In this chapter we discuss the most important second-order effect on CMB anisotropies and polarization. Patches of higher or lower CMB temperature are modified and polarization patterns are distorted when they propagate through an inhomogeneous gravitational field. The content of this chapter is strongly inspired by the excellent review by Lewis & Challinor (2006) on the subject.
An introduction to lensing
On their path from the last scattering surface into our antennas, the CMB photons are deflected by the perturbed gravitational field. If the CMB were perfectly isotropic, the net effect of this deflection would vanish, since, by the conservation of photon number, as many photons would be deflected out of a small solid angle as into it. On the other hand, if there is no perturbation in the gravitational field, the latter is perfectly isotropic and the effect also vanishes. Hence, gravitational lensing of the CMB is a second-order effect and we have not discussed it within linear perturbation theory.
To estimate the effect let us consider the CMB temperature in a point n in the sky, T(n). If the direction n is deflected by a small angle α, we receive the temperature T(n) from the direction n′ = n + α. Note that, since α is a vector normal to n also n′ is a unit vector to first order in α.