Long-wavelength non-linear perturbations of a complex scalar field
  We study the evolution of nonlinear superhorizon perturbations in a universe
dominated by a complex scalar field. The analysis is performed adopting the
gradient expansion approach, in the constant mean curvature slicing. We derive
general solutions valid to second order in the ratio $H^{-1}/L$ for scalar
field inhomogeneities of size $L$ subject to an arbitrary canonical potential.
We work out explicit solutions for the quadratic and the quartic potentials,
and discuss their relevance in setting initial conditions required for the
simulations of Primordial Black Hole formation.
