This routine extracts the alm coefficients out of maps of spin s and -s. A (complex) map S of spin s is a linear combination of the spin weighted harmonics _sY_lm

$\begin{displaymath} {_s}S(p) = \sum_{lm} {_s}a_{lm}\ \ {_s}Y_{lm}(p) \end{displaymath}$ (8)

for $l \ge \vert m\vert, l \ge \vert s\vert$ , and is such that _sS^* = _-sS.
The two (real) input maps for map2alm_spin* are defined respectively as

_|s|S⁺ $\textstyle \myequal$ (_|s|S + _-|s|S)/2 (9)

_|s|S^- $\textstyle \myequal$ (_|s|S - _-|s|S)/(2i). (10)

map2alm_spin* outputs the alm coefficients defined as

html_comment_mark>362 _|s|a⁺_lm $\textstyle \myequal$ - ( _|s|a_lm + (-1)^s _-|s|a_lm )/2 (11)

_|s|a^-_lm $\textstyle \myequal$ - ( _|s|a_lm - (-1)^s _-|s|a_lm )/(2i) (12)

for $m\ge 0$ , knowing that, just as for spin 0 maps, the coefficients for m<0 are given by

_|s|a⁺_l-m $\textstyle \myequal$ (-1)^m _|s|a^+_lm, (13)

_|s|a^-_l-m $\textstyle \myequal$ (-1)^m _|s|a^-_lm. (14)

With these definitions, ₂a⁺, ₂a^-, ₂S⁺ and ₂S^- match HEALPix polarization a^E, a^B, Q and U respectively. However, for s=0, $\ _{0}a^+_{lm} = -a^T_{lm}$ , $\ _{0}a^-_{lm} = 0$ , $\ {_0}S^+ = T$ , $\ {_0}S^- = 0.$

Location in HEALPix directory tree: src/f90/mod/alm_tools.f90

FORMAT

call map2alm_spin*( nsmax, nlmax, nmmax, spin, map, alm [, zbounds, w8ring_TQU] )

ARGUMENTS

name & dimensionality	kind	in/out	description

nsmax	I4B	IN	the N_side value of the map to analyse.
nlmax	I4B	IN	the maximum l value for the analysis.
nmmax	I4B	IN	the maximum m value for the analysis.
spin	I4B	IN	the spin s of the maps to be analysed (only its absolute value is relevant).
map(0:12nsmax*2-1, 1:2)	SP/ DP	IN	_\|s\|S⁺ and _\|s\|S^- input maps
alm(1:2, 0:nlmax, 0:nmmax)	SPC/ DPC	OUT	The _\|s\|a⁺_lm and _\|s\|a^-_lm output values.
zbounds(1:2), OPTIONAL	DP	IN	section of the map on which to perform the a_lm analysis, expressed in terms of $z=\sin({\rm latitude}) = \cos(\theta).$ If zbounds(1)<zbounds(2), the analysis is performed on the strip zbounds(1)<z<zbounds(2); if not, it is performed outside of the strip zbounds(2)<z<zbounds(1).
w8ring(1:2*nsmax,1:2), OPTIONAL	DP	IN	ring weights for quadrature corrections. If ring weights are not used, this array should be 1 everywhere.

EXAMPLE:

use healpix_types use alm_tools use pix_tools integer(i4b) :: nside, lmax, spin real(sp), allocatable, dimension(:,:) :: map complex(spc), allocatable, dimension(:,:,:) :: alm nside = 256 lmax = 512 spin = 5 allocate(map(0:nside2npix(nside)-1,1:2)) allocate(alm(1:2, 0:lmax, 0:lmax) ... call map2alm_spin(nside, lmax, lmax, spin, map, alm)

Analyses spin 5 and -5 maps. The maps have an N_side of 256, and the analysis is performed up to 512 in l and m. The resulting a_lm coefficients for are returned in alm.

MODULES & ROUTINES

This section lists the modules and routines used by map2alm_spin*.

ring_analysis: Performs FFT for the ring analysis.
compute_lam_mm, get_pixel_layout,
gen_lamfac_der, gen_mfac,
gen_recfac, init_rescale, l_min_ylm: Ancillary routines used for $\ {_s}Y_{lm}$ recursion
misc_util: module, containing:
assert_alloc: routine to print error message when an array is not properly allocated

RELATED ROUTINES

This section lists the routines related to map2alm_spin*

alm2map_spin: routine performing the inverse transform of map2alm_spin*.
map2alm: routine analyzing temperature and polarization maps

Eric Hivon 2010-06-18

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