Shape of Universe & Cosmological Principle

[ConfusedMonkey]

Let me preface this post by saying that I only have a very cursory understanding of general relativity.

I happen to know that if we assume the cosmological principle, then the hypersurface $\Sigma_t$ of the spacetime manifold $M$, for any positive $t$, is either a 3-sphere, a 3-hyperboloid or just flat 3-space depending on the value of $k$ in the FRW-metric. Now, the assumptions of homogeneity and isotropy are made very mathematically precise. However, we know that the universe is only approximately homogeneous and isotropic, so our mathematical notions of homogeneous and isotropic do not exactly reflect what we observe in the universe. What consequence does this have for the shape of $\Sigma_t$. Will it only be approximately a 3-sphere, for example? Perhaps a 3-spheroid? Or is the shape completely different and/or indeterminable?

EDIT: If the answer is that indeed the shape is approximately a 3-sphere, for example, how do we go about proving this? I get that the universe is almost an FLRW spacetime, but that does not mean that its shape has to be as if it were exactly an FLRW spacetime, does it?

 

[timmdeeg]

The data we have indicate that the universe is spatially flat. The sphere can't be excluded though but seems quite unlikely.
And further the global shape whatsoever does not depend on local inhomogeneities, because it is assumed that the cosmological principle holds on large scales.

 

[Dale]

I thought that I saw a recent paper on this topic. Let me dig a bit

Edit: I think this is the paper I was thinking of https://arxiv.org/abs/1511.01105