By Mark Trodden | July 30, 2011 7:21 pm

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.

  1. 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.
  2. 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.
  3. 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.

  • Jolyon

    How do you counter the claim that Gallileon theories generically allow for superluminal propagation?

  • Sili

    attended by the majority (but not all, unfortunately) of people in the world working on these ideas.

    What do you mean “unfortunately”?

    Imagine if the roof had collapsed. This is why royal families, say, never travel together.

  • Fireworks below 1Tev

    Isn’t this an incredibly complicated and unappealing way to obtain something that is already part of Einstein’s gravity, namely cosmic acceleration (due to CC)? DGP actions appear extremely fine tuned, it is unclear if they are well behaved quantum mechanically, do not address the smallness of the CC problem, have no UV completion, and lead to violations of causality. On the other hand, GR with a CC is weakly coupled, quantum mechanically well behaved as a low energy EFT, has no known obstruction to UV completion (and is realized in string theory), is manifestly causal, is in agreement with all observations, is simple and elegant, and appears the only sensible theory of gravity that can emerge as a low energy effective theory from any plausible theory of quantum gravity (and is even dual to other manifestly well behaved theories, such as gauge theories etc via holography).

    The only problem for GR + CC is that the CC is so small (which is perhaps ameliorated by anthropics anyhow). But I’ll take one aesthetic problem over 10 major physical problems any day. And besides, DGP does not address the smallness of CC problem either. So I don’t quite get the fuss over these bizarre modified gravity models. I think that most of the people who work on them are focussed on making cosmological predictions, while sweeping most of the major fundamental issues under the rug. On the other hand, the people who spend their time focussed on fundamental issues, such as black hole information paradox, quantum gravity, eternal inflation, string theory, loop quantum gravity, Wheeler de Witt equation, holography, duality, gauge theory, etc, have never found DGP or Galileons ever being relevant or useful at all.

  • Mark

    Hi Jolyon, I think it remains to be seen whether this is generically the case. My own interest here is in trying to map out the possible 4d effective field theories that one can discover by going through the probe brane construction. Clearly some of these, with some couplings to matter, have problems. However, precisely the point of exploring the more general models we have looked at is to see whether there are new consistent constructions. We’ll see!

  • Mark

    Hi Fireworks. I’m not specifically focused on these for cosmic acceleration, although it might indeed be interesting if one finds a novel way of getting it out of them. My main interests are outlined in my response to Jolyon. Regarding modified gravity models, I think the general question of whether there exist sensible infrared modifications of GR that can yield interesting cosmology is a good one. However, through a lot of work, it has become clear that the models people have proposed are very tightly constrained by observations and issues of theoretical consistency, such as ghosts. This is how some progress is made in science.

    DGP was not invented to describe cosmic acceleration – but indeed many people have worked on it for that, and it certainly is an interesting question. The work I’m describing above is about 4d effective field theories, not really about modified gravity at all. One can consider the galileons, and other similar fields, as arising from higher dimensional physics, or just use that idea merely as a mathematical tool to discover new interesting 4d field theories. The properties I outlined in the post were unexpected and are, I think, interesting enough to warrant some further study. My interests are precisely in seeing whether they constitute sensible theories, which make sense technically. If the answer is yes, I’ll remain interested; if not, then I’ll shift this part of my research interests elsewhere.

  • Mark F.


    This does sound interesting, but do these theories have certain restrictions on the size of the extra dimensions and, thus, have the potential to be probed by the LHC? The LHC has ruled out certain extra dimensional theories. Do the theories you’re working on fall under this umbrella?

    Also, is there any connection between these theories and f(R) gravity? Thanks!

  • Peter Mortensen

    “but there equations” -> “but their equations”. “the such theories” -> “that such theories”

  • Mark

    Peter – oops! Thanks – fixed.

    Mark F. The theories I’m writing about are 4d effective field theories, and so on their own don’t face such constraints. However, the original Galileon, and the ones I’ve discussed, certainly can find their origins in higher dimensional implementations. In that case, the extra dimensions are infinite. So far these theories face their most stringent tests from internal consistency (ghost free conditions, lack of superluminal modes, etc.) and from consistency with solar system constraints (such as the Shapiro time delay) rather than from collider measurements.

    To the best of my knowledge there is no connection to f(R) gravity, beyond the fact that both have been invoked as possible explanations for cosmic acceleration.

  • Rich

    One more: “It is much to early to know” -> “It is much to0 early to know”

    Interesting article (even though it goes right over my head) – thanks!

  • Mark

    Thanks again – fixed!

  • Fireworks Below 1TeV

    Not sure I understand Mark’s motivation or claims here.

    Firstly, Mark says in #4 and #5 that he is not really interested in the 5d brane theories per se, they are just a device to construct interesting 4d effective theories. But if one is only interested in 4d effective actions, then one is free to make up any 4d action you want. In fact if the 4d EFT is the only thing of interest, then the right way to study them is what EFTheorists have been doing for decades, which is to parameterize all possible theories in the EFT framework (possibly assuming some symmetries, field content, etc), rather than restricting to some special subset of EFTs. The only legitimate reason why an EFTheorist would look at a special subset of 4d EFTs, derived from some 5d action, is because they really care about the underlying 5d microphysical theory.

    Secondly, Mark claims in #8 that since he only studies the 4d EFTs, then they are not subject to the same experimental constraints that have been found for the 5d theories. But this seems to be a misunderstanding of what an EFT is. If the underlying 5d theory breaks down at an energy scale of, say M, then the 4d EFT will break down at a scale M at the most (and possibly at a lower scale). For scales less than M, the EFT knows about all the details of the full 5d theory through the coefficients in an infinite expansion of operators. So any constraint that the full theory is subject to, is a constraint the EFT is subject to also. For instance, the chiral Lagrangian of QCD does not tip-toe its way around constraints on QCD – it too is subject to all of them (and works, of course). But breaks down as we go up to the QCD scale. In short, the EFT is subject to all the constraints of the underlying microphysical theory, plus more constraints (from a limited range of validity), not less.

  • Orlando

    Hi Mark,
    I recently read some stuff by Kurt Hinterbichler about Galileon ideas and found it pretty interesting.
    Many aspects are just too complex for me but I found your article very helpful to understand what’s all about.
    Thanks for your work.


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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.


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