Carroll and Tam have written a nice and very readable paper revisiting Penrose’s entropy problem as well as work related to Liouville’s theorem in the context of inflation. They look at the claim that inflation evolves from generic initial conditions. This often cited claim is radically at odds with Liouville’s theorem, which requires a given number of states of the universe to evolve into the same number of states at a later time. In other words, it is not possible for a large number of (highly curved and inhomogeneous) initial states to evolve into a small number of smooth and homogeneous “final” states. Thus it is not clear whether the main argument by which inflation is usually motivated actually holds up! (Of course inflation has the appealing byproduct that it can produce density perturbations along the way.)
The paper does not provide a resolution of the problem, but is interesting in that it highlights an often overlooked issue that is certainly in need of a solution, if we want to claim that inflation explains the initial conditions of the hot big bang cosmology.
[1007.1417] Unitary Evolution and Cosmological FineTuning
Authors:  Sean M. Carroll, Heywood Tam 
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 finetuning. The most natural interpretation of the measure is the flatness problem does not exist; almost all RobertsonWalker cosmologies are spatially flat. The homogeneity of the early universe, however, does represent a substantial finetuning; the horizon problem is real. When perturbations are taken into account, inflation only occurs in a negligibly small fraction of cosmological histories, less than $10^{6.6\times 10^7}$. 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 subPlanckian curvature to grow into reasonable universes. 
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 Joined: February 18 2010
 Affiliation: Princeton Center for Theoretical Science