Photon and Lepton Decoupling 133streaming photons from this era form the CMB radiation and their point of last
contact with matter forms an isotropiclast scattering surface(LSS). The era of recom-
bination provides a crucial observational limit beyond which we cannot hope to see
using electromagnetic radiation. The LSS is not a sharp boundary and does not exist
at a unique redshift. Once the conditions for recombination have been met in one por-
tion they should be also met in any other—thus isotropicity. The reason for seeing this
as a theoretical spherical shell is that only one specific shell at one specific distance
can be seen by any specific observer at any specific time. The photons from the LSS
preserve the polarization they incurred in the last Thomson scattering. This remain-
ing primordial polarization is an interesting detectable signal, albeit much weaker
than the intensity of the thermalized radiation.
The LSS of the Universe has an exact analogue in the surface of the Sun. Photons
inside the Sun are continuously scattered, so it takes millions of years for some pho-
tons to reach the surface. But once they do not scatter any more they continue in
straight lines (really on geodesics) towards us. Therefore, we can see the surface of
the Sun, which is the LSS of the solar photons, but we cannot see the solar interior.
We can also observe that sunlight is linearly polarized. In contrast, neutrinos hardly
scatter at all in the Sun, thus neutrino radiation brings us a clear (albeit faint with
present neutrino detectors) picture of the interior of the Sun.
Equilibrium Theory. An analysis based on thermal equilibrium, theSaha equa-
tion, implies that the temperature must fall to about 0.3eV before the proportion of
high-energy photons falls sufficiently to allow recombination to occur. The Saha anal-
ysis also implies that the time (or energy or redshift) for decoupling and last scattering
depend on cosmological parameters such as the total cosmic density parameter훺 0 ,
the baryon density훺b, and the Hubble parameter. However, a second feature of the
physics of recombination implies that the equilibrium analysis itself is not sufficient.
The Saha analysis describes the initial phase of departure from complete ionization
but, as recombination proceeds, the assumption of equilibrium ceases to be appro-
priate [see, e.g., Equation (6.9)]. Paradoxically, the problem is that electromagnetic
interactions are too fast (in contrast with the weak interaction that freezes out from
equilibrium because of a small cross-section). A single recombination directly to the
ground state would produce a photon with energy greater than the 13.59eV binding
energy and this photon would travel until it encountered a neutral atom and ionized
it. This implies that recombination in an infinite static universe would have to proceed
by smaller intermediate steps (thus not directly to the ground state).
In fact the situation is even worse, because reaching the ground state by single pho-
ton emission requires transition from the 2P to 1S levels and thus production of pho-
tons with energy at least 10.2eV (Lyman훼with휆=1216Å). As these photons become
abundant they will re-ionize any neutral hydrogen through multiple absorption and so
it would seem that recombination will be, at a minimum, severely impeded. (Recom-
bination in a finite HII region is different because the Ly훼photons can escape.)
There is an alternative path, however. Two-photon emission generated by the 2S→
1S transition produces lower-energy photons. The process is slow (with a lifetime
of approximately 0.1s), so recombination proceeds at a rate quite different from the