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 &#x2113;-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 &#x2113;-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 [math]C_{XX}^{map}(\ell)[/math], to the true underlying sky angular power spectrum [math]C_{XX}^{sky}(\ell)[/math] (assumed to have isotropic statistical properties, as is the case for the CMB).


QuickPol effective beam window products[edit]

They are available in two forms:

Beam window functions[edit]

[math]b_{T}(\ell), b_{E}(\ell), b_{B}(\ell),[/math] such that
[math]C_{XX}^{map}(\ell)\quad = \quad b_{X}^2(\ell) \quad w_{pix}^2(\ell) \quad 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).

They are provided for each multipole [math] 0 \le \ell \le \ell_\text{max}[/math], in FITS format files compatible with HEALPix tools such as synfast and smoothing, as well as with PolSpice.

Beam matrices[edit]

[math]W_{XY,X'Y'}(\ell)[/math], such that
[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]
for X,Y,X',Y'= T, E or B.

They are provided in FITS files, containing 4 extensions each:

  1. 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 [math]W_{TT}(\ell) = B_T(\ell)^2,[/math] with [math]B_T(\ell=0)=1.[/math]
  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, ...
    EE_2_EE is the usual [math]W_{EE}(\ell) = B_E(\ell)^2.[/math]
  3. 3rd extension: 'BB' with 'BB_2_TT', ...
  4. 4th extension: 'TE' with 'TE_2_TT', ...
  5. 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 [math]C^*(\ell)[/math] are then related to the sky ones [math]C(\ell)[/math] 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

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]

  1. 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
  2. Planck collaboration, 2018, Planck 2016 results. V. Legacy Power Spectra and Likelihoods

Cosmic Microwave background

(Planck) High Frequency Instrument

Flexible Image Transfer Specification