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## What is the redshift of the Cosmic Microwave Background (CMB)?

Last week, as I mentioned in this blog here, I had an article on the Cosmic Microwave Background’s accidental discovery in 1965 published in The Conversation. Here is a link to the article. As of writing this, there have been two questions/comments. One was from what I, quite frankly, refer to as a religious nutter, although that may be a bit harsh! But, the second comment/question by a Mark Robson was very interesting, so I thought I would blog the answer here.

This article on the Cosmic Microwave Background was published in The Conversation last Thursday (23rd July 2015)

Mark asked how we know the redshift of the CMB if it has no emission or absorption lines, which is the usual way to determine redshifts of e.g. stars and galaxies. I decided that the answer deserves its own blogpost – so here it is.

## How does the CMB come about

As I explain in more detail in my book on the CMB, the origin of the CMB is from the time that the Universe had cooled enough so that hydrogen atoms could form from the sea of protons and electrons that existed in the early Universe. Prior to when the CMB was “created”, the temperature was too high for hydrogen atoms to exist; electrons were prevented from combining with protons to form atoms because the energy of the photons in the Universe’s radiation (given by $E=h \nu$ where $\nu$ is the frequency) and of the thermal energy of the electrons was high enough to ionise any hydrogen atoms that did form. But, as the Universe expanded it cooled.

In fact, the relationship between the Universe’s size and its temperature is very simple; if $a(t)$ represents the size of the Universe at time $t$, then the temperature $T$ at time $t$ is just given by

$T(t) \propto \frac{ 1 }{ a(t) }$

This means that, as the Universe expands, the temperature just decreases in inverse proportion to its size. Double the size of the Universe, and the temperature will halve.

When the Universe had cooled to about 3,000K it was cool enough for the electrons to finally combine with the protons and form neutral hydrogen. At this temperature the photons were not energetic enough to ionise any hydrogen atoms, and the electrons had lost enough thermal energy that they too could not ionise electrons bound to protons. Finally, for the first time in the Universe’s history, neutral hydrogen atoms could form.

For reasons that I have never properly understood, astronomers and cosmologists tend to call this event recombination, although really it was combination, without the ‘re’ as it was happening for the first time. A term I prefer more is decoupling, it is when matter and radiation in the Universe decoupled, and the radiation was free to travel through the Universe. Before decoupling, the photons could not travel very far before they scattered off free electrons; after decoupling they were free to travel and this is the radiation we see as the CMB.

## The current temperature of the CMB

It was shown by Richard Tolman in 1934 in a book entitled Relativity, Thermodynamics, and Cosmology that a blackbody will retain its blackbody spectrum as the Universe expands; so the blackbody produced at the time of decoupling will have retained its blackbody spectrum through to the current epoch. But, because the Universe has expanded, the peak of the spectrum will have been stretched by the expansion of space (so it is not correct to think of the CMB spectrum as having cooled down, rather than space has expanded and stretched its peak emission to a lower temperature). The peak of a blackbody spectrum is related to its temperature in a very precise way, it is given by Wien’s displacement law, which I blogged about here.

In 1990 the FIRAS instrument on the NASA satellite COBE (COsmic Background Explorer) measured the spectrum of the CMB to high precision, and found it to be currently at a temperature of $2.725 \text{ Kelvin}$ (as an aside, the spectrum measured by FIRAS was the most perfect blackbody spectrum ever observed in nature).

The spectrum of the CMB as measured by the FIRAS instrument on COBE in 1990. It is the most perfect blackbody spectrum in nature ever observed. The error bars are four hundred times larger than normal, just so one can see them!

It is thus easy to calculate the current redshift of the CMB, it is given by

$z \text{ (redshift)} = \frac{3000}{2.725} = 1100$

and “voilà”, that is the redshift of the CMB.  Simples 😉

### 29 Responses

1. Reblogged this on johngribbinscience and commented:
Another nice explanation from Rhodri.

2. Thanks John!

3. It’s a pity you couldn’t quote, or indeed answer, my actual question.

• I thought I had. Explain to me how I haven’t answered it.

• This was my question, ”Given that the CMB contains no emission or absorption lines, what features does it contain that allow you to calculate its redshift to confirm that it was emitted when claimed ?” But you seem to have answered a question like, ‘If we assume a hotter earlier universe, and assume that the CMB must have been emitted then (and at no other time by no other source) and has cooled, what redshift must we assign it to fit in with these assumptions?’ I want to know what unique data the CMB possesses which allows you to calculate its age or the temperature at which it was emitted.

• Ok, that is NOT what you asked in your original question

• I will answer that in this comment section tomorrow or Monday. But, suffice it to say that there are very sound answers to your questions, and I will enumerate them in my answer.

• So I’ll assume you are busy and not avoiding my question.

• Yes, very busy, but not avoiding your question. I hope to find time to answer it next week.

Why are you so rude?

• Rude ? You’re the one misquoting me and not answering my question. I just wanted to remind you that I am still waiting – more than 3 weeks on.

• I will answer your question, but you are not my highest priority. I have other, more pressing, things to finish. If you think three weeks is a long time to get an answer then try emailing your local college physics professor and see how long he/she takes.

4. […] of the Cosmic Microwave Background (CMB)?” appeared on my blog on the 30th of August, here is a link to that blogpost. However, it would seem that Mark Robson was not happy with my answer, and commented that I had not […]

5. on 08/06/2016 at 18:27 | Reply CosmoCurious

RhEvans, You say “When the Universe had cooled to about 3,000K [~ 0.3 eV] it was cool enough for the electrons to finally combine with the protons and form neutral hydrogen.” But the Hydrogen ionization energy is ~ 13.6 eV.

So, I think that Mark Robson’s question is actually:
Why the temperature [energy] of photons should have dropped two orders of magnitude below the H ionization energy to ensure decoupling?
In short: why 0.3 eV instead of 13.6 eV?

Thanks and I also would appreciate an answer.

• I have written the blog today explaining why the recombination/decoupling temperature is about 3000K – it will go up on Tuesday 14 June. Your simple reasoning that it is related to the thermal energy of electrons is wrong.

• Mr(?) CosmoCurious. No, that is not and should not be my question. Don’t try to speak for me as your assumptions are wrong. My question was (and is) as stated, and it remains unanswered as I don’t count introducing another unverified assumption (”recombination”) as valid. Though I’m sure now you may tell me of another unverified assumption to back up recombination – even though this recombination has not been detected.

• It’s Doctor to you, you rude twat. Doctor as in PhD. Mr. Robson.

• Nice name calling there Evans, but you might want to try reading my comment – I wasn’t addressing you, I was addressing CosmoCurious – the clue was at the start when I used his(?) pseudonym.

• Just go and learn some basic physics. And manners. Learn some manners first; although physics is important you’re desperately in need of acquiring some manners. And then go and annoy one of your local colleges / universities, but stop posting your rudeness here.

• And, it’s “Doctor Evans, PhD” to you!

• Go and annoy someone else. Or try (a) learning some manners and (b) some basic physics.

• What the hell do you think the black body spectrum observed by COBE is? That, you idiot, is the detection of recombination. Go and bother someone else with your rudeness and stupidity.

• on 13/12/2018 at 18:27 Ricardo Gomez

Dr. Evans, I have the same question that Mr. Robson has, and I don’t believe your other blog posts address the issue either. Did you ever figure it out?
I don’t understand this part of your comment:
“What the hell do you think the black body spectrum observed by COBE is? That, you idiot, is the detection of recombination.”
The original question (that sparked the blog post) asks how we can determine the redshift of the CMB without first assuming that recombination must have happened (although it might have not been explicitly stated, it seems clear from the context).

6. […] week someone (“Cosm”) posted an interesting comment/question on my post “What is the redshift of the Cosmic Microwave Background (CMB)?” . The question asked how we can say that the temperature of the Universe at recombination is about […]

7. on 18/03/2017 at 22:50 | Reply Robert Williams

I once thought of red shifts as Doppler shifts and those greater than one as caused by relativistic effects. Recently I’ve tryed to think of the very large redshift of the cosmic background as being caused by seeing almost all the way out to our horizon where space itself is moving away at the speed of light. Is this valid ?….also, hasn’t space expanded by much more than a factor of 1100 in the 13.9997 billion years since the decoupling ?

• As always, my suggestion is to read Edward Harrison’s book Cosmology: The Science of the Universe. No-one explains the redshift better than he does.

8. Is there any detectable “doppler shift” of the CMB in any particular direction, from which we can determine the earth’s or sol’s actual absolute velocity and vector? And if so is there any possible use for that information?

If it can’t be determined by CMB red shift discrepancy, can it be determined by sending out probes with atomic clocks to six cardinal directions? (north pole, south pole, and every third sign of the zodiac?) Ones sent backwards from the earth’s absolute vector would tick slower, allowing us to measure (perhaps with more probes) exactly how fast the earth is moving WRT the universe?

• Yes, check out the COBE result showing Earth’s motion relative to the CMB

• Thx Rh! I see from http://www.astro.ucla.edu/~wright/CMB-DT.html that we’re travelling a mere 368km/sec vs. the CMB, and the direction can be pinpointed to within a tenth of a degree. That’s incredible. But what are the chances after all the crazy things leading up to our stardust becoming the solar system, this solar system is so slow? Or would most observers in most star systems we see, even extremely distant ones, think they’re travelling at these slow speeds?