Vacuum stability

It’s not so hard to write down a model of phantom energy: just invent an ordinary scalar field, but with a negative kinetic energy. Left to its own devices, such a field will gradually increase its potential energy, leading to a net increase in the energy density, so cosmologists would measure the equation-of-state parameter w to be less than -1.

But just because you can write a model down doesn’t mean it makes sense. Remember that, in a model-independent sense, there was a good argument against w<-1: it violates the Dominant Energy Condition, which is the requirement that assures us that energy doesn't propagate faster than light. So could their be something fundamentally sick about theories of phantom energy? Well, yes. In particular, the energy density is not bounded below -- as the field vibrates more and more, you can create a negative energy that is as large as you like. This means that the theory is not stable. Ordinarily in field theory, we like to invent models that have a unique "vacuum state," the state of absolutely lowest energy; all other states are excitations of the vacuum, with an unambiguously larger energy. Then we can be assured that the dynamics don't go crazy; systems will tend to oscillate around the vacuum, or (in the presence of friction or other damping forces) gradually wind down to the vacuum. But if there is no vacuum, the system can simply run away, like a ball rolling down a hill with no bottom. This possibility is not so horrifying in principle, but conflicts with the apparent stability that we observe around us in Nature. Think of it in terms of particles. The world is made of fields, but quantum fields, not classical ones. When you quantize a field, and then observe it, you see particles. For a phantom field, the negative kinetic energy implies that the particle excitations have a negative mass. (In contrast to tachyons, which have an imaginary mass.) This helps us to see why there is an instability: starting from a purported “vacuum” state of completely empty space at zero energy, we can imagine processes that conserve energy while creating large numbers of positive-mass ordinary particles plus compensating numbers of negative-mass phantom particles. Empty space itself is liable to dissolve into a bath of billions of particles!

This is why most particle physicists just laugh at the idea of phantom energy: it seems ruled out before you even start. But because there is so little we know about dark energy, it’s a good idea to keep our options open. In collaboration with Mark Hoffman and Mark Trodden, I wrote a paper examining whether the idea of phantom energy could be part of a larger scheme that was not obviously ruled out. The idea is a very common one in field theory: you have some model (an “effective field theory“) that describes everything perfectly well, but only at energies below some cutoff where unknown physics kicks in. This is an interesting feature about quantum field theory; the effect of high-energy processes is to change the parameters (the coupling constants) of your low-energy theory, but not to produce qualitatively new phenomena. In other words, dramatic new physics at high energies is basically hidden from our sight, subsumed in the quantitative behavior of the low-energy physics we can actually observe. (That’s why the best way to learn new particle physics is to build particle accelerators of ever-higher energy, and also why it’s so damned difficult to get any direct experimental handle on string theory or other models of quantum gravity, which live way up at the Planck energy.)

So we asked the question: could phantom energy be right, if only as an effective field theory valid below certain energies? If the phantom theory were valid up to arbitrarily large energies, not only would the vacuum be unstable, the decay rate would be infinite! What we found is that you can indeed imagine that there is a cutoff beyond which the phantom description doesn’t apply, and if that cutoff is awfully low (about a milli-electron-volt) the field would be essentially stable over the lifetime of the universe. (Some of our numbers have been brought into question in a paper by Cline, Jeon, and Moore.) An explicit example of the kind of cutoff we were proposing was later investigated by Arkani-Hamed et al.

There’s an additional possibility, that I’ve investigated more recently with Trodden and Antonio DeFelice. This is that there is no phantom energy, and the real equation-of-state parameter w is -1 or larger, but that we can be tricked into thinking that we’ve measured w to be less than -1. That’s because we never really measure w; what we measure is the expansion of the universe, and use general relativity to infer the properties of the dark energy. But general relativity might not be right. We investigated the specific example of a scalar-tensor theory, where some scalar field was causing the value of Newton’s gravitational constant to gradually vary with time. Then what you’re measuring in cosmology isn’t simply the behavior of the dark energy, but some combination of the dark energy and the gravitational scalar field. We found that you can indeed be tricked into thinking that w is less than -1, but only with a very unnatural behavior for the scalar field; most of the time, we would have already detected the variation of Newton’s constant right here in the Solar System long before you would measure some unusual behavior of the expansion of the universe.

The lesson is simply this: don’t be too dogmatic, but do be honest, when you are inventing theories of something you are as clueless about as the dark energy. Theorists need to be quite careful; if they are going to propose models of phantom energy and so forth, they need to do the hard work of determining whether their models are stable and otherwise well-defined. But observers shouldn’t take too seriously the grandiose claims of theorists about what is and is not possible; they should do their experiments and see what the data imply. It would be a shame to miss out on a fantastic discovery because you believed some theorist who told you it couldn’t possibly be there.

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