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 1022, which is about e50. 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 Many e-Folds Should We Expect from High-Scale Inflation?
Grant N. Remmen, Sean M. Carroll
We address the issue of how many e-folds we would naturally expect if inflation occurred at an energy scale of order 1016 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)=m2ϕ2/2 with m=2×1013 GeV and cutoff at MPl=2.4×1018 GeV, we find an expectation value of 2×1010 e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ4[1−cos(ϕ/f)] with f=1.5×1019 GeV, we find that the expected total number of e-folds is 50, which would just satisfy the observed requirements of our own Universe; if f is larger, more than 50 e-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. Remember that the inflaton field acts just like a ball rolling down a hill, except that there is an effective “friction” from the expansion of the universe. That friction becomes greater as the expansion rate is higher, which for inflation happens when the field is high on the potential. So in the quadratic case, even though the slope of the potential (and therefore the force pushing the field downwards) grows quite large when the field is high up, the friction is also very large, and it’s actually the increased friction that dominates in this case. So the field rolls slowly (and the universe inflates) at large values, while inflation stops at smaller values. But there is a cutoff, since we can’t let the potential grow larger than the Planck scale. So the quadratic model allows for a large but finite amount of inflation.
The cosine model, on the other hand, allows for a potentially infinite amount of inflation. That’s because the potential has a maximum at the top of the hill. In principle, there is a trajectory where the field simply sits there and inflates forever. Slightly more realistically, there are other trajectories that start (with zero velocity) very close to the top of the hill, and take an arbitrarily long time to roll down. There are also, of course, trajectories that have a substantial velocity near the top of the hill, which would inflate for a relatively short period of time. (Inflation only happens when the energy density is mostly in the form of the potential itself, with a minimal contribution from the kinetic energy caused by the field velocity.) So there is an interesting tradeoff, and we would like to know which effect wins.
The answer that Grant and I derived is: in the quadratic potential, we generically expect a huge amount of inflation, while in the cosine potential, we expect barely enough (fairly close to the hoped-for 50 e-folds). Although you can in principle inflate forever near a hilltop, such behavior is non-generic; you really need to fine-tune the initial conditions to make it happen. In the quadratic potential, by contrast, getting a buttload of inflation is no problem at all.
Of course, this entire analysis is done using the classical measure on trajectories through phase space. The actual world is not classical, nor is there any strong reason to expect that the initial conditions for the universe are randomly chosen with respect to the Liouville measure. (Indeed, we’re pretty sure that they are not.)
So this study has a certain aspect of looking-under-the-lamppost. We consider the naturalness of cosmological histories with respect to the conserved measure on classical phase space because that’s what we can do. If we had a finished theory of quantum gravity and knew the wave function of the universe, we would just look at that.
We’re not looking for blatant contradictions with data, we’re looking for clues that can help us move forward. The way to interpret our result is to say that, if the universe has a field with a potential like the quadratic inflation model (and the initial conditions are sufficiently smooth to allow inflation at all), then it’s completely natural to get more than enough inflation. If we have the cosine potential, on the other hand, than getting enough inflation is possible, but far from a sure thing. Which might be very good news — it’s generally thought that getting precisely “just enough” inflation is annoying finely-tuned, but here it seems quite plausible. That might suggest that we could observe remnants of the pre-inflationary universe at very large scales today.