Frequency maps angular power spectra
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]
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 $Y$, the unbinned $C^{XY}(\ell)$ in units of $\mu$K is available for all $\ell$ from 0 to 3508.
The $\langle\Delta C^{XY}(\ell)\Delta C^{XY}(\ell')\rangle$ 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. 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 correlattion 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 "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 .
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]
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:
| 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 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
