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I seem to remember reading somewhere that within the lifetime of a closed, recollapsing Friedman-Lemaitre-Robertson-Walker universe light could travel around the space precisely once.

Could someone please enlighten me, or provide a negative answer if I'm "off my rocker"? I would imagine things like matter v.s. radiation dominance would matter (pun intended). Please note this is plain old GR, so no cosmological constant here.

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It is true, but the matter to radiation ratio is not important, only the fact that it is closed and first expands and then recollapses is important, that alone is enough to discover that the statement is correct for purely geometric reasons.

If the circumference (2π times the radius of curvature) and/or Hubble parameter is small enough though light might also circle around multiple times, but at least once, since at the latest during the collapse at one point it must reach the half circumference, from where it is nearer to the looping point at the full circumference in front than in the back.

Then during the further collapse the rest distance to the looping point in front also continuously decreases as space shrinks, so no later than when the distances have all shrunk to 0 at the big crunch it must have made all the way around.

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