The CMB radiation field is described by aintensity tensor Iij [Chandrasekhar (1960)]. The Stokes parameters Q and U are defined as Q=(I11-I22)/4 and U=I12/2, while the temperature anisotropy is given by T=(I11+I22)/4. The fourth Stokes parameter V that describes circular polarization is not necessary in standard cosmological models because it cannot be generated through the process of Thomson scattering. While the temperature is a scalar quantity Q and U are not. They depend on the direction of observation
and on the two axis
perpendicular to
used to define them. If for a given
the axes
are rotated by an angle
such that
and
the Stokes parameters change as
To analyze the CMB temperature on the sky, it is natural to expand it in spherical harmonics. These are not appropriate for polarization, because the two combinationsare quantities of spin
[Goldberg (1967)]. They should be expanded in spin-weighted harmonics
[Zaldarriaga & Seljak (1997)], [Seljak & Zaldarriaga (1997)],
To perform this expansion, Q and U in equation (6) are measured relative to, the unit vectors of the spherical coordinate system. Where
is tangent to the local meridian and directed from North to South, and
is tangent to the local parallel, and directed from West to East. The coefficients
are observable on the sky and their power spectra can be predicted for different cosmological models. Instead of
it is convenient to use their linear combinations
which transform differently under parity. Four power spectra are needed to characterize fluctuations in a gaussian theory, the autocorrelation between T, E and B and the cross correlation of E and T. Because of parity considerations the cross-correlations between B and the other quantities vanish and one is left with
where X stands for T, E or B,means ensemble average and
is the Kronecker delta.
We can rewrite equation (6) as
where we have introducedand
. They satisfy Y*lm = (-1)m Y*l-m, X*1,lm=(-1)m X1,l-m and X*2,lm=(-1)m+1X2,l-m which together with aT,lm=(-1)m aT,l-m*, aE,lm=(-1)m aE,l-m* and aB,lm=(-1)m aB,l-m* make T, Q and U real.
In factand
have the form,
and
,
can be calculated in terms of Legendre polynomials [Kamionkowski et al (1997)]
where
Note thatif m=0, as it must to make the Stokes parameters real.
The correlation functions between 2 points on the sky (noted 1 and 2) separated by an anglecan be calculated using equations (8) and (A.3.1). However, as pointed out in [Kamionkowski et al (1997)], the natural coordinate system to express the correlations is one in which
vectors at each point are tangent to the great circle connecting these 2 points, with the
vectors being perpendicular to the
vectors. With this choice of reference frames, and using the addition theorem for the spin harmonics [Hu & White (1997)],
we have [Kamionkowski et al (1997)]
The subscript r here indicate that the Stokes parameters are measured in this particular coordinate system. We can use the transformation laws in equation (5) to write (Q,U) in terms of (Qr,Ur).
The definitions above imply that the variances of the temperature and polarization are related to the power spectra by
It is also worth noting that with these conventions, the cross power CCl for scalar perturbations must be positive at low l, in order to produce at large scales a radial pattern of polarization around cold temperature spots (and a tangential pattern around hot spots) as it is expected from scalar perturbations [Crittenden et al (1995)].
Finally, with these conventions, a polarization with (Q>0,U=0) will be along the North-South axis, and (Q=0,U>0) will be along a North-West to South-East axis (see Fig. [])
Eric Hivon 2005-08-31