How to concile flat spacetime and big bang?

Question

After reading How do we resolve a flat spacetime and the cosmological principle? I still remain perplex.
Please excuse my ignorance and try explaining to me :

I thought that basically, when we rewind back to the big bang, we get down to planck's dimension (something like 10exp-35) which is small and therefore (?) finite. (i acknowledge we have yet no theory beyond that).
Since :
big bang => small
small => finite
finite * whatever_expansion = finite
finite ~> curved (but see below point #2)
I derive :
big bang ~> should still be curved

So, just like @adam asked (see link above), how can spacetime be said to be flat now ?
May be my question simply gets down to clarify :
When experts say "flat", do they mean :

1. strictly flat whatever the geometry (and then i am lost)
2. strictly flat, but in the sens of specific geometry like "Flat universe ... In three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable" as mentionned in [wikipedia Shape_of_the_Universe] (http://en.wikipedia.org/wiki/Shape_of_the_Universe)
3. or : nearly flat only, as we can observe, (but can't be strictly, because ... see above my reasoning).
4. other ? (please elaborate ...)

Answer 1

Your reasoning contains some errors and unjustified assumptions.

The first error is to think that the universe started out with a size of about the Plank length. This may not be the case. If it is flat and infinite now then it would always have been flat and infinite, even at the beginning, or at least as far back as the point where it makes sense to talk about space-time in such terms. It is true that the observable universe would have started from a very small point, but the whole universe is likely to be much bigger. Even if the universe is curved and finite in size its initial size could have been anything from much smaller than the plank size to much larger.

Your second error is to think that a flat universe has to be infinite. It is true that a universe with constant positive curvature over space must be finite, but the converse is not true. A flat universe or even one with negative curvature can be finite if it repeats with periodic boundary conditions. For a flat space the simplest topology that can have this property is the 3-torus.

You are also making the assumption that the cosmological principle holds on all scales no matter how large. Our observations of the observable universe suggest that this principle is quite reasonable on scales up to billions of light years, but we can't say anything for sure about what the universe is like on much larger scales. The curvature of space may vary in quite dramatic ways beyond the horizon that limits how far we can see due to the finite speed of light.

All four of your options are possibilities and there are too many in the "other" category to elaborate.

Answer 2

During the earliest moments of the universe, energy densities were high enough such that a complete understanding of the physics at that time would require an understanding of how the fabric of spacetime behaves on the scales of the planck length and the planck time. This does not necessarily mean that space was finite in size. It is difficult to say much of anything about the universe until the energy densities dropped low enough such that the standard picture of smooth spacetime, operating according to the principles of GR, becomes applicable. If the universe today is flat and infinite, it would also have been flat and infinite at that earliest time.

One way to define what experts mean when the say 'flat' is that angles drawn between the sides of a triangle add up to 180 degrees, no matter how long the sides of the triangle are. A universe that has positive curvature (angles add up to more than 180 degrees) will be finite in size, but a flat universe can be finite in size. Right now all we know about our own universe is that it appears to be flat or nearly flat. We don't know whether it is finite or infinite in extent, but both are possible.