alm2map_spin*

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 $\ {_s}Y_{\ell m}$

$${_s}S(p) = \sum_{\ell m} {_s}a_{\ell m}\ \ {_s}Y_{\ell m}(p)$$ (3)

for $\ell \ge \vert m\vert, \ell \ge \vert s\vert$, and is such that _sS^* = _-sS.
The usual phase convention for the spin weighted harmonics is ${_s}Y_{\ell m}^* = (-1)^{s+m} {_{-s}}Y_{\ell -m}$ and therefore ${_s}a_{\ell m}^* = (-1)^{s+m} {_{-s}}a_{\ell -m}$.

alm2map_spin* expects the alm coefficients to be provided as

$${_{\vert s\vert}}a^{+}_{\ell m} \myequal - ( {_{\vert s\vert}}a_{\ell m} + (-1)^s {_{-\vert s\vert}}a_{\ell m} )/2$$ (4)

$${_{\vert s\vert}}a^{-}_{\ell m} \myequal - ( {_{\vert s\vert}}a_{\ell m} - (-1)^s {_{-\vert s\vert}}a_{\ell m} )/(2i)$$ (5)

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

$${_{\vert s\vert}}a^{+}_{\ell-m} \myequal (-1)^m {_{\vert s\vert}}a^{+*}_{\ell m},$$ (6)

$${_{\vert s\vert}}a^{-}_{\ell-m} \myequal (-1)^m {_{\vert s\vert}}a^{-*}_{\ell m}.$$ (7)

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

$$\myequal$$ _|s|S⁺ (_|s|S + _-|s|S)/2 (8)

$$\myequal$$ _|s|S^- (_|s|S - _-|s|S)/(2i). (9)

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

Location in HEALPix directory tree: src/f90/mod/alm_tools.F90 

FORMAT

call alm2map_spin*( nsmax, nlmax, nmmax, spin, alm, map[, zbounds=] )

ARGUMENTS

name & dimensionality	kind	in/out	description
nsmax	I4B	IN	the Nside value of the map to synthesize.
nlmax	I4B	IN	the maximum $\ell$ value used for the $a_{\ell m}$.
nmmax	I4B	IN	the maximum m value used for the $a_{\ell m}$.
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 ${_{\vert s\vert}}a^+_{\ell m}$ and ${_{\vert s\vert}}a^-_{\ell m}$ values to make the map from.
map(0:12*nsmax**2-1, 1:2)	SP/ DP	OUT	|s|S+ and |s|S- output maps
zbounds(1:2), OPTIONAL	DP	IN	section of the sphere on which to perform the map synthesis, expressed in terms of $z=\sin(\mathrm{latitude}) = \cos(\theta).$ If zbounds(1)<zbounds(2), it is performed on the strip zbounds(1)<z<zbounds(2); if not, it is performed outside the strip zbounds(2)$\le z \le$zbounds(1). If absent, the whole map is processed.

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 $a_{\ell m}$ passed in alm. The maps have N_side of 256, and are constructed from $a_{\ell m}$ values up to 512 in $\ell$ 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 $Y_{\ell m}$ recursion

misc_utils

module, containing:

assert_alloc

routine to print error message, when an array can not be allocated properly

Note: Starting with version 3.10, libsharp routines will be called if $0 < \vert s\vert \le 100$.

RELATED ROUTINES

This section lists the routines related to alm2map_spin*

alm2map

routine generating maps of temperature and polarisation from their $a_{\ell m}$

alm2map_der

routine generating maps of temperature and polarisation, and their spatial derivatives, from their $a_{\ell m}$

map2alm_spin

routine performing the inverse transform of alm2map.

create_alm

routine to generate randomly distributed $a_{\ell m}$ coefficients according to a given power spectrum