Exploring the Relationship between CMB Radiation and Blackbody Temperature

[artis]

Here is something that struck a note to me, they give the CMB radiation in it's frequency which is in Ghz as the name "microwave" implies and then they also give a temperature in Kelvin.
But how can light aka EM radiation have a temperature? I thought only matter with mass can have a temperature which is then a measure of the average particle/atom energy.
Does the CMB temp. refers to the temp a ideal black body would radiate outwards if struck by the CMB radiation?

 

[vanhees71]

It's radiation in thermal equilibrium. If you have a cavity with walls at a constant temperature that means that inside this material a lot of charged particles are in thermal motion and since they are accelerated they emit em. radiation (if the body is hot enough part of this radiation is in the visible spectrum and you see it glow in different colors depending on the temperature). On the other hand the em. radiation in the cavity is also wiggling around the charged particles in the wall and thus absorbed. The walls and the radiation in the cavity are in thermal equilibrium as soon as as much radiation gets absorbed as emitted.

The fascinating thing with this thermal radiation is that it is completely independent on the material the cavity walls consist of. Its spectrum is a universal law, only dependent on temperature as a parameter and the rest are fundamental constants.

The CMB radiation is a remnant of radiation in thermal equilibrium with the matter in the universe until about 380000 years after the big bang. At this moment the universe has cooled down to a temperature, where the matter formed electrically neutral atoms. Before it has been a plasma, consisting of charged particles. The light is much less scattered around by neutral atoms forming a very dilute gas than by the charged particles in the plasma. That's why since then the electromagnetic radiation is decoupled from the medium and just moves as free electromagnetic waves.

Now another fundamental aspect becomes important: As far as we know, electromagnetic fields do not define any energy scale, because it's what's called a massless field. The only scale is the temperature of the radiation at decoupling defining a black-body spectrum at the temparature at some 1000 K. But since this time the universe has still Hubble-expanded and the wavelength of the light has been expanded following the only length scale around, which is the scale factor of the spacetime metric describing the large-scale structure of space-time (the socalled Robertson-Walker-Friedmann-Lemaitre solutions of general relativity). That's why today this radiation looks to us still like a perfect black-body spectrum but with a temperature of around 2.75 K with typical wavelengths in the microwave region.

 

[artis]

@vanhees71 Ok, so far so good, the early universe being in a plasma state can be considered a black body and the EM radiation was in equilibrium with the energy of this black body ? Photons got absorbed and re-emitted constantly and their energy/wavelength represented the energy of the plasma ?

Now if I'm not mistaken plasma absorbs and emits EM radiation much better than gas which is less dense and cooler so after "recombination" the leftover photons don't have the same capacity to interact with nearby matter so now they just "fly off" into space, or should I say they can escape as the chance of them getting absorded has decreased significatnly ?
Now when you say

The only scale is the temperature of the radiation at decoupling defining a black-body spectrum at the temparature at some 1000 K.

I take that as that in case of Em field temperature is equal to the frequency/wavelength of the em radiation/photon?
So by looking at an emitted radiation we can tell the temperature of the object that emitted it, only in the case of CMB we must also account for expanding space otherwise it's current temp would indicate a very low energy object.

Oh and is the expansion of space the only factor that has decreased the CMB frequency or does it get also absorbed by matter in space occasionally ?

 

[vanhees71]

Sure, the em. radiation gets absorbed by usual matter too, but one should note that our universe is very empty. On a large scale there's almost nothing for the photons to interact with, and that's why we still see this radiation from the hot and dense soup the matter formed shortly after the big bang until it got combined into neutral atoms. Then the radiation is almost free and for the reason that the em. field is massless the only thing that happens due to the Hubble expansion is that the wavelengths get longer in proprotion to the scale factor describing the Hubble expansion. The net effect is that you still see a black-body radiation Planck Law but with a much smaller temperature than the decoupling temperature. Instead of some 1000K at the decoupling (around 380000 years after the big bang) today we observe a temperature of around 2.75 K (around $14\cdot 10^9$ years after the big bang).

 

[artis]

@vanhees71 Well the universe is pretty empty but I guess one additional factor is not only the increased wavelength due to expansion but also the "spreading out" or decrease in density of the CMB ?

Here is also one part I don't quite get, the frequency of the CMB is in the Ghz yet the temperature is barely above absolute zero, doesn't seem proportional to me , is this due to the very low density of this radiation? Like since space has expanded there are very few photons per say cubic metre to fill?

 

[jbriggs444]

Here is also one part I don't quite get, the frequency of the CMB is in the Ghz yet the temperature is barely above absolute zero, doesn't seem proportional to me , is this due to the very low density of this radiation? Like since space has expanded there are very few photons per say cubic metre to fill?

The frequency of visible light (e.g. the glow of plasma at 1000K) is in the hundreds of terahertz. Direct proportionality seems quite plausible to me.

 

[gmax137]

But how can light aka EM radiation have a temperature?

Check the wiki page on Planck's Law

There isn't a specific frequency (or wavelength) associated with blackbody radiation at a given temperature, rather there is a distribution of frequencies. The Wiki shows plots at various temperatures (3000, 4000, 5000 K). You can see, the peak of these curves moves towards longer wavelengths as the temperature decreases.

If I plotted the Planck's Law equation correctly, at 2.75K the peak is around 1.8*10-3 meter (right in the microwave band). The corresponding frequency (=c/wavelength) is ~170 Gigahertz.

So I think when they say "2.75K" and provide a frequency, they're talking about the frequency at the peak. If that's not right, maybe someone more knowledgeable will chime in and correct me.

 

[phyzguy]

@vanhees71
Here is also one part I don't quite get, the frequency of the CMB is in the Ghz yet the temperature is barely above absolute zero, doesn't seem proportional to me , is this due to the very low density of this radiation? Like since space has expanded there are very few photons per say cubic metre to fill?

Actually, there are approximately 1 billion CMB photons per cubic meter.

 

[vanhees71]

The black-body radiation is a continuum in frequency. You can characterize a typical wavelength/frequency by looking at the maximum of the spectrum
$$f_{\text{max}}=A T$$
[EDIT: corrected typo in view of #10]
with the constant $A \simeq 5.897 \cdot 10^{10} \; \text{Hz}/\text{K}$. That's one form of Wien's displacement law.

Again, it's easy to see, why $f_{\text{max}}$ must be proportional to $T$: Since the em. field is massless the black-body spectrum must be a function of $T$ (despite its dependence on the natural constants $\hbar$, $c$, $k_{\text{B}}$, which however are just due to our choice of units). The only dimensionful quantity thus is $T$, and since frequency is related to the energy of photons by $E=\hbar \omega=h f$ the only way a frequency can get the right dimension of energy (up to the unit-conversion factor $\hbar$) is that it is proportional to $T$.

https://en.wikipedia.org/wiki/Black-body_radiation#Wien's_displacement_law

 

[gmax137]

5.879E10 Hz/K :)

 

[vanhees71]

Thanks for pointing out this 10-orders-of magnitude typo!

 

[artis]

hmm ok it sounds logical that the frequency of the em should be proportional to the temp of the object emitting it because the Em field has certain energy and that energy needed to come from somewhere as it cannot just arise out of nowhere. So the average kinetic energy of the atoms/molecules within the blackbody is the source.

Ok you guys said the peak EM frequency represents the temp of the blackbody , but this is why I feel I don't get it because the said temp of the CMB is almost absolute zero which would imply there is almost no energy left in the system, yet frequency in Ghz seems rather high. I mean if one took a Ghz EM source with sufficient intensity one could cook say food up to 300+K easily. In fact that is what we do all the time.

So this is why I asked about what is to blame why the CMB having a rather high frequency is of such low temperature? One explanation would be it's low left over intensity as the universe expanded the original CMB field stretched out and so the photon intensity decreased as no new photons were created the already existing ones now had more space between them, is this correct or am I wrong here ?

 

[jbriggs444]

hmm ok it sounds logical that the frequency of the em should be proportional to the temp of the object emitting it because the Em field has certain energy and that energy needed to come from somewhere as it cannot just arise out of nowhere. So the average kinetic energy of the atoms/molecules within the blackbody is the source.

Ok you guys said the peak EM frequency represents the temp of the blackbody , but this is why I feel I don't get it because the said temp of the CMB is almost absolute zero which would imply there is almost no energy left in the system, yet frequency in Ghz seems rather high. I mean if one took a Ghz EM source with sufficient intensity one could cook say food up to 300+K easily. In fact that is what we do all the time.

So this is why I asked about what is to blame why the CMB having a rather high frequency is of such low temperature? One explanation would be it's low left over intensity as the universe expanded the original CMB field stretched out and so the photon intensity decreased as no new photons were created the already existing ones now had more space between them, is this correct or am I wrong here ?

The graph of the black body spectrum has a specific shape that is independent of temperature. If you make the temperature higher, the graph is scaled to higher frequencies. If you make the temperature lower, the graph is scaled to lower frequencies.

Here is the wiki link.

In an equilibrium situation, black body radiation is omni-directional. It comes from every direction and is radiated evenly in every direction. In this situation, the intensity, the frequency and equilibrium temperature are all related. Not as a direct proportion. Intensity (power emitted or absorbed divided by surface area) scales as the fourth power of temperature while frequency is directly proportional to temperature.

A microwave oven is not an equilibrium situation. It is not produced by a black body. It is produced by a magnetron. It is not omnidirectional. The intensity does not correlate with the [fixed] spectral distribution. It is nothing at all like the CMBR.

Edit: found this article which indicates that the Cosmic Microwave Background has a black body distribution. [Something I'd assumed but had not verified until just now]

 

[artis]

Hmm okay, so if frequency of emitted photons is directly proportional to blackbody temperature in this case the temp. of CMB then could it be that the temp. relation to EM frequency is not linear?

Because 2.7K temperature/KE wise is very low, in fact so low that we can barely freeze something down to such a temp, yet about 170 Ghz in the EM spectrum , although nothing exotic is pretty high in terms of frequency.

My reasoning just goes like this , if 2.7K is about 170Ghz peak then what is the gradation below 2.7K?
Like what would be the frequency of 1.7K, 170 Ghz divided by 2? And if yes is the answer then even at 0.1K where things should for all practical reasons "stand still" we would get something in the Khz range ?
This just seems weird to me

 

[gmax137]

Maybe you are mixing the frequency of the emission with the power being emitted? The flux (watts/second/meter^2) can be very low regardless of how high the frequency seems.

Blacksmiths endure the heat of working indoors because if they carry their red-hot steel (2200 F, 1475K) outdoors they can't see the color. It just isn't very "bright."

 

[artis]

well I do get that frequency is the wavelength of the photons while power is frequency times intensity and intensity wise the CMB is very low.

But I am looking more from a particle point of view. Because the blacbody photons that are emitted correspond to the energy of the blackbody particles that emit those photons right?
So at absolute zero there is no kinetic energy , okay we cannot reach absolute zero fully but let's just say we are next to it, what would be the frequency of a blackbody at say 0.1K?

I think the answer to this question would somewhat also answer my curiosity

 

[jbriggs444]

Hmm okay, so if frequency of emitted photons is directly proportional to blackbody temperature in this case the temp. of CMB then could it be that the temp. relation to EM frequency is not linear?

It is linear.

Because 2.7K temperature/KE wise is very low, in fact so low that we can barely freeze something down to such a temp, yet about 170 Ghz in the EM spectrum , although nothing exotic is pretty high in terms of frequency.

Your evaluations of these numbers as "low" or "high" is meaningless. 2.7 kelvins is "low" relative to the room I am sitting in. But that means nothing. 170 Ghz is high relative to the rate at which I can count to ten. But that also means nothing.

My reasoning just goes like this , if 2.7K is about 170Ghz peak then what is the gradation below 2.7K?
Like what would be the frequency of 1.7K, 170 Ghz divided by 2? And if yes is the answer then even at 0.1K where things should for all practical reasons "stand still" we would get something in the Khz range ?

That is not valid reasoning. "For all practical reasons stand still" is not a quantitative measure of anything.

The temperature of the CMBR has decreased by a factor of roughly 1000 since it was generated.
The peak frequency of the CMBR has decreased by a factor of roughly 1000 since it was generated.

This is a direct proportionality. Wien's law says that it is a direct proportionality.

 

[gmax137]

photons

Well that leaves me out of this discussion. I don't know enough about QM to say anything about photons.

 

[Ibix]

what would be the frequency of a blackbody at say 0.1K?

See post #9.

 

[artis]

Ok , I get the idea ,
I used an online calculator for the displacement law , here
https://www.omnicalculator.com/physics/wiens-law

So by putting in say 1K and 0.5K I can see how the wavelength increases by half so as @jbriggs444 and @vanhees71 have already pointed to me the temp to frequency relation is linear.

I guess one could say that even when everything is at it's maximum entropy and lowest possible energy state that even then matter still has some EM field emitted from it , so the universe will never cease to have a CMB , just that it will get lower yet to a certain point where it then couldn't get any lower , ?

 

[artis]

0

 

[artis]

oops I just noticed this is in classical physics so I guess should have better been asked in cosmology, pardon