# The Eternally Existing, Self-Reproducing, Frequently Puzzling Inflationary Universe

My inaugural column for *Discover* discussed the lighting-rod topic of the inflationary multiverse. But there’s only so much you can cover in 1500 words, and there are a number of foundational issues regarding inflation that are keeping cosmologists up at night these days. We have a guest post or two coming up that will highlight some of these issues, so I thought it would be useful to lay a little groundwork. (Post title paraphrased from Andrei Linde.)

This summer I helped organize a conference at the Perimeter Institute on Challenges for Early Universe Cosmology. The talks are online here — have a look, there are a number of really good ones, by the established giants of the field as well as by hungry young up-and-comers. There was also one by me, which starts out okay but got a little rushed at the end.

What kinds of challenges for early universe cosmology are we talking about? Paul Steinhardt pointed out an interesting sociological fact: twenty years ago, you had a coterie of theoretical early-universe cosmologists who had come from a particle/field-theory background, almost all of whom thought that the inflationary universe scenario was the right answer to our problems. (For an intro to inflation, see this paper by Alan Guth, or lecture 5 here.) Meanwhile, you had a bunch of working observational astrophysicists, who didn’t see any evidence for a flat universe (as inflation predicts) and weren’t sure there were any other observational predictions, and were consequently extremely skeptical. Nowadays, on the other hand, cosmologists who work closely with data (collecting it or analyzing it) tend to take for granted that inflation is right, and talk about constraining its parameters to ever-higher precision. Among the more abstract theorists, however, doubt has begun to creep in. Inflation, for all its virtues, has some skeletons in the closet. Either we have to exterminate the skeletons, or get a new closet.

Inflation is a simple idea: imagine that the universe begins in a tiny patch of space dominated by the potential energy of some scalar field, a kind of super-dense dark energy. This causes that patch to expand at a terrifically accelerated rate, smoothing out the density and diluting away any unwanted relics. Eventually the scalar field decays into ordinary matter and radiation, reheating the universe into a conventional Big Bang state, after which things proceed as normal.

Note that the *entire point* of inflation is to make the initial conditions of our observable universe seem more “natural.” Inflation is a process, not a law of nature. If you don’t care about naturalness, and are willing to say “things just happened that way,” there is absolutely no reason to ever think about inflation. So the success or failure of inflation as a scenario depends on how natural it really is.

This raises a problem, as Roger Penrose has been arguing for years, with people like me occasionally backing him up. Although inflation does seem to create a universe like ours, it needs to start in a very particular kind of state. If the laws of physics are “unitary” (reversible, preserving information over time), then the number of states that would begin to inflate is actually much smaller than the number of states that just look like the hot Big Bang in the first place. So inflation seems to replace a fine-tuning of initial conditions with an even greater fine-tuning.

One possible response to this is to admit that inflation by itself is not the final answer, and we need a theory of why inflation started. Here, it is crucial to note that in conventional non-inflationary cosmology, our current observable universe was about a centimeter across at the Planck time. That’s a huge size by particle physics standards. In inflation, by contrast, the whole universe could have fit into a Planck volume, 10^{-33} centimeters across, much tinier indeed. So for some people (like me), the benefit of inflation isn’t that it’s more “natural,” it’s that it presents an easier target for a true theory of initial conditions, even if we don’t have such a theory yet.

But there’s another possible response, which is to appeal to eternal inflation. The point here is that most — “essentially all” — models of inflation lead to the prediction that inflation never completely ends. The vicissitudes of quantum fluctuations imply that even inflation doesn’t smooth out everything perfectly. As a result, inflation will end in some places, but in other places it keeps going. Where it keeps going, space expands at a fantastic rate. In some parts of that region, inflation eventually ends, but in others it keeps going. And that process continues forever, with some part of the universe perpetually undergoing inflation. That’s how the multiverse gets off the ground — we’re left with a chaotic jumble consisting of numerous “pocket universes” separated by regions of inflating spacetime.

It’s therefore possible to respond to the “inflation requires even more finely-tuned initial conditions than the ordinary Big Bang” critique by saying “sure, but once it starts, it creates an infinite number of smooth `universes,’ so as long as it starts at least once we win.” A small number (the probability of inflation starting somewhere) times infinity (the number of universes you make each time it starts) is still infinity.

But if eternal inflation offers solutions, it also presents problems, which might be worse than the original disease. These problems are at the heart of the worries that Steinhardt mentioned. Let me just mention three of them.

The one I fret about the most is the “unitarity” or “Liouville” problem. This is essentially Penrose’s original critique, updated to eternal inflation. Liouville’s Theorem in classical mechanics states that if you take a certain number of states and evolve them forward in time, you will end up with precisely the same number of states you started with; states aren’t created or destroyed. So imagine that there is some number of states which qualify as “initial conditions for inflation.” Then eternal inflation says we can evolve them forward and get a collection of universes that grows with time. The problem is that, as this collection grows, there is an increasing number of states that look identical to them, but which *didn’t* begin with a single tiny inflating patch at all. (Just like an ice cube in a glass of water will evolve to a glass of cooler water, but most glasses of cool water didn’t start with an ice cube in them.) So while it might be true that you can generate an infinite number of universes, at the same time the fraction of such states that actually began in a single inflating patch goes to zero just as quickly. It is far from clear that this picture actually increases the probability that a universe like ours started from inflation.

There is an obvious way out of this challenge, which is to say that all of these “numbers of states” are simply infinite, and this purported calculation just divides infinity by infinity and gets nonsense. And that’s very plausibly true! But if you reject the argument that universes beginning with inflation are an infinitesimally small fraction of all the universes, you are not allowed to accept the argument that there’s some small probability inflation starts and once it does it makes an infinite number of universes. All you can really do is say “we can’t calculate anything.” Which is fine, but we are left without a firm reason for believing that inflation actually solves the naturalness problems it was intended to solve.

A second problem, much more celebrated in the recent cosmological literature and closely related to the first, is known as the measure problem. (Not to be confused with the “measurement problem” in quantum mechanics, which is completely different.) The measure problem isn’t about the probability that inflation starts; it assumes so, and tries to calculate probabilities within the infinite ensemble of universes that eternal inflation creates. The problem is that we would like to calculate probabilities by simply counting the fraction of things that have a certain property — but here we aren’t sure what the “things” are that we should be counting, and even worse we don’t know how to calculate the fraction. Say there are an infinite number of universes in which George W. Bush became President in 2000, and also an infinite number in which Al Gore became President in 2000. To calculate the fraction N(Bush)/N(Gore), we need to have a measure — a way of taming those infinities. Usually this is done by “regularization.” We start with a small piece of universe where all the numbers are finite, calculate the fraction, and then let our piece get bigger, and calculate the limit that our fraction approaches. The problem is that the answer seems to depend very sensitively on how we do that procedure, and we don’t really have any justification at all for preferring one procedure over another. Therefore, in the context of eternal inflation, it’s very hard to predict anything at all.

This quick summary is somewhat unfair, as a number of smart people have tried very hard to propose well-defined measures and use them to calculate within eternal inflation. It may be that one of these measures is simply correct, and there’s actually no problem. Or it may be that the measure problem is a hint that eternal inflation just isn’t on the right track.

The final problem is what we might call the holography/complementarity problem. As I explained a while ago, thinking about black hole entropy has led physicists to propose something called “horizon complementarity” — the idea that one observer can’t sensibly talk about things that are happening outside their horizon. When applied to cosmology, this means we should think locally: talk about one or another pocket universe, but not all of them at the same time. In a very real sense, the implication of complementarity is that things outside our horizon aren’t actually real — all that exists, from our point of view, are degrees of freedom inside the horizon, and on the horizon itself.

If something like that is remotely true, the conventional story of eternal inflation is dramatically off track. There isn’t really an infinite ensemble of pocket universes — or at least, not from the point of view of any single observer, which is all that matters. This helps with the measure problem, obviously, since we don’t have to take fractions over infinitely big ensembles. But one would be right to worry that it brings us back to where we started, wondering why inflation really helps us solve naturalness problems at all.

Personally I suspect (i.e. would happily bet at even money, if there were some way to actually settle the bet) that inflation will turn out to be “right,” in the sense that it will be an ingredient in the final story. But these concerns should help drive home how far away we are from actually telling that story in a complete and compelling way. That should hardly come as a surprise, given the remoteness from our view of the events we’re trying to describe. But the combination of logical consistency and known physics is extremely powerful, and I think there’s a good chance that we’re making legitimate progress toward understanding the origin of the universe.

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