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Matthew O'Dowd specifically and repeatedly states in the PBS Space Time video called, "Why the Universe Needs Dark Energy", that even without taking an expansionary field into account an under-dense Universe can, and probably will, expand forever...

I have read this elsewhere, as well...

How is this possible?

If the space of the universe is not curved outwards (hyperbolic curvature), and does not contain a cosmological constant (or eternal inflation, or dark energy, etc.), how can it not slow down and eventually stop? And then start to shrink?

All theories of gravity claim it has infinite reach, correct?

I am deeply confused....

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Without dark energy, the expansion of the universe will slow down (because of gravity). But it need not stop expanding and contract again. That's because it's possible the force of gravity is not strong enough to reverse the rate of expansion.

As Greg Miller points out in a comment, it's similar to the escape velocity. If you throw something fast enough, it'll leave Earth forever and never return. Earth's gravity will slow it down, but not by enough for it to turn around.

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