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There are more than one way to view the description of the universe as flat. There is the description of an open, flat or closed universe in terms of it's fate, expansion forever away from gravity, or a big crunch.

Then there is the description of our observable universe being flat in it's geometry, or is it that there is no way to measure the tiny curve of space in our part of the sphere.

Another probably wrong idea is that the entire universe is flat. Are these things confused, even by the experts?

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    $\begingroup$ Are experts the ones that don't confuse such words? $\endgroup$
    – BMS
    Commented Apr 2, 2014 at 4:59
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    $\begingroup$ Diagram describing various properties and behaviour of universes with matter & cosmological constant (i.e. radiation energy density negligible). "Flat" is any universe on the line between open and closed (notice that this is NOT the same as will/will not recollapse!!!). Negative cosmological constants are allowed in this diagram, and behave as a constant attractive term instead of a constant repulsive one. astro.uni-wuppertal.de/~kampert/Kosmologie-Bilder/… $\endgroup$
    – Kyle Oman
    Commented Apr 2, 2014 at 20:01

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It is only in the absence of dark energy that the correspondence between geometrical curvature and the ultimate fate of the universe is as straightforward as you describe.

Measurements (primarily of the cosmic microwave background) indicate that our universe is flat or very nearly so, which should be interpreted geometrically (i.e. in terms of the sum of the angles of a geodesic triangle). In the absence of dark energy, this would correspond to a scenario in which the universe continued to expand but asymptotically approached zero expansion velocity.

However, the concurrent measurements of the presence dark energy suggest that our geometrically flat universe will continue to undergo accelerated expansion.

The influence of dark energy is sometimes neglected in popular accounts, leading to much confusion among non-experts. Keep in mind, though, that it was only in the last 15 years or so that scientists had any direct evidence for the presence of dark energy, so it might be understandable why it was put aside in earlier simplified explanations.

A discussion of the ultimate fate of the universe, its connection to geometrical curvature, and the role of dark energy, is found in the wiki article.

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  • $\begingroup$ OK, but even with the presence of dark energy, it would depend on whether it remained at constant density, despite the expansion, which causes a lower density of every other type of energy, and whether dark energy has a point when it stops pushing, and starts pulling with it's own gravity. I find it hard to believe they know anything about it, based on observations, as the vast majority of the universe can't be seen, and what we can see, we interpret with a fairly large margin of error like intergalactic distance measurements. The simplest solution is usually true. $\endgroup$
    – rowanman28
    Commented Apr 2, 2014 at 6:35
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    $\begingroup$ First, every time I speak of the "universe," I implicitly mean the observable universe, which we see all the way out to 13 billion light years via the CMB. Second, you are correct that we don't understand dark energy in detail. Many of these details will affect the exact rate at which the universe will expand in the future, which remains uncertain. But the fact that the expansion is accelerating has been measured quite robustly, even when accounting for probable sources of error. This accelerated expansion calls for some new piece of physics beyond matter and radiation energy. $\endgroup$
    – kleingordon
    Commented Apr 2, 2014 at 6:59
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There's a difference between curvature of spacetime and curvature of space.

Extrapolating from what we can see around us and assuming the cosmological constant lives up to its name, spacetime will eventually approach curved de Sitter geometry, in contrast to flat Minkowski geometry or anti-de Sitter geometry of opposite curvature. This is something of an idealization: Obviously, we can't really know what goes on beyond the cosmic event horizon, and the cosmological constant might not deserve its name.

Taking the preferred spatial slicing at constant cosmological time into account, we can also talk about spatial curvature. It turns out the universe is spatially flat (or very nearly so). However, this does not tell us if the universe is spatially compact or infinite - eg a torus can be equipped with a flat connection in spite of being compact.

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  • $\begingroup$ Christoph: "eg a torus can be equipped with a flat connection" -- On any usual torus surface e.g. there are easily 4 points, $A$, $B$, $J$, $K$ such that $AB = JK$ and $AJ = AK = BJ = BK$. (In the sketch e.g. $A$ and $B$ on the red ring symmetrically above and below the outer black circumference; together with $J$, $K$ on the inner black ring left and right of the red ring.) Those are explicitly not plane to each other: their Cayley-Menger determinant doesn't vanish. (So: the universe is flat?? ...) $\endgroup$
    – user12262
    Commented Apr 2, 2014 at 21:01
  • $\begingroup$ @user12262: see en.wikipedia.org/wiki/Torus#Flat_torus $\endgroup$
    – Christoph
    Commented Apr 2, 2014 at 21:17
  • $\begingroup$ Christoph: First I have to correct: The relations $AB = JK$ and $AJ = AK = BJ = BK$ are of course also satisfied by those four points being "corners of a square" (with "diagonals" $AB$ and $JK$) which is of course plane since precisely $AB = JK = \sqrt{2} \, AJ$. My point was then that on a torus surface (or even on a cylinder) four points can be easily found such that $AB = JK$ and $AJ = AK = BJ = BK$ and $AB < AJ$. "en.wikipedia.org/wiki/Torus#Flat_torus" -- Indeed, one more reason to be extremely wary to use coordinates in cosmology, or in physics overall. $\endgroup$
    – user12262
    Commented Apr 3, 2014 at 5:40
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I think this is an example,universe is rotating about its own central axis.if this is the case,take a curve beaker and a flat plate with some water in them.First shake clockwise the curved beaker and then flat plate.In which,beaker or plate did the circular motion was seen about its center?I guess curved one.So,our universe is curved.

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