Unitary Evolution and Cosmological Fine-Tuning
  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 $10^{-6.6\times 10^7}$. 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 sub-Planckian curvature to grow into reasonable universes.
