Big bang cosmology, as far as I understand it, says that the universe was super hot and super dense and super small. It looks like that all the current matter, seen and unseen, were compressed to infinitesimal distance, which means it was a black hole.

1- Is the big bang, and our universe expansion in this case, hence is nothing but a black evaporation via Hawking radiation?

2- Are we living inside that primordial black hole explosion?

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Here is a copy of an answer I wrote some time ago for the Physics FAQ

Is the Big Bang a black hole?

This question can be made into several more specific questions with different answers.

Why did the universe not collapse and form a black hole at the beginning?

Sometimes people find it hard to understand why the Big Bang is not a black hole. After all, the density of matter in the first fraction of a second was much higher than that found in any star, and dense matter is supposed to curve spacetime strongly. At sufficient density there must be matter contained within a region smaller than the Schwarzschild radius for its mass. Nevertheless, the Big Bang manages to avoid being trapped inside a black hole of its own making and paradoxically the space near the singularity is actually flat rather than curving tightly. How can this be?

The short answer is that the Big Bang gets away with it because it is expanding rapidly near the beginning and the rate of expansion is slowing down. Space can be flat even when spacetime is not. Spacetime's curvature can come from the temporal parts of the spacetime metric which measures the deceleration of the expansion of the universe. So the total curvature of spacetime is related to the density of matter, but there is a contribution to curvature from the expansion as well as from any curvature of space. The Schwarzschild solution of the gravitational equations is static and demonstrates the limits placed on a static spherical body before it must collapse to a black hole. The Schwarzschild limit does not apply to rapidly expanding matter.

What is the distinction between the Big Bang model and a black hole?

The standard Big Bang models are the Friedmann-Robertson-Walker (FRW) solutions of the gravitational field equations of general relativity. These can describe open or closed universes. All of these FRW universes have a singularity at their beginning, which represents the Big Bang. Black holes also have singularities. Furthermore, in the case of a closed universe no light can escape, which is just the common definition of a black hole. So what is the difference?

The first clear difference is that the Big Bang singularity of the FRW models lies in the past of all events in the universe, whereas the singularity of a black hole lies in the future. The Big Bang is therefore more like a "white hole": the time-reversed version of a black hole. According to classical general relativity white holes should not exist, since they cannot be created for the same (time-reversed) reasons that black holes cannot be destroyed. But this might not apply if they have always existed.

But the standard FRW Big Bang models are also different from a white hole. A white hole has an event horizon that is the reverse of a black hole event horizon. Nothing can pass into this horizon, just as nothing can escape from a black hole horizon. Roughly speaking, this is the definition of a white hole. Notice that it would have been easy to show that the FRW model is different from a standard black- or white hole solution such as the static Schwarzschild solutions or rotating Kerr solutions, but it is more difficult to demonstrate the difference from a more general black- or white hole. The real difference is that the FRW models do not have the same type of event horizon as a black- or white hole. Outside a white hole event horizon there are world lines that can be traced back into the past indefinitely without ever meeting the white hole singularity, whereas in an FRW cosmology all worldlines originate at the singularity.

Even so, could the Big Bang be a black- or white hole?

In the previous answer I was careful only to argue that the standard FRW Big Bang model is distinct from a black- or white hole. The real universe may be different from the FRW universe, so can we rule out the possibility that it is a black- or white hole? I am not going to enter into such issues as to whether there was actually a singularity, and I will assume here that general relativity is correct.

The previous argument against the Big Bang's being a black hole still applies. The black hole singularity always lies on the future light cone, whereas astronomical observations clearly indicate a hot Big Bang in the past. The possibility that the Big Bang is actually a white hole remains.

The major assumption of the FRW cosmologies is that the universe is homogeneous and isotropic on large scales. That is, it looks the same everywhere and in every direction at any given time. There is good astronomical evidence that the distribution of galaxies is fairly homogeneous and isotropic on scales larger than a few hundred million light years. The high level of isotropy of the cosmic background radiation is strong supporting evidence for homogeneity. However, the size of the observable universe is limited by the speed of light and the age of the universe. We see only as far as about ten to twenty thousand million light years, which is about 100 times larger than the scales on which structure is seen in galaxy distributions.

Homogeniety has always been a debated topic. The universe itself may well be many orders of magnitude larger than what we can observe, or it may even be infinite. Astronomer Martin Rees compares our view with looking out to sea from a ship in the middle of the ocean. As we look out beyond the local disturbances of the waves, we see an apparently endless and featureless seascape. From a ship the horizon will be only a few miles away, and the ocean may stretch for hundreds of miles before there is land. When we look out into space with our largest telescopes, our view is also limited to a finite distance. No matter how smooth it seems, we cannot assume that it continues like that beyond what we can see. So homogeneity is not certain on scales much larger than the observable universe. We might argue in favour of it on philosophical grounds, but we cannot prove it.

In that case, we must ask if there is a white hole model for the universe that would be as consistent with observations as the FRW models. Some people initially think that the answer must be no, because white holes (like black holes) produce tidal forces that stretch and compress in different directions. Hence they are quite different from what we observe. This is not conclusive, because it applies only to the spacetime of a black hole in the absence of matter. Inside a star the tidal forces can be absent.

A white hole model that fits cosmological observations would have to be the time reverse of a star collapsing to form a black hole. To a good approximation, we could ignore pressure and treat it like a spherical cloud of dust with no internal forces other than gravity. Stellar collapse has been intensively studied since the seminal work of Snyder and Oppenheimer in 1939 and this simple case is well understood. It is possible to construct an exact model of stellar collapse in the absence of pressure by gluing together any FRW solution inside the spherical star and a Schwarzschild solution outside. Spacetime within the star remains homogeneous and isotropic during the collapse.

It follows that the time reversal of this model for a collapsing sphere of dust is indistinguishable from the FRW models if the dust sphere is larger than the observable universe. In other words, we cannot rule out the possibility that the universe is a very large white hole. Only by waiting many billions of years until the edge of the sphere comes into view could we know.

It has to be admitted that if we drop the assumptions of homogeneity and isotropy then there are many other possible cosmological models, including many with non-trivial topologies. This makes it difficult to derive anything concrete from such theories. But this has not stopped some brave and imaginative cosmologists thinking about them. One of the most exciting possibilities was considered by C. Hellaby in 1987, who envisaged the universe being created as a string of beads of isolated while holes that explode independently and coalesce into one universe at a certain moment. This is all described by a single exact solution of general relativity.

There is one final twist in the answer to this question. It has been suggested by Stephen Hawking that once quantum effects are accounted for, the distinction between black holes and white holes might not be as clear as it first seems. This is due to "Hawking radiation", a mechanism by which black holes can lose matter. (See the relativity FAQ article on Hawking radiation.) A black hole in thermal equilibrium with surrounding radiation might have to be time symmetric, in which case it would be the same as a white hole. This idea is controversial, but if true it would mean that the universe could be both a white hole and a black hole at the same time. Perhaps the truth is even stranger. In other words, who knows?

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