This thesis presents results 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 revisit the results of previous work concerning scaling-laws, noting 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.
The specification of the growing-mode perturbations in the above work is approximate and in
the later part of the thesis, we introduce a more sophisticated and elegant formulation in terms
of curvature perturbations. This allows a direct connection to be made with the spectrum
of perturbations coming from inflation and also, using this, we find that there is no longer
evidence of shock production in connection with primordial black hole formation. Introducing
adaptive mesh refinement into our code, we are able to follow black hole formation nearer to
the critical limit and find evidence suggesting that scaling laws may continue down to very
small n1asses, in contrast with previous suggestions in the literature.
