# HFI detset maps power spectra

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

## Product description

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.

For each pair of detectors $X$ and $Y$, are provided,

• the unbinned power spectrum $C^{XY}_\ell$ for all $\ell$ from 0 to 3508, as well as
• the unbinned covariance matrix

\begin{align}

  M^{XY}_{\ell \ell'} \equiv \langle\Delta C^{XY}_\ell\Delta C^{XY}_{\ell'}\rangle
\label{eq:covmatCl}


\end{align} for all $\ell$ on the same range.

Note that $M$ only describes the statistical covariance of the power spectrum induced by the signal and noise of the input map on the cut sky begin analyzed. Most sources of systematic effects (such as uncertainty on the beam modeling) as well as post-processing steps (such as foreground subtraction) will increase the covariance. In the particular case of the uncertainty on the beam window functions $B(l)$, the RIMO provides for each pair XY a set of eigen-vectors $\boldsymbol{\mathrm{E}}$ of $B^{XY}_{\ell}$ (see "HFI time response and beams paper"planck2013-p03c), $E$ is a matrix with 5 rows and 2501 to 4001 columns depending on the frequency uncertainty on the beam window functions, the relevant beam information should be read from the RIMO and from the .

## Production process

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. The different steps of the calculation are

• computation of the Spherical Harmonics coefficients $a_{\ell m}$ of the masked input maps and of the input mask,
• the sky (cross or auto) pseudo-power spectrum and mask power spectrum are computed from the $a_{\ell m}$,
• the sky and mask angular correlation function are computed from the respective power spectra,
• the ratio of the sky angular correlation by the mask correlation provides the cut sky corrected angular correlation,
• the sky angular correlation function which is then turned into a angular power spectrum,
• the resulting power spectrum is corrected from the nominal beam and average pixel window functions.

The covariance matrix is computed by PolSpice using the formalism described in Efstathiou (2004), also sketched in the appendix of "CMB power spectra and likelihood paper"planck2013-p08, assuming the instrumental noise to be white and uniform.

The products described here, spectra $C(l)$ and covariances $M_{\ell \ell'}$ can be used to estimate the high-$\ell$ likelihood of a given theoretical model given the data available.

## Inputs

Input data:

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 planck2013-p08). 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 planck2013-p08 and provided in the HFI RIMO.

## Related products

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

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

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

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

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

## Product description

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 .

## Production process

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

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):
• 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

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

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

## Meta Data

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:

Column Name Data Type Units Description
L Integer*4 ell parameter
TEMP_CL Real*8 uK (temperature)
TEMP_CL_ERR Real*8 uK 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 Used FebeCoP 30 GHz wls
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
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
COORDSYS= 'GALACTIC'
COMMENT ---------------------------------------------------------------------
COMMENT Combined galactic mask 0.6 sky fraction
COMMENT Objects used:
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
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
COORDSYS= 'GALACTIC'
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 2048.00_DX9_nonblind_holesize3.fits
COMMENT ---------------------------------------------------------------------
END


# References

<biblio force=false>

1. References

</biblio>

(Planck) High Frequency Instrument

Cosmic Microwave background

Flexible Image Transfer Specification

(Planck) Low Frequency Instrument

Full-Width-at-Half-Maximum