# Difference between revisions of "CMB spectrum & Likelihood Code"

Line 28: | Line 28: | ||

The <math>\ell \ge 30</math> 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 {{PlanckPapers|planck2013-p08}} and {{PlanckPapers|planck2014-a13}}. 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 {{PlanckPapers|planck2014-a15}} and in {{PlanckPapers|planck2014-a13}}. 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 {{PlanckPapers|planck2014-a15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-a13}}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The <math>\ell \ge 30</math> CMB TT spectrum and associated covariance matrix are available in two formats: | The <math>\ell \ge 30</math> 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 {{PlanckPapers|planck2013-p08}} and {{PlanckPapers|planck2014-a13}}. 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 {{PlanckPapers|planck2014-a15}} and in {{PlanckPapers|planck2014-a13}}. 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 {{PlanckPapers|planck2014-a15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-a13}}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The <math>\ell \ge 30</math> CMB TT spectrum and associated covariance matrix are available in two formats: | ||

#Unbinned. TT: 2479 bandpowers (<math>\ell=30-2508</math>); TE or EE: 1697 bandpowers (<math>\ell=30-1996</math>). | #Unbinned. TT: 2479 bandpowers (<math>\ell=30-2508</math>); TE or EE: 1697 bandpowers (<math>\ell=30-1996</math>). | ||

− | #Binned, in bins of <math> \Delta\ell=30 </math>. TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b} w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_b\ell}=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\</math> Equivalently, using the matrix formalism, we can construct the binning matrix B as: <math>\\ B_{\ell_b \ell}=w_{\ell_b\ell} \\ </math> where B is a <math> n_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus: <math> \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C}_{binned}\, (\vec{C}) </math> is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathrm{cov} </math> is the covariance matrix and <math>\ell_b</math> is the weighted average multipole in each bin. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\frac{\ell_b (\ell_b+1)}{2\pi} C_{\ell_b} </math>. | + | #Binned, in bins of <math> \Delta\ell=30 </math>. TT: 83 bandpowers. TE or EE: 66 bandpowers. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b} w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_b\ell}=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\</math> Equivalently, using the matrix formalism, we can construct the binning matrix B as: <math>\\ B_{\ell_b \ell}=w_{\ell_b\ell} \\ </math> where B is a <math> n_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus: <math> \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C}_{binned}\, (\vec{C}) </math> is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathrm{cov} </math> is the covariance matrix and <math>\ell_b</math> is the weighted average multipole in each bin. Note that following this definition, <math>\ell_b</math> can be a non-integer. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\frac{\ell_b (\ell_b+1)}{2\pi} C_{\ell_b} </math>. |

===Inputs=== | ===Inputs=== | ||

Line 38: | Line 38: | ||

* compact sources catalog | * compact sources catalog | ||

− | ; High-l spectrum (<math>30 \ | + | ; High-l spectrum (<math>30 \ge \ell \le 2500</math>): |

* 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 {{PlanckPapers|planck2014-a15}}) | * 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 {{PlanckPapers|planck2014-a15}}) | ||

Line 59: | Line 59: | ||

# ''ERRDOWN'' (float): the downward uncertainty | # ''ERRDOWN'' (float): the downward uncertainty | ||

− | ; BINNED HIGH-ELL (BINTABLE) | + | ; TT BINNED HIGH-ELL (BINTABLE) |

: with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-2499\ </math> in bins of width <math>l=30</math> (with the exception of the last bin that is smaller). The table columns are as follows: | : with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-2499\ </math> in bins of width <math>l=30</math> (with the exception of the last bin that is smaller). The table columns are as follows: | ||

# ''ELL'' (integer): mean multipole number of bin | # ''ELL'' (integer): mean multipole number of bin | ||

Line 67: | Line 67: | ||

# ''ERR'' (float): the uncertainty | # ''ERR'' (float): the uncertainty | ||

− | ; UNBINNED HIGH-ELL (BINTABLE) | + | ; TT UNBINNED HIGH-ELL (BINTABLE) |

: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-2508\ </math>. The table columns are as follows: | : with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-2508\ </math>. The table columns are as follows: | ||

# ''ELL'' (integer): multipole | # ''ELL'' (integer): multipole | ||

Line 73: | Line 73: | ||

# ''ERR'' (float): the uncertainty | # ''ERR'' (float): the uncertainty | ||

+ | ; TE BINNED HIGH-ELL (BINTABLE) | ||

+ | : with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-1988\ </math> in bins of width <math>l=30</math> (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): <math>D_\ell</math> as described above | ||

+ | # ''ERR'' (float): the uncertainty | ||

+ | |||

+ | ; TE UNBINNED HIGH-ELL (BINTABLE) | ||

+ | : with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-1996\ </math>. The table columns are as follows: | ||

+ | # ''ELL'' (integer): multipole | ||

+ | # ''D_ELL'' (float): <math>D_\ell</math> as described above | ||

+ | # ''ERR'' (float): the uncertainty | ||

+ | |||

+ | ; EE BINNED HIGH-ELL (BINTABLE) | ||

+ | : with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-1988\ </math> in bins of width <math>l=30</math> (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): <math>D_\ell</math> as described above | ||

+ | # ''ERR'' (float): the uncertainty | ||

+ | |||

+ | ; EE UNBINNED HIGH-ELL (BINTABLE) | ||

+ | : with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-1996\ </math>. The table columns are as follows: | ||

+ | # ''ELL'' (integer): multipole | ||

+ | # ''D_ELL'' (float): <math>D_\ell</math> as described above | ||

+ | # ''ERR'' (float): the uncertainty | ||

− | The spectra give <math>D_\ell = \ell(\ell+1)C_\ell / 2\pi </math> in units of <math>\mu\, K^2</math>. | + | The spectra give <math>D_\ell = \ell(\ell+1)C_\ell / 2\pi </math> in units of <math>\mu\, K^2</math>. The covariance matrices of the spectra will be released in a second moment. |

<!-- | <!-- | ||

## Revision as of 17:22, 4 February 2015

## 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 *Commander*: applied to maps in the frequency range 30–353 GHz over 91% of the sky 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,

For multipoles equal or greater than *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+lowP CDM run. Associated 1-sigma errors include beam uncertainties. Both *Commander* and *Plik* are described in more details in the sections below.

#### TE 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]

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:

- 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]

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

- TT 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

- TT 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

- TE 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

- TE 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

- EE 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

- EE 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 . 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]

- ↑
^{1.0}^{1.1}^{1.2}**Planck 2013 results. XI. Component separation**, Planck Collaboration, 2014, A&A, 571, A11 - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}**Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of cosmological parameters**, Planck Collaboration, 2016, A&A, 594, A11. - ↑
^{3.0}^{3.1}^{3.2}^{3.3}^{3.4}^{3.5}^{3.6}**Planck 2015 results. XIII. Cosmological parameters**, Planck Collaboration, 2016, A&A, 594, A13. - ↑
^{4.0}^{4.1}**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