The problems with branes
As an explanation of the weakness of gravity, the world-as-brane idea is striking. However, there are a number of problems and constraints, many of which were first pointed out by Arkani-Hamed, Dimopoulos and Dvali. Many have to do with cosmology, some with astrophysics, and some problems are more aesthetic.
First, the naive statement that two extra dimensions, 0.2 mm in size, are allowed is not quite correct. Such large dimensions would significantly affect the behaviour of astrophysical objects, such as supernovae, because they would cause the object to lose energy by emitting gravitons into the bulk. This graviton emission would show up as an anomalous cooling of the objects' interiors. A precise calculation shows that the two extra dimensions must be smaller than the sub-millimetre size currently accessible in tabletop experiments.
However, the bulk almost certainly has other fields besides gravity. For example, if there are gauge fields in the bulk that are associated with "new forces", then their strength is predicted to be roughly a million times stronger than the gravitational force. It would therefore be possible to detect these stronger forces in tabletop experiments. In addition, gauge forces between like-charged objects are naturally repulsive, so we may even find that gravity seems to become repulsive on sub-millimetre distance scales.
Secondly, although it is inspired by particle physics, the world-as-a-brane picture has dramatic implications for the early evolution of the universe. Conversely, cosmology can place severe constraints on the brane picture. To understand why, recall that in the traditional cosmological view, what we see when we look at the sky today is the remnant of an earlier epoch when the universe was much smaller and hotter. Moreover, the traditional picture of the universe's evolution since the big bang is remarkably successful in many details. For example, it is possible to calculate the synthesis of the light elements - hydrogen, helium, deuterium, lithium and beryllium - using physics that is very well understood. The relative abundances of these light elements agree with measurements, provided that the universe evolved in a conventional way from temperatures below about 3 MeV. (Note that 1 MeV is approximately 1010 kelvin.) This poses a potential problem for the brane-world idea because of a striking new effect that limits how far back our universe can evolve normally.
If our universe, with its three spatial dimensions, is trapped on a brane then it could cool by emitting gravitons into the higher-dimensional bulk, just like a hot object - such as an ember from a fire - typically cools through the emission of infrared radiation in our 3-D world. In the conventional picture, this process does not occur since there is no space "outside" our universe into which the radiation can evaporate. However, for a brane world there are now two processes by which our world with its three spatial dimensions can cool: expansion, plus evaporation into the bulk.
Our conventional view of the evolution means that the first form of cooling should dominate. However, evaporative cooling prevailed at early times when the universe was very hot. Consequently there is a maximum temperature, T*, above which the universe would have evolved in an unconventional way. Calculations show that T* varies from about 1 MeV to 500 MeV as the number of extra dimensions increases from two to six. For two extra dimensions, this is below the temperature at which nucleosynthesis begins, which leads to an unacceptable modification of the light-element abundances.
One way around this is to raise the new fundamental scale of gravity above 1000 GeV, in which case the modification of the evolution of our universe is pushed to higher, and safer, temperatures.
Furthermore, such evaporation is dangerous for another reason. It fills the bulk with energetic gravitons, which can later decay into energetic photons on the brane, thus leading to an unacceptable distortion of the diffuse gamma-ray background that astronomers observe.
The upshot of this analysis is that the universe should never have had a temperature that exceeded about 1 GeV. Moreover, it is difficult, but not impossible, to accommodate the other necessary cosmological ingredients - including inflation and baryogenesis - in such a constrained scenario.
The quest for unification
The third problem is more aesthetic and has to do with the unification of the electromagnetic, weak and strong forces into a single force. One of the most successful and appealing aspects of the traditional view of the world at energy scales above 1000 GeV is that the full unification of these forces comes almost for free. Using conventional physics in 4-D space-time, we can predict how the strengths of the forces change as we increase the energy of an interaction. For example, in the supersymmetric version of the Standard Model - a collection of theories that describes our current understanding of the building blocks of matter and their interactions - the strengths of the three gauge forces become identical (or unify) when we extrapolate the energy to 1016 GeV.
In addition, this unification satisfies a number of non-trivial theoretical and experimental consistency tests. For example, it predicts one of the most important parameters of the Standard Model - the ratio of the strengths of weak interactions to electromagnetic ones. Furthermore, the scale of unification is high enough to prevent the decay of protons.
There are a number of extensions and refinements to this theory that also work well, and it is rather hard to give up this success. Does the would-be new paradigm do as well in this regard? Unfortunately, at the moment, the answer is no, but there are glimmers of hope.
At first glance, the success of the unification of forces seems to be absolutely destroyed by the world-on-a-brane picture. According to this theory, our usual description of the world would break down above 1000 GeV, the new fundamental scale of gravity, and the strengths of the forces would no longer evolve in a way that leads to successful unification.
One possibility emerged a few months after the appearance of Arkani-Hamed, Dimopoulos and Dvali's paper. Keith Dienes, Emilian Dudas and Tony Gherghetta, then all based at the CERN laboratory in Geneva, suggested that the gauge forces can feel some extra dimensions, but not the very large ones that explain the weakness of gravity. They showed that in some cases it was possible to regain a different form of unification that now occurred close to the fundamental scale of gravity of 1000 GeV or above. The concern with this approach is that the calculations, and the possibilities for proton decay, are now very sensitive to the exact theory at the new fundamental scale of gravity, so reliable predictions are difficult to obtain. Also there was no explanation of why the unification of forces in the standard 4-D world was so successful. We are forced to assume that its success was just a lucky accident.
Another approach was initiated at roughly the same time by Antoniadis and by Costas Bachas at the Ecole Normale Supérieure in Paris, and later developed by Arkani-Hamed, Dimopoulos and one of us (JMR). The idea uses some special features of two large extra dimensions. The strengths of the gauge forces on a string-theory brane depend on the properties of the bulk. For two extra dimensions, the variation of this strength can mimic the way that the electromagnetic, weak and strong forces vary with energy in the supersymmetric version of the Standard Model. Thus in Antoniadis and Bachas's approach it might be possible to keep the attractive unification prediction of the standard approach and explain its success. However, no model has been constructed that is successful in detail.
New solutions to old problems
As well as these difficulties, however, the brane-world picture offers new solutions to old problems. One example is the dark-matter problem - why does most of the matter that gravitates in the universe seem to be invisible? (see Smith and Spooner in further reading). An interesting possibility raised by the brane-world proposal is that this mysterious form of matter is trapped on another brane. Such matter would be invisible since it can only communicate to us through the bulk via gravity. In particular, matter on a different brane cannot emit photons by which we could observe it. The existence of other parallel branes in the bulk is very natural, and indeed string theories typically require multiple sets of such branes.
The brane-world picture also offers an intriguing explanation for why the fundamental particles vary so widely in mass. Neutrinos, for example, seem to weigh less than a few electronvolts while other particles are over a billion times heavier. These ideas were originally suggested by Arkani-Hamed, Dimopoulos, Dvali and one of us (JMR) and also by Dienes, Dudas and Gherghetta. In these scenarios, the large size of the extra-dimensional bulk suppresses the interactions that give rise to particle masses. This suppression is possible if there are new fields, in addition to the graviton, that propagate in the bulk and do not feel the influence of the electromagnetic, weak and strong forces. In this picture, the observed neutrinos have such a small mass for precisely the same reason that gravity is very weak.
Finally, the most serious of all problems in particle physics and cosmology is the cosmological constant (see the article by Caldwell and Steinhardt, this issue). This term in Einstein's equations of general relativity is roughly a measure of the mass density of the vacuum. Although the cosmological constant is predicted by our current theories and by world-on-a-brane scenarios to be very large, nature appears to have tuned it to be incredibly small. In fact, the existence of a large long-lived universe demands that the cosmological constant is tiny. Consequently this number is the most constrained and the smallest constant in nature.
Explaining why the cosmological constant is so small has occupied cosmologists and particle physicists ever since Einstein first introduced it. Many proponents of the brane-world picture are tackling this problem again. One recent approach, motivated by a variation of the brane-world idea developed by Lisa Randall of the Massachusetts Institute of Technology and Raman Sundrum of Boston University, is to look at branes in which the bulk dimensions are extremely curved or "warped", but not necessarily compactified. By warping the extra dimensions in the right way, it may be possible to explain why the cosmological constant appears to be so small.
Kaluza-Klein gravitons and black holes
What other experimental signatures might arise from our world being a brane embedded in a higher-dimensional space? One possibility is the appearance of new states, called Kaluza-Klein excitations, at high-energy colliders. These excited states are a feature of models with compactified dimensions, and can be understood by drawing an analogy with water.
Imagine a swimming pool that is infinitely long and just 1 mm wide. Not much use for swimming in admittedly, but the infinitely large side is a good analogy for the large dimensions we experience every day, while the short side is like a compactified dimension. Waves moving in the long direction can have any wavelength, and this is analogous to particles being able to take any energy. However, it is much harder to excite waves in the short direction. In fact, the waves must be smaller than 1 mm to exist at all. Shorter waves are more energetic, so a single wavelength of a 1 mm wave corresponds to the first Kaluza-Klein state, the next state has two 0.5 mm wavelengths and so on.
The large extra dimensions that are felt only by gravity can reveal themselves through the emission of gravitational Kaluza-Klein states into the bulk. This emission is another way of describing the process of graviton "evaporation". Moreover, because of the relatively large size of the extra dimension, the mass difference between one Kaluza-Klein state and the next is very small. There is therefore a huge number of such Kaluza-Klein excitations below the new fundamental scale of gravity. The combined effect of these excitations might be observable close to the new fundamental energy scale. If this fundamental scale is about 1000 GeV then we could see evidence for Kaluza-Klein states in experiments at the Tevatron collider at Fermilab or at the Large Hadron Collider (LHC) at CERN, which will be completed in 2005.
A typical process might involve a proton and antiproton colliding to produce a single spray or jet of particles plus a graviton, which is emitted into the bulk. Since the energy of the graviton would be lost from our 4-D world, the telltale sign for such a process would be an excess of collisions with one jet and "missing" energy above the expectations of the Standard Model.
The particles that are confined to the brane also have Kaluza-Klein or higher string-excitation states, but for them the relevant scale (i.e. the width of the pool) is either the brane thickness or the new fundamental string scale. Both of these scales should correspond in energy to the new gravity scale of 1000 GeV or higher. The LHC could well produce fundamental string or brane relations of our familiar particles. For example, whole towers of Kaluza-Klein states that look like very heavy versions of electrons, photons and so on could be produced. Since these states feel the forces of the Standard Model they would be easy to detect, giving dramatic signals.
As yet, however, there is no evidence for Kaluza-Klein states up to energies of roughly 1000 GeV from high-energy colliders. And this is how we know that the strong, weak and electromagnetic forces do not feel extra dimensions.
Even more strikingly, due to the now much stronger gravitational interactions at short distances, there is also a slight possibility that microscopic black holes could be produced. Fortunately, such small black holes would quickly evaporate and would not be dangerous. In fact they would resemble exotic particles that decayed quite quickly. Nevertheless, it would be truly extraordinary if nature gave us the chance to study objects such as black holes directly in the laboratory.