Difference between revisions of "Beam Window Functions"
| Line 16: | Line 16: | ||
<math>C_{XX}^{map}(\ell) = b_{X}^2(\ell) w_{pix}^2(\ell) C_{XX}^{sky}(\ell)</math> | <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). | 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 [http://healpix.sourceforge.net HEALPix] | They are provided in FITS files, in a format compatible with [http://healpix.sourceforge.net HEALPix] | ||
tools such as synfast and smoothing, as well as with PolSpice. | tools such as synfast and smoothing, as well as with PolSpice. | ||
| Line 23: | Line 24: | ||
such that | 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> | <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 | + | 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 W'_TT_'(l) = B'_T_'(l)'^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, ... | ||
| + | ### 3rd extension: 'BB' with 'BB_2_TT', ... | ||
| + | ### 4th extension: 'TE' with 'TE_2_TT', ... | ||
| + | ### %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 [[http://wiki.planck.fr/index.php/Proc/InstrumentModels#tools]] | ||
| + | |||
=== References === | === References === | ||
<references /> | <references /> | ||
Revision as of 09:27, 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 in FITS files, in a format 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 W'_TT_'(l) = B'_T_'(l)'^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, ...
- 3rd extension: 'BB' with 'BB_2_TT', ...
- 4th extension: 'TE' with 'TE_2_TT', ...
- %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]]
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
