CMB spectrum and likelihood code
Contents
CMB spectra[edit]
General description[edit]
TT[edit]
The Planck best-fit CMB temperature power spectrum, shown in the figure below, covers the wide range of multipoles = 2-2508. Over the multipole range = 2–29, the power spectrum is derived from the Commander: component separation algorithm applied to the combination of Planck 2015 temperature data between 30 and 857 GHz, the 9-year WMAP sky maps, and the 408 MHz Haslam et al. (1982) survey, including 93% of the sky Planck-2015-A10[1] . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction.
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 a Planck TT+lowP CDM 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 CDM run.
Production process[edit]
The Planck-2015-A10[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-2015-A10[1].
< 30 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 parametersThe 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. Note that following this definition, can be a non-integer. The binned power spectrum is then calculated as: .
Inputs[edit]
- Low-l spectrum ( )
- Planck 30 and 44 GHz frequency maps
- Planck 70 to 857 GHz detector and detector set maps
- 9-year WMAP temperature sky maps between 23 and 94 GHz
- 408 MHz survey by Haslam et al. (1982)
- Commander Planck-2015-A10[1] based LM93 confidence mask
- 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]
The CMB spectrum and its covariance matrix are distributed in a single FITS file named
- COM_PowerSpect_CMB_R2.nn.fits
which contains 7 BINTABLE extensions
- 1. TT low-ell, unbinned (TTLOLUNB)
- with the low ell part of the spectrum, not binned, and for l=2-29. The table columns are
- ELL (integer): multipole number
- D_ELL (float): as described above
- ERRUP (float): the upward uncertainty
- ERRDOWN (float): the downward uncertainty
- 2. TT high-ell, binned (TTHILBIN)
- 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
- 3. TT high-ell unbinned (TTHILUNB)
- 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
- 4. TE high-ell, binned (TEHILBIN)
- 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
- 5. TE high-ell, unbinned (TEHILUNB)
- 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
- 6. EE high-ell, binned (EELOLBIN)
- 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
- 7. EE high-ell, unbinned (EEHILUNB)
- 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 . The covariance matrices of the spectra will be released in a second moment.
Likelihood[edit]
The likelihood will soon be released with an accompanying paper and an Explanatory Supplement update.
References[edit]
- ↑ 1.01.11.21.3 Planck 2015 results. X. Diffuse component separation: Foreground maps, Planck Collaboration, 2016, A&A, 594, A10.
- ↑ 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.
- ↑ 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