I don't understand this answer. By GR there is no possible flat space-time around a dense mass no? BC the energy will curve the space-time. Saying that the space-time was expanding very quickly is also describing the shape of the space-time. Isn't it kind of circular to say that big bang doesn't end in a singularity b/c it is curved out? You can still ask why it's curved out with so much energy and whether it is compatible with GR? But I guess the answer if GR was holding near big bang must just be that there's some solution which is compatible with GR with so much energy in a small place which doesn't end in singularity.
The Schwarzschild solution is the unique distribution in GR for nonrotating mass in a small area, in a universe that is asymptotically flat at a long distance from the mass. This is not a flat universe, but most of it is pretty darned close to flat.
As for describing the shape of space-time, that's what GR does. What we can think of as the "shape" is actually described by something called the metric. GR says that the metric satisfies a differential equation. If the universe starts close to flat, things are moving slowly, and there is a low density of mass, the solutions to this equation create an effect that, to first order, matches Newtonian gravity. But the full theory has solutions with all sorts of bizarre things in it, like waves traveling through space, made up of fluctuations in the very structure of space-time. We call those gravity waves.
And yes, those solutions do include things like expanding universes. And the effect of gravity within an expanding universe is to slow the rate of expansion.
The metric you're referring to, oddly enough is a mapping from flat spacetime to curved. This is why the Schwarzschild and Kerr solutions to the EFE have `r` values that are in flat spacetime and yield spacetime intervals. The metric is symmetric, so you can also map back from curved to flat spacetime.
> By GR there is no possible flat space-time around a dense mass no?
In standard cosmology in the super early universe there wasn't _a_ mass, like a point mass -- there was lots of mass-energy everywhere (not a point anything but a huge swath of space, and very dense), pulling on everything, yes, but at the same time stuff was flying apart with more momentum than the gravity of all the stuff because that gravity was pulling in all directions (therefore causing the gravitational potential to be huge but the net gravitational pull in any direction to be zero) but the pressure was pushing in all directions, so it all could fly apart after all.
