I watched with interest the NOVA Black Hole Apocalypse which was fun. They had a secondary list of vids of one was entitled Four Types of Black Holes.
Now I thought that black holes don't necessarily have to be dense, if they are very large. The Swartzchild radius is
$$ r_\text{s} = \frac{2 M G}{c^2} $$
so if the mass-to-radius ratio exceeds
$$ \frac{M}{r} > \frac{c^2}{2 G} $$
you gotta black hole. But it's only a single power of $r$ in the denominator. This means, assuming a sphere, if the density exceeds
$$ \frac{M}{\tfrac{4}{3}\pi r^3} > \frac{3 c^2}{8 \pi G r^2} $$
you gotta black hole. With $r^2$ in the denominator, it appears you wouldn't need a lotta density if the radius is large enough. With
$ \frac{c^2}{2 G} = 6.7332 × 10^{26} \frac{\text{kg}}{\text{m}} $
and $ \frac{M}{r} = 4.5455 × 10^{27} \frac{\text{kg}}{\text{m}} $
(Numbers come from Wikipedia. This is including non-baryonic matter and energy, all which should have gravitational effect.) It looks pretty close, but with these numbers the entire mass and energy of the observable Universe is contained in a radius smaller than the Swartzchild radius. 45.455 is bigger than 6.7332.
Couldn't an entire observeable universe be a black hole? Wouldn't this be a different "type", a fifth type of black hole than the four described in the NOVA vid?
