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This question hit me when I was watching 'Cosmos by Neil deGrasse Tyson' explaining how the universe is expanding and how it accelerates because of dark energy and dark matter, then I asked myself what if there were only ordinary matter in the universe?

Why am I here
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If there is no dark energy, the fate of the universe depends on whether the matter density exceeds the critical density. If it does then the universe recollapses. If it doesn't then it expands forever.

Even if the universe expands forever, the expansion always decelerates if there is no dark energy. (Compare this to the motion of a rock thrown upward in a gravitational field. If its speed exceeds the escape velocity then it continues to rise forever, even though it is always decelerating from the downward gravitational force.)

Dark matter behaves like ordinary matter in cosmology, so there is no need to exclude it from these models.

You can find a discussion of these models in any textbook or popular book about cosmology that was published before c.2001, when the first strong evidence for dark energy was found.

benrg
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    For a more recent than 2001 pop-cosmology book that discusses some of this, try "The End of the Everything (Astrophysically Speaking)" by Katie Mack. It discusses various ways the universe could end, including big crunch, heat death, and big rip. It also discusses how the relative make-up of the universe (matter, dark energy) leads to different outcomes. – Paul T. Aug 29 '22 at 15:33
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The amount of ordinary matter in the universe, is less than 5% of the amount that would be required to ultimately halt the expansion of the universe (Planck 2018 results).

Even if you were to add in the "dark matter" this fraction would only creep up to 31.5%.

Thus without dark energy, the fate of the universe would be to expand at a decelerating rate, but never coming to a halt. A bit like throwing a ball away from the Earth at greater than the escape speed.

The addition of dark energy means that the expansion is now (and in the future, as far as we know) accelerating.

ProfRob
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