  Results are presented from general relativistic numerical computations of
primordial black-hole formation during the radiation-dominated era of the
universe. Growing-mode perturbations are specified within the linear regime and
their subsequent evolution is followed as they become nonlinear. We use a
spherically symmetric Lagrangian code and study both super-critical
perturbations, which go on to produce black holes, and sub-critical
perturbations, for which the overdensity eventually disperses into the
background medium. For super-critical perturbations, we confirm the results of
previous work concerning scaling-laws but note that the threshold amplitude for
a perturbation to lead to black-hole formation is substantially reduced when
the initial conditions are taken to represent purely growing modes. For
sub-critical cases, where an initial collapse is followed by a subsequent
re-expansion, strong compressions and rarefactions are seen for perturbation
amplitudes near to the threshold. We have also investigated the effect of
including a significant component of vacuum energy and have calculated the
resulting changes in the threshold and in the slope of the scaling law.
