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 = k \rho ( 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 \leq 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 .