Beam Window Functions

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Beam window functions have computed with the Febecop Pipeline (as described there), and the QuickPol pipeline (see Hivon et al, 2017[1], and the Planck 2016 Likelihood paper[2]).

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[edit]

They are available in two forms:

Beam window functions[edit]

[math]b_{T}, b_{E}, b_{B},[/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 in FITS files, in a format compatible with HEALPix tools such as synfast and smoothing, as well as with PolSpice.

Beam matrices[edit]

[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 W'_TT_'(l) = B'_T_'(l)'^2^'.

  1. 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, ...
  2. 3rd extension: 'BB' with 'BB_2_TT', ...
  3. 4th extension: 'TE' with 'TE_2_TT', ...
  4.  %item value=0% (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 C'^*^'(l) are then related to the sky ones C(l)\\

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) \\ ...

    • These FITS files contain the same information as the raw npz files above, with the added value that the matrix elements \\

have been re-scaled so that B'_T_'(l=0) = 1 (a relative shift < 10'^-3^' in power).

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


  1. Hivon E., Mottet, S. & Ponthieu N., 2017 QuickPol: Fast calculation of effective beam matrices for CMB polarization A&A 598, A25, 2017A&A...598A..25H
  2. Planck collaboration, 2018, Planck 2016 results. V. Legacy Power Spectra and Likelihoods

Cosmic Microwave background

(Planck) High Frequency Instrument

Flexible Image Transfer Specification