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?
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.
I don't think that the question "what does the universe look like from the outside?" is very meaningful. Just because there is not outside for the universe. As for the black hole why should high density i.e. a lot of mass in little volume, cause the creation of a black hole? If you are thinking about the Schwarzschild solution (and radius), it describes a spherical object outside of which the space is empty, and as I said there is no outside for the universe.
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.
This is a FAQ entry written by the following members of physicsforums.com: bcrowell George Jones jim mcnamara marcus PAllen tiny-tim vela
The standard ΛCDM model of the Big Bang fits obsersvations to the Friedmann-Robertson-Walker solutions of general relativity, which do not form black holes. Intuitively, the initial expansion is great enough to counteract the usual tendency of matter to gravitationally collapse. As far as we know, the universe looks about the same from every point on the large scale. It is a built-in assumption of the FRW family solutions, and sometimes called the "Copernican principle."
It doesn't absolutely have to be right, of course, though in a sense it is the simplest possible empirically adequate model, and so is favored by Ockham's razor. There have been attempts to fit the astronomical observations to an isotropic and inhomogeneous solution of GTR (meaning, we would be near "the center"), but to my knowledge they have been less than conclusive.
There is an oversimplified model of spherical stellar collapse assumes that the star has uniform density and no pressure, the interior of which comes out to be equivalent to the k = +1 (positive curvature, closed) contracting FRW universe. The interior is smoothly patched to a Schwarzschild exterior. The k = 0 (flat) and k = -1 (open) cases can be thought of as the interior of such a star in the limit of infinite radius, collapsing from rest and with some finite velocity, respectively. They too can be smoothly patched to a Schwarzschild exterior.
Our the observed universe is expanding, but we still can say that's it's possible for the isotropic and homogeneous region we observe to have an edge, or perhaps even be the the interior of a time-reversed black hole. But it should be emphasized that we have no empirical reason to believe that it's anything more exotic than a plain FRW universe. Though on a more serious alternatives, some models of cosmic inflation have our observed universe as one of many "bubbles" in an inflating background.