Physical Foundations of Cosmology

9.10 Polarization of the cosmic microwave background 399

n

l

x

z

y

e

Fig. 9.5.

components of the electric field along vectoreaare

E ̃a=E ̃·ea=AE·ea. (9.142)

If incoming light arriving from directionlis completely unpolarized, then the
resulting polarization tensor can be calculated using (9.142) and averaging over all
directions ofEperpendicular tol.

Problem 9.20Show that in this case
〈 ̃
EaE ̃b

〉

=^12 A^2

〈

E^2

〉

(gab−(l·ea)(l·eb)), (9.143)

and

I=

〈 ̃

EaE ̃a

〉

=^12 A^2

〈

E^2

〉(

1 +(l·n)^2

)

, (9.144)

where

〈

E^2

〉

is the average of the squared electric field in the incident unpolarized
beam. (HintJustify and use the following formula for averaging over directions of
the electric field in the incident beam:
〈
EiEj

〉

=^12

〈

E^2

〉(

δij−lilj

)

, (9.145)

whereEi,li(i= 1 , 2 ,3) are the components of the appropriate 3-vectors in some
orthonormal basis.)

Write down the polarization tensor and verify that fa=(l·ea)is an eigenvector
ofPbawith negative eigenvalue. Show that the polarization vectorpais the vector
perpendicular tofawith norm

p^2 =

1 −(l·n)^2

1 +(l·n)^2

. (9.146)