Friday 5 October 2007


Last time I scolded the speaker for giving an utterly unattractive seminar title. Degravitation - which is the title of Gia Dvali's talk two weeks ago - is on the other hand very catchy and will certainly attract many Roswell aficionados to my blog. But this post, I'm afraid, is not about classified experiments with gravity performed here at CERN but about a new interesting approach to solving the cosmological constant problem. Gia is going to be around for some time, so you may expect more posts with weird titles in future.

The cosmological constant problem is usually phrased as the question why the vacuum energy is so small. Formulated that way, it is very hard to solve, given large existing contributions (zero-point oscillations, vacuum condensates) and vicious no-go theorems set up by Weinberg. The problem has ruined many lives and transformed some weaker spirits into anthropic believers. Gia does not give up and attempts to tackle the problem from a different angle. He tries to construct a theory where the vacuum energy may be large but it does not induce large effects on the gravitational field. This is of course impossible in Einstein gravity where all forms of energy gravitate. The idea can be realized, however, in certain modified gravity theories.

Gia pursues theories where gravity is strongly modified at large distances, above some distance scale L usually assumed to be of similar size as the observable universe. The idea is to modify the equations of gravity so as to filter out sources whose characteristic length is larger than L. The gravitational field would then ignore the existence of a cosmological constant, which uniformly fills the entire universe.

On a slightly more formal level, Gia advocates a quite general approach where the equations for the gravitational fields can be written as
$ ( 1 - \frac{m^2(p^2)}{p^2} ) G_{\mu \nu} = \frac{1} {2} T_{\mu \nu}$
where, as usual, $G$ is the Einstein tensor and $T$ is the energy-momentum tensor. Deviations from the Einstein theory are parameterized by $m^2(p^2)$ which is a function of momentum (or a funtion of derivatives in the position-space picture). For $m^2=0$, the familiar Einstein equations are recovered. The effects of $m^2$ set in at large distance scales.
At low momenta/large distances one assumes $m^2 \sim L^{-2} (p^2 L^2)^\alpha$ with $0 <= \alpha < 1$. The case $\alpha = 0$ corresponds to adding the graviton mass, the case $\alpha = 1/2$ corresponds to a certain 5D framework called the DGP model (where Gia is the D). In fact, the latter case is the only one for which the full, non-linear, generally covariant completion is known. Other values of $\alpha$ may or may not correspond to a sensible non-linear theory.

Gia argues that any consistent theory effectively described by this kind of filter equations has to be a theory of a massive or resonance graviton. This means that the graviton propagates 5 degrees of freedom and not 2 as in the Einstein theory. In addition to 2 tensor polarizations, there are 2 vector and 1 scalar polarization. The additional polarizations also couple to massive sources and their exchange contributes to the gravitational potential.

Everybody who ever played with modified gravity knows well that Einstein gravity reacts histerically to all manipulations and often breaks down. In the present case what happens is that, once the theory is extended beyond the linear approximation, the scalar polarization gets strongly coupled far below the Planck scale. But Gia argues that one can live with it and, in fact, the strong coupling saves the theory. It is well known since ages that the massive gravity suffers from the so-called van Dam--Veltman discontinuity: the potential between two sources is different than in Einstein gravity, even in the zero-mass limit. The responsible for that is precisely the scalar polarization. The predictions from massive gravity are at odds with precise tests of gravity, for example with observations of the light-bending by the Sun. These predictions, however, are derived using the linear approximation which breaks down near massive sources. Gia argues that the effect of the strong coupling is to suppress the scalar polarization exchange near massive sources and there is no contradiction with experiment.

So the picture of the gravitational field around a massive source in massive or resonance gravity is more complex, as shown to the right. Apart from the Schwarzschild radius, there are two other scales. One is the scale L above which gravity shuts off. The other is the r* scale where the scalar polarization gets strongly coupled. At scales larger than r* we have a sort of scalar-tensor gravity that differs ifrom Einstein gravity. At scales shorter than r* Einstein gravity is approximately recovered up to small corrections. Gia estimates that these latter corrections can be measured in future by the lunar laser ranging experiment if $\alpha$ is of order 1/2.

Coming back to the cosmological constant problem, the analysis is complicated and depends on the non-linear completion of the theory. Gia's analysis shows that this class of theories can indeed degravitate the cosmological constant when $\alpha < 1/2$. I'm not sure if this conclusion is bulletproof since it is derived in a special limit where the equations for the tensor and scalar polarizations decouple. What is certain is that the complete non-linear DGP model (corresponding to $\alpha = 1/2$) does not enjoy the mechanism of degravitation. The hope is that theories with $\alpha < 1/2$ do exist and that a full non-linear analysis will demonstrate one day that the cosmological constant problem is solved.

Slides available here. The paper has been out for 6 months now. It is worth looking at the previous paper of Gia, where the strong coupling phenomenon is discussed at more length. Try also to google degravitation to see how amazing paths the human mind may wander.


Unknown said...

The problem has ruined many lives and transformed some weaker spirits into anthropic believers.

I look at this a little differently:

I see "anthropic selection" as a weaker persons bail-out on first principles, but I think that it requires an even weaker "soul" to reject the very obvious anthropic constraint on the forces as a plausible answer to the problem from first principles.

In other words, Jester, you act like the universe isn't observed to be anthropically constrained, so like Peter Woit, you sit in denial of the observed fact, and I'll bet that you know about as much about the anthropic physics as he does, which ain't much.

In other, other words.... we have David Gross and others who cry about how the biggest failure of science in the last twenty years is its inability to produce a dynamic stabilty principle, and yet... they refuse to recognize evidence that we might be directly related to it.

For twenty straight years the freaking math whizzes can't even add one plus one...

Sabine Hossenfelder said...

Yes, its an interesting idea, isn't it? I've had a post on that as well, see Filtering Gravity. I hope there is some follow-up work on this out soon.