Primordial black holes in linear and non-linear regimes
  Using the concept of apparent horizon for dynamical black holes, we revisit
the formation of primordial black holes (PBH) in the early universe for both
linear and non-linear regimes. First, we develop the perturbation theory for
spherically symmetric spacetimes to study the formation of spherical PBHs in
linear regime and we fix two gauges. We also introduce a well defined gauge
invariant quantity for the expansion. Using this quantity, we argue that PBHs
do not form in the linear regime. Finally, we study the non-linear regime. We
adopt the spherical collapse picture by taking a closed FRW model in the
radiation dominated era to investigate PBH formation. Taking the initial
condition of the spherical collapse from the linear theory of perturbations, we
allow for both density and velocity perturbations. Our model gives a constraint
on the velocity perturbation. This model also predicts that the apparent
horizon of PBHs forms when $\delta > 3$. Applying the sound horizon constraint,
we have shown the threshold value of density perturbations at horizon re-entry
must be larger than $\delta_{th} > 0.7$ to overcome the pressure gradients.
