cl2map

Description

cl2map synthesizes CMB temperature maps from spherical harmonic coefficients $a_{{\ell}m}$.

$$\Delta T(\theta, \phi) = \sum_{{\ell}=2}^{\max}\sum^{{\ell}}_{m=-{\ell}} a_{{\ell}m}Y_{{\ell}m}(\theta,\phi)$$ (3)

Inversely, cl2map can decompose an input CMB temperature map into its spherical harmonic coefficients $a_{{\ell}m}$ and/or power spectrum:

$$C_{{\ell}}=\frac{1}{2{\ell}+1}\sum_{m=-{\ell}}^{{\ell}} \vert a_{{\ell}m}\vert^2.$$ (4)

$$D_{{\ell}}=\frac{ {\ell}({\ell}+1)C_{{\ell}}}{2 \pi T_0^2}.$$ (5)

where $T_0 = 2.726$ K. If necessary, dipole can be calculated and included.

To synthesize a map, cl2map can process an $a_{{\ell}m}$ file either in FITS Binary Extension-like or in ASCII format, or can process ASCII $a_{{\ell}m}$ strings.

If the input is a power spectrum $C_{{\ell}}$ or $D_{{\ell}}$, cl2map 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 decompose an input map for its $a_{{\ell}m}$ coefficients, cl2map processes the FITS Binary map with position and temperature $T(\theta, \phi)$ with 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. Synthesizing a map from given spherical harmonic coefficients:
   • cl2map -falm alm.fits -o map.fits -nx 250 -np 500 -lmax 100
     synthesizes the map from input FITS file ' alm.fits' and outputs to the file ' map.fits' with resolution (250,500). The maximum multipole number ${\ell}$ is 100.

   • cl2map -alm "2,1,1,0" -o 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 outputs to the file ' map.y21.fits'. The resolution of the map is (100,200). The maximum multipole number ${\ell}$ is 4.

   • cl2map -falm alm.txt -dipole -o map.dipole.fits -nx 50 -np 100 -lmax 10
     synthesizes the dipole map from the input ASCII file ' alm.txt' and outputs to the file ' map.dipole.fits'. The resolution of this dipole map is (50, 100).

   • cl2map -falm alm.fits -di+ -o map.dip.fits -lmax 50
     synthesizes the map (up to ${\ell}=50$ including the dipole) from the input FITS file ' alm.fits' and outputs to the file ' map.dip.fits'. The resolution of the map is the default value, i.e. (201,402).

   • cl2map -falm alm.fits -lmin 5 -lmax 5 -o map.fits
     synthesizes the map for ${\ell}=5$ from the input FITS file ' alm.fits' and outputs to the file ' map.fits'. The resolution is the default value (201,402).

   • cl2map -falm alm.fits -mmin 2 -mmax 10 -nx 400 -np 800 -o map.fits -lmax 50
     synthesizes the map up to ${\ell}=50$ for all $m$ from 2 to 10 from the input FITS file ' alm.fits' and outputs to the file ' map.fits'. The resolution is (400,800).

2. Gaussian random map simulation from a given angular power spectrum:
   • cl2map -dl dl.lcdm.txt -o 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.

   • cl2map -cl cl.txt -o map.fits -nx 300 -np 600 -lmax 50 -gs A
     generates a Gaussian random map from the input ASCII $C_{{\ell}}$ file ' cl.txt' and outputs 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. All the pixel have quasi-equal areas.

   • cl2map -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. Map decomposition for its angular power spectrum and/or spherical harmonic coefficients:
   • cl2map -map map.fits -o cl.txt -lmax 200 -ao alm.fits
     processes the map ' map.fits' and outputs its $C_{{\ell}}$ up to ${\ell}=200$ to file ' cl.txt' and its $a_{{\ell}m}$ to ' alm.fits'.

   • cl2map -map map.fits -od dl.txt -lmax 1300 -ao alm.fits
     processes the map ' map.fits' and outputs up to ${\ell}=1300$ its $D_{{\ell}}$ to file ' dl.txt' and its $a_{{\ell}m}$ to the file ' alm.fits'.