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.