Question

Wasn't the density of the universe at the moment after the Big Bang so great as to create a black hole? If the answer is that the universe/space-time can expand anyway what does it imply about what our universe looks like from the outside?

Answer 1

A high enough energy density is a necessary condition but not a sufficient condition for black holes to form: one needs to have a center which will ultimately become the center of the black holes; one needs the matter that collapses to the black hole to have a low enough velocity so that gravity may squeeze it before the matter manages to fly away and dilute the density.

The latter two conditions are usually almost trivially satisfied for ordinary chunks of matter peacefully sitting at some place of the Universe; but they're almost maximally violated by the matter density right after the Big Bang. This matter has no center - it is almost uniform throughout space - and has high enough velocity (away from itself) that the density eventually gets diluted. And indeed, we know that it did get diluted.

In other words, a collapse of matter (e.g. a star) into a black hole is an idealized calculation that makes certain assumptions about the initial state of the matter. These assumptions are clearly not satisfied by matter after the Big Bang. Instead of a collapse of a star, you should use another simplified version of Einstein's equations of general relativity - namely the Friedmann equations for cosmology. You will get the FRW metric as a solution. When it is uniform to start with, it will pretty much stay uniform.

The visible Universe is, in some sense, analogous to a black hole. There exists a cosmic horizon and we can't see behind it. However, it is more correct to imagine that the interior of the visible space - that increasingly resembles de Sitter space because the cosmological constant increasingly dominates the energy density - should be viewed as an analogy to the exterior of a black hole. And it's the exterior of the visible de Sitter space that plays the role of the interior of a black hole.

The relationship between (namely the ratio of) the mass and the radius for the visible Universe is not too far from the relationship between (or ratio of) the black hole mass and radius of the same size. However, it's not accurate, and it is not supposed to be accurate. The mass/radius ratio is only universal for static (and neutral) black holes localized in an external flat space and our Universe is clearly not one of them