Calc. Temperature of CMB | Textbook Info

[micomaco86572]

Which textbook does contain some accessible details of the calculation of the temperature of CMB?

Thx!

 

[Wallace]

Any cosmology textbook written in the last 30 years or so should have the details you are after. Which book is best for you depends on your level. In general, I would recommend the textbook written by Peacock and the one by Linder, but that's a personal preference and I'm sure others could reccomend similar books. It really depends what you have access to. If you can borrow books from a library that has comology textbooks you should be able to pick any of them up to find this info.

Alternatively check out some websites. Wayne Hu has an excellent intro to the CMB (with lots of nice animations, even if they look a little dated by todays standards) at this site http://groundup.uchicago.edu/~whu/beginners/introduction.html

 

[micomaco86572]

Thx, Wallace

 

[micomaco86572]

I have read the content of Peacock's book roughly. It seems that we cannot know the original temperature of the CMB when these photon were emitted. What we can do is just using the temperature at present and tracing it back to the original temperature. Can we just obtain the original temperature directly?

forgive me poor English, lol!

 

[Wallace]

Don't worry, your English is fine!

Yes and no. We can't directly 'know' what the temperature was, because we can't go back in time and there is then the degeneracy between the temperature we observed and the redshift of the CMB.

That being said, what we can do is use physics that we know in order to calculate the temperature of the Universe at the time in which the CMB photons were emmitted (or technically, when they were last scattered). We can do this because we know the temperature at which neutral hydrogen ionises into free protons and electrons. When the Universe was above that temperature the ionised plasma that filled the Universe did not allow photons to travel very far (it was 'opaque'). When the Universe cooler enough (through expansion) to go below this temperature, the plasma was able to form neutral Hydrogen. At this point the photons around at that time became able to travel (since the neutral hydrogen does not absorb these photons anymore) thus the black body temperature of the CMB *at that time* corresponds to the temperature at which neutral hydrogen could form.

I don't remember exactly what that temperature is. I would have thought it would be in a textbook somewhere, or someone else will probably chime in with the answer.

The full story is a little more complicated than I suggested, because to be very accurate you need to consider that not all the original material in the Universe was Hydrogen, there was also some Helium and some heavy elements and this makes a difference to the result.

 

[nicksauce]

Wallace, does the temperature at last scattering, though, not depend on the ratio of the number of photons to the number of electrons (or protons), which we cannot determine a priori (unless we know the CMB temperature today, or from something else such as BBN)? I have always been under the impression that we can't calculate what the CMB temperature should be today unless you know this ratio.

 

[twofish-quant]

No the temperature of last scattering is about 3000K. Above 3000K, then hydrogen atoms break up into protons and electrons. Once you have charged particles then compton scattering vastly increases the opacity of the universe.

The electron/photon ratio matters for a lot of things such as BBN, but the temperature of last scattering is pretty insensitive to it. I've always thought of the CMB as the "wall of fire".

 

[Wallace]

The baryon/photon ratio changes the sound speed of the plasma, which in turn determines (in part) the location of the peaks in the angular power spectrum. I've never actually thought much about whether this changes the actual temperature at which re-combination occurs, most of the stuff I know about the CMB is much more related to the power spectrum than the temperature of re-combination.

I'm guessing now, but I think nicksauce must be right at some level, though how much of a difference it makes I'm not sure? Thinking about it, the pressure as well as the temperature should determine at what point neutral hydrogen can form, and the baryon/photon ratio will determine in part the pressure (which is why it changes the sounds speed and hence the accoustic peaks). As I say though, I'm thinking out loud here.

 

[twofish-quant]

Thinking about it, the pressure as well as the temperature should determine at what point neutral hydrogen can form, and the baryon/photon ratio will determine in part the pressure (which is why it changes the sounds speed and hence the accoustic peaks). As I say though, I'm thinking out loud here.

If I recall correctly, the temperature at which neutral hydrogen forms is rather insensitive to the pressure. What you need is enough energy to start knocking electrons out of protons and this is a function of the amount of radiation which is very, very sensitive to temperature (i.e. T^4), but not sensitive to pressure. Pressure would impact interactions between the hydrogen atoms, but for recombination, that doesn't matter.

As long as everything is a gas, I don't see how pressure can influence the process of recombination very much.

 

[twofish-quant]

FYI, the Saha equation which determines ionization doesn't include the pressure at all.

http://en.wikipedia.org/wiki/Saha_ionization_equation

 

[micomaco86572]

Gentlemen, I have an additional question: Can we calculate the time of the moment of photon's decoupling, exactly? Thx!

 

[Wallace]

In principle yes, and essentially you get the answer to that question 'for free' if you have solved cosmology. What we do know is the redshift of the CMB (by comparing the observed black body temperature to the one we can calculate from theory). If we are confident in our cosmological model, then we can accurately translate between redshift and time, but that is model dependant so if our model is wrong then we would get that answer wrong as well.

Not to critise the question, but it actually doesn't matter what 'time' this happened in terms of what information we can pull out of the CMB. It's one of those derived quantities like say the age of the Universe, it's nice to know, but doesn't tell you anything since it's just a consuqence of your particular cosmological model rather than something that helps to constrain said model. I hope that makes sense!

 

[marcus]

Micomaco,
Wallace and others have already answered in a completely satisfactory way, so I'll just throw in some extra non-essential detail.

You asked about how the original temp at last scatter is estimated. I've seen a complete derivation of that and it's a fairly involved estimation involving variables like the percent ionization of the gas and the mean free path of the photons.

The last scattering "surface" is actually not a mathematical surface. It has finite thickness. "Recombination" is not an instantaneous event. The redshift at which it occurred, approx z = 1090, is an "around that time" approximation.

The point is that yes as somebody already said the temp is estimated to be about 3000 kelvin. That corresponds to a particle energy of 0.26 eV. But the ionization energy of hydrogen is 13.6 eV. So at 3000 kelvin only a small percentage of the hydrogen is ionized.

That corresponds to a large but finite mean free path. If the universe were not expanding and thinning out and cooling, then a photon would travel only that far on average before getting scattered again. Like in a fog there would not be perfect transparency, you couldn't "see forever", it would be like "visibility one mile" that you hear aircraft pilots say.

The mathematical condition that determines that you get effective transparency at 3000 kelvin is that the mean free path by then is then long enough and the universe is still expanding fast enough, that by the time the photon can expect to hit charge and be scattered again, the gas has thinned out some more and it doesn't get scattered.

Like if medical science were always increasing your life expectancy which at any given time always finite, but if progress were fast enough so you would be effectively immortal because, by the time you reach the original limit, they have improved stuff and you have a new extended expectancy.

So the photon's mean free path is finite, but it is increasing fast enough that most photons make it to infinity.

That means that the estimate of 3000 K depends on a calculation, and slightly somewhat on which model of expansion is used. But it's still pretty reliable---it's robust, not too dependent on details.

 

[twofish-quant]

Cool. Can you point me to a page on the web where someone has worked this out?

 

[marcus]

Cool. Can you point me to a page on the web where someone has worked this out?

Not off-hand. I read it in a book over at the astro department of the local campus. I went in the building wondering about this and found two graduate students hanging out up on the 4th floor. It was over 10 years ago before I knew how to google things and use wikipedia. So I had to ask graduate students. One of them made me sit down with a book, open to a certain chapter, and said to read. Nice guys. Didn't waste anybody's time.

My guess is that it is standard cosmo textbook material. I could probably find a section of a chapter of the new book by Weinberg.

If somebody else finds a webpage about the actual nittygrit of the last scattering surface, and how the temperature is calculated, I hope they post the link for us. I looked several times and didn't find anything. But not recently. I suppose "mean free path" could be one of the keywords you could use. Maybe "scattering surface temperature".