CMB spectrum and likelihood code
Contents
CMB spectra[edit]
General description[edit]
TT[edit]
The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles = 2-2508. UPDATE COMMANDER: Over the multipole range Planck-2013-XII[1] . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction. = 2–29, the power spectrum is derived from a component-separation algorithm, Commander: applied to maps in the frequency range 30–353 GHz over 91% of the sky
For multipoles equal or greater than Planck-2015-A11[2] by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from Planck TT+lowP LCDM run. Associated 1-sigma errors include beam uncertainties. Both Commander and Plik are described in more details in the sections below.
, instead, the spectrum is derived from the Plik likelihoodTE and EE[edit]
The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range = 30-1996. The data points relative to the multipole range = 2-29 will be released in a second moment. Analogously to the TT case, the spectrum is derived from the Plik likelihood Planck-2015-A11[2] by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds using the best-fit foreground solution from a Planck TT,TE,EE+lowP LCDM run.
Production process[edit]
UPDATE COMMANDER The Planck-2013-XII[1]. The power spectrum at any multipole is given as the maximum probability point for the posterior distribution, marginalized over the other multipoles, and the error bars are 68% confidence level Planck-2013-XV[4]. < 50 part of the Planck TT power spectrum is derived from the Commander approach, which implements Bayesian component separation in pixel space, fitting a parametric model to the data by sampling the posterior distribution for the model parameters
The Planck-2013-XV[4] and Planck-2015-A11[2]. Frequency spectra are computed as cross-spectra between half-mission maps. Mask and multipole range choices for each frequency spectrum are summarized in Section 3.3 of Planck-2015-A13[3] and in Planck-2015-A11[2]. The final power spectrum is an optimal combination of the 100, 143, 143x217 and 217 GHz spectra, corrected for the best-fit unresolved foregrounds and inter-frequency calibration factors, as derived from the full likelihood analysis (for TT we use the best-fit solutions for the nuisance parameters from the Planck+TT+lowP data combination, while for TE and EE we use the best fit from Planck+TT+lowP, cf Table 3 of Planck-2015-A13[3]). A thorough description of the models of unresolved foregrounds is given in Planck-2015-A11[2]. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The CMB TT spectrum and associated covariance matrix are available in two formats:
part of the TT, TE and EE power spectra have been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of- Unbinned. TT: 2479 bandpowers ( ); TE or EE: 1697 bandpowers ( ).
- Binned, in bins of . TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the power spectrum with a weight proportional to , so that the binned bandpower centered in is: Equivalently, using the matrix formalism, we can construct the binning matrix B as: where B is a matrix, with the number of bins and the number of unbinned multipoles. Thus: Here, is the vector containing all the binned (unbinned) bandpowers, is the covariance matrix and is the weighted average multipole in each bin. The binned power spectrum is then calculated as: .
Inputs[edit]
UPDATE COMMANDER
- Low-l spectrum ( )
- frequency maps from 30–353 GHz
- common mask Planck-2013-XII[1]
- compact sources catalog
- High-l spectrum ( )
- 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 Planck-2015-A13[3])
- best-fit foreground templates and inter-frequency calibration factors (Table 3 of Planck-2015-A13[3])
- Beam transfer function uncertainties Planck-2015-A07[5]
File names and Meta data[edit]
CHECK EXTENSION NAMES
The CMB spectrum and its covariance matrix are distributed in a single FITS file named
- COM_PowerSpect_CMB_R2.nn.fits
which contains 5 extensions
- LOW-ELL (BINTABLE)
- with the low ell part of the spectrum, not binned, and for l=2-49. The table columns are
- ELL (integer): multipole number
- D_ELL (float): as described above
- ERRUP (float): the upward uncertainty
- ERRDOWN (float): the downward uncertainty
- BINNED HIGH-ELL (BINTABLE)
- with the high-ell part of the spectrum, binned into 83 bins covering in bins of width (with the exception of the last bin that is smaller). The table columns are as follows:
- ELL (integer): mean multipole number of bin
- L_MIN (integer): lowest multipole of bin
- L_MAX (integer): highest multipole of bin
- D_ELL (float): as described above
- ERR (float): the uncertainty
- UNBINNED HIGH-ELL (BINTABLE)
- with the high-ell part of the spectrum, unbinned, in 2979 bins covering . The table columns are as follows:
- ELL (integer): multipole
- D_ELL (float): as described above
- ERR (float): the uncertainty
The spectra give
in units of .Likelihood[edit]
The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.
References[edit]
- ↑ 1.01.11.2 Planck 2013 results. XI. Component separation, Planck Collaboration, 2014, A&A, 571, A11.
- ↑ 2.02.12.22.32.4 Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of cosmological parameters, Planck Collaboration, 2016, A&A, 594, A11.
- ↑ 3.03.13.23.33.43.53.6 Planck 2015 results. XIII. Cosmological parameters, Planck Collaboration, 2016, A&A, 594, A13.
- ↑ 4.04.1 Planck 2013 results. XV. CMB power spectra and likelihood, Planck Collaboration, 2014, A&A, 571, A15.
- ↑ Planck 2015 results. VII. High Frequency Instrument data processing: Time-ordered information and beam processing, Planck Collaboration, 2016, A&A, 594, A7.
Cosmic Microwave background
Flexible Image Transfer Specification