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

$\begin{displaymath} {_s}S(p) = \sum_{lm} {_s}a_{lm}\ \ {_s}Y_{lm}(p) \end{displaymath}$ (1)

for $l \ge \vert m\vert, l \ge \vert s\vert$ , and is such that _sS^* = _-sS.
alm2map_spin* expects the alm coefficients to be provided as

html_comment_mark>57 _|s|a⁺_lm $\textstyle \myequal$ - ( _|s|a_lm + (-1)^s _-|s|a_lm )/2 (2)

_|s|a^-_lm $\textstyle \myequal$ - ( _|s|a_lm - (-1)^s _-|s|a_lm )/(2i) (3)

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, (4)

_|s|a^-_l-m $\textstyle \myequal$ (-1)^m _|s|a^-_lm. (5)

The two (real) maps produced by alm2map_spin* are �defined respectively as

_|s|S⁺ $\textstyle \myequal$ (_|s|S + _-|s|S)/2 (6)

_|s|S^- $\textstyle \myequal$ (_|s|S - _-|s|S)/(2i). (7)

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 alm2map_spin*( nsmax, nlmax, nmmax, spin, alm, map )

ARGUMENTS

name & dimensionality	kind	in/out	description

nsmax	I4B	IN	the N_side value of the map to synthesize.
nlmax	I4B	IN	the maximum l value used for the a_lm.
nmmax	I4B	IN	the maximum m value used for the a_lm.
spin	I4B	IN	spin s of the maps to be generated (only its absolute value is relevant).
alm(1:2, 0:nlmax, 0:nmmax)	SPC/ DPC	IN	The _\|s\|a⁺_lm and _\|s\|a^-_lm values to make the map from.
map(0:12nsmax*2-1, 1:2)	SP/ DP	OUT	_\|s\|S⁺ and _\|s\|S^- output maps

EXAMPLE:

use healpix_types use pix_tools, only : nside2npix use alm_tools, only : alm2map_spin integer(I4B) :: nside, lmax, mmax, npix, spin real(SP), dimension(:,:), allocatable :: map complex(SPC), dimension(:,:,:), allocatable :: alm ... nside=256 ; lmax=512 ; mmax=lmax ; spin=4 npix=nside2npix(nside) allocate(alm(1:2,0:lmax,0:mmax)) allocate(map(0:npix-1,1:2)) ... call alm2map_spin(nside, lmax, mmax, spin, alm, map)

Make spin-4 maps from the a_lm passed in alm. The maps have N_side of 256, and are constructed from a_lm values up to 512 in l and m.

MODULES & ROUTINES

This section lists the modules and routines used by alm2map_spin*.

ring_synthesis: Performs FFT over m for synthesis of the rings.
compute_lam_mm, get_pixel_layout,
gen_lamfac_der, gen_mfac, gen_mfac_spin, do_lam_lm_spin,
gen_recfac, gen_recfac_spin, init_rescale, l_min_ylm: Ancillary routines used for Y_lm recursion
misc_utils: module, containing:
assert_alloc: routine to print error message, when an array can not be allocated properly

RELATED ROUTINES

This section lists the routines related to alm2map_spin*

alm2map: routine generating maps of temperature and polarisation from their a_lm
alm2map_der: routine generating maps of temperature and polarisation, and their spatial derivatives, from their a_lm
map2alm_spin: routine performing the inverse transform of alm2map.
create_alm: routine to generate randomly distributed a_lm coefficients according to a given power spectrum

Eric Hivon 2010-06-18

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