by Brian Greene
Physicists have shown that precisely this problem arises in the standard big bang model. Detailed calculations show that there is no way for regions of space that are currently widely separated to have had the exchange of heat energy that would explain their having the same temperature. As the word horizon refers to how far we can see—how far light can travel, so to speak—physicists call the unexplained uniformity of temperature throughout the vast expanse of the cosmos the "horizon problem." The puzzle does not mean the standard cosmological theory is wrong. But the uniformity of temperature does strongly suggest that we are missing an important part of the cosmological story. In 1979, the physicist Alan Guth, now of the Massachusetts Institute of Technology, wrote the missing chapter.
Inflation
The root of the horizon problem is that in order to get two widely separated regions of the universe close together, we have to run the cosmic film way back toward the beginning of time. So far back, in fact, that there is not enough time for any physical influence to have traveled from one region to the other. The difficulty, therefore, is that as we run the cosmological film backward and approach the big bang, the universe does not shrink at a fast enough rate.
Well, that's the rough idea, but it's worthwhile sharpening the description a bit. The horizon problem stems from the fact that like a ball tossed upward, the dragging pull of gravity causes the expansion rate of the universe to slow down. This means that, for example, to halve the separation between two locations in the cosmos we must run the film back more than halfway toward its beginning. In turn, we see that to halve the separation we must more than halve the time since the big bang. Less time since the bang—proportionally speaking—means it is harder for the two regions to communicate, even though they get closer.
Guth's resolution of the horizon problem is now simple to state. He found another solution to Einstein's equations in which the very early universe undergoes a brief period of enormously fast expansion—a period during which, in fact, it "inflates" in size at an unheralded exponential expansion rate. Unlike the case of a ball that slows down after being tossed upward, exponential expansion gets faster as it proceeds. When we run the cosmic film in reverse, rapid accelerating expansion turns into rapid decelerating contraction. This means that to halve the separation between two locations in the cosmos (during the exponential epoch) we need run the film back less than halfway—much less, in fact. Running the film back less implies that the two regions will have had more time to communicate thermally and, like hot soup and air, they will have had ample time to come to the same temperature.
Through Guth's discovery and later important refinements made by André Linde, now of Stanford University, Paul Steinhardt and Andreas Albrecht, then of the University of Pennsylvania, and many others, the standard cosmological model was revamped into the inflationary cosmological model. In this framework, the standard cosmological model is modified during a tiny window of time—around 10-36 to 10-34 seconds ATB—in which the universe expanded by a colossal factor of at least 1030, compared with a factor of about a hundred during the same time interval in the standard scenario. This means that in a brief flicker of time, about a trillionth of a trillionth of a trillionth of a second ATB, the size of the universe increased by a greater percentage than it has in the 15 billion years since. Before this expansion, matter that is now in far-flung regions of the cosmos was much closer together than in the standard cosmological model, making it possible for a common temperature to be easily established. Then, through Guth's momentary burst of cosmological inflation—followed by the more usual expansion of the standard cosmological model—these regions of space were able to become separated by the vast distances we witness currently. And so, the brief but profound inflationary modification of the standard cosmological model solves the horizon problem (as well as a number of other important problems we have not discussed) and has gained wide acceptance among cosmologists.3
We summarize the history of the universe from just after the Planck time to the present, according to the current theory, in Figure 14.1.
Cosmology and Superstring Theory
There remains a sliver of Figure 14.1, between the big bang and the Planck time, that we have not yet discussed. By blindly applying the equations of general relativity to that region, physicists have found that the universe continues to get ever smaller, ever hotter, and ever denser, as we move backward in time toward the bang. At time zero, as the size of the universe vanishes, the temperature and density soar to infinity, giving us the most extreme signal that this theoretical model of the universe, firmly rooted in the classical gravitational framework of general relativity, has completely broken down.
Nature is telling us emphatically that under such conditions we must merge general relativity and quantum mechanics—in other words, we must make use of string theory. Currently, research on the implications of string theory for cosmology is at an early stage of development. Perturbative methods can, at best, give skeletal insights, since the extremes of energy, temperature, and density require precision analysis. Although the second superstring revolution has provided some nonperturbative techniques, it will be some time before they are honed for the kinds of calculations required in a cosmological setting. Nevertheless, as we now discuss, during the last decade or so, physicists have taken the first steps toward understanding string cosmology. Here is what they have found.
It appears that there are three essential ways in which string theory modifies the standard cosmological model. First, in a manner that current research continues to clarify, string theory implies that the universe has what amounts to a smallest possible size. This has profound consequences for our understanding of the universe at the moment of the bang itself, when the standard theory claims that its size has shrunk all the way to zero. Second, string theory has a small-radius/large-radius duality (intimately related to its having a smallest possible size), which also has deep cosmological significance, as we will see in a moment. Finally, string theory has more than four spacetime dimensions, and from a cosmological standpoint, we must address the evolution of them all. Let's discuss these points in greater detail.
In the Beginning There Was a Planck-Sized Nugget
In the late 1980s, Robert Brandenberger and Cumrun Vafa made the first important strides toward understanding how the application of these string theoretic features modifies the conclusions of the standard cosmological framework. They came to two important realizations. First, as we run the clock backward in time toward the beginning, the temperature continues to rise until the size of the universe is about the Planck length in all directions. But then, the temperature hits a maximum and begins to decrease. The intuitive reason behind this is not hard to come by. Imagine for simplicity (as Brandenberger and Vafa did) that all of the space dimensions of the universe are circular. As we run the clock backward and the radius of each of these circles shrinks, the temperature of the universe increases. But as each of the radii collapses toward and then through the Planck length, we know that, within string theory, this is physically identical to the radii shrinking to the Planck length and then bouncing back toward increasing size. Since temperature goes down as the universe expands, we would expect that the futile attempt to squeeze the universe to sub-Planck size means that the temperature stops rising, hits a maximum, and then begins to decrease. Through detailed calculations, Brandenberger and Vafa explicitly verified that indeed this is the case.
This led Brandenberger and Vafa to the following cosmological picture. In the beginning, all of the spatial dimensions of string theory are tightly curled up to their smallest possible extent, which is roughly the Planck length. The temperature and energy are high, but not infinite, since string theory has avoided the conundrums of an infinitely compressed zero-size starting point. At this beginning moment of the universe, all the spatial dimensions of string theory are on completely equal footing—they are completely symmetric—all curled up into a multidimensional, Planck-sized nugget. Then, acco
rding to Brandenberger and Vafa, the universe goes through its first stage of symmetry reduction when, at about the Planck time, three of the spatial dimensions are singled out for expansion, while all others retain their initial Planck-scale size. These three space dimensions are then identified with those in the inflationary cosmological scenario, the post-Planck-time evolution summarized in Figure 14.1 takes over, and these three dimensions expand to their currently observed form.
Why Three?
An immediate question is, What drives the symmetry reduction that singles out precisely three spatial dimensions for expansion? That is, beyond the experimental fact that only three of the space dimensions have expanded to observably large size, does string theory provide a fundamental reason for why some other number (four, five, six, and so on) or, even more symmetrically, all of the space dimensions don't expand as well? Brandenberger and Vafa came up with a possible explanation. Remember that the small-radius/large-radius duality of string theory rests upon the fact that when a dimension is curled up like a circle, a string can wrap around it. Brandenberger and Vafa realized that, like rubber bands wrapped around a bicycle tire inner tube, such wrapped strings tend to constrict the dimensions they encircle, keeping them from expanding. At first sight, this would seem to mean that each of the dimensions will be constricted, since the strings can and do wrap them all. The loophole is that if a wrapped string and its antistring partner (roughly, a string that wraps the dimension in the opposite direction) should come into contact, they will swiftly annihilate one other, producing an unwrapped string. If these processes happen with sufficient rapidity and efficiency, enough of the rubber band-like constriction will be eliminated, allowing the dimensions to expand. Brandenberger and Vafa suggested that this reduction in the choking effect of wrapped strings will happen in only three of the spatial dimensions. Here's why.
Imagine two point particles rolling along a one-dimensional line such as the spatial extent of Lineland. Unless they happen to have identical velocities, sooner or later one will overtake the other, and they will collide. Notice, however, that if these same point particles are randomly rolling around on a two-dimensional plane such as the spatial extent of Flatland, it is likely that they will never collide. The second spatial dimension opens up a new world of trajectories for each particle, most of which do not cross each other at the same point at the same time. In three, four, or any higher number of dimensions, it gets increasingly unlikely that the two particles will ever meet. Brandenberger and Vafa realized that an analogous idea holds if we replace point particles with loops of string, wrapped around spatial dimensions. Although it's significantly harder to see, if there are three (or fewer) circular spatial dimensions, two wrapped strings will likely collide with one another—the analog of what happens for two particles moving in one dimension. But in four or more space dimensions, wrapped strings are less and less likely ever to collide—the analog of what happens for point particles in two or more dimensions.4
And so we have the following picture. In the first moment of the universe, the tumult from the high, but finite, temperature drives all of the circular dimensions to try to expand. As they do, the wrapped strings constrict the expansion, driving the dimensions back to their original Planck-size radii. But, sooner or later a random thermal fluctuation will drive three dimensions momentarily to grow larger than the others, and our discussion then shows that strings which wrap these dimensions are highly likely to collide. About half of the collisions will involve string/antistring pairs, leading to annihilations that continually lessen the constriction, allowing these three dimensions to continue to expand. The more they expand, the less likely it is for other strings to get entangled around them since it takes more energy for a string to wrap around a larger dimension. And so, the expansion feeds on itself, becoming ever less constricted as the dimensions get ever larger. We can now imagine that these three spatial dimensions continue to evolve in the manner described in the previous sections, and expand to a size as large as or larger than the currently observable universe.
Cosmology and Calabi-Yau Shapes
For simplicity, Brandenberger and Vafa imagined that all of the spatial dimensions are circular. In fact, as noted in Chapter 8, so long as the circular dimensions are large enough that they curve back on themselves only beyond the range of our current observational capacity, a circular shape is consistent with the universe we observe. But for dimensions that stay small, a more realistic scenario is one in which they are curled up into a more intricate Calabi-Yau space. Of course, the key question is, Which Calabi-Yau space? How is this particular space determined? No one has been able to answer this question. But by combining the drastic topology-changing results described in the preceding chapter with these cosmological insights, we can suggest a framework for doing so. Through the space-tearing conifold transitions, we now know that any Calabi-Yau shape can evolve into any other. So, we can imagine that in the tumultuous, hot moments after the bang, the curled-up Calabi-Yau component of space stays small, but goes through a frenetic dance in which its fabric rips apart and reconnects over and over again, rapidly taking us through a long sequence of different Calabi-Yau shapes. As the universe cools and three of the spatial dimensions get large, the transitions from one Calabi-Yau to another slow down, with the extra dimensions ultimately settling into a Calabi-Yau shape that, optimistically, gives rise to the physical features we observe in the world around us. The challenge facing physicists is to understand, in detail, the evolution of the Calabi-Yau component of space so that its present form can be predicted from theoretical principles. With the newfound ability of one Calabi-Yau to change smoothly into another, we see that the issue of selecting one, Calabi-Yau shape from the many may in fact be reduced to a problem of cosmology.5
Before the Beginning?
Lacking the exact equations of string theory, Brandenberger and Vafa were forced to make numerous approximations and assumptions in their cosmological studies. As Vafa recently said,
Our work highlights the new way in which string theory allows us to start addressing persistent problems in the standard approach to cosmology. We see, for example, that the whole notion of an initial singularity may be completely avoided by string theory. But, because of difficulties in performing fully trustworthy calculations in such extreme situations with our present understanding of string theory, our work only provides a first look into string cosmology, and is very far from the final word.6
Since their work, physicists have made steady progress in furthering the understanding of string cosmology, spearheaded by, among others, Gabriele Veneziano and his collaborator Maurizio Gasperini of the University of Torino. Gasperini and Veneziano have, in fact, come up with their own intriguing version of string cosmology that shares certain features with the scenario described above, but also differs in significant ways. As in the Brandenberger and Vafa work, they too rely on string theory's having a minimal length in order to avoid the infinite temperature and energy density that arises in the standard and inflationary cosmological theories. But rather than concluding that this means the universe begins as an extremely hot Planck-size nugget, Gasperini and Veneziano suggest that there may be a whole prehistory to the universe—starting long before what we have so far been calling time zero—that leads up to the Planckian cosmic embryo.
In this so-called pre-big bang scenario, the universe began in a vastly different state than it does in the big bang framework. Gasperini and Veneziano's work suggests that rather than being enormously hot and tightly curled into a tiny spatial speck, the universe started out as cold and essentially infinite in spatial extent. The equations of string theory then indicate that—somewhat as in Guth's inflationary epoch—an instability kicked in, driving every point in the universe to rush rapidly away from every other. Gasperini and Veneziano show that this caused space to become increasingly curved and results in a dramatic increase in temperature and energy density.7 After some time, a millimeter-sized three-dimensional region w
ithin this vast expanse could look just like the superhot and dense patch emerging from Guth's inflationary expansion. Then, through the standard expansion of ordinary big bang cosmology, this patch can account for the whole of the universe with which we are familiar. Moreover, because the pre-big bang epoch involves its own inflationary expansion, Guth's solution to the horizon problem is automatically built into the pre-big bang cosmological scenario. As Veneziano has said, "String theory offers us a version of inflationary cosmology on a silver platter."8
The study of superstring cosmology is rapidly becoming an active and fertile arena of research. The pre-big bang scenario, for example, has already generated a significant amount of heated, yet fruitful debate, and it is far from clear what role it will have in the cosmological framework that will ultimately emerge from string theory. Achieving these cosmological insights will, no doubt, rely heavily on the ability of physicists to come to grips with all aspects of the second superstring revolution. What, for example, are the cosmological consequences of the existence of fundamental higher-dimensional branes? How do the cosmological properties we have discussed change if string theory happens to have a coupling constant whose value places us more toward the center of Figure 12.11 rather than in one of the peninsular regions? That is, what is the impact of full-fledged M-theory on the earliest moments of the universe? These central questions are now being studied vigorously. Already, one important insight has emerged.
M-Theory and the Merging of All Forces
