Difference between revisions of "Beam Window Functions"
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==== Beam window functions ==== | ==== Beam window functions ==== | ||
<math>b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),</math> such that <br /> | <math>b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),</math> such that <br /> | ||
| − | <math>C_{XX}^{map}(\ell) = b_{X}^2(\ell) w_{pix}^2(\ell) C_{XX}^{sky}(\ell)</math> <br /> | + | <math>C_{XX}^{map}(\ell)\quad = \quad b_{X}^2(\ell) \quad w_{pix}^2(\ell) \quad C_{XX}^{sky}(\ell)</math> <br /> |
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). | 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). | ||
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==== Beam matrices ==== | ==== Beam matrices ==== | ||
<math>W_{XY,X'Y'}(\ell)</math>, such that <br /> | <math>W_{XY,X'Y'}(\ell)</math>, such that <br /> | ||
| − | <math>C_{XY}^{map}(\ell) = \sum_{X',Y'} W_{XY,X'Y'}(\ell) w_{pix}^2(\ell) C_{X'Y'}^{sky}(\ell)</math> <br /> | + | <math>C_{XY}^{map}(\ell) \quad = \quad \sum_{X',Y'} W_{XY,X'Y'}(\ell) \quad w_{pix}^2(\ell) \quad C_{X'Y'}^{sky}(\ell)</math> <br /> |
for X,Y,X',Y'= T, E or B. | for X,Y,X',Y'= T, E or B. | ||
They are provided in FITS files, containing 4 extensions each: | 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'<br /> describing the ℓ-dependent leakage template of TT towards TT, EE, BB, ... respectively.<br /> TT_2_TT is the usual <math>W_{TT}(\ell) = B_T(\ell)^2.</math> | + | # 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'<br /> describing the ℓ-dependent leakage template of TT towards TT, EE, BB, ... respectively.<br /> TT_2_TT is the usual <math>W_{TT}(\ell) = B_T(\ell)^2,</math> with <math>B_T(\ell=0)=1.</math> |
# 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, ...<br />EE_2_EE is the usual <math>W_{EE}(\ell) = B_E(\ell)^2.</math> | # 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, ...<br />EE_2_EE is the usual <math>W_{EE}(\ell) = B_E(\ell)^2.</math> | ||
# 3rd extension: 'BB' with 'BB_2_TT', ... | # 3rd extension: 'BB' with 'BB_2_TT', ... | ||
Revision as of 10:23, 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 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 , to the true underlying sky angular power spectrum (assumed to have isotropic statistical properties, as is the case for the CMB).
Contents
QuickPol effective beam window products[edit]
They are available in two forms:
Beam window functions[edit]
such that
for X=T, E or B, and where is the pixel window function, which depends on the resolution parameter Nside (=2048 for Planck HFI maps).
They are provided for each multipole , in FITS format files compatible with HEALPix tools such as synfast and smoothing, as well as with PolSpice.
Beam matrices[edit]
, such that
for X,Y,X',Y'= T, E or B.
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
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)
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)
- These FITS files contain the same information as the raw npz files above, with the added value that the matrix elements
- 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]]
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
(Planck) High Frequency Instrument
Flexible Image Transfer Specification
