Physical Foundations of Cosmology

9.10 Polarization of the cosmic microwave background 401

that at conformal time ̃ηLthe probability of last scattering is given by (9.47) and

J(η ̃L,l)∝(T 0 +δT(η ̃L,l))^4 , (9.150)

we obtain (to leading order) from (9.147):

Pab(n)= 3

∫ [

1

2

gab

(

1 −(l·n)^2

)

−(l·ea)(l·eb)

]

×

δT

T 0

(η ̃L,l)μ′(η ̃L)e−μ(η ̃L)dη ̃L

d^2 l

4 π

. (9.151)

Hence, the polarization should be proportional to the quadrupole temperature fluc-
tuations generated during the delayed recombination. To calculateδT/T 0 (η ̃L,l)at
the point of scatteringx, resulting from the scalar metric perturbations, we can use
(9.49) together with (9.48), where one has to replaceη 0 by ̃ηLand integrate over
the time interval ̃ηL>ηL>0, that is,

δT

T 0

(η ̃L,l)=

∫ ∫η ̃L

0

( +

δ

4

−

3 δ′

4 k^2

∂

∂η ̃L

)

ηL

eik·[x+l(ηL−η ̃L)]

×μ′(ηL)e−μ(ηL)dηL

d^3 k

( 2 π)^3 /^2

. (9.152)

Because we will be content with only a rough estimate of the expected polarization
we note that the visibility functionμ′(η ̃L)e−μ(η ̃L)in (9.151) has a sharp maximum
at ̃ηL=ηrcorresponding tozr 1050 .Thus the polarization should be about
the quadrupole temperature fluctuation at this time. As we have seen, the main
contribution to the quadrupole component comes from perturbations with the scales
comparable to the horizon scale, that is, withkηr∼ 1 .One can get an idea of the
amplitude of this component by noting that the quadrupole is proportional to terms
quadratic inlarising from the expansion of

exp[ik·l(ηL−η ̃L)]∼exp[ik·l(ηL−ηr)]

in powers oflin (9.152) (note that the higher multipoles also contain terms quadratic
inl). Because the visibility function has a sharp peak of widthη∼σηr(see (9.55),
(9.57) and (9.60)), we can estimatek·l(ηL−ηr)asσ forkηr∼ 1 .Therefore,
the quadrupole components atηr,and hence the expected polarization should be
aboutO( 1 )σ∼ 10 −^2 –10−^1 times the temperature fluctuations observed today on
angular scales corresponding to the recombination horizon. Polarization, then, is
proportional to the duration of recombination and vanishes if recombination is
instantaneous. Numerical calculations show that polarization never exceeds 10%
of the temperature fluctuations on any angular scales.