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For instance, at the scale factor $a=0.5$ and matter dominated era, what was the size of the observable universe and matter density?

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    $\begingroup$ You can find the answers here, if you're willing to squint at the plots closely enough: astronomy.stackexchange.com/a/53121/47607 ($a=0.5$ corresponds to a time of about 6 billion years or $2\times 10^{17}$ seconds). $\endgroup$
    – Sten
    Commented Aug 16, 2023 at 5:18

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