Beam Window Functions
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 CMB)., to the true underlying sky angular power spectrum (assumed to have isotropic statistical properties, as is the case for the
- 1 QuickPol effective beam window products
- 2 Availability
- 3 References
QuickPol effective beam window products
The effective beam products are defined for each multipole. They are available in three forms:
Beam window functions for unpolarized detectors (HFI SWB)
The temperature beam window function
where is the pixel window function, parameterized by the HEALPix resolution parameter (=2048 for Planck HFI maps).
Beam window functions for polarized detectors (HFI PSB)
The temperature and polarization beam window functions
for X=T, E or B, and where is defined above.
- These polarized and unpolarized effective beam window functions are provided in FITS format files compatible with HEALPix tools for map synthesis (such as
syn_alm_cxx) or map smoothing (such as
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
for X,Y,X',Y'= T, E or B, and where is defined above.
The non-diagonal terms of the matrix (CMB polarization, which is done mostly at 100, 143 and 217GHz for HFI.), 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
- They are provided in FITS files, containing 4 extensions each:
- 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 with
- 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
- 3rd extension: 'BB' with 'BB_2_TT', ...
- 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 are then related to the sky ones via (ignoring the pixel window function )
CTT*(ℓ) = CTT(ℓ) TT_2_TT(ℓ) + CEE(ℓ) EE_2_TT(ℓ) + CBB(ℓ) BB_2_TT(ℓ) + CTE(ℓ) TE_2_TT(ℓ)
CEE*(ℓ) = CTT(ℓ) TT_2_EE(ℓ) + CEE(ℓ) EE_2_EE(ℓ) + CBB(ℓ) BB_2_EE(ℓ) + CTE(ℓ) TE_2_EE(ℓ)
CBB*(ℓ) = CTT(ℓ) TT_2_BB(ℓ) + CEE(ℓ) EE_2_BB(ℓ) + CBB(ℓ) BB_2_BB(ℓ) + CTE(ℓ) TE_2_BB(ℓ)
CTE*(ℓ) = CTT(ℓ) TT_2_TE(ℓ) + CEE(ℓ) EE_2_TE(ℓ) + CBB(ℓ) BB_2_TE(ℓ) + CTE(ℓ) TE_2_TE(ℓ)
CET*(ℓ) = CTT(ℓ) TT_2_ET(ℓ) + CEE(ℓ) EE_2_ET(ℓ) + CBB(ℓ) BB_2_ET(ℓ) + CTE(ℓ) TE_2_ET(ℓ)
T or TEB window functions
- 100 to 857GHz for full sky maps:
- full mission
- odd and even rings
Polarized beam matrices
- 100, 143 and 217GHz, full sky maps
- 100, 143 and 217GHz, Plik-like masks
- full mission ?
- odd and even rings
- To read these FITS file in IDL or python, see []
- Hivon E., Mottet, S. & Ponthieu N., 2017 QuickPol: Fast calculation of effective beam matrices for CMB polarization A&A 598, A25
- Planck collaboration, 2018, Planck 2016 results. V. Legacy Power Spectra and Likelihoods
Cosmic Microwave background
(Planck) High Frequency Instrument
(Hierarchical Equal Area isoLatitude Pixelation of a sphere, <ref name="Template:Gorski2005">HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, K. M. Górski, E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, M. Bartelmann, ApJ, 622, 759-771, (2005).
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