Have CMB photons “cooled” or been “stretched”?

Introductory texts and popular accounts of why we see the "once hot" CMB as microwaves nearly always say something about the photons "cooling" since the Big Bang. But isn't that misleading? Don't those photons have long ("cool") wavelengths because space expanded since they were emitted. There's no separate "cooling" process, is there?

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Maybe they have been ironed flat? –  Georg Apr 16 '11 at 16:51

Cooling and stretching essentially mean the same thing here. The temperature of any blackbody radiation is related to the peak wavelength by Wien's Law $$\lambda_{\mathrm{max}} = \frac{b}{T}$$ Therefore as the universe expands, all of the photon wavelengths get stretched out and so does the peak wavelength. $$\lambda \propto a(t)$$ This decreases the temperature of the radiation by the same factor that expands the wavelength of the photons. $$T \propto \frac{1}{a(t)}$$ This factor $a(t)$ is the scale factor of the universe and increases with time for an expanding universe.

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I think the best way to think about it is that the sentence "the photons have cooled" is simply describing a fact, not explaining that fact. At early times, the photons at any given location had a thermal (blackbody) distribution corresponding to a high temperature (as measured by observers at rest in the natural, comoving reference frame). At later times, the photons at any given location had a thermal distribution corresponding to a lower temperature. That's what we mean when we say that they "cooled."

Of course, it's then very natural to ask why they cooled. That's where the "stretching of space" explanation comes in. I think that that explanation is problematic, as I wrote at great length here, but others disagree.

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Isn't it a pretty big difference between the "stretching" of an individual photon (resulting in a lower detected quanta of energy on reception) and a change in overall wavelength distribution (corresponding to a shift in apparent blackbody temperature)? –  BjornW Apr 16 '11 at 15:02
I don't think I understand this. The "facts of the matter" are that observers at one time measure photons whose wavelength distribution is centered around short wavelengths, observers at a later time measure a distribution centered around long wavelengths, and the latter observers measure a lower density of photons. All of those facts can be summarized in the word "cooled" (since those facts precisely describe the difference between a high-temperature and a low-temperature blackbody spectrum). Some people then like to "explain" that fact with the "stretching" language. –  Ted Bunn Apr 16 '11 at 15:06
I was thinking more of the special case with a single photon of a sufficiently specified energy being released 3 billion years ago and its energy measured now and found to be reduced. The black-body stuff is at another level of analysis on top of this in't it, by discussing interpretations of an entire distribution of photons? –  BjornW Apr 16 '11 at 15:16
It's true that it wouldn't be very natural to use the word "cooled" to describe a single photon, especially one not drawn from a thermal distribution. –  Ted Bunn Apr 16 '11 at 15:53