Primordial black hole formation in the early universe: critical
  behaviour and self-similarity
  Following on after three previous papers discussing the formation of
primordial black holes during the radiative era of the early universe, we
present here a further investigation of the critical nature of the process
involved, aimed at making contact with some of the basic underlying ideas from
the literature on critical collapse. We focus on the intermediate state, which
we have found appearing in cases with perturbations close to the critical
limit, and examine the connection between this and the similarity solutions
which play a fundamental role in the standard picture of critical collapse. We
have derived a set of self-similar equations for the null-slicing form of the
metric which we are using for our numerical calculations, and have then
compared the results obtained by integrating these with the ones coming from
our simulations for collapse of cosmological perturbations within an expanding
universe. We find that the similarity solution is asymptotically approached in
a region which grows to cover both the contracting matter and part of the
semi-void which forms outside it. Our main interest is in the situation
relevant for primordial black hole formation in the radiative era of the early
universe, where the relation between the pressure $p$ and the energy density
$e$ can be reasonably approximated by an expression of the form $p = we$ with
$w=1/3$. However, we have also looked at other values of $w$, both because
these have been considered in previous literature and also because they can be
helpful for giving further insight into situations relevant for primordial
black hole formation. As in our previous work, we have started our simulations
with initial supra-horizon scale perturbations of a type which could have come
from inflation.
