Zaldarriaga (1998)]). How this relates to
what was used in previous releases is exposed in A.3.2.
The CMB radiation field is described by a intensity 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 combinations are 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 introduced and . They satisfy Y*lm = (-1)m Yl-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 fact and have the form, and , can be calculated in terms of Legendre polynomials [Kamionkowski et al (1997)]
Note that if 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 angle can be calculated using equations (8) and (9). 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)].
Note that Eq. (9) implies that, if the Stokes parameters are
rotated everywhere via
then the polarized alm coefficients are submittted to the same rotation
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. 5)
Eric Hivon 2010-06-18