Spin harmonics   

• Introduction & notation;
• Transforms;

Introduction & notation

The core of the formalism implemented for the (un)polarized transforms is inherited from the relevant HEALpix routines, the notation of which was originally based on the paper by M. Kamionkowski, A. Kosowsky, & A. Stebbins, (Phys. Rev. D., 71, 083008, (1998), hereafter KKS98).

Below we however follow the naming convention of U. Seljak & M. Zaldarriaga, (Phys. Rev. D, 55, 1830, (1997), hereafter, SZ97) and the presentation of the paper by A. Lewis, (Phys. Rev. D., 71, 083008, (2005), hereafter Lewis05).

The polarized routines compute or use as an input, (map2alm and alm2map routines, respectively) the total intensity scalar harmonic coefficients, $I_{\ell m}$, and, optionally if requested, the electric ($_2E_{\ell m}$) and magnetic ($_2B_{\ell m}$) representation of the polarized harmonic coefficients (Lewis05), where hereafter for shortness we drop the subscript $2$. Those are defined as, (e.g., Eqs. (A4) of Lewis05),

$$\begin{array}{c} \medskip {\displaystyle I_{\ell m} \equiv a... ...left( _2a_{\ell m} \, - \,_{-2}a_{\ell m}\right)} \end{array},$$ (1)

where,

$$\begin{array}{l c l} \medskip {\displaystyle I} & {\displays... ...\sum_{\ell m}\, _{-2}a_{\ell m}\,_{-2}Y_{\ell m}} \end{array}.$$

The parameter $H$ sets the polarization to be right handed or left handed set wrt to the incoming photons. Note that the original HEALpix choice for $H$ is one and was also adopted as a default in S2HAT. It is decided by a value of the parameter SPIN_CONV_SIGN, corresponding to (-1)^H, and defined in the file s2hat_defs.f90 (and duplicated for consistency in s2hat_defs.h). The default value is: SPIN_CONV_SIGN=-1 corresponding to the HEALPix convention.

The polarized routines process or compute maps of I, Q, and U.

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The transforms

We first define (KKS98, Lewis05, Eq. (A6)),


$$_2F_{\ell m} \equiv \,_2Y_{\ell m}\left(\theta, \phi\right) \, + \,_{-2}Y_{\ell m}\left(\theta, \phi\right),$$ (2)

$$_{-2}F_{\ell m} \equiv \,_2Y_{\ell m}\left(\theta, \phi\right) \, - \,_{-2}Y_{\ell m}\left(\theta, \phi\right).$$ (3)

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

    $$I\left(\theta,\phi\right) = \sum_{\ell, m}\;I_{\ell m}\,Y_{\ell m}\left(\theta\right)$$ (4)

    $$Q\left(\theta, \phi\right) = \left(-1\right)^H\,\sum_{\ell m}\,\left(_2F^{+}_{\ell m}\left(\th... ... \iota\,_2F^-_{\ell m}\left(\theta\right)\,B_{\ell m}\right)\,e^{\iota\,m\phi},$$ (5)

    $$U\left(\theta, \phi\right) = (-1)^H\,\sum_{\ell m}\,\left(_2 F^{+}_{\ell m}\left(\theta\right)... ...iota\,_2F^-_{\ell m}\left(\theta\right)\,E_{\ell m}\right) \, e^{\iota\,m\phi},$$ (6)

    
    

  • for the Southern Hemisphere,
    

    $$I\left(\pi-\theta,\phi\right) = \sum_{\ell, m}\;(-1)^{\ell+m}\;I_{\ell m}\,Y_{\ell m}\left(\theta\right)$$ (7)

    $$Q\left(\pi-\theta, \phi\right) = \left(-1\right)^{H}\,\sum_{\ell m}\,\left(-1\right)^{\ell+m}\, \l... ... \iota\,_2F^-_{\ell m}\left(\theta\right)\,B_{\ell m}\right)\,e^{\iota\,m\phi},$$ (8)

    $$U\left(\pi-\theta, \phi\right) = \left(-1\right)^{H}\,\sum_{\ell m}\,\left(-1\right)^{\ell+m}\, \l... ...ell m}\left(\theta\right)\,E_{\ell m}\right)\, e^{\iota\,m\phi}. \ \ \ \ \ \ \$$ (9)

    
    

• Inverse transforms:
  The inverse transform implement the following expressions:
  • Scalar (total intensity) component:
    

    $$I_{\ell m} = \sum_{\theta \le {\pi\over2} \atop \phi}\;\left... ...\pi-\theta, \phi\right)\right)\,Y_{\ell m}\left(\theta\right).$$ (10)

    
    

  • Electric component:
    

    $$E_{\ell m} = (-1)^H\,{1\over 2} \, \sum_{\theta \le {\pi\over2}\atop \phi}\, \... ...(\pi - \theta, \phi\right)\right) \,_2F^{+}_{\ell m}\left(\theta\right) \right.$$

    $$- \left. \iota\,(-1)^H\,\left(U\left(\theta, \phi\right) - (-1)^{\e... ...(\pi - \theta, \phi\right)\right) \,_2F^{-}_{\ell m}\left(\theta\right) \right]$$ (11)

    
    

  • Magnetic component:
    

    $$B_{\ell m} = (-1)^H\,{1\over 2} \, \sum_{\theta \le {\pi\over2} \atop \phi}\, ... ...(\pi - \theta, \phi\right)\right) \,_2F^{-}_{\ell m}\left(\theta\right) \right.$$

    $$- \left. (-1)^H\,\left(U\left(\theta, \phi\right) + (-1)^{\ell+m}\,... ...\pi - \theta, \phi\right)\right) \,_2F^{+}_{\ell m}\left(\theta\right) \right].$$ (12)

    
    

  NB. Note that the sign for the scalar component (total intensity) transforms, as defined above here, is not consistent with the overall convention for an arbitrary spin field transform implemented by the spin routines map2alm_spin and alm2map_spin described in the following.

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