CMB spectrum and likelihood code
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 . 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 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 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 likelihood
TE and EE
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 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,TE,EE+lowP run.
UPDATE COMMANDER The Planck-2013-XII. 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. < 50 part of the Planck 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 CMB temperature power spectrum has been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of Planck-2013-XV. 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 and in Planck-2015-A11. 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 (cf Planck+TT+lowP in Table 3 of Planck-2015-A13). A thorough description of the models of unresolved foregrounds is given in Planck-2015-A11. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The > 30 CMB TT spectrum and associated covariance matrix are available in two formats:> 30 part of the
- Unbinned, with 2479 bandpowers ( ).
- Binned, in bins of , with 83 bandpowers in total. 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: .
- Low-l spectrum ( )
- High-l spectrum ( )
- 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 Planck-2015-A13)
- best-fit foreground templates and inter-frequency calibration factors (Table 3 of Planck-2015-A13)
- Beam transfer function uncertainties Planck-2015-A07
File names and Meta data
CHECK EXTENSION NAMES
The CMB spectrum and its covariance matrix are distributed in a single FITS file named
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 below
- 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): , not of the .
- 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): $D_l$ as described below
- ERR (float): the uncertainty
- UNBINNED COV-MAT (IMAGE)
- with the covariance matrix of the high-ell part of the spectrum in a 2979x2979 pixel image, i.e., covering the same bins as the HIGH-ELL table. Note that this is the covariance matrix of the , not of the .
The spectra give $D_\ell = \ell(\ell+1)C_\ell / 2\pi$ in units of $\mu\, K^2$, and the covariance matrix is in units of $\mu\, K^4$ (NOTE that the covariance matrix is for the , not for the ).
The CMB spectrum is also given in a simple text comma-separated file:
The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.
- Planck 2013 results. XI. Component separation, Planck Collaboration, 2014, A&A, 571, A11
- Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of cosmological parameters, Planck Collaboration, 2016, A&A, 594, A11.
- Planck 2015 results. XIII. Cosmological parameters, Planck Collaboration, 2016, A&A, 594, A13.
- 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