Over the last year or so I’ve been devoting quite a bit of my time to exploring the origins and implications of a relatively new class of models known as Galileons. These may turn out to be nothing but mathematical curiosities, but while they’re still interesting I thought I’d try to explain what these theories are. This will be a little more technical than typical posts, but I’m hoping to get across the main reasons people are interested in these ideas even if the technicalities become a little much for some readers.
The resurgence of interest in extra dimensional models of particle physics and gravity during the last thirteen years has led to a number of novel approaches to cosmology, one of which is the fascinating idea of Dvali, Gabadadze and Porrati (DGP). In this picture, one begins by thinking of our four-dimensional world as residing on a brane floating in one extra dimension. The difference between this and other extra-dimensional models is that one imagines gravity as being described by a sum of the action for general relativity in 5d, and a 4d version just defined on the brane. This is rather technical, I know, but the main point is that gravity is described by an unusual but deceptively simple action. We, of course measure our world to be four-dimensional, and so the relevant question to ask of theories like this one is how the extra-dimensional physics manifests itself in the four-dimensional world.
As you might expect, this is a complicated issue. There is of course, the way in which the dynamics of four-dimensional gravity differ from that one would expect from pure General Relativity (GR). Furthermore, there are parts of the five dimensional gravity theory that manifest themselves as fields other than the graviton in four dimensions. One of these is a scalar field that can be interpreted as describing the bending of the brane in the extra dimension, and whose dynamics are bound up with those of the graviton in a complex way.
Now, surprisingly, one can learn quite a lot about this theory of modified gravity by doing away with the gravitational interactions all together! This so-called decoupling limit happens by taking the masses describing the strength of the gravitational interactions to infinity, while keeping a special combination of them – the one describing the strength of the self-interactions of the scalar field – constant. This limit is interesting because it isolates the dynamics of the scalar field, and nothing else. Given that what remains is a scalar field theory in four dimensions, one might guess that a host of possible terms would be allowed, and that their behavior would be well-understood; after all, scalar field theories have been studied for a long time and in great detail. However, it turns out that the symmetries of the DGP model, from which this theory originates, lead to an extremely restrictive form of the action – a scalar field theory with a single complicated derivative interaction, obeying the galilean symmetry under which the action is invariant when derivatives of the field are shifted by a constant vector.
I could go on to discuss only this theory, as a way to learn more about the DGP model. However, the realization that there existed a previously unconsidered symmetry of scalar field theories led Nicolis, Rattazzi and Trincherini to consider abstracting the symmetry, and asking what other terms may be allowed for scalar fields. And, remarkably, there turn out to only be five! In this abstracted scalar field theory we refer to these terms as the galileon terms, and to the scalar field itself as the Galileon. In a very nice paper, de Rham and Tolley later showed how these extra terms can also arise from their own actions for a brane living in a flat five-dimensional space. But for now, let’s just focus on the Galileons as interesting new four-dimensional scalar field theories.
I’m not going to write down the mathematical form of these terms here, but there are a number of properties they have that should illustrate why a number of people in the community have found them sufficiently interesting to warrant further study.
- The Galileon terms involve higher derivatives, but their equations of motion are only second order in time, and hence they avoid some well-known proofs of instability that plague a lot of higher derivative systems.
- There exists a range of energy scales over which the Galileon terms are important, and hence higher derivatives are important, yet quantum mechanical effects are irrelevant, and classical physics holds.
- The Galileon terms are unrenormalized! Their coefficients pick up no modifications from quantum corrections arising from other Galileon terms!
This last feature hints at a number of possible applications in cosmology. For example, cosmic acceleration, either in the early or the late universe, typically requires scalar fields with dynamics that are finely tuned, and hence are easily perturbed by quantum corrections. There is therefore the possibility that Galileons may lead to a natural way to achieve such behavior.
A number of authors have begun exploring these possibilities, and my collaborators and I have our eyes on them also. Before that, however, we’ve been spending a significant amount of time trying to understand how the Galileon idea might fit into more general frameworks. We’ve explored multi-Galileon theories, that may arise from the types of gravity action I described earlier, but with more than a single extra dimension. And more recently we’ve expanded on the idea that such theories arise from branes floating in a flat five-dimensional spacetime to show how entirely new Galileon-like theories arise whenever we have the same types of actions for a brane floating in a more general bulk with a number of special symmetries.
Back in April, we held a mini-workshop at the Center for Particle Cosmology at Penn, attended by the majority (but not all, unfortunately) of people in the world working on these ideas. We left that meeting with a bunch of new ideas, working on which has occupied much of my summer. When they get worked out, I’ll tell you more about them.
It is much too early to know if the Galileon idea will help with any of the cosmological and particle physics problems it may be suited to. They’ve been turning up in a variety of surprising and fascinating ways even since our workshop, but that doesn’t necessarily mean anything. But whatever the answer is, we’re learning things, and the process is wonderful fun.