# If photon decoupling was a departure from equilibrium why does CMB exhibit a blackbody spectrum?

Decoupling of photons from the cosmic plasma or thermal bath is said to be a point of departure from thermal equilibrium. But we know that the cosmic microwave background (CMB), consisting of the decoupled photons, fits a blackbody distribution curve to an astonishing accuracy of $1\%$. But thermal radiation can have blackbody spectrum only if the radiation is in equilibrium with the cavity.

So if the photons are not in equilibrium, why do we have blackbody distribution for the CMB?

• I see it as if the universe was a BB at decoupling, or just prior to it.. I am a naive as I am not a specialist. But am I overly naive? – Alchimista Mar 12 '18 at 10:56

I'm not exactly sure what you mean by "equilibrium with the cavity", but radiation and matter was in thermal equilibrium until decoupling 379,000 yr after Big Bang, i.e. photons and particles shared the same distribution in energy. This energy was given by the temperature, so the spectrum of the photons was a Planck spectrum, characteristic of a blackbody.

The "recombination" of protons and electrons, together with the expansion of the Universe, depleted space of its main scattering agent, electrons, over a rather short period of time. In a few $10^4$ yr, the scattering rate of photons went from being significantly shorter than the expansion rate of the Universe, to being significantly longer. The is equivalent to saying that the photons' mean free path went from being shorter than the size of the observable Universe, to being longer.

After this, (most of) the photons haven't been interacting with anything, so they are still exactly the same photons that were released at that time, only they have redshifted on their way through the expanding Universe. That means that they still exhibit the same Planck spectrum as then, only redshifted.

You might think that because the photons didn't decouple exactly simultaneously, the spectrum shouldn't be an exact Planck spectrum, but be "smeared out" a little. But the photons have always been subjected to a redshift, also prior to, and during, decoupling. When the Universe was, say, 370,000 yr old, it was 2987 K hot (using a Planck 2016 cosmology), and so its spectrum peaked at $$\lambda_\mathrm{peak,370\,kyr} = \frac{b}{2\,987\,\mathrm{K}} = 970\,\mathrm{nm},$$ where $b$ is Wien's displacement constant. On the other hand, photons that decoupled late, say when the Universe was 390,000 yr and the temperature had fallen to 2895 K, would peak at $$\lambda_\mathrm{peak,390\,kyr} = \frac{b}{2\,895\,\mathrm{K}} = 1\,001\,\mathrm{nm}.$$ But in the 20,000 yr that went from photon A decoupled with $\lambda = 970\,\mathrm{nm}$ till photon B decoupled with $\lambda = 1001\,\mathrm{nm}$, photon A has redshifted to $\lambda = 1001\,\mathrm{nm}$, so it is indistinguishable from a photon B.

This is why the CMB is described by such a perfect blackbody curve.

• I'm not exactly sure what you mean by "equilibrium with the cavity"... The discussion of BB spectrum assumes that there is some thermal radiation inside a metallic cavity maintained at a fixed temperature T. It's with the interaction with the atoms of the walls of the cavity that causes the thermal radiation to equilibrate at the temperature T and attain BB spectrum. @pela – SRS Mar 14 '18 at 5:04
• @SRS There are several ways to derive the functional form of a Planck spectrum. The "wave in a box" is a heuristic, classical-ish way of deriving, I think. But both photons and other particles have a distribution in energy, even without being enclosed in a box. For a cloud of particles, the fast-moving particles may escape, so the remaining ensemble has a non-Planckian distribution. But if the cloud is infinitely large — as in the case of the Universe — there is nowhere to escape, so the distribution doesn't change. – pela Mar 14 '18 at 8:04
• But where is the departure from equilibrium? @pela – SRS Mar 14 '18 at 15:39
• @SRS If I understand you correctly, the departure from equilibrium that you're referring to, is the fact that after decoupling, the radiation no longer interacts with matter, so radiation and matter is no longer bound to be in equilibrium. – pela Mar 14 '18 at 17:06
• Since photons don't interact with each other, if by magic you removed all photons below a certain energy threshold, they would keep missing from the spectrum. In contrast, if you did that before decoupling, they would re-appear, because they were in equilibrium with the matter, so they kept exchanging energy. – pela Mar 14 '18 at 17:06