Beam Window Functions

Beam window functions have been computed with the Febecop Pipeline (as described there), and the QuickPol pipeline (see Hivon et al, 2017, and the Planck 2016 Likelihood paper).

The beam window function relates (over the full or masked sky) the angular power spectrum measured (in the absence of noise) on a map produced by a set of detectors C^XX_map(ℓ), to the true underlying sky angular power spectrum C^XX_sky(ℓ), assumed to have isotropic statistical properties, as is the case for the CMB.

QuickPol effective beam window products

The effective beam products are defined for each multipole 0 ≤ ℓ ≤ ℓ_max = 4000.

They are available in three forms:

Temperature beam window functions for polarized and unpolarized detectors (HFI PSB and SWBs)

The temperature beam window function b_T(ℓ), is such that

[math]C^{TT}_\text{map}(\ell)\, = \, b_{T}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{TT}_\text{sky}(\ell)[/math],

where w_pix is the pixel window function, parameterized by the HEALPix resolution parameter N_side (=2048 for Planck HFI maps). Their availability is described in the Availability section.

Temperature and polarized beam window functions for polarized detectors (HFI PSBs)

The temperature and polarization beam window functions b_T(ℓ), b_E(ℓ), and b_T(ℓ) are such that

[math]C^{XX}_\text{map}(\ell)\, = \, b_{X}^2(\ell) \, w_\text{pix}^2(\ell) \, C^{XX}_\text{sky}(\ell)[/math]

for X = T, E, or B, and where w_pix is defined above. Their availability is described in the Availability section.

• These polarized and unpolarized effective beam window functions are provided in FITS format files compatible with HEALPix tools for map synthesis (such as synfast or syn_alm_cxx) or map smoothing (such as smoothing or smoothing_cxx), as well as with map analysis tools such as PolSpice.

Beam matrices for polarized detectors

The beam matrices W_XY,X'Y'(ℓ) are such that

[math]C^{XY}_\text{map}(\ell) \, = \, \sum_{X',Y'} W_{XY,X'Y'}(\ell) \, w_\text{pix}^2(\ell) \, C^{X'Y'}_\text{sky}(\ell)[/math]

for X,Y,X',Y' = T, E, or B, and where w_pix is defined above.

The non-diagonal terms of the matrix (XY ≠ X'Y'), not present in the usual beam window function defined in the previous section, describe the power spectra cross-talk induced by the non-circularity of the scanning beam and the inter-detector beam mismatch; they are important for the high-ℓ cosmological analysis of the CMB polarization, which is done mostly at 100, 143, and 217GHz for HFI.

• They are provided in FITS files, containing four extensions each:

1. the first one, named "TT", contains the nine 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 V-dependent leakage template of TT towards TT, EE, BB, ... respectively. Here "TT_2_TT" is the usual W_{TT}(ℓ) = b_T²(ℓ), with b_T(ℓ=0)=1;
2. the second extension, named "EE", contains the nine fields "EE_2_TT", "EE_2_EE", "EE_2_BB", ... for leakage of EE towards TT, EE, BB, ..., with

"EE_2_EE" being the usual W_{EE}(ℓ) = b_E²(ℓ)

1. the 3rd extension is "BB", with "BB_2_TT", ...
   BB_2_BB is the usual [math]W_{BB}(\ell) = b_B^2(\ell).[/math]
2. the 4th extension is "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 [math]C^*(\ell)[/math] are then related to the sky ones [math]C(\ell)[/math] via (ignoring the pixel window function [math]w_\text{pix}^2(\ell)[/math])

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^BB*(ℓ) = C^TT(ℓ) TT_2_BB(ℓ) + C^EE(ℓ) EE_2_BB(ℓ) + C^BB(ℓ) BB_2_BB(ℓ) + C^TE(ℓ) TE_2_BB(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)

• Their availability is described in the Availability section

Availability

The beam window function products are all contained in the same directory BeamWF_HFI_R3.00, shipped in the tar ball BeamWF_HFI_R3.00.tar.gz (or BeamWF_HFI_R3.00.tar.bz)

They are systematically provided for the (co-)analysis of the

• full mission,
• half-mission1 / half-mission2, and
• even-rings / odd-rings

HFI frequency maps described there.

Those products are identified by a string of the form map1xmap2 where map1 and map2 each take the form freqtype with

• freq being the map frequency in GHz among 100, 143, 217, 353 (PSB+SWB), 353p (PSB only), 545, 857 (with freq1 [math]\le[/math] freq2) and
• type being either empty (full mission map), hm1 or hm2 (for 1st and 2nd half-missions), or even or odd (for maps made of even or odd rings)
• examples of map1xmap2 values
  • 143x217 (for full mission maps),
  • 100hm1x353phm2 (correlation of 100GHz 1st Half mission map, with 353GHz PSB only 2nd Half mission map),
  • 217oddx217odd (auto-correlation of 217GHz odd-rings map), ...

T or TEB window functions

• T only: 100 to 857GHz for full sky maps:
  • Bl_T_R3.00_fullsky_map1xmap2.fits
• TEB: 100 to 353GHz for full sky maps:
  • Bl_TEB_R3.00_fullsky_map1xmap2.fits

Polarized beam matrices

• 100, 143, 217 and 353 GHz, full sky maps
  • Wl_R3.00_fullsky_map1xmap2.fits
• 100, 143 and 217GHz, Plik-like masks
  • valid for analysis done on maps masked with the Plik masks described ??
  • Wl_R3.00_plikmask_map1xmap2.fits

FITS parsing

These FITS files can be read in IDL and python, as follows

• IDL:

fits_info, FITSfile,extname=extname ; print list of extensions found in FITSfile, and store their names in extname

data=mrdfits(FITSfile, 10, header)  ; read extension #10 (data and header)

data=mrdfits(FITSfile, 'ABC', header)  ; read extension having EXTNAME='ABC' (data and header)

print,header  ; print header

print,tag_names(data)  ; print column names

plot,data.(0)  ; plot 1st column of binary table

Note: this requires recent versions of fits_info, mrdfits and their supporting routines, all available at http://idlastro.gsfc.nasa.gov .

• python2:

import astropy.io.fits as pf

import pylab # only to produce plots

pf.info(FITSfile) # print list of extensions found in FITSfile

data, header = pf.getdata(FITSfile, 10, header=True) # read extension #10 (data and header)

data, header = pf.getdata(FITSfile, 'ABC', header=True) # read extension having EXTNAME='ABC' (data and header)

print header # print header

print data.names # print column names

pylab.plot( data.field(0).flatten() ) # plot 1st column of binary table

Note: astropy.io.fits replaces the older pyfits

• python3:

same as python2, replacing   print x   with   print (x)

References

--Ehivon (talk) 15:30, 16 February 2018 (CET)
--Ehivon (talk) 18:05, 19 February 2018 (CET)
--Ehivon (talk) 18:55, 29 March 2018 (CEST)