We have seen how the inhomogeneities of the matter and photon densities evolve, from early times to the present epoch. Now we will do the same thing for the anisotropy of the photon distribution function. This anisotropy is observed today as the cosmic microwave background (CMB) anisotropy.

Photons in the early Universe are in thermal equilibrium, with the blackbody distribution of momenta. As the epoch of photon decoupling is approached, the distribution begins to fall out of equilibrium, developing anisotropy which is different for the two polarization states. After decoupling at z ~ 1000, the redshifting of the photons through the inhomogeneous gravitational field generates more anisotropy, without affecting the polarization. Then reionization at z ~ 10 generates further anisotropy and polarization. Finally, the photons are observed at the present epoch, as the CMB anisotropy. The anisotropy is characterized by a perturbation in the intensity, which corresponds to a perturbation in the temperature of the blackbody distribution, and by two polarization parameters. In this chapter we study the temperature perturbation.

We are going to see that the CMB anisotropy is observable only on rather large scales, corresponding to comoving wavenumber k ≲ (10 Mpc)−1. In this regime, first-order (linear) cosmological perturbation theory is almost always a good approximation, failing on the smallest scales and only around the present epoch. In the latter regime, the dominant effect is the thermal Sunyaev–Zel'dovich effect, which occurs when a galaxy cluster is in the line of sight.