polmap

Description
polmap synthesizes CMB temperature and polarization maps from spherical harmonic coefficients $a_{{\ell}m}$ and polarization mode coefficients $e_{{\ell}m}$ and $b_{{\ell}m}$.

$$T(\theta, \phi) = \sum_{{\ell}=0}^{\max}\sum^{{\ell}}_{-{\ell}} a_{{\ell}m}Y_{{\ell}m}(\theta,\phi)$$ (6)

To synthesize a map, polmap can process an $a_{{\ell}m}$ file either in FITS Binary Extension-like or in ASCII format, or can read an ASCII string of $a_{{\ell}m}$.

If the input is a power spectrum $C_{{\ell}}$ or $D_{{\ell}}$ (where the $C_{{\ell}}$ ASCII data have to be in units of $T^2$, and $D_{{\ell}}$ unitless), polmap generates a Gaussian map of CMB anisotropies by assigning the $a_{{\ell}m}$ = COMPLEX($\Re$, $\Im$) mutually independent Gaussian-distributed real ($\Re$) and imaginary part ($\Im$), both with a standard deviation $\sqrt{C_{{\ell}}/2}$ at each ${\ell}$ harmonic.

To synthesize polarization maps of Stokes-parameters Q and U, polmap can process 3-alm file containing $a_{{\ell}m}$, $b_{{\ell}m}$ and $e_{{\ell}m}$ data, separate $b_{{\ell}m}$ and $e_{{\ell}m}$ files and file either in FITS Binary Extension-like or in ASCII format, or can read an ASCII strings of $e_{{\ell}m}$ and $b_{{\ell}m}$.

Maps of the Stokes parameters $Q$ and $U$ are calculated with the potentials $S$ and $P$ as follows:

$$Q=D_1S-D_2P,\quad U=D_2S+D_1P,$$ (7)

and operators $D_1$ and $D_2$ are:

$$D_1={1\over 2}\left({\partial^2\over\partial\theta^2}-cot\th... ... x^2}- {1\over 1-x^2}{\partial^2\over\partial\varphi^2}\right)$$ (8)

$$D_2=\left({1\over\sin\theta}{\partial^2\over\partial\theta ... ...arphi}+{x\over 1-x^2}{\partial\over\partial\varphi} \right) .$$

where $x=cos\theta$.

$$S={1\over\sqrt{2\pi}}\sum_{l=2}^{l_{max}}\left(s_{l0} f^0_l(... ...(x)[s_{lm}^c\cos(m\varphi)- s_{lm}^s\sin(m\varphi)]\right) ,$$ (9)

$$P={1\over\sqrt{2\pi}}\sum_{l=2}^{l_{max}}\left( f^0_l(x)+2\s... ...(x)[p_{lm}^c\cos(m\varphi)- p_{lm}^s\sin(m\varphi)]\right) ,$$ (10)

$$f_l^m=\sqrt{(l+0.5){(l-m)!\over( l+m)!}}P_l^m,\quad 0\leq m\leq l\leq l_{max} ,$$ (11)

Here $P_l^m(x)$ and $f_l^m(x)$ are the associated Legendre functions (ordinary and normalized) and $s_{lm}$ and $p_{lm}$ (of B and E-modes) are the coefficients of decomposition characterizing properties of polarization. See other details in the correpsnoding paper of GLESP 2.0.

Examples

1. Synthesizing an anisotropy map from given spherical harmonic coefficients (-a, -as)
   • polmap -a alm.fits -t map.fits -nx 250 -np 500 -lmax 100
     synthesizes the map from input FITS file 'alm.fits' and writes to the file ' map.fits' with resolution (250,500). The maximum multipole number ${\ell}$ is 100.

   • polmap -as "2,1,1,0" -t map.y21.fits -nx 100 -np 200 -lmax 4
     synthesizes a map from the $a_{{\ell}m}$ string '"2,1,1,0"' ( $a_{21}=(1,0)$ ) and writes to the file ' map.y21.fits'. The resolution of the map is (100,200). The maximum multipole number ${\ell}$ is 4.

   • polmap -a alm.txt -L 1 -t map.dipole.fits -nx 50 -np 100
     synthesizes the dipole map from the input ASCII file ' alm.txt' and writes to the file ' map.dipole.fits'. The resolution of this dipole map is (50, 100).

2. Gaussian random map simulation from a given angular power spectrum (-Dl, -Cl):
   • polmap -Dl dl.lcdm.txt -t map.lcdm.fits -nx 1200 -np 2400 -r 3456 -ao alm.lcdm.fits -lmax 500 -fw 10
     generates a Gaussian random map from the input ASCII $D_{{\ell}}$ file ' dl.lcdm.txt'. The resolution is (1200,2400). The random seed is 3456. The map is convolved with a Gaussian beam with FWHM 10 arcmin. The map output is written in ' map.lcdm.fits' and its $a_{{\ell}m}$ in ' alm.lcdm.fits'. The maximum multipole number ${\ell}$ for the map simulation is 500.

   • polmap -Cl cl.txt -t map.fits -nx 300 -np 600 -lmax 50 -gr A
     generates a Gaussian random map from the input ASCII $C_{{\ell}}$ file ' cl.txt' and writes to the file ' map.fits'. The map is produced with resolution (300,600). The maximum l-multipole number from ' dl.txt' for map simulation is 50. The map has quasi-equal pixels similar to those in GLESP 1.0.

   • polmap -Cl cl.txt -ao alm.fits -nx 700 -np 1400 -lmax 300 -fw 5
     generates a Gaussian random map from the input ASCII $C_{{\ell}}$ file ' cl.txt'. The map is convolved with a Gaussian beam with FWHM 5 arcmin.

3. Synthesizing maps of Q and U Stokes parameters unisg given scalar and pseudosacalar harmonic coefficients (E and B-modes)
   • polmap 3alm.fits -o 3map.fits -nx 250 -np 500 -lmax 100
     synthesizes T, Q, and U-maps from input FITS file ' 3alm.fits' containing $a_{{\ell}m}$, $b_{{\ell}m}$, and $e_{{\ell}m}$ modes, and writes them to the file ' 3-map.fits' with resolution (250,500). The maximum multipole number ${\ell}$ is 100.

   • polmap -a alm.fits -b blm.fits -e blm.fits -lmin 5 -lmax 5 -t tmap.fits -u umap.fits -q qmap.fits
     synthesizes the map for ${\ell}=5$ from the input FITS file ' alm.fits' containing temperature anisotropy coefficients and writes to the file ' map.fits' containing a temperature anisotropy map. And the program takes E,B-mode coefficients from files ' blm.fits' ' blm.fits' and writes Q and U-polarization maps to files ' umap.fits' and ' qmap.fits', respectively. The resolution is the default value (201,402).