The inflation paradigm is frequently invoked to explain the
flatness () of the Universe, attributing it to an era of exponential
expansion at about
seconds after the Big Bang. The mechanism of inflation is
generally taken to be an effective cosmological constant from the
dominant stress-energy of the inflation scalar field that
regulates GUT symmetry breaking, particle creation, and the
reheating of the Universe. In order to study whether inflation
can occur for arbitrary anisotropic and inhomogeneous data, many
numerical simulations have been carried out with different
symmetries, matter types and perturbations. A sample of such
calculations are described in the following paragraphs.

Kurki-Suonio et al. [41] extended the planar cosmology code of Centrella and Wilson [24,
25] (see §
2.5) to include a scalar field and simulate the onset of inflation
in the early Universe starting with an inhomogeneous Higgs field.
Their results suggest that whether inflation occurs or not can be
sensitive to the shape of the potential. In particular, if the
shape is flat enough, even large initial fluctuations of the
Higgs field do not prevent inflation. They considered two
different forms of the potential: a flat Coleman-Weinberg type
which is very flat close to the false vacuum and does inflate;
and a rounder ``
'' type which, for their parameter combinations, does not.

Goldwirth and Piran [33] studied the onset of inflation with inhomogeneous initial
conditions for closed, spherically symmetric spacetimes
containing a massive scalar field and radiation field sources
(described by a massless scalar field). In all the cases they
considered, the radiation field damps quickly and only an
inhomogeneous massive scalar field remains to inflate the
Universe. They find that small inhomogeneities tend to reduce the
amount of inflation and large initial inhomogeneities can even
suppress the onset of inflation. Their calculations indicate that
the scalar field must have ``suitable'' initial values over a
domain of several horizon lengths in order for inflation to
begin.

Anninos et al. [5] investigated the simplest Bianchi model (type V) background
that admits velocities or tilt in order to address the question
of how the Universe can choose a uniform reference frame at the
exit from inflation, since the deSitter metric does not have a
preferred frame. They find that inflation does not isotropize the
Universe in the short wavelength limit. However if inflation
persists, the wave behavior eventually freezes in and all
velocities go to zero at least as fast as
, where
is the relativistic tilt angle (a measure of velocity), and
R
is a typical scale associated with the radius of the Universe.
Their results indicate that the velocities eventually go to zero
as inflation carries all spatial variations outside the horizon,
and that the answer to the posed question is that memory is
retained and the Universe is never really deSitter.

In addition to the inflation field, one can consider other
sources of inhomogeneity, such as gravitational waves. Although
linear waves in deSitter space will decay exponentially and
disappear, it is unclear what will happen if strong waves exist.
Shinkai & Maeda [58] investigated the cosmic no-hair conjecture with gravitational
waves and a cosmological constant in 1D plane symmetric vacuum
spacetimes, setting up Gaussian pulse wave data with amplitudes
and widths
, where
I
is the invariant constructed from the 3-Riemann tensor and
is the horizon scale. They also considered colliding plane waves
with amplitudes
and widths
. They find that for any large amplitude and/or small width
inhomogeneity in their parameter sets, the nonlinearity of
gravity has little effect and the spacetime always evolves into a
deSitter spacetime.

The previous paragraphs discussed results from highly symmetric
spacetimes, but the possibility of inflation remains to be
established for more general inhomogeneous and nonperturbative
data. To this end, Kurki-Suonio et al. [42] investigated fully three-dimensional inhomogeneous spacetimes
with a chaotic inflationary potential
. They considered basically two types of runs: small and large
scale. In the small scale run, the grid length was initially set
equal to the Hubble length so the inhomogeneities are well inside
the horizon and the dynamical time scale is shorter than the
expansion or Hubble time. As a result, the perturbations
oscillate and damp while the field evolves and the spacetime
inflates. In the large scale run, the inhomogeneities are outside
the horizon and they do not oscillate. They maintain their shape
without damping and, because larger values of
lead to faster expansion, the inhomogeneity in the expansion
becomes steeper in time since the regions of large
and high inflation stay correlated. Both runs have sufficient
inflation to solve the flatness problem.