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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 😉