Self-consistent initial conditions for primordial black hole formation
  For an arbitrary strong, spherically symmetric super-horizon curvature
perturbation, we present analytical solutions of the Einstein equations in
terms of asymptotic expansion over the ratio of the Hubble radius to the
length-scale of the curvature perturbation under consideration. To obtain this
solution we develop a recursive method of quasi-linearization which reduces the
problem to a system of coupled ordinary differential equations for the $N$-th
order terms in the asymptotic expansion with sources consisting of a non-linear
combination of the lower order terms. We use this solution for setting initial
conditions for subsequent numerical computations. For an arbitrary precision
requirement predetermined by the intended accuracy and stability of the
computer code, our analytical solution yields optimal truncated asymptotic
expansion which can be used to find the upper limit on the moment of time when
the initial conditions expressed in terms of such truncated expansion should be
set. Examples of how these truncated (up to eighth order) solutions provide
initial conditions with given accuracy for different radial profiles of
curvature perturbations are presented.
