However, the Big Bang theory suggests that the universe was in a very compact form about 13 billion years ago. If you work out the Schwarzchild radius of this mass, we could well be inside a *universe*-sized black hole.

Hi evan_au,

The curvature of the observable universe is currently measured to be flat, or very close to it. It appears that the density is at the so the called critical value. As a consequence of flatness, not a black hole, the Schwarzchild radius of the universe equals its age multiplied by the speed of light(which is the observable radius in a flat universe). This is a mathematical identity. When you calculate the Schwarzchild radius, and assume a flat universe, ie., one whose critical density equals its actual density, the schwarzchild radius will yield an answer equal to the radius of the observable universe. No black hole here. I can show my work if you like.

Incidentally, in Friedman-Lemaitre-Robertson-Walker spacetime, the metric is not the same as the schwarzchild metric. The Schwarzchild equation for a black hole is not typically used to calculate a solution to a metric of time dependent, matter filled, homogeneous and isotropic conditions.