force, only they’d be heavier by virtue of the extra momentum they carry. One way to pick out such gravitons amid the vast sea of particles produced at LHC is to look not only at the particle’s mass but also at its spin. Fermions, such as electrons, have a certain amount of angular momentum that we classify as spin-½. Bosons, like photons and gluons, have somewhat more angular momentum and are classified as spin-1. Any particles with spin-2 detected at LHC are likely to be Kaluza-Klein gravitons.

Such a detection would be momentous indeed, for physicists not only would have caught the first glimpse of a long-sought particle, but also would have obtained strong evidence of extra dimensions themselves. Showing the existence of at least one extra dimension would be a breathtaking find in itself, but Shiu and his colleagues would like to go even further, gaining hints about the geometry of that extra-dimensional space. In a 2008 paper Shiu wrote with Underwood, Devin Walker of the University of California, Berkeley, and Kathryn Zurek of the University of Wisconsin, the team found that small changes in the shape of the extra dimensions would cause big changes—on the order of 50 to 100 percent—in both the mass and the interactivity of Kaluza-Klein gravitons. “When we changed the geometry a little bit, the numbers changed dramatically,” Underwood notes.25

Though it’s a far cry from being able to say anything conclusive about the shape of inner space or to specify the exact Calabi-Yau geometry, the analysis by Shiu et al. offers some hope of using experiments to “reduce the class of allowed shapes to a smaller range of possibilities. The power lies in cross-correlating between different types of experiments in both cosmology and high-energy physics,” Shiu says.26

The mass of the particles, as determined at LHC, would also provide hints about the size of the extra dimensions. For particles that venture into the higher-dimensional domain, the smaller those dimensions are, the heavier the particles will be. As for why that’s the case, you could ask how much energy it takes to stroll down a short hallway. Probably not much, you figure. But what if the hallway weren’t short but was instead very narrow? Getting through that tunnel will involve a struggle every inch of the way—accompanied, no doubt, by a string of curses and dietary vows—while requiring a bigger expenditure of energy. That’s roughly what’s going on here, but in more technical terms, it really comes down to the Heisenberg uncertainty principle, which states that the momentum of a particle is inversely proportional to the accuracy of its position measurement. Phrased another way, if a wave or particle is confined to a tiny, tiny space, where its position is thus highly constrained, it will have tremendous momentum and a correspondingly high mass. Conversely, if the extra dimensions are large, the wave or particle will have more room to move in and correspondingly less momentum, and will therefore be lighter.

There is a catch, however: The LHC will only detect things like Kaluza-Klein gravitons if these particles are far, far lighter than we’ve traditionally expected, which is another way of saying that either the extra dimensions must be extremely warped or they must be much larger than the Planck-scale range traditionally assumed in string theory. In the Randall-Sundrum picture of warping, for instance, the extra-dimensional space is bounded by two branes, with the spacetime between them everywhere curved. On one brane, which exists at a high-energy scale, gravity is strong; on the other brane—the lower-energy one on which we live—gravity is feeble. An effect of that arrangement is that mass and energy values change drastically, depending on one’s position with respect to the two branes. This means that the mass of fundamental particles, which we ordinarily held to be close to the Planck scale (on the order of 1028 electron volts), would instead be “rescaled” to something closer to 1012 electron volts, or 1 tera-electron volt (1 TeV), which might place them within the range of the LHC. The size of the extra dimensions in this picture could be as small as in conventional string theory models (though that is not required), whereas particles themselves would appear to be much lighter (and therefore lower-energy) than is typically assumed.

Another novel approach under consideration today was first proposed in 1998 by the physicists Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali when they were all at Stanford. Challenging Oskar Klein’s assertion that we can’t see any extra dimensions because they are so small, Arkani-Hamed, Dimopoulos, and Dvali—a trio often abbreviated as ADD—claimed that the extra dimensions could be much larger than the Planck scale, at least 10-12 cm and possibly as big as 10-1 cm (a millimeter). This would be possible, they said, if our universe is stuck on a three-dimensional brane (with an additional dimension thrown in for time) and if that three-dimensional world is all we can see.

This might seem like an odd argument to make—after all, the idea that the extra dimensions are minuscule is the premise on which the vast majority of string theory models have been built. But it turns out that the size of Calabi-Yau space, which is often considered a given, “is still an open question,” according to Polchinski. “For mathematicians, the size of a space is about the least interesting thing. If you double the size of something, mathematically that’s trivial. But to a physicist, the size is anything but trivial because it tells us how much energy you need to see it.”27

The ADD scenario does more than just expand the size of the extra dimensions; it lowers the energy scale at which gravity and the other forces become unified, and in the process lowers the Planck scale as well. If Arkani-Hamed and his colleagues are right, energy generated through particle collisions at the LHC may seep out into higher dimensions, showing up as an ostensible violation of energy conservation laws. And in their picture, even strings themselves, the basic unit of string theory, might become big enough to see—something that had never been deemed possible before. The ADD team was motivated, in part, to account for the apparent weakness of gravity in comparison with the other forces, given that a compelling explanation for that disparity had not yet been put forth. ADD theory suggests a novel answer: Gravity is not weaker than the other forces, but only appears to be weaker because, unlike the other forces, it “leaks” into the other dimensions so that we feel just a tiny fraction of its true strength. The same sort of thing happens when balls smack into each other on a pool table; some of the kinetic energy, which had been bound up in the moving balls and thus confined to the table’s two-dimensional surface, escapes in the form of sound waves into the third dimension.

The details of how this might work suggest possible observational strategies: Gravity as we know it in four-dimensional spacetime obeys an inverse square law. The gravitational influence of a body drops off with the square of the distance from it. But if we added another dimension, gravity would drop off as the cube of distance. With ten dimensions, as is posited in string theory, gravity would drop off according to the eighth power of distance. In other words, the more extra dimensions, the weaker gravity would appear as measured from our four-dimensional perspective. (The electrostatic force, similarly, is inversely proportional to the square of the distance between two point charges in four-dimensional spacetime and inversely proportional to the eighth power of the distance in ten-dimensional spacetime.) In thinking about gravity over big distances, as we do in astronomy and cosmology, the inverse square law works perfectly well because those interactions take place in the space of three giant dimensions plus time. We wouldn’t notice a gravitational pull in a strange new direction (corresponding to a hidden internal dimension) until we got down to a scale small enough to move around in those dimensions. And since we’re physically barred from doing that, our best and perhaps only hope is to look for hints of the extra dimensions in the form of deviations from the inverse square law. That is precisely the effect that physicists at the University of Washington, the University of Colorado, Stanford University, and elsewhere are looking for through short-distance gravity measurements.

While these researchers have different experimental apparatus at their disposal, their goals are the same nevertheless: to measure the strength of gravity on a small scale at precisions never before dreamed of. Eric Adelberger’s team at the University of Washington, for instance, performs “torsion balance” experiments that are similar in spirit to those conducted in 1798 by Henry Cavendish. The basic approach is to infer the strength of gravity by measuring the torque on a suspended pendulum. Adelberger’s group employs a small metal pendulum dangling above a pair of metal disks, which exert a gravitational pull on the pendulum. The attractive forces from the two disks are balanced in such a way that if Newton’s inverse square law holds, the pendulum will not twist at all.


12.3—Minute rotations induced by gravitational attraction are measured at short distances, and with great precision, by the Mark VI pendulum designed and operated by the Eöt-Wash research group at the University of Washington. If gravity were observed to behave differently at close range and to deviate from the inverse square law well established in classical physics, it could signal the presence of the extra dimensions predicted by string theory. (University of Washington/Mary Levin)

In the experiments performed to date, the pendulum has shown no sign of twisting, as measured to an accuracy of a tenth of a millionth of a degree. By placing the pendulum ever closer to the disks, the researchers have ruled out the existence of dimensions larger than about 40 microns (or micrometers) in radius. In future experiments, Adelberger aims to test gravity at an even smaller scale, securing measurements down to a dimension size of about 20 microns. But that may be the limit, he says. In order to look at ever smaller scales, a different technological approach will probably be needed to test the large-dimension hypothesis.

Adelberger considers the idea of large extra dimensions revolutionary, but says that this doesn’t make it right.28 We not only need new tactics for probing the large-dimension question, but also need new tactics for dealing with more general questions about the existence of extra dimensions and the veracity of string theory.

So that’s where things stand today, with various leads being chased down—only a handful of which have been discussed here—and no sensational results to speak of yet. Looking ahead, Shamit Kachru, for one, is hopeful that the range of experiments under way, planned, or yet to be devised will afford many opportunities to see new things. Nevertheless, he admits that a less rosy scenario is always possible, in the event that we live in a frustrating universe that offers little, if anything, in the way of empirical clues. “If we see nothing in cosmology, nothing in accelerator experiments, and nothing in lab experiments, then we’re basically stuck,” says Kachru. Although he considers such an outcome unlikely, he says this kind of situation is in no way peculiar to string theory or cosmology, as the dearth of data would affect other branches of science in the same way.29

What we do next, after coming up empty-handed in every avenue we set out, will be an even bigger test than looking for gravitational waves in the CMB or infinitesimal twists in torsion-balance measurements. For that would be a test of our intellectual mettle. When that happens, when every idea goes south and every road leads to a dead end, you either give up or try to think of other questions you can ask—questions for which there might be some answers.

Edward Witten, who, if anything, tends to be conservative in his pronouncements, is optimistic in the long run, feeling that string theory is too good not to be true. Though in the short run, he admits, it’s going to be difficult to know exactly where we stand. “To test string theory, we will probably have to be lucky,” he says. That might sound like a slender thread upon which to pin one’s dreams for a theory of everything—almost as slender as a cosmic string itself. But fortunately, says Witten, “in physics, there are many ways of being lucky.”30

I have no quarrel with that statement and, more often than not, tend to agree with Witten, as I’ve generally found this to be a wise policy. But if the physicists find their luck running dry, they might want to turn to their mathematical colleagues, who have enjoyed their fair share of that commodity as well.