The first thing to understand is that the Big Bang was not an explosion that happened at one place in a preexisting, empty space. The Big Bang happened everywhere at once, so there is no location that would be the place where we would expect a black hole’s singularity to form. Cosmological models are either exactly or approximately homogeneous. In a homogeneous cosmology, symmetry guarantees that tidal forces vanish everywhere, and that any observer at rest relative to the average motion of matter will measure zero gravitational field. Based on these considerations, it’s actually a little surprising that the universe ever developed any structure at all. The only kind of collapse that can occur in a purely homogeneous model is the recollapse of the entire universe in a “Big Crunch,” and this happens only for matter densities and values of the cosmological constant that are different from what we actually observe.
A black hole is defined as a region of space from which light rays can’t escape to infinity. “To infinity” can be defined in a formal mathematical way,[HE] but this definition requires the assumption that spacetime is asymptotically flat. To see why this is required, imagine a black hole in a universe that is spatially closed. Such a cosmology is spatially finite, so there is no sensible way to define what is meant by escaping “to infinity.” In cases of actual astrophysical interest, such as Cygnus X-1 and Sagittarius A*, the black hole is surrounded by a fairly large region of fairly empty interstellar space, so even though our universe isn’t asymptotically flat, we can still use a portion of an infinite and asymptotically flat spacetime as an approximate description of that region. But if one wants to ask whether the entire universe is a black hole, or could have become a black hole, then there is no way to even approximately talk about asymptotic flatness, so the standard definition of a black hole doesn’t even give a yes-no answer. It’s like asking whether beauty is a U.S. citizen; beauty isn’t a person, and wasn’t born, so we can’t decide whether beauty was born in the U.S.
Black holes can be classified, and we know, based on something called a no-hair theorem, that all static black holes fall within a family of solutions to the Einstein field equations called Kerr-Newman black holes. (Non-static black holes settle down quickly to become static black holes.) Kerr-Newman black holes have a singularity at the center, are surrounded by a vacuum, and have nonzero tidal forces everywhere. The singularity is a point at which the world-lines only extend a finite amount of time into the future. In our universe, we observe that space is not a vacuum, and tidal forces are nearly zero on cosmological distance scales (because the universe is homogeneous on these scales). Although cosmological models do have a Big Bang singularity in them, it is not a singularity into which future world-lines terminate in finite time, it’s a singularity from which world-lines emerged at a finite time in the past.
A more detailed and technical discussion is given in [Gibbs]. [HE] Hawking and Ellis, The large-scale structure of spacetime, p. 315. [Gibbs] http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/universe.html
The following forum members have contributed to this FAQ: bcrowell, George Jones, jim mcnamara, marcus, PAllen, tiny-tim, vela