Does Flat Universe mean even distribution?

Question

I've read through all the other related answers to "flatness" questions but I need a bit more clarification.

I understand that a triangle in a 2D universe would not equal 180 degrees in both a closed and open universe. If I am correct, not being flat during the initial conditions would mean exponential expansion or collapse as time went by.

Given the above, could we explain flatness as uniform density of the universe, meaning all of the universe is the same approximate "consistency" on a large scale (homogeneity).

Answer 1

No, being flat and being homogeneous is not equivalent$^\dagger\!\!\!$.

Flatness refers to the geometry, which depends on the total energy density $\rho$; if it is above or below a certain critical threshold $\rho_\mathrm{cr}$, we call the Universe "closed" or "open", respectively, while if $\rho$ is exactly equal to $\rho_\mathrm{cr}$, we call it "flat".

Homogeneity refers, as you say, to matter being evenly distributed (on large scales). If the Universe is not only homogeneous, but also isotropic (i.e. looks the same in all directions), then there are three different possible geometries, namely flat, open, or closed. But a homogeneous universe doesn't have to be isotropic, and both open and closed universes can have evenly distributed matter.

Thus, $$ \mathrm{homogeneity} \nRightarrow \mathrm{flatness} $$

Whereas the local geometry depends on the local density$^\ddagger$, the global geometry does not depend on how the matter is distributed. For instance, you could in principle have a universe with no upper limit of structure size. In our Universe, we find observationally that the largest structures have sizes of roughly half a gigalightyear. Above this scale, it is homogeneous (although it could be inhomogeneous on scales larger than the particle horizon). But another universe might have structure on all scales and thus not be homogeneous, but still meet the criterion $\rho = \rho_\mathrm{cr}$ and thus be flat. You could also, as discussed in this answer about homogeneity and isotropy, imagine a universe originating at a central point away from which the density always decreases (i.e. is isotropic around this point), but which still has $\rho = \rho_\mathrm{cr}$.

Thus, $$ \mathrm{flatness} \nRightarrow \mathrm{homogeneity} $$

Note that while your statement about the geometry determining the fate of the Universe would be correct if it contained only matter and radiation, it seems that $\sim70$% of the density making it flat has the annoying propety of accelerating the expansion of the Universe. Thus, whereas a flat, matter-dominated universe would expand asymptotically toward a finite size, our Universe is dominated, it seems, by something dubbed dark energy making it expand exponentially despite being flat.

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$^\dagger$_{For a universe, at least. For a roadkill, it might be equivalent.}

$^\ddagger$_{For instance, space "bends" enough around a massive cluster of galaxies to make gravitational lenses.}

Answer 2

Does a flat universe mean even distribution?

Strictly speaking no. But there's no actual evidence for any kind of "higher dimensional curvature". See https://arxiv.org/abs/1303.5086 where the Planck mission reported on the Background geometry and topology of the Universe. They found no evidence for any kind of "asteroids" toroidal universe.

I understand that a triangle in a 2D universe would not equal 180 degrees in both a closed and open universe. If I am correct, not being flat during the initial conditions would mean exponential expansion or collapse as time went by.

That's what people tend to say, but IMHO you shouldn't treat it as settled science. See "The Waters I am Entering No One yet Has Crossed": Alexander Friedman and the Origins of Modern Cosmology. It’s by Ari Belenkiy, who credits Friedmann with the expanding universe. He also refers to Friedmann's m-for-monotonic M1 universe which expands from a singularity and has an inflexion point, his M2 universe which expands from a non-zero radius, and his p-for-periodic P universe which expands and contracts. The thing is, that we don't actually know for sure that any of these options properly describe the universe. Particularly since we have good evidence that the universe was once very dense, but nevertheless expanded.

Given the above, could we explain flatness as uniform density of the universe, meaning all of the universe is the same approximate "consistency" on a large scale (homogeneity).

I would say this: given that we don't know that Friedmann cosmology is correct, we might explain flatness as uniform density of the universe.