Upper limits on the size of a primordial black hole

We provide precise constraints on the size of any black holes forming in the early Universe for a variety of formation scenarios. In particular, we prove that the size of the apparent horizon of a primordial black hole formed by causal processes in a flat Friedmann universe is considerably smaller than the cosmological apparent horizon size for an equation of state p=kρ (1/3<k<1). This also applies for a stiff equation of state (k=1) or for a massless scalar field. The apparent horizon of a primordial black hole formed through hydrodynamical processes is also considerably smaller than the cosmological apparent horizon for 0<k≤1. We derive an expression for the maximum size which an overdense region can have without being a separate closed universe rather than part of our own. Newtonian argument shows that a black hole smaller than the cosmological horizon can never accrete much.