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Relation to previous releases

Even though it was stated otherwise in the documention, HEALPix used a different convention for the polarization in its previous releases. The tensor harmonics approach ([Kamionkowski et al (1997)], hereafter KKS) was used, instead of the current spin weighted spherical harmonics. These two approaches differ by the normalisation and sign of the basis functions used, which in turns change the normalisation of the power spectra. Table 1 summarize the relations between the CMB power spectra in the different releases. See § A.2 about the interface between HEALPix and CMBFAST.

Table 1: Relation between CMB power spectra conventions used in HEALPix, CMBFAST and KKS. The power spectra on the same row are equal.
Component HEALPix $\ge$ 1.21 CMBFAST KKS HEALPix $\le$ 1.12
Temperature $C_{l}^{\rm {TEMP}} $ $C_{\rm {T},l} $ $C_{l}^{\rm T} $ $C_{l}^{\rm {TEMP}} $
Electric or Gradient $C_{l}^{\rm {GRAD}} $ $C_{\rm {E},l} $ $2C_{l}^{\rm G} $ $2C_{l}^{\rm {GRAD}} $
Magnetic or Curl $C_{l}^{\rm {CURL}} $ $C_{\rm {B},l} $ $2C_{l}^{\rm C} $ $2C_{l}^{\rm {CURL}} $
Temp.-Electric cross correlation $C_{l}^{\rm {T-GRAD}}\rule[.3cm]{0cm}{.2cm}$ $C_{\rm {C},l} $ $ -\sqrt{2}$ $C_{l}^{\rm TG} $ $\sqrt{2}C_{l}^{\rm {T-GRAD}} $
1 Version 1.2 (Feb 2003) or more recent of HEALPix package
2 Version 1.1 or older of HEALPix package

Introducing the matrices

Mlm $\textstyle \myequal $ $\displaystyle \left(
\begin{array}{cc} X_{1,lm} & i X_{2,lm} \\
-i X_{2,lm} & X_{1,lm}
\end{array} \right)$ (17)

where the basis functions X1 and X2 have been defined in Eqs. (10) and above, the decomposition in spherical harmonics coefficients (9) of a given map of the Stokes parameter Q and U can be written in the case of HEALPix 1.2 as
$\displaystyle \phantom{1.2}{
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\...
...}{.2cm}\\  U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}$ $\textstyle \myequal $ $\displaystyle \sum_{lm} M_{lm} {
\begin{array}{c} -a_{lm}^{\rm GRAD} \r...
...^{\rm CURL} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}.$ (18)

For KKS, with the same definition of M, the decomposition reads

$\displaystyle {
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\  -U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}$ $\textstyle \myequal $ $\displaystyle \sum_{lm} M_{lm} {
\begin{array}{c} \sqrt{2}a_{{\rm E},lm...
...{{\rm B},lm} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}$ (19)

whereas in HEALPix 1.1 it was

$\displaystyle \phantom{1.1}{
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\...
...}{.2cm}\\  U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}$ $\textstyle \myequal $ $\displaystyle \sum_{lm} M_{lm} {
\begin{array}{c} -\sqrt{2}a_{lm}^{\rm ...
...^{\rm CURL} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}.$ (20)

The difference between KKS and 1.1 was due to an error of sign on one the basis functions.

Eric Hivon 2010-06-18
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