MODERN COSMOLOGY

230 The cosmic microwave background

as well, although quite generically the polarization fluctuations are expected to
be significantly smaller than the temperature fluctuations. This section reviews
the physics of polarization generation and its description. For a more detailed
pedagogical discussion of microwave background polarization, see Kosowsky
(1999), from which this section is excerpted.

7.3.1 Stokes parameters

Polarized light is conventionally described in terms of the Stokes parameters,
which are presented in any optics text. If a monochromatic electromagnetic wave
propagating in thez-direction has an electric field vector at a given point in space
given by

Ex=ax(t)cos[ω 0 t−θx(t)], Ey=ay(t)cos[ω 0 t−θy(t)], (7.10)

then the Stokes parameters are defined as the following time averages:

I≡〈a^2 x〉+〈a^2 y〉; (7.11)

Q≡〈a^2 x〉−〈a^2 y〉; (7.12)

U≡〈 2 axaycos(θx−θy)〉; (7.13)

V≡〈 2 axaysin(θx−θy)〉. (7.14)

The averages are over times long compared to the inverse frequency of the wave.
The parameterIgives the intensity of the radiation which is always positive and is
equivalent to the temperature for blackbody radiation. The other three parameters
define the polarization state of the wave and can have either sign. Unpolarized
radiation, or ‘natural light’, is described byQ=U=V=0.
The parameters I and V are physical observables independent of the
coordinate system, butQandUdepend on the orientation of thexandyaxes. If
a given wave is described by the parametersQandUfor a certain orientation of
the coordinate system, then after a rotation of thex–yplane through an angleφ,it
is straightforward to verify that the same wave is now described by the parameters

Q′=Qcos( 2 φ)+Usin( 2 φ),

U′=−Qsin( 2 φ)+Ucos( 2 φ). (7.15)

From this transformation it is easy to see that the quantityP^2 ≡Q^2 +U^2 is
invariant under rotation of the axes, and the angle

α≡

1

2

tan−^1

U

Q

(7.16)

defines a constant orientation parallel to the electric field of the wave. The Stokes
parameters are a useful description of polarization because they areadditivefor
incoherent superposition of radiation; note this is not true for the magnitude or