polalm

description
polalm decomposes an input CMB temperature map into its spherical harmonic coefficients $a_{{\ell}m}$ and/or power spectrum:

$$C_{{\ell}}=\frac{1}{2{\ell}+1}\sum_{m=-{\ell}}^{l} \vert a_{{\ell}m}\vert^2.$$ (12)

$$D_{{\ell}}=\frac{ {\ell}({\ell}+1)C_{{\ell}}}{2 \pi T_0^2}.$$ (13)

where $T_0 = 2.726$ K. and calculates decomposition coefficients of scalar and pseudoscalar potentials (E and B-modes) for Q and U-maps:

$$s_{lm}^c=\int_{-1}^1dx[\Phi^m_l(x)U_s^m(x)+F^m_l(x)Q_c^m(x)] ,$$ (14)

$$s^s_{lm}=\int_{-1}^1dx[\Phi^m_l(x)U_c^m(x)-F^m_l(x)Q_s^m(x)] ,$$ (15)

$$p_{lm}^c=\int_{-1}^1dx[F^m_l(x)U_c^m(x)-\Phi^m_l(x)Q_s^m(x)] ,$$ (16)

$$p^s_{lm}=\int_{-1}^1dx[-F^m_l(x)U_s^m(x)-\Phi^m_l(x)Q_c^m(x)] ,$$ (17)

where

$$F^m_m={m-1\over 2M_m}m{1+x^2\over 1-x^2}f^m_m,\quad F^m_{m+... ...m+3}\over M_{m+1}} \left[{m-1\over 1-x^2}-{m+1\over 2}\right],$$

$$\Phi_m^m=-{m(m-1)\over M_m(1-x^2)}xf^m_m,\quad \Phi^m_{m+1}=mf^m_m{\sqrt{2m+3}\over M_{m+1}} {1-mx^2\over 1-x^2} ,$$ (18)

See other details in the correpsnoding paper of GLESP 2.0.

To decompose an input maps for harmonic coefficients, polalm processes the FITS Binary map with position and temperature, or Q and U Stokes parameters ( $T(\theta, \phi)$, $Q(\theta, \phi)$, $U(\theta, \phi)$) in 3 sections:

1. $x =\cos(\theta)$, the center-position of the rings as a function of the polar angle (nx positions),

2. nx numbers with the number of pixels, nphi, in the corresponding ring,

3. $T(\theta, \phi)$ organized in one row using Point 1 as the major index and Point 2 as the minor index.

Examples

1. Map decomposition for its angular power spectrum and/or spherical harmonic coefficients (-map):
   • polalm -t map.fits -dl dl.txt -lmax 200 -a alm.fits
     processes the map ' map.fits' and writes its $D_{{\ell}}$ up to ${\ell}=200$ to the file ' dl.txt' and its $a_{{\ell}m}$ to ' alm.fits'.

   • polalm 3map.fits -lmax 1300 -o 3alm.fits
     processes the map ' 3-map.fits' containing 3 maps (anisotropy, Q and U-stokes parameters) and writes $a_{{\ell}m}$, $b_{{\ell}m}$ and $e_{{\ell}m}$ up to ${\ell}=1300$ to the file ' 3alm.fits'.

   • polalm -t map.fits -dl dl.txt -lmax 200 -a alm.fits
     processes the map ' map.fits' and writes its $D_{{\ell}}$ up to ${\ell}=200$ to the file ' dl.txt' and its $a_{{\ell}m}$ to ' alm.fits'.

   • polalm -t map.fts -u umap.fts -q qmap.fts -a alm.fts -b blm.fts -e elm.fts -lmax 1300
     processes the maps of anisotrupy ' map.fts' and Stokes parameters ' umap.fts' and ' qmap.fts and writes $a_{{\ell}m}$, $b_{{\ell}m}$ and $e_{{\ell}m}$ up to ${\ell}=1300$ to the files ' alm.fts', ' blm.fts' and ' elm.fts', respectively.