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Cosmic microwave background radiation is 2.7 K today. When would it have been between 0 and 100 C?

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    $\begingroup$ Note that at this time — in the so-called "dark ages" — stars hadn't formed yet, so there was no oxygen and hence no water. $\endgroup$ – pela May 10 '18 at 15:15
  • $\begingroup$ @pela Except... nature.com/news/life-possible-in-the-early-universe-1.14341 $\endgroup$ – Rob Jeffries May 10 '18 at 15:39
  • $\begingroup$ @pela Photino birds don't need no steenkin' water $\endgroup$ – Carl Witthoft May 10 '18 at 16:54
  • $\begingroup$ @RobJeffries Of course, Avi Loeb is always ready with a craz… I mean with an interesting hypothesis. $\endgroup$ – pela May 10 '18 at 20:30
  • $\begingroup$ Sorry Rob. I upvote almost everything you enter on this site, but I'm with pela on this one. $\endgroup$ – Jack R. Woods May 20 '18 at 17:06
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The temperature of the cosmic microwave background scales as the inverse of the cosmic scale factor $a$. i.e. When everything was at half the separation it is now, then the CMB was twice the (absolute) temperature. The scale factor in turn is reciprocally related to the redshift by $a/a_0 = (1 +z)^{-1}$, where $z$ is the redshift and $a_0$ is the present-day scale factor, usually taken to be 1.

From this we see that $$T(z) = T_0 (1+z),$$ where $T(z)$ is the temperature at redshift $z$ at some time in the past and $T_0 = 2.73$K is the temperature of the CMB now. For the range of temperatures you specify requires redshifts in the range $99< z< 136$.

Getting from a redshift to a time in the universe since the big bang is not so simple. The answer depends on what you take to be the "cosmological parameters" - i.e. the values of the cosmic matter density, the dark energy density and so on.

However, we can avoid looking "under the hood" and use a cosmology calculator. The one I have linked to is for a "flat" universe and has default values for what are currently good estimates for the matter density and present-day Hubble parameter. For the range of $z$ I found above, this corresponds to an age of between 10.8 and 17.3 million years after the big bang.

I guess you are perhaps thinking about these ideas that life could have been around about 15 million years after the big bang.

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    $\begingroup$ The wording "2 times hotter" is confusing. If something is 2 times hotter, then it's 3 times as hot, not 2 times as hot. $\endgroup$ – Monty Harder May 10 '18 at 18:56
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    $\begingroup$ @MontyHarder And then there's the phrase "Twice as small". $\endgroup$ – Acccumulation May 10 '18 at 21:43
  • $\begingroup$ @MontyHarder Going down the grammatical rabbit hole, you can't have any "N-times hotter" . You can have "N-times as hot" , or you can have "much hotter." But at least we can agree (?!?!) that "hot" means a temperature and "hotter" means a higher temperature, even tho' in colloquial speech we use those to describe the rate of energy transfer from an object to our about-to-be-burned fingertips. $\endgroup$ – Carl Witthoft May 11 '18 at 12:27

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