<p>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.</p>

<p>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 $99&lt; z&lt; 136$. </p>

<p>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.</p>

<p>However, we can avoid looking "under the hood" and use a <a href="http://home.fnal.gov/~gnedin/cc/" rel="noreferrer">cosmology calculator</a>. 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 <strong>between 10.8 and 17.3 million years after the big bang</strong>.</p>

<p>I guess you are perhaps thinking about <a href="https://www.nature.com/news/life-possible-in-the-early-universe-1.14341" rel="noreferrer">these ideas</a> that life could have been around about 15 million years after the big bang.</p>

CC BY-SA 4.0
2018-05-10T18:56:09.260Z
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.
Monty Harder
CC BY-SA 4.0
2018-05-10T21:43:07.307Z
@MontyHarder And then there's the phrase "Twice as small".
Acccumulation
CC BY-SA 4.0
2018-05-11T12:27:11.767Z
@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.
Carl Witthoft
CC BY-SA 4.0
2018-05-10T11:45:03.330Z
2018-05-10T20:58:36.847Z
2018-05-10T20:58:36.847Z
ProfRob
<p>Cosmic microwave background radiation is 2.7 K today. When would it have been between 0 and 100 C?</p>

CC BY-SA 4.0
2018-05-10T15:15:40.223Z
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.
pela
CC BY-SA 4.0
2018-05-10T15:39:51.633Z
@pela  Except... https://www.nature.com/news/life-possible-in-the-early-universe-1.14341
ProfRob
CC BY-SA 4.0
2018-05-10T16:54:28.310Z
@pela  Photino birds don't need no steenkin' water
Carl Witthoft
CC BY-SA 4.0
2018-05-10T20:30:35.780Z
@RobJeffries Of course, Avi Loeb is always ready with a craz… I mean with an interesting hypothesis.
pela
CC BY-SA 4.0
2018-05-20T17:06:49.497Z
Sorry Rob. I upvote almost everything you enter on this site, but I'm with pela on this one.
Jack R. Woods
<cosmic-microwave-background>
How long after the Big Bang would CMB radiation have been between 273 and 373 K?
user332336
# How long after the Big Bang would CMB radiation have been between 273 and 373 K?

(Asked by: username_0 on 2018-05-10)

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

**Tags:** cosmic-microwave-background
> **Comments:**
> **username_1**: 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.  
> **username_2**: @pela  Except... https://www.nature.com/news/life-possible-in-the-early-universe-1.14341  
> **username_3**: @pela  Photino birds don't need no steenkin' water  
> **username_1**: @RobJeffries Of course, Avi Loeb is always ready with a craz… I mean with an interesting hypothesis.  
> **username_4**: Sorry Rob. I upvote almost everything you enter on this site, but I'm with pela on this one.  

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## 1 answers

### Answer by username_2 on 2018-05-10. Score: 17 | Accepted answer

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](http://home.fnal.gov/~gnedin/cc/). 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](https://www.nature.com/news/life-possible-in-the-early-universe-1.14341) that life could have been around about 15 million years after the big bang.

> **Comments:**
> **username_5**: 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.  
> **username_6**: @MontyHarder And then there's the phrase "Twice as small".  
> **username_3**: @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.  

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