Relation to previous releases

This page contains many equations that may not show up correctly in HTML. We recommend that the Postscript document be used instead.
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 ([11], 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 ClTEMP CT, l ClT ClTEMP
Electric or Gradient ClGRAD CE, l 2ClG 2ClGRAD
Magnetic or Curl ClCURL CB, l 2ClC 2ClCURL
Temp.-Electric cross correlation ClT-GRAD CC, l $-\sqrt{2}$ ClTG $ \sqrt{2}$ClT-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 \begin{displaymath}M_{lm} = \left(
\begin{array}{cc} X_{1,lm} & i X_{2,lm} \\
-i X_{2,lm} & X_{1,lm}
\end{array}\right)
\end{displaymath}
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 \begin{displaymath}{
\left(
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3c...
...[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}.
\end{displaymath}

For KKS, with the same definition of M, the decomposition reads \begin{displaymath}{
\left(
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3c...
...e[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}
\end{displaymath}

whereas in HEALPix 1.1 it was \begin{displaymath}{
\left(
\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3c...
...[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right)
}.
\end{displaymath}
The difference between KKS and 1.1 was due to an error of sign on one the basis functions.

Eric Hivon 2003-02-07