Wednesday, 28 March 2012

Inflation and quantum gravity

Today's arXiv listing brought an interesting new paper by Joe Conlon of Oxford. In it he discusses constraints on inflation models coming from general principles of quantum gravity.

Inflation in short is the idea that very early in its history, the universe underwent rapid expansion by a factor of something like a billion billion billion (or about 60 'e-folds' in the jargon of the field). This solves certain problems of cosmology, which I don't want to go into here. In the context of general relativity, inflation can be achieved by a scalar field $\phi$ slowly rolling down a potential; inflation stops when it reaches its minimum. This scalar field is called the inflaton. Note that I am deliberately ignoring the fact that multiple fields can play important roles in inflation; this doesn't really matter for what I want to discuss, although see the caveat at the end of section 2 of Joe's paper.

One variable feature of inflation models is the total distance (in field space) which is traversed by the inflaton during inflation. There has been some controversy about whether this can consistently be greater than the Planck scale, because standard effective field theory arguments break down in this case (we can, and probably should, add arbitrary operators $\phi^k/M_P^{k-4}$ to the Lagrangian of the theory, and these all become important if $\phi \sim M_P$, rendering the theory meaningless). This is the sort of problem that one might hope to make some progress on by turning to string theory…

Most string theory compactifications abound with scalar fields (the infamous moduli), and various scenarios have been cooked up in which one of these can play the role of the inflaton. However, obtaining trans-Planckian field excursions, as defined above, is impossible in many scenarios. For example, moduli spaces of Calabi-Yau manifolds are of 'finite size' (you might rightly point out that in fact the volume modulus can take values all the way up to infinity, but this is special: it corresponds to decompactification of the internal dimensions, and we no longer have a four-dimensional theory at all!), and physically, this comes out to be the Planck scale or lower. One might suspect that string theory is trying to tell us that trans-Planckian fields don't make sense.

Joe's idea is to tackle these issues by appealing to the connection between principles of thermodynamics and the geometry of spacetime. This has proven to be the most robust and plausible feature of various approaches to quantum gravity, and is widely believed to be 'true'. In particular, inflation is described by the geometry of de Sitter spacetime, and this has a finite horizon. The covariant entropy bound applied to this situation states roughly that the maximum entropy of the interior of the spacetime is given by the area of this horizon (in appropriate units). But what is the entropy? In systems with a finite number of degrees of freedom, it is simply the logarithm of the number of microstates which give rise to the same macroscopic physics. Here the 'macroscopic physics' is simply the particular semi-classical de Sitter spacetime in which we are interested. Counting the microstates in a sensible way takes up a good part of the paper, and I will leave you to read about it yourself. The conclusion is that allowing trans-Planckian field values will lead to an entropy which is too large, and therefore such theories cannot be embedded in a consistent theory of quantum gravity.

I see two ways to look at the results of this paper:

  • Effective field theory models of inflation are only interesting if they can ultimately be embedded in a complete theory of quantum gravity. One of the few things we think we know about such a theory is the relationship between entropy and horizons, so this is a sensible consistency criterion to impose.
  • It gives a sanity-check on complicated string theory compactifications. The consistency of backgrounds which give rise to de Sitter spacetime is often rather precarious, and if the resulting four-dimensional theories violate basic principles of quantum gravity, one might suspect that some element of the construction is inconsistent.
An example of the latter is a neat idea which seems to go back to this paper, and has had quite a lot of attention (the paper has 100 citations). The authors consider string compactifications in which there is a five-brane wrapping some surface in the compact dimensions. The presence of the five-brane has a dramatic effect on one particular scalar field: the axionic part of the Kähler modulus associated to the surface. In the absence of the five-brane, this axion takes values on a circle, the size of which is less than the Planck scale. With the brane present, this circle unwraps, and the axion can take on much larger values. The jargon for this is 'axion monodromy'.

In a previous paper, Joe had pointed out quite a specific problem with models of axion monodromy inflation, but this more general argument is perhaps even more damaging. It will be interesting to observe how the community reacts to it.

Edit: Joe was kind enough to comment on what I had written here, and tells me that actually his first paper was much stronger on the issue of axion monodromy inflation. He showed that the backreaction of the five-branes is much larger than first anticipated, and ruins the consistency of the scenario.


  1. I stayed up late the other night drinking whiskey and reading this paper. To me it seems quite profound. Although the arguments are focussed on inflation they should apply to any theory/model with large field values and so it seems this might be applicable to a host of other cosmological/particle concepts. Lovely stuff.

  2. Nice overview, thanks! I don't understand why it states on page 18 that the bound m_s<M_p/60 is not satisfied. It seems to be that it can certainly be satisfied by e.g. tuning the fluxes to make g_s small enough.

    1. It's not obvious to me either; I don't know the details of those models well enough.

    2. The point is that g_s also enters the instanton/gaugino condensation term (recall the action of a D brane has 1/g_s in front), which is e^{-2 \pi Vol(\Sigma_4)/N g_s}, where the volume is in string units. So as you tune g_s to be small you simultaneously kill the instanton - or force the volume to be small and uncontrolled.

  3. Thanks, Joe! I see your point. I forgot about the 1/g_s factor because people usually rescale the moduli to switch to the Einstein frame once the axio-dilaton is stabilized. So, as I understand it, one can't simultaneously make g_s small, keep the non-rescaled volume the same to satisfy the supergravity regime and maintain the non-perturbative term large enough. That's true in Type IIB but might not be true for other compactifications, e.g. in the M-theory on G2 case there is no 1/g_s factor so this restriction should loosen up, I think.


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