Nothing focuses the mind like a hanging, and nothing focuses the science like an unexpected experimental result. The BICEP2 claimed discovery of gravitational waves in the cosmic microwave background — although we still don’t know whether it will hold up — has prompted cosmologists to think hard about the possibility that inflation happened at a very high energy scale. The BICEP2 paper has over 600 citations already, or more than 3/day since it was released. And hey, I’m a cosmologist! (At times.) So I am as susceptible to being prompted as anyone.

Cosmic inflation, a period of super-fast accelerated expansion in the early universe, was initially invented to help explain why the universe is so flat and smooth. (Whether this is a good motivation is another issue, one I hope to talk about soon.) In order to address these problems, the universe has to inflate by a sufficiently large amount. In particular, we have to have enough inflation so that the universe expands by a factor of more than 10^{22}, which is about *e*^{50}. Since physicists think in exponentials and logarithms, we usually just say “inflation needs to last for over 50 *e*-folds” for short.

So Grant Remmen, a grad student here at Caltech, and I have been considering a pretty obvious question to ask: if we assume that there was cosmic inflation, how much inflation do we actually expect to have occurred? In other words, given a certain inflationary model (some set of fields and potential energies), is it most likely that we get lots and lots of inflation, or would it be more likely to get just a little bit? Everyone who invents inflationary models is careful enough to check that their proposals *allow* for sufficient inflation, but that’s a bit different from asking whether it’s likely.

The result of our cogitations appeared on arxiv recently:

How Manye-Folds Should We Expect from High-Scale Inflation?

Grant N. Remmen, Sean M. CarrollWe address the issue of how many

e-folds we would naturally expect if inflation occurred at an energy scale of order 10^{16}GeV. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential V(ϕ)=m^{2}ϕ^{2}/2 withm=2×10^{13}GeV and cutoff atM_{Pl}=2.4×10^{18}GeV, we find an expectation value of 2×10^{10}e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ^{4}[1−cos(ϕ/f)] withf=1.5×10^{19}GeV, we find that the expected total number ofe-folds is 50, which would just satisfy the observed requirements of our own Universe; iffis larger, more than 50e-folds are generically attained. We conclude that one should expect a large amount of inflation in large-field models and more limited inflation in small-field (hilltop) scenarios.

As should be evident, this builds on the previous paper Grant and I wrote about cosmological attractors. We have a technique for finding a measure on the space of cosmological histories, so it is natural to apply that measure to different versions of inflation. The result tells us — at least as far as the classical dynamics of inflation are concerned — how much inflation one would “naturally” expect in a given model.

The results were interesting. For definiteness we looked at two specific simple models: quadratic inflation, where the potential for the inflaton ϕ is simply a parabola, and cosine (or “natural“) inflation, where the potential is — wait for it — a cosine. There are many models one might consider (one recent paper looks at 193 possible versions of inflation), but we weren’t trying to be comprehensive, merely illustrative. And these are two nice examples of two different kinds of potentials: “large-field” models where the potential grows without bound (or at least until you reach the Planck scale), and “small-field” models where the inflaton can sit near the top of a hill.

Think for a moment about how much inflation *can* occur (rather than “probably does”) in these models. Continue reading