• Introduction & notation;
• Transforms;

Introduction and notation

The following formalism and notation is based on the paper by A. Lewis, (Phys. Rev. D., 71, 083008, (2005), hereafter Lewis05).

The field of a spin, $s$, defined on a sphere ($\theta, \phi$) is represented as,

$$\,_s\eta\left(\theta, \phi\right) \equiv R\left(\theta, \phi\right) + \iota \, I\left( \theta, \phi\right),$$ (1)

where the two real fields (or map components as referred to in the documentation) are on the input or the output of the transform routines implemented in the S$^2$HAT library.

Both positive and negative spins are accepted by the routines. However, for the negative spin, $-s$, we first make use of the fact that

$$\;_{-s}\eta^{*} = \,_s\eta, \ \ \ \ \ \ \ \ \ \hbox{\rm$s \ge 0$},$$ (2)

and proceed with the non-negative spin field, $\;_s\eta$, such as,

$$_s\eta = R\left(\theta, \phi\right) + \iota \left( - I\left(\theta, \phi\right)\right),$$ (3)

where the change of the sign of the imaginary part, $I\left(\theta,\phi\right)$, is performed by the routines. As a result the routines effectively perform on transforms on the non-negative spin fields.

Consequently, in the following equations we will always assume that spin is non-negative ($s \ge 0$).

The routines compute or use the electric ($_sE_{\ell m}$) and magnetic ($_sB_{\ell m}$) representation of the harmonic space coefficients (Lewis05). These are defined as, (e.g., Eqs. (A4) of Lewis05),

$$\begin{array}{c} \medskip {\displaystyle _sE_{\ell m} \equiv... ...t( _sa_{\ell m} - (-1)^s\,_{-s}a_{\ell m}\right)} \end{array},$$ (4)

where $s \ge 0$ and

$$_s\eta = \sum_{\ell m}\, _sa_{\ell m}\,_sY_{\ell m} \ \ \ \ ... ... _{-s}\eta = \sum_{\ell m}\, _{-s}a_{\ell m}\,_{-s}Y_{\ell m}.$$

The parameter $H$ sets the polarization (right handed or left handed set wrt to the incoming photons). The parameter $(-1)^H$ is defined in the Library as a constant SPIN_SIGN_CONV and set by default to $-1$, i.e., consistent with the HEALpix choice. It is defined in the files s2hat_defs.f90 and s2hat_defs.h.

The complex map produced or proccesed by the spin tranform routines, alm2map_spin and map2alm_spin, as two real maps, which for spin s=2 correspond to the Stokes Q an U parameter maps.

Top of the page



Spin transforms

We first define (Lewis05, Eq. (A6)),

$$_sF^{+}_{\ell m} \equiv \,_sY_{\ell m}\left(\theta, \phi\right) + (-1)^s\, \,_{-s}Y_{\ell m}\left(\theta, \phi\right),$$ (5)

$$_sF^{-}_{\ell m} \equiv \,_sY_{\ell m}\left(\theta, \phi\right) - (-1)^s\, \,_{-s}Y_{\ell m}\left(\theta, \phi\right).$$ (6)

• Direct transforms:
  The direct transform spin routine computes the following expressions:
  • for the Northern Hemisphere,
    

    $$R\left(\theta, \phi\right) = \left(-1\right)^H\,\sum_{\ell m}\,\left(_s F^{+}_{\ell m}\left(\t... ...iota\,_sF^-_{\ell m}\left(\theta\right)\,_sB_{\ell m}\right)\,e^{\iota\,m\phi},$$ (7)

    $$I\left(\theta, \phi\right) = (-1)^H\,\sum_{\ell m}\,\left(_s F^{+}_{\ell m}\left(\theta\right)... ...a\,_sF^-_{\ell m}\left(\theta\right)\,_sE_{\ell m}\right)) \, e^{\iota\,m\phi},$$ (8)

    
    

  • for the Southern Hemisphere,
    

    $$R\left(\pi-\theta, \phi\right) = \left(-1\right)^{H+s}\,\sum_{\ell m}\,\left(-1\right)^{\ell+m}\, ... ...iota\,_sF^-_{\ell m}\left(\theta\right)\,_sB_{\ell m}\right)\,e^{\iota\,m\phi},$$ (9)

    $$I\left(\pi-\theta, \phi\right) = \left(-1\right)^{H+s}\,\sum_{\ell m}\,\left(-1\right)^{\ell+m}\, ... ...l m}\left(\theta\right)\,_sE_{\ell m}\right)\, e^{\iota\,m\phi}, \ \ \ \ \ \ \$$ (10)

    
    

  (where these latter Equations are hoped to correct some typos in Lewis05 Eqs. (A10) while not introducing any new ones).

  If requested spin is negative, the imaginary part of the field is subsequently multiplied by $-1$.

• Inverse transforms:
  The inverse transform implement the following expressions:
  • Electric component:
    

    $$_sE_{\ell m} = (-1)^H\,{1\over 2} \, \sum_{\theta \le {\pi\over2} \atop \phi}\, ... ...(\pi - \theta, \phi\right)\right)\, _sF^{+}_{\ell m}\left(\theta\right) \right.$$

    $$- \left. \iota\,(-1)^H\,\left(I\left(\theta, \phi\right) - (-1)^{\e... ...(\pi - \theta, \phi\right)\right)\, _sF^{-}_{\ell m}\left(\theta\right) \right]$$ (11)

    
    

  • Magnetic component:
    

    $$_sB_{\ell m} = (-1)^H\,{1\over 2} \, \sum_{\theta \le {\pi\over2} \atop \phi}\, ... ...(\pi - \theta, \phi\right)\right) \,_sF^{-}_{\ell m}\left(\theta\right) \right.$$

    $$- \left. (-1)^H\,\left(I\left(\theta, \phi\right) + (-1)^{\ell+m+s}... ...\pi - \theta, \phi\right)\right) \,_sF^{+}_{\ell m}\left(\theta\right) \right].$$ (12)

    
    

  If requested spin is negative, the imaginary part of the field is multiplied by $-1$ prior to the transform calculations.

Top of the page