Difference between revisions of "Beam Window Functions"
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C<sup>TT*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_TT(ℓ) + C<sup>EE</sup>(ℓ) EE_2_TT(ℓ) + C<sup>BB</sup>(ℓ) BB_2_TT(ℓ) + C<sup>TE</sup>(ℓ) TE_2_TT(ℓ) <br /> | C<sup>TT*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_TT(ℓ) + C<sup>EE</sup>(ℓ) EE_2_TT(ℓ) + C<sup>BB</sup>(ℓ) BB_2_TT(ℓ) + C<sup>TE</sup>(ℓ) TE_2_TT(ℓ) <br /> | ||
C<sup>EE*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_EE(ℓ) + C<sup>EE</sup>(ℓ) EE_2_EE(ℓ) + C<sup>BB</sup>(ℓ) BB_2_EE(ℓ) + C<sup>TE</sup>(ℓ) TE_2_EE(ℓ) <br /> | C<sup>EE*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_EE(ℓ) + C<sup>EE</sup>(ℓ) EE_2_EE(ℓ) + C<sup>BB</sup>(ℓ) BB_2_EE(ℓ) + C<sup>TE</sup>(ℓ) TE_2_EE(ℓ) <br /> | ||
+ | C<sup>BB*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_BB(ℓ) + C<sup>EE</sup>(ℓ) EE_2_BB(ℓ) + C<sup>BB</sup>(ℓ) BB_2_BB(ℓ) + C<sup>TE</sup>(ℓ) TE_2_BB(ℓ) <br /> | ||
C<sup>TE*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_TE(ℓ) + C<sup>EE</sup>(ℓ) EE_2_TE(ℓ) + C<sup>BB</sup>(ℓ) BB_2_TE(ℓ) + C<sup>TE</sup>(ℓ) TE_2_TE(ℓ) <br /> | C<sup>TE*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_TE(ℓ) + C<sup>EE</sup>(ℓ) EE_2_TE(ℓ) + C<sup>BB</sup>(ℓ) BB_2_TE(ℓ) + C<sup>TE</sup>(ℓ) TE_2_TE(ℓ) <br /> | ||
C<sup>ET*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_ET(ℓ) + C<sup>EE</sup>(ℓ) EE_2_ET(ℓ) + C<sup>BB</sup>(ℓ) BB_2_ET(ℓ) + C<sup>TE</sup>(ℓ) TE_2_ET(ℓ) <br /> | C<sup>ET*</sup>(ℓ) = C<sup>TT</sup>(ℓ) TT_2_ET(ℓ) + C<sup>EE</sup>(ℓ) EE_2_ET(ℓ) + C<sup>BB</sup>(ℓ) BB_2_ET(ℓ) + C<sup>TE</sup>(ℓ) TE_2_ET(ℓ) <br /> |
Revision as of 13:45, 16 February 2018
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 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 theContents
QuickPol effective beam window products[edit]
They are available in two forms:
Beam window functions[edit]
The beam window functions
for X=T, E or B, and where is the pixel window function, parameterized by the HEALPix resolution parameter (=2048 for Planck HFI maps).
They are provided for each multipole FITS format files compatible with HEALPix tools (such as synfast, smoothing, syn_alm_cxx, smoothing_cxx), as well as with PolSpice.
, inBeam matrices[edit]
The beam matrices
for X,Y,X',Y'= T, E or B, and where is defined above.
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.
They are provided for each multipole .
The measured
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)
- To read these FITS file in IDL or python, see [[1]]
References[edit]
- ↑ 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
- ↑ Planck collaboration, 2018, Planck 2016 results. V. Legacy Power Spectra and Likelihoods
Cosmic Microwave background
(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).
(Planck) High Frequency Instrument
Flexible Image Transfer Specification