Measure and Probability in Cosmology
  General relativity has a Hamiltonian formulation, which formally provides a
canonical (Liouville) measure on the space of solutions. In ordinary
statistical physics, the Liouville measure is used to compute probabilities of
macrostates, and it would seem natural to use the similar measure arising in
general relativity to compute probabilities in cosmology, such as the
probability that the universe underwent an era of inflation. Indeed, a number
of authors have used the restriction of this measure to the space of
homogeneous and isotropic universes with scalar field matter
(minisuperspace)---namely, the Gibbons-Hawking-Stewart measure---to make
arguments about the likelihood of inflation. We argue here that there are at
least four major difficulties with using the measure of general relativity to
make probability arguments in cosmology: (1) Equilibration does not occur on
cosmological length scales. (2) Even in the minisuperspace case, the measure of
phase space is infinite and the computation of probabilities depends very
strongly on how the infinity is regulated. (3) The inhomogeneous degrees of
freedom must be taken into account (we illustrate how) even if one is
interested only in universes that are very nearly homogeneous. The measure
depends upon how the infinite number of degrees of freedom are truncated, and
how one defines "nearly homogeneous." (4) In a universe where the second law of
thermodynamics holds, one cannot make use of our knowledge of the present state
of the universe to "retrodict" the likelihood of past conditions.
