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It makes sense to me that the universe would eventually stop expanding, as bodies of mass attract each other, and I understand that dark energy can counteract this attraction. However, I don't understand how an open universe can expand forever without dark energy. Why is a closed universe any different? What about negatively-curved space allows the universe to maintain a fast enough expansion velocity?

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An open universe is simply a universe which has too high an expansion rate and too little gravity to slow it to a point where it would begin to collapse. As you say, dark energy is believed to be speeding up the expansion and the universe will go on expanding forever, but don't worry if you don't understand what dark energy is or where it came from. Neither does anyone else. Measurements of the redshift of very distant type 1a supernovae suggest that the universe is expanding faster than we thought, but these measurements are extremely difficult to make, so there is room for error. We will just have to wait to see if further investigation confirms or revises this evidence, or puts a new interpretation on it. A one-off universe without a cause and which finally fades away into a black void poses several problems, as does the dark energy which makes it possible.

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    $\begingroup$ But how does the high expansion rate relate to negatively-curved space? $\endgroup$ – Stym Jul 30 '19 at 14:41
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    $\begingroup$ This answer doesn't address the question at all. Your first sentence just restates what the OP asked about. The rest is a tangent about dark energy when the question specifically asked about a non-dark energy universe. $\endgroup$ – Sir Cumference Jul 31 '19 at 1:35

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