Frequency maps angular power spectra

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HFI detset maps power spectra[edit]



Angular power spectra of cut sky CMB dominated maps are provided to allow independent cosmological analysis at high ell.

Product description[edit]

For HFI, the auto and cross-spectra of the 13 detector (set) maps at 100, 143 and 217GHz, all analyzed on the same 42.8% of the sky, are provided. The mask used is apodized to reduce the power leakage from large scale to small scale (see input section). Except for the removal of the most contaminated pixels through masking, no attempt at astrophysical components separation is performed. The unbinned [math]C[/math]XY[math](l)[/math] in units of μK are available for all [math]l[/math] from 0 to 3508.

The [math]<ΔC[/math]XY[math](l)ΔC[/math]XY[math](l')>[/math] covariance matrix also is provided in order to perform correct model fitting and cosmological parameter estimation. It only describes the statistical correlation induced by the sample variance and cut sky. For correlations induced by the uncertainty on the beam window functions, the relevant beam information should be read from the RIMO and from the "Planck 2013 results. VII. HFI time response and beams" paper.

Production process[edit]

The spectra computed up to ell=3508 using PolSpice (http://prof.planck.fr/article141.html, Chon et al (2004)) are corrected from the effect of the cut sky, and from the nominal beam window function and average pixel function.

  • one
  • two


The covariance matrix is computed by PolSpice using the formalism described in Efstathiou (2004), also sketched in the appendix of "Planck 2013 results. XV. CMB power spectra and likelihood" paper, assuming the instrumental noise to be white and uniform.

The products described here (spectra and covariances) only are adapted for high-ell study of the CMB fluctuations.

Inputs[edit]

Input data:

  • Mask:

All maps were analyzed on the fsky=42.8% of the sky defined by the apodized CL43 mask, which masks out Galactic and point sources contamination (see Planck 2013 results. XV. paper). It is available as a FITS file under the name HFI_PowerSpect_Mask_2048_R1.10.fits

  • Maps

The input maps are the 13 HFI detector (set) maps available at 100, 143 and 217GHz. These are the same as the ones used for high-ell part of the Planck Likelihood Code, but that code applies different masks for each cross-spectra in order to minimize further the foreground contamination.

  • Beam Window Function

The beam window functions B(l), and their uncertainties, are the ones used in the high-ell likelihood analysis, described in section 6.1 "Error Eigenmodes" of the Planck 2013 results. VII. paper and provided in the RIMO HFI_RIMO_R1.10.fits

Related products[edit]

A description of other products that are related and share some commonalities with the product being described here. E.g. if the description is of a generic product (e.g. frequency maps), all the products falling into that type should be listed and referenced.

If none, please delete this section

File names and structure[edit]

Power spectra are provided for the auto and cross products built from the 13 detsets available at 100, 143 and 217 GHz, namely:

  • 100-ds1, 100-ds2,
  • 143-ds1, 143-ds2, 143-5, 143-6, 143-7,
  • 217-ds1, 217-ds2, 217-1, 217-2, 217-3, 217-4

which makes 13*(13+1)/2 = 91 spectra. Filenames for the auto spectra are HFI_PowerSPect_{detset}_Relnum.fits and HFI_PowerSPect_{detset1}x{detset2}_Relnum.fits for the auto- and cross-spectra, respectively. The list of the 91 files is given below. Each files contains 2 BINTABLE extensions:


Column Name Data Type Units Description
1. EXTNAME = 'POW-SPEC' : Data columns
TEMP_CL Real*4 microKcmb2 the power spectrum
TEMP_CL_ERR Real*4 microKcmb2 estimate of the uncertainty in the power spectrum
Keywords
LMIN Integer 0 First value of ell (origin 0)
LMAX Integer value Last value of ell (origin 0)
2. EXTNAME = 'PSCOVMAT' : Data columns
COVMAT Real*4 microKcmb4 the covariance matrix
Keywords
TDIM1 Integer (dim1, dim2) matrix dimensions


Ext POW-SPEC[edit]

A BINTABLE extension with two columns, containing the spectrum and the estimated uncertainty on the spectrum, which is simply the diagonal of the covariance matrix. The length of both vectors is LMAX+1, which is the number of table rows (this is also the NAXIS keyword).

Extension PSCOVMAT[edit]

A BINTABLE extension for the covariance matrix. It consists of a single column of LMAX+1 cells, each cell containing the LMAX+1 elements which make up one row of the matrix.

List of filenames[edit]

HFI_PowerSpect_100-ds1_R1.00.fits
HFI_PowerSpect_100-ds1x100-ds2_R1.00.fits
HFI_PowerSpect_100-ds1x143-5_R1.00.fits
HFI_PowerSpect_100-ds1x143-6_R1.00.fits
HFI_PowerSpect_100-ds1x143-7_R1.00.fits
HFI_PowerSpect_100-ds1x143-ds1_R1.00.fits
HFI_PowerSpect_100-ds1x143-ds2_R1.00.fits
HFI_PowerSpect_100-ds1x217-1_R1.00.fits
HFI_PowerSpect_100-ds1x217-2_R1.00.fits
HFI_PowerSpect_100-ds1x217-3_R1.00.fits
HFI_PowerSpect_100-ds1x217-4_R1.00.fits
HFI_PowerSpect_100-ds1x217-ds1_R1.00.fits
HFI_PowerSpect_100-ds1x217-ds2_R1.00.fits
HFI_PowerSpect_100-ds2_R1.00.fits
HFI_PowerSpect_100-ds2x143-5_R1.00.fits
HFI_PowerSpect_100-ds2x143-6_R1.00.fits
HFI_PowerSpect_100-ds2x143-7_R1.00.fits
HFI_PowerSpect_100-ds2x143-ds1_R1.00.fits
HFI_PowerSpect_100-ds2x143-ds2_R1.00.fits
HFI_PowerSpect_100-ds2x217-1_R1.00.fits
HFI_PowerSpect_100-ds2x217-2_R1.00.fits
HFI_PowerSpect_100-ds2x217-3_R1.00.fits
HFI_PowerSpect_100-ds2x217-4_R1.00.fits
HFI_PowerSpect_100-ds2x217-ds1_R1.00.fits
HFI_PowerSpect_100-ds2x217-ds2_R1.00.fits
HFI_PowerSpect_143-5_R1.00.fits
HFI_PowerSpect_143-5x143-6_R1.00.fits
HFI_PowerSpect_143-5x143-7_R1.00.fits
HFI_PowerSpect_143-5x217-1_R1.00.fits
HFI_PowerSpect_143-5x217-2_R1.00.fits
HFI_PowerSpect_143-5x217-3_R1.00.fits
HFI_PowerSpect_143-5x217-4_R1.00.fits
HFI_PowerSpect_143-5x217-ds1_R1.00.fits
HFI_PowerSpect_143-5x217-ds2_R1.00.fits
HFI_PowerSpect_143-6_R1.00.fits
HFI_PowerSpect_143-6x143-7_R1.00.fits
HFI_PowerSpect_143-6x217-1_R1.00.fits
HFI_PowerSpect_143-6x217-2_R1.00.fits
HFI_PowerSpect_143-6x217-3_R1.00.fits
HFI_PowerSpect_143-6x217-4_R1.00.fits
HFI_PowerSpect_143-6x217-ds1_R1.00.fits
HFI_PowerSpect_143-6x217-ds2_R1.00.fits
HFI_PowerSpect_143-7_R1.00.fits
HFI_PowerSpect_143-7x217-1_R1.00.fits
HFI_PowerSpect_143-7x217-2_R1.00.fits
HFI_PowerSpect_143-7x217-3_R1.00.fits
HFI_PowerSpect_143-7x217-4_R1.00.fits
HFI_PowerSpect_143-7x217-ds1_R1.00.fits
HFI_PowerSpect_143-7x217-ds2_R1.00.fits
HFI_PowerSpect_143-ds1_R1.00.fits
HFI_PowerSpect_143-ds1x143-5_R1.00.fits
HFI_PowerSpect_143-ds1x143-6_R1.00.fits
HFI_PowerSpect_143-ds1x143-7_R1.00.fits
HFI_PowerSpect_143-ds1x143-ds2_R1.00.fits
HFI_PowerSpect_143-ds1x217-1_R1.00.fits
HFI_PowerSpect_143-ds1x217-2_R1.00.fits
HFI_PowerSpect_143-ds1x217-3_R1.00.fits
HFI_PowerSpect_143-ds1x217-4_R1.00.fits
HFI_PowerSpect_143-ds1x217-ds1_R1.00.fits
HFI_PowerSpect_143-ds1x217-ds2_R1.00.fits
HFI_PowerSpect_143-ds2_R1.00.fits
HFI_PowerSpect_143-ds2x143-5_R1.00.fits
HFI_PowerSpect_143-ds2x143-6_R1.00.fits
HFI_PowerSpect_143-ds2x143-7_R1.00.fits
HFI_PowerSpect_143-ds2x217-1_R1.00.fits
HFI_PowerSpect_143-ds2x217-2_R1.00.fits
HFI_PowerSpect_143-ds2x217-3_R1.00.fits
HFI_PowerSpect_143-ds2x217-4_R1.00.fits
HFI_PowerSpect_143-ds2x217-ds1_R1.00.fits
HFI_PowerSpect_143-ds2x217-ds2_R1.00.fits
HFI_PowerSpect_217-1_R1.00.fits
HFI_PowerSpect_217-1x217-2_R1.00.fits
HFI_PowerSpect_217-1x217-3_R1.00.fits
HFI_PowerSpect_217-1x217-4_R1.00.fits
HFI_PowerSpect_217-1x217-ds1_R1.00.fits
HFI_PowerSpect_217-1x217-ds2_R1.00.fits
HFI_PowerSpect_217-2_R1.00.fits
HFI_PowerSpect_217-2x217-3_R1.00.fits
HFI_PowerSpect_217-2x217-4_R1.00.fits
HFI_PowerSpect_217-2x217-ds1_R1.00.fits
HFI_PowerSpect_217-2x217-ds2_R1.00.fits
HFI_PowerSpect_217-3_R1.00.fits
HFI_PowerSpect_217-3x217-4_R1.00.fits
HFI_PowerSpect_217-3x217-ds1_R1.00.fits
HFI_PowerSpect_217-3x217-ds2_R1.00.fits
HFI_PowerSpect_217-4_R1.00.fits
HFI_PowerSpect_217-4x217-ds1_R1.00.fits
HFI_PowerSpect_217-4x217-ds2_R1.00.fits
HFI_PowerSpect_217-ds1_R1.00.fits
HFI_PowerSpect_217-ds1x217-ds2_R1.00.fits
HFI_PowerSpect_217-ds2_R1.00.fits

LFI frequency maps power spectra[edit]


Product description[edit]

The angular power spectrum provides information about the distribution of power on the sky map at the various angular scales. It is especially important for CMB, because it is characterized by a number of features, most notably the acoustic peaks, that encode the dependence from cosmological parameters. Therefore, angular power spectra are the basic inputs for the Planck Likelihood Code, and for estimation of cosmological parameters in general.

For this release we have computed only temperature power spectra. Polarization is not included.

Please note that these spectra come from frequency maps. No component separation has been applied, and we have only masked Galactic Plane and detected point sources. Units are [math] \mu K^2_{CMB} [/math].

Production process[edit]

Spectra are computed using cROMAster, a implementation of the pseudo-Cl method described in Hivon et al, 2002. In addition to the original approach, our implementation allows for estimation of cross-power spectra from two or more maps (see Polenta et al, 2005, for details). The software package uses HEALPix modules for spherical harmonic transform and Cl calculation. The schematic of the estimation process is as follows:

  • computing the a_lm coefficients from the input temperature map after masking Galactic Plane and point sources.
  • computing the pseudo power spectrum from the alms.
  • estimating the bias due to the noise power spectrum of the map from noise-only Monte Carlo simulations based on detector noise properties
  • correcting for the effect of the adopted mask by computing the mode-mode coupling kernel corresponding to that mask
  • deconvolving the effect due to the finite angular resolution of the telescope by using the beam window function
  • deconvolving the effect due to the finite size of the pixel in the map by using a pixel window function
  • binning the power spectrum from individual multipoles into bandpowers
  • estimating error bars on bandpowers from signal plus noise Monte Carlo simulations, where signal simulations include only CMB anisotropies.

Inputs[edit]

The inputs are the following:

  • LFI Frequency Maps
  • Point source and galactic plane masks (the name being specified in the comment keyword in the header, see Note in Meta Data section below):
Point source masks
LFI_MASK_030-ps_2048_R1.00.fits
LFI_MASK_044-ps_2048_R1.00.fits
LFI_MASK_070-ps_2048_R1.00.fits
Galactic plane masks
COM_MASK_gal-06_2048_R1.00.fits
COM_MASK_gal-07_2048_R1.00.fits
  • Beam window functions
  • Monte Carlo simulations
  • binning scheme (3 columns: central l parameter, first l, last l):
     2.0           2           2
     3.0           3           3
     4.0           4           4
     5.0           5           5
     6.0           6           6
     7.0           7           7
     8.0           8           8
     9.0           9           9
    10.0          10          10
    11.0          11          11
    12.0          12          12
    13.0          13          13
    14.0          14          14
    15.0          15          15
    16.0          16          16
    17.0          17          17
    18.0          18          18
    19.0          19          19
    20.0          20          20
    21.0          21          21
    22.0          22          22
    23.0          23          23
    24.0          24          24
    25.0          25          25
    26.0          26          26
    27.0          27          27
    28.0          28          28
    29.0          29          29
    30.0          30          30
    31.0          31          31
    32.0          32          32
    33.0          33          33
    34.0          34          34
    35.0          35          35
    36.0          36          36
    37.0          37          37
    38.0          38          38
    39.5          39          40
    41.5          41          42
    43.5          43          44
    45.5          45          46
    47.5          47          48
    49.5          49          50
    51.5          51          52
    53.5          53          54
    55.5          55          56
    57.5          57          58
    59.5          59          60
    61.5          61          62
    63.5          63          64
    65.5          65          66
    67.5          67          68
    69.5          69          70
    71.5          71          72
    73.5          73          74
    75.5          75          76
    77.5          77          78
    80.0          79          81
    83.0          82          84
    86.0          85          87
    89.0          88          90
    92.0          91          93
    95.0          94          96
    98.0          97          99
   101.0         100         102
   104.0         103         105
   107.0         106         108
   110.0         109         111
   113.0         112         114
   116.0         115         117
   119.0         118         120
   122.5         121         124
   126.5         125         128
   130.5         129         132
   134.5         133         136
   138.5         137         140
   142.5         141         144
   146.5         145         148
   150.5         149         152
   154.5         153         156
   158.5         157         160
   163.0         161         165
   168.0         166         170
   173.0         171         175
   178.0         176         180
   183.0         181         185
   188.0         186         190
   193.0         191         195
   198.0         196         200
   203.5         201         206
   209.5         207         212
   215.5         213         218
   221.5         219         224
   227.5         225         230
   233.5         231         236
   239.5         237         242
   246.0         243         249
   253.0         250         256
   260.0         257         263
   267.0         264         270
   274.0         271         277
   281.0         278         284
   288.5         285         292
   296.5         293         300
   304.5         301         308
   312.5         309         316
   320.5         317         324
   329.0         325         333
   338.0         334         342
   347.0         343         351
   356.0         352         360
   365.5         361         370
   375.5         371         380
   385.5         381         390
   395.5         391         400
   406.0         401         411
   417.0         412         422
   428.0         423         433
   439.0         434         444
   450.5         445         456
   462.5         457         468
   474.5         469         480
   487.0         481         493
   500.0         494         506
   513.0         507         519
   526.5         520         533
   540.5         534         547
   554.5         548         561
   569.0         562         576
   584.0         577         591
   599.0         592         606
   614.5         607         622
   630.5         623         638
   647.0         639         655
   664.0         656         672
   681.0         673         689
   698.5         690         707
   716.5         708         725
   735.0         726         744
   754.0         745         763
   773.5         764         783
   793.5         784         803
   814.0         804         824
   835.0         825         845
   856.5         846         867
   878.5         868         889
   901.0         890         912
   924.0         913         935
   947.5         936         959
   972.0         960         984
   997.0         985        1009
  1022.5        1010        1035
  1048.5        1036        1061
  1075.0        1062        1088
  1102.5        1089        1116
  1130.5        1117        1144
  1159.0        1145        1173
  1188.5        1174        1203
  1219.0        1204        1234
  1250.0        1235        1265
  1281.5        1266        1297
  1314.0        1298        1330
  1347.5        1331        1364
  1382.0        1365        1399
  1417.5        1400        1435
  1453.5        1436        1471
  1490.0        1472        1508
  1527.5        1509        1546
  1566.0        1547        1585
  1605.5        1586        1625
  1646.0        1626        1666
  1687.5        1667        1708
  1730.0        1709        1751
  1773.5        1752        1795
  1818.0        1796        1840
  1864.0        1841        1887
  1911.5        1888        1935
  1960.0        1936        1984
  2009.5        1985        2034
  2060.0        2035        2085
  2112.0        2086        2138
  2165.5        2139        2192
  2220.0        2193        2247
  2276.0        2248        2304
  2333.5        2305        2362
  2392.5        2363        2422
  2453.0        2423        2483
  2515.0        2484        2546
  2578.5        2547        2610
  2643.5        2611        2676
  2710.0        2677        2743
  2778.0        2744        2812
  2848.0        2813        2883
  2920.0        2884        2956
  2993.5        2957        3030

Related products[edit]

A description of other products that are related and share some commonalities with the product being described here. E.g. if the description is of a generic product (e.g. frequency maps), all the products falling into that type should be listed and referenced.

File Names[edit]

LFI_PowerSpect_030_R1.10.fits
LFI_PowerSpect_044_R1.10.fits
LFI_PowerSpect_070_R1.10.fits

Meta Data[edit]

The angular power spectra source list in each frequency is structured as a FITS binary table. The Fits extension is composed by the columns described below:


FITS header
Column Name Data Type Units Description
L Integer*4 ell parameter
TEMP_CL Real*8 uK[math]_{CMB}^2[/math] [math]C_l[/math] (temperature)
TEMP_CL_ERR Real*8 uK[math]_{CMB}^2[/math] [math]C_l[/math] error


Note.- in the comment keyword in the header, the galactic and point source maps used to generate the angular spectra are specified (LFI_MASK_030-ps_2048_R1.00.fits and COM_MASK_gal-06_2048_R1.00.fits in the example below). Note also that, due to an oversight, the mask description related to COM_MASK_gal-xxx is wrong and should refer to the galactic mask.

Below an example of the header.

XTENSION= 'BINTABLE'           /Written by IDL:  Sat Feb 16 00:44:22 2013
BITPIX  =                    8 /
NAXIS   =                    2 /Binary table
NAXIS1  =                   20 /Number of bytes per row
NAXIS2  =                  130 /Number of rows
PCOUNT  =                    0 /Random parameter count
GCOUNT  =                    1 /Group count
TFIELDS =                    3 /Number of columns
TFORM1  = '1J      '           /Integer*4 (long integer)
TTYPE1  = 'L       '           /
TFORM2  = '1D      '           /Real*8 (double precision)
TTYPE2  = 'TEMP_CL '           /
TFORM3  = '1D      '           /Real*8 (double precision)
TTYPE3  = 'TEMP_CL_ERR'        /
EXTNAME = 'POW-SPEC'           / Extension name
EXTVER  =                    1 /Extension version
DATE    = '2013-02-15'         /Creation date
TUNIT2  = 'uK_CMB^2'           /
TUNIT3  = 'uK_CMB^2'           /
FILENAME= 'LFI_PowerSpect_030_R1.00.fits' /
PROCVER = 'Dx9_delta'          /
COMMENT ---------------------------------------------
COMMENT     Original Inputs
COMMENT ---------------------------------------------
COMMENT TT_30GHz_maskCS0.60_PS30GHzdet_febecopWls
COMMENT Used Point source Mask LFI_MASK_030-ps_2048_R1.00.fits
COMMENT Used Point source Mask COM_MASK_gal-06_2048_R1.00.fits
COMMENT Used FebeCoP 30 GHz wls
END

Below an example of the header of two masks used as input: COM_MASK_gal-06_2048_R1.00.fits and LFI_MASK_030-ps_2048_R1.00.fits:

XTENSION= 'BINTABLE'           / binary table extension
BITPIX  =                    8 / 8-bit bytes
NAXIS   =                    2 / 2-dimensional binary table
NAXIS1  =                    4 / width of table in bytes
NAXIS2  =             50331648 / number of rows in table
PCOUNT  =                    0 / size of special data area
GCOUNT  =                    1 / one data group (required keyword)
TFIELDS =                    1 / number of fields in each row
TTYPE1  = 'Mask    '           / label for field   1
TFORM1  = 'E       '           / data format of field: 4-byte REAL
TUNIT1  = 'none    '           / physical unit of field
EXTNAME = '06-GalMask'
DATE    = '2013-02-16T11:07:42' / file creation date (YYYY-MM-DDThh:mm:ss UT)
CHECKSUM= 'NaGVNZGUNaGUNYGU'   / HDU checksum updated 2013-02-16T11:07:43
DATASUM = '2540860986'         / data unit checksum updated 2013-02-16T11:07:43
COMMENT
COMMENT *** Planck params ***
COMMENT
PIXTYPE = 'HEALPIX '           / HEALPIX pixelisation
ORDERING= 'NESTED  '           / Pixel ordering scheme, either RING or NESTED
NSIDE   =                 2048 / Resolution parameter for HEALPIX
FIRSTPIX=                    0 / First pixel # (0 based)
LASTPIX =             50331647 / Last pixel # (0 based)
INDXSCHM= 'IMPLICIT'           / Indexing: IMPLICIT or EXPLICIT
OBJECT  = 'FULLSKY '           / Sky coverage, either FULLSKY or PARTIAL
BAD_DATA=          -1.6375E+30
COORDSYS= 'GALACTIC'
FILENAME= 'COM_MASK_gal-06_2048_R1.00.fits'
COMMENT ---------------------------------------------------------------------
COMMENT Combined galactic mask 0.6 sky fraction
COMMENT Objects used:
COMMENT /sci_planck/lfi_dpc_test/ashdown/repository/masks/component_separation/d
COMMENT x9/combined_mask_0.60_sky_fraction.fits
COMMENT ---------------------------------------------------------------------
END
XTENSION= 'BINTABLE'           / binary table extension
BITPIX  =                    8 / 8-bit bytes
NAXIS   =                    2 / 2-dimensional binary table
NAXIS1  =                    4 / width of table in bytes
NAXIS2  =             50331648 / number of rows in table
PCOUNT  =                    0 / size of special data area
GCOUNT  =                    1 / one data group (required keyword)
TFIELDS =                    1 / number of fields in each row
TTYPE1  = 'Mask    '           / label for field   1
TFORM1  = 'E       '           / data format of field: 4-byte REAL
TUNIT1  = 'none    '           / physical unit of field
EXTNAME = '030-PSMask'
DATE    = '2013-02-16T11:03:20' / file creation date (YYYY-MM-DDThh:mm:ss UT)
CHECKSUM= 'fR7ThO7RfO7RfO7R'   / HDU checksum updated 2013-02-16T11:03:21
DATASUM = '3828742620'         / data unit checksum updated 2013-02-16T11:03:21
COMMENT
COMMENT *** Planck params ***
COMMENT
PIXTYPE = 'HEALPIX '           / HEALPIX pixelisation
ORDERING= 'NESTED  '           / Pixel ordering scheme, either RING or NESTED
NSIDE   =                 2048 / Resolution parameter for HEALPIX
FIRSTPIX=                    0 / First pixel # (0 based)
LASTPIX =             50331647 / Last pixel # (0 based)
INDXSCHM= 'IMPLICIT'           / Indexing: IMPLICIT or EXPLICIT
OBJECT  = 'FULLSKY '           / Sky coverage, either FULLSKY or PARTIAL
BAD_DATA=          -1.6375E+30
COORDSYS= 'GALACTIC'
FILENAME= 'LFI_MASK_030-ps_2048_R1.00.fits'
COMMENT ---------------------------------------------------------------------
COMMENT The radius of the holes is 3 times the sigma of the beam at the correspo
COMMENT nding frequency and sigma is FWHM/(2*sqrt(2ln2))
COMMENT FWHM at 30GHz used = 33.158 arcmin
COMMENT Objects used:
COMMENT /planck/sci_ops1/LFI_MAPs/DX9_Delta/MASKs/mask_ps_30GHz_beam33amin_nside
COMMENT 2048.00_DX9_nonblind_holesize3.fits
COMMENT ---------------------------------------------------------------------
END

(Planck) High Frequency Instrument

Cosmic Microwave background

Flexible Image Transfer Specification

reduced IMO

(Planck) Low Frequency Instrument

Full-Width-at-Half-Maximum