Breaking radio silence here to report on some of the actual work I've been able to complete: a new paper with Heywood Tam.
Unitary Evolution and Cosmological Fine-Tuning Authors: Sean M. Carroll, Heywood Tam (Submitted on 8 Jul 2010) Abstract: Inflationary cosmology attempts to provide a natural explanation for the flatness and homogeneity of the observable universe. In the context of reversible (unitary) evolution, this goal is difficult to satisfy, as Liouville's theorem implies that no dynamical process can evolve a large number of initial states into a small number of final states. We use the invariant measure on solutions to Einstein's equation to quantify the problems of cosmological fine-tuning. The most natural interpretation of the measure is the flatness problem does not exist; almost all Robertson-Walker cosmologies are spatially flat. The homogeneity of the early universe, however, does represent a substantial fine-tuning; the horizon problem is real. When perturbations are taken into account, inflation only occurs in a negligibly small fraction of cosmological histories, less than 10^-6.6x10^7. We argue that while inflation does not affect the number of initial conditions that evolve into a late universe like our own, it nevertheless provides an appealing target for true theories of initial conditions, by allowing for small patches of space with sub-Planckian curvature to grow into reasonable universes.
In English: our universe looks very unusual. You might think we have nothing to compare it to, but that's not quite right; given the particles that make up the universe (or the quantum degrees of freedom, to be technical about it), we can compare their actual configuration to all the possible configurations they could have been in. The answer is, our observed universe is highly non-generic, and in the past it was even more non-generic, or "finely tuned." One way of describing this state of affairs is to say that the early universe had a very low entropy. We don't know why; that's an important puzzle, worth writing books about. Part of the motivation of this paper was to put some quantitative meat on some ideas I discussed in my book. The basic argument is an old one, going back to Roger Penrose in the late 1970's. The advent of inflation in the early 1980's seemed to change things -- it showed how to get a universe just like ours starting from a tiny region of space dominated by "false vacuum energy." But a more careful analysis shows that inflation doesn't really change the underlying problem -- sure, you can get our universe if you start in the right state, but that state is even more finely-tuned than the conventional Big Bang beginning. We revisit this question, bringing to bear some mathematical heavy machinery developed in the 1980's by Gary Gibbons, Stephen Hawking, and John Stewart. Previous discussions have invoked general ideas of entropy or reversibility, but we were able to do a relatively down-to-earth calculation using conventional cosmological models. And we tried our best to explicitly list all of the caveats of the argument, which is important in a context like this where we don't know all the rules. We find that inflation is very unlikely, in the sense that a negligibly small fraction of possible universes experience a period of inflation. On the other hand, our universe is unlikely, by exactly the same criterion. So the observable universe didn't "just happen"; it is either picked out by some general principle, perhaps something to do with the wave function of the universe, or it's generated dynamically by some process within a larger multiverse. And inflation might end up playing a crucial role in the story. We don't know yet, but it's important to lay out the options to help us find our way.