Setting Initial Conditions for Inflation with Reaction-Diffusion
  Equation
  We discuss the issue of setting appropriate initial conditions for inflation.
Specifically, we consider natural inflation model and discuss the fine tuning
required for setting almost homogeneous initial conditions over a region of
order several times the Hubble size which is orders of magnitude larger than
any relevant correlation length for field fluctuations. We then propose to use
the special propagating front solutions of reaction-diffusion equations for
localized field domains of smaller sizes. Due to very small velocities of these
propagating fronts we find that the inflaton field in such a ${\it field
domain}$ changes very slowly, contrary to naive expectation of rapid roll down
to the true vacuum. Continued expansion leads to the energy density in the
Hubble region being dominated by the vacuum energy, thereby beginning the
inflationary phase. Our results show that inflation can occur even with a
single localized field domain of size smaller than the Hubble size. We discuss
possible extensions of our results for different inflationary models, as well
as various limitations of our analysis (e.g. neglecting self gravity of the
localized field domain).
