# 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)\ | + | <math>C_{XX}^{map}(\ell)\, = \, b_{X}^2(\ell) \, w_{pix}^2(\ell) \, 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) \ | + | <math>C_{XY}^{map}(\ell) \, = \, \sum_{X',Y'} W_{XY,X'Y'}(\ell) \, w_{pix}^2(\ell) \, 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. | ||

## Revision as of 10:24, 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 CMB).

, to the true underlying sky angular power spectrum (assumed to have isotropic statistical properties, as is the case for the## Contents

### QuickPol effective beam window products[edit]

They are available in two forms:

#### Beam window functions[edit]

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 FITS format files compatible with HEALPix tools such as synfast and smoothing, as well as with PolSpice.

, in#### Beam matrices[edit]

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

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

- 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