Inhomogeneous Anisotropic Cosmology
  In homogeneous and isotropic Friedmann-Robertson-Walker cosmology, the
topology of the universe determines its ultimate fate. If the Weak Energy
Condition is satisfied, open and flat universes must expand forever, while
closed cosmologies can recollapse to a Big Crunch. A similar statement holds
for homogeneous but anisotropic (Bianchi) universes. Here, we prove that
$arbitrarily$ inhomogeneous and anisotropic cosmologies with "flat" (including
toroidal) and "open" (including compact hyperbolic) spatial topology that are
initially expanding must continue to expand forever at least in some region at
a rate bounded from below by a positive number, despite the presence of
arbitrarily large density fluctuations and/or the formation of black holes.
Because the set of 3-manifold topologies is countable, a single integer
determines the ultimate fate of the universe, and, in a specific sense, most
3-manifolds are "flat" or "open". Our result has important implications for
inflation: if there is a positive cosmological constant (or suitable
inflationary potential) and initial conditions for the inflaton, cosmologies
with "flat" or "open" topology must expand forever in some region at least as
fast as de Sitter space, and are therefore very likely to begin inflationary
expansion eventually, regardless of the scale of the inflationary energy or the
spectrum and amplitude of initial inhomogeneities and gravitational waves. Our
result is also significant for numerical general relativity, which often makes
use of periodic (toroidal) boundary conditions.
