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 function relates, 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}_\text{map}(\ell)[/math], to the true underlying sky angular power spectrum [math]C^{XX}_\text{sky}(\ell)[/math] (assumed to have isotropic statistical properties, as is the case for the CMB).

QuickPol effective beam window products

The effective beam products are defined for each multipole [math] 0 \le \ell \le \ell_\text{max} = 4000[/math].
Since they are relevant for the (co-)analysis of a single or tow frequency map(s), among those produced by Planck HFI, and for a given fraction of the sky (either full, or masked as in the Plik likelihood analysis), they are identified by a string of the form det1xdet2 where det1 and det2 each take the form freqtype with

• freq being the detector frequency in GHz among 100, 143, 217, 353 (PSB+SWB), 353p (PSB only), 545, 857, and
• type being either empty (full mission map), hm1 or hm2 (for 1st and 2nd half-missions), or odd or even (for maps made of even or odd rings)

They are available in three forms:

Temperature beam window functions for polarized and unpolarized detectors (HFI PSB and SWB)

The temperature beam window function [math]b_{T}(\ell),[/math] is such that
[math]C^{TT}_\text{map}(\ell)\, = \, b_{T}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{TT}_\text{sky}(\ell)[/math]
where [math]w_\text{pix}[/math] is the pixel window function, parameterized by the HEALPix resolution parameter [math]N_\text{side}[/math] (=2048 for Planck HFI maps).
They are provided in FITS files named Bl_R3.00_full_det1xdet2.fits

• det1 and det2 each take the form freqtype where

* freq is the detector frequency in GHz among 100, 143, 217, 353 (PSB+SWB), 353p (PSB only), 545, 857, and

* type is either empty (full mission map), hm1 or hm2 (for 1st and 2nd half-missions), or odd or even (for maps made of even or odd rings)

Temperature and polarized beam window functions for polarized detectors (HFI PSB)

The temperature and polarization beam window functions [math]b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),[/math] are such that
[math]C^{XX}_\text{map}(\ell)\, = \, b_{X}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{XX}_\text{sky}(\ell)[/math]
for X=T, E or B, and where [math]w_\text{pix}[/math] is defined above.
They are provided in FITS files named Bl_TEB_R3.00_full_det1xdet2.fits

• These polarized and unpolarized effective beam window functions are provided in FITS format files
  compatible with HEALPix tools for map synthesis (such as synfast or syn_alm_cxx)
  or map smoothing (such as smoothing or smoothing_cxx), as well as with map analysis tools such as PolSpice.
• Their availability is described in the Availability section

Beam matrices for polarized detectors

The beam matrices [math]W_{XY,X'Y'}(\ell)[/math] are such that
[math]C^{XY}_\text{map}(\ell) \, = \, \sum_{X',Y'} W_{XY,X'Y'}(\ell) \, w_\text{pix}^2(\ell) \, C^{X'Y'}_\text{sky}(\ell)[/math]
for X,Y,X',Y'= T, E or B, and where [math]w_\text{pix}[/math] is defined above.

The non-diagonal terms of the matrix ([math]XY \ne X'Y'[/math]), not present in the usual beam window function defined in the previous section, describe the power spectra cross-talk induced by the scanning beam non-circularity and inter-detector beam mismatch, and are important for the high-ℓ cosmological analysis of the CMB polarization, which is done mostly at 100, 143 and 217GHz for HFI.

• 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^2(\ell),[/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^2(\ell).[/math]
3. 3rd extension: 'BB' with 'BB_2_TT', ...
   BB_2_BB is the usual [math]W_{BB}(\ell) = b_B^2(\ell).[/math]
4. 4th extension: 'TE' with 'TE_2_TT', ...
   

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 (ignoring the pixel window function [math]w_\text{pix}^2(\ell)[/math])

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^BB*(ℓ) = C^TT(ℓ) TT_2_BB(ℓ) + C^EE(ℓ) EE_2_BB(ℓ) + C^BB(ℓ) BB_2_BB(ℓ) + C^TE(ℓ) TE_2_BB(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)

• The beam matrices are in FITS files named Wl_R3.00_full_det1xdet2.fits or Wl_R3.00_plik_det1xdet2.fits
• Their availability is described in the Availability section

Availability

T or TEB window functions

• 100 to 857GHz for full sky maps:
  • full mission
  • half-mission
  • odd and even rings

Polarized beam matrices

• 100, 143, 217 and 353 GHz, full sky maps
  • full mission
  • half-mission
  • odd and even rings
• 100, 143 and 217GHz, Plik-like masks
  • full mission
  • half-mission
  • odd and even rings

FITS parsing

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

References

--Ehivon (talk) 15:30, 16 February 2018 (CET)
--Ehivon (talk) 18:05, 19 February 2018 (CET)