Why did the universe not collapse to a black hole shortly after the big bang? 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: 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]
Is the Big Bang a black hole?
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A: 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.
A: 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.
A: 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. 
A: In many ways, the early universe was very similar in structure to a black hole, if one takes the singularity picture seriously. And even then, singularity free models still exist of black holes, so maybe the early universe does not require one either. 
Anyway, this isn't important, what is important is that mathematics supports strongly an early universe with a structure similar to a black hole and in the later epoch where the universe has sufficiently cooled down and got large enough, seems to preserve the weak equivalence principle. (if you want more information on this I will elaborate).
It is possible, that these analogies to be taken seriously enough to speculate we live in a black-hole-like structure. Certainly there is a lot of arguments which attempt to support it. For instance, The radius of a black hole is found directly proportional to its mass $R \propto m$. The density of a black hole is given by its mass divided by its volume $\rho = \frac{m}{V}$ and since the volume is proportional to the radius of the black hole to the power of three $V \propto R^3$ then the density of a black hole is inversely proportional to its mass radius by the second power $\rho \propto m^2$)
What does all this mean? It means that if a black hole has a large enough mass then it does not appear to be very dense, which is more or less the description of our own vacuum: it has a lot of matter, around $3 \times 10^{80}$ particles give or take a few power of tens of atoms in spacetime alone, the factor of $3$ to account how many spacetime dimensions there are - this is certainly not an infinite amount of matter, but it is arguably a lot yet, our universe does not appear very dense at all. 
Early rotation properties resulting in centrifugal and torsion (the latter here to prevent singularities forming) as corrections to cosmology (if our universe is not a black hole analogy) could explain how a universe can break free from a dense Planck epoch (according to Arun and Sivaram). A lot of misconceptions concerning primordial rotation, exists even today. 
Instead of going into great deal concerning equations I studied, I will give a summary of what I learned from it:
Hoyle and Narlikar showed that rotation exponentially decays with the linear expansion of a universe (this solves nicely why we cannot detect the background radiation ‘’axis of evil’’ expected to be like a finger print of rotation in the background temperatures).
Dark flow, an unusual flow that seems to show that galaxies are drifting in some particular direction at a very slow speed, could be the existence of a residual spin that has been left over.
Rotation explains cosmic expansion as a centrifigal force. Arun and Sivaram made a calculation from an expanding model.
Because rotation is suggested to slow down, it would seem then at odds to why the universe is now accelerating. There may be two ways out of this problem. The light we detect from the further galaxies tend to tell us something about the past, not something about the present moment in that region of spacetime. What appears to be accelerating, maybe the light from an early universe when it was accelerating. This would explain nicely the Hubble recession in which the further the galaxy, the faster it appears to receed. A second option comes from recent studies, which it has been stated that cosmologists are pretty sure the universe is expanding, but they are no longer sure at what speed.
If particle production happened while the universe expanded due to centrifugal acceleration, then there is no need for inflation to explain why matter appears to be evenly distributed (as noticed by Hoyle). In fact, Inflation doesn’t answer for anything, according to Penrose because it requires a fine tuning. Though this bit is quite speculative, I have wondered whether the spin has ''taken'' the bulk energy of the vacuum in an attempt to explain the quantum discrepency, dubbed the ''worst prediction'' ever made. 
The fact the universe could have a rotational property, would explain why there is an excess of matter over antimatter because the universe would possess a particular handednesss (chirality) - there is also a bulk excess of a particular rotation property observed in a large collection of galaxies with odds ranging between 1 to a million by chance. 
But most importantly of all (and related to the previous statement), it suggests that there is in fact a preferred frame in the universe so long as it rotates. This will imply a Lorentz violating theory but one that satisfies the full Poincare group of symmetries. According to Sean Carrol, Lorentz violating theories will involve absolute acceleration. 
Some people might say ''dark energy is responsible,'' and there would have been a time I would have disagreed with this, since dark energy only becomes significant when a universe gets sufficiently large enough - it's effects are apparent because we believe the universe is accelerating now. 
But I noticed a while back, that this is not the case if the impetus of a universe was constant, but strong enough to break free from dense fields. The difference here, is that scientists tend to think of this situation as one where the cosmological constant is not really a constant - but I tend to think now that it is a constant and the real dynamic effect giving rise to acceleration, is a weakening of gravity as it gets larger. 
A: The OP's question may have concerned the "direct collapse" of matter into a black hole, which has been astronomically verified on one recent occasion (in 2008), as discussed on the Astronomy Stack Exchange in 2018.  However, for reasons I'll put forward here, it may also relate to stellar collapse, which is much more common.
On more than 90 occasions, clear evidence of the "stellar collapse" of matter into black holes has been found:  Because all known stars rotate, and most stars are partners in binary pairs, most such evidence consists of a partner continuing to follow the elliptical orbit both had shared prior to the other partner's collapse, under its own weight, after the conversion of most of its nuclear fuel to radiation had left it without the internal radiation pressure adequate to prevent such a collapse.
The effects of each such collapse remain as permanent as anything we can detect:  Theories of particulate decay that have been endorsed by Stephen Hawking's frequent collaborator, the mathematical physicist Sir Roger Penrose, predict that any particles within black holes will be the last to decay within any locality where particles might be observed.  Although faint radiation outward from black holes has been hypothesized by Hawking, its observation is not, under such theories, expected to occur while any life that has originated on earth, or even any cybernetic descendants of it that might be composed of subatomic particles as we know them, remain viable.  Except in some purely idealistic (and consequently imaginary) sense, the duration of BHs may, consequently, be considered infinite.
Virtual particles and their anti-particle partners are necessarily separated from each other (by at least the Compton wavelength, during at least the Compton time) by the event horizon of any black hole, during that horizon's extremely rapid propagation outward from the center of the volume which had been occupied by the collapsing star. The fermions among those separated particles on the inboard side of that horizon are materialized, in a reversal of the processes which can convert intense concentrations of heavy matter into nuclear energy.
Like all fermions, the newly-materialized ones spin, and the interaction of their spin with that of the star's own (vastly larger) fermions reverses and accelerates their trajectories outward, thereby forming a local universe (shaped approximately like the thick skin of a basketball), in a phenomenon that can be described as an expansion of the space within the region that has been causally-separated from the larger LU which had been its "parent".
After an inflationary (asymptotically exponential) expansion, the newer LU continues expanding quasi-inertially.  As in the other inflationary cosmologies, the spatial expansion of the local universe continues into an infinite future.
(Readers finding this description too mechanistic for something as ethereal-sounding as spatial expansion might want to consider Rebhan's 2012 paper at https://arxiv.org/pdf/1211.1006.pdf, whose conclusions point out the fact that descriptions of such expansion are fundamentally identical to those describing an explosion, with their consideration as the latter being more appropriate for such outside observers as ourselves, given the fact that the causal separation between ourselves and the "parenting" LU would combine with the continuing expansion of that parent to render astronomical observations of it impossible.)
The process described above is described more formally in Nikodem J. Poplawski's "Cosmology with torsion", the first of many papers he's written about his past- and future-eternal cosmological model between 2010 and 2020:  They're available free on the Arxiv site, and are also found in articles put out by Elsevier and other highly-reputed publishers of scientific material.  His cosmology is based on Einstein-Cartan Theory, which was worked out through conversations between Einstein and the mathematician Cartan, 14 years after Einstein's publication of General Relativity.  Although Poplawski's 2010 "Cosmology with torsion" sketches his model as an "alternative" to cosmic inflation, it's now more generally considered to be a version of inflation, and perhaps the only one that does not rely upon a hypothetical field of "inflaton" particles.
With their expansion continuing inertially, the local universes of Poplawski's multiverse, evolving on sequentially smaller scales of spacetime, might eventually contain black holes of their own, although, due to local limitations on the time and energy available for use in magnification, their inhabitants might be able to observe any one of the black holes of any other sequence (even through the indirect process I described earlier) only if it would happen to be one that's on a scale approximately corresponding to the scale of their own astronomical surroundings:  With expansion differing from relative motion and consequently not subject to the speed of light (which might itself vary between such causally-separated regions), the only exception might be their own LU, whose outermost spatial surface they would see simply as those parts of the night sky which are not occupied by stars.
Consequently, Poplawski's relativistic cosmology provides the simplest explanation for Olbers' Paradox, answering the question as to why the sky is not a lethal sheet of fire everywhere.
It has two other advantages:  First, as detailed by the Rutgers philosopher Paul Linford in the last two paragraphs of Section 4.2 in his draft at https://arxiv.org/pdf/2006.07748.pdf, the time-asymmetric boundary and entropic conditions of "offspring" black holes clearly allow for the past-eternality often favored in "naturalistic" theories of reality.  Second, applications of his cosmology, even by civilizations not much more advanced than our own, might provide for "artificial" additions to reality on scales of unprecedented size: The addition of a mass as small as "an apple" to a star about to collapse into a neutron star might allow its collapse into a black hole instead, starting a sequence of local universes on Poplawski's model, as described recently at   https://www.scientificamerican.com/article/mystery-object-blurs-line-between-neutron-stars-and-black-holes/s .
A: Yes, it is dense, but at that "time" space and time, is undefined, so the two statements would be non-sense. Plus, the inflation field has way more power than the gravitational pull, so the universe bursts in just a fraction of a second.  
Brief story of the Big Bang and Before it
Before the Big Bang, there was this field called the inflation field. Which consists of repulsive particles, called the inflatons. Theoretically, the inflation field is considered to be the reason of the creation of a new universe. Every time the vacuum of the field gets excited, it bursts, forming a new universe.
And the field would stay quiet to regain its energy for the next universe to be born.
And the inflation field might explain the repulsive dark energy. (we don't know yet)  
How to get a peak at our universe
Under our best understanding, the only way possible, regardless of being smaller than the plank scale(which will never be possible), is String Theory.
String Theory says that all matters are made out of strings, and the strings are made out of extra dimensions(10D or 11D). The stings live in a 10 ten dimensional world. It is from the six extra dimensions that made up all the matters!!. Physicists have calculated just how many ways the extra dimensions can interweave each other to form a new string, and the result they found was impressive: one followed by 500 zeros. All of them potentially able to give rise to a universe!
Stepping outside
To see our entire universe, first we need to shrink. To the size of about a billion billion billion times smaller than an actual human hair. Now, you need to slip in one of the strings, into the extra dimensions, in order to see your entire universe from above. There you will see the entire universe in front of you, along with other parallel universes, called Branes. These universe comes with different sizes, different dimensions.
And that's how you can actually see the whole universe from "above". Theoretically, it works, but in real life, its not quite practical.
