400 Cosmic microwave background anisotropies
It follows from (9.146) that the incoming unpolarized radiation scattered at right
angles (l·n=0) comes out completely polarized in the direction perpendicular
to the plane containing the vectorslandnas, for instance, the sunlight from the
horizon is linearly polarized parallel to the horizon at midday.
Now, if we generalize to the case of an incoming unpolarized radiation field with
intensityJ(l)∝
〈
E^2 (l)〉
and take into account that〈E ̃aE ̃b〉andIare additive for the
incoherent light, we obtain
Pab(n)=∫ [
1
2
gab(
1 −(l·n)^2)
−(l·ea)(l·eb)]
J(l)d^2 l
∫ [
1 +(l·n)^2]
J(l)d^2 l(9.147)
If the incoming radiation is isotropic (J(l)=const), then integrating overl,we
find thatPab(n)= 0 ,that is, the scattered radiation remains unpolarized. The ex-
pression inside the integrand is quadratic inland can be expressed in terms of
the quadrupole spherical harmonics. Thus, scattering induces polarization in ini-
tially unpolarized radiation only if the incident radiation is anisotropic and only the
quadrupole component of the anisotropy contributes to the polarization.
Problem 9.21If the vectorseaare orthonormal, then the triplete 1 ,e 2 andnforms
an orthonormal basis in three-dimensional space and one can specify the direction
of the incident vectorlby the Euler anglesθandφ. Verify that in this case the
polarization tensor components given in (9.147) are
P 11 =−P 22 =
1
̃I
∫ √
3
40
ReY 22 Jd
,P 12 =
1
̃I
∫ √
3
40
ImY 22 Jd
,(9.148)
where
I ̃=∫(
Y 00 −
1
2
√
5
Y 20
)
Jd
, (9.149)andYlm(θ,φ)are the appropriate spherical functions.
9.10.3 Delayed recombination and polarization
Before recombination begins, the radiation field has only dipole anisotropy (see
(9.31)) and therefore polarization cannot be generated. If recombination were in-
stantaneous, then after recombination the photons would propagate without further
scattering and no polarization would be generated. Therefore, the background ra-
diation becomes polarized only if recombination is delayed. Taking into account