What is the temperature of the CMBR?

[photonkid]

I've read that the temperature of the CMBR is 2.7 degrees K or so and is much cooler than when it started out, due to the expansion of the universe making the wavelength longer.

What is meant by the temperature of the CMBR? How do you measure it? Is it determines purely by the wavelength?

Is the "photon density" a factor in the temperature?

Is the photon density of the CMBR changing over time? i.e. since the "surface of last scattering" is increasing in size, shouldn't the amount of photons that reach us in a given time period be increasing?

The observable universe (including the CMBR) is said to homogeneous and isotropic. Is it possible that the CMBR we see is not actually isotropic but the differences are too small to be detectable and that there actually is a "center" to the universe somewhere, outside the observable universe.

 

[Chronos]

The CMB temperature is affirmed in various ways. One of the more compelling is molecular excitation in the intergalactic medium. We have detected these at great distances and they confirm theoretical CMB temperature projections. See http://arxiv.org/abs/1212.5456, A precise and accurate determination of the cosmic microwave background temperature at z=0.89.

 

[marcus]

...What is meant by the temperature of the CMBR? How do you measure it? Is it determines purely by the wavelength? ...

Here is a picture of black body radiation curve at various temperatures 3000K, 4000K, 5000K
http://en.wikipedia.org/wiki/File:Black_body.svg

There is a particular "lopsided bell-curve" shape to power per wavelength distribution curve that you find in the heat glow from a warm or hot object.

Where the PEAK power comes depends on the temperature. You can tell the temperature of a generic dull-finish surface by finding the peak wavelength. The interval of wavelengths where you get the most energy per unit time, the most "wattage".

I think you may know this because you say

How do you measure it? Is it determine purely by the wavelength?

the answer is basically YES, they measure the whole power spectrum, how much wattage in each wavelength bracket. They find they get a PERFECT FIT to the black body curve for a certain temperature, the right peak, the right slopes the right tails, everything fits!

From there it is an easy step to start talking about the "temperature" of the CMB light.

The estimated REDSHIFT z, or let me say the wave stretch factor z+1, is about 1090. Wavelengths and distances have expanded by a factor of 1090 since the light was emitted.

If you divided every wavelength by 1090 and look at the power spectrum then, you see the spectrum of heat glow of an object at about 3000K, like in the diagram.

So the temperature of the gas, when the CMB was emitted, that we are now getting, was about 3000K. And the "temperature of the ancient light" at that time of emission was 3000K.
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You talk about "photon density". That goes along with this in a consistent way. As distances and wavelengths have expanded by a factor of about 1000, volumes have expanded by a factor of 1,000,000,000. So densities have gone down by a factor of a billion.
It doesn't matter about the area of the "last scattering" increasing. As there get to be more units of area shooting, we get to look like a smaller and smaller target, and it cancels. What matters is that space is filled with a certain density of ancient light, coming from wherever, and as volumes expand the number density of photons declines.

Also the energy of each photon falls off as wavelength increases. So energy per volume falls off as the fourth power of the length scale.

But this is routine. Just how thermal radiation behaves. Energy per volume falls off as the fourth power of temperature. (Temperature is what falls off as the length scale increases)
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So yes to your question. The temperature of the CMB now (and also the temperature back then when it was emitted from the hot gas) is told from the curve that shows how power is distributed over the range of wavelengths---the so called "spectrum".

You can get a rough idea just from the peak power wavelength bracket. But for more precision they measure and plot the whole curve over many brackets.