Beam Window Functions

Beam window functions have computed with the Febecop Pipeline (as described there), and the QuickPol pipeline (see Hivon et al, 2017, and the Planck 2016 Likelihood paper).

The beam window functions relate, over the full sky or over a masked sky, the angular power spectrum measured (in the absence of noise) on a map produced by a set of detectors [math]C_{XX}^{map}(\ell)[/math], to the true underlying sky angular power spectrum [math]C_{XX}^{sky}(\ell)[/math] (assumed to have isotropic statistical properties, as is the case for the CMB).

QuickPol effective beam window products

They are available in two forms:

Beam window functions

[math]b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),[/math] such that
[math]C_{XX}^{map}(\ell)\, = \, b_{X}^2(\ell) \, w_{pix}^2(\ell) \, C_{XX}^{sky}(\ell)[/math]
for X=T, E or B, and where [math]w_{pix}[/math] is the pixel window function, which depends on the resolution parameter Nside (=2048 for Planck HFI maps).

They are provided for each multipole [math] 0 \le \ell \le \ell_\text{max} = 4000[/math], in FITS format files compatible with HEALPix tools such as synfast and smoothing, as well as with PolSpice.

Beam matrices

[math]W_{XY,X'Y'}(\ell)[/math], such that
[math]C_{XY}^{map}(\ell) \, = \, \sum_{X',Y'} W_{XY,X'Y'}(\ell) \, w_{pix}^2(\ell) \, C_{X'Y'}^{sky}(\ell)[/math]
for X,Y,X',Y'= T, E or B.

They are provided in FITS files, containing 4 extensions each:

1. first one, named 'TT', contains the 9 fields: 'TT_2_TT', 'TT_2_EE', 'TT_2_BB', 'TT_2_TE', 'TT_2_TB', 'TT_2_EB', 'TT_2_ET', 'TT_2_BT', 'TT_2_BE'
   describing the ℓ-dependent leakage template of TT towards TT, EE, BB, ... respectively.
   TT_2_TT is the usual [math]W_{TT}(\ell) = b_T(\ell)^2,[/math] with [math]b_T(\ell=0)=1.[/math]
2. second extension, named 'EE', contains the 9 fields 'EE_2_TT', 'EE_2_EE', 'EE_2_BB', ... for leakage of EE towards TT, EE, BB, ...
   EE_2_EE is the usual [math]W_{EE}(\ell) = b_E(\ell)^2.[/math]
3. 3rd extension: 'BB' with 'BB_2_TT', ...
4. 4th extension: 'TE' with 'TE_2_TT', ...
5. Beware: there is no extension #5 nor 6, corresponding to TB and EB, since these terms are unlikely to be major sources of contamination for the other spectra
   

The measured [math]C^*(\ell)[/math] are then related to the sky ones [math]C(\ell)[/math] via
C^TT*(ℓ) = C^TT(ℓ) TT_2_TT(ℓ) + C^EE(ℓ) EE_2_TT(ℓ) + C^BB(ℓ) BB_2_TT(ℓ) + C^TE(ℓ) TE_2_TT(ℓ)
C^EE*(ℓ) = C^TT(ℓ) TT_2_EE(ℓ) + C^EE(ℓ) EE_2_EE(ℓ) + C^BB(ℓ) BB_2_EE(ℓ) + C^TE(ℓ) TE_2_EE(ℓ)
C^TE*(ℓ) = C^TT(ℓ) TT_2_TE(ℓ) + C^EE(ℓ) EE_2_TE(ℓ) + C^BB(ℓ) BB_2_TE(ℓ) + C^TE(ℓ) TE_2_TE(ℓ)
C^ET*(ℓ) = C^TT(ℓ) TT_2_ET(ℓ) + C^EE(ℓ) EE_2_ET(ℓ) + C^BB(ℓ) BB_2_ET(ℓ) + C^TE(ℓ) TE_2_ET(ℓ)

• To read these FITS file in IDL or python, see [[1]]